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ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND QUANTUM EFFECTS IN OPTOMECHANICAL SYSTEMS By SUMEI HUANG Bachelor of Science in Physics Education Fujian Normal University Fuzhou, Fujian, China 2001 Master of Science in Theoretical Physics Fujian Normal University Fuzhou, Fujian, China 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2011 COPYRIGHT c⃝ By SUMEI HUANG December, 2011 ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND QUANTUM EFFECTS IN OPTOMECHANICAL SYSTEMS Dissertation Approved: Dr. Girish S. Agarwal Dissertation advisor Dr. Xincheng Xie Dr. Albert T. Rosenberger Dr. Yin Guo Dr. Daniel Grischkowsky Dr. Jacques H. H. Perk Dr. Sheryl A. Tucker Dean of the Graduate College iii iv TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Optomechanical System . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The Dispersive Optomechanical System . . . . . . . . . . . . . 3 1.1.3 The Reactive Optomechanical System . . . . . . . . . . . . . . 5 1.2 Sideband Cooling of the Nano Mechanical Mirror . . . . . . . . . . . 6 1.3 Degenerate Parametric Amplification . . . . . . . . . . . . . . . . . . 12 1.4 Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Electromagnetically Induced Transparency . . . . . . . . . . . . . . . 16 1.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 ENHANCEMENT OF CAVITY COOLING OF A MICROMECHAN ICAL MIRROR USING PARAMETRIC INTERACTIONS 27 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 31 2.4 Cooling Mirror to About SubKelvin Temperatures . . . . . . . . . . 35 2.4.1 From Room Temperature (T=300 K) to About SubKelvin Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 From 1 K to Millikelvin Temperatures . . . . . . . . . . . . . 39 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 v 3 NORMAL MODE SPLITTING IN A COUPLED SYSTEM OF A NANOMECHANICAL OSCILLATOR AND A PARAMETRIC AM PLIFIER CAVITY 43 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 47 3.4 Normal Mode Splitting and the Eigenvalues of the Matrix A . . . . . 50 3.5 The Spectra of the Output Field . . . . . . . . . . . . . . . . . . . . . 51 3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 SQUEEZING OF A NANOMECHANICAL OSCILLATOR 60 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 64 4.4 Squeezing of the Movable Mirror . . . . . . . . . . . . . . . . . . . . . 68 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 ENTANGLING NANOMECHANICAL OSCILLATORS IN A RING CAVITY BY FEEDING SQUEEZED LIGHT 74 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 80 5.4 Entanglement of the Two Movable Mirrors . . . . . . . . . . . . . . . 84 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6 NORMALMODE SPLITTING AND ANTIBUNCHING IN STOKES AND ANTISTOKES PROCESSES IN CAVITY OPTOMECHAN ICS: RADIATIONPRESSUREINDUCED FOURWAVEMIXING vi CAVITY OPTOMECHANICS 90 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 Model: Stimulated Generation of Stokes and AntiStokes fields . . . . 91 6.3 The Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.4 Normalmode Splittings in the Output Fields . . . . . . . . . . . . . . 97 6.5 Spontaneous Generation of Stokes and Antistokes Photons: Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7 THE ELECTROMAGNETICALLY INDUCED TRANSPARENCY IN MECHANICAL EFFECTS OF LIGHT 105 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 EIT in the Out Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8 REACTIVECOUPLINGINDUCED NORMAL MODE SPLITTINGS IN MICRODISK RESONATORS COUPLED TO WAVEGUIDES 114 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.3 Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.4 Normal Mode Splitting In Output Fields . . . . . . . . . . . . . . . . 120 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9 CAN REACTIVE COUPLING BEAT MOTIONAL QUANTUM LIMIT OF NANO WAVEGUIDES COUPLED TO MICRODISK RESONATOR 125 9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 vii 9.3 Beating the Motional Quantum Limit for the Waveguide . . . . . . . 128 9.4 Numerical Results for Nano Waveguide Fluctuations below Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10 ELECTROMAGNETICALLY INDUCED TRANSPARENCY FROM TWO PHOTON PROCESSES IN QUADRATICALLY COUPLED MEMBRANES 135 10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.3 EIT in the Output Field . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11 ELECTROMAGNETICALLY INDUCED TRANSPARENCY WITH QUANTIZED FIELDS IN OPTOCAVITY MECHANICS 148 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.3 The Output Field and its Measurement . . . . . . . . . . . . . . . . . 152 11.4 EIT in the Homodyne Spectrum of the Output Quantized Field . . . 155 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12 OPTOMECHANICAL SYSTEMS AS SINGLE PHOTON ROUTERS160 12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.3 EIT in the Reflection Spectrum of the Single Photon . . . . . . . . . 166 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 13 SUMMARY AND FUTURE DIRECTIONS 170 13.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 viii 13.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 BIBLIOGRAPHY 174 A THE VARIANCE OF MOMENTUMDERIVATION OF EQUA TION EQ. (9.18) 191 B RELATION BETWEEN THE QUANTUM FLUCTUATIONS OF NANO WAVEGUIDE AND THE OUTPUT FIELD 193 ix LIST OF TABLES Table Page x LIST OF FIGURES Figure Page 1.1 A FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror. . . . . . . . . . . . . . . . . . . 3 1.2 The optomechaical system that consists of a microdisk resonator coupled to a waveguide (from Ref.[67]). . . . . . . . . . . . . . . . . . . . 5 1.3 The effective temperature Teff (mK) of the movable mirror as a function of the laser power ℘ (μW). The initial temperature is taken to be 1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Parametric amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Balanced homodyne detection. PD:photodetector. . . . . . . . . . . . 15 1.6 A threelevel Λtype atomic system, where the probe field at frequency ν couples levels b⟩ and a⟩, while the coupling field at frequency νμ couples levels c⟩ and a⟩. . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 The real part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. . . . . . . . . . . . . . . . . . . . . . . . 21 1.8 The imaginary part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. . . . . . . . . . . . . . . . . . . 22 xi 2.1 Sketch of the cavity used to cool a micromechanical mirror. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 The dotted curve indicates the χqs (106 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G=0. . . . . . . . . . . . . . . . . . . . . 36 2.3 The dotted curve indicates the χqs (107 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5×107s−1, parametric phase θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 The behavior of χqs (107 s−1) shown as a function of the detuning Δ0 (107 s−1). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. . . . . . . . . . . . . . . 38 xii 2.5 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1). Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. . . . . . . . . . . . . . . 39 2.6 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 0. . . . . . . . . . 40 2.7 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107 s−1, parametric phase θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Sketch of the studied system. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. . . . . . . 45 3.2 The roots of d(ω) in the domain Re(ω) > 0 as a function of parametric gain. ℘ = 6.9 mW (dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. . . . . . . . . . . . . . . . . 52 3.3 The imaginary parts of the roots of d(ω) as a function of parametric gain. ℘ = 6.9 mW ( dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. . . . . . . . . . . . . . . . . 52 xiii 3.4 The scaled spectrum SQ(ω)×γm versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 The spectrum Scout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6 The spectrum Sxout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 The spectrum Syout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.8 The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = √ ω2m + 4G2, parametric gain G = 1.3κ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.9 The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = ωm, parametric gain G = 0. 59 xiv 4.1 Sketch of the studied system. A laser with frequency ωL and squeezed vacuum light with frequency ωS enter the cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1) for different values of the squeezing of the input field. r = 0 (red, big dashed line), r = 0.5 (green, small dashed line), r = 1 (black, solid curve), r = 1.5 (blue, dotdashed curve), r = 2 (brown, solid curve). The minimum values of ⟨δ ˜ P2⟩ are 1.071 (r=0), 0.467 (r=0.5), 0.319 (r=1), 0.468 (r=1.5), 1.078 (r=2). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the temperature of the environment T = 1 mK, the laser power ℘ = 6.9 mW. . . . . . 69 4.3 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (blue, solid curve), 1 mK (red, small dashed curve), 5 mK (brown, big dashed curve), 10 mK (green, dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.252 (T=0 K), 0.611 (T=1 mK), 2.082 (T=5 mK), 3.919 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 0.6 mW. . . . . . . . . . . . . . . . . . . . 71 4.4 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (solid curve), 1 mK (dashed curve), 10 mK (dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.261 (T=0 K), 0.330 (T=1 mK), 0.968 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 3.8 mW. . . . . . . . . . . . . . . . . . . . . . . . 72 xv 4.5 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (solid curve), 1 mK (dashed curve), 10 mK (dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.275 (T=0 K), 0.319 (T=1 mK), 0.731 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 Sketch of the studied system. A laser with frequency ωL and a squeezed vacuum light with frequency ωS enter the ring cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 The mean square fluctuations ⟨δ ˜ P2− ⟩ versus the detuning Δ/ωm for different values of the squeezing of the input field. r = 0 (red, big dashed line), r = 0.5 (green, small dashed line), r = 1 (black, solid curve), r = 1.5 (blue, dotdashed curve), r = 2 (brown, solid curve). The minimum values of ⟨δ ˜ P2− ⟩ are 1.027 (r=0), 0.422 (r=0.5), 0.271(r=1), 0.412 (r=1.5), 0.999 (r=2). The flat dotted line represents ⟨δ ˜ P2− ⟩=1. Parameters: the temperature of the environment T = 41.4 μK, the laser power ℘ = 3.8 mW. . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 The mean square fluctuations ⟨δ ˜ P2−⟩ versus the detuning Δ/ωm, each curve corresponds to a different laser power. ℘=0.6 mW (red, big dashed curve), 3.8 mW (green, small dashed curve), 6.9 mW (black, solid curve), 10.7 mW (blue, dotdashed curve). The minimum values of ⟨δ ˜ P2− ⟩ are 0.257 (℘=0.6 mW), 0.271 (℘=3.8 mW), 0.291 (℘=6.9 mW), 0.315 (℘=10.7 mW). The flat dotted line represents ⟨δ ˜ P2− ⟩=1. Parameters: the squeezing parameter r = 1, the temperature of the environment T = 41.4 μK. . . . . . . . . . . . . . . . . . . . . . . . . 86 xvi 5.4 The value of ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩ versus the temperature of the environment T (μK). The minimum value of ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩ is 0.135 at T = 0 K. The flat dotted line represents ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩=1. Parameters: the squeezing parameter r = 1, the laser power ℘ = 3.8 mW, the detuning Δ = 0.965ωm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.5 Sketch of 4mirror ring cavity. A laser with frequency ωL and squeezed vacuum light with frequency ωS = ωL + ωm enter the ring cavity through the partially transmitting fixed mirror 1. The fixed mirror 2 and the two identical movable mirrors are perfectly reflecting. . . . 88 6.1 Sketch of the studied system. A pump field with frequency ωl and a Stokes field with frequency ωs enter the cavity through the partially transmitting mirror. The output fields cout have three components (ωl, ωs, 2ωl − ωs). No vacuum fields are shown here because we are examining only the mean response. . . . . . . . . . . . . . . . . . . . 92 6.2 The roots of d(ωs −ωl) in the domain Re(ωs −ωl) > 0 as a function of the pump power ℘. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 The imaginary parts of the roots of d(ωs − ωl) as a function of the pump power ℘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 The normalized quadrature vs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . . . . 98 6.5 The normalized quadrature ˜vs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . . . . 99 xvii 6.6 The normalized output power Gs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . 99 6.7 The normalized output power Gas plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . 100 6.8 The normalized secondorder correlation function g(2)(τ ) as a function of the time delay τ (μs) for different pump powers at T = 0K. ℘=1 mW (solid curve), and 4 mW(dotted curve). . . . . . . . . . . . . . . . . . 103 7.1 Sketch of the optomechanical system coupled to a highquality cavity via radiation pressure effects. . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Quadrature of the output field υp (solid black curve) and the different contributions to it: the real parts of A+ x−x+ (dotted red curve) and A− x−x− (dashed green curve) as a function of the normalized frequency x/ωm for input coupling laser power ℘c = 1 mW. The dotdashed blue curve is υp in the absence of the coupling laser. . . . . . . . . . . . . . . . . 110 7.3 Quadrature of the output field ˜υp (solid black curve) and the different contributions to it: the imaginary parts of A+ x−x+ (dotted red curve) and A− x−x− (dashed green curve) as a function of the normalized frequency x/ωm for input coupling laser power ℘c = 1 mW. The dotdashed blue curve is ˜υp in the absence of the coupling laser. . . . . . . . . . . . . 111 7.4 Same as in Fig. 7.2 except the input coupling laser power ℘c = 6.9 mW and ℘c = 0 case is not shown. . . . . . . . . . . . . . . . . . . . 111 7.5 Same as in Fig. 7.3 except the input coupling laser power ℘c = 6.9 mW and ℘c = 0 case is not shown. . . . . . . . . . . . . . . . . . . . 112 xviii 8.1 Sketch of the studied system (from Ref.[67]). The microdisk cavity is driven by a pump field and a Stokes field. The nonlinearity of the interaction also generates antiStokes field. . . . . . . . . . . . . . . . 115 8.2 The real roots of d(δ) in the domain Re(δ) > 0 as a function of the pump power ℘l for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.3 Imaginary parts of the roots of d(δ) as a function of the pump power ℘l for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.4 The lower two curves show the normalized quadrature vs as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π×26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized quadrature vs+1.5 for pump power ℘l = 200 μW. . . . 121 8.5 The lower two curves show the normalized output power Gs as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π×26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized output power Gs+1.5 for pump power ℘l = 200 μW. . 122 8.6 The lower two curves show the normalized output power Gas as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized output power Gas+0.15 for pump power ℘l = 200 μW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 xix 9.1 The variance of momentum ⟨δP2⟩ as a function of the detuning Δ (2π×106Hz) for different temperatures of the environment: T = 1 mK (red solid), T = 10 mK (blue dotted), T = 50 mK (purple dashed), and T = 100 mK (green dotdashed). The horizontal dotted line represents the standard quantum limit (⟨δP2⟩=1). The parameters: the pump power ℘l = 20 μW, r = 1. . . . . . . . . . . . . . . . . . . . . . . . . 132 9.2 The variance of momentum ⟨δP2⟩ as a function of the pump power (μW) for different temperatures of the environment: T = 1 mK (red solid) and T = 20 mK (green dotdashed). The horizontal dotted line represents the standard quantum limit (⟨δP2⟩=1). The parameters: Δ = ωm, r = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1 Sketch of the studied system. A strong coupling field at frequency ωc and a weak probe field at frequency ωp are injected into the cavity through the left mirror. A membrane with finite reflectivity is located at the middle position of the cavity. After the interaction between the cavity field and the membrane, the output field will contain three frequencies (ωc, ωp, and 2ωc − ωp). . . . . . . . . . . . . . . . . . . . . 137 10.2 Sketch of twophonon process. For a onephonon case the corresponding condition on frequencies will be ωc + ωm = ωp ≈ ω0. . . . . . . . . 138 10.3 Level diagram for the atomic EIT. For optocavity mechanics, 1⟩ ↔ 3⟩ would be the excitation at cavity frequency; 2⟩ ↔ 3⟩ would be the excitation of the mechanical oscillator. For the quadratically coupled membrane, 2⟩ → 3⟩ would be the twophonon excitation which makes ⟨q⟩ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 xx 10.4 Quadrature of the output field υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.45, ℘c = 90 μW, T = 90 K. The inset zooms the EITlike dip. . . . . . . 145 10.5 Quadrature of the output field υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.81, ℘c = 20 μW, T = 90 K. The inset zooms the EITlike dip. . . . . . . 146 10.6 Quadrature of the output field ˜υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.81, ℘c = 20 μW, T = 90 K. The inset zooms the change in the dispersion produced by the coupling field. . . . . . . . . . . . . . . . . . . . . . 146 11.1 Sketch of the studied system. A coherent coupling field at frequency ωc and a squeezed vacuum at frequency ωp enter the cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . 149 11.2 Sketch of the measurement of the output field. The output field ˜cout(t) is mixed with a strong local field clo(t) centered around the probe frequency ωp at a beam splitter, where ˜cout(t) is defined as the sum of the output field cout(t) from the cavity and the input quantized field cin(t). BS, 50:50 beam splitter; PD, photodetector; SA, spectrum analyzer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xxi 11.3 Homodyne spectrum X(ω) as a function of ω/ωm for N = 5 in the absence (dotted curve) and the presence (solid, dotdashed, and dashed curves) of the coupling field for the temperature of the environment T = 20 mK. The solid curve is for ℘ = 10 mW and M = √ N(N + 1), the dotdashed curve is for ℘ = 20 mW and M = √ N(N + 1), and the dashed curve is for ℘ = 20 mW and M = 0. . . . . . . . . . . . . . . 156 11.4 Homodyne spectrum X(ω) as a function of ω/ωm for different values of the parameter N and M = √ N(N + 1) in the absence (dotted curves) and the presence (solid curves) of a coupling field with power ℘ = 10 mW and temperature of the environment T = 100 mK. The upper two curves are for N = 5, and the lower two curves are for N = 1. . . . . 158 11.5 As in Fig. 11.4 but now the parameters used are from Ref. [50]. . . . 159 12.1 A doubleended cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.2 A doubleended cavity with a moving nanomechanical mirror as a single photon router. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 12.3 The reflection spectrum R(ω) of the single photon as a function of the normalized frequency ω/ωm without and with the coupling field. ℘ = 0 (solid), 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . 167 12.4 The transmission spectrum T(ω) of the single photon as a function of the normalized frequency ω/ωm without and with the coupling field. ℘ = 0 (solid), 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . 167 12.5 The vacuum noise spectrum S(v)(ω) as a function of the normalized frequency ω/ωm with the coupling field. ℘ = 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 12.6 The thermal noise spectrum S(T)(ω) as a function of the normalized frequency ω/ωm with the coupling field for T = 20 mK. ℘ = 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . . . . . . . . . 169 xxii CHAPTER 1 INTRODUCTION 1.1 Optomechanical System 1.1.1 Overview Radiation pressure force, due to the momentum carried by light, has received considerable attention since Kepler proposed that the tail of a comet was caused by the force exerted by the sunlight in the 16th century. It was deduced theoretically by J. C. Maxwell in 1871, and first observed experimentally [1, 2] in the early 1900s. With the invention of lasers in the 1970s, it has been shown that radiation pressure force can be used to manipulate atoms [3, 4, 5] i.e., to slow them, cool them, or trap them, owing to the relatively large power of the laser fields. In 2004, it was first demonstrated experimentally that radiation pressure force exerted by the light stored inside an optical cavity can be use to cool the motion of a mechanical oscillator made of roughly 1015 atoms in a cavity optomechanical system which parametrically couples an optical cavity and a mechanical resonator through radiation pressure [6]. Due to rapid advances in micro and nanofabrication techniques, various geometries of the optomechanical system have been developed, such as a FabryPerot cavity with mirrored microcantilevels [7], or with one movable end mirror [8, 9], or with a movable semitransparent membrane in the middle of the cavity [10, 11, 12, 13], or with a BoseEinstein condensate [14], or with a trapped macroscopic ensemble of ultracold atoms [15], radially vibrating microspheres [16], radially vibrating microtoroids [17, 18], GaAs nanooptomechanical disk resonator [19], and optomechanical 1 crystals [20]. Meanwhile, the optomechanical coupling idea has been extended to nanoelectromechanical systems, formed by a nanomechanical resonator capacitively coupled to a superconducting microwave cavity [21, 22, 23, 24]. The major challenge in all of these setups is to achieve simultaneously a high optical finesse (currently in the range from 103 to 105) and a high mechanical quality factor (currently in the range from 103 to 105). It has been shown theoretically and experimentally that such optomechanical systems at macroscopic scale can exhibit a very rich quantum effects, which usually exist in the microscopic system. For example, squeezing of the light field [25, 26], superposition state [27, 28], quantum nondemolition measurements of photon numbers [29, 30], the preparation of a mechanical oscillator in a squeezed state of motion [31, 32, 33], the creation of entangled photon pairs [34], the entanglement between the light and mechanical mode [35, 36], entangling two mechanical oscillators [37, 38, 39], and Fock state detection [10]. Moreover, the optomechanical coupling in such systems induces nonlinear behaviors, including an optical spring effect [40, 41], bistability [41, 42], multistability [43], selfinduced oscillations [44, 45, 46], optomechanical normal mode splitting [22, 47, 48, 49, 50], and optomechanically induced transparency [51, 52, 53, 54, 55]. Due to unavoidable coupling of the mechanical oscillator to its surrounding thermal environment, the random, thermal motion associated with mechanical dissipation mask the quantum behaviors. To see quantum effects in large objects, they must be cooled down to its quantum ground state. The ground state cooling requires that the mechanical oscillator’s temperature T must be reduced so that T ≪ ¯hωm kB , where ¯h is Planck’s constant h divided by 2π, kB is Boltzmann’s constant, ωm is the resonance frequency of the mechanical oscillator, typically between a few kilohertz and a few hundred megahertz. For a mechanical oscillator with a resonance frequency of 1kHz (100MHz), the ground state cooling requires ¯hωm/kB=50 nK (5 mK), which 2 are below those achievable with standard cryogenic cooling. So far, significant effort has been devoted to developing alternative cooling techniques. In the past few years, extraordinary progress has been made in cooling a mechanical resonator down to its quantum ground state [6, 7, 8, 9, 10, 56]. In 2009, the minimum achievable phonon number of the mechanical oscillator is 63 in a toroidal microresonator [57], 37 in a microsphere resonator [58], and 35 in a FabryPerot cavity [59]. In 2010, the preparation of mechanical resonator with the final phonon number below 10 was reported in Refs. [23, 60]. Recent work has shown experimentally that laser cooling can reduce the average occupancy of the mechanical oscillator below unity [61, 62, 63]. However, the ground state cooling has so far not been reached experimentally. in c out c fixed mirror movable mirror cavity axis Figure 1.1: A FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror. 1.1.2 The Dispersive Optomechanical System The canonical optomechanical system is a FabryPerot cavity with one heavy, fixed partially transmitting mirror and one light, movable totally reflecting mirror of effective mass m (typically in the micro or nanogram range), as shown in Fig. 1.1. The system is driven by an external laser at frequency ωl, then the circulating photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer from the intracavity photons to the movable mirror. Here, the movable mirror is modeled as a single mode quantum harmonic oscillator. Moreover, when the 3 mechanical frequency ωm is much smaller than the cavity free spectral range (c/2L), where L is the initial cavity length, the input laser drives only one cavity mode ωc and scattering of photons from the driven mode into other cavity modes is negligible [64]. During the cavity roundtrip time t = 2L/c, there are n photons hitting on the surfaces of the movable mirrors, the momentum transferred to the movable mirror will be P = 2n¯hωc/c, hence the radiation pressure force acting on the movable mirror would be F = P/t = n¯hωc L . The force is proportional to the instantaneous photon number in the cavity. Moreover, the movable mirror is in thermal equilibrium with its environment at temperature T. Thus the mirror can move under the influence of the radiation pressure and in the same time undergoes Brownian motion as a result of its interaction with the environment. In turn, the movable mirror’s small oscillation changes the length of the cavity and shifts the cavity resonance frequency so that the phase and amplitude of the cavity field are changed. This in turn changes the radiation pressure force experienced by the mirror such that the optical and mechanical dynamics are coupled. Thus the cavity resonance frequency depends on the displacement q of the movable mirror, represented by ωc(q) = nπc L+q , where n is the mode number in the cavity, c is the light speed in vacuum, L is the initial cavity length. For small displacements of the mirror, q << L, the frequency ωc(q) can be approximated to the first order of q ωc(q) ≈ ωc + gq, (1.1) where ωc = nπc L , g = −ωc/L is the linear coupling constant between the cavity field and the movable mirror, the minus sign in g implies that the cavity resonance frequency decreases when increasing the displacement q of the mirror elongates the cavity. What we discussed previously is the linear optomechanical coupling case, i.e., the frequency shift of the cavity field depends linearly on the displacement of the 4 mechanical oscillator. However, in a Fabry Perot cavity with a vibrating membrane in the middle of the optical cavity [10, 11, 12, 13], if the membrane is positioned at an antinode of the intracavity standing wave, the optomechanical coupling is quadratic i.e., the frequency shift of the cavity field depends quadratically on the displacement of the mechanical oscillator. If we expand the cavity frequency ωc(q) about the antinode point q0, then ωc(q) = ωc(q0) + ∂ωc(q) ∂q q=q0 q + 1 2 ∂2ωc(q) ∂q2 q=q0 q2 + · · · ≈ ωc(q0) + 1 2 ∂2ωc(q) ∂q2 q=q0 q2, (1.2) since at the antinode ∂ωc(q) ∂q q0 = 0. Compared to the linear optomechanical coupling system, the quadratic optomechanical system has the advantage in the quantum nondemolition measurement of mechanical energy quantization [10, 11, 12]. Note that the cavity decay rate only depends on the transmission of the fixed mirror, and is unrelated to the mechanical motion. Therefore, the optomechanical coupling via radiation pressure is dispersive. Figure 1.2: The optomechaical system that consists of a microdisk resonator coupled to a waveguide (from Ref.[67]). 1.1.3 The Reactive Optomechanical System In other optomechanical devices, the optomechanical coupling is induced by optical gradient force such as in silicon waveguide evanescently coupled to a microdisk 5 resonator [67], suspended silicon photonic waveguides [68, 69], SiN nanowire evanescently coupled to a microtoroidal resonator [70], and in ”zipper” cavities formed by two adjacent photonic crystal wires [71]. In this thesis, we focus on the optomechanical design proposed by [67], as shown in Fig. 1.2. The freestanding silicon waveguide with 10 μm length, 300 nm height, and 300 nm width is supported by two singlesided photonic crystal waveguide structures. The microdisk resonator with a radius of 40 μm is placed in close to the waveguide with a gap of 250 nm. A laser is injected into the waveguide, then light is coupled into and out of the microdisk through the evanescent fields from the waveguide and microdisk in the air gap, which decay exponentially with the distance from their geometric boundaries. And the dipoles in the waveguide induced by the evanescent field from the microdisk in turn interacts with the evanescent field from the microdisk and generate a gradient optical force. Under the action of this force, the waveguide is attracted toward the microdisk. Further the displacement of the waveguide modifies the resonance frequency of the microdisk resonator and the extrinsic photon decay rate of the microdisk resonator. Thus the coupling between the waveguide and the microdisk resonator is dispersive and reactive. 1.2 Sideband Cooling of the Nano Mechanical Mirror Recent experiments have demonstrated that the mechanical mirror can be cooled by the dynamical backaction of radiation pressure [7, 8, 9]. And it is possible to cool the mechanical mirror to the quantum ground state by resolved sideband cooling as first shown theoretically in Refs. [47, 66]. Sideband cooling was demonstrated experimentally by Kippenberg [57] and by Wang [58]. Both these experiments started the system at about 1.5K and showed cooling down to about 200 mK. The amount of cooling depends on the system parameters and the laser power. Harris et. al. has shown that the lowest temperature achieved is 6.82 mK in an optical cavity with a 6 vibrating membrane [10]. Before we give details of the theoretical discussion of sideband cooling, we discuss the physics which shows why sideband cooling results in cooling. When the pump field with frequency ωl interacts with the mechanical mirror with frequency ωm, absorption and emission of phonons create the Stokes field (ωl + ωm) and the antiStokes field (ωl − ωm). During the Stokes process, the pump field extracts a quantum of energy ¯hωm from the movable mirror, leading to the cooling of the movable mirror. While during the antiStokes process, the pump field emits a quantum of energy ¯hωm to the movable mirror, leading to the heating of the movable mirror. If the pump frequency is detuned below the cavity resonance frequency by an amount ωm, the amplitude of the Stokes field is resonantly enhanced, since the frequency of the Stokes field is close to the cavity resonance frequency ωc; however, the antiStokes field is suppressed since its frequency is far away from the cavity resonance frequency, thus the optomechanical coupling causes the cooling of the mirror. Further in the resolved sideband limit, the cavity amplitude decay rate κ is much less than the mechanical oscillation frequency ωm. In this case, the linewidth κ of the cavity field is much smaller than the frequency spacing 2ωm between the Stokes field and the antiStokes field, thus the amplitude of the antiStokes field is close to zero, ground state cooling becomes possible. We now develop the theoretical treatment of sideband cooling. The studied system is a FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror of effective mass m and damping rate γm, as shown in Fig. 1.1. The Hamiltonian of the system in a rotating frame with respect to the laser frequency ωl is given by H = ¯h(ωc − ωl)c†c − ¯hωmχc†cQ + ¯hωm 4 (Q2 + P2) + i¯hε(c† − c). (1.3) In Eq. (1.3), the first term is the energy of the cavity field, c and c† are the annihilation and creation operators for the cavity field satisfying the commutation relation [c, c†] = 1. The second term describes the interaction of the movable mirror with 7 the cavity field, the dimensionless parameter χ = 1 ωm ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term gives the energy of the movable mirror, described by the dimensionless position and momentum operators Q and P, defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with commutation relation [Q, P] = 2i. The fourth term describes the cavity driven by a laser with power ℘, and ε = √ 2κ℘ ¯hωl . The time evolution of the system operators can be derived by using the Heisenberg equations of motion and adding the corresponding damping and noise terms. We find a set of nonlinear quantum Langevin equations as follows, ˙Q = ωmP, ˙P = 2ωmχc†c − ωmQ − γmP + ξ, ˙ c = −i(ωc − ωl − ωmχQ)c + ε − κc + √ 2κcin, ˙ c† = i(ωc − ωl − ωmχQ)c† + ε − κc† + √ 2κc † in. (1.4) Here cin is the input vacuum noise operator with zero mean value and nonzero correlation function in the time domain ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′). (1.5) The force ξ is the Brownian noise operator associated with the mechanical damping, whose mean value is zero, and its correlation function reads ⟨ξ(t)ξ(t ′ )⟩ = 1 π γm ωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (1.6) where kB is the Boltzmann constant and T is the thermal bath temperature. The steadystate solution to Eq. (1.4) can be obtained by setting all the time derivatives in Eq. (1.4) to zero. They are Ps = 0, Qs = 2χcs2, cs = ε κ + iΔ , (1.7) 8 where Δ = ωc − ωl − ωmχQs (1.8) is the effective cavity detuning, in which the term −ωmχQs is the cavity resonance frequency shift due to radiation pressure. The Qs denotes the steadystate position of the movable mirror. And cs represents the steadystate amplitude of the cavity field. In order to investigate cooling of the movable mirror, we need to calculate the fluctuations of the system. We linearize the nonlinear equation (1.4) by writing each operator of the system as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (1.9) Inserting Eq. (1.9) into Eq. (1.4), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators take the form δ ˙Q = ωmδP, δ ˙P = 2ωmχ(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + iωmχcsδQ + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − iωmχc∗ sδQ + √ 2κδc † in. (1.10) We transform Eq. (1.10) to the frequency domain by using f(t) = 1 2π ∫ +∞ −∞ f(ω)e−iωtdω and f†(t) = 1 2π ∫ +∞ −∞ f†(−ω)e−iωtdω, where f†(−ω) = [f(−ω)]†, and solve it, we obtain the position fluctuations of the movable mirror δQ(ω) = − ωm d(ω) [2 √ 2κωmχ{[κ − i(Δ + ω)]c∗ sδcin(ω) + [κ + i(Δ − ω)]csδc † in(−ω)} +[(κ − iω)2 + Δ2]ξ(ω)], (1.11) where d(ω) = 4ω3m χ2Δcs2 + (ω2 − ω2m + iγmω)[(κ − iω)2 + Δ2]. (1.12) 9 In Eq. (1.11), the first term proportional to χ is the contribution of radiation pressure, while the second term involving ξ(ω) is the contribution of the thermal noise. In the absence of the cavity field, the movable mirror will make Brownian motion, δQ(ω) = ωmξ(ω)/(ω2m − ω2 − iγmω), whose susceptibility has a Lorentzian shape centered at frequency ωm with full width at half maximum γm. The twotime correlation function of the fluctuations in position of the movable mirror is given by 1 2 (⟨δQ(t)δQ(t + τ )⟩ + ⟨δQ(t + τ )δQ(t)⟩) = 1 2π ∫ +∞ −∞ dωSQ(ω)eiωτ , (1.13) in which SQ(ω) is the spectrum of fluctuations in position of the movable mirror, defined by 1 2 (⟨δQ(ω)δQ(Ω)⟩ + ⟨δQ(Ω)δQ(ω)⟩) = 2πSQ(ω)δ(ω + Ω). (1.14) By aid of the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(−Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 4π γm ωm ω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (1.15) we obtain the spectrum of fluctuations in position of the movable mirror SQ(ω) = ω2m d(ω)2 {8ω2m χ2κ(κ2 + ω2 + Δ2)cs2 + 2 γm ωm ω[(Δ2 + κ2 − ω2)2 +4κ2ω2] coth( ¯hω 2kBT )}. (1.16) In Eq. (1.16), the first term involving χ arises from radiation pressure, while the second term originates from the thermal noise. So the spectrum SQ(ω) of the movable mirror depends on radiation pressure and the thermal noise. Then Fourier transforming δ ˙Q = ωmδP in Eq. (1.10), we obtain δP(ω) = − iω ωm δQ(ω), which leads to the spectrum of fluctuations in momentum of the movable mirror SP (ω) = ω2 ω2m SQ(ω). (1.17) 10 The phonon number n in the movable mirror can be calculated from the total energy of the movable mirror ¯hωm 4 (⟨δQ2⟩ + ⟨δP2⟩) = ¯hωm ( n + 1 2 ) , n = [exp(¯hωm/(kBT)) − 1]−1, (1.18) where the variances of position and momentum are ⟨δQ2⟩ = 1 2π ∫ +∞ −∞ SQ(ω)dω and ⟨δP2⟩ = 1 2π ∫ +∞ −∞ SP (ω)dω. Then the effective temperature Teff of the movable mirror can be determined from the phonon number nin the movable mirror, which is Teff = ¯hωm kB ln(1 + 1 n) . (1.19) The parameters used are from an experimental paper on optomechanical normal mode splitting [50]: the wavelength of the laser λ = 2πc/ωl = 1064 nm, L = 25 mm, m = 145 ng, ωm = 2π × 947 × 103 Hz, the mechanical quality factor Q ′ = ωm/γm = 6700, κ = 2π × 215 × 103 Hz, κ/ωm ≈ 0.23, thus the system is operating in resolved sideband regime. And in the high temperature limit kBT ≫ ¯hωm, the approximation coth(¯hω/2kBT) ≈ 2kBT/¯hω can be made. The laser is detuned below the cavity resonance frequency by an amount Δ = ωm. We work in the stable regime. 0 200 400 600 800 1000 0 10 20 30 40 50 PHmWL Teff HmKL 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 PHmWL Teff HKL Figure 1.3: The effective temperature Teff (mK) of the movable mirror as a function of the laser power ℘ (μW). The initial temperature is taken to be 1 K. Figure 1.3 shows the variation of the effective temperature Teff of the movable mirror with the laser power ℘. It is clear to see that the effective temperature Teff of 11 the movable mirror decreases with increases the laser power ℘. When ℘ = 100 μW, the movable mirror can be cooled to about 50 mK, a factor of 20 below the starting temperature of 1 K [57, 58]. If the laser power is further increased to 1 mW, the movable mirror can be cooled to about 6 mK. Therefore the movable mirror can be effectively cooled in the resolved sideband limit. 1.3 Degenerate Parametric Ampli cation Nonlinear Crystal ! " # Figure 1.4: Parametric amplifier. In a parametric amplifier [72], a pump beam at higher frequency ωp interacts with a nonlinear crystal, a signal and idler modes at lower frequencies ωs and ωi would be generated, as shown in Fig. 1.4. During the nonlinear optical process, the energy is conserved ωp = ωs + ωi. If the signal and the idler modes have identical frequencies, such a parametric amplifier is called a degenerate parametric amplifier. In the following, we will show that the degenerate parametric amplifier can be used as a generator of a singlemode squeezed state. The Hamiltonian for degenerate parametric amplification, in the interaction picture, is Hint = ¯hμ(a†2b + a2b†), (1.20) where b and a are the annihilation operators for the pump and signal modes, respectively, and μ is a coupling strength between the pump field and the nonlinear crystal, and it is related to the secondorder nonlinear susceptibility. Assuming that 12 the pump field is a strong coherent classical field and pump depletion is neglected, thus the operators b and b† can be represented by βe−iϕ and βeiϕ, where β and ϕ are the real amplitude and phase of the coherent pump field. Hence the Hamiltonian (1.20) becomes Hint = ¯hμβ(a†2e−iϕ + a2eiϕ), (1.21) The time evolution of the signal mode can be derived by the Heisenberg equation of motion, which yields ˙a = −iΩa†e−iϕ, ˙a† = iΩaeiϕ. (1.22) Here Ω = 2μβ is the effective Rabi frequency. The solution to Eq. (1.22) is a(t) = a0 cosh(Ωt) − ia † 0 sinh(Ωt)e−iϕ, a†(t) = a † 0 cosh(Ωt) + ia0 sinh(Ωt)eiϕ, (1.23) where a0 = a(0). For ϕ = π/2, when the signal initially is in a vacuum state, the variances in the two quadratures X1 = (a + a†)/2 and X2 = (a − a†)/2i are given by (ΔX1)2t = 1 4 e−2u, (ΔX2)2t = 1 4 e2u, (1.24) where u = Ωt is the effective squeezing parameter. Eq. (1.24) shows the output from the degenerate parametric amplifier can be squeezed state, and the squeezing exists in the X1 quadrature. 1.4 Standard Quantum Limit For a onedimensional harmonic oscillator with mass m and frequency ωm, its Hamiltonian is H0 = p2 2m+1 2mω2m q2, where p is the momentum operator and q is the position operator, satisfying the commutation relation [q, p] = i¯h. In the ground state, the 13 fluctuations in the position and the momentum are not equal to zero due to the zeropoint energy. They are δq = √ ¯h 2mωm , δp = √ m¯hωm 2 , (1.25) respectively, which are called the standard quantum limit. These fluctuations have no classical analog. If we write the position operator q and the momentum operator p in terms of the dimensionless position operator Q and momentum operator P, q = √ ¯h 2mωm Q and p = √ m¯hωm 2 P, then the standard quantum limit would be δQ = δP = 1, (1.26) thus the fluctuations in the two dimensionless quadratures are identical, each of them is equal to unity. For very highprecision interferometers, the standard quantum limit limits their sensitivity. To improve their sensitivity, this limit need to be beaten, which means that the fluctuations need to be reduced below the standard quantum limit. According to the Heisenberg uncertainty principle ΔAΔB ≥ 1 2 ⟨[A,B]⟩, where ΔA = (⟨A2⟩ − ⟨A⟩2)1/2 and similarly for ΔB, the fluctuations in position and momentum should satisfy the inequality δQδP ≥ 1, (1.27) thus the fluctuations in the position and momentum could not be reduced below unity simultaneously. If the fluctuations in position is less than unity, the fluctuations in momentum should be larger than unity, or vice versa. Moreover, the harmonic oscillator is said to be squeezed if either δQ < 1 or δP < 1. Therefore, as the standard quantum limit is beaten, the harmonic oscillator is quadrature squeezed. 1.5 Homodyne Detection Homodyne detection is usually used to measure the amplitude and the phase quadrature components of the light field. In this section, we describe balanced homodyne detection [73]. 14 ! " # $% & $' PD PD Figure 1.5: Balanced homodyne detection. PD:photodetector. Figure 1.5 schematically shows a balanced homodyne detection setup. The signal light and a strong local laser light, described by the annihilation operators a and b, respectively, are mixed on a 50/50 beam splitter. The two output fields c and d can be obtained through the relation c = √1 2 (a + ib), d = √1 2 (b + ia). (1.28) The two output fields c and d are detected individually by two photodetectors. Then the two intensities Ic = ⟨c†c⟩ and Id = ⟨d†d⟩ measured by the two photodetectors are subtracted each other, the result is Ic − Id = ⟨ncd⟩ = ⟨c†c − d†d⟩, = i⟨a†b − ab†⟩. (1.29) Assuming the b mode to be in the coherent state βe−iωt⟩, and β = βe−iψ, the operator b can be replaced by βe−i(ωt+ψ), we obtain ⟨ncd⟩ = β[aeiωte−iθ + a†e−iωteiθ], (1.30) 15 where θ = ψ +π/2. Assuming that the signal mode a has the same frequency as that of the local oscillator b, thus a = a0e−iωt, Eq. (1.30) reduces to ⟨ncd⟩ = 2β⟨X(θ)⟩, (1.31) where ⟨X(θ)⟩ = 1 2 (a0e−iθ + a0eiθ) is the field quadrature operator at the angle θ. By changing θ, which can be done by changing the phase ψ of the local oscillator, an arbitrary quadrature component of the signal field can be measured. Moreover, the balanced homodyne detection can be used to detect the squeezed state. The variance of the output signal can be found to be ⟨(Δncd)2⟩ = 4β2⟨(ΔX(θ))2⟩, (1.32) The squeezing condition for the signal is ⟨(ΔX(θ))2⟩ < 1 4 , we have ⟨(Δncd)2⟩ < β2. 1.6 Electromagnetically Induced Transparency Generally, if a laser light passes through a twolevel atomic system whose atoms are all in the ground state, the light will be strongly absorbed if the laser field is near resonant with the atomic transition. However, for a threelevel atomic system whose atoms are all in the lowestenergy state, the atomic system becomes transparent for a weak probe field tuned to an atomic transition resonance when a strong coupling field is applied to the other atomic transition. This phenomenon is called as electromagnetically induced transparency (EIT). The effect of EIT allows a weak signal field to propagate without being absorbed by the atomic medium. It was theoretically proposed in 1989 [74] and first experimentally demonstrated in 1991 [75]. Meanwhile, the phenomenon of EIT [76] is accompanied by a sharp dispersion change in the transmitted probe field on resonance, which leads to the generation of ultrafast light [77, 78] and ultraslow light [79, 80, 81]. Accordingly considerable interest has been dedicated to EIT due to its potential applications in an optical switch [82], optical storage [83, 84, 85, 86]. 16 a μ n coupling n probe b c Figure 1.6: A threelevel Λtype atomic system, where the probe field at frequency ν couples levels b⟩ and a⟩, while the coupling field at frequency νμ couples levels c⟩ and a⟩. We consider a threelevel Λtype atomic system [72], as shown in Fig. 1.6. The atoms have one upper level a⟩ and two lower levels b⟩ and c⟩ with energies ¯hωa, ¯hωb, and ¯hωc, where the transitions b⟩ → a⟩ and c⟩ → a⟩ are dipole allowed, but the transition b⟩ → c⟩ is dipole forbidden since c⟩ is a metastable state. The levels a⟩ and b⟩ are coupled by a weak probe field of amplitude ε at frequency ν, while the levels a⟩ and c⟩ are coupled by a strong coupling field at frequency νμ. The coupling strength of the probe field to the atomic transition b⟩ → a⟩ is described by the Rabi frequency ℘abε/¯h, where ℘ab is the electricdipole transition matrix element, and it is assumed to be real. The interaction strength between the coupling field and the c⟩ → a⟩ transition is characterized by the complex Rabi frequency Ωμ exp(−iϕμ), and Ωμ is assumed to be real. The state of the atom can be written as a linear combination of states a⟩, b⟩, and c⟩, i.e., Ψ⟩ = Ca(t)a⟩ + Cb(t)b⟩ + Cc(t)c⟩. Here, Ca(t), Cb(t), and Cc(t) are the probability amplitudes corresponding to the three atomic levels a⟩, b⟩, and c⟩, respectively. The density matrix operator of the atom takes form ρ = Ψ⟩⟨Ψ = [Ca(t)a⟩ + Cb(t)b⟩ + Cc(t)c⟩][C∗ a(t)⟨a + C∗ b (t)⟨b + C∗ c (t)⟨c] 17 = Ca(t)2a⟩⟨a + Ca(t)C∗ b (t)a⟩⟨b + Ca(t)C∗ c (t)a⟩⟨c +Cb(t)C∗ a(t)b⟩⟨a + Cb(t)2b⟩⟨b + Cb(t)C∗ c (t)b⟩⟨c +Cc(t)C∗ a(t)c⟩⟨a + Cc(t)C∗ b (t)c⟩⟨b + Cc(t)2c⟩⟨c. (1.33) Taking the matrix elements, we get ρaa = ⟨aρa⟩ = Ca(t)2, ρab = ⟨aρb⟩ = Ca(t)C∗ b (t), ρac = ⟨aρc⟩ = Ca(t)C∗ c (t), ρba = ⟨bρa⟩ = Cb(t)C∗ a(t), ρbb = ⟨bρb⟩ = Cb(t)2, ρbc = ⟨bρc⟩ = Cb(t)C∗ c (t), ρca = ⟨cρa⟩ = Cc(t)C∗ a(t), ρcb = ⟨cρb⟩ = Cc(t)C∗ b (t), ρcc = ⟨cρc⟩ = Cc(t)2. (1.34) Hence, the threelevel atom can be described by the 3 × 3 density matrix ρ, ρ = ρaa ρab ρac ρba ρbb ρbc ρca ρcb ρcc , (1.35) where the diagonal elements ρii = ⟨iρi⟩ (i = a, b, c) describe the populations in the three levels, respectively, and the offdiagonal elements ρij = ⟨iρj⟩ (i, j = a, b, c and i ̸= j) represent the atomic coherence between levels. The density matrix is a Hermitian operator satisfying ρ = ρ†.The offdiagonal decay rates for ρab, ρac, and ρcb are denoted by γ1, γ2, and γ3, respectively. Since the level c⟩ is assumed to be a metastable state, γ3 << γ1. In the rotatingwave approximation, the Hamiltonian of the system is given by H = ¯hωaa⟩⟨a + ¯hωbb⟩⟨b + ¯hωcc⟩⟨c 18 +[ − ¯h 2 ( ℘abε ¯h e−iνta⟩⟨b + Ωμe−iϕ e−iν ta⟩⟨c) + H.C.], (1.36) where the first three terms are the free energies of the atomic three levels, and the last four terms gives the interactions of the threelevel atoms with the probe field and the coupling field. The time evolution for the density matrix elements ρab, ρcb, and ρac can be derived by using the Liouville equation ρ˙ij = −i ¯h[H, ρij ] and considering the corresponding damping term, which yields ρ˙ab = −(iωab + γ1)ρab − i 2 ℘abε ¯h e−iνt(ρaa − ρbb) + i 2 Ωμe−iϕ e−iν tρcb, ρ˙cb = −(iωcb + γ3)ρcb − i 2 ℘abε ¯h e−iνtρca + i 2 Ωμeiϕ eiν tρab, ρ˙ac = −(iωac + γ2)ρac − i 2 Ωμe−iϕ e−iν t(ρaa − ρcc) + i 2 ℘abε ¯h e−iνtρbc, (1.37) where ωab, ωcb, and ωac are the Bohr frequencies, ωab = ωa − ωb, ωcb = ωc − ωb, and ωac = ωa − ωc. We assume all atoms are initially in the lowestenergy state b⟩, ρbb(0) = 1, ρaa(0) = ρcc(0) = ρac(0) = 0. (1.38) Since the probe field is very weak, most of the atoms keep staying in the lowestenergy state b⟩ at any time so that the atomic population in level b⟩ is close to unity. Thus we can adopt the approximation condition ρbb(t) ≈ 1, ρaa(t) ≈ ρcc(t) ≈ ρac(t) ≈ 0. (1.39) Thus Eq. (1.37) reduces to ρ˙ab = −(iωab + γ1)ρab + i 2 ℘abε ¯h e−iνt + i 2 Ωμe−iϕ e−iν tρcb, ρ˙cb = −(iωcb + γ3)ρcb + i 2 Ωμeiϕ eiν tρab. (1.40) Then we convert the usual densitymatrix elements ρij to slowly varying variables ˜ρij in order to remove the fast optical oscillation by using the following transformations ρab = ˜ρabe−iνt, ρcb = ˜ρcbe−i(ν+ωca)t, (1.41) 19 thus the time evolution of the slowly varying densitymatrix elements ˜ρab and ˜ρcb is given by ˙˜ρab = −(γ1 − iΔ)˜ρab + i 2 ℘abε ¯h + i 2 Ωμe−iϕ ˜ρcb, ˙˜ρcb = −(γ3 − iΔ)˜ρcb + i 2 Ωμeiϕ ˜ρab, (1.42) where Δ = ν −ωab is the detuning of the probe frequency ν from the frequency ωab of the b⟩ → a⟩ transition, and we assume that the coupling field is resonant with the c⟩ → a⟩ transition, i.e., νμ = ωac. We write Eq. (1.42) in the matrix form as ˙R = −MR + A, (1.43) where R = ˜ρab ˜ρcb , M = γ1 − iΔ −i 2Ωμe−iϕ −i 2Ωμeiϕ γ3 − iΔ , A = i℘abε 2¯h 0 , (1.44) then integrating R(t) = ∫ t −∞ e−M(t−t′)Adt′ = M−1A, (1.45) we obtain ρab(t) = i℘abεe−iνt(γ3 − iΔ) 2¯h [ (γ1 − iΔ)(γ3 − iΔ) + Ω2 4 ] . (1.46) The dielectric response of the atomic system to the probe field is determined by the electric polarization P. The polarization of an ensemble of identical atoms will be P = 2℘abρab(t)eiνtNa, where Na is the atom number density for the threelevel atoms. In addition, the linear polarization is related to the amplitude ε of the probe field through P = ϵ0χε, where ε0 is the electric permittivity of free space and χ is the 20 electric susceptibility of the atomic system. Hence, the susceptibility of the Λ system is given by χ = Na℘2 ab ϵ0¯h i(γ3 − iΔ) (γ1 − iΔ)(γ3 − iΔ) + Ω2 4 , = χ′ + iχ′′, (1.47) where χ′ and χ′′ are the real and imaginary parts of the complex susceptibility χ of the atomic system. The χ′ and χ′′ determine the dispersion and absorption of the probe field, respectively. It is seen that from Eq. (1.47), on resonance, if there is a coupling field, i.e., Ωμ ̸= 0, χ′ = 0 and χ′′ = Na℘2 ab ϵ0¯h γ3 γ1γ3+ Ω2 4 , which is proportional to γ3. If the decay rate γ3 is very small (or approaching zero), the imaginary part of the electric susceptibility would be negligibly small. We plot the real and imaginary parts of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 without and with the coupling field, as shown in Figs. 1.7 and 1.8. In the 2 1 0 1 2 0.4 0.2 0.0 0.2 0.4 D Γ1 Γ1 Ε0 Ñ ReIΧM Na P ab 2 Figure 1.7: The real part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. absence of the coupling field, Ωμ = 0, the curve χ′′ has a Lorentzian lineshape, and the curve χ′ exhibits the anomalous dispersion since the slope of χ′ at the line center 21 2 1 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 D Γ1 Γ1 Ε0 Ñ ImIΧM Na P ab 2 Figure 1.8: The imaginary part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. is less than zero. In the presence of the coupling field, Ωμ = 2γ1, and γ1 >> γ3(γ3 = 10−4γ1), when Δ = 0, ωab = ν, the probe field is in resonance with the b⟩ → a⟩ atomic transition, we can see χ′′ ≈ 0, the medium becomes completely transparent for the probe field, thus the probe field can propagate through the atoms without any absorption even with most of the atoms in the lowestenergy state b⟩. It has been calculated that the width of the transparency window depends on the Rabi frequency Ωμ, which is related to the power of the coupling field. And increasing the power of the coupling field, the EIT dip becomes wider due to power broadening. We also note χ′ = 0 as Δ = 0, hence the refractive index of the medium is equal to unity since the refractive index is related to the susceptibility by n(ν) = [1+χ′(ν)+iχ′′(ν)]0.5. Thus the phase velocity of the probe field propagating through the medium is equal to that in vacuum. Moreover, the slope of the curve χ′ at the line center is larger than zero, thus the curve χ′ exhibits the normal dispersion. And the steepness of the curve χ′ where the absorption vanishes depends on the power of the coupling field, i.e., the curve χ′ becomes steeper at the line center by decreasing the power of the coupling 22 field, implying that the group velocity can be dramatically reduced, and even can be reduced to zero such that the probe field can be completely stopped and stored within the atomic medium. In summary, when the coupling field resonant with the c⟩ → a⟩ atomic transition is applied, the interaction of a threelevel Λtype atomic system with a weak probe field depends on the frequency of the probe field. If the frequency of the probe field matches the frequency of the b⟩ → a⟩ transition, the EIT phenomenon occurs, the effect of the atomic system on the probe field can be eliminated. 1.7 Organization Chapter 2 shows that an optical parametric amplifier inside a cavity can considerably improve the cooling of the micromechanical mirror by radiation pressure. The micromechanical mirror can be cooled from room temperature 300 K to subKelvin temperatures, which is much lower than what is achievable in the absence of the parametric amplifier. This is further illustrated in case of a precooled mirror, where one can reach millikelvin temperatures starting with about 1 K. Our work demonstrates the fundamental dependence of radiation pressure effects on photon statistics. Chapter 3 discusses how an optical parametric amplifier inside the cavity can affect the normalmode splitting behavior of the coupled movable mirror and the cavity field. We work in the resolved sideband regime. The spectra exhibit a doublepeak structure as the parametric gain is increased. Moreover, for a fixed parametric gain, the doublepeak structure of the spectrum is more pronounced with increasing the input laser power. We give results for mode splitting. The widths of the split lines are sensitive to parametric gain. Chapter 4 presents that squeezing of a nanomechanical mirror can be generated by injecting broad band squeezed vacuum light and laser light into the cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum 23 momentum squeezing of the movable mirror, the squeezing parameter of the input light should be about 1. We can obtain more than 70% squeezing. Besides, for a fixed squeezing parameter, decreasing the temperature of the environment or increasing the laser power increases the momentum squeezing. We find very large squeezing with respect to thermal fluctuations, for instance at 1 mK, the momentum fluctuations go down by a factor more than one hundred. Chapter 5 presents a scheme for entangling two separated nanomechanical oscillators by injecting broad band squeezed vacuum light and laser light into the ring cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum entanglement of the two oscillators, the squeezing parameter of the input light should be about 1. We report significant entanglement over a very wide range of power levels of the pump and temperatures of the environment. Chapter 6 discusses Stokes and antiStokes processes in cavity optomechanics in the regime of strong coupling. The Stokes and antiStokes signals exhibit prominently the normalmode splitting. We report gain for the Stokes signal. We also report lifetime splitting when the pump power is less than the critical power for normalmode splitting. The nonlinear Stokes processes provide a useful method for studying the strongcoupling regime of cavity optomechanics. We also investigate the correlations between the Stokes and the antiStokes photons produced spontaneously by the optomechanical system. At zero temperature, our nanomechanical system leads to the correlations between the spontaneously generated photons exhibiting photon antibunching and those violating the CauchySchwartz inequality. Chapter 7 discusses the dynamical behavior of a nanomechanical mirror in a highquality cavity under the action of a coupling laser and a probe laser. We demonstrate the existence of the analog of electromagnetically induced transparency (EIT) in the output field at the probe frequency. Our calculations show explicitly the origin of EITlike dips as well as the characteristic changes in dispersion from anomalous to 24 normal in the range where EIT dips occur. Remarkably the pumpprobe response for the optomechanical system shares all the features of the Λ system as discovered by Harris and collaborators. Chapter 8 studies the optomechanical design introduced by M. Li et. al. [Phys. Rev. Lett. 103, 223901 (2009)], which is very effective for investigation of the effects of reactive coupling. We show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling. We suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactivecouplinginduced normal mode splitting. Chapter 9 shows that dissipatively coupled nanosystems can be prepared in states which beat the standard quantum limit of the mechanical motion. We show that the reactive coupling between the waveguide and the microdisk resonator can generate the squeezing of the waveguide by injecting a quantum field and laser into the resonator through the waveguide. The waveguide can show about 7075% of maximal squeezing for temperature about 110 mK. The maximum squeezing can be achieved with an incident pump power of only 12 μW for a temperature of about 1 mK. Even for temperatures of 20 mK, achievable by dilution refrigerators, the maximum squeezing is about 60%. Chapter 10 describes how electromagnetically induced transparency can arise in quadratically coupled optomechanical systems. Due to quadratic coupling, the underlying optical process involves a twophonon process in an optomechanical system, and this twophonon process makes the mean displacement, which plays the role of atomic coherence in traditional electromagnetically induced transparency (EIT), zero. We show how the fluctuation in displacement can play a role similar to atomic coherence and can lead to EITlike effects in quadratically coupled optomechanical systems. We show how such effects can be studied using the existing optomechanical systems. Chapter 11 discusses electromagnetically induced transparency (EIT) using quan 25 tized fields in optomechanical systems. The weak probe field is a narrowband squeezed field. We present a homodyne detection of EIT in the output quantum field. We find that the EIT dip exists even though the photon number in the squeezed vacuum is at the singlephoton level. The EIT with quantized fields can be seen even at temperatures on the order of 100 mK, thus paving the way for using optomechanical systems as memory elements. Chapter 12 demonstrate theoretically the possibility of using nano mechanical systems as single photon routers. We show how EIT in cavity optomechanical systems can be used to produce a switch for a probe field in a single photon Fock state using very low pumping powers of few microwatt. We present estimates of vacuum and thermal noise and show the optimal performance of the single photon switch is deteriorated by only few percent even at temperatures of the order of 20 mK. Chapter 13 gives the summary of what we have done in this thesis and the direction of the future work. 26 CHAPTER 2 ENHANCEMENT OF CAVITY COOLING OF A MICROMECHANICAL MIRROR USING PARAMETRIC INTERACTIONS 2.1 Overview Recently there is considerable interest in micromechanical mirrors. These are macroscopic quantum mechanical systems and the important question is how to reach their quantum characteristics [8, 87, 88, 89]. The thermal noise limits many highly sensitive optical measurements [90, 91]. We also note that there has been considerable interest in using micromirrors for producing superpositions of macroscopic quantum states if such micromirrors can be cooled to their quantum ground states [27, 28]. Thus cooling of micromechanical resonators becomes a necessary prerequisite for all such studies. So far two different ways to cool a mechanical resonator mode have been proposed. One is the active feedback scheme [7, 92, 93, 116], where a viscous force is fed back to the movable mirror to decrease its Brownian motion. The other is the passive feedback scheme [6, 8, 9, 56, 95, 159], in which the Brownian motion of the movable mirror is damped by the radiation pressure force exerted by photons in an appropriately detuned optical cavity. Clearly we need to think of methods which can cool the micromirror toward its ground state. Since radiation pressure depends on the number of photons, one would think that the cooling of the micromirror can be manipulated by using effects of the photon statistics. In this chapter, we propose and analyze a method to achieve cooling of a movable mirror to subKelvin temperatures by using a type I optical 27 parametric amplifier inside a cavity. We remind the reader of the great success of cavities with parametric amplifiers in the production of nonclassical light [97, 98, 99]. The movable mirror can reach a minimum temperature of about a few hundred mK, a factor of 500 below room temperature 300 K. The lowering of the temperature is achieved by changes in photon statistics due to parametric interactions [100, 101, 102, 103, 104, 105]. Note that if the mirror is already precooled to say about 1 K, then we show that by using an optical parametric amplifier we can cool to about millikelvin temperatures or less. The chapter is organized as follows. In Sec. II we describe the model and derive the quantum Langevin equations. In Sec. III we obtain the stability conditions, calculate the spectrum of fluctuations in position and momentum of the movable mirror, and define the effective temperature of the movable mirror. In Sec. IV we show how the movable mirror can be effectively cooled by using the parametric amplifier inside the cavity. 2.2 Model in c out c movable mirror OPA cavity axis fixed mirror Figure 2.1: Sketch of the cavity used to cool a micromechanical mirror. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. We consider a degenerate optical parametric amplifier (OPA) inside a FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror in contact with a thermal bath in equilibrium at temperature T, as shown 28 in Fig. 2.1. The movable mirror is free to move along the cavity axis and is treated as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm, and energy decay rate γm. The effect of the thermal bath can be modeled by a Langevin force. The cavity field is driven by an input laser field with frequency ωL and positive amplitude related to the input laser power P by ˜ε = √ P/(¯hωL). When photons in the cavity reflect off the surface of the movable mirror, the movable mirror will receive the action of the radiation pressure force, which is proportional to the instantaneous photon number inside the cavity. So the mirror can oscillate under the effects of the thermal Langevin force and the radiation pressure force. Meanwhile, the movable mirrors motion changes the length of the cavity; hence the movable mirror displacement from its equilibrium position will induce a phase shift on the cavity field. Here we assume the system is in the adiabatic limit, which means ωL ≪ πc/L; c is the speed of light in vacuum and L is the cavity length in the absence of the cavity field. We assume that the motion of the mirror is so slow that the scattering of photons to other cavity modes can be ignored, thus we can consider one cavity mode only [64, 106], say, ωc. Moreover, in the adiabatic limit, the number of photons generated by the Casimir effect [107], retardation, and Doppler effects is negligible [26, 92, 108]. Under these conditions, the total Hamiltonian for the system in a frame rotating at the laser frequency ωL can be written as H = ¯h(ωc − ωL)nc − ¯hχncq + 1 2 ( p2 m + mω2m q2) +i¯hε(c† − c) + i¯hG(eiθc†2 − e−iθc2). (2.1) Here c and c† are the annihilation and creation operators for the field inside the cavity, respectively; nc = c†c is the number of the photons inside the cavity; and q and p are the position and momentum operators for the movable mirror. The parameter χ = ωc/L is the coupling constant between the cavity and the movable mirror; and 29 ε = √ 2κ˜ε. Note that κ is the photon decay rate due to the photon leakage through the fixed partially transmitting mirror. Further κ = πc/(2FL), where F is the cavity finesse. In Eq. (2.1), G is the nonlinear gain of the OPA, and θ is the phase of the field driving the OPA. The parameter G is proportional to the pump driving the OPA. In Eq. (2.1), the first term corresponds to the energy of the cavity field, the second term arises from the coupling of the movable mirror to the cavity field via radiation pressure, the third term gives the energy of the movable mirror, the fourth term describes the coupling between the input laser field and the cavity field, and the last term is the coupling between the OPA and the cavity field. The motion of the system can be described by the Heisenberg equations of motion and adding the corresponding damping and noise terms, which leads to the following quantum Langevin equations: q˙ = p m, ˙ p = −mω2m q + ¯hχnc − γmp + ξ, ˙ c = i(ωL − ωc + χq)c + ε + 2Geiθc† − κc + √ 2κcin, ˙ c† = −i(ωL − ωc + χq)c† + ε + 2Ge−iθc − κc† + √ 2κc † in. (2.2) Here cin is the input vacuum noise operator with zero mean value; its correlation function is [141] ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = ⟨δc † in(t)δcin(t′)⟩ = 0. (2.3) The force ξ is the Brownian noise operator resulting from the coupling of the movable mirror to the thermal bath, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = ¯hγm 2π m ∫ ωe−iω(t−t ′ ) [ coth( ¯hω 2kBT ) + 1 ] dω, (2.4) 30 where kB is the Boltzmann constant and T is the thermal bath temperature. In order to analyze Eq. (2.2), we use standard methods from quantum optics [110]. A detailed calculation of the temperature for G = 0 is given by Paternostro et al. [35]. By setting all the time derivatives in Eq. (2.2) to zero, we obtain the steadystate mean values ps = 0, qs = ¯hχcs2 mω2m , cs = κ − iΔ + 2Geiθ κ2 + Δ2 − 4G2 ε, (2.5) where Δ = ωc − ωL − χqs = Δ0 − χqs = Δ0 − ¯hχ2cs2 mω2m (2.6) is the effective cavity detuning, including the radiation pressure effects. The modification of the detuning by the χqs term depends on the range of parameters. The qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. Note that qs and cs can display optical multistable behavior, which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. Mathematically this is contained in the dependence of the detuning parameter Δ on the mirrors amplitude qs. It is evident from Eqs. (2.5) and (2.6) that Δ satisfies a fifthorder equation and in principle can have five real solutions implying multistability. Generally, in this case, at most three solutions would be stable. The bistable behavior is reported in Refs. [41, 42]. 2.3 Radiation Pressure and Quantum Fluctuations In order to determine the cooling of the mirror, we need to find out the fluctuations in the mirrors amplitude. Since the problem is nonlinear, we assume that the nonlinearity is weak. We are thus interested in the dynamics of small fluctuations around the steady state of the system. Such a linearized analysis is quite common in quantum optics [110, 111]. So we write each operator of the system as the sum of its steady 31 state mean value and a small fluctuation with zero mean value, q = qs + δq, p = ps + δp, c = cs + δc. (2.7) Inserting Eq. (2.7) into Eq. (2.2), then assuming cs ≫ 1, we get the linearized quantum Langevin equations for the fluctuation operators δq˙ = δp m, δ ˙ p = −mω2m δq + ¯hχ(csδc† + c∗ sδc) − γmδp + ξ, δ ˙ c = −iΔδc + iχcsδq + 2Geiθδc† − κδc + √ 2κδcin, δc˙† = iΔδc† − iχc∗ sδq + 2Ge−iθδc − κδc† + √ 2κδc † in. (2.8) Introducing the cavity field quadratures δx = δc† + δc and δy = i(δc† − δc), and the input noise quadratures δxin = δc † in +δcin and δyin = i(δc † in −δcin), Eq. (2.8) can be written in the matrix form f˙ = Af(t) + η(t), (2.9) where f(t) is the column vector of the fluctuations, and η(t) is the column vector of the noise sources. For the sake of simplicity, their transposes are f(t)T = (δq, δp, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (2.10) and the matrix A is given by A = 0 1 m 0 0 −mω2m −γm ¯hχcs+c∗ s 2 ¯hχcs−c∗ s 2i iχ(cs − c∗ s) 0 2Gcos θ − κ Δ + 2Gsin θ χ(cs + c∗ s) 0 2Gsin θ − Δ −(κ + 2Gcos θ) . (2.11) The solutions to Eq. (2.9) are stable only if all the eigenvalues of the matrix A have negative real parts. Applying the RouthHurwitz criterion [112, 113], we get the 32 stability conditions 2κ(κ2 − 4G2 + Δ2 + 2κγm) + γm(2κγm + ω2m ) > 0, (2κ + γm)2[ 2¯hχ2cs2 m Δ + 2¯hχ2(c2s + c∗2 s )Gsin θ m + 2i¯hχ2(c2s − c∗2 s )Gcos θ m ] + 2κγm{(κ2 − 4G2 + Δ2)2 +(2κγm + γ2m )(κ2 − 4G2 + Δ2) +ω2m [2(κ2 + 4G2 − Δ2) + ω2m + 2κγm]} > 0, ω2m (κ2 − 4G2 + Δ2) − 2¯hχ2cs2 m Δ −2¯hχ2(c2s + c∗2 s )Gsin θ m − 2i¯hχ2(c2s − c∗2 s )Gcos θ m > 0. (2.12) Note that in the absence of coupling χ, the conditions (2.12) become equivalent to κ2 − 4G2 + Δ2 > 0 (2.13) The condition for the threshold for parametric oscillations is κ2 −4G2 +Δ2 = 0. We always would work under the condition that (2.13) is satisfied. Further for χ ̸= 0 we would do numerical simulations using parameters so that conditions (2.12) are satisfied. On Fourier transforming all operators and noise sources in Eq. (2.8) and solving it in the frequency domain, the position fluctuations of the movable mirror are given by δq(ω) = − 1 d(ω)([Δ2 + (κ − iω)2 − 4G2]ξ(ω) −i¯h √ 2κχ{[(ω + iκ − Δ)cs + 2iGeiθc∗ s]δc † in(ω) +[(ω + iκ + Δ)c∗ s + 2iGe−iθcs]δcin(ω)}), (2.14) where d(ω) = 2¯hχ2(Δcs2+iGe−iθc2s −iGeiθc∗2 s )+m(ω2−ω2m +iωγm)[Δ2+(κ−iω)2− 4G2]. In Eq. (2.14), the first term proportional to ξ(ω) originates from the thermal noise, while the second term proportional to χ arises from radiation pressure. So the position fluctuations of the movable mirror are now determined by the thermal noise 33 and radiation pressure. Notice that if there is no radiation pressure, the movable mirror will make Brownian motion, δq(ω) = −ξ(ω)/[m(ω2 − ω2m + iωγm)], whose susceptibility has a Lorentzian shape centered at frequency ωm with width γm. The spectrum of fluctuations in position of the movable mirror is defined by Sq(ω) = 1 4π ∫ dΩe−i(ω+Ω)t⟨δq(ω)δq(Ω) + δq(Ω)δq(ω)⟩. (2.15) To calculate the spectrum, we need the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 2π¯hγmmω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (2.16) Substituting Eq. (2.14) and Eq. (2.16) into Eq. (2.15), we obtain the spectrum of fluctuations in position of the movable mirror Sq(ω) = ¯h d(ω)2 {2κ¯hχ2[(κ2 + ω2 + Δ2 + 4G2)cs2 +2Geiθc∗2 s (κ − iΔ) + 2Ge−iθc2s (κ + iΔ)] +mγmω[(Δ2 + κ2 − ω2 − 4G2)2 + 4κ2ω2] ×coth( ¯hω 2kBT )}. (2.17) In Eq. (2.17), the first term is the radiation pressure contribution, whereas the second term corresponds to the thermal noise contribution. Then Fourier transforming q˙ = δp/m in Eq. (2.8), we obtain δp(ω) = −imωδq(ω), which leads to the spectrum of fluctuations in momentum of the movable mirror Sp(ω) = m2ω2Sq(ω). (2.18) For a system in thermal equilibrium, we can use the equipartition theorem to define temperature 1 2mω2m ⟨q2⟩ = ⟨p2⟩ 2m = 1 2kBTeff , where ⟨q2⟩ = 1 2π ∫ +∞ −∞ Sq(ω)dω, and ⟨p2⟩ = 1 2π ∫ +∞ −∞ Sp(ω)dω. However, here we are dealing with a driven system and 1 2mω2m ⟨q2⟩ ̸= 34 ⟨p2⟩ 2m , hence the question is how to define temperature. We use an effective temperature defined by the total energy of the movable mirror kBTeff = 1 2mω2m ⟨q2⟩ + ⟨p2⟩ 2m . We also introduce the parameter r = m2ω2m ⟨q2⟩/⟨p2⟩.This parameter gives us the relative importance of fluctuations in position and momentum of the mirror. We mention that one can calculate the quantum state of the oscillator and we find that the Wigner function is Gaussian. Equation (2.17) is our key result which tells how the temperature of the micromirror would depend on the parameters of the cavity: κ, gain of the OPA, external laser power, etc. We specifically investigate the dependence of the temperature on the gain G and the phase θ associated with the parametric amplification process. In the limit of G → 0, the result (2.17) reduces to the one derived by Paternostro et al.[35]. 2.4 Cooling Mirror to About SubKelvin Temperatures In this section, we present the possibility of cooling the micromirror to temperatures of about subKelvin by using parametric amplifiers inside cavities. In all the numerical calculations we choose the values of the parameters which are similar to those used in recent experiments: λL = 2πc/ωL = 1064 nm, L = 25 mm, P = 4 mW, m = 15 ng, ωm/(2π) = 275 kHz, and the mechanical quality factor Q = ωm/γm = 2.1 × 104. Further in the hightemperature limit kBT ≫ ¯hω, we have coth(¯hω/2kBT) ≈ 2kBT/¯hω. 2.4.1 From Room Temperature (T=300 K) to About SubKelvin Tem peratures If we choose κ = 108 s−1, F = 188.4, G = 0 to satisfy the stability conditions (2.12), the detuning must satisfy Δ0 ≥ 4 × 106s−1. Figure 2.2 gives the variations of the χqs, the effective temperature Teff , and the parameter r with the detuning Δ0. It should be borne in mind that for the range of the detuning shown in Fig. 2.2, 35 2 4 6 8 10 0 5 10 15 20 25 30 0. 1. 2. 3. 4. 5. 6. D0 H107 s1L T eff HKL , r+5 Χqs H10 6 s1L Figure 2.2: The dotted curve indicates the χqs (106 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G=0. Δ = Δ0 − χqs ≈ Δ0. We find the χqs is single valued, so the movable mirror is monostable. Note that the parameter r is very close to unity, 1 2mω2m ⟨q2⟩ ≈ ⟨p2⟩ 2m ; the mirror is thus in nearly thermal equilibrium. Figure 2.2 shows the possibility of cooling the mirror to a temperature of 15.23 K for Δ0 = 4.9 × 107 s−1, which is in agreement with the previous calculation [35]. Now we keep the values of κ and F the same as in Fig. 2.2, and we choose parametric gain G = 3.5 × 107 s−1 and parametric phase θ = 0; the detuning must satisfy Δ0 ≥ 5.7 × 107 s−1. If Δ0 < 5.7 × 107 s−1 and for fixed κ and G, the system will be unstable. The threshold for unstable behavior occurs when any of the three conditions (2.12) is not satisfied. It may be noted that the threshold for parametric oscillation has been of great importance in connection with the production 36 6 7 8 9 10 0 1 2 3 4 5 6 0 1 2 3 4 5 6 D0 H107 s1 L T eff HKL , r Χqs H107 s1 L Figure 2.3: The dotted curve indicates the χqs (107 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107s−1, parametric phase θ = 0. of nonclassicalsqueezed light. Near the parametric thresholds but under (2.13), large degrees of squeezing were produced [97, 98]. Thus it would be advantageous to work near the threshold of instability but below the instability point. Figure 2.3 shows the variations of the χqs, the effective temperature Teff , and the parameter r with the detuning Δ0. We find the χqs is still single valued, so the movable mirror is still monostable. The minimum temperature reached is 0.65 K for Δ0 = 6.7 × 107 s−1. Thus, with the parametric amplifier the minimum temperature is about a factor of 20 lower than the one without parametric interaction. Note that the parameter r is always larger than 1, implying that momentum fluctuations are suppressed over 37 2.0 2.2 2.4 2.6 2.8 3.0 0 1 2 3 4 5 D0 H107 s 1 L Χq s H107 s 1 L Figure 2.4: The behavior of χqs (107 s−1) shown as a function of the detuning Δ0 (107 s−1). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. position fluctuations. Note that as one moves away from the threshold for parametric instability, the minimum temperature does not rise sharply which is in contrast to the behavior in Fig. 2.2, and is advantageous in giving one flexibility about the choice of the detuning parameter. We next examine the case when the behavior of the system is multistable. For this purpose, we choose the cavity to have the higher quality factor. We choose κ = 107 s−1, F = 1884, G = 5×106 s−1 and θ = 3π/4; then to satisfy the stability conditions (2.12), the detuning must satisfy Δ0 ≥ 1.847×107 s−1. Figure 2.4 gives the behavior of χqs as a function of the detuning Δ0. We find the χqs is multivalued, so the movable 38 2.0 2.2 2.4 2.6 2.8 3.0 0.0 0.5 1.0 1.5 2.0 D0 H107 s1 L Teff HKL , r Figure 2.5: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1). Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5×106s−1, parametric phase θ = 3π/4. mirror is multistable. By use of the lowest curve of the χqs, we obtain the variations of the effective temperature Teff and the parameter r with the detuning Δ0, as shown in Fig. 2.5. We choose that the range of the detuning is 2.0 × 107 s−1 − 3.0 × 107 s−1. The minimum temperature achieved is 0.265 K for Δ0 = 2.0 × 107 s−1. Note that r is close to unity but larger than unity. The general trend is clear. By playing around with various parameters such as laser power, cavity finesse, and parametric gain, one can achieve a variety of different temperatures. As another example, if we choose κ = 5 × 106 s−1, F = 3768, G = 107 s−1 and θ = 0.2467 + π/2, then we find that the minimum temperature is 0.092 K for Δ0 = 2.13 × 107s−1. 2.4.2 From 1 K to Millikelvin Temperatures If the thermal bath is cryogenically cooled down to a temperature of 1 K and the mirror is initially thermalized, then we can use radiation pressure effects and photon 39 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0. 0.5 1. 1.5 2. 2.5 3. D0 H107 s 1L Teff HKL r Figure 2.6: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 0. statistics to reach millikelvin or even lower temperatures. If we choose κ = 108 s−1, F = 188.4, G = 0, the effective temperature Teff with the detuning Δ0 is shown in Fig. 2.6. The minimum temperature reached is 0.051 K for Δ0 = 4.9 × 107 s−1. Next we examine how the effective temperature changes by the parametric interactions inside the cavity. We keep all other parameters as in Fig. 6 and choose parametric gain G = 3.5 × 107 s−1 and phase θ = 0. Then the effective temperature Teff with the detuning Δ0 exhibits behavior as shown in Fig. 2.7. The minimum temperature achieved is 0.0044 K for Δ0 = 7.9 × 107 s−1, a factor of 12 lower than the one without parametric interaction. Finally it should be borne in mind that the radiation pressure depends on the number operator and then it is sensitive to the photon statistics of the field in the cavity. The photon statistics can be calculated from the quantum Langevin equations (2.8). It can be proved that the Wigner function W of the field in the cavity is 40 6 7 8 9 10 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0. 1. 2. 3. 4. D0 H107 s 1L Teff HKL r Figure 2.7: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107 s−1, parametric phase θ = 0. Gaussian of the form exp[μ(α − cs)2 + ν(α∗ − c∗ s)2 + λ(α − cs)(α∗ − c∗ s)] with μ, ν, λ determined by κ, Δ, G, θ, etc. The photon number distribution [103] associated with such a Gaussian Wigner function depends in an important way on the parameter μ and the inequality of μ and ν. The latter depend on G ̸= 0 or on the presence of OPA in the cavity. 2.5 Conclusions In conclusion, we have demonstrated how the addition of a parametric amplifier in a cavity can lead to cooling of the micromirror to a temperature; which is much lower than what is achieved in an identical experiment without the use of a parametric amplifier. The parametric processes inside the cavity change the quantum statistics of the field in the cavity. This change leads to lower cooling since the radiation pressure 41 effects are dependent on the photon number. Thus photon statistics becomes central to achieve lower cooling temperatures. The use of parametric processes could provide us with a way to cool the mirror to its quantum ground state or even squeeze it. The content of this chapter has been published in Phys. Rev. A 79, 013821 (2009). 42 CHAPTER 3 NORMAL MODE SPLITTING IN A COUPLED SYSTEM OF A NANOMECHANICAL OSCILLATOR AND A PARAMETRIC AMPLIFIER CAVITY 3.1 Overview Recently there has been a major effort in applying many of the well tested ideas from quantum optics such as squeezing, quantum entanglement to optomechanical systems which are macroscopic systems. Thus observation of entanglement [28, 35, 36, 38, 114, 115], squeezing [25, 26] etc in optomechanical systems would enable one to study quantum behavior at macroscopic scale. This of course requires cooling such systems to their ground state and significant advances have been made in cooling the mechanical mirror to far below the temperature of the environment [7, 8, 9, 116, 117, 118, 119]. Further it has been pointed out that using optical back action one can possibly achieve the ground state cooling in the resolved sideband regime where the frequency of the mechanical mirror is much larger than the cavity decay rate, that is ωm ≫ κ [47, 66, 120]. Another key idea from quantum optics is the vacuum Rabi splitting [121, 122] which is due to strong interaction between the atoms and the cavity mode. The experimentalists have worked hard over the years to produce stronger and stronger couplings to produce larger and larger splittings [123, 124, 213]. Application of these ideas to macroscopic systems is challenging as well. In a recent paper Kippenberg et al. [48] proposed the possibility of normal mode splitting in the resolved sideband regime using optomechanical oscillators. In this chapter, we propose placing a type 43 I optical parametric amplifier inside the cavity to increase the coupling between the movable mirror and the cavity field, and this should make the observation of the normal mode splitting of the movable mirror and the output field more accessible. The chapter is structured as follows. In Sec. II we present the model, derive the quantum Langevin equations, and give the steadystate mean values. In Sec. III we present solution to the linearized Langevin equations and give the spectrum of the movable mirror. In Sec. IV we analyse and estimate the amount of the normal mode splitting of the spectra. In Sec. V we calculate the spectra of the output field. In Sec. VI we discuss the mode splitting of the spectra of the movable mirror and the output field. 3.2 Model The system under consideration, sketched in Fig. 3.1, is an optical parametric amplifier (OPA) placed within a FabryPerot cavity formed by one fixed partially transmitting mirror and one movable perfectly reflecting mirror in equilibrium with its environment at a low temperature. The movable mirror is treated as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm, and energy decay rate γm. An external laser enters the cavity through the fixed mirror, then the photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer. This force is proportional to the instantaneous photon number in the cavity. In the adiabatic limit, the frequency ωm of the movable mirror is much smaller than the free spectral range of the cavity c 2L (c is the speed of light in vacuum and L is the cavity length), the scattering of photons to other cavity modes can be ignored, thus only one cavity mode ωc is considered [64, 106]. The Hamiltonian for the system in a frame rotating at the laser frequency ωL can be written as H = ¯h(ωc − ωL)nc − ¯hωmχncQ + ¯hωm 4 (Q2 + P2) 44 in c out c movable mirror OPA cavity axis fixed mirror Figure 3.1: Sketch of the studied system. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. +i¯hε(c† − c) + i¯hG(eiθc†2 − e−iθc2). (3.1) Here Q and P are the dimensionless position and momentum operators for the movable mirror, defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with [Q, P] = 2i. In Eq. (3.1), the first term is the energy of the cavity field, nc = c†c is the number of the photons inside the cavity, c and c† are the annihilation and creation operators for the cavity field satisfying the commutation relation [c, c†] = 1. The second term comes from the coupling of the movable mirror to the cavity field via radiation pressure, the dimensionless parameter χ = 1 ωm ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term corresponds the energy of the movable mirror. The fourth term describes the coupling between the input laser field and the cavity field, ε is related to the input laser power ℘ by ε = √ 2κ℘ ¯hωL , where κ is the cavity decay rate. The last term is the coupling between the OPA and the cavity field, G is the nonlinear gain of the OPA, and θ is the phase of the field driving the OPA. The parameter G is proportional to the pump driving the OPA. Using the Heisenberg equations of motion and adding the corresponding damping 45 and noise terms, we obtain the quantum Langevin equations as follows, ˙Q = ωmP, ˙P = 2ωmχnc − ωmQ − γmP + ξ, ˙ c = −i(ωc − ωL − ωmχQ)c + ε + 2Geiθc† − κc + √ 2κcin, ˙ c† = i(ωc − ωL − ωmχQ)c† + ε + 2Ge−iθc − κc† + √ 2κc † in. (3.2) Here we have introduced the input vacuum noise operator cin with zero mean value, which obeys the correlation function in the time domain [141] ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = ⟨δc † in(t)δcin(t′)⟩ = 0. (3.3) The force ξ is the Brownian noise operator resulting from the coupling of the movable mirror to the thermal bath, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = 1 π γm ωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (3.4) where kB is the Boltzmann constant and T is the thermal bath temperature. Following standard methods from quantum optics [110], we derive the steadystate solution to Eq. (3.2) by setting all the time derivatives in Eq. (3.2) to zero. They are Ps = 0, Qs = 2χcs2, cs = κ − iΔ + 2Geiθ κ2 + Δ2 − 4G2 ε, (3.5) where Δ = ωc − ωL − ωmχQs (3.6) is the effective cavity detuning, depending on Qs. The Qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. From Eq. (3.5) and Eq. (3.6), we can see Qs satisfies a fifth order equation, it can at most have five real 46 solutions. Therefore, the movable mirror displays an optical multistable behavior [41, 42, 43], which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. 3.3 Radiation Pressure and Quantum Fluctuations In order to investigate the normal mode splitting of the movable mirror and the output field, we need to calculate the fluctuations of the system. Since the problem is nonlinear, we assume that the nonlinearity is weak. Thus we can focus on the dynamics of small fluctuations around the steady state of the system. Each operator of the system can be written as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (3.7) Inserting Eq. (3.7) into Eq. (3.2), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators take the form δ ˙Q = ωmδP, δ ˙P = 2ωmχ(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + iωmχcsδQ + 2Geiθδc† + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − iωmχc∗ sδQ + 2Ge−iθδc + √ 2κδc † in. (3.8) Introducing the cavity field quadratures δx = δc + δc† and δy = i(δc† − δc), and the input noise quadratures δxin = δcin + δc † in and δyin = i(δc † in − δcin), Eq. (3.8) can be rewritten in the matrix form f˙(t) = Af(t) + η(t), (3.9) 47 in which f(t) is the column vector of the fluctuations, η(t) is the column vector of the noise sources. Their transposes are f(t)T = (δQ, δP, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (3.10) and the matrix A is given by A = 0 ωm 0 0 −ωm −γm ωmχ(cs + c∗ s) −iωmχ(cs − c∗ s) iωmχ(cs − c∗ s) 0 2Gcos θ − κ 2Gsin θ + Δ ωmχ(cs + c∗ s) 0 2Gsin θ − Δ −(2Gcos θ + κ) . (3.11) The system is stable only if all the eigenvalues of the matrix A have negative real parts. The stability conditions for the system can be derived by applying the Routh Hurwitz criterion [112, 113]. This gives 2κ(κ2 − 4G2 + Δ2 + 2κγm) + γm(2κγm + ω2m ) > 0, 2ω3m χ2(2κ + γm)2[cs2Δ + iG(c2s e−iθ − c∗2 s eiθ)] +κγm{(κ2 − 4G2 + Δ2)2 + (2κγm + γ2m ) ×(κ2 − 4G2 + Δ2) + ω2m [2(κ2 + 4G2 − Δ2) +ω2m + 2κγm]} > 0, κ2 − 4G2 + Δ2 − 4ωmχ2[cs2Δ + iG(c2s e−iθ − c∗2 s eiθ)] > 0. (3.12) All the external parameters must be chosen to satisfy the stability conditions (3.12). Taking Fourier transform of Eq. (3.8) by using f(t) = 1 2π ∫ +∞ −∞ f(ω)e−iωtdω and f†(t) = 1 2π ∫ +∞ −∞ f†(−ω)e−iωtdω, where f†(−ω) = [f(−ω)]†, then solving it, we obtain 48 the position fluctuations of the movable mirror δQ(ω) = − ωm d(ω) [2 √ 2κωmχ{[(κ − i(Δ + ω))c∗ s + 2Ge−iθcs]δcin(ω) +[(κ + i(Δ − ω))cs + 2Geiθc∗ s]δc † in(−ω)} +[(κ − iω)2 + Δ2 − 4G2]ξ(ω)], (3.13) where d(ω) = 4ω3m χ2[Δcs2 + iG(c2s e−iθ − c∗2 s eiθ)] +(ω2 − ω2m + iγmω)[(κ − iω)2 + Δ2 − 4G2]. (3.14) In Eq. (3.13), the first term proportional to χ originates from radiation pressure, while the second term involving ξ(ω) is from the thermal noise. So the position fluctuations of the movable mirror are now determined by radiation pressure and the thermal noise. In the case of no coupling with the cavity field, the movable mirror will make Brownian motion, δQ(ω) = ωmξ(ω)/(ω2m −ω2−iγmω), whose susceptibility has a Lorentzian shape centered at frequency ωm with width γm. The spectrum of fluctuations in position of the movable mirror is defined by 1 2 (⟨δQ(ω)δQ(Ω)⟩ + ⟨δQ(Ω)δQ(ω)⟩) = 2πSQ(ω)δ(ω + Ω). (3.15) To calculate the spectrum, we require the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(−Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 4π γm ωm ω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (3.16) Substituting Eq. (3.13) and Eq. (3.16) into Eq. (3.15), we obtain the spectrum of fluctuations in position of the movable mirror [126] SQ(ω) = ω2m d(ω)2 {8ω2m χ2κ[(κ2 + ω2 + Δ2 + 4G2)cs2 +2Geiθc∗2 s (κ − iΔ) + 2Ge−iθc2s (κ + iΔ)] +2γm ωm ω[(Δ2 + κ2 − ω2 − 4G2)2 + 4κ2ω2] ×coth( ¯hω 2kBT )}. (3.17) 49 In Eq. (3.17), the first term involving χ arises from radiation pressure, while the second term originates from the thermal noise. So the spectrum SQ(ω) of the movable mirror depends on radiation pressure and the thermal noise. 3.4 Normal Mode Splitting and the Eigenvalues of the Matrix A The structure of all the spectra is determined by the eigenvalues of iA (Eq. (3.11)) or the complex zeroes of the function d(ω) defined by Eq. (3.14). Clearly we need the eigenvalues of iA as the solution of (Eq. (3.9)) in Fourier domain is f(ω) = i(ω − iA)−1η(ω). Let us analyse the eigenvalues of Eq. (3.11). Note that in the absence of the coupling χ=0, the eigenvalues of iA are ± √ ω2m − γ2m 4 − iγm 2 ;± √ Δ2 − 4G2 − iκ. (3.18) Thus the positive frequencies of the normal modes are given by √ Δ2 − 4G2, √ ω2m − γ2m 4 (Δ > 2G, ωm > γm 2 ). The case that we consider in this chapter corresponds to ωm ≫ γm 2 ;Δ > 2G; κ ≫ γm; ωm > κ. (3.19) The coupling between the normal modes would be most efficient in the degenerate case i.e. when ωm = √ Δ2 − 4G2. It is known from cavity QED that the normal mode splitting leads to symmetric (asymmetric) spectra in the degenerate (nondegenerate) case, provided that the dampings of the individual modes are much smaller than the coupling constant. Thus the mechanical oscillator is like the atomic oscillator, the cavity mode in the rotating frame acquires the effective frequency √ Δ2 − 4G2 which is dependent on the parametric coupling. All this applies provided that damping terms do not mix the modes significantly. An estimate of the splitting can be made by using the approximations given by Eq. (3.19) and the zeroes of d(ω). We find that the frequency splitting is given by [127] ω2± ∼= ω2m +Δ2−4G2 2 ± √ (ω2m −Δ2+4G2 2 )2 + 4ω2m g2, (3.20) 50 where we have defined g2 = ωmχ2cs2[Δ + 2Gsin(θ − 2φ)], e2iφ = c2s /cs2. (3.21) It should be borne in mind that cs is dependent on the parametric coupling G. The splitting is determined by the pump power, the couplings χ and G. The parameters used are the same as those in the recent successful experiment on optomechanical normal mode splitting [50]: the wavelength of the laser λ = 2πc/ωL = 1064 nm, L = 25 mm, m = 145 ng, κ = 2π × 215 × 103 Hz, ωm = 2π × 947 × 103 Hz, T = 300 mK, the mechanical quality factor Q ′ = ωm/γm = 6700, parametric phase θ = π/4. And in the high temperature limit kBT ≫ ¯hωm, we have coth(¯hω/2kBT) ≈ 2kBT/¯hω. Figure 3.2 shows the roots of d(ω) in the domain Re(ω) > 0 for different values of G. Figure 3.3 shows imaginary parts of the roots of d(ω) for different values of G. The parametric coupling affects the width of the lines and this for certain range of parameters aids in producing well split lines. One root broadens and the other root narrows. The root that broadens is the one that moves further away from the position for G = 0. 3.5 The Spectra of the Output Field In this section, we would like to calculate the spectra of the output field. The fluctuations δc(ω) of the cavity field can be obtained from Eq. (3.8). Further using the inputoutput relation [128] cout(ω) = √ 2κc(ω)−cin(ω), the fluctuations of the output field are given by δcout(ω) = V (ω)ξ(ω) + E(ω)δcin(ω) + F(ω)δc † in(−ω), (3.22) 51 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 G k ReHwL wm Figure 3.2: The roots of d(ω) in the domain Re(ω) > 0 as a function of parametric gain. ℘ = 6.9 mW (dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 G Κ ImHwL wm Figure 3.3: The imaginary parts of the roots of d(ω) as a function of parametric gain. ℘ = 6.9 mW ( dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. 52 where V (ω) = − √ 2κω2m χ d(ω) i{[κ − i(ω + Δ)]cs − 2Geiθc∗ s }, E(ω) = 2κ (κ−iω)2+Δ2−4G2 [−2ω3m χ2 d(ω) i{[κ − i(ω + Δ)]cs −2Geiθc∗ s }{[κ − i(ω + Δ)]c∗ s + 2Ge−iθcs} +κ − i(ω + Δ)] − 1, F(ω) = 2κ (κ−iω)2+Δ2−4G2 [−2ω3m χ2 d(ω) i{[κ − i(ω + Δ)]cs −2Geiθc∗ s }{[κ − i(ω − Δ)]cs + 2Geiθc∗ s } +2Geiθ]. (3.23) In Eq. (3.22), the first term associated with ξ(ω) stems from the thermal noise of the mechanical oscillator, while the other two terms are from the input vacuum noise. So the fluctuations of the output field are influenced by the thermal noise and the input vacuum noise. The spectra of the output field are defined as ⟨δc † out(−Ω)δcout(ω)⟩ = 2πScout(ω)δ(ω + Ω), ⟨δxout(Ω)δxout(ω)⟩ = 2πSxout(ω)δ(ω + Ω), ⟨δyout(Ω)δyout(ω)⟩ = 2πSyout(ω)δ(ω + Ω). (3.24) where δxout(ω) and δyout(ω) are the Fourier transform of the fluctuations δxout(t) and δyout(t) of the output field , which are defined by δxout(t) = δcout(t) + δc † out(t) and δyout(t) = i[δc † out(t) − δcout(t)] [110]. Here Scout(ω) denotes the spectral density of the output field, Sxout(ω) means the spectrum of fluctuations in the x quadrature of the output field, and Syout(ω) is the spectrum of fluctuations in the y quadrature of the output field. Combining Eq. (3.16), Eq. (3.22), and Eq. (3.24), we obtain the spectra of the 53 output field Scout(ω) = V ∗(ω)V (ω) × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] + F∗(ω)F(ω), Sxout(ω) = [V (−ω) + V ∗(ω)][V (ω) + V ∗(−ω)] × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] +[E(−ω) + F∗(ω)][F(ω) + E∗(−ω)], Syout(ω) = −[V ∗(ω) − V (−ω)][V ∗(−ω) − V (ω)] × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] −[F∗(ω) − E(−ω)][E∗(−ω) − F(ω)]. (3.25) From Eq. (3.25), it is seen that any spectrum of the output field includes two terms, the first term is from the contribution of the thermal noise of the mechanical oscillator, the second term is from the contribution of the input vacuum noise. We note that the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) are determined by the detuning Δ, parametric gain G, parametric phase θ, input laser power ℘, and cavity length L. In the following we will concentrate on discussing the dependence of the spectra on parametric gain and input laser power. 3.6 Numerical Results In this section, we numerically evaluate the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) given by Eq. (3.17) and Eq. (3.25) to show the effect of an OPA in the cavity on the normal mode splitting of the movable mirror and the output field. We typically imagine a setup like in the original squeezing experiment [97] where the experiment is done, for different levels of the pumping of OPA i.e., we start with G = 0 and then increase it to a value consistent with the stability requirements. We consider the degenerate case Δ = ωm for G = 0, and choose ℘ = 6.9 mW. In order to satisfy the stability conditions (3.12), parametric gain must satisfy G ≤ 1.62κ. The figures 3.4 – 3.7 show the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) as a function of the normalized frequency ω/ωm for various values of parametric gain. When the OPA is absent (G = 0), the spectra barely show the normal mode splitting. As 54 0.0 0.5 1.0 1.5 2.0 0.000 0.005 0.010 0.015 0.020 w wm SQHwLΓm G=1.45Κ G=1.3Κ G=0 Figure 3.4: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 w wm Scout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.5: The spectrum Scout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. 55 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 w wm Sxout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.6: The spectrum Sxout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. parametric gain is increased, the normal mode splitting becomes observable. This is due to significant changes in the line widths and position. When G = 1.3κ, two peaks can be found in the spectra. According to the numerical calculations of Figs. 3.2 and 3.3, these roots in units of ωm are at (A) G = 0: 0.885−0.113i, 1.091−0.113i for 6.9 mW pump power and 0.826 − 0.113i, 1.136 − 0.113i for 10.7 mW pump power. (B) G = 1.3κ: 0.596−0.156i, 1.129−0.070i for 6.9 mW pump power and 0.490−0.148i, 1.178−0.079i for 10.7 mW pump power. We see that the line width of the two peaks is approximately same for G = 0 but for two different power levels. The line widths change significantly for G ̸= 0. Note that the separation between two peaks becomes larger as parametric gain increases. The reason is that increasing the parametric gain causes a stronger coupling between the movable mirror and the cavity field due to an increase in the photon number in the cavity. The values of intracavity photon number cs2 are 2.68 × 109, 4.30 × 109, 5.65 × 109 for G = 0, 1.3κ, and 1.45κ respectively. 56 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 w wm Syout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.7: The spectrum Syout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. We have examined the contributions of various terms in Eq. (3.25) to the output spectrum. The dominant contribution comes from the mechanical oscillator. Note further the similarity [50] of the spectrum of the output quadrature y (Fig. 3.7) to the spectrum of the mechanical oscillator (Fig. 3.4). It should be borne in mind that the strong asymmetries in the spectra for G ̸= 0 arise from the fact that by fixing Δ at ωm, the frequencies of the cavity mode and the mechanical oscillator do not coincide if G ̸= 0; χ = 0. Besides the damping term κ, being not negligible compared to Δ, also contributes to asymmetries. Now we fix parametric gain G = 1.3κ, and choose Δ = √ ω2m + 4G2, the input laser power must satisfy ℘ ≤ 55 mW. The spectrum SQ(ω) as a function of the normalized frequency ω/ωm for increasing the input laser power is shown in Fig. 3.8. As we increase the laser power from 0.6 mW to 10.7 mW, the spectrum exhibits a doublet and the peak separation is proportional to the laser power, because the 57 coupling between the movable mirror and the cavity field for a given parametric gain G is increased with increasing the input laser power due to an increase in photon number. 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0. 0.005 0.01 0.015 0.02 0.025 0.03 w wm SQHwLΓm Ã=10.7 mW Ã=6.9 mW Ã=0.6 mW Figure 3.8: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = √ ω2m + 4G2, parametric gain G = 1.3κ. For comparison, we also consider the case of the cavity without OPA (G = 0), the spectrum SQ(ω) as a function of the normalized frequency ω/ωm for increasing the input laser power at Δ = ωm is plotted in Fig. 3.9. We can see if the laser power is increased from 0.6 mW to 10.7 mW, the spectrum also displays normal mode splitting. However the normal mode with OPA (Fig. 3.8) are more pronounced than that in the absence of OPA (Fig. 3.9). 58 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 w wm SQHwLΓm Ã=10.7 mW Ã=6.9 mW Ã=0.6 mW Figure 3.9: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = ωm, parametric gain G = 0. 3.7 Conclusions In conclusion, we have shown how the normal mode splitting behavior of the movable mirror and the output field is affected by the OPA in the cavity. We work in the resolved sideband regime and operate under the stability conditions (3.12). We find that increasing parametric gain can make the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) evolve from a single peak to two peaks. Furthermore, for a given parametric gain, increasing input laser power can increase the amount of normal mode splitting of the movable mirror due to the stronger coupling between the movable mirror and the cavity field. The content of this chapter has been published in Phys. Rev. A 80, 033807 (2009). 59 CHAPTER 4 SQUEEZING OF A NANOMECHANICAL OSCILLATOR 4.1 Overview The optomechanical system has attracted much attention because of its potential applications in high precision measurements and quantum information processing [28, 35, 36, 37, 88, 90, 129, 130, 131]. Meanwhile, it provides a means of probing quantum behavior of a macroscopic object if a nanomechanical oscillator can be cooled down to near its quantum ground state [38, 115]. Many of these applications are becoming possible due to advances in cooling the mirror [6, 7, 8, 9, 10, 56, 118]. Further as pointed out in Refs [47, 66, 120], the ground state cooling can be achieved in the resolved sideband regime where the frequency of the mechanical mirror is much larger than the cavity decay rate. Squeezing of a nanomechanical oscillator plays a vital role in highsensitive detection of position and force due to its less noise in one quadrature than the coherent state. A number of different methods have been developed to generate and enhance squeezing of a nanomechanical oscillator, such as coupling a nanomechanical oscillator to an atomic gas [132], a Cooper pair box [133], a SQUID device [215], using threewave mixing [135] or Circuit QED [136], or by means of quantum measurement and feedback schemes [137, 138, 139, 140]. A recent paper [32] reports squeezed state of a mechanical mirror can be created by transfer of squeezing from a squeezed vacuum to a membrane within an optical cavity under the conditions of ground state cooling. We previously considered the possibility of using an OPA inside the cavity for changing the nature of the statistical fluctuations [126]. 60 In this chapter, we propose a scheme that is capable of generating squeezing of the movable mirror by feeding broad band squeezed vacuum light along with the laser light. The achieved squeezing of the mirror depends on the temperature of the mirror, the laser power, and degree of squeezing of the input light. One can obtain squeezing which could be more than 70%. The chapter is structured as follows. In Sec. II we describe the model, give the quantum Langevin equations, and obtain the steadystate mean values. In Sec. III we derive the stability conditions, calculate the mean square fluctuations in position and momentum of the movable mirror. In Sec. IV we analyze how the momentum squeezing of the movable mirror is affected by the squeezing parameter, the temperature of the environment, and the laser power. We also compare the momentum fluctuations of the movable mirror in the presence of the coupling to the cavity field with that in the absence of the coupling to cavity field. We find very large squeezing with respect to thermal fluctuations, for instance at 1 mK, the momentum fluctuations go down by a factor more than one hundred. Our predictions of squeezing are based on the parameters used in a recent experiment on normal mode splitting in a nanomechanical oscillator [50]. 4.2 Model The system to be considered, sketched in Fig. 4.1, is a FabryPerot cavity with one fixed partially transmitting mirror and one movable perfectly reflecting mirror in thermal equilibrium with its environment at a low temperature. The cavity with length L is driven by a laser with frequency ωL, then the photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer. This force is proportional to the instantaneous photon number in the cavity. The mirror also undergoes thermal fluctuations due to environment. Under the effects of the two forces, the movable mirror makes oscillation around its equilibrium position. 61 Here we treat the movable mirror as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm and momentum decay rate γm. We further assume that the cavity is fed with squeezed light at frequency ωS. Figure 4.1: Sketch of the studied system. A laser with frequency ωL and squeezed vacuum light with frequency ωS enter the cavity through the partially transmitting mirror. In the adiabatic limit, ωm ≪ c 2L ( c is the speed of light in vacuum), we ignore the scattering of photons to other cavity modes, thus only one cavity mode ωc is considered [64, 106]. In a frame rotating at the laser frequency, the Hamiltonian for the system can be written as H = ¯h(ωc − ωL)nc − ¯hgncQ + ¯hωm 4 (Q2 + P2) + i¯hε(c† − c), (4.1) we have used the normalized coordinates for the oscillator defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with [Q, P] = 2i. This normalization implies that in the ground state of the nanomechanical mirror ⟨Q2⟩ = ⟨P2⟩ = 1. Further in Eq. (4.1) the first term is the energy of the cavity field, nc = c†c is the number of the photons inside the cavity, c and c† are the annihilation and creation operators for the cavity field with [c, c†] = 1. The second term comes from the coupling of the movable mirror to the cavity field via radiation pressure, the parameter g = ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term corresponds the energy of the movable mirror. The fourth term describes 62 the coupling between the input laser field and the cavity field, ε is related to the input laser power ℘ by ε = √ 2κ℘ ¯hωL , where κ is the cavity decay rate associated with the transmission loss of the fixed mirror. The equations of motion of the system can be derived by the Heisenberg equations of motion and adding the corresponding noise terms, this gives the quantum Langevin equations ˙Q = ωmP, ˙P = 2gnc − ωmQ − γmP + ξ, ˙ c = i(ωL − ωc + gQ)c + ε − κc + √ 2κcin, ˙ c† = −i(ωL − ωc + gQ)c† + ε − κc† + √ 2κc † in. (4.2) Here we have introduced the input squeezed vacuum noise operator cin with frequency ωS = ωL + ωm. It has zero mean value, and nonzero timedomain correlation functions [141] ⟨δc † in(t)δcin(t′)⟩ = Nδ(t − t′), ⟨δcin(t)δc † in(t′)⟩ = (N + 1)δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = Me−iωm(t+t′)δ(t − t′), ⟨δc † in(t)δc † in(t′)⟩ = M∗eiωm(t+t′)δ(t − t′). (4.3) where N = sinh2(r), M = sinh(r) cosh(r)eiφ, r is the squeezing parameter of the squeezed vacuum light, and φ is the phase of the squeezed vacuum light. For simplicity, we choose φ = 0. The force ξ is the thermal Langevin force resulting from the coupling of the movable mirror to the environment, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = γm πωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (4.4) where kB is the Boltzmann constant and T is the temperature of the environment. By using standard methods [110], setting all the time derivatives in Eq. (4.2) to 63 zero, and solving it, we obtain the steadystate mean values Ps = 0, Qs = 2gcs2 ωm , cs = ε κ + iΔ , (4.5) where Δ = ωc − ωL − gQs = Δ0 − gQs = Δ0 − 2g2cs2 ωm (4.6) is the effective cavity detuning, depending on Qs. The Qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. From Eq. (4.5) and Eq. (4.6), we can see Qs satisfies a third order equation. For a given detuning Δ0, Qs will at most have three real values. Therefore, Qs and cs display an optical multistable behavior [41, 42, 43], which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. 4.3 Radiation Pressure and Quantum Fluctuations To study squeezing of the movable mirror, we need to calculate the fluctuations in the mirror’s amplitude. Assuming that the nonlinear coupling between the cavity field and the movable mirror is weak, the fluctuation of each operator is much smaller than the corresponding steadystate mean value, thus we can linearize the system around the steady state. Writing each operator of the system as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (4.7) 64 Inserting Eq. (4.7) into Eq. (4.2), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators can be expressed as follows, δ ˙Q = ωmδP, δ ˙P = 2g(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + igcsδQ + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − igc∗ sδQ + √ 2κδc † in. (4.8) Introducing the cavity field quadratures δx = δc + δc† and δy = i(δc† − δc), and the input noise quadratures δxin = δcin + δc † in and δyin = i(δc † in − δcin), Eq. (4.8) can be rewritten in the matrix form f˙(t) = Af(t) + η(t), (4.9) in which f(t) is the column vector of the fluctuations, η(t) is the column vector of the noise sources. Their transposes are f(t)T = (δQ, δP, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (4.10) and the matrix A is given by A = 0 ωm 0 0 −ωm −γm g(cs + c∗ s) −ig(cs − c∗ s) ig(cs − c∗ s) 0 −κ Δ g(cs + c∗ s) 0 −Δ −κ . (4.11) The system is stable only if the real parts of all the eigenvalues of the matrix A are negative. The stability conditions for the system can be derived by applying the 65 RouthHurwitz criterion [112, 113], we get κγm[(κ2 + Δ2)2 + (2κγm + γ2m − 2ω2m )(κ2 + Δ2)
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Title  Electromagnetically Induced Transparency and Quantum Effects in Optomechanical Systems 
Date  20111201 
Author  Huang, Sumei 
Keywords  Electromagnetically induced transparency, Entanglement, Normal mode splitting, Quantum fluctuations, Radiation pressure, Squeezing 
Department  Physics 
Document Type  
Full Text Type  Open Access 
Abstract  In this thesis, we study cooling, normal mode splitting, squeezing, entanglement, and electromagnetically induced transparency in the macroscopic optomechancial system. We show that a type I optical parametric amplifier in a FabryPerot cavity can considerably improve cooling of a mechanical oscillator, and it also can affect the normalmode splitting behavior of the movable mirror and the output field. Next, we discover that squeezing of a mechanical oscillator in a FabryPerot cavity and the entanglement between two mechanical oscillators in a ring cavity can be generated by injecting a broadband squeezed vacuum along with laser light. Then, we study that the strong dispersive optomechanical coupling between the optical and mechanical oscillators can induce the normal mode splitting in the two quadratures of the output fields by the injection of coherent coupling and probe fields. Further, we study that electromagnetically induced transparency can occur in the output field at the probe frequency in the linear optomechanical coupling system which is driven by a weak probe field in the presence of a strongcoupling field. Next we present that the strong reactive optomechanical coupling between the optical resonator and the waveguide can also induce the normal mode splitting in the two quadratures of the output fields by the injection of coherent coupling and probe fields. In addition, We discuss that the standard quantum limit of the waveguide can be beaten by inputting a narrowband squeezed vacuum and laser light into a reactive optomechanical coupling system. Further, we study that electromagnetically induced transparency also can occur in the output field at the probe frequency in the quadratic optomechanical coupling system. We also investigate that electromagnetically induced transparency can arise in the homodyne spectrum of the output field by use of a weak probe field in a squeezed vacuum state in the presence of a strong coupling field. Finally, we show that the optomechanical systems can serve as single photon routers. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND QUANTUM EFFECTS IN OPTOMECHANICAL SYSTEMS By SUMEI HUANG Bachelor of Science in Physics Education Fujian Normal University Fuzhou, Fujian, China 2001 Master of Science in Theoretical Physics Fujian Normal University Fuzhou, Fujian, China 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2011 COPYRIGHT c⃝ By SUMEI HUANG December, 2011 ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND QUANTUM EFFECTS IN OPTOMECHANICAL SYSTEMS Dissertation Approved: Dr. Girish S. Agarwal Dissertation advisor Dr. Xincheng Xie Dr. Albert T. Rosenberger Dr. Yin Guo Dr. Daniel Grischkowsky Dr. Jacques H. H. Perk Dr. Sheryl A. Tucker Dean of the Graduate College iii iv TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Optomechanical System . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The Dispersive Optomechanical System . . . . . . . . . . . . . 3 1.1.3 The Reactive Optomechanical System . . . . . . . . . . . . . . 5 1.2 Sideband Cooling of the Nano Mechanical Mirror . . . . . . . . . . . 6 1.3 Degenerate Parametric Amplification . . . . . . . . . . . . . . . . . . 12 1.4 Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Electromagnetically Induced Transparency . . . . . . . . . . . . . . . 16 1.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 ENHANCEMENT OF CAVITY COOLING OF A MICROMECHAN ICAL MIRROR USING PARAMETRIC INTERACTIONS 27 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 31 2.4 Cooling Mirror to About SubKelvin Temperatures . . . . . . . . . . 35 2.4.1 From Room Temperature (T=300 K) to About SubKelvin Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 From 1 K to Millikelvin Temperatures . . . . . . . . . . . . . 39 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 v 3 NORMAL MODE SPLITTING IN A COUPLED SYSTEM OF A NANOMECHANICAL OSCILLATOR AND A PARAMETRIC AM PLIFIER CAVITY 43 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 47 3.4 Normal Mode Splitting and the Eigenvalues of the Matrix A . . . . . 50 3.5 The Spectra of the Output Field . . . . . . . . . . . . . . . . . . . . . 51 3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 SQUEEZING OF A NANOMECHANICAL OSCILLATOR 60 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 64 4.4 Squeezing of the Movable Mirror . . . . . . . . . . . . . . . . . . . . . 68 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 ENTANGLING NANOMECHANICAL OSCILLATORS IN A RING CAVITY BY FEEDING SQUEEZED LIGHT 74 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Radiation Pressure and Quantum Fluctuations . . . . . . . . . . . . . 80 5.4 Entanglement of the Two Movable Mirrors . . . . . . . . . . . . . . . 84 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6 NORMALMODE SPLITTING AND ANTIBUNCHING IN STOKES AND ANTISTOKES PROCESSES IN CAVITY OPTOMECHAN ICS: RADIATIONPRESSUREINDUCED FOURWAVEMIXING vi CAVITY OPTOMECHANICS 90 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 Model: Stimulated Generation of Stokes and AntiStokes fields . . . . 91 6.3 The Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.4 Normalmode Splittings in the Output Fields . . . . . . . . . . . . . . 97 6.5 Spontaneous Generation of Stokes and Antistokes Photons: Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7 THE ELECTROMAGNETICALLY INDUCED TRANSPARENCY IN MECHANICAL EFFECTS OF LIGHT 105 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 EIT in the Out Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8 REACTIVECOUPLINGINDUCED NORMAL MODE SPLITTINGS IN MICRODISK RESONATORS COUPLED TO WAVEGUIDES 114 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.3 Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.4 Normal Mode Splitting In Output Fields . . . . . . . . . . . . . . . . 120 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9 CAN REACTIVE COUPLING BEAT MOTIONAL QUANTUM LIMIT OF NANO WAVEGUIDES COUPLED TO MICRODISK RESONATOR 125 9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 vii 9.3 Beating the Motional Quantum Limit for the Waveguide . . . . . . . 128 9.4 Numerical Results for Nano Waveguide Fluctuations below Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10 ELECTROMAGNETICALLY INDUCED TRANSPARENCY FROM TWO PHOTON PROCESSES IN QUADRATICALLY COUPLED MEMBRANES 135 10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.3 EIT in the Output Field . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11 ELECTROMAGNETICALLY INDUCED TRANSPARENCY WITH QUANTIZED FIELDS IN OPTOCAVITY MECHANICS 148 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.3 The Output Field and its Measurement . . . . . . . . . . . . . . . . . 152 11.4 EIT in the Homodyne Spectrum of the Output Quantized Field . . . 155 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12 OPTOMECHANICAL SYSTEMS AS SINGLE PHOTON ROUTERS160 12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.3 EIT in the Reflection Spectrum of the Single Photon . . . . . . . . . 166 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 13 SUMMARY AND FUTURE DIRECTIONS 170 13.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 viii 13.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 BIBLIOGRAPHY 174 A THE VARIANCE OF MOMENTUMDERIVATION OF EQUA TION EQ. (9.18) 191 B RELATION BETWEEN THE QUANTUM FLUCTUATIONS OF NANO WAVEGUIDE AND THE OUTPUT FIELD 193 ix LIST OF TABLES Table Page x LIST OF FIGURES Figure Page 1.1 A FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror. . . . . . . . . . . . . . . . . . . 3 1.2 The optomechaical system that consists of a microdisk resonator coupled to a waveguide (from Ref.[67]). . . . . . . . . . . . . . . . . . . . 5 1.3 The effective temperature Teff (mK) of the movable mirror as a function of the laser power ℘ (μW). The initial temperature is taken to be 1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Parametric amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Balanced homodyne detection. PD:photodetector. . . . . . . . . . . . 15 1.6 A threelevel Λtype atomic system, where the probe field at frequency ν couples levels b⟩ and a⟩, while the coupling field at frequency νμ couples levels c⟩ and a⟩. . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 The real part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. . . . . . . . . . . . . . . . . . . . . . . . 21 1.8 The imaginary part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. . . . . . . . . . . . . . . . . . . 22 xi 2.1 Sketch of the cavity used to cool a micromechanical mirror. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 The dotted curve indicates the χqs (106 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G=0. . . . . . . . . . . . . . . . . . . . . 36 2.3 The dotted curve indicates the χqs (107 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5×107s−1, parametric phase θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 The behavior of χqs (107 s−1) shown as a function of the detuning Δ0 (107 s−1). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. . . . . . . . . . . . . . . 38 xii 2.5 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1). Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. . . . . . . . . . . . . . . 39 2.6 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 0. . . . . . . . . . 40 2.7 The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107 s−1, parametric phase θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Sketch of the studied system. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. . . . . . . 45 3.2 The roots of d(ω) in the domain Re(ω) > 0 as a function of parametric gain. ℘ = 6.9 mW (dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. . . . . . . . . . . . . . . . . 52 3.3 The imaginary parts of the roots of d(ω) as a function of parametric gain. ℘ = 6.9 mW ( dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. . . . . . . . . . . . . . . . . 52 xiii 3.4 The scaled spectrum SQ(ω)×γm versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 The spectrum Scout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6 The spectrum Sxout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 The spectrum Syout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.8 The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = √ ω2m + 4G2, parametric gain G = 1.3κ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.9 The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = ωm, parametric gain G = 0. 59 xiv 4.1 Sketch of the studied system. A laser with frequency ωL and squeezed vacuum light with frequency ωS enter the cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1) for different values of the squeezing of the input field. r = 0 (red, big dashed line), r = 0.5 (green, small dashed line), r = 1 (black, solid curve), r = 1.5 (blue, dotdashed curve), r = 2 (brown, solid curve). The minimum values of ⟨δ ˜ P2⟩ are 1.071 (r=0), 0.467 (r=0.5), 0.319 (r=1), 0.468 (r=1.5), 1.078 (r=2). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the temperature of the environment T = 1 mK, the laser power ℘ = 6.9 mW. . . . . . 69 4.3 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (blue, solid curve), 1 mK (red, small dashed curve), 5 mK (brown, big dashed curve), 10 mK (green, dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.252 (T=0 K), 0.611 (T=1 mK), 2.082 (T=5 mK), 3.919 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 0.6 mW. . . . . . . . . . . . . . . . . . . . 71 4.4 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (solid curve), 1 mK (dashed curve), 10 mK (dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.261 (T=0 K), 0.330 (T=1 mK), 0.968 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 3.8 mW. . . . . . . . . . . . . . . . . . . . . . . . 72 xv 4.5 The mean square fluctuations ⟨δ ˜ P2⟩ versus the detuning Δ0 (106 s−1), each curve corresponds to a different temperature of the environment. T=0 K (solid curve), 1 mK (dashed curve), 10 mK (dotdashed curve). The minimum values of ⟨δ ˜ P2⟩ are 0.275 (T=0 K), 0.319 (T=1 mK), 0.731 (T=10 mK). The flat dotted line represents the variance of the coherent light (⟨δ ˜ P2⟩=1). Parameters: the squeezing parameter r = 1, the laser power ℘ = 6.9 mW. . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 Sketch of the studied system. A laser with frequency ωL and a squeezed vacuum light with frequency ωS enter the ring cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 The mean square fluctuations ⟨δ ˜ P2− ⟩ versus the detuning Δ/ωm for different values of the squeezing of the input field. r = 0 (red, big dashed line), r = 0.5 (green, small dashed line), r = 1 (black, solid curve), r = 1.5 (blue, dotdashed curve), r = 2 (brown, solid curve). The minimum values of ⟨δ ˜ P2− ⟩ are 1.027 (r=0), 0.422 (r=0.5), 0.271(r=1), 0.412 (r=1.5), 0.999 (r=2). The flat dotted line represents ⟨δ ˜ P2− ⟩=1. Parameters: the temperature of the environment T = 41.4 μK, the laser power ℘ = 3.8 mW. . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 The mean square fluctuations ⟨δ ˜ P2−⟩ versus the detuning Δ/ωm, each curve corresponds to a different laser power. ℘=0.6 mW (red, big dashed curve), 3.8 mW (green, small dashed curve), 6.9 mW (black, solid curve), 10.7 mW (blue, dotdashed curve). The minimum values of ⟨δ ˜ P2− ⟩ are 0.257 (℘=0.6 mW), 0.271 (℘=3.8 mW), 0.291 (℘=6.9 mW), 0.315 (℘=10.7 mW). The flat dotted line represents ⟨δ ˜ P2− ⟩=1. Parameters: the squeezing parameter r = 1, the temperature of the environment T = 41.4 μK. . . . . . . . . . . . . . . . . . . . . . . . . 86 xvi 5.4 The value of ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩ versus the temperature of the environment T (μK). The minimum value of ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩ is 0.135 at T = 0 K. The flat dotted line represents ⟨δQ2 + ⟩⟨δ ˜ P2− ⟩=1. Parameters: the squeezing parameter r = 1, the laser power ℘ = 3.8 mW, the detuning Δ = 0.965ωm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.5 Sketch of 4mirror ring cavity. A laser with frequency ωL and squeezed vacuum light with frequency ωS = ωL + ωm enter the ring cavity through the partially transmitting fixed mirror 1. The fixed mirror 2 and the two identical movable mirrors are perfectly reflecting. . . . 88 6.1 Sketch of the studied system. A pump field with frequency ωl and a Stokes field with frequency ωs enter the cavity through the partially transmitting mirror. The output fields cout have three components (ωl, ωs, 2ωl − ωs). No vacuum fields are shown here because we are examining only the mean response. . . . . . . . . . . . . . . . . . . . 92 6.2 The roots of d(ωs −ωl) in the domain Re(ωs −ωl) > 0 as a function of the pump power ℘. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 The imaginary parts of the roots of d(ωs − ωl) as a function of the pump power ℘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 The normalized quadrature vs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . . . . 98 6.5 The normalized quadrature ˜vs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . . . . 99 xvii 6.6 The normalized output power Gs plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . . 99 6.7 The normalized output power Gas plotted as a function of the normalized frequency (ωs − ωl)/ωm for different pump power. ℘ = 1 mW (solid curve), 6.9 mW (dotted curve), and 20 mW (dashed curve). . 100 6.8 The normalized secondorder correlation function g(2)(τ ) as a function of the time delay τ (μs) for different pump powers at T = 0K. ℘=1 mW (solid curve), and 4 mW(dotted curve). . . . . . . . . . . . . . . . . . 103 7.1 Sketch of the optomechanical system coupled to a highquality cavity via radiation pressure effects. . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Quadrature of the output field υp (solid black curve) and the different contributions to it: the real parts of A+ x−x+ (dotted red curve) and A− x−x− (dashed green curve) as a function of the normalized frequency x/ωm for input coupling laser power ℘c = 1 mW. The dotdashed blue curve is υp in the absence of the coupling laser. . . . . . . . . . . . . . . . . 110 7.3 Quadrature of the output field ˜υp (solid black curve) and the different contributions to it: the imaginary parts of A+ x−x+ (dotted red curve) and A− x−x− (dashed green curve) as a function of the normalized frequency x/ωm for input coupling laser power ℘c = 1 mW. The dotdashed blue curve is ˜υp in the absence of the coupling laser. . . . . . . . . . . . . 111 7.4 Same as in Fig. 7.2 except the input coupling laser power ℘c = 6.9 mW and ℘c = 0 case is not shown. . . . . . . . . . . . . . . . . . . . 111 7.5 Same as in Fig. 7.3 except the input coupling laser power ℘c = 6.9 mW and ℘c = 0 case is not shown. . . . . . . . . . . . . . . . . . . . 112 xviii 8.1 Sketch of the studied system (from Ref.[67]). The microdisk cavity is driven by a pump field and a Stokes field. The nonlinearity of the interaction also generates antiStokes field. . . . . . . . . . . . . . . . 115 8.2 The real roots of d(δ) in the domain Re(δ) > 0 as a function of the pump power ℘l for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.3 Imaginary parts of the roots of d(δ) as a function of the pump power ℘l for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.4 The lower two curves show the normalized quadrature vs as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π×26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized quadrature vs+1.5 for pump power ℘l = 200 μW. . . . 121 8.5 The lower two curves show the normalized output power Gs as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π×26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized output power Gs+1.5 for pump power ℘l = 200 μW. . 122 8.6 The lower two curves show the normalized output power Gas as a function of the normalized detuning between the Stokes field and the pump field, δ/ωm for κom = 0 (dotted curve) and κom = −2π × 26.6 MHz/nm (solid curve) for pump power ℘l = 20 μW. The upper two curves give the normalized output power Gas+0.15 for pump power ℘l = 200 μW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 xix 9.1 The variance of momentum ⟨δP2⟩ as a function of the detuning Δ (2π×106Hz) for different temperatures of the environment: T = 1 mK (red solid), T = 10 mK (blue dotted), T = 50 mK (purple dashed), and T = 100 mK (green dotdashed). The horizontal dotted line represents the standard quantum limit (⟨δP2⟩=1). The parameters: the pump power ℘l = 20 μW, r = 1. . . . . . . . . . . . . . . . . . . . . . . . . 132 9.2 The variance of momentum ⟨δP2⟩ as a function of the pump power (μW) for different temperatures of the environment: T = 1 mK (red solid) and T = 20 mK (green dotdashed). The horizontal dotted line represents the standard quantum limit (⟨δP2⟩=1). The parameters: Δ = ωm, r = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1 Sketch of the studied system. A strong coupling field at frequency ωc and a weak probe field at frequency ωp are injected into the cavity through the left mirror. A membrane with finite reflectivity is located at the middle position of the cavity. After the interaction between the cavity field and the membrane, the output field will contain three frequencies (ωc, ωp, and 2ωc − ωp). . . . . . . . . . . . . . . . . . . . . 137 10.2 Sketch of twophonon process. For a onephonon case the corresponding condition on frequencies will be ωc + ωm = ωp ≈ ω0. . . . . . . . . 138 10.3 Level diagram for the atomic EIT. For optocavity mechanics, 1⟩ ↔ 3⟩ would be the excitation at cavity frequency; 2⟩ ↔ 3⟩ would be the excitation of the mechanical oscillator. For the quadratically coupled membrane, 2⟩ → 3⟩ would be the twophonon excitation which makes ⟨q⟩ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 xx 10.4 Quadrature of the output field υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.45, ℘c = 90 μW, T = 90 K. The inset zooms the EITlike dip. . . . . . . 145 10.5 Quadrature of the output field υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.81, ℘c = 20 μW, T = 90 K. The inset zooms the EITlike dip. . . . . . . 146 10.6 Quadrature of the output field ˜υp as a function of the normalized frequency δ/ωm in the absence (red dotted line) and presence (blue solid line) of the quadratic coupling. Parameters are as follows: R = 0.81, ℘c = 20 μW, T = 90 K. The inset zooms the change in the dispersion produced by the coupling field. . . . . . . . . . . . . . . . . . . . . . 146 11.1 Sketch of the studied system. A coherent coupling field at frequency ωc and a squeezed vacuum at frequency ωp enter the cavity through the partially transmitting mirror. . . . . . . . . . . . . . . . . . . . . 149 11.2 Sketch of the measurement of the output field. The output field ˜cout(t) is mixed with a strong local field clo(t) centered around the probe frequency ωp at a beam splitter, where ˜cout(t) is defined as the sum of the output field cout(t) from the cavity and the input quantized field cin(t). BS, 50:50 beam splitter; PD, photodetector; SA, spectrum analyzer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xxi 11.3 Homodyne spectrum X(ω) as a function of ω/ωm for N = 5 in the absence (dotted curve) and the presence (solid, dotdashed, and dashed curves) of the coupling field for the temperature of the environment T = 20 mK. The solid curve is for ℘ = 10 mW and M = √ N(N + 1), the dotdashed curve is for ℘ = 20 mW and M = √ N(N + 1), and the dashed curve is for ℘ = 20 mW and M = 0. . . . . . . . . . . . . . . 156 11.4 Homodyne spectrum X(ω) as a function of ω/ωm for different values of the parameter N and M = √ N(N + 1) in the absence (dotted curves) and the presence (solid curves) of a coupling field with power ℘ = 10 mW and temperature of the environment T = 100 mK. The upper two curves are for N = 5, and the lower two curves are for N = 1. . . . . 158 11.5 As in Fig. 11.4 but now the parameters used are from Ref. [50]. . . . 159 12.1 A doubleended cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.2 A doubleended cavity with a moving nanomechanical mirror as a single photon router. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 12.3 The reflection spectrum R(ω) of the single photon as a function of the normalized frequency ω/ωm without and with the coupling field. ℘ = 0 (solid), 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . 167 12.4 The transmission spectrum T(ω) of the single photon as a function of the normalized frequency ω/ωm without and with the coupling field. ℘ = 0 (solid), 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . 167 12.5 The vacuum noise spectrum S(v)(ω) as a function of the normalized frequency ω/ωm with the coupling field. ℘ = 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 12.6 The thermal noise spectrum S(T)(ω) as a function of the normalized frequency ω/ωm with the coupling field for T = 20 mK. ℘ = 5 μW (dotted), 20 μW (dashed). . . . . . . . . . . . . . . . . . . . . . . . . 169 xxii CHAPTER 1 INTRODUCTION 1.1 Optomechanical System 1.1.1 Overview Radiation pressure force, due to the momentum carried by light, has received considerable attention since Kepler proposed that the tail of a comet was caused by the force exerted by the sunlight in the 16th century. It was deduced theoretically by J. C. Maxwell in 1871, and first observed experimentally [1, 2] in the early 1900s. With the invention of lasers in the 1970s, it has been shown that radiation pressure force can be used to manipulate atoms [3, 4, 5] i.e., to slow them, cool them, or trap them, owing to the relatively large power of the laser fields. In 2004, it was first demonstrated experimentally that radiation pressure force exerted by the light stored inside an optical cavity can be use to cool the motion of a mechanical oscillator made of roughly 1015 atoms in a cavity optomechanical system which parametrically couples an optical cavity and a mechanical resonator through radiation pressure [6]. Due to rapid advances in micro and nanofabrication techniques, various geometries of the optomechanical system have been developed, such as a FabryPerot cavity with mirrored microcantilevels [7], or with one movable end mirror [8, 9], or with a movable semitransparent membrane in the middle of the cavity [10, 11, 12, 13], or with a BoseEinstein condensate [14], or with a trapped macroscopic ensemble of ultracold atoms [15], radially vibrating microspheres [16], radially vibrating microtoroids [17, 18], GaAs nanooptomechanical disk resonator [19], and optomechanical 1 crystals [20]. Meanwhile, the optomechanical coupling idea has been extended to nanoelectromechanical systems, formed by a nanomechanical resonator capacitively coupled to a superconducting microwave cavity [21, 22, 23, 24]. The major challenge in all of these setups is to achieve simultaneously a high optical finesse (currently in the range from 103 to 105) and a high mechanical quality factor (currently in the range from 103 to 105). It has been shown theoretically and experimentally that such optomechanical systems at macroscopic scale can exhibit a very rich quantum effects, which usually exist in the microscopic system. For example, squeezing of the light field [25, 26], superposition state [27, 28], quantum nondemolition measurements of photon numbers [29, 30], the preparation of a mechanical oscillator in a squeezed state of motion [31, 32, 33], the creation of entangled photon pairs [34], the entanglement between the light and mechanical mode [35, 36], entangling two mechanical oscillators [37, 38, 39], and Fock state detection [10]. Moreover, the optomechanical coupling in such systems induces nonlinear behaviors, including an optical spring effect [40, 41], bistability [41, 42], multistability [43], selfinduced oscillations [44, 45, 46], optomechanical normal mode splitting [22, 47, 48, 49, 50], and optomechanically induced transparency [51, 52, 53, 54, 55]. Due to unavoidable coupling of the mechanical oscillator to its surrounding thermal environment, the random, thermal motion associated with mechanical dissipation mask the quantum behaviors. To see quantum effects in large objects, they must be cooled down to its quantum ground state. The ground state cooling requires that the mechanical oscillator’s temperature T must be reduced so that T ≪ ¯hωm kB , where ¯h is Planck’s constant h divided by 2π, kB is Boltzmann’s constant, ωm is the resonance frequency of the mechanical oscillator, typically between a few kilohertz and a few hundred megahertz. For a mechanical oscillator with a resonance frequency of 1kHz (100MHz), the ground state cooling requires ¯hωm/kB=50 nK (5 mK), which 2 are below those achievable with standard cryogenic cooling. So far, significant effort has been devoted to developing alternative cooling techniques. In the past few years, extraordinary progress has been made in cooling a mechanical resonator down to its quantum ground state [6, 7, 8, 9, 10, 56]. In 2009, the minimum achievable phonon number of the mechanical oscillator is 63 in a toroidal microresonator [57], 37 in a microsphere resonator [58], and 35 in a FabryPerot cavity [59]. In 2010, the preparation of mechanical resonator with the final phonon number below 10 was reported in Refs. [23, 60]. Recent work has shown experimentally that laser cooling can reduce the average occupancy of the mechanical oscillator below unity [61, 62, 63]. However, the ground state cooling has so far not been reached experimentally. in c out c fixed mirror movable mirror cavity axis Figure 1.1: A FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror. 1.1.2 The Dispersive Optomechanical System The canonical optomechanical system is a FabryPerot cavity with one heavy, fixed partially transmitting mirror and one light, movable totally reflecting mirror of effective mass m (typically in the micro or nanogram range), as shown in Fig. 1.1. The system is driven by an external laser at frequency ωl, then the circulating photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer from the intracavity photons to the movable mirror. Here, the movable mirror is modeled as a single mode quantum harmonic oscillator. Moreover, when the 3 mechanical frequency ωm is much smaller than the cavity free spectral range (c/2L), where L is the initial cavity length, the input laser drives only one cavity mode ωc and scattering of photons from the driven mode into other cavity modes is negligible [64]. During the cavity roundtrip time t = 2L/c, there are n photons hitting on the surfaces of the movable mirrors, the momentum transferred to the movable mirror will be P = 2n¯hωc/c, hence the radiation pressure force acting on the movable mirror would be F = P/t = n¯hωc L . The force is proportional to the instantaneous photon number in the cavity. Moreover, the movable mirror is in thermal equilibrium with its environment at temperature T. Thus the mirror can move under the influence of the radiation pressure and in the same time undergoes Brownian motion as a result of its interaction with the environment. In turn, the movable mirror’s small oscillation changes the length of the cavity and shifts the cavity resonance frequency so that the phase and amplitude of the cavity field are changed. This in turn changes the radiation pressure force experienced by the mirror such that the optical and mechanical dynamics are coupled. Thus the cavity resonance frequency depends on the displacement q of the movable mirror, represented by ωc(q) = nπc L+q , where n is the mode number in the cavity, c is the light speed in vacuum, L is the initial cavity length. For small displacements of the mirror, q << L, the frequency ωc(q) can be approximated to the first order of q ωc(q) ≈ ωc + gq, (1.1) where ωc = nπc L , g = −ωc/L is the linear coupling constant between the cavity field and the movable mirror, the minus sign in g implies that the cavity resonance frequency decreases when increasing the displacement q of the mirror elongates the cavity. What we discussed previously is the linear optomechanical coupling case, i.e., the frequency shift of the cavity field depends linearly on the displacement of the 4 mechanical oscillator. However, in a Fabry Perot cavity with a vibrating membrane in the middle of the optical cavity [10, 11, 12, 13], if the membrane is positioned at an antinode of the intracavity standing wave, the optomechanical coupling is quadratic i.e., the frequency shift of the cavity field depends quadratically on the displacement of the mechanical oscillator. If we expand the cavity frequency ωc(q) about the antinode point q0, then ωc(q) = ωc(q0) + ∂ωc(q) ∂q q=q0 q + 1 2 ∂2ωc(q) ∂q2 q=q0 q2 + · · · ≈ ωc(q0) + 1 2 ∂2ωc(q) ∂q2 q=q0 q2, (1.2) since at the antinode ∂ωc(q) ∂q q0 = 0. Compared to the linear optomechanical coupling system, the quadratic optomechanical system has the advantage in the quantum nondemolition measurement of mechanical energy quantization [10, 11, 12]. Note that the cavity decay rate only depends on the transmission of the fixed mirror, and is unrelated to the mechanical motion. Therefore, the optomechanical coupling via radiation pressure is dispersive. Figure 1.2: The optomechaical system that consists of a microdisk resonator coupled to a waveguide (from Ref.[67]). 1.1.3 The Reactive Optomechanical System In other optomechanical devices, the optomechanical coupling is induced by optical gradient force such as in silicon waveguide evanescently coupled to a microdisk 5 resonator [67], suspended silicon photonic waveguides [68, 69], SiN nanowire evanescently coupled to a microtoroidal resonator [70], and in ”zipper” cavities formed by two adjacent photonic crystal wires [71]. In this thesis, we focus on the optomechanical design proposed by [67], as shown in Fig. 1.2. The freestanding silicon waveguide with 10 μm length, 300 nm height, and 300 nm width is supported by two singlesided photonic crystal waveguide structures. The microdisk resonator with a radius of 40 μm is placed in close to the waveguide with a gap of 250 nm. A laser is injected into the waveguide, then light is coupled into and out of the microdisk through the evanescent fields from the waveguide and microdisk in the air gap, which decay exponentially with the distance from their geometric boundaries. And the dipoles in the waveguide induced by the evanescent field from the microdisk in turn interacts with the evanescent field from the microdisk and generate a gradient optical force. Under the action of this force, the waveguide is attracted toward the microdisk. Further the displacement of the waveguide modifies the resonance frequency of the microdisk resonator and the extrinsic photon decay rate of the microdisk resonator. Thus the coupling between the waveguide and the microdisk resonator is dispersive and reactive. 1.2 Sideband Cooling of the Nano Mechanical Mirror Recent experiments have demonstrated that the mechanical mirror can be cooled by the dynamical backaction of radiation pressure [7, 8, 9]. And it is possible to cool the mechanical mirror to the quantum ground state by resolved sideband cooling as first shown theoretically in Refs. [47, 66]. Sideband cooling was demonstrated experimentally by Kippenberg [57] and by Wang [58]. Both these experiments started the system at about 1.5K and showed cooling down to about 200 mK. The amount of cooling depends on the system parameters and the laser power. Harris et. al. has shown that the lowest temperature achieved is 6.82 mK in an optical cavity with a 6 vibrating membrane [10]. Before we give details of the theoretical discussion of sideband cooling, we discuss the physics which shows why sideband cooling results in cooling. When the pump field with frequency ωl interacts with the mechanical mirror with frequency ωm, absorption and emission of phonons create the Stokes field (ωl + ωm) and the antiStokes field (ωl − ωm). During the Stokes process, the pump field extracts a quantum of energy ¯hωm from the movable mirror, leading to the cooling of the movable mirror. While during the antiStokes process, the pump field emits a quantum of energy ¯hωm to the movable mirror, leading to the heating of the movable mirror. If the pump frequency is detuned below the cavity resonance frequency by an amount ωm, the amplitude of the Stokes field is resonantly enhanced, since the frequency of the Stokes field is close to the cavity resonance frequency ωc; however, the antiStokes field is suppressed since its frequency is far away from the cavity resonance frequency, thus the optomechanical coupling causes the cooling of the mirror. Further in the resolved sideband limit, the cavity amplitude decay rate κ is much less than the mechanical oscillation frequency ωm. In this case, the linewidth κ of the cavity field is much smaller than the frequency spacing 2ωm between the Stokes field and the antiStokes field, thus the amplitude of the antiStokes field is close to zero, ground state cooling becomes possible. We now develop the theoretical treatment of sideband cooling. The studied system is a FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror of effective mass m and damping rate γm, as shown in Fig. 1.1. The Hamiltonian of the system in a rotating frame with respect to the laser frequency ωl is given by H = ¯h(ωc − ωl)c†c − ¯hωmχc†cQ + ¯hωm 4 (Q2 + P2) + i¯hε(c† − c). (1.3) In Eq. (1.3), the first term is the energy of the cavity field, c and c† are the annihilation and creation operators for the cavity field satisfying the commutation relation [c, c†] = 1. The second term describes the interaction of the movable mirror with 7 the cavity field, the dimensionless parameter χ = 1 ωm ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term gives the energy of the movable mirror, described by the dimensionless position and momentum operators Q and P, defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with commutation relation [Q, P] = 2i. The fourth term describes the cavity driven by a laser with power ℘, and ε = √ 2κ℘ ¯hωl . The time evolution of the system operators can be derived by using the Heisenberg equations of motion and adding the corresponding damping and noise terms. We find a set of nonlinear quantum Langevin equations as follows, ˙Q = ωmP, ˙P = 2ωmχc†c − ωmQ − γmP + ξ, ˙ c = −i(ωc − ωl − ωmχQ)c + ε − κc + √ 2κcin, ˙ c† = i(ωc − ωl − ωmχQ)c† + ε − κc† + √ 2κc † in. (1.4) Here cin is the input vacuum noise operator with zero mean value and nonzero correlation function in the time domain ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′). (1.5) The force ξ is the Brownian noise operator associated with the mechanical damping, whose mean value is zero, and its correlation function reads ⟨ξ(t)ξ(t ′ )⟩ = 1 π γm ωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (1.6) where kB is the Boltzmann constant and T is the thermal bath temperature. The steadystate solution to Eq. (1.4) can be obtained by setting all the time derivatives in Eq. (1.4) to zero. They are Ps = 0, Qs = 2χcs2, cs = ε κ + iΔ , (1.7) 8 where Δ = ωc − ωl − ωmχQs (1.8) is the effective cavity detuning, in which the term −ωmχQs is the cavity resonance frequency shift due to radiation pressure. The Qs denotes the steadystate position of the movable mirror. And cs represents the steadystate amplitude of the cavity field. In order to investigate cooling of the movable mirror, we need to calculate the fluctuations of the system. We linearize the nonlinear equation (1.4) by writing each operator of the system as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (1.9) Inserting Eq. (1.9) into Eq. (1.4), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators take the form δ ˙Q = ωmδP, δ ˙P = 2ωmχ(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + iωmχcsδQ + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − iωmχc∗ sδQ + √ 2κδc † in. (1.10) We transform Eq. (1.10) to the frequency domain by using f(t) = 1 2π ∫ +∞ −∞ f(ω)e−iωtdω and f†(t) = 1 2π ∫ +∞ −∞ f†(−ω)e−iωtdω, where f†(−ω) = [f(−ω)]†, and solve it, we obtain the position fluctuations of the movable mirror δQ(ω) = − ωm d(ω) [2 √ 2κωmχ{[κ − i(Δ + ω)]c∗ sδcin(ω) + [κ + i(Δ − ω)]csδc † in(−ω)} +[(κ − iω)2 + Δ2]ξ(ω)], (1.11) where d(ω) = 4ω3m χ2Δcs2 + (ω2 − ω2m + iγmω)[(κ − iω)2 + Δ2]. (1.12) 9 In Eq. (1.11), the first term proportional to χ is the contribution of radiation pressure, while the second term involving ξ(ω) is the contribution of the thermal noise. In the absence of the cavity field, the movable mirror will make Brownian motion, δQ(ω) = ωmξ(ω)/(ω2m − ω2 − iγmω), whose susceptibility has a Lorentzian shape centered at frequency ωm with full width at half maximum γm. The twotime correlation function of the fluctuations in position of the movable mirror is given by 1 2 (⟨δQ(t)δQ(t + τ )⟩ + ⟨δQ(t + τ )δQ(t)⟩) = 1 2π ∫ +∞ −∞ dωSQ(ω)eiωτ , (1.13) in which SQ(ω) is the spectrum of fluctuations in position of the movable mirror, defined by 1 2 (⟨δQ(ω)δQ(Ω)⟩ + ⟨δQ(Ω)δQ(ω)⟩) = 2πSQ(ω)δ(ω + Ω). (1.14) By aid of the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(−Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 4π γm ωm ω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (1.15) we obtain the spectrum of fluctuations in position of the movable mirror SQ(ω) = ω2m d(ω)2 {8ω2m χ2κ(κ2 + ω2 + Δ2)cs2 + 2 γm ωm ω[(Δ2 + κ2 − ω2)2 +4κ2ω2] coth( ¯hω 2kBT )}. (1.16) In Eq. (1.16), the first term involving χ arises from radiation pressure, while the second term originates from the thermal noise. So the spectrum SQ(ω) of the movable mirror depends on radiation pressure and the thermal noise. Then Fourier transforming δ ˙Q = ωmδP in Eq. (1.10), we obtain δP(ω) = − iω ωm δQ(ω), which leads to the spectrum of fluctuations in momentum of the movable mirror SP (ω) = ω2 ω2m SQ(ω). (1.17) 10 The phonon number n in the movable mirror can be calculated from the total energy of the movable mirror ¯hωm 4 (⟨δQ2⟩ + ⟨δP2⟩) = ¯hωm ( n + 1 2 ) , n = [exp(¯hωm/(kBT)) − 1]−1, (1.18) where the variances of position and momentum are ⟨δQ2⟩ = 1 2π ∫ +∞ −∞ SQ(ω)dω and ⟨δP2⟩ = 1 2π ∫ +∞ −∞ SP (ω)dω. Then the effective temperature Teff of the movable mirror can be determined from the phonon number nin the movable mirror, which is Teff = ¯hωm kB ln(1 + 1 n) . (1.19) The parameters used are from an experimental paper on optomechanical normal mode splitting [50]: the wavelength of the laser λ = 2πc/ωl = 1064 nm, L = 25 mm, m = 145 ng, ωm = 2π × 947 × 103 Hz, the mechanical quality factor Q ′ = ωm/γm = 6700, κ = 2π × 215 × 103 Hz, κ/ωm ≈ 0.23, thus the system is operating in resolved sideband regime. And in the high temperature limit kBT ≫ ¯hωm, the approximation coth(¯hω/2kBT) ≈ 2kBT/¯hω can be made. The laser is detuned below the cavity resonance frequency by an amount Δ = ωm. We work in the stable regime. 0 200 400 600 800 1000 0 10 20 30 40 50 PHmWL Teff HmKL 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 PHmWL Teff HKL Figure 1.3: The effective temperature Teff (mK) of the movable mirror as a function of the laser power ℘ (μW). The initial temperature is taken to be 1 K. Figure 1.3 shows the variation of the effective temperature Teff of the movable mirror with the laser power ℘. It is clear to see that the effective temperature Teff of 11 the movable mirror decreases with increases the laser power ℘. When ℘ = 100 μW, the movable mirror can be cooled to about 50 mK, a factor of 20 below the starting temperature of 1 K [57, 58]. If the laser power is further increased to 1 mW, the movable mirror can be cooled to about 6 mK. Therefore the movable mirror can be effectively cooled in the resolved sideband limit. 1.3 Degenerate Parametric Ampli cation Nonlinear Crystal ! " # Figure 1.4: Parametric amplifier. In a parametric amplifier [72], a pump beam at higher frequency ωp interacts with a nonlinear crystal, a signal and idler modes at lower frequencies ωs and ωi would be generated, as shown in Fig. 1.4. During the nonlinear optical process, the energy is conserved ωp = ωs + ωi. If the signal and the idler modes have identical frequencies, such a parametric amplifier is called a degenerate parametric amplifier. In the following, we will show that the degenerate parametric amplifier can be used as a generator of a singlemode squeezed state. The Hamiltonian for degenerate parametric amplification, in the interaction picture, is Hint = ¯hμ(a†2b + a2b†), (1.20) where b and a are the annihilation operators for the pump and signal modes, respectively, and μ is a coupling strength between the pump field and the nonlinear crystal, and it is related to the secondorder nonlinear susceptibility. Assuming that 12 the pump field is a strong coherent classical field and pump depletion is neglected, thus the operators b and b† can be represented by βe−iϕ and βeiϕ, where β and ϕ are the real amplitude and phase of the coherent pump field. Hence the Hamiltonian (1.20) becomes Hint = ¯hμβ(a†2e−iϕ + a2eiϕ), (1.21) The time evolution of the signal mode can be derived by the Heisenberg equation of motion, which yields ˙a = −iΩa†e−iϕ, ˙a† = iΩaeiϕ. (1.22) Here Ω = 2μβ is the effective Rabi frequency. The solution to Eq. (1.22) is a(t) = a0 cosh(Ωt) − ia † 0 sinh(Ωt)e−iϕ, a†(t) = a † 0 cosh(Ωt) + ia0 sinh(Ωt)eiϕ, (1.23) where a0 = a(0). For ϕ = π/2, when the signal initially is in a vacuum state, the variances in the two quadratures X1 = (a + a†)/2 and X2 = (a − a†)/2i are given by (ΔX1)2t = 1 4 e−2u, (ΔX2)2t = 1 4 e2u, (1.24) where u = Ωt is the effective squeezing parameter. Eq. (1.24) shows the output from the degenerate parametric amplifier can be squeezed state, and the squeezing exists in the X1 quadrature. 1.4 Standard Quantum Limit For a onedimensional harmonic oscillator with mass m and frequency ωm, its Hamiltonian is H0 = p2 2m+1 2mω2m q2, where p is the momentum operator and q is the position operator, satisfying the commutation relation [q, p] = i¯h. In the ground state, the 13 fluctuations in the position and the momentum are not equal to zero due to the zeropoint energy. They are δq = √ ¯h 2mωm , δp = √ m¯hωm 2 , (1.25) respectively, which are called the standard quantum limit. These fluctuations have no classical analog. If we write the position operator q and the momentum operator p in terms of the dimensionless position operator Q and momentum operator P, q = √ ¯h 2mωm Q and p = √ m¯hωm 2 P, then the standard quantum limit would be δQ = δP = 1, (1.26) thus the fluctuations in the two dimensionless quadratures are identical, each of them is equal to unity. For very highprecision interferometers, the standard quantum limit limits their sensitivity. To improve their sensitivity, this limit need to be beaten, which means that the fluctuations need to be reduced below the standard quantum limit. According to the Heisenberg uncertainty principle ΔAΔB ≥ 1 2 ⟨[A,B]⟩, where ΔA = (⟨A2⟩ − ⟨A⟩2)1/2 and similarly for ΔB, the fluctuations in position and momentum should satisfy the inequality δQδP ≥ 1, (1.27) thus the fluctuations in the position and momentum could not be reduced below unity simultaneously. If the fluctuations in position is less than unity, the fluctuations in momentum should be larger than unity, or vice versa. Moreover, the harmonic oscillator is said to be squeezed if either δQ < 1 or δP < 1. Therefore, as the standard quantum limit is beaten, the harmonic oscillator is quadrature squeezed. 1.5 Homodyne Detection Homodyne detection is usually used to measure the amplitude and the phase quadrature components of the light field. In this section, we describe balanced homodyne detection [73]. 14 ! " # $% & $' PD PD Figure 1.5: Balanced homodyne detection. PD:photodetector. Figure 1.5 schematically shows a balanced homodyne detection setup. The signal light and a strong local laser light, described by the annihilation operators a and b, respectively, are mixed on a 50/50 beam splitter. The two output fields c and d can be obtained through the relation c = √1 2 (a + ib), d = √1 2 (b + ia). (1.28) The two output fields c and d are detected individually by two photodetectors. Then the two intensities Ic = ⟨c†c⟩ and Id = ⟨d†d⟩ measured by the two photodetectors are subtracted each other, the result is Ic − Id = ⟨ncd⟩ = ⟨c†c − d†d⟩, = i⟨a†b − ab†⟩. (1.29) Assuming the b mode to be in the coherent state βe−iωt⟩, and β = βe−iψ, the operator b can be replaced by βe−i(ωt+ψ), we obtain ⟨ncd⟩ = β[aeiωte−iθ + a†e−iωteiθ], (1.30) 15 where θ = ψ +π/2. Assuming that the signal mode a has the same frequency as that of the local oscillator b, thus a = a0e−iωt, Eq. (1.30) reduces to ⟨ncd⟩ = 2β⟨X(θ)⟩, (1.31) where ⟨X(θ)⟩ = 1 2 (a0e−iθ + a0eiθ) is the field quadrature operator at the angle θ. By changing θ, which can be done by changing the phase ψ of the local oscillator, an arbitrary quadrature component of the signal field can be measured. Moreover, the balanced homodyne detection can be used to detect the squeezed state. The variance of the output signal can be found to be ⟨(Δncd)2⟩ = 4β2⟨(ΔX(θ))2⟩, (1.32) The squeezing condition for the signal is ⟨(ΔX(θ))2⟩ < 1 4 , we have ⟨(Δncd)2⟩ < β2. 1.6 Electromagnetically Induced Transparency Generally, if a laser light passes through a twolevel atomic system whose atoms are all in the ground state, the light will be strongly absorbed if the laser field is near resonant with the atomic transition. However, for a threelevel atomic system whose atoms are all in the lowestenergy state, the atomic system becomes transparent for a weak probe field tuned to an atomic transition resonance when a strong coupling field is applied to the other atomic transition. This phenomenon is called as electromagnetically induced transparency (EIT). The effect of EIT allows a weak signal field to propagate without being absorbed by the atomic medium. It was theoretically proposed in 1989 [74] and first experimentally demonstrated in 1991 [75]. Meanwhile, the phenomenon of EIT [76] is accompanied by a sharp dispersion change in the transmitted probe field on resonance, which leads to the generation of ultrafast light [77, 78] and ultraslow light [79, 80, 81]. Accordingly considerable interest has been dedicated to EIT due to its potential applications in an optical switch [82], optical storage [83, 84, 85, 86]. 16 a μ n coupling n probe b c Figure 1.6: A threelevel Λtype atomic system, where the probe field at frequency ν couples levels b⟩ and a⟩, while the coupling field at frequency νμ couples levels c⟩ and a⟩. We consider a threelevel Λtype atomic system [72], as shown in Fig. 1.6. The atoms have one upper level a⟩ and two lower levels b⟩ and c⟩ with energies ¯hωa, ¯hωb, and ¯hωc, where the transitions b⟩ → a⟩ and c⟩ → a⟩ are dipole allowed, but the transition b⟩ → c⟩ is dipole forbidden since c⟩ is a metastable state. The levels a⟩ and b⟩ are coupled by a weak probe field of amplitude ε at frequency ν, while the levels a⟩ and c⟩ are coupled by a strong coupling field at frequency νμ. The coupling strength of the probe field to the atomic transition b⟩ → a⟩ is described by the Rabi frequency ℘abε/¯h, where ℘ab is the electricdipole transition matrix element, and it is assumed to be real. The interaction strength between the coupling field and the c⟩ → a⟩ transition is characterized by the complex Rabi frequency Ωμ exp(−iϕμ), and Ωμ is assumed to be real. The state of the atom can be written as a linear combination of states a⟩, b⟩, and c⟩, i.e., Ψ⟩ = Ca(t)a⟩ + Cb(t)b⟩ + Cc(t)c⟩. Here, Ca(t), Cb(t), and Cc(t) are the probability amplitudes corresponding to the three atomic levels a⟩, b⟩, and c⟩, respectively. The density matrix operator of the atom takes form ρ = Ψ⟩⟨Ψ = [Ca(t)a⟩ + Cb(t)b⟩ + Cc(t)c⟩][C∗ a(t)⟨a + C∗ b (t)⟨b + C∗ c (t)⟨c] 17 = Ca(t)2a⟩⟨a + Ca(t)C∗ b (t)a⟩⟨b + Ca(t)C∗ c (t)a⟩⟨c +Cb(t)C∗ a(t)b⟩⟨a + Cb(t)2b⟩⟨b + Cb(t)C∗ c (t)b⟩⟨c +Cc(t)C∗ a(t)c⟩⟨a + Cc(t)C∗ b (t)c⟩⟨b + Cc(t)2c⟩⟨c. (1.33) Taking the matrix elements, we get ρaa = ⟨aρa⟩ = Ca(t)2, ρab = ⟨aρb⟩ = Ca(t)C∗ b (t), ρac = ⟨aρc⟩ = Ca(t)C∗ c (t), ρba = ⟨bρa⟩ = Cb(t)C∗ a(t), ρbb = ⟨bρb⟩ = Cb(t)2, ρbc = ⟨bρc⟩ = Cb(t)C∗ c (t), ρca = ⟨cρa⟩ = Cc(t)C∗ a(t), ρcb = ⟨cρb⟩ = Cc(t)C∗ b (t), ρcc = ⟨cρc⟩ = Cc(t)2. (1.34) Hence, the threelevel atom can be described by the 3 × 3 density matrix ρ, ρ = ρaa ρab ρac ρba ρbb ρbc ρca ρcb ρcc , (1.35) where the diagonal elements ρii = ⟨iρi⟩ (i = a, b, c) describe the populations in the three levels, respectively, and the offdiagonal elements ρij = ⟨iρj⟩ (i, j = a, b, c and i ̸= j) represent the atomic coherence between levels. The density matrix is a Hermitian operator satisfying ρ = ρ†.The offdiagonal decay rates for ρab, ρac, and ρcb are denoted by γ1, γ2, and γ3, respectively. Since the level c⟩ is assumed to be a metastable state, γ3 << γ1. In the rotatingwave approximation, the Hamiltonian of the system is given by H = ¯hωaa⟩⟨a + ¯hωbb⟩⟨b + ¯hωcc⟩⟨c 18 +[ − ¯h 2 ( ℘abε ¯h e−iνta⟩⟨b + Ωμe−iϕ e−iν ta⟩⟨c) + H.C.], (1.36) where the first three terms are the free energies of the atomic three levels, and the last four terms gives the interactions of the threelevel atoms with the probe field and the coupling field. The time evolution for the density matrix elements ρab, ρcb, and ρac can be derived by using the Liouville equation ρ˙ij = −i ¯h[H, ρij ] and considering the corresponding damping term, which yields ρ˙ab = −(iωab + γ1)ρab − i 2 ℘abε ¯h e−iνt(ρaa − ρbb) + i 2 Ωμe−iϕ e−iν tρcb, ρ˙cb = −(iωcb + γ3)ρcb − i 2 ℘abε ¯h e−iνtρca + i 2 Ωμeiϕ eiν tρab, ρ˙ac = −(iωac + γ2)ρac − i 2 Ωμe−iϕ e−iν t(ρaa − ρcc) + i 2 ℘abε ¯h e−iνtρbc, (1.37) where ωab, ωcb, and ωac are the Bohr frequencies, ωab = ωa − ωb, ωcb = ωc − ωb, and ωac = ωa − ωc. We assume all atoms are initially in the lowestenergy state b⟩, ρbb(0) = 1, ρaa(0) = ρcc(0) = ρac(0) = 0. (1.38) Since the probe field is very weak, most of the atoms keep staying in the lowestenergy state b⟩ at any time so that the atomic population in level b⟩ is close to unity. Thus we can adopt the approximation condition ρbb(t) ≈ 1, ρaa(t) ≈ ρcc(t) ≈ ρac(t) ≈ 0. (1.39) Thus Eq. (1.37) reduces to ρ˙ab = −(iωab + γ1)ρab + i 2 ℘abε ¯h e−iνt + i 2 Ωμe−iϕ e−iν tρcb, ρ˙cb = −(iωcb + γ3)ρcb + i 2 Ωμeiϕ eiν tρab. (1.40) Then we convert the usual densitymatrix elements ρij to slowly varying variables ˜ρij in order to remove the fast optical oscillation by using the following transformations ρab = ˜ρabe−iνt, ρcb = ˜ρcbe−i(ν+ωca)t, (1.41) 19 thus the time evolution of the slowly varying densitymatrix elements ˜ρab and ˜ρcb is given by ˙˜ρab = −(γ1 − iΔ)˜ρab + i 2 ℘abε ¯h + i 2 Ωμe−iϕ ˜ρcb, ˙˜ρcb = −(γ3 − iΔ)˜ρcb + i 2 Ωμeiϕ ˜ρab, (1.42) where Δ = ν −ωab is the detuning of the probe frequency ν from the frequency ωab of the b⟩ → a⟩ transition, and we assume that the coupling field is resonant with the c⟩ → a⟩ transition, i.e., νμ = ωac. We write Eq. (1.42) in the matrix form as ˙R = −MR + A, (1.43) where R = ˜ρab ˜ρcb , M = γ1 − iΔ −i 2Ωμe−iϕ −i 2Ωμeiϕ γ3 − iΔ , A = i℘abε 2¯h 0 , (1.44) then integrating R(t) = ∫ t −∞ e−M(t−t′)Adt′ = M−1A, (1.45) we obtain ρab(t) = i℘abεe−iνt(γ3 − iΔ) 2¯h [ (γ1 − iΔ)(γ3 − iΔ) + Ω2 4 ] . (1.46) The dielectric response of the atomic system to the probe field is determined by the electric polarization P. The polarization of an ensemble of identical atoms will be P = 2℘abρab(t)eiνtNa, where Na is the atom number density for the threelevel atoms. In addition, the linear polarization is related to the amplitude ε of the probe field through P = ϵ0χε, where ε0 is the electric permittivity of free space and χ is the 20 electric susceptibility of the atomic system. Hence, the susceptibility of the Λ system is given by χ = Na℘2 ab ϵ0¯h i(γ3 − iΔ) (γ1 − iΔ)(γ3 − iΔ) + Ω2 4 , = χ′ + iχ′′, (1.47) where χ′ and χ′′ are the real and imaginary parts of the complex susceptibility χ of the atomic system. The χ′ and χ′′ determine the dispersion and absorption of the probe field, respectively. It is seen that from Eq. (1.47), on resonance, if there is a coupling field, i.e., Ωμ ̸= 0, χ′ = 0 and χ′′ = Na℘2 ab ϵ0¯h γ3 γ1γ3+ Ω2 4 , which is proportional to γ3. If the decay rate γ3 is very small (or approaching zero), the imaginary part of the electric susceptibility would be negligibly small. We plot the real and imaginary parts of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 without and with the coupling field, as shown in Figs. 1.7 and 1.8. In the 2 1 0 1 2 0.4 0.2 0.0 0.2 0.4 D Γ1 Γ1 Ε0 Ñ ReIΧM Na P ab 2 Figure 1.7: The real part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. absence of the coupling field, Ωμ = 0, the curve χ′′ has a Lorentzian lineshape, and the curve χ′ exhibits the anomalous dispersion since the slope of χ′ at the line center 21 2 1 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 D Γ1 Γ1 Ε0 Ñ ImIΧM Na P ab 2 Figure 1.8: The imaginary part of the susceptibility in units of Na℘2 ab γ1ϵ0¯h as a function of the normalized detuning Δ/γ1 in the absence (dotted) and in the presence (solid) of the coupling field. is less than zero. In the presence of the coupling field, Ωμ = 2γ1, and γ1 >> γ3(γ3 = 10−4γ1), when Δ = 0, ωab = ν, the probe field is in resonance with the b⟩ → a⟩ atomic transition, we can see χ′′ ≈ 0, the medium becomes completely transparent for the probe field, thus the probe field can propagate through the atoms without any absorption even with most of the atoms in the lowestenergy state b⟩. It has been calculated that the width of the transparency window depends on the Rabi frequency Ωμ, which is related to the power of the coupling field. And increasing the power of the coupling field, the EIT dip becomes wider due to power broadening. We also note χ′ = 0 as Δ = 0, hence the refractive index of the medium is equal to unity since the refractive index is related to the susceptibility by n(ν) = [1+χ′(ν)+iχ′′(ν)]0.5. Thus the phase velocity of the probe field propagating through the medium is equal to that in vacuum. Moreover, the slope of the curve χ′ at the line center is larger than zero, thus the curve χ′ exhibits the normal dispersion. And the steepness of the curve χ′ where the absorption vanishes depends on the power of the coupling field, i.e., the curve χ′ becomes steeper at the line center by decreasing the power of the coupling 22 field, implying that the group velocity can be dramatically reduced, and even can be reduced to zero such that the probe field can be completely stopped and stored within the atomic medium. In summary, when the coupling field resonant with the c⟩ → a⟩ atomic transition is applied, the interaction of a threelevel Λtype atomic system with a weak probe field depends on the frequency of the probe field. If the frequency of the probe field matches the frequency of the b⟩ → a⟩ transition, the EIT phenomenon occurs, the effect of the atomic system on the probe field can be eliminated. 1.7 Organization Chapter 2 shows that an optical parametric amplifier inside a cavity can considerably improve the cooling of the micromechanical mirror by radiation pressure. The micromechanical mirror can be cooled from room temperature 300 K to subKelvin temperatures, which is much lower than what is achievable in the absence of the parametric amplifier. This is further illustrated in case of a precooled mirror, where one can reach millikelvin temperatures starting with about 1 K. Our work demonstrates the fundamental dependence of radiation pressure effects on photon statistics. Chapter 3 discusses how an optical parametric amplifier inside the cavity can affect the normalmode splitting behavior of the coupled movable mirror and the cavity field. We work in the resolved sideband regime. The spectra exhibit a doublepeak structure as the parametric gain is increased. Moreover, for a fixed parametric gain, the doublepeak structure of the spectrum is more pronounced with increasing the input laser power. We give results for mode splitting. The widths of the split lines are sensitive to parametric gain. Chapter 4 presents that squeezing of a nanomechanical mirror can be generated by injecting broad band squeezed vacuum light and laser light into the cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum 23 momentum squeezing of the movable mirror, the squeezing parameter of the input light should be about 1. We can obtain more than 70% squeezing. Besides, for a fixed squeezing parameter, decreasing the temperature of the environment or increasing the laser power increases the momentum squeezing. We find very large squeezing with respect to thermal fluctuations, for instance at 1 mK, the momentum fluctuations go down by a factor more than one hundred. Chapter 5 presents a scheme for entangling two separated nanomechanical oscillators by injecting broad band squeezed vacuum light and laser light into the ring cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum entanglement of the two oscillators, the squeezing parameter of the input light should be about 1. We report significant entanglement over a very wide range of power levels of the pump and temperatures of the environment. Chapter 6 discusses Stokes and antiStokes processes in cavity optomechanics in the regime of strong coupling. The Stokes and antiStokes signals exhibit prominently the normalmode splitting. We report gain for the Stokes signal. We also report lifetime splitting when the pump power is less than the critical power for normalmode splitting. The nonlinear Stokes processes provide a useful method for studying the strongcoupling regime of cavity optomechanics. We also investigate the correlations between the Stokes and the antiStokes photons produced spontaneously by the optomechanical system. At zero temperature, our nanomechanical system leads to the correlations between the spontaneously generated photons exhibiting photon antibunching and those violating the CauchySchwartz inequality. Chapter 7 discusses the dynamical behavior of a nanomechanical mirror in a highquality cavity under the action of a coupling laser and a probe laser. We demonstrate the existence of the analog of electromagnetically induced transparency (EIT) in the output field at the probe frequency. Our calculations show explicitly the origin of EITlike dips as well as the characteristic changes in dispersion from anomalous to 24 normal in the range where EIT dips occur. Remarkably the pumpprobe response for the optomechanical system shares all the features of the Λ system as discovered by Harris and collaborators. Chapter 8 studies the optomechanical design introduced by M. Li et. al. [Phys. Rev. Lett. 103, 223901 (2009)], which is very effective for investigation of the effects of reactive coupling. We show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling. We suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactivecouplinginduced normal mode splitting. Chapter 9 shows that dissipatively coupled nanosystems can be prepared in states which beat the standard quantum limit of the mechanical motion. We show that the reactive coupling between the waveguide and the microdisk resonator can generate the squeezing of the waveguide by injecting a quantum field and laser into the resonator through the waveguide. The waveguide can show about 7075% of maximal squeezing for temperature about 110 mK. The maximum squeezing can be achieved with an incident pump power of only 12 μW for a temperature of about 1 mK. Even for temperatures of 20 mK, achievable by dilution refrigerators, the maximum squeezing is about 60%. Chapter 10 describes how electromagnetically induced transparency can arise in quadratically coupled optomechanical systems. Due to quadratic coupling, the underlying optical process involves a twophonon process in an optomechanical system, and this twophonon process makes the mean displacement, which plays the role of atomic coherence in traditional electromagnetically induced transparency (EIT), zero. We show how the fluctuation in displacement can play a role similar to atomic coherence and can lead to EITlike effects in quadratically coupled optomechanical systems. We show how such effects can be studied using the existing optomechanical systems. Chapter 11 discusses electromagnetically induced transparency (EIT) using quan 25 tized fields in optomechanical systems. The weak probe field is a narrowband squeezed field. We present a homodyne detection of EIT in the output quantum field. We find that the EIT dip exists even though the photon number in the squeezed vacuum is at the singlephoton level. The EIT with quantized fields can be seen even at temperatures on the order of 100 mK, thus paving the way for using optomechanical systems as memory elements. Chapter 12 demonstrate theoretically the possibility of using nano mechanical systems as single photon routers. We show how EIT in cavity optomechanical systems can be used to produce a switch for a probe field in a single photon Fock state using very low pumping powers of few microwatt. We present estimates of vacuum and thermal noise and show the optimal performance of the single photon switch is deteriorated by only few percent even at temperatures of the order of 20 mK. Chapter 13 gives the summary of what we have done in this thesis and the direction of the future work. 26 CHAPTER 2 ENHANCEMENT OF CAVITY COOLING OF A MICROMECHANICAL MIRROR USING PARAMETRIC INTERACTIONS 2.1 Overview Recently there is considerable interest in micromechanical mirrors. These are macroscopic quantum mechanical systems and the important question is how to reach their quantum characteristics [8, 87, 88, 89]. The thermal noise limits many highly sensitive optical measurements [90, 91]. We also note that there has been considerable interest in using micromirrors for producing superpositions of macroscopic quantum states if such micromirrors can be cooled to their quantum ground states [27, 28]. Thus cooling of micromechanical resonators becomes a necessary prerequisite for all such studies. So far two different ways to cool a mechanical resonator mode have been proposed. One is the active feedback scheme [7, 92, 93, 116], where a viscous force is fed back to the movable mirror to decrease its Brownian motion. The other is the passive feedback scheme [6, 8, 9, 56, 95, 159], in which the Brownian motion of the movable mirror is damped by the radiation pressure force exerted by photons in an appropriately detuned optical cavity. Clearly we need to think of methods which can cool the micromirror toward its ground state. Since radiation pressure depends on the number of photons, one would think that the cooling of the micromirror can be manipulated by using effects of the photon statistics. In this chapter, we propose and analyze a method to achieve cooling of a movable mirror to subKelvin temperatures by using a type I optical 27 parametric amplifier inside a cavity. We remind the reader of the great success of cavities with parametric amplifiers in the production of nonclassical light [97, 98, 99]. The movable mirror can reach a minimum temperature of about a few hundred mK, a factor of 500 below room temperature 300 K. The lowering of the temperature is achieved by changes in photon statistics due to parametric interactions [100, 101, 102, 103, 104, 105]. Note that if the mirror is already precooled to say about 1 K, then we show that by using an optical parametric amplifier we can cool to about millikelvin temperatures or less. The chapter is organized as follows. In Sec. II we describe the model and derive the quantum Langevin equations. In Sec. III we obtain the stability conditions, calculate the spectrum of fluctuations in position and momentum of the movable mirror, and define the effective temperature of the movable mirror. In Sec. IV we show how the movable mirror can be effectively cooled by using the parametric amplifier inside the cavity. 2.2 Model in c out c movable mirror OPA cavity axis fixed mirror Figure 2.1: Sketch of the cavity used to cool a micromechanical mirror. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. We consider a degenerate optical parametric amplifier (OPA) inside a FabryPerot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror in contact with a thermal bath in equilibrium at temperature T, as shown 28 in Fig. 2.1. The movable mirror is free to move along the cavity axis and is treated as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm, and energy decay rate γm. The effect of the thermal bath can be modeled by a Langevin force. The cavity field is driven by an input laser field with frequency ωL and positive amplitude related to the input laser power P by ˜ε = √ P/(¯hωL). When photons in the cavity reflect off the surface of the movable mirror, the movable mirror will receive the action of the radiation pressure force, which is proportional to the instantaneous photon number inside the cavity. So the mirror can oscillate under the effects of the thermal Langevin force and the radiation pressure force. Meanwhile, the movable mirrors motion changes the length of the cavity; hence the movable mirror displacement from its equilibrium position will induce a phase shift on the cavity field. Here we assume the system is in the adiabatic limit, which means ωL ≪ πc/L; c is the speed of light in vacuum and L is the cavity length in the absence of the cavity field. We assume that the motion of the mirror is so slow that the scattering of photons to other cavity modes can be ignored, thus we can consider one cavity mode only [64, 106], say, ωc. Moreover, in the adiabatic limit, the number of photons generated by the Casimir effect [107], retardation, and Doppler effects is negligible [26, 92, 108]. Under these conditions, the total Hamiltonian for the system in a frame rotating at the laser frequency ωL can be written as H = ¯h(ωc − ωL)nc − ¯hχncq + 1 2 ( p2 m + mω2m q2) +i¯hε(c† − c) + i¯hG(eiθc†2 − e−iθc2). (2.1) Here c and c† are the annihilation and creation operators for the field inside the cavity, respectively; nc = c†c is the number of the photons inside the cavity; and q and p are the position and momentum operators for the movable mirror. The parameter χ = ωc/L is the coupling constant between the cavity and the movable mirror; and 29 ε = √ 2κ˜ε. Note that κ is the photon decay rate due to the photon leakage through the fixed partially transmitting mirror. Further κ = πc/(2FL), where F is the cavity finesse. In Eq. (2.1), G is the nonlinear gain of the OPA, and θ is the phase of the field driving the OPA. The parameter G is proportional to the pump driving the OPA. In Eq. (2.1), the first term corresponds to the energy of the cavity field, the second term arises from the coupling of the movable mirror to the cavity field via radiation pressure, the third term gives the energy of the movable mirror, the fourth term describes the coupling between the input laser field and the cavity field, and the last term is the coupling between the OPA and the cavity field. The motion of the system can be described by the Heisenberg equations of motion and adding the corresponding damping and noise terms, which leads to the following quantum Langevin equations: q˙ = p m, ˙ p = −mω2m q + ¯hχnc − γmp + ξ, ˙ c = i(ωL − ωc + χq)c + ε + 2Geiθc† − κc + √ 2κcin, ˙ c† = −i(ωL − ωc + χq)c† + ε + 2Ge−iθc − κc† + √ 2κc † in. (2.2) Here cin is the input vacuum noise operator with zero mean value; its correlation function is [141] ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = ⟨δc † in(t)δcin(t′)⟩ = 0. (2.3) The force ξ is the Brownian noise operator resulting from the coupling of the movable mirror to the thermal bath, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = ¯hγm 2π m ∫ ωe−iω(t−t ′ ) [ coth( ¯hω 2kBT ) + 1 ] dω, (2.4) 30 where kB is the Boltzmann constant and T is the thermal bath temperature. In order to analyze Eq. (2.2), we use standard methods from quantum optics [110]. A detailed calculation of the temperature for G = 0 is given by Paternostro et al. [35]. By setting all the time derivatives in Eq. (2.2) to zero, we obtain the steadystate mean values ps = 0, qs = ¯hχcs2 mω2m , cs = κ − iΔ + 2Geiθ κ2 + Δ2 − 4G2 ε, (2.5) where Δ = ωc − ωL − χqs = Δ0 − χqs = Δ0 − ¯hχ2cs2 mω2m (2.6) is the effective cavity detuning, including the radiation pressure effects. The modification of the detuning by the χqs term depends on the range of parameters. The qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. Note that qs and cs can display optical multistable behavior, which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. Mathematically this is contained in the dependence of the detuning parameter Δ on the mirrors amplitude qs. It is evident from Eqs. (2.5) and (2.6) that Δ satisfies a fifthorder equation and in principle can have five real solutions implying multistability. Generally, in this case, at most three solutions would be stable. The bistable behavior is reported in Refs. [41, 42]. 2.3 Radiation Pressure and Quantum Fluctuations In order to determine the cooling of the mirror, we need to find out the fluctuations in the mirrors amplitude. Since the problem is nonlinear, we assume that the nonlinearity is weak. We are thus interested in the dynamics of small fluctuations around the steady state of the system. Such a linearized analysis is quite common in quantum optics [110, 111]. So we write each operator of the system as the sum of its steady 31 state mean value and a small fluctuation with zero mean value, q = qs + δq, p = ps + δp, c = cs + δc. (2.7) Inserting Eq. (2.7) into Eq. (2.2), then assuming cs ≫ 1, we get the linearized quantum Langevin equations for the fluctuation operators δq˙ = δp m, δ ˙ p = −mω2m δq + ¯hχ(csδc† + c∗ sδc) − γmδp + ξ, δ ˙ c = −iΔδc + iχcsδq + 2Geiθδc† − κδc + √ 2κδcin, δc˙† = iΔδc† − iχc∗ sδq + 2Ge−iθδc − κδc† + √ 2κδc † in. (2.8) Introducing the cavity field quadratures δx = δc† + δc and δy = i(δc† − δc), and the input noise quadratures δxin = δc † in +δcin and δyin = i(δc † in −δcin), Eq. (2.8) can be written in the matrix form f˙ = Af(t) + η(t), (2.9) where f(t) is the column vector of the fluctuations, and η(t) is the column vector of the noise sources. For the sake of simplicity, their transposes are f(t)T = (δq, δp, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (2.10) and the matrix A is given by A = 0 1 m 0 0 −mω2m −γm ¯hχcs+c∗ s 2 ¯hχcs−c∗ s 2i iχ(cs − c∗ s) 0 2Gcos θ − κ Δ + 2Gsin θ χ(cs + c∗ s) 0 2Gsin θ − Δ −(κ + 2Gcos θ) . (2.11) The solutions to Eq. (2.9) are stable only if all the eigenvalues of the matrix A have negative real parts. Applying the RouthHurwitz criterion [112, 113], we get the 32 stability conditions 2κ(κ2 − 4G2 + Δ2 + 2κγm) + γm(2κγm + ω2m ) > 0, (2κ + γm)2[ 2¯hχ2cs2 m Δ + 2¯hχ2(c2s + c∗2 s )Gsin θ m + 2i¯hχ2(c2s − c∗2 s )Gcos θ m ] + 2κγm{(κ2 − 4G2 + Δ2)2 +(2κγm + γ2m )(κ2 − 4G2 + Δ2) +ω2m [2(κ2 + 4G2 − Δ2) + ω2m + 2κγm]} > 0, ω2m (κ2 − 4G2 + Δ2) − 2¯hχ2cs2 m Δ −2¯hχ2(c2s + c∗2 s )Gsin θ m − 2i¯hχ2(c2s − c∗2 s )Gcos θ m > 0. (2.12) Note that in the absence of coupling χ, the conditions (2.12) become equivalent to κ2 − 4G2 + Δ2 > 0 (2.13) The condition for the threshold for parametric oscillations is κ2 −4G2 +Δ2 = 0. We always would work under the condition that (2.13) is satisfied. Further for χ ̸= 0 we would do numerical simulations using parameters so that conditions (2.12) are satisfied. On Fourier transforming all operators and noise sources in Eq. (2.8) and solving it in the frequency domain, the position fluctuations of the movable mirror are given by δq(ω) = − 1 d(ω)([Δ2 + (κ − iω)2 − 4G2]ξ(ω) −i¯h √ 2κχ{[(ω + iκ − Δ)cs + 2iGeiθc∗ s]δc † in(ω) +[(ω + iκ + Δ)c∗ s + 2iGe−iθcs]δcin(ω)}), (2.14) where d(ω) = 2¯hχ2(Δcs2+iGe−iθc2s −iGeiθc∗2 s )+m(ω2−ω2m +iωγm)[Δ2+(κ−iω)2− 4G2]. In Eq. (2.14), the first term proportional to ξ(ω) originates from the thermal noise, while the second term proportional to χ arises from radiation pressure. So the position fluctuations of the movable mirror are now determined by the thermal noise 33 and radiation pressure. Notice that if there is no radiation pressure, the movable mirror will make Brownian motion, δq(ω) = −ξ(ω)/[m(ω2 − ω2m + iωγm)], whose susceptibility has a Lorentzian shape centered at frequency ωm with width γm. The spectrum of fluctuations in position of the movable mirror is defined by Sq(ω) = 1 4π ∫ dΩe−i(ω+Ω)t⟨δq(ω)δq(Ω) + δq(Ω)δq(ω)⟩. (2.15) To calculate the spectrum, we need the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 2π¯hγmmω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (2.16) Substituting Eq. (2.14) and Eq. (2.16) into Eq. (2.15), we obtain the spectrum of fluctuations in position of the movable mirror Sq(ω) = ¯h d(ω)2 {2κ¯hχ2[(κ2 + ω2 + Δ2 + 4G2)cs2 +2Geiθc∗2 s (κ − iΔ) + 2Ge−iθc2s (κ + iΔ)] +mγmω[(Δ2 + κ2 − ω2 − 4G2)2 + 4κ2ω2] ×coth( ¯hω 2kBT )}. (2.17) In Eq. (2.17), the first term is the radiation pressure contribution, whereas the second term corresponds to the thermal noise contribution. Then Fourier transforming q˙ = δp/m in Eq. (2.8), we obtain δp(ω) = −imωδq(ω), which leads to the spectrum of fluctuations in momentum of the movable mirror Sp(ω) = m2ω2Sq(ω). (2.18) For a system in thermal equilibrium, we can use the equipartition theorem to define temperature 1 2mω2m ⟨q2⟩ = ⟨p2⟩ 2m = 1 2kBTeff , where ⟨q2⟩ = 1 2π ∫ +∞ −∞ Sq(ω)dω, and ⟨p2⟩ = 1 2π ∫ +∞ −∞ Sp(ω)dω. However, here we are dealing with a driven system and 1 2mω2m ⟨q2⟩ ̸= 34 ⟨p2⟩ 2m , hence the question is how to define temperature. We use an effective temperature defined by the total energy of the movable mirror kBTeff = 1 2mω2m ⟨q2⟩ + ⟨p2⟩ 2m . We also introduce the parameter r = m2ω2m ⟨q2⟩/⟨p2⟩.This parameter gives us the relative importance of fluctuations in position and momentum of the mirror. We mention that one can calculate the quantum state of the oscillator and we find that the Wigner function is Gaussian. Equation (2.17) is our key result which tells how the temperature of the micromirror would depend on the parameters of the cavity: κ, gain of the OPA, external laser power, etc. We specifically investigate the dependence of the temperature on the gain G and the phase θ associated with the parametric amplification process. In the limit of G → 0, the result (2.17) reduces to the one derived by Paternostro et al.[35]. 2.4 Cooling Mirror to About SubKelvin Temperatures In this section, we present the possibility of cooling the micromirror to temperatures of about subKelvin by using parametric amplifiers inside cavities. In all the numerical calculations we choose the values of the parameters which are similar to those used in recent experiments: λL = 2πc/ωL = 1064 nm, L = 25 mm, P = 4 mW, m = 15 ng, ωm/(2π) = 275 kHz, and the mechanical quality factor Q = ωm/γm = 2.1 × 104. Further in the hightemperature limit kBT ≫ ¯hω, we have coth(¯hω/2kBT) ≈ 2kBT/¯hω. 2.4.1 From Room Temperature (T=300 K) to About SubKelvin Tem peratures If we choose κ = 108 s−1, F = 188.4, G = 0 to satisfy the stability conditions (2.12), the detuning must satisfy Δ0 ≥ 4 × 106s−1. Figure 2.2 gives the variations of the χqs, the effective temperature Teff , and the parameter r with the detuning Δ0. It should be borne in mind that for the range of the detuning shown in Fig. 2.2, 35 2 4 6 8 10 0 5 10 15 20 25 30 0. 1. 2. 3. 4. 5. 6. D0 H107 s1L T eff HKL , r+5 Χqs H10 6 s1L Figure 2.2: The dotted curve indicates the χqs (106 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G=0. Δ = Δ0 − χqs ≈ Δ0. We find the χqs is single valued, so the movable mirror is monostable. Note that the parameter r is very close to unity, 1 2mω2m ⟨q2⟩ ≈ ⟨p2⟩ 2m ; the mirror is thus in nearly thermal equilibrium. Figure 2.2 shows the possibility of cooling the mirror to a temperature of 15.23 K for Δ0 = 4.9 × 107 s−1, which is in agreement with the previous calculation [35]. Now we keep the values of κ and F the same as in Fig. 2.2, and we choose parametric gain G = 3.5 × 107 s−1 and parametric phase θ = 0; the detuning must satisfy Δ0 ≥ 5.7 × 107 s−1. If Δ0 < 5.7 × 107 s−1 and for fixed κ and G, the system will be unstable. The threshold for unstable behavior occurs when any of the three conditions (2.12) is not satisfied. It may be noted that the threshold for parametric oscillation has been of great importance in connection with the production 36 6 7 8 9 10 0 1 2 3 4 5 6 0 1 2 3 4 5 6 D0 H107 s1 L T eff HKL , r Χqs H107 s1 L Figure 2.3: The dotted curve indicates the χqs (107 s−1) as a function of the detuning Δ0 (107 s−1) (rightmost vertical scale). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1) (leftmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107s−1, parametric phase θ = 0. of nonclassicalsqueezed light. Near the parametric thresholds but under (2.13), large degrees of squeezing were produced [97, 98]. Thus it would be advantageous to work near the threshold of instability but below the instability point. Figure 2.3 shows the variations of the χqs, the effective temperature Teff , and the parameter r with the detuning Δ0. We find the χqs is still single valued, so the movable mirror is still monostable. The minimum temperature reached is 0.65 K for Δ0 = 6.7 × 107 s−1. Thus, with the parametric amplifier the minimum temperature is about a factor of 20 lower than the one without parametric interaction. Note that the parameter r is always larger than 1, implying that momentum fluctuations are suppressed over 37 2.0 2.2 2.4 2.6 2.8 3.0 0 1 2 3 4 5 D0 H107 s 1 L Χq s H107 s 1 L Figure 2.4: The behavior of χqs (107 s−1) shown as a function of the detuning Δ0 (107 s−1). The position that corresponds to the minimum effective temperature reached is indicated by the arrow. Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5 × 106s−1, parametric phase θ = 3π/4. position fluctuations. Note that as one moves away from the threshold for parametric instability, the minimum temperature does not rise sharply which is in contrast to the behavior in Fig. 2.2, and is advantageous in giving one flexibility about the choice of the detuning parameter. We next examine the case when the behavior of the system is multistable. For this purpose, we choose the cavity to have the higher quality factor. We choose κ = 107 s−1, F = 1884, G = 5×106 s−1 and θ = 3π/4; then to satisfy the stability conditions (2.12), the detuning must satisfy Δ0 ≥ 1.847×107 s−1. Figure 2.4 gives the behavior of χqs as a function of the detuning Δ0. We find the χqs is multivalued, so the movable 38 2.0 2.2 2.4 2.6 2.8 3.0 0.0 0.5 1.0 1.5 2.0 D0 H107 s1 L Teff HKL , r Figure 2.5: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1). Parameters: cavity decay rate κ = 107 s−1, cavity finesse F = 1884, parametric gain G = 5×106s−1, parametric phase θ = 3π/4. mirror is multistable. By use of the lowest curve of the χqs, we obtain the variations of the effective temperature Teff and the parameter r with the detuning Δ0, as shown in Fig. 2.5. We choose that the range of the detuning is 2.0 × 107 s−1 − 3.0 × 107 s−1. The minimum temperature achieved is 0.265 K for Δ0 = 2.0 × 107 s−1. Note that r is close to unity but larger than unity. The general trend is clear. By playing around with various parameters such as laser power, cavity finesse, and parametric gain, one can achieve a variety of different temperatures. As another example, if we choose κ = 5 × 106 s−1, F = 3768, G = 107 s−1 and θ = 0.2467 + π/2, then we find that the minimum temperature is 0.092 K for Δ0 = 2.13 × 107s−1. 2.4.2 From 1 K to Millikelvin Temperatures If the thermal bath is cryogenically cooled down to a temperature of 1 K and the mirror is initially thermalized, then we can use radiation pressure effects and photon 39 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0. 0.5 1. 1.5 2. 2.5 3. D0 H107 s 1L Teff HKL r Figure 2.6: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 0. statistics to reach millikelvin or even lower temperatures. If we choose κ = 108 s−1, F = 188.4, G = 0, the effective temperature Teff with the detuning Δ0 is shown in Fig. 2.6. The minimum temperature reached is 0.051 K for Δ0 = 4.9 × 107 s−1. Next we examine how the effective temperature changes by the parametric interactions inside the cavity. We keep all other parameters as in Fig. 6 and choose parametric gain G = 3.5 × 107 s−1 and phase θ = 0. Then the effective temperature Teff with the detuning Δ0 exhibits behavior as shown in Fig. 2.7. The minimum temperature achieved is 0.0044 K for Δ0 = 7.9 × 107 s−1, a factor of 12 lower than the one without parametric interaction. Finally it should be borne in mind that the radiation pressure depends on the number operator and then it is sensitive to the photon statistics of the field in the cavity. The photon statistics can be calculated from the quantum Langevin equations (2.8). It can be proved that the Wigner function W of the field in the cavity is 40 6 7 8 9 10 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0. 1. 2. 3. 4. D0 H107 s 1L Teff HKL r Figure 2.7: The solid curve shows the effective temperature Teff(K) as a function of the detuning Δ0 (107 s−1)(leftmost vertical scale). The dashed curve represents the parameter r as a function of the detuning Δ0 (107 s−1)(rightmost vertical scale). Parameters: cavity decay rate κ = 108 s−1, cavity finesse F = 188.4, parametric gain G = 3.5 × 107 s−1, parametric phase θ = 0. Gaussian of the form exp[μ(α − cs)2 + ν(α∗ − c∗ s)2 + λ(α − cs)(α∗ − c∗ s)] with μ, ν, λ determined by κ, Δ, G, θ, etc. The photon number distribution [103] associated with such a Gaussian Wigner function depends in an important way on the parameter μ and the inequality of μ and ν. The latter depend on G ̸= 0 or on the presence of OPA in the cavity. 2.5 Conclusions In conclusion, we have demonstrated how the addition of a parametric amplifier in a cavity can lead to cooling of the micromirror to a temperature; which is much lower than what is achieved in an identical experiment without the use of a parametric amplifier. The parametric processes inside the cavity change the quantum statistics of the field in the cavity. This change leads to lower cooling since the radiation pressure 41 effects are dependent on the photon number. Thus photon statistics becomes central to achieve lower cooling temperatures. The use of parametric processes could provide us with a way to cool the mirror to its quantum ground state or even squeeze it. The content of this chapter has been published in Phys. Rev. A 79, 013821 (2009). 42 CHAPTER 3 NORMAL MODE SPLITTING IN A COUPLED SYSTEM OF A NANOMECHANICAL OSCILLATOR AND A PARAMETRIC AMPLIFIER CAVITY 3.1 Overview Recently there has been a major effort in applying many of the well tested ideas from quantum optics such as squeezing, quantum entanglement to optomechanical systems which are macroscopic systems. Thus observation of entanglement [28, 35, 36, 38, 114, 115], squeezing [25, 26] etc in optomechanical systems would enable one to study quantum behavior at macroscopic scale. This of course requires cooling such systems to their ground state and significant advances have been made in cooling the mechanical mirror to far below the temperature of the environment [7, 8, 9, 116, 117, 118, 119]. Further it has been pointed out that using optical back action one can possibly achieve the ground state cooling in the resolved sideband regime where the frequency of the mechanical mirror is much larger than the cavity decay rate, that is ωm ≫ κ [47, 66, 120]. Another key idea from quantum optics is the vacuum Rabi splitting [121, 122] which is due to strong interaction between the atoms and the cavity mode. The experimentalists have worked hard over the years to produce stronger and stronger couplings to produce larger and larger splittings [123, 124, 213]. Application of these ideas to macroscopic systems is challenging as well. In a recent paper Kippenberg et al. [48] proposed the possibility of normal mode splitting in the resolved sideband regime using optomechanical oscillators. In this chapter, we propose placing a type 43 I optical parametric amplifier inside the cavity to increase the coupling between the movable mirror and the cavity field, and this should make the observation of the normal mode splitting of the movable mirror and the output field more accessible. The chapter is structured as follows. In Sec. II we present the model, derive the quantum Langevin equations, and give the steadystate mean values. In Sec. III we present solution to the linearized Langevin equations and give the spectrum of the movable mirror. In Sec. IV we analyse and estimate the amount of the normal mode splitting of the spectra. In Sec. V we calculate the spectra of the output field. In Sec. VI we discuss the mode splitting of the spectra of the movable mirror and the output field. 3.2 Model The system under consideration, sketched in Fig. 3.1, is an optical parametric amplifier (OPA) placed within a FabryPerot cavity formed by one fixed partially transmitting mirror and one movable perfectly reflecting mirror in equilibrium with its environment at a low temperature. The movable mirror is treated as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm, and energy decay rate γm. An external laser enters the cavity through the fixed mirror, then the photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer. This force is proportional to the instantaneous photon number in the cavity. In the adiabatic limit, the frequency ωm of the movable mirror is much smaller than the free spectral range of the cavity c 2L (c is the speed of light in vacuum and L is the cavity length), the scattering of photons to other cavity modes can be ignored, thus only one cavity mode ωc is considered [64, 106]. The Hamiltonian for the system in a frame rotating at the laser frequency ωL can be written as H = ¯h(ωc − ωL)nc − ¯hωmχncQ + ¯hωm 4 (Q2 + P2) 44 in c out c movable mirror OPA cavity axis fixed mirror Figure 3.1: Sketch of the studied system. The cavity contains a nonlinear crystal which is pumped by a laser (not shown) to produce parametric amplification and to change photon statistics in the cavity. +i¯hε(c† − c) + i¯hG(eiθc†2 − e−iθc2). (3.1) Here Q and P are the dimensionless position and momentum operators for the movable mirror, defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with [Q, P] = 2i. In Eq. (3.1), the first term is the energy of the cavity field, nc = c†c is the number of the photons inside the cavity, c and c† are the annihilation and creation operators for the cavity field satisfying the commutation relation [c, c†] = 1. The second term comes from the coupling of the movable mirror to the cavity field via radiation pressure, the dimensionless parameter χ = 1 ωm ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term corresponds the energy of the movable mirror. The fourth term describes the coupling between the input laser field and the cavity field, ε is related to the input laser power ℘ by ε = √ 2κ℘ ¯hωL , where κ is the cavity decay rate. The last term is the coupling between the OPA and the cavity field, G is the nonlinear gain of the OPA, and θ is the phase of the field driving the OPA. The parameter G is proportional to the pump driving the OPA. Using the Heisenberg equations of motion and adding the corresponding damping 45 and noise terms, we obtain the quantum Langevin equations as follows, ˙Q = ωmP, ˙P = 2ωmχnc − ωmQ − γmP + ξ, ˙ c = −i(ωc − ωL − ωmχQ)c + ε + 2Geiθc† − κc + √ 2κcin, ˙ c† = i(ωc − ωL − ωmχQ)c† + ε + 2Ge−iθc − κc† + √ 2κc † in. (3.2) Here we have introduced the input vacuum noise operator cin with zero mean value, which obeys the correlation function in the time domain [141] ⟨δcin(t)δc † in(t′)⟩ = δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = ⟨δc † in(t)δcin(t′)⟩ = 0. (3.3) The force ξ is the Brownian noise operator resulting from the coupling of the movable mirror to the thermal bath, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = 1 π γm ωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (3.4) where kB is the Boltzmann constant and T is the thermal bath temperature. Following standard methods from quantum optics [110], we derive the steadystate solution to Eq. (3.2) by setting all the time derivatives in Eq. (3.2) to zero. They are Ps = 0, Qs = 2χcs2, cs = κ − iΔ + 2Geiθ κ2 + Δ2 − 4G2 ε, (3.5) where Δ = ωc − ωL − ωmχQs (3.6) is the effective cavity detuning, depending on Qs. The Qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. From Eq. (3.5) and Eq. (3.6), we can see Qs satisfies a fifth order equation, it can at most have five real 46 solutions. Therefore, the movable mirror displays an optical multistable behavior [41, 42, 43], which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. 3.3 Radiation Pressure and Quantum Fluctuations In order to investigate the normal mode splitting of the movable mirror and the output field, we need to calculate the fluctuations of the system. Since the problem is nonlinear, we assume that the nonlinearity is weak. Thus we can focus on the dynamics of small fluctuations around the steady state of the system. Each operator of the system can be written as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (3.7) Inserting Eq. (3.7) into Eq. (3.2), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators take the form δ ˙Q = ωmδP, δ ˙P = 2ωmχ(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + iωmχcsδQ + 2Geiθδc† + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − iωmχc∗ sδQ + 2Ge−iθδc + √ 2κδc † in. (3.8) Introducing the cavity field quadratures δx = δc + δc† and δy = i(δc† − δc), and the input noise quadratures δxin = δcin + δc † in and δyin = i(δc † in − δcin), Eq. (3.8) can be rewritten in the matrix form f˙(t) = Af(t) + η(t), (3.9) 47 in which f(t) is the column vector of the fluctuations, η(t) is the column vector of the noise sources. Their transposes are f(t)T = (δQ, δP, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (3.10) and the matrix A is given by A = 0 ωm 0 0 −ωm −γm ωmχ(cs + c∗ s) −iωmχ(cs − c∗ s) iωmχ(cs − c∗ s) 0 2Gcos θ − κ 2Gsin θ + Δ ωmχ(cs + c∗ s) 0 2Gsin θ − Δ −(2Gcos θ + κ) . (3.11) The system is stable only if all the eigenvalues of the matrix A have negative real parts. The stability conditions for the system can be derived by applying the Routh Hurwitz criterion [112, 113]. This gives 2κ(κ2 − 4G2 + Δ2 + 2κγm) + γm(2κγm + ω2m ) > 0, 2ω3m χ2(2κ + γm)2[cs2Δ + iG(c2s e−iθ − c∗2 s eiθ)] +κγm{(κ2 − 4G2 + Δ2)2 + (2κγm + γ2m ) ×(κ2 − 4G2 + Δ2) + ω2m [2(κ2 + 4G2 − Δ2) +ω2m + 2κγm]} > 0, κ2 − 4G2 + Δ2 − 4ωmχ2[cs2Δ + iG(c2s e−iθ − c∗2 s eiθ)] > 0. (3.12) All the external parameters must be chosen to satisfy the stability conditions (3.12). Taking Fourier transform of Eq. (3.8) by using f(t) = 1 2π ∫ +∞ −∞ f(ω)e−iωtdω and f†(t) = 1 2π ∫ +∞ −∞ f†(−ω)e−iωtdω, where f†(−ω) = [f(−ω)]†, then solving it, we obtain 48 the position fluctuations of the movable mirror δQ(ω) = − ωm d(ω) [2 √ 2κωmχ{[(κ − i(Δ + ω))c∗ s + 2Ge−iθcs]δcin(ω) +[(κ + i(Δ − ω))cs + 2Geiθc∗ s]δc † in(−ω)} +[(κ − iω)2 + Δ2 − 4G2]ξ(ω)], (3.13) where d(ω) = 4ω3m χ2[Δcs2 + iG(c2s e−iθ − c∗2 s eiθ)] +(ω2 − ω2m + iγmω)[(κ − iω)2 + Δ2 − 4G2]. (3.14) In Eq. (3.13), the first term proportional to χ originates from radiation pressure, while the second term involving ξ(ω) is from the thermal noise. So the position fluctuations of the movable mirror are now determined by radiation pressure and the thermal noise. In the case of no coupling with the cavity field, the movable mirror will make Brownian motion, δQ(ω) = ωmξ(ω)/(ω2m −ω2−iγmω), whose susceptibility has a Lorentzian shape centered at frequency ωm with width γm. The spectrum of fluctuations in position of the movable mirror is defined by 1 2 (⟨δQ(ω)δQ(Ω)⟩ + ⟨δQ(Ω)δQ(ω)⟩) = 2πSQ(ω)δ(ω + Ω). (3.15) To calculate the spectrum, we require the correlation functions of the noise sources in the frequency domain, ⟨δcin(ω)δc † in(−Ω)⟩ = 2πδ(ω + Ω), ⟨ξ(ω)ξ(Ω)⟩ = 4π γm ωm ω [ 1 + coth( ¯hω 2kBT ) ] δ(ω + Ω). (3.16) Substituting Eq. (3.13) and Eq. (3.16) into Eq. (3.15), we obtain the spectrum of fluctuations in position of the movable mirror [126] SQ(ω) = ω2m d(ω)2 {8ω2m χ2κ[(κ2 + ω2 + Δ2 + 4G2)cs2 +2Geiθc∗2 s (κ − iΔ) + 2Ge−iθc2s (κ + iΔ)] +2γm ωm ω[(Δ2 + κ2 − ω2 − 4G2)2 + 4κ2ω2] ×coth( ¯hω 2kBT )}. (3.17) 49 In Eq. (3.17), the first term involving χ arises from radiation pressure, while the second term originates from the thermal noise. So the spectrum SQ(ω) of the movable mirror depends on radiation pressure and the thermal noise. 3.4 Normal Mode Splitting and the Eigenvalues of the Matrix A The structure of all the spectra is determined by the eigenvalues of iA (Eq. (3.11)) or the complex zeroes of the function d(ω) defined by Eq. (3.14). Clearly we need the eigenvalues of iA as the solution of (Eq. (3.9)) in Fourier domain is f(ω) = i(ω − iA)−1η(ω). Let us analyse the eigenvalues of Eq. (3.11). Note that in the absence of the coupling χ=0, the eigenvalues of iA are ± √ ω2m − γ2m 4 − iγm 2 ;± √ Δ2 − 4G2 − iκ. (3.18) Thus the positive frequencies of the normal modes are given by √ Δ2 − 4G2, √ ω2m − γ2m 4 (Δ > 2G, ωm > γm 2 ). The case that we consider in this chapter corresponds to ωm ≫ γm 2 ;Δ > 2G; κ ≫ γm; ωm > κ. (3.19) The coupling between the normal modes would be most efficient in the degenerate case i.e. when ωm = √ Δ2 − 4G2. It is known from cavity QED that the normal mode splitting leads to symmetric (asymmetric) spectra in the degenerate (nondegenerate) case, provided that the dampings of the individual modes are much smaller than the coupling constant. Thus the mechanical oscillator is like the atomic oscillator, the cavity mode in the rotating frame acquires the effective frequency √ Δ2 − 4G2 which is dependent on the parametric coupling. All this applies provided that damping terms do not mix the modes significantly. An estimate of the splitting can be made by using the approximations given by Eq. (3.19) and the zeroes of d(ω). We find that the frequency splitting is given by [127] ω2± ∼= ω2m +Δ2−4G2 2 ± √ (ω2m −Δ2+4G2 2 )2 + 4ω2m g2, (3.20) 50 where we have defined g2 = ωmχ2cs2[Δ + 2Gsin(θ − 2φ)], e2iφ = c2s /cs2. (3.21) It should be borne in mind that cs is dependent on the parametric coupling G. The splitting is determined by the pump power, the couplings χ and G. The parameters used are the same as those in the recent successful experiment on optomechanical normal mode splitting [50]: the wavelength of the laser λ = 2πc/ωL = 1064 nm, L = 25 mm, m = 145 ng, κ = 2π × 215 × 103 Hz, ωm = 2π × 947 × 103 Hz, T = 300 mK, the mechanical quality factor Q ′ = ωm/γm = 6700, parametric phase θ = π/4. And in the high temperature limit kBT ≫ ¯hωm, we have coth(¯hω/2kBT) ≈ 2kBT/¯hω. Figure 3.2 shows the roots of d(ω) in the domain Re(ω) > 0 for different values of G. Figure 3.3 shows imaginary parts of the roots of d(ω) for different values of G. The parametric coupling affects the width of the lines and this for certain range of parameters aids in producing well split lines. One root broadens and the other root narrows. The root that broadens is the one that moves further away from the position for G = 0. 3.5 The Spectra of the Output Field In this section, we would like to calculate the spectra of the output field. The fluctuations δc(ω) of the cavity field can be obtained from Eq. (3.8). Further using the inputoutput relation [128] cout(ω) = √ 2κc(ω)−cin(ω), the fluctuations of the output field are given by δcout(ω) = V (ω)ξ(ω) + E(ω)δcin(ω) + F(ω)δc † in(−ω), (3.22) 51 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 G k ReHwL wm Figure 3.2: The roots of d(ω) in the domain Re(ω) > 0 as a function of parametric gain. ℘ = 6.9 mW (dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 G Κ ImHwL wm Figure 3.3: The imaginary parts of the roots of d(ω) as a function of parametric gain. ℘ = 6.9 mW ( dotted line), ℘ = 10.7 mW (dashed line). Parameters: the cavity detuning Δ = ωm. 52 where V (ω) = − √ 2κω2m χ d(ω) i{[κ − i(ω + Δ)]cs − 2Geiθc∗ s }, E(ω) = 2κ (κ−iω)2+Δ2−4G2 [−2ω3m χ2 d(ω) i{[κ − i(ω + Δ)]cs −2Geiθc∗ s }{[κ − i(ω + Δ)]c∗ s + 2Ge−iθcs} +κ − i(ω + Δ)] − 1, F(ω) = 2κ (κ−iω)2+Δ2−4G2 [−2ω3m χ2 d(ω) i{[κ − i(ω + Δ)]cs −2Geiθc∗ s }{[κ − i(ω − Δ)]cs + 2Geiθc∗ s } +2Geiθ]. (3.23) In Eq. (3.22), the first term associated with ξ(ω) stems from the thermal noise of the mechanical oscillator, while the other two terms are from the input vacuum noise. So the fluctuations of the output field are influenced by the thermal noise and the input vacuum noise. The spectra of the output field are defined as ⟨δc † out(−Ω)δcout(ω)⟩ = 2πScout(ω)δ(ω + Ω), ⟨δxout(Ω)δxout(ω)⟩ = 2πSxout(ω)δ(ω + Ω), ⟨δyout(Ω)δyout(ω)⟩ = 2πSyout(ω)δ(ω + Ω). (3.24) where δxout(ω) and δyout(ω) are the Fourier transform of the fluctuations δxout(t) and δyout(t) of the output field , which are defined by δxout(t) = δcout(t) + δc † out(t) and δyout(t) = i[δc † out(t) − δcout(t)] [110]. Here Scout(ω) denotes the spectral density of the output field, Sxout(ω) means the spectrum of fluctuations in the x quadrature of the output field, and Syout(ω) is the spectrum of fluctuations in the y quadrature of the output field. Combining Eq. (3.16), Eq. (3.22), and Eq. (3.24), we obtain the spectra of the 53 output field Scout(ω) = V ∗(ω)V (ω) × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] + F∗(ω)F(ω), Sxout(ω) = [V (−ω) + V ∗(ω)][V (ω) + V ∗(−ω)] × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] +[E(−ω) + F∗(ω)][F(ω) + E∗(−ω)], Syout(ω) = −[V ∗(ω) − V (−ω)][V ∗(−ω) − V (ω)] × 2 γm ωm ω[−1 + coth( ¯hω 2kBT )] −[F∗(ω) − E(−ω)][E∗(−ω) − F(ω)]. (3.25) From Eq. (3.25), it is seen that any spectrum of the output field includes two terms, the first term is from the contribution of the thermal noise of the mechanical oscillator, the second term is from the contribution of the input vacuum noise. We note that the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) are determined by the detuning Δ, parametric gain G, parametric phase θ, input laser power ℘, and cavity length L. In the following we will concentrate on discussing the dependence of the spectra on parametric gain and input laser power. 3.6 Numerical Results In this section, we numerically evaluate the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) given by Eq. (3.17) and Eq. (3.25) to show the effect of an OPA in the cavity on the normal mode splitting of the movable mirror and the output field. We typically imagine a setup like in the original squeezing experiment [97] where the experiment is done, for different levels of the pumping of OPA i.e., we start with G = 0 and then increase it to a value consistent with the stability requirements. We consider the degenerate case Δ = ωm for G = 0, and choose ℘ = 6.9 mW. In order to satisfy the stability conditions (3.12), parametric gain must satisfy G ≤ 1.62κ. The figures 3.4 – 3.7 show the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) as a function of the normalized frequency ω/ωm for various values of parametric gain. When the OPA is absent (G = 0), the spectra barely show the normal mode splitting. As 54 0.0 0.5 1.0 1.5 2.0 0.000 0.005 0.010 0.015 0.020 w wm SQHwLΓm G=1.45Κ G=1.3Κ G=0 Figure 3.4: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 w wm Scout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.5: The spectrum Scout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. 55 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 w wm Sxout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.6: The spectrum Sxout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. parametric gain is increased, the normal mode splitting becomes observable. This is due to significant changes in the line widths and position. When G = 1.3κ, two peaks can be found in the spectra. According to the numerical calculations of Figs. 3.2 and 3.3, these roots in units of ωm are at (A) G = 0: 0.885−0.113i, 1.091−0.113i for 6.9 mW pump power and 0.826 − 0.113i, 1.136 − 0.113i for 10.7 mW pump power. (B) G = 1.3κ: 0.596−0.156i, 1.129−0.070i for 6.9 mW pump power and 0.490−0.148i, 1.178−0.079i for 10.7 mW pump power. We see that the line width of the two peaks is approximately same for G = 0 but for two different power levels. The line widths change significantly for G ̸= 0. Note that the separation between two peaks becomes larger as parametric gain increases. The reason is that increasing the parametric gain causes a stronger coupling between the movable mirror and the cavity field due to an increase in the photon number in the cavity. The values of intracavity photon number cs2 are 2.68 × 109, 4.30 × 109, 5.65 × 109 for G = 0, 1.3κ, and 1.45κ respectively. 56 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 w wm Syout HwL G=1.45Κ G=1.3Κ G=0 Figure 3.7: The spectrum Syout(ω) versus the normalized frequency ω/ωm for different parametric gain. G= 0 (solid curve), 1.3κ (dotted curve), 1.45κ (dashed curve). Parameters: the cavity detuning Δ = ωm, the laser power ℘ = 6.9 mW. We have examined the contributions of various terms in Eq. (3.25) to the output spectrum. The dominant contribution comes from the mechanical oscillator. Note further the similarity [50] of the spectrum of the output quadrature y (Fig. 3.7) to the spectrum of the mechanical oscillator (Fig. 3.4). It should be borne in mind that the strong asymmetries in the spectra for G ̸= 0 arise from the fact that by fixing Δ at ωm, the frequencies of the cavity mode and the mechanical oscillator do not coincide if G ̸= 0; χ = 0. Besides the damping term κ, being not negligible compared to Δ, also contributes to asymmetries. Now we fix parametric gain G = 1.3κ, and choose Δ = √ ω2m + 4G2, the input laser power must satisfy ℘ ≤ 55 mW. The spectrum SQ(ω) as a function of the normalized frequency ω/ωm for increasing the input laser power is shown in Fig. 3.8. As we increase the laser power from 0.6 mW to 10.7 mW, the spectrum exhibits a doublet and the peak separation is proportional to the laser power, because the 57 coupling between the movable mirror and the cavity field for a given parametric gain G is increased with increasing the input laser power due to an increase in photon number. 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0. 0.005 0.01 0.015 0.02 0.025 0.03 w wm SQHwLΓm Ã=10.7 mW Ã=6.9 mW Ã=0.6 mW Figure 3.8: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = √ ω2m + 4G2, parametric gain G = 1.3κ. For comparison, we also consider the case of the cavity without OPA (G = 0), the spectrum SQ(ω) as a function of the normalized frequency ω/ωm for increasing the input laser power at Δ = ωm is plotted in Fig. 3.9. We can see if the laser power is increased from 0.6 mW to 10.7 mW, the spectrum also displays normal mode splitting. However the normal mode with OPA (Fig. 3.8) are more pronounced than that in the absence of OPA (Fig. 3.9). 58 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 w wm SQHwLΓm Ã=10.7 mW Ã=6.9 mW Ã=0.6 mW Figure 3.9: The scaled spectrum SQ(ω) × γm versus the normalized frequency ω/ωm, each curve corresponds to a different input laser power. ℘= 0.6 mW (solid curve, leftmost vertical scale), 6.9 mW (dotted curve, rightmost vertical scale), 10.7 mW (dashed curve, rightmost vertical scale). Parameters: the cavity detuning Δ = ωm, parametric gain G = 0. 3.7 Conclusions In conclusion, we have shown how the normal mode splitting behavior of the movable mirror and the output field is affected by the OPA in the cavity. We work in the resolved sideband regime and operate under the stability conditions (3.12). We find that increasing parametric gain can make the spectra SQ(ω), Scout(ω), Sxout(ω), and Syout(ω) evolve from a single peak to two peaks. Furthermore, for a given parametric gain, increasing input laser power can increase the amount of normal mode splitting of the movable mirror due to the stronger coupling between the movable mirror and the cavity field. The content of this chapter has been published in Phys. Rev. A 80, 033807 (2009). 59 CHAPTER 4 SQUEEZING OF A NANOMECHANICAL OSCILLATOR 4.1 Overview The optomechanical system has attracted much attention because of its potential applications in high precision measurements and quantum information processing [28, 35, 36, 37, 88, 90, 129, 130, 131]. Meanwhile, it provides a means of probing quantum behavior of a macroscopic object if a nanomechanical oscillator can be cooled down to near its quantum ground state [38, 115]. Many of these applications are becoming possible due to advances in cooling the mirror [6, 7, 8, 9, 10, 56, 118]. Further as pointed out in Refs [47, 66, 120], the ground state cooling can be achieved in the resolved sideband regime where the frequency of the mechanical mirror is much larger than the cavity decay rate. Squeezing of a nanomechanical oscillator plays a vital role in highsensitive detection of position and force due to its less noise in one quadrature than the coherent state. A number of different methods have been developed to generate and enhance squeezing of a nanomechanical oscillator, such as coupling a nanomechanical oscillator to an atomic gas [132], a Cooper pair box [133], a SQUID device [215], using threewave mixing [135] or Circuit QED [136], or by means of quantum measurement and feedback schemes [137, 138, 139, 140]. A recent paper [32] reports squeezed state of a mechanical mirror can be created by transfer of squeezing from a squeezed vacuum to a membrane within an optical cavity under the conditions of ground state cooling. We previously considered the possibility of using an OPA inside the cavity for changing the nature of the statistical fluctuations [126]. 60 In this chapter, we propose a scheme that is capable of generating squeezing of the movable mirror by feeding broad band squeezed vacuum light along with the laser light. The achieved squeezing of the mirror depends on the temperature of the mirror, the laser power, and degree of squeezing of the input light. One can obtain squeezing which could be more than 70%. The chapter is structured as follows. In Sec. II we describe the model, give the quantum Langevin equations, and obtain the steadystate mean values. In Sec. III we derive the stability conditions, calculate the mean square fluctuations in position and momentum of the movable mirror. In Sec. IV we analyze how the momentum squeezing of the movable mirror is affected by the squeezing parameter, the temperature of the environment, and the laser power. We also compare the momentum fluctuations of the movable mirror in the presence of the coupling to the cavity field with that in the absence of the coupling to cavity field. We find very large squeezing with respect to thermal fluctuations, for instance at 1 mK, the momentum fluctuations go down by a factor more than one hundred. Our predictions of squeezing are based on the parameters used in a recent experiment on normal mode splitting in a nanomechanical oscillator [50]. 4.2 Model The system to be considered, sketched in Fig. 4.1, is a FabryPerot cavity with one fixed partially transmitting mirror and one movable perfectly reflecting mirror in thermal equilibrium with its environment at a low temperature. The cavity with length L is driven by a laser with frequency ωL, then the photons in the cavity will exert a radiation pressure force on the movable mirror due to momentum transfer. This force is proportional to the instantaneous photon number in the cavity. The mirror also undergoes thermal fluctuations due to environment. Under the effects of the two forces, the movable mirror makes oscillation around its equilibrium position. 61 Here we treat the movable mirror as a quantum mechanical harmonic oscillator with effective mass m, frequency ωm and momentum decay rate γm. We further assume that the cavity is fed with squeezed light at frequency ωS. Figure 4.1: Sketch of the studied system. A laser with frequency ωL and squeezed vacuum light with frequency ωS enter the cavity through the partially transmitting mirror. In the adiabatic limit, ωm ≪ c 2L ( c is the speed of light in vacuum), we ignore the scattering of photons to other cavity modes, thus only one cavity mode ωc is considered [64, 106]. In a frame rotating at the laser frequency, the Hamiltonian for the system can be written as H = ¯h(ωc − ωL)nc − ¯hgncQ + ¯hωm 4 (Q2 + P2) + i¯hε(c† − c), (4.1) we have used the normalized coordinates for the oscillator defined by Q = √ 2mωm ¯h q and P = √ 2 m¯hωm p with [Q, P] = 2i. This normalization implies that in the ground state of the nanomechanical mirror ⟨Q2⟩ = ⟨P2⟩ = 1. Further in Eq. (4.1) the first term is the energy of the cavity field, nc = c†c is the number of the photons inside the cavity, c and c† are the annihilation and creation operators for the cavity field with [c, c†] = 1. The second term comes from the coupling of the movable mirror to the cavity field via radiation pressure, the parameter g = ωc L √ ¯h 2mωm is the optomechanical coupling constant between the cavity and the movable mirror. The third term corresponds the energy of the movable mirror. The fourth term describes 62 the coupling between the input laser field and the cavity field, ε is related to the input laser power ℘ by ε = √ 2κ℘ ¯hωL , where κ is the cavity decay rate associated with the transmission loss of the fixed mirror. The equations of motion of the system can be derived by the Heisenberg equations of motion and adding the corresponding noise terms, this gives the quantum Langevin equations ˙Q = ωmP, ˙P = 2gnc − ωmQ − γmP + ξ, ˙ c = i(ωL − ωc + gQ)c + ε − κc + √ 2κcin, ˙ c† = −i(ωL − ωc + gQ)c† + ε − κc† + √ 2κc † in. (4.2) Here we have introduced the input squeezed vacuum noise operator cin with frequency ωS = ωL + ωm. It has zero mean value, and nonzero timedomain correlation functions [141] ⟨δc † in(t)δcin(t′)⟩ = Nδ(t − t′), ⟨δcin(t)δc † in(t′)⟩ = (N + 1)δ(t − t′), ⟨δcin(t)δcin(t′)⟩ = Me−iωm(t+t′)δ(t − t′), ⟨δc † in(t)δc † in(t′)⟩ = M∗eiωm(t+t′)δ(t − t′). (4.3) where N = sinh2(r), M = sinh(r) cosh(r)eiφ, r is the squeezing parameter of the squeezed vacuum light, and φ is the phase of the squeezed vacuum light. For simplicity, we choose φ = 0. The force ξ is the thermal Langevin force resulting from the coupling of the movable mirror to the environment, whose mean value is zero, and it has the following correlation function at temperature T [108]: ⟨ξ(t)ξ(t ′ )⟩ = γm πωm ∫ ωe−iω(t−t ′ ) [ 1 + coth( ¯hω 2kBT ) ] dω, (4.4) where kB is the Boltzmann constant and T is the temperature of the environment. By using standard methods [110], setting all the time derivatives in Eq. (4.2) to 63 zero, and solving it, we obtain the steadystate mean values Ps = 0, Qs = 2gcs2 ωm , cs = ε κ + iΔ , (4.5) where Δ = ωc − ωL − gQs = Δ0 − gQs = Δ0 − 2g2cs2 ωm (4.6) is the effective cavity detuning, depending on Qs. The Qs denotes the new equilibrium position of the movable mirror relative to that without the driving field. Further cs represents the steadystate amplitude of the cavity field. From Eq. (4.5) and Eq. (4.6), we can see Qs satisfies a third order equation. For a given detuning Δ0, Qs will at most have three real values. Therefore, Qs and cs display an optical multistable behavior [41, 42, 43], which is a nonlinear effect induced by the radiationpressure coupling of the movable mirror to the cavity field. 4.3 Radiation Pressure and Quantum Fluctuations To study squeezing of the movable mirror, we need to calculate the fluctuations in the mirror’s amplitude. Assuming that the nonlinear coupling between the cavity field and the movable mirror is weak, the fluctuation of each operator is much smaller than the corresponding steadystate mean value, thus we can linearize the system around the steady state. Writing each operator of the system as the sum of its steadystate mean value and a small fluctuation with zero mean value, Q = Qs + δQ, P = Ps + δP, c = cs + δc. (4.7) 64 Inserting Eq. (4.7) into Eq. (4.2), then assuming cs ≫ 1, the linearized quantum Langevin equations for the fluctuation operators can be expressed as follows, δ ˙Q = ωmδP, δ ˙P = 2g(c∗ sδc + csδc†) − ωmδQ − γmδP + ξ, δ ˙ c = −(κ + iΔ)δc + igcsδQ + √ 2κδcin, δ ˙ c† = −(κ − iΔ)δc† − igc∗ sδQ + √ 2κδc † in. (4.8) Introducing the cavity field quadratures δx = δc + δc† and δy = i(δc† − δc), and the input noise quadratures δxin = δcin + δc † in and δyin = i(δc † in − δcin), Eq. (4.8) can be rewritten in the matrix form f˙(t) = Af(t) + η(t), (4.9) in which f(t) is the column vector of the fluctuations, η(t) is the column vector of the noise sources. Their transposes are f(t)T = (δQ, δP, δx, δy), η(t)T = (0, ξ, √ 2κδxin, √ 2κδyin); (4.10) and the matrix A is given by A = 0 ωm 0 0 −ωm −γm g(cs + c∗ s) −ig(cs − c∗ s) ig(cs − c∗ s) 0 −κ Δ g(cs + c∗ s) 0 −Δ −κ . (4.11) The system is stable only if the real parts of all the eigenvalues of the matrix A are negative. The stability conditions for the system can be derived by applying the 65 RouthHurwitz criterion [112, 113], we get κγm[(κ2 + Δ2)2 + (2κγm + γ2m − 2ω2m )(κ2 + Δ2) 



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