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TRANSPORT AND RESONANCES IN KICKED BOSEEINSTEIN CONDENSATES By ISHAN TALUKDAR Master of Science University of Delhi Delhi, India 2003 Master of Science Oklahoma State University Stillwater, Oklahoma, USA 2006 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, 2010 COPYRIGHT c⃝ By ISHAN TALUKDAR December, 2010 TRANSPORT AND RESONANCES IN KICKED BOSEEINSTEIN CONDENSATES Dissertation Approved: Dr. Gil S. Summy Dissertation Advisor Dr. Girish S. Agarwal Dr. John W. Mintmire Dr. Nicholas F. Materer Dr. Mark E. Payton Dean of the Graduate College iii ACKNOWLEDGMENTS I would like to begin by expressing my gratitude to my advisor Gil Summy. Gil has taught me much of what I know of this field today, patiently answering my questions and showing me new, creative ways of looking at scientific problems. His enthusiasm and words of encouragement guided me during the difficult times and will remain invaluable in the years to come. In the lab it has always amazed and inspired me how Gil finds a way to simply make things work. I would like to thank my committee members, Prof. G. Agarwal, Prof. J. Mintmire, and Prof. N. Materer for their time, help and advice. Without the love and continued support from my parents Kanteswar and Mohini Talukdar, I would not be where I am today. They showed me the importance of aiming high and working hard. I wish to thank them and my sister Shruti for her cheer and motivation. I have had the good fortune of meeting some incredible people during my stay at OSU. Peyman had defended his thesis when I joined the group. An exceptional physicist and a great individual, he has remained a good friend and mentor throughout these years. Ghazal cheerfully introduced me to the experiments and helped me adjust to my new lab. Vijay, who was my roommate as well, explained the nitty gritties of the experiments and gladly fielded my incessant questions. It has been a nice experience working with Raj who joined our lab later and brought a refreshing way of doing things. Part of my physics family were our department staff members, Susan, Cindy, Stephanie, Danyelle, and recently Tamara. Besides their helpful and supportive attiiv tude, they are among the nicest people I have met in my life. Melissa has been a kind and understanding labcoordinator. Thanks also to Warren, our technical support. He is a one man army, handling all kinds of requests from the students. Mike and his staff at the machine shop have the special ability to bring any model design to life, only better. I would like to thank the physics faculty for providing me with the knowledge and guidance that has helped me pursue my research. Especially, I wish to thank Prof.Paul Westhaus, our former graduate coordinator and Prof. R. Hauenstein, our current graduate coordinator for their support during my years at the physics department. Thanks are also due to my former professors at the University of Delhi, Prof. Seshadri and Prof. Annapoorni for having shown me the path of research. Finally, I would like to express my thanks to all my friends at Stillwater, especially DJ (Deok Jin Yu), Prem and Amit. v TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 DeltaKicked Rotor : Theory and Experiments 5 2.1 Classical kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Atom optics δkicked rotor . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Quantum resonances and antiresonances . . . . . . . . . . . . 9 2.2.2 Dynamical Localization . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 BoseEinstein Condensation . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . 15 2.3.2 Limits of laser cooling . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . 19 3 Quantum transport with a kicked BEC 27 3.1 Highorder resonances of a quantum accelerator mode . . . . . . . . . 27 3.1.1 Rephasing theory . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.2 ϵclassical theory . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 A quantum ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 vi 4 SubFourier resonances of the kicked rotor 48 4.1 A fidelity measurement on the QDKR . . . . . . . . . . . . . . . . . . 49 4.1.1 Effect of ‘gravity’ . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Experimental Configuration and Results . . . . . . . . . . . . . . . . 53 5 Photoassociation of a 87Rb BEC 64 5.1 Photoassociative spectroscopy . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Ultracold Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 Scattering length and Feshbach Resonances . . . . . . . . . . . . . . . 69 5.3.1 Optical Feshbach Resonance . . . . . . . . . . . . . . . . . . . 73 5.4 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 CONCLUSIONS 91 6.0.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.0.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 BIBLIOGRAPHY 94 A Publications 108 vii LIST OF FIGURES Figure Page 2.1 Classical δkicked rotor phase space for different kick strengths K. Onset of stochastic regions can be seen in (c) while (d) is predominantly chaotic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 (a) and (b), A quadratic growth in the mean energy at a quantum resonance at the Talbot time, and (c) and (d), an oscillatory mean energy at an antiresonance at T = T1/2. . . . . . . . . . . . . . . . . 11 2.3 Quantum suppression of classical chaos. Simulation of the QDKR near the Talbot time with ϕd=3.0 shows the onset of dynamical localization after six kicks. Also shown is the classical momentum diffusion (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Evolution of thermodynamic quantities as a function of the trap truncation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 A schematic of the rubidium D2 transition. . . . . . . . . . . . . . . . 22 2.6 Setup of the optical table for the BEC experiments. Not shown is a final 80 MHz AOM for both the MOT Slave beams before the optical fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Alignment of beams for the MagnetoOptic Trap and the FORT inside the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 viii 3.1 Phase space of quantum accelerator modes generated by the map of (3.22) for τ = 5.744 and ϕd = 1.4. Mode (a) with (p, j) = (1, 0) for η = 2.1459. is a primary QAM. Higher order modes are seen in (b) with (p, j) = (2, 1), η = 2.766, and (c) (p, j) = (5, 1), η = 4.1801. . . . . 32 3.2 Setup for the kicking experiments. Two counterpropagating beams formed the standing wave oriented at an angle of 52◦ to the vertical. AOM2 was driven by an RF signal with a fixed frequency, ω/2π = 40MHz. 34 3.3 Quantum accelerator modes at (a) T=22.68 μs close to (2/3)T1/2, (b) T=17.1 μs which is close to (1/2)T1/2, and (c) T=72.4 μs close to 2T1/2. a value of g′=6 ms−2 was used in these scans. The arrows in (a) and (b) show orders separated by b~G which participate in the QAMs. Dashed lines correspond to the ϵclassical theory of Eq. (3.27) . . . . 36 3.4 Horizontally stacked momentum distributions across (a) (1/2)T1/2 for 40 kicks, ϕd = 1.4 and effective acceleration g′ = 6 ms−2; and (b) (1/3)T1/2 for 30 kicks, ϕd = 1.8, and g′=4.5 ms−2. The dashed curve is a fit to the theory in Eq. (3.27) . . . . . . . . . . . . . . . . . . . . 38 3.5 Initial momentum scans for QAMs near (a) (1/2)T1/2 (T=17.1 μs) for 30 kicks and g′=6 ms−2; and (b) (2/3)T1/2 (T=22.53 μs) with 40 kicks and g′=4.5 ms−2. The dashes indicate QAMs at the resonant β. . . . 39 3.6 Dependence of the mean momentum of the quantum ratchet on the offset angle γ for 5 kicks and β = 0.5. The dashed and solid lines represent Eqs. (3.33) and (3.36) respectively. The inset shows the offset γ created between the symmetry centers of the initial distribution (blue curve) and the kicking potential V (red curve). . . . . . . . . . . . . . 43 ix 3.7 The ratchet effect. This time of flight image shows growth of mean momentum with each standing wave pulse applied with a period of T1/2 and a maximum offset (γ = π/2) between the standing wave and the initial state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.8 Mean momentum as a function of kicks. The data and error bars are from experiments with ϕd = 1.4, γ = π/s and β = 0.5. The solid line is Eq. (3.33) while the dashed line corresponds to Eq. (3.36). . . . . 45 3.9 Change in mean momentum vs the quasimomentum β for ϕd = 1.4 and (a) γ = −π/2, (b) γ = π/2. Shown are the fits of Eqs. (3.33), dashed line and (3.36), solid line respectively to the experimental data (filled circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Momentum distributions of a sequence of 8 kicks of strength ϕd followed by a final πphase reversed kick of strength Nϕd, with a time period equal to the Talbot time 106.5μs. . . . . . . . . . . . . . . . . . . . . 54 4.2 (a) Horizontally stacked timeofflight images of a fidelity scan around the Talbot time. Each TOF image was the result of 5 kicks with ϕd = 0.8 followed by a πphase shifted kick at 5ϕd. (b) Mean energy distribution of the 5 kick rotor with the same ϕd. (c) The measured fidelity distribution (circles) from (a). The mean energy of the scan in (b) is shown by the triangles. Numerical simulations of the experiment for a condensate with momentum width 0.06 ~G are also plotted for fidelity (bluedashed line) and mean energy (redsolid line). The amplitude and offset of the simulated fidelity were adjusted to account for the experimentally imperfect reversal phase. . . . . . . . . . . . . . . 56 x 4.3 Experimentally measured fidelity (circles) and mean energy (triangles) widths (FWHM) as a function of (a) the number of pulses, and (b) the kicking strength ϕ˜d scaled to ϕd of the first data point. In (a), the data are for 4 to 9 kicks in units normalized to the 4th kick. Error bars in (a) are over three sets of experiments and in (b) 1σ of a Gaussian fit to the distributions. Dashed lines are linear fits to the data. Stars are numerical simulations for an initial state with a momentum width of 0.06~G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Variation of the fidelity peak width around β=0 as a function of kick number N(N +1)s = N(N +1)/20 scaled to the 4th kick. The straight line is a linear fit to the data with a slope of −0.92 ± 0.06. Error bars as in Fig. 2(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.5 (a) Momentum width of the reversed zeroth order state as a function of kick number. Error bars are an average over three experiments. (b) Optical density plots for the initial state (red,solid) and kick numbers 2 (magenta,dotdashed),4 (black,dotted), and 6 (blue,dashed) after summation of the timeofflight image along the axis perpendicular to the standing wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.6 Dependence of the acceleration resonance peak width as a function of the kick number in units scaled to the 4th kick. Error bars are over three sets of experiments. . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1 Schematic of a photoassociation process. Two atoms colliding along the ground state potential (S+S) absorb a photon and get excited to the (S+P) molecular potential. The excited molecule can subsequently decay to free atoms or a ground state molecule. . . . . . . . . . . . . 66 xi 5.2 Centrifugal energy term ~2l(l + 1)/2μr2 of the Hamiltonian for three partial waves, l=0,1,2. For low energy scattering all partial waves l >0 are blocked by the centrifugal barrier. . . . . . . . . . . . . . . . . . . 70 5.3 Variation of the scattering length a as a function of λ = √ mV0/~2. As the well depth V0 increases ((a) to (c)) a bound state is formed (dashed line) and the scattering length passes through a divergence and changes sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4 A Feshbach resonance occurs when an excited state has a bound state close to the collisional threshold. Changing the detuning Δ by an external field can couple the collisional to the bound state and change the scattering length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.5 Schematic of the optical setup for the photoassociation light. . . . . 77 5.6 Flowchart of the locking for the photoassociation master laser. . . . . 78 5.7 Experimental configuration for photoassociation. Shown are the CO2 laser FORT and the photoassociation beams. The Bragg (kicking) beams were aligned such that a horizontal standing wave was created along the long axis of the FORT. . . . . . . . . . . . . . . . . . . . . 84 5.8 Photoassociation spectrum of the 1g(P3/2), v = 152 state. 0 MHz on this scale corresponds to a point 713 GHz below the 87Rb D2 line. The states a through e correspond to 2,−2, 3,−3⟩, 3,−2, 3,−3⟩, 1,−1, 3,−2⟩, 2,−1, 3,−2⟩, 1, 0, 1,−1⟩ of Ref. [118] labeled by the F, f, I, i⟩ quantum numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.9 Photoassociation of the 0− g (∼ S1/2+P3/2), v = 1 state showing the J=0 and J=2 rotational levels. Each point is separated by 5 MHz. . . . . 86 xii 5.10 (a) First order Bragg diffraction as a function of the frequency difference between the beams used to create the standing wave. (b) Percentage of Bragg diffracted atoms for two photoassociation light detunings, Δ/2π =+10MHz (red) and 10MHz (blue). . . . . . . . . . . . . . . . 87 5.11 Measured inelastic collision rate coefficients for the 0− g (P3/2), v = 1, J = 2 state. Also shown is a lorentzian fit to the data from which values of Γspon and Γstim are obtained. . . . . . . . . . . . . . . . . . . . . . . 88 5.12 Plot of the scattering length a in units of a0 = 0.53˚A , the Bohr radius, calculated from the data in Fig. 5.11. The dashed line is the background scattering length of 100a0 and the detuning is with respect to the 0− g (P3/2), v = 1, J = 2 state. . . . . . . . . . . . . . . . . . . . . . 89 5.13 Interference between condensates as a measure of mean field energy. Please refer to text for details. . . . . . . . . . . . . . . . . . . . . . 90 xiii CHAPTER 1 INTRODUCTION Since its birth, quantum mechanics has stood the test of experiment. It has made remarkable progresses from theory to practice in the form of devices like atom interferometers, scanning tunneling microscopes and atomic clocks, to name a few. Despite the success of quantum theory, subtle issues still remain in our understanding. According to Bohr’s Correspondence Principle quantum mechanics reduces to Newtonian dynamics when the classical unit of action S becomes larger than the Planck’s constant ~. However, the exact nature and causes for this transition is an open subject for discussion. This is perhaps best seen in the problem of quantizing chaos, noticed by none other than Einstein [1]. The existence of classical chaos was first shown by the French mathematicianphysicist Henri Poincar´e. Attempting to find an analytical solution to the dynamics of three gravitationally interacting bodies, he discovered the possibility of irregular motion, where slight changes in initial conditions could lead to vastly different trajectories. This was chaos in a deterministic system, without any random parameters. The problem was revisited in the midtwentieth century. The result was the KolmogorovArnoldMoser (KAM) theorem [2]. It quantified the amount of perturbation necessary for an integrable system to develop chaotic motion. Carrying this concept of chaos over to the quantum realm runs into difficulties however. The overlap integral of two initially close wave packets undergoes unitary evolution to remain preserved over time. A better way of finding the quantum mechanical equivalent of chaos might be to look at the overlap of two similar states 1 evolving under slightly different Hamiltonians, as suggested by Peres [3]. These strategies for governing the evolution in phase space necessitate a deeper understanding of quantum transport in a classically chaotic system. One of the widely researched systems of quantum transport in momentum or energy space is the multiphoton ionization of Rydberg atoms under periodic microwave driving [4, 5]. This was the first demonstration of the δkicked rotor, a paradigm for studying classical and quantum chaos. Raizen’s group at Austin demonstrated a much more experimentally manageable version with atom optics. Here, cold atoms were subjected to periodic flashes from an optical standing wave [6, 7]. In general, onedimensional autonomous Hamiltonian systems are integrable as a result of the conservation of energy. There is therefore no chaos and the dynamics is regular. However, the energy periodically pumped into the kicked rotor breaks the timeinvariance of the Hamiltonian and makes chaotic dynamics possible in this one dimensional classical system. Surprisingly, the quantum kicked rotor, despite the inherent nonintegrability, was found to suppress classical chaotic motion. This was termed dynamical localization [6, 8–11], since it was reminiscent of Anderson localization in disordered solids. Decoherence introduced into the system for instance via spontaneous emissions or noise added to the kicking strength was found to destroy dynamical and restore classical diffusion [12–15]. Dynamical localization to this day remains an invaluable resource in investigating quantumclassical correspondence. Such atom optical systems have also proved their versatility in modeling other physical phenomena in the spirit of Feynman’s quantum simulator [16]. Cold atoms in accelerating optical lattices were found to exhibit Bloch oscillations, Wannier Stark ladders and tunneling, transport behavior normally associated with solid state physics [17–22]. Variants of the kicked rotor have enabled the demonstrations of a quantum accelerator and quantum ratchet [23–27]. The sensitivity of momentum transport to the underlying classical chaos in this system also offers possibilities for 2 precision measurements. A group in France observed subFourier resonances in a kicked rotor subject to two kicking frequencies [28]. Another study looked at the prospect of exploiting quantum rotor resonances to improve atom interferometric measurements of physical constants like the photon recoil frequency [29, 30]. The creation of a BoseEinstein condensate by Cornell and Wieman in 1995 heralded a new era of experiments [31, 32]. Unprecedented control over quantum state preparation and manipulation was now possible. A BEC has a momentum width less than that of a photon recoil. In the context of the quantum kicked rotor and accelerator, it allowed a close look at their phase spaces. The first BEC was achieved by evaporative cooling of already cold atoms in a magnetic trap. Condensates in our lab are produced in an optical trap, which has proved to be a simple yet robust method [33]. 1.1 Organization Two important aspects of the atom optics kicked rotor are investigated in this thesis  transport and resonance width scaling. Experiments were also conducted to observe an Optical Feshbach resonance in a 87Rb BoseEinstein Condensate, which could be useful for further kicked rotor research. To provide a theoretical and experimental backdrop for these studies, we begin with Chapter 2 which introduces the quantum deltakicked rotor and discusses properties exhibited by it and its variants. The laboratory realization of the quantum rotor is in the form of a BoseEinstein Condensate subjected to a periodically pulsed standing wave of light detuned far from any atomic resonances. An account of the theoretical background, from laser cooling techniques to evaporative cooling, necessary in achieving a phase space transition to a BEC is given. This is followed by a description of the experimental techniques which we employ in order to create such a condensate and perform experiments on it. Chapter 3 reviews two kinds of quantum transport behavior made possible by this 3 system. The first one details the observation and properties of quantum accelerator modes, specifically higher order resonances of such a mode. Two models, an interference model based on the rephasing of wavefronts, and a classicallike model can explain the occurrences of such modes and are discussed. The momentum evolution predicted by the two theories is then compared with experiment. Next we see how preparation of a special initial state of the rotor can establish a momentum current even in the absence of a net bias force. The directed current appears in the absence of dissipation and is an example of a quantum ratchet. Its realization will be described next. The effect of the finite momentum width of a condensate on the ratchet current is also investigated. The relatively nascent topic of applying the quantum kicked rotor towards precision measurements is the subject of Chapter 4. This system exhibits resonances in the mean energy at specific combinations of the kicking period and initial momentum. It is shown that the fidelity or quantum mechanical overlap of an offresonant state with one that is on resonance scales at a subFourier rate, with respect to the measurement time. The existence of resonances at particular values of the acceleration of the standing wave is also analyzed and verified by experiment. These resonances are also shown to have subFourier widths. Interactions among atoms in a BoseEinstein condensate can be strongly tuned near a Feshbach resonance. Presence of such interactions can introduce nonlinearity into the kicked rotor, thereby enabling a probe at quantum chaos. Chapter 5 details photoassociation spectroscopy performed on a BoseEinstein condensate for the 1g and 0− g long range molecular states. Bragg spectroscopy was employed to detect optically induced changes in the elastic scattering length near one of these molecular states, the sign of an Optical Feshbach resonance. Finally, the conclusions are laid out in chapter 6 including the outlook for future research. 4 CHAPTER 2 DeltaKicked Rotor : Theory and Experiments For more than three decades the delta kicked rotor has been at the centerstage in the study of quantum chaos. The relative ease with which it can be handled analytically has drawn many researchers to it. It is the basic model for studying dynamical chaos [34] and is described classically by the Standard Map. In this chapter we review this model and its quantum analogue. Along the way we examine some of the rich features this simple system has revealed so far. Finally, we look at the implementation of this model in our lab in an atom optics setting. 2.1 Classical kicked rotor The kicked rotor is a pendulum exposed to a periodic constant force like gravity. Its Hamiltonian is H = p2 2I + V0 cos(θ) Σ N δ(t − NT) (2.1) p is the angular momentum, θ the angular displacement, I the moment of inertia and V0 the kick strength. t is the time and N counts the kicks which are switched on at time intervals of T. Choosing I = 1, we can write Hamilton’s equation’s of motion as ∂H ∂θ = −p˙ = −V0 sin(θ)δ(t − NT) (2.2) ∂H ∂p = θ˙ = p Integrating Eqs. (2.2) over one period between t = N and t = N + 1, pN+1 = pN + V0 sin(θN) (2.3) θN+1 = θN + pN+1T 5 Using the rescaled variables ρ = pT and K = V0T, (2.3) can be written as ρN+1 = ρN + K sin(θN) (2.4) θN+1 = θN + ρN+1 This is known as the Standard or TaylorChirikov Map where the dynamics is completely determined by the ‘stochasticity parameter’ K [35]. As seen in the ρ − θ phase space (modulo 2π) of Fig. 2.1, the islands of stability at small K give way to chaotic regions with increasing K. Numerical analysis showed that the transition to global stochasticity occurs at a value of K ≈ 0.9716 [36]. At the end of ‘t’ kicks, ⟨(ρt − ρ0)2⟩ = K2 Σt−1 q,r (sin θq sin θr) (2.5) Ignoring the correlation terms for large K, we get ⟨ρ2t ⟩ ∼= 1 2 K2t (2.6) Thus the chaotic domain of the classical δkicked rotor is characterized by a diffusive growth of the mean energy. 2.2 Atom optics δkicked rotor Study of the quantum version of the δkicked rotor was made possible by remarkable advances in atom optics. Its experimental realization can be described in terms of the center of mass Hamiltonian of an atom of mass M exposed to short periodic pulses of far detuned light [6, 7, 37], ˆH = ˆ P2 2M + ~ϕd cos(G ˆX ) Σ∞ N=−∞ δ(t′ − NT) (2.7) G = 4π/λ is the grating vector of the standing wave formed from a laser of wavelength λ. It is convenient to convert this Hamiltonian to dimensionless units. This can be 6 −pi 0 pi −pi 0 pi K=0.5 q r −pi 0 pi −pi 0 pi K=0.9 q r −pi 0 pi −pi 0 pi K=1.5 q r −pi 0 pi −pi 0 pi K=5 q r (a) (c) (b) (d) Figure 2.1: Classical δkicked rotor phase space for different kick strengths K. Onset of stochastic regions can be seen in (c) while (d) is predominantly chaotic. 7 done by expressing position in units of G−1, ˆx = G ˆX and momentum in units of ~G, ˆp = ˆ P/~G. Hˆ = ˆp2 2 + ϕd cos(ˆx) Σ N δ(t − Nτ ) (2.8) The scaled period is τ = 2πT/T1/2, time is measured in t = 2πt′/T1/2 and Hˆ = (M/(~G)2) ˆH . T1/2 = 2πM/~G2 is known as the halfTalbot time, the physical significance of which we shall soon see. The periodicity of the potential provides the connection between the particle propagating along it and the kicked rotor. The position of the particle can be folded into an angular coordinate θ = x mod(2π). From a quantum mechanical perspective, the periodicity of the potential allows us to use Bloch’s theorem on the atomic deBroglie wave. The solutions are then invariant under translations by one period of the potential [38]. A result of this invariance is the conservation of the particle quasimomentum. In terms of photon exchange, each atom absorbs a photon from one of the standing wave beams followed by its subsequent stimulated emission into the other beam. The net result is that the atom momentum changes by two photon recoils (2~k or ~G) while leaving the fractional part of its momentum (or the quasimomentum, in units of ~G) unchanged. The light shift of the atomic ground state in the presence of this optical potential is ΔEg = ~Ω2 4δ cos2(x/2) (2.9) where Ω ≡ −eE0 ~ ⟨erg⟩ is the Rabi frequency between the excited and ground states. E0 is the amplitude of the potential and it is detuned δ from the atomic transition. This sinusoidal energy shift leads to the standing wave acting as a phase grating on the atomic wave function. The strength of the kicks is therefore given by the phase modulation depth parameter, ϕd = Ω2Δt/8δ. Change in the kinetic energy of the atoms is negligible during the short time the potential is on (RamanNath regime). In this limit, the imprint of the thin phasegrating on an incident plane wave 0⟩ 8 results in a final state ψ(Δt)⟩ = e−iϕd cos(ˆx)0⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)e−inˆx0⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)n⟩ (2.10) where the second line derives from the JacobiAnger expansion, Jn is the nth order Bessel function of the first kind, and n⟩ is a momentum order along the grating in units of ~G. After a free evolution for a period T, the wave function is ψ(t)⟩ = Σ n (−i)nJn(ϕd)e−i ^p2 2 τ n⟩ (2.11) 2.2.1 Quantum resonances and antiresonances We can now write the one period evolution operator as ˆU = e−iϕd cos(ˆx)e−i ^p2 2 τ . (2.12) When the time period τ is an integer multiple of 4π, the free evolution factor is unity. N consecutive kicks of strength ϕd are then equivalent to one kick of strength Nϕd. That is, ˆU N = e−iNϕd cos(ˆx). (2.13) From the initial state 0⟩, we then have a momentum distribution pn = ⟨nUN0⟩2 = J2 n(Nϕd) (2.14) The mean energy at the end of N kicks is then ⟨E⟩ = Σ∞ n=−∞ n2pn = Σ n n2J2 n(Nϕd) = 1 2 N2ϕ2 d (2.15) 9 This uniquely quantum effect, characterized by the quadratic growth of energy, is termed a quantum rotor resonance. It is the temporal equivalent of the Talbot effect in optics and the resonant time period is therefore called the Talbot time, TT = 4πM/~G2 [6, 39]. Next we look at a zero initial momentum state, exposed to a kick, and allowed to evolve for half of the Talbot time (T1/2). Now, τ = 2π (or any odd integer multiple of it), and the phases acquired by the momentum orders are +1 or 1 depending on whether the order is even or odd. The state at the end of the halfTalbot time is, ψ(t = T1/2)⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)e−iπn2 e−inˆxψ0⟩ = eiϕd cos(ˆx)ψ0⟩ (2.16) where we have used the property, e−iπn2 = e−iπn. A kick applied at this point will cancel out the spatial variation due to the first kick recreating the original state. This phenomenon is known as a quantum antiresonance. A numerical simulation of these two effects can be performed using a technique shown in Ref. [39]. Beginning with a zero momentum initial state (convoluted by a finite width Gaussian to account for the BEC momentum spread), the quantum deltakicked rotor is realized by repeated application of the one period evolution operator of (2.12). Figure 2.2 shows a simulated quantum resonance and antiresonance. 2.2.2 Dynamical Localization We now turn our attention to the generic behavior of the quantum delta kicked rotor, away from these resonances, that is when τ/2π is an irrational number. In Section 2.1 we saw how strong chaos dictates a diffusive energy growth of the classical kicked rotor. In order to investigate the behavior of its quantum counterpart, Casati, Chirikov, Ford and Izrailev [40, 41] in 1979 simulated the quantum kicked rotor. Contrary to expectations, however, they were surprised when the quantum rotor showed corre 10 0 5 10 −30 −20 −10 0 10 20 30 kicks Momentum Resonance 0 5 10 0 20 40 60 80 100 kicks Mean Energy 0 5 10 −30 −20 −10 0 10 20 30 kicks Momentum Anti−resonance 0 5 10 0 0.2 0.4 0.6 0.8 1 kicks Mean Energy (a) (b) (c) (d) Figure 2.2: (a) and (b), A quadratic growth in the mean energy at a quantum resonance at the Talbot time, and (c) and (d), an oscillatory mean energy at an antiresonance at T = T1/2. 11 spondence with the classical case only upto a certain time even in the deeply chaotic domain (ϕd ≫1). After this, quantum interference was found to suppress any further growth of momentum [42]. This phenomenon came to be known as dynamical localization, and was shown to be analogous to Anderson localization in disordered solids [34, 43]. In the rotor, destructive interference between momentum orders separated by irrational multiples of 2π leads to a final eigenstate exponentially localized in momentum. 2.2.3 Quantum Transport While the mean energy of the QDKR increases ballistically with kicks at a quantum resonance, the mean momentum ⟨p⟩ remains fixed. Presence of a linear potential like gravity breaks the symmetry of the system. The momentum acquired by the qth order at the end of N kicks is mvi +q~G+mgNT. The phase acquired by the momentum state q⟩ therefore has a gravity dependent term, ϕq = viGTq + ~G2 2m Tq2 + gGT2Nq. When the phase difference between q⟩ and q − 1⟩, ϕq − ϕq−1 = ~G2 2m T(2q − 1) + viGT + gGT2N (2.17) is a multiple of 2π, order q − 1⟩ can be perfectly coupled to q⟩ by the next kick. This imposes two conditions on rephasing, one which depends on the kick number and another which does not. ~G2 2m T2q + gGT2N = 2πql′ (2.18) viGT − ~G2 2m T = 2πl (2.19) Equation (2.18) can be solved to find the momentum at the end of N kicks, q = N γ α2 l′ − α (2.20) where α = T/T1/2 and γ = ~2G3/2πm2g. Therefore, near a resonance, a fixed momentum can be imparted to a section of the atoms by each kick. This is a quan tum accelerator mode created with the quantum delta kicked accelerator (QDKA). 12 0 2 4 6 8 10 0 5 10 15 20 25 30 35 40 45 kicks Mean energy (Er) Figure 2.3: Quantum suppression of classical chaos. Simulation of the QDKR near the Talbot time with ϕd=3.0 shows the onset of dynamical localization after six kicks. Also shown is the classical momentum diffusion (solid line). 13 Among many possibilities, its use as a coherent beam splitter for an interferometer has brought it a great deal of attention among researchers since its discovery almost a decade ago in Oxford [44]. Another topic of recent focus in rectified transport is a ratchet, where a directed current of particles along a periodic potential can be established even in the absence of a biased force. A pure quantum ratchet was first demonstrated at a QDKR resonance by Sadgrove et al. [45]. It was realized with an initial state prepared in a superposition of a zeroth and first order momentum states (in units of ~G), ψ⟩i = √1 2 [0⟩ + 1⟩]. (2.21) After a time ‘t’ of free evolution, the first order accumulates a phase θ, ψ(t)⟩ = √1 2 [0⟩ + eiθ1⟩]. (2.22) The state after application of a kick become, ψ(t+)⟩ = e−iϕd cos(ˆx)ψ⟩i = √1 2 Σ q (−i)q[Jq(ϕd) + eiθJq−1(ϕd)]q⟩ (2.23) with a distribution, pq = 1 2 [Jq(ϕd)2 + Jq−1(ϕd)2 + 2 cos θJq(ϕd)Jq−1(ϕd)] (2.24) Recalling the Bessel function property, J−n(x) = (−1)nJn(x), we notice that the momentum distribution is asymmetric whether q is odd or even. This is a ratchet current, the strength of which can be tuned by the relative phase θ. Experiments on these and other exotic properties of the δkicked rotor depend on an initial momentum state with a spread that is less than one photon recoil. The ideal atomic physics system for this is a BoseEinstein condensate. The next sections outline the stages in reaching this quantum state of matter and its realization in our lab. 14 2.3 BoseEinstein Condensation In 1924, Satyendranath Bose proposed a statistical technique to evaluate the photon black body spectrum [46]. Einstein extended the theory to the general case of identical particles leading to the birth of BoseEinstein statistics [47]. The distribution function for particles obeying this statistics is N(E) = 1 eβ(E−μ) − 1 (2.25) where β = 1/kBT and μ is the chemical potential. Einstein noticed a peculiarity, that below a certain critical temperature, these bosonic atoms would accumulate in the lowest energy quantum state: the onset of the BoseEinstein Condensate (BEC) phase. In terms of the phase space density, ρ = nλ3 dB, where n is the particle number density, a BEC phase transition happens when ρ = ζ( 3 2 ) = 2.612. In other words once a particle’s thermal deBroglie wavelength λdB = ( 2π~2 mkBT )1/2 becomes greater than the interparticle separation, λdB > n−1 3 , this macroscopic quantum state starts appearing. A subject of academic curiosity for more than half a century, achievement of a BEC was given serious thought when laser cooling of atoms was realized. This cooling scheme is based on using optical forces to reduce the thermal velocity distribution of atoms. An indepth discussion of BEC in the Summy lab at OSU can be found in the theses of Ahmadi, Timmons, and Behin Aein [48–50]. I shall therefore only outline the general principles involved. 2.3.1 Laser Cooling and Trapping The light force on an atom in general comprises of two parts, F = Fdip + Fsc, a conservative dipole force, Fdip and a dissipative scattering force Fsc. The science of laser cooling was built primarily on the second kind, which is a result of absorption 15 of photons by atoms followed by spontaneous emission. This force is expressed as Fsc = ~kγρee (2.26) where ~k is the momentum transferred by a photon, γ is the rate of decay of the excited state of the atom and ρee is the probability for the atoms to be in the excited state. Evaluating ρee using the optical Bloch equations, one can show that, Fsc = ~k s0γ/2 1 + s0 + (2δ/γ)2 . (2.27) Here, s0 = 2Ω2/γ2 is the onresonance saturation parameter and δ = ωl − ωa is the detuning of the light from the atomic transition. We now consider the case of an atom with velocity ⃗v placed in the light field of two beams counterpropagating along the zaxis. In addition we arrange a linearly inhomogeneous magnetic field B = Az formed by a magnetic quadrupole field. In the limit of low light intensity, the total force on the atom due to the two beams is, ⃗F = ⃗F+ + ⃗F−, where ⃗F± = ~⃗k s0γ/2 1 + s0 + (2δ±/γ)2 . (2.28) The detuning is δ± = δ ∓ ωD ± ωZ, where we now have to include the Doppler shift ωD = ⃗k.⃗v and the Zeeman shift ωZ = μ′B/~, μ′ being an effective magnetic moment [51]. In the limit of Doppler and Zeeman shifts that are small compared to δ, we arrive at the total force, ⃗F = −β⃗v − μ′A ~k β⃗r (2.29) which is the motion of a damped harmonic oscillator with the damping constant β = − 8~k2δs0 γ{1+s0+(2δ/γ)2}2 . Thus with a configuration of three retroreflecting beams tuned below the atomic resonance, the Zeeman shift provides a confining potential for the atoms creating what is known as the MagnetoOptic Trap (MOT). The presence of a viscous damping force due to the Doppler effect leads to a significant reduction in the velocity distribution of the atoms forming an “Optical Molasses” [52, 53]. 16 2.3.2 Limits of laser cooling There is a limit to the temperatures that can be obtained in an optical molasses due to recoil heating. This effect results from a diffusion of the atoms in momentum space set off by the random nature of the photon scattering events. A steady state is reached when molasses cooling equals recoil heating which determines the limiting Doppler temperature, TD, TD = ~γ 2kB . (2.30) For 87Rb the Doppler temperature is 146 μK. Surprisingly, in one of the early experiments with Na atoms, temperatures 10 times lower than the Doppler temperature were observed [54]. Later, a theory which included the multilevel structure of the atomic states and the optical pumping among these sublevels was able to explain these subDoppler temperatures [55, 56]. This process became known as polarization gradient cooling. The next limit to the laser cooling temperature is set by the energy associated with a photon recoil, Er = ~2k2/2m. The recoil limit temperature, Tr = ~2k2 mkB (2.31) has a value of 360 nK for 87Rb. 2.3.3 Evaporative cooling The maximum phase space density possible with laser cooling is 10−5 − 10−4. Cold atom physicists soon realized that in order to increase it any further, evaporative cooling would be the way to go. Originally proposed by Hess [57] for atomic hydrogen, the method is based on the preferential removal of high energy atoms from a confined sample, followed by rethermalization of the remaining atoms by elastic collisions. The simultaneous decrease in the temperature and the volume leads to an increase in the phase space density. 17 Several models have been developed to explain the process of evaporative cooling. These include analytical and numerical treatments by Doyle and coworkers [58], Luiten et al., and Wu and Foot [60] among a few. We shall here focus on a simple yet highly instructive analytical model due to Davis et al. [59]. In this model the trap depth is lowered in one single step to a finite value ηkBT and the effect of removal of high energy atoms on the thermodynamical quantities is calculated. The remaining fraction of atoms is ν = N′/N. The decrease in temperature caused by the release of the hot atoms can be defined by the quantity, γ = log(T′/T ) log ν . (2.32) In a ddimensional potential, U(r) ∝ rd/ξ and the volume V ∝ Tξ [61]. The value of ξ describes the type of the potential. For a linear potential like a spherical quadrupole trap, ξ = 3 and for a harmonic potential as in an optical trap, ξ = 3/2. We thus have the scaling of the important thermodynamic quantities, N′ = Nν, T′ = Tνγ, and V ′ = V νγξ. The phase space density ρ = nλ3 dB scales as ρ′ = ρν1−γ(ξ+3/2). With the knowledge of ξ, ν(η), and γ(η) one can track the evolution of these quantities with the lowering of the trap depth. The density of states for atoms in a trapping potential U(x, y, z) is D(E) = 2π(2M)3/2 ~3 ∫ V √ E − U(x, y, z)d3r (2.33) The fraction of atoms which remain in the trap after its depth has been decreased to ηkBT is ν(η) = 1 N ∫ ηkBT 0 D(E)e−(E−μ)/kBTdE. (2.34) The occupation number is given by the MaxwellBoltzmann distribution e−(E−μ)/kBT since the effects of quantum statistics can be neglected for a dilute gas. μ is the chemical potential of the gas. We can write Eq.(2.34) as, ν(η) = ∫ η 0 Δ(ϵ)e−ϵdϵ (2.35) 18 where ϵ = E/kBT is the reduced energy and the reduced density of states is Δ(ϵ) = ϵ1/2+ξ Γ(3/2 + ξ) (2.36) After truncation, the total energy of the atoms is α(η)NkBT where α(η) = ∫ η 0 ϵ Δ(ϵ)e−ϵdϵ. (2.37) Therefore the average total energy per atom(in units of kBT) is α(η)/ν(η). Before truncation (η → ∞) this quantity is α(∞)/ν(∞) = (3/2 + ξ)/1. The decrease in temperature is thus T′ T = α(η)/ν(η) α(∞)/ν(∞) . (2.38) Using Eq.(2.32) we now have, γ(η) = log( α(η) ν(η)α(∞) ) log[ν(η)] . (2.39) Solving Eqs. (2.35), (2.36), and (2.37) for a specific form of potential, one can determine γ(η). For ξ = 3/2, ν(η) = 1 − 2+2η+η2 2eη and α(η) = 3 − 6+6η+3η2+η3 2eη . Figure 2.4 shows the dependence of the number of atoms, temperature, density and phase space density on the normalized truncation parameter ˜η = η 3/2+ξ for ξ = 3/2 and ξ = 3. It can be seen that for the same truncation, ˜η, a higher phase space density is achieved with a larger ξ due to a faster shrinking of volume with decreasing temperature (V ∝ Tξ). 2.4 Experimental Configuration Ever since the first BoseEinstein condensate [31], rubidium 87 has been an atom of choice, due to its large elastic cross sections and the convenience of trapping it with inexpensive diode lasers. The light needed to trap and cool these atoms is tuned close to the Rb87 F=2 → F′=3 transition shown in the D2 hyperfine structure in Fig. 2.5. In our experiment this light was derived from a grating stabilized Toptica 19 0 1 2 3 0 2 4 6 8 10 ˜´ phase space density linear harmonic 0 1 2 3 0 0.5 1 1.5 2 2.5 ˜´ density (n¢/n) 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ˜´ number of atoms (N¢/N) 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ˜´ temperature (T¢/T) Figure 2.4: Evolution of thermodynamic quantities as a function of the trap truncation parameter. 20 DL100 diode laser with a 1 MHz linewidth and 15 mW of output power. We call this our master laser. Using saturated absorption spectroscopy it was locked to the F=2 → F′=2 and F′=3 crossover line. This light was injected into a homemade temperature stabilized slave laser with an output of 110 mW as shown in Fig. 2.6. Three frequency detunings of the light are needed in the experiment: 20 MHz for the MOT, 90 MHz for molasses cooling, and 0 MHz for imaging. To enable fast switching between these, the output from this main slave was sent through an AcoustoOptic Modulator (AOM) in a doublepass configuration. The frequency adjusted light was finally injected into a series of two slave lasers used for the actual trapping and cooling of atoms. Due to nonresonant excitation to the F′=2 state, some of the atoms can fall down to the F=1 ground state and get out of the trapping cycle. To bring them back into the MOT, a repumping laser tuned to the F=1→F′=2 state is used. Control over the intensity of this light is also crucial during the FORT loading stage and is adjusted by an AOM. All of these optics are placed in an isolated table as seen in Fig. 2.6. The trapping and repump beams are coupled through two fibers to the table containing the vacuum chamber. The vacuum chamber consisted of an octagonal multiport chamber attached to a flange of a sixway cross. An ion pump attached to another flange kept the vacuum pressure at 10−11 Torr. After exiting the fiber on the second table, the MOT light is divided into three beams, circularly polarized and diameters expanded to 2.2 cm. These are then sent into the vacuum chamber from three orthogonal directions and then retroreflected with opposite circular polarization. The MOT is formed at the intersection of these six beams. The inhomogeneous magnetic field required to create the confining potential of the MOT was formed using two sets of water cooled coils arranged in an anti Helmholtz configuration. A magnetic field gradient of 10 Gauss/cm along the axial direction was obtained with the coil geometry of our experiment. Around 10 million 21 Figure 2.5: A schematic of the rubidium D2 transition. 22 Figure 2.6: Setup of the optical table for the BEC experiments. Not shown is a final 80 MHz AOM for both the MOT Slave beams before the optical fibers. 23 atoms were trapped in this MagnetoOptic trap. Finally, stray magnetic fields in the region of the MOT were removed with three pairs of Helmholtz coils, commonly known as the nulling coils. A 50W CO2 laser formed the Far OffResonant Trap (FORT) for the evaporative cooling stage. With a wavelength of 10.6μm, the laser is far detuned from the atomic resonance and its effect on the atoms can be considered as that of a static electric field. An atom with a ground state polarizability αg placed in the electric field E(x, y, z) of the Gaussian beam given by E2 = E2 0 exp[ −2(x2+y2) w2 0(1+( z zR )2) ] 1 + ( z zR )2 . (2.40) experiences a potential U = 1 2αgE2. Here w0 is the beam waist and zR = πw2 0/λ is the Rayleigh length. The focus of the beam where z << zR, serves as a harmonic trapping potential U = −1 2 αgE2 0(1 − 2x2 w2 0 − 2y2 w2 0 − z2 z2R ). (2.41) The output of the CO2 laser passed through a 40 MHz RFdriven AOM. The 35W first order from the AOM overlapped the MOT for a loading time of typically 20 seconds. The MOT light detuning was then increased to 90 MHz for optical molasses cooling. This was followed by reduction of the repump intensity to produce a temporal dark MOT. During this crucial step, atoms start entering into a state that is “dark” to the cooling light. The resultant decrease in the recoil heating and excited state collisions leads to an increase in the phase space density. After around 100ms, the MOT and repump beams were extinguised and the magnetic coil current switched off. The FORT beam was transported into the vacuum chamber by a three lens assembly: the first two lenses formed a 2x beam expander followed by a third focussing lens mounted inside the chamber. The final spot size at the center of the MOT was w0 = λf/(πR), where f = 38.1mm is the focal length of the third lens and R is the 24 radius of the beam incident on it. The second lens of the beam expander was mounted on a translation stage (Fig. 2.7). Moving this by 15 mm in 1 s increased the beam size R thereby compressing the FORT. A tightly focussed trap was essential to increase the elastic collision rate and enhance the evaporative cooling. The cloud of around a million ultracold atoms was then subjected to forced evaporative cooling, where the trap depth was lowered at an exponential rate by reducing the FORT beam intensity. This was done by decreasing the RF power driving the CO2 AOM. After 5 s and at a final laser power of 40 mW, a pure condensate of around 30,000 atoms in the 5S1/2, F=1 state was obtained. To image the BEC, it was released from the FORT and pulsed with the repump beam at full intensity to pump the atoms to the F=2 state. After 8 ms of expansion, a 100ns pulse of light tuned to the resonant F=2 → F′=3 transition cast a shadow of the falling condensate on a CCD camera. Subtracting a reference image (without the condensate) from this gave the final distribution of the BEC. Before we conclude it must be mentioned that aspects of the experimental setup for kicking and photoassociation are explained in their respective chapters. 25 Figure 2.7: Alignment of beams for the MagnetoOptic Trap and the FORT inside the vacuum chamber. 26 CHAPTER 3 Quantum transport with a kicked BEC Unraveling the details of transport in solid state systems is important for fundamental physics and is becoming particularly relevant for new nanoscopic devices [62, 63]. Atom optic systems offer an easily configurable and ‘clean’ system to study many aspects of it. In this chapter we investigate two models of directed transport in a quantum Hamiltonian. Section 3.1 is on higher order resonances of a quantum accelerator mode, which was discussed in the previous chapter. First, two theoretical approaches of explaining these resonances are described. These are then used to analyze experimental results. In section 3.2, the concept of a quantumresonance ratchet is discussed. An expression of generalized momentum current is derived. Our realization of such a ratchet is then compared to this theory and the effect of the various experimental parameters studied. 3.1 Highorder resonances of a quantum accelerator mode 3.1.1 Rephasing theory Presence of a linear potential like gravity along the grating leads to the quantum kicked accelerator. In dimensionless units, ˆH = ˆp2 2 − η τ ˆx + ϕd cos(ˆx) Σ∞ Np=0 δ(t − Npτ ). (3.1) Here, η = mg′T/~G is the unitless ‘gravity’ g′. As seen in the previous chapter, near multiples of the halfTalbot time T/T1/2, a group of atoms can acquire a fixed momentum with each kick. In general, such accelerator modes can exist near rational 27 fractions of the Talbot time, that is when τ = 2π(a/b). As in section 2.3, and using the definitions of τ and η, the phase acquired by the qth order is ϕq = τ q(pi + ηNp + 1 2 q) (3.2) where pi = Pi/~G is the scaled initial velocity. For these higher order resonances, rephasing occurs between momentum orders separated by b~G [64]. We can again divide the phase difference between these orders, ϕq − ϕq−b = τ b[pi + ηNp + 1 2 (2q − b)] (3.3) into two parts, τ b(q + ηNp) = 2πqa (3.4) τ b(pi − b/2) = 2πl (3.5) the first of which is dependent on the pulse number. Solution of Eq. (3.4) gives the momentum acquired by the accelerator mode, q = − ητ τ − 2πa b Np (3.6) For a primary resonance, (a = l′b), the above expression reduces to Eq. (2.20) as expected. Equation (3.5) also imposes a condition on the initial momentum of the atoms in the mode, pi = 2πl τ b + b 2 (3.7) 3.1.2 ϵclassical theory Fishman, Guarneri, and Rebuzzini (FGR) in 2002 proposed a theory where the detuning, ϵ, of the pulse period from resonance plays the role of Planck’s constant. The map obtained in the classical (or correctly, ϵclassical, ϵ → 0) limit of this system was able to successfully explain quantum accelerator modes. It further predicted the existence of higher order modes and spurred the experimental search for them [68]. 28 We review this theory here. The one period evolution operator or the Floquet operator for the Hamiltonian (3.1) can be written as ˆU = ˆK ˆ F = e−iϕd cos(ˆx) ˆ F (3.8) The matrix elements of this operator are evaluated in a basis of eigenfunctions of a particle falling under gravity, uE(p) = ⟨pE⟩, uE(p) = ( τ 2πη )1/2 ei τ η (Ep−p3 6 ) (3.9) The matrix elements of the free evolution operator are, ⟨p′ ˆ Fp′′⟩ = ∫ dEe−EτuE(p′)u∗ E(p′′) = δ(p′ − p′′ − η)e−i τ 2 (p′−η 2 )2 (3.10) The kick operator ˆK reads, ⟨pe−iϕd cos(ˆx)p′⟩ = Σ∞ n=−∞ (−i)nJnδ(p − p′ − n) (3.11) Thus we have the propagator, (ˆU ψ)(p) = ∫ dp′⟨pˆU p′⟩⟨p′ψ⟩ = Σ∞ n=−∞ (−i)nJne−i τ 2 (p−n−η 2 )2 ψ(p − n − η) (3.12) The presence of the linear term in the Hamiltonian (3.1) breaks the kicking potential’s periodicity, and quasimomentum is no longer conserved. By means of a gauge transformation, spatial periodicity and therefore conservation of the particle quasimomentum, β, can be restored, ˆH g(t) = 1 2 ( ˆp + η τ t )2 + ϕd cos(ˆx) Σ∞ Np=−∞ δ(t − Npτ ). (3.13) 29 This amounts to writing the state in a momentum basis falling freely with the particle, ψg(p,Np) = ⟨p + ηNpˆU Np ψ⟩ (3.14) where ηNp is the momentum gained due to gravity. The propagator (3.12) in this frame reads, ˆU g(Np) = e−iϕd cos(ˆx)e−i τ 2 (ˆp+ηNp+η 2 )2 (3.15) The kickedrotor We can write the Bloch eigenstates for the above periodic potential as eiβxψβ(x), where ψβ(x) is 2π periodic in x. Introducing an angular coordinate θ = x mod(2π), ψβ(θ) can be considered a fictitious rotor (also called a βrotor). The angular momentum representation is related to the kicked particle by, ⟨nψβ⟩ = ⟨n + βψ⟩ (3.16) In the θrepresentation, ⟨θψβ⟩ = √1 2π Σ n ⟨n + βψ⟩einθ (3.17) The one kick evolution operator for ψβ⟩ is ˆU β = e−iϕd cos(ˆθ)e−i τ 2 (Nˆ+β+ηNp+η 2 )2 (3.18) where the angular momentum operator Nˆ = −i d dθ is related to the particle momentum by pˆ = Nˆ + β. Dynamics near a Resonance: ϵClassical treatment Let us now investigate the rotor dynamics near a resonance and define ϵ = τ − 2πl (ϵ << 1). With ˜k = ϵϕd, and using e−iπln2 = e−iπln, the evolution operator (3.18) becomes, ˆU β(Np) = e −i ~k ϵ cos ˆθe − i ϵ Hˆβ(ˆI,Np). (3.19) 30 Here we have defined Hˆβ(ˆI, t) = 1 2 ϵ ϵ (ϵ)ˆI2+ ˆI[πl+τ (β+Npη+ η 2 )] with Iˆ= ϵNˆ = −iϵ d dθ (3.20) If ϵ in Eq. (3.19) is treated as a Planck’s constant, (3.19) would follow from quantization of the following classical (timedependent) map, INp+1 = INp + ˜k sin(θNp+1) θNp+1 = θNp + ϵ ϵINp + πl + τ (β + Npη + η 2 ) (3.21) In other words, as ϵ → 0, the quantum rotor dynamics can be described in terms of ‘classical rays’ along trajectories of (3.21). This limit has been dubbed “ϵclassical” to distinguish it from the actual ~ → 0 classical limit. We can define JNp = INp + ϵ ϵ [πl + τ (β + Npη + η 2 )] and remove the explicit time dependence, JNp+1 = JNp + ˜k sin(θNp+1) + ϵ ϵτη θNp+1 = θNp + ϵ ϵJNp (3.22) These describe stable periodic orbits on a 2torus parametrized by J and θ (mod2π). If the orbit has periodicity p (measured in kicks), then Jp = J0 + 2πj, θp = θ0 + 2πn (3.23) j, called the jumping index, gives the number of times the orbit has wound around itself in the J direction. Figure shows phase space maps of Eq. (3.23) for a primary and two higher order modes. After Np cycles, the physical momentum qNp = I/ϵ along this orbit is [67] qNp ≃ n0 − ητ ϵ Np + 2π j pϵNp (3.24) where we have used Eq. (3.23) and the definition of J. These are trajectories surrounding a ‘stable fixed point’. Together they form an island of stability in the phase space. A wave packet starting with a sizable overlap with such an island grows linearly in momentum with the pulse number (3.24), and forms an ‘accelerator mode’. The mode is characterised by the parameters (p, j). 31 Figure 3.1: Phase space of quantum accelerator modes generated by the map of (3.22) for τ = 5.744 and ϕd = 1.4. Mode (a) with (p, j) = (1, 0) for η = 2.1459. is a primary QAM. Higher order modes are seen in (b) with (p, j) = (2, 1), η = 2.766, and (c) (p, j) = (5, 1), η = 4.1801. 32 ϵclassical theory of higher order resonances Higher order resonances appear near τ = 2πa/b where a, b are coprime integers. Near a higher order resonance, we define ϵ = τ −2πa/b. However, in general, no ϵclassical limit exists near these higher order times. Nevertheless, it was shown in Ref. [65] that the QAM’s can still be generated, not by a single classical map, but by bundles of “classical rays” which follow the trajectories, θNp+1 = θNp + INp + τ (δβ + Npη + η 2 ) + 2πSNp/b INp+1 = INp − ˜k sin(θNp+1 + 1) (3.25) where δβ = β − βr gives the closeness to resonant βr. The integers SNp can take integer values between 1 and b and are arbitrary. The average physical momentum is then, qNp ≃ −ητ ϵ Np + ( 2π j p − TΣ′−1 r=0 ΔSr ) Np T′ϵ (3.26) Here, ΔSr = 2π(Sr+1 − Sr)/b and T′ satisfies ΔSr+T′ = ΔSr. For a primary QAM which we investigate experimentally, j = 0. Also with the simple case of ΔSr = 0, Eq. (3.26) reduces to, qNp ≃ −ητ ϵ Np (3.27) identical to Eq. (3.6) obtained from the interference model. 3.1.3 Experiment To experimentally realize such higher order resonances, an initial state was produced in the form of a BoseEinstein condensate of around 30,000 atoms of Rb87 in the F = 1, 5S1/2 state. The CO2 beam forming the optical trap was then extinguished and the condensate released from the trap. The kicking beams were derived from a slave laser injection locked to the F = 2 → F′ = 3 transition frequency of the master laser. It was thus 6.8 GHz reddetuned with respect to the condensate atoms 33 AOM2: Asin(wt) AOM1: Asin[(w+w )t] D Kicking beams Figure 3.2: Setup for the kicking experiments. Two counterpropagating beams formed the standing wave oriented at an angle of 52◦ to the vertical. AOM2 was driven by an RF signal with a fixed frequency, ω/2π = 40MHz. 34 (Fig. 2.5). An optical fiber transported this beam from the optical table to the BEC table where it was divided into two by a 5050 beam splitter cube. Each of these beams had 25 mW of power and was directed into an acoustooptic modulater (Isomet model 40N). The frequency of the first order beam diffracted by the AOM is Doppler shifted by an amount equal to the frequency of the acoustic wave in the AOM. The acoustic wave was generated by an RF electrical signal supplied by an arbitrary waveform generator HP8770A after amplification. One of the AOMs was driven at a fixed 40 MHz while the other had a variable frequency input from a second HP8770A which was phaselocked to the first generator. By adjusting the variable part of this frequency, ωD = 2π T1/2 β + 1 2Gat, the initial momentum β, and acceleration a, of the standing wave relative to the condensate could be changed. A schematic of the kicking setup is shown in Fig. 3.2. After the application of the kicking sequence, the condensate was allowed to expand for 8 ms to allow the diffracted momentum orders to separate before imaging. Pulse lengths of 1.8 μs were used for the kicks. The probability of the nth momentum order getting populated is Jn(ϕd)2. Thus, by examining a distribution after one kick, the strength ϕd of a kick was inferred to be ≈ 1.5. Figure 3.3 depicts the evolution of a QAM as a function of kicks. The plots show horizontally stacked distributions with increasing pulse number near a resonance time. Figure 3.3(a) is a kick scan performed at T = 22.68 μs, close to the resonance at (2/3)T1/2 [70]. The linear growth of a QAM with orders separated by 3~G can be seen. This behavior where states separated by b~G rephase to form the bth order resonance is expected from the fractional Talbot effect [71]. The effect is clearer in Fig. 3.3(b) which is near the resonance at (1/2)T1/2 with a pulse separation of T = 17.1 μs. Figure 3.3(c) is at the Talbot time (2T1/2), where the QAM can be seen to comprise of neighboring momentum states. Next, time scans across two highorder resonances at (1/2)T1/2 and (1/3)T1/2 are 35 20 Figure 3.3: Quantum accelerator modes at (a) T=22.68 μs close to (2/3)T1/2, (b) T=17.1 μs which is close to (1/2)T1/2, and (c) T=72.4 μs close to 2T1/2. a value of g′=6 ms−2 was used in these scans. The arrows in (a) and (b) show orders separated by b~G which participate in the QAMs. Dashed lines correspond to the ϵclassical theory of Eq. (3.27) 36 shown in Figs. 3.4(a) and (b) respectively. The dashed curves are Eq. (3) demonstrating a good fit with the ϵclassical theory. The relatively weak experimental signal for the resonance at (1/3)T1/2 is also noticed in Fig. 3.4. This difficulty in creating a highorder QAM stems from the high kicking strength ϕd needed to populate states comparable to b~G. It can be seen from the fact that the population of the state b~G is proportional to Jb(ϕd)2 with a maximum at b ∼ ϕd. This high value of ϕd has the unwanted effect of increasing the distribution of states not participating in the QAM, thereby masking the presence of a QAM. Finally, the dependence of QAM on the initial momentum of the BEC was investigated by moving the standing wave using the variable frequency of a kicking AOM. The initial momentum scans performed near resonances at (1/2)T1/2 and (2/3)T1/2 are shown in Fig. 3.5(a) and (b) respectively. The QAM appears once every 1~G of initial momentum at (1/2)T1/2 and twice at (2/3)T1/2. To understand this, we can use Eq. (3.7) to obtain the separation in initial momenta for a QAM to be Δpi = 2π/τb ≈ 1/a. Thus our experimental results bear out the predictions of the theory of highorder resonances. 37 Figure 3.4: Horizontally stacked momentum distributions across (a) (1/2)T1/2 for 40 kicks, ϕd = 1.4 and effective acceleration g′ = 6 ms−2; and (b) (1/3)T1/2 for 30 kicks, ϕd = 1.8, and g′=4.5 ms−2. The dashed curve is a fit to the theory in Eq. (3.27) 38 Figure 3.5: Initial momentum scans for QAMs near (a) (1/2)T1/2 (T=17.1 μs) for 30 kicks and g′=6 ms−2; and (b) (2/3)T1/2 (T=22.53 μs) with 40 kicks and g′=4.5 ms−2. The dashes indicate QAMs at the resonant β. 39 3.2 A quantum ratchet Feynmann in his classic Lectures on Physics [72] discussed the concept of a ratchet, and demonstrated how directed motion could not be extracted from a spatially asymmetric system kept in thermal equilibrium. However away from equilibrium, a ratchet current, defined as directed transport in a spatially periodic system with no bias field, becomes a possibility [73]. The best examples of this are seen in nature in the form of biological motors which can use thermal fluctuations to establish a particulate current. Hamiltonian ratchets, where deterministic chaos replaces dissipation to drive motion, have recently gathered theoretical and experimental interest. For instance, asymmetric dynamical localization led to a unidirectional current in a kicked rotor with spatiotemporal asymmetry [75]. The possibility of using a resonance of a kicked rotor with a broken spatial symmetry was shown in [74] and subsequently observed experimentally [26]. We realize such a quantum ratchet at arbitrary initial momenta (quasimomentum) and investigate its various parameter dependencies. The initial state is prepared as a superposition of the momentum states ψ0⟩ = √1 2 [0⟩ + 1⟩] (3.28) With Eq. (3.17), we can express this as a rotor state, ψ0(θ) = √1 4π [1 + eiθ] (3.29) After a time T, the 1st order picks up a phase γ = (~G)2 2m~ T. Equivalently, the free evolution can be substituted by a translation of the grating from cos(θ) → cos(θ − γ). We shall adopt this latter approach in the analysis below, and the experiments thereafter, where an offset kicking potential V (θ) = cos(θ − γ) is applied on the initial state. The one period evolution operator for the Hamiltonian ˆH = ˆp2 2 + ϕd cos(ˆx) Σ∞ Np=0 δ(t − Npτ ) is given by (Eq. 3.18 with η = 0), ˆU β = e−iϕdV (ˆθ)e−iτ(Nˆ+β)2/2 40 = e−iϕd cos(θˆ−γ)e−iNˆ τβ (3.30) Here we have ignored an irrelevant phase factor and τβ = πl0(2β + 1) with l0 an integer. After Np kicks, ψNp(θ) = ˆU Np β ψ0(θ) = ˆU Np−1 β e−iϕd cos(ˆθ−γ)ψ0(θ − τβ) = ˆU Np−2 β e−iϕd cos(ˆθ−γ)e−iϕd cos(ˆθ−γ−τβ)ψ0(θ − 2τβ) = e−iϕd ΣNp−1 s=0 cos(ˆθ−γ−sτβ)ψ0(θ − Npτβ) = e [−iϕd sin(τβNp/2) sin(τβ/2) cos{θ−(Np−1) τβ 2 −γ}] ψ0(θ − Npτβ) (3.31) where we have used the relation Σt−1 s=0 eas = (1 − eat)/(1 − ea) in the last step. The momentum current at time Np is, ⟨Nˆ ⟩Np ≡ ⟨ψNp Nˆ ψNp ⟩ = −i ∫ 2π 0 dθψ∗ Np(θ) dψNp(θ) dθ = −i ∫ 2π 0 dθ [iϕd sin(τβNp/2) sin(τβ/2) sin{θ − (Np − 1) τβ 2 − γ}ψ0(θ − Npτβ)2 +ψ∗ 0(θ − Npτβ)ψ′ 0(θ − Npτβ)] (3.32) Here the prime denotes a derivative with respect to θ. Using the initial state (3.29) and identifying the second term as the mean momentum of the initial state, the change in the momentum current is [76], Δ⟨ˆp⟩Np = ⟨Nˆ ⟩Np − ⟨Nˆ ⟩0 = ϕd 2 sin(τβNp/2) sin(τβ/2) sin[(Np + 1)τβ/2 − γ] (3.33) When τβ → 2rπ, where r is an integer, we obtain a linear growth in the current, a ratchet acceleration: Δ⟨ˆp⟩Np,r = −ϕd 2 sin(γ)Np. (3.34) 41 At this point we would like to note that for a plane wave initial state, the integral of the first term in (3.32) vanishes and there is no change in the momentum current from its initial value. This explains our choice of the special superposition initial state (3.29) to create a ratchet. Of course one has to account for the finite initial momentum width of the condensate in an experiment. Towards this end we consider a condensate with a Gaussian distribution, √ 1 2π(Δβ)2 exp−(β′−β)2 2(Δβ)2 , where β and Δβ are the average and standard deviation respectively. The average of the momentum current (3.33) over β = β′ is [27, 77], ⟨Δ⟨ˆp⟩Np ⟩Δβ = ϕd 2 ΣNp s=1 sin(τβs − γ) exp[−2(πl0Δβs)2] (3.35) On resonance, that is when τβ = 2rπ, Eq. (3.35) reduces to ⟨Δ⟨ˆp⟩Np ⟩Δβ = −ϕd 2 sin(γ) ΣNp s=1 exp[−2(πl0Δβs)2] (3.36) 3.2.1 Experiment The experimental configuration was similar to that used to observe quantum accelerator modes but with η = 0. The initial state was prepared by applying a weak but long pulse (duration of 38 μs) to the released condensate. This resulted in a Bragg diffraction of the atoms into an equal superposition of the 0~G⟩ and 1~G⟩ states. This was followed by short (2 μs) kicks of the rotor, each with a strength ϕd ∼ 1.4. To produce an offset γ, the standing wave was shifted by changing the phase of the variable kicking AOM. A systematic study was done to experimentally determine the parameter dependence of a quantum ratchet. We worked at the first quantum resonance where τβ = 2π, corresponding to the halfTalbot time, T1/2. This gives a resonant β of 0.5 (for l0 = 1). Figure 3.6 shows the mean momentum as a function of the offset phase, γ, after 5 kicks on the initial state. The largest change in momentum was at γ = π/2, 3π/2 etc. This can be seen classically as a maximum gradient of the optical potential acting on 42 Figure 3.6: Dependence of the mean momentum of the quantum ratchet on the offset angle γ for 5 kicks and β = 0.5. The dashed and solid lines represent Eqs. (3.33) and (3.36) respectively. The inset shows the offset γ created between the symmetry centers of the initial distribution (blue curve) and the kicking potential V (red curve). 43 Figure 3.7: The ratchet effect. This time of flight image shows growth of mean momentum with each standing wave pulse applied with a period of T1/2 and a maximum offset (γ = π/2) between the standing wave and the initial state. 44 Figure 3.8: Mean momentum as a function of kicks. The data and error bars are from experiments with ϕd = 1.4, γ = π/s and β = 0.5. The solid line is Eq. (3.33) while the dashed line corresponds to Eq. (3.36). 45 the atomic distribution when it is displaced by an integer multiple of γ = π/2 (Fig. 3.6). The ratchet effect can be seen clearly in the time of flight image of Fig. 3.7 at γ = π/2. The mean momentum as a function of kicks is plotted in Fig. 3.8, and exhibits a saturation of the ratchet acceleration. A fit of the mean momentum in Figs. 3.6 and 3.8 to Eq. (3.36) yields a value of Δβ = 0.056. This value is consistent with a condensate momentum width measured using a timeofflight technique. Finally, with γ = ±π/2, the dependence of the mean momentum on β is depicted in Fig. 3.9. As expected, a pronounced ratchet effect is seen at the resonant β = 0.5 but is suppressed overall due to the finite Δβ. Thus, we realized a quantumresonance ratchet by applying periodic kicks from a symmetric optical potential on a superposition state. An asymmetry introduced between the centers of the initial state and the kicking potential created a ratchet current (Fig. 3.6), with a maximum magnitude at γ = ±π/2. A linear growth in momentum was observed at the resonant β = 0.5, but was found to be suppressed by the finite momentum width of the BEC. It is therefore desirable to reduce Δβ to obtain a higher ratchet current. 46 0 0.2 0.4 0.6 0.8 1 −4 −2 0 Quasimomentum (b) Mean−momentum change (2 photon recoils) 0 2 4 (a) (b) Figure 3.9: Change in mean momentum vs the quasimomentum β for ϕd = 1.4 and (a) γ = −π/2, (b) γ = π/2. Shown are the fits of Eqs. (3.33), dashed line and (3.36), solid line respectively to the experimental data (filled circles). 47 CHAPTER 4 SubFourier resonances of the kicked rotor A discrete Fourier transform of input test signals was recently performed with a BEC in an optical lattice [78]. The resultant frequency spectra were resolved more accurately than possible by classical Fourier analysis. Such examples exploit the nonlinear response of the quantum system to an excitation. For instance, the linewidth of the nth harmonic of a multiphoton Raman resonance was shown to be nfold narrower than the Fourier transform linewidth of the driving optical pulse [79]. In another demonstration, dynamical localization in the classically chaotic regime of the quantum kicked rotor was utilized to discriminate between two driving frequencies with a sub Fourier resolution [28]. At a primary quantum resonance at the Talbot time, adjacent momentum orders evolve in phases which are integer multiples of 2π. As we saw in chapter 2, this leads to a quadratic growth in mean energy, ⟨E⟩ = 2πErϕ2 dN2 where Er = ~2G2/8M is the photon recoil energy. The width of mean energy distribution around the resonance time was found to decrease with kicks and kicking strength as 1/(N2ϕd) [91]. This subFourier behavior was attributed to the nonlinear nature of the quantum delta kicked rotor and explained using the ϵclassical theory [80]. Highprecision measurements using quantum mechanical principles have been carried with atom interferometers for many years [81]. Such devices were used to determine the Earth’s gravitational acceleration [82–84], fine structure constant α [86–88], and the Newtonian constant of gravity [89]. The promise of the QDKR as a candidate for making these challenging measurements has begun to be realized [29]. Recently a scheme was proposed for measuring the overlap or fidelity between a nearresonant 48 δkicked rotor state and a resonant state via application of a tailored pulse at the end of a rotor pulse sequence [90]. It predicted a 1/N3 scaling of the temporal width of the fidelity peak. In this chapter we describe our observation of such fidelity resonance peaks and their subFourier nature. In section 4.1 we discuss the theory. We also calculate the effect of gravity on such a measurement. Section 5.7 details the experimental configuration and the results. 4.1 A fidelity measurement on the QDKR We begin with the evolution of a state ψ0⟩ due to N kicks at a period near the Talbot time, i.e., τ = 4π + ϵ, ψf ⟩ = ˆU Nψ0⟩ = Σ n cnn⟩ (4.1) where the n⟩ are momentum eigenstates in units of ~G. The effect of the deviation from resonance (to the first order in ϵ), results in a change of the phase of the complex coefficients cn = An(ϵ) exp{iθn(ϵ)}, cn(ϵ) ≃ cn(0) exp ( i ∂θn ∂ϵ ϵ=0ϵ ) = (−i)nJn(ϕd) exp ( i ∂θn ∂ϵ ϵ=0ϵ ) (4.2) cn(0) are the resonant coefficients as seen in Eq. (2.10). We notice that a measurement of the mean energy ⟨E⟩ = 1 2 Σ n n2pn for the distribution pn = ⟨nψ⟩f 2 will not detect these phases. To probe the effect of these phase changes, Ref. [90] proposed a measurement of the projection of this final offresonant state onto the resonant state. To this end one can define a “fidelity”, F = ⟨ψ0ˆU rˆU Nψ0⟩2 (4.3) 49 where ˆU = exp(−i τ 2 ˆp2) exp[−iϕd cos(ˆx)] and ˆU r = exp[iNϕd cos(ˆx)]. Experimentally, ˆU r can be implemented with a kick changed in phase by π and carrying a strength of Nϕd. Note that on resonance, exp(−i τ 2 ˆp2) = 1 and the final state is the same as the initial. F therefore measures the probability of revival of the initial state as a function of these deviations. Using Eqs. (4.1) and (4.2), the fidelity can be expressed as, F ≃  Σ n J2 n(Nϕd) exp(iΘn)2 (4.4) Here we have introduced Θn = ∂θn ∂ϵ ϵ=β=η=0ϵ + ∂θn ∂β ϵ=β=η=0β + ∂θn ∂η ϵ=β=η=0η (4.5) to account for the general case of phases accruing from deviations in resonant time (ϵ), initial momentum (β) or a nonzero gravity (η). This perturbative treatment was used to show that near the Talbot time, τ = 4π, the fidelity is [90] F(ϵ, β = 0, η = 0) ≃ J2 0 ( 1 12 N3ϕ2 dϵ ) (4.6) The width of such a peak in ϵ therefore changes as 1/(N3ϕ3 d), displaying a subFourier dependence on the measurement time, expressed here in units of kicks, N. We now investigate the behavior of the fidelity in the presence of a linear potential like gravity. 4.1.1 Effect of ‘gravity’ For a perturbation only due to gravity, Θn = ∂θn ∂η ϵ=β=η=0η (4.7) Equation (4.2) can be rearranged as, ∂θn ∂η (ϵ=β=η=0) = ∂cn ∂η (ϵ=β=η=0) icn(0, 0, 0) (4.8) 50 Coefficients cn are cn(ϵ, β, η) = ⟨n + βUˆgN ...Uˆg2 Uˆg1 β⟩ (4.9) Uˆgt = exp[−i τ 2 (ˆp + tη + η 2 )2] exp[−iϕd cos(ˆx)] is the tth kick evolution operator in the freely falling frame which was introduced in section (3.1.2) obtained after a gauge transformation of the QDKA Hamiltonian. After 3 kicks on the initial zero momentum eigenstate, ∂cn ∂η (ϵ=β=η=0) = ⟨n∂ ˆ Ug3 ∂η Uˆg2 Uˆg1 + Uˆg3 ∂Uˆg2 ∂η Uˆg1 + Uˆg3 Uˆg2 ∂Uˆg1 ∂η 0⟩(ϵ=β=η=0) = −2iτ 2 ⟨n(ˆp + 3η + η 2 )(3 + 1 2 ) exp(−iϕd cos xˆ)Uˆg2 Uˆg1 + Uˆg3(ˆp + 2η + η 2 )(2 + 1 2 ) exp(−iϕd cos xˆ)Uˆg1 + Uˆg3 Uˆg2(ˆp + 1η + η 2 )(1 + 1 2 ) exp(−iϕd cos ˆx)0⟩(ϵ=β=η=0) = −iτ ⟨ne−iϕd(1−1) cos ˆx ˆp (3 + 1 2 )e−iϕd(3+1−1) cos ˆx + e−iϕd(2−1) cos ˆx ˆp (2 + 1 2 )e−iϕd(3+1−2) cos ˆx + e−iϕd(3−1) cos ˆx ˆp (1 + 1 2 )e−iϕd(3+1−3) cos ˆx0⟩ Therefore for N kicks, ∂cn ∂η (ϵ=β=η=0) = −iτ ΣN r=1 bn,r (4.10) where bn,r = ⟨ne−iϕd(r−1) cos ˆx ˆp (N − r + 3 2 )e−iϕd(N+1−r) cos ˆx0⟩ (4.11) We make an observation here. Equation (4.11) can be expanded as bn,r = (N − r + 3 2 ) Σ∞ m=−∞m2Jm[(N + 1 − r)ϕd]Jm−n[(r − 1)ϕd] (−1)m (−i)n . This expression and Eq. (4.10) show that, a first order perturbation in η can be viewed as a rotation in the complex plane (real→imaginary and vice versa). Similar results hold for deviations from resonant ϵ and β and explain the basis of assumption (4.2). We now return back to evaluate (4.10) with (4.11), ∂cn ∂η (ϵ=β=η=0) = −iτϕd ΣN r=1 (N − r + 3 2 )(N + 1 − r)⟨ne−iϕd(r−1) cos ˆx sin ˆx e−iϕd(N−1+r) cos ˆx0⟩ 51 = −iτϕd⟨n sin ˆx e−iNϕd cos ˆx0⟩ ΣN r=1 (N − r + 3 2 )(N + 1 − r) = iτ 2i ϕd⟨n(eiˆx − e−iˆx) e−iNϕd cos ˆx0⟩ ΣN r=1 (N − r + 3 2 )(N + 1 − r) = τ 2 ϕd(⟨n + 1 − ⟨n − 1){ Σ l (−i)lJl(Nϕd)l⟩} ΣN r=1 (N − r + 3 2 )(N + 1 − r) = τ 2 ϕd[(−i)n+1Jn+1(Nϕd) − (−i)n−1Jn−1(Nϕd)] ΣN r=1 (N − r + 3 2 )(N + 1 − r) = −iτn N (−i)nJn(Nϕd) ΣN r=1 (N − r + 3 2 )(N + 1 − r) (4.12) using Jn+1(x) = 2n x Jn(x) − Jn−1(x). This gives us, ∂θn ∂η (ϵ=β=η=0) = ∂cn ∂η (ϵ=β=η=0) icn(0, 0, 0) = −τn[N2 + 5 2 N + 3 2 + (N + 1) 2 {2N + 1 3 − (2N + 5 2 )}] Keeping only terms in N2, ∂θn ∂η (ϵ=β=η=0) = −τn N2 3 = −4π 3 nN2 (4.13) Therefore, the fidelity is F(η, ϵ = β = 0) ≃  Σ n J2 n(Nϕd) exp(i ∂θn ∂η η)2 = Σ n J−n(Nϕd)Jn(Nϕd) exp{i(−Δ + π)n}2 where we have used result (4.13) to define Δ = 4π 3 N2η and used the Bessel function property, J−m(x) = (−1)mJm(x). Finally with the help of Graf’s identity, Σ l J−l+m(A)Jl(B) eilϕ = A+Beiϕ A+Be−iϕ Jm[(A2 + B2 + 2AB cos ϕ)1/2], we can simplify this 52 to F(η, ϵ = β = 0) = Nϕd(1 + e−iΔ) Nϕd(1 + eiΔ) J0[Nϕd(2 − 2 cos Δ)1/2] 2 = J2 0 (NϕdΔ) = J2 0 ( 4π 3 N3ϕdη) (4.14) keeping the lowest terms in η. Thus the width of a fidelity peak centered at the resonant zero acceleration should drop as 1/N3. A similar procedure gives the fidelity near the resonant(β = 0) initial momentum [90], F(β, ϵ = η = 0) = J2 0 (2πϕdN(N + 1)β) (4.15) 4.2 Experimental Configuration and Results These experiments were performed with a BoseEinstein condensate of around 20,000 87Rb atoms in the 5S1/2, F = 1,mF = 0 level. The selection of the mF = 0 Zeeman sublevel was done by keeping the MOT coils on during the evaporative cooling. Only the mF = 0 atoms which are insensitive to the magnetic field gradient (10G/cm) of the MOT coils could then undergo a stable evaporation [94], producing a pure mF = 0 condensate. After being released from the trap, the condensate was exposed to a horizontal optical grating created by two beams of wavelength λ = 780 nm light detuned 6.8 GHz to the red of the atomic transition. The wave vector of each beam was aligned θ = 52◦ to the vertical. This formed a horizontal standing wave at the point of intersection with a wavelength of λG = λ/2 sin θ. With this grating vector G = 2π/λG the resonant Talbot time is TT = 4πM/~G2 = 106.5 μs. The kicking pulse length was Δt = 0.8 μs with a measured ϕd ≈ 0.6. To create the reversal operator ˆU r in Eq. (4.3), a final kick of strength Nϕd was applied at the end of the N rotor kicks. We did this by increasing the intensity rather than the pulse length in order to keep this 53 Figure 4.1: Momentum distributions of a sequence of 8 kicks of strength ϕd followed by a final πphase reversed kick of strength Nϕd, with a time period equal to the Talbot time 106.5μs. 54 final kick within the RamanNath regime. Moreover, the standing wave for this kick was shifted by half a wavelength by changing the phase of one of the kicking AOM’s rf driving signal. A timeofflight image of a fidelity kicking sequence on resonance is depicted in Fig. 4.1. The sensitivity of the phases in this experiment to external perturbations made it necessary to identify and minimize such sources. Calculating the photon scattering rate γp = Ω2 δ2 γ for our kicking strength ϕd(= Ω2Δt 8δ ), we find that decoherence due to spontaneous emission is expected to be important only beyond ∼ 35 kicks. Dephasing primarily due to vibrations made the reversal process inconsistent for N > 6. To reduce this, the standing wave at each kick was shifted by half a wavelength with respect to the previous kick. This had the effect of shifting the Talbot time resonance to T1/2. Consequently the reduced experimental time led to much improved results. After 8 ms of expansion, the separated momentum orders were absorption imaged. In the experiment fidelity is measured by the fraction of atoms which have returned to the initial zero momentum state. That is we measure F = P0/ Σ n Pn where Pn is the number of atoms in the nth momentum order obtained from the timeofflight images. To facilitate the analysis of the data, all of the resonance widths (δϵ) were scaled to that at a reference kick number of N = 4. That is we define a scaled fidelity width Δϵ = δϵ/δϵN=4 for each scaled kick number Ns = N/4 and recover logΔϵ = −3 logNs using Eq. (4.6). For each kick, a scan is performed around the resonance time. To ensure the best possible fit of the central peak of the fidelity spectrum to a gaussian, the time is scanned between values which make the argument of J2 0 of Eq. (4.6) ≈ 2.4 so that the first side lobes are only just beginning to appear. Figure 4.3(a) plots the logarithm of the FWHM for 4 to 9 kicks scaled to the fourth kick. A linear fit to the data gives a slope of −2.73±0.19 giving a reasonable agreement with the predicted value of 3 within the experimental error [93]. As seen in the same figure, the results are close to the numerical simulations which take into account the 55 Figure 4.2: (a) Horizontally stacked timeofflight images of a fidelity scan around the Talbot time. Each TOF image was the result of 5 kicks with ϕd = 0.8 followed by a πphase shifted kick at 5ϕd. (b) Mean energy distribution of the 5 kick rotor with the same ϕd. (c) The measured fidelity distribution (circles) from (a). The mean energy of the scan in (b) is shown by the triangles. Numerical simulations of the experiment for a condensate with momentum width 0.06 ~G are also plotted for fidelity (bluedashed line) and mean energy (redsolid line). The amplitude and offset of the simulated fidelity were adjusted to account for the experimentally imperfect reversal phase. 56 finite width of the initial state of 0.06 ~G [27]. We also compared the resonance widths of the kickedrotor mean energy ⟨E⟩ to that of the fidelity widths. As in the fidelity, the plotted values Δ⟨E⟩ have been normalized to that of the fourth kick. On the log scale, the width of each peak gets narrower with the kick number with a slope of −1.93±0.21 (Fig. 4.3(a)) in agreement with previous results [92, 95]. The relative scalings of the fidelity and mean energy distributions can be seen clearly in Fig. 4.2. It can be seen that even for relatively few kicks the fidelity peak can be significantly narrower. As a further test of Eq. (4.6), the variation in the widths of the fidelity and mean energy peaks were studied as a function of ϕd. Figure 4.3(b) shows the fidelity width changing with a slope of −1.96 ± 0.30, close to the predicted value of 2. This is again a faster scaling compared to the mean energy width which decreases with a slope of −0.88 ± 0.24 (the theoretical value being 1). The resonances studied here appear for pulses separated by the Talbot time and an initial momentum state of β = 0. As seen in Eq. (4.15), the peak width in β space is expected to change as 1/[N(N + 1)] around this resonant, as against a 1/N scaling of the mean energy width [92]. To verify this, the initial momentum of the condensate with respect to the standing wave was varied and the kicking sequence applied. The experimentally measured widths Δβ = δβ/δβN=4 in Fig. 4.4 display a scaling of Δβ _ [N(N + 1)]−0.92 close to the theoretical value. For an initial state β+n⟩, the wave function acquires a nonzero phase during the free evolution even at the Talbot time. Therefore the final kick performs a velocity selective reversal, preferentially bringing back atoms closer to an initial momentum of β = 0. This is similar to the timereversed Loschmidt cooling process proposed in Refs. [96, 97], although in that technique a forward and reverse path situated on either side of the resonant time was used in order to benefit from the chaotic dynamics. To observe this effect the current scheme offers experimental advantage in terms of 57 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −2 −1.5 −1 −0.5 0 (a) log ¢², log ¢hEi log Ns 0 0.1 0.2 0.3 0.4 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (b) log ¢², log ¢hEi log ˜ Ád Figure 4.3: Experimentally measured fidelity (circles) and mean energy (triangles) widths (FWHM) as a function of (a) the number of pulses, and (b) the kicking strength ϕ˜d scaled to ϕd of the first data point. In (a), the data are for 4 to 9 kicks in units normalized to the 4th kick. Error bars in (a) are over three sets of experiments and in (b) 1σ of a Gaussian fit to the distributions. Dashed lines are linear fits to the data. Stars are numerical simulations for an initial state with a momentum width of 0.06~G. 58 0 0.5 1 1.5 −1.5 −1 −0.5 0 log ¢¯ log N(N +1)s Figure 4.4: Variation of the fidelity peak width around β=0 as a function of kick number N(N +1)s = N(N +1)/20 scaled to the 4th kick. The straight line is a linear fit to the data with a slope of −0.92 ± 0.06. Error bars as in Fig. 2(b). 59 0 1 2 3 4 5 6 7 8 9 0.12 0.13 0.14 0.15 0.16 (a) momentum width Kicks (¯h G) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 (¯h G) Optical density (b) momentum N=0 N=2 N=4 N=6 Figure 4.5: (a) Momentum width of the reversed zeroth order state as a function of kick number. Error bars are an average over three experiments. (b) Optical density plots for the initial state (red,solid) and kick numbers 2 (magenta,dotdashed),4 (black,dotted), and 6 (blue,dashed) after summation of the timeofflight image along the axis perpendicular to the standing wave. 60 stability due to the reduced length of the pulse sequence. Here, only a single pulse performs the velocity selection at the end, whereas in the Loschmidt technique N phase reversed kicks separated by a finite time are used. Figure 4.5 demonstrates the reduction of the momentum distribution width. Accompanying this decrease is a drop in the peak height. Our simulations and the results of Ref. [97] predict that for the case of a noninteracting condensate this should remain constant. In addition to interactions we expect experimental imperfections in the fidelity sequence to play a role in the smaller peak densities with increasing kick numbers. We performed the same experiment 4.5 ms after the BEC was released from the trap when the mean field energy had mostly been transformed to kinetic energy in the expanding condensate. A similar reduction in the momentum width of the reversed state alongwith a decrease in the peak density was observed. Finally to investigate the sensitivity of the fidelity resonance to gravity, the standing wave was accelerated during the application of the pulses. This acceleration was scanned across the resonant zero value and readings of the fidelity collected. Since a typical value of the halfwidth at half maximum is η = 0.05 for N=4 (corresponding to an acceleration of 4m/s2), the perturbative treatment of acceleration on fidelity used above is justified. Figure 4.6 plots the experimental data for 4 to 9 kicks, where the widths of the peaks decrease with a slope of −3.00 ± 0.23 in excellent agreement with the theory. In conclusion, we performed experimental measurements of the fidelity widths of a δkicked rotor state near a quantum resonance. The width of these peaks centered at the Talbot time decreased at a rate of N−2.73 comparable to the predicted exponent of −3. By comparison, the mean energy widths was found to reduce only as N−1.93. Furthermore, the fidelity peaks in momentum space changed as (N(N +1))−0.92, also consistent with theory. The reversal process used in the fidelity experiments led to a decrease in the momentum distribution of the final zeroth order state by ∼ 25% (for N=9) from the initial width. The subFourier dependencies of the 61 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −2.5 −2 −1.5 −1 −0.5 0 log ¢g log Ns Figure 4.6: Dependence of the acceleration resonance peak width as a function of the kick number in units scaled to the 4th kick. Error bars are over three sets of experiments. 62 mean energy and fidelity observed here are characteristic of the dynamical quantum system that is the QDKR [90]. The narrower resonances of the fidelity scheme could be exploited in locating the resonance frequency with a resolution below the limit imposed by the fourier relation. This can help determine the photon recoil frequency (ωr = Er/~) which together with the photon wavelength enables measurement of the fine structure constant with a high degree of precision [29, 86–88]. We also demonstrated a N−3 dependence of the resonance width in acceleration space in accordance with the extended theory. The sensitivity of an atom interferometer based gravimeter scales as the square of the loop time, hence the pursuit of large area interferometers to improve accuracy [82–84]. By comparison, the fidelity is responsive to the gravitational acceleration g with the cube of the ‘time’ N, leading to the possibility of higher precision measurements. One could perform a fidelity measurement on a freely falling condensate exposed to kicks accelerating at the local value of g (to realize η ≪ 1). Variation in g would then manifest itself as a shift of the resonant acceleration. A parts per billion precision [85] would require a judicious selection of the parameters (N, ϕd, T), for instance (150, 10, 16T1/2). Such a resolution, though not feasible in the current setup without addressing stability related issues, could be possible with future refinements, for instance through active stabilization measures. 63 CHAPTER 5 Photoassociation of a 87Rb BEC Our studies on kicked atoms have so far employed the linear kicked rotor model. A nonlinearity can be introduced via interactions among the condensate atoms. Such interactioninduced nonlinearity in the kicked rotor was shown to lead to a shift in the resonance and a striking cutoff at the maximum, due to coupling between phonon modes [98]. Another system studied in this context is the quantum δkicked harmonic oscillator. It is the realized with a kicked rotor in a harmonic trap and exhibits resonances like the rotor. Dephasing and destruction of the resonant motion were found to occur in the presence of interactions [99]. Furthermore, the nonlinear version remains an excellent candidate to study quantumclassical correspondence [100]. To pave the way for such studies, it is necessary to tune the interactions on demand. Magnetic Feshbach resonances, and more recently Optical Feshbach resonances, have enabled such control over the condensate interaction. In this chapter, we describe our experiments on photoassociation spectroscopy performed near long range molecular states of a 87Rb BEC to realize an Optical Feshbach resonance. In section 5.1 we discuss such spectroscopy. This is followed by a review of the theory ultracold collisions and Feshbach resonances in sections 5.2 and 5.3 respectively. We then discuss the experimental configuration and results in section 5.4, and the conclusion in section 5.5. 64 5.1 Photoassociative spectroscopy Photoassociation (PA) is a process in which two atoms colliding in the presence of a light field absorb a photon to form a bound, excited molecule. This effect was first observed for the H2 molecule and rare gas halide molecules [101,102]. With the advent of laser cooling, photoassociation of samples of cold atoms became a high resolution spectroscopic technique as a result of the very low energy spread of the atoms [103]. It has since been used to precisely measure atomic lifetimes, map ground state collisional wavefunctions, probe long range and ‘pure long range’ molecular states and to produce translationally cold molecules from cold atoms [104–107]. Furthermore, it was discovered that the collisional state scattering length can be altered near a photoassociation resonance [111]. This effect has been termed as an optically induced Feshbach resonance and has interesting implications for the dynamics of BoseEinstein condensates [112–114]. Experimental results of photoassociation of a 87 BoseEinstein condensate and the effect on the scattering length will be outlined in this chapter. Figure 1 depicts a typical photoassociative process. The interaction of the groundstate atoms (S+S) occurs along a Vander Waals potential Vg(R) = −C6/R6 at long ranges. For short internuclear distances the interaction is described by a 1/R12 strong electron exchange repulsion. Atoms colliding in the presence of light tuned close to the SP transition can absorb a photon and form an electronically excited molecule. The interaction between the two atoms in the excited molecule is determined by a resonant dipoledipole interaction, Ve = −C3/R3, at long ranges. Many of the bound states that are formed have energies close to the threshold and are therefore easier to access via freebound photoassociation compared to traditional boundbound spectroscopy [104]. The absorption takes place at an internuclear separation called the Condon point RC where the laser photon energy is equal to the difference between the ground the excited state potentials. The stimulated rate of transition from the 65 Figure 5.1: Schematic of a photoassociation process. Two atoms colliding along the ground state potential (S+S) absorb a photon and get excited to the (S+P) molecular potential. The excited molecule can subsequently decay to free atoms or a ground state molecule. 66 free to bound state is given by Γstim = 4π2Id2 M c ⟨eg⟩2, (5.1) where I is the laser intensity, dM is the molecular dipole transition moment, g⟩ is the ground state collisional wavefunction and e⟩ the bound, excited state [112]. The decay of the excited state molecule to either ground state molecules or to hot atoms leads to loss from the trap containing the molecules. Monitoring this loss as a function of the PA laser frequency results in a spectrum of the freebound transitions. This technique is generally known as photoassociation spectroscopy. This is a very high resolution technique resulting from the very low energy spread of ultracold collisions (a few MHz at μK temperatures, which is comparable to the natural molecular linewidths) [104]. As can be seen from Eqn. (5.1), the transition rate depends on the FranckCondon overlap between the ground and excited state wavefunctions. With larger collisional wavefunctions at greater internuclear distances, ultracold photoassociation is ideal for probing longrange excited molecular states [104]. The molecular states investigated using photoassociation in this study were the 1g and the 0− g pure long range state. These are labeled by Hund’s case (c). The total electron orbital angular momentum ⃗l couples strongly with the total spin ⃗s to form ⃗j = ⃗l + ⃗s. The projection of ⃗j along the internuclear axis Ω is conserved due to cylindrical symmetry of the molecule. The molecular states are labeled as Ωg/u. The g/u denotes the inversion symmetry of the electronic wavefunction. The 1g state has Ω=1 and is symmetric with respect to a reflection about the center of the internuclear axis(a gerade state, g). The 0− g state has an additional reflection symmetry and changes sign upon reflection in a plane containing the internuclear axis (the negative sign) [106]. 67 5.2 Ultracold Collisions Elastic twobody collisions are central to the description of BoseEinstein condensates as they determine the behavior as well as the stability of the condensates. The following discussion reviews basic elastic collision theory. One begins by looking for solutions to the timeindependent Schr¨odinger equation [ ˆp2 2μ + V (r)]ψ(r) = Eψ(r) (5.2) which describes the motion of two atoms with relative momentum p = ~k colliding in the potential V (r). μ is the reduced mass of the particles and E = ~2k2/2μ is the collision energy. At large r, where the potential is negligible, the scattering wave function satisfies the freeparticle Schr¨odinger equation and the solution can be written as ψk(r) −−→ r→∞ A[exp(ik.r) + f(k, θ, ϕ) exp(ikr) r ] (5.3) That is, the scattering wave function at large distances is a superposition of an incident plane wave and an outgoing spherical wave. A is a normalization constant. To calculate the scattering amplitude f, we consider the case of a central potential V (r). For such a potential the Hamiltonian commutes with the total angular momentum, L2 and its projection Lz. Also the scattering is symmetric about the incident direction and therefore independent of ϕ. One can thus separate the solutions in terms of radial components Rl(k, r) and Legendre polynomials Pl(cosθ) with contributions of l different partial waves. ψk(k, r, θ) = Σ∞ l=0 Rl(k, r)Pl(cosθ) (5.4) The Schr¨odinger equation for the radial part of the wavefunction becomes [ d2 dr2 − l(l + 1) r2 − U(r) + k2]ul(k, r) = 0 (5.5) 68 where ul(k, r) = rRl(k, r) and U(r) = 2μV (r)/~2. The total scattering cross section is defined as σ = ∫ f2dΩ, where the integration is over solid angle, can be shown to be σ = 4π k2 Σ l (2l + 1)sin2δl (5.6) where δl is the phase shift of the lth partial wave as a result of the collision. For identical bosons, only even partial waves contribute, σ = 8π k2 Σ l even (2l + 1)sin2δl (5.7) In the ultracold regime the collisional energy is too small to overcome the centrifugal barrier ~2l(l + 1)/2μr2. As can be seen from Fig. 5.2, any partial wave with l>0 cannot enter inside the potential. Therefore only l=0 (i.e. swaves) need to be considered for ultracold collisions. From Eq. 5.7 the scattering crosssection for the l=0 wave is σl=0 → 8πa2 (5.8) where the scattering length a has been defined as a = − lim k→0 tanδ0 k (5.9) Physically, the scattering length can be understood as the intercept of the unperturbed collisional wavefunction with the internuclear axis. 5.3 Scattering length and Feshbach Resonances The dynamics of an interacting BoseEinstein condensate are described by the Gross Pitaevskii equation [− ~2 2m ∇2 + U(r) + gψ(r)2]ψ(r) = −i~ ∂ ∂t ψ(r) (5.10) 69 Figure 5.2: Centrifugal energy term ~2l(l + 1)/2μr2 of the Hamiltonian for three partial waves, l=0,1,2. For low energy scattering all partial waves l >0 are blocked by the centrifugal barrier. 70 ψ(r) is the condensate wave function. The first two terms in the Hamiltonian are the kinetic energy and the trapping potential U(r). It also contains a nonlinear mean field energy term gψ(r)2 which describes the interation energy of an atom in the meanfield produced by the other bosons. It is proportional to the condensate density n = ψ(r)2 and the interaction coefficient g = 4π~2a/m for twobody elastic collisions between the bosons. The elastic scattering length, a, thus determines the interactions. For a > 0 the interactions are repulsive, and a < 0 leads to a condensate with attractive interactions stable only below a certain critical density [108]. Changing the scattering length has therefore received much attention and has been made possible by the use of Feshbach resonances. The scattering length depends strongly on the nature of the interatomic potential. This is best illustrated by the textbook problem of scattering by a squarewell potential: V (r) = −V0 if r < R = 0 if r > R Upon solving the radial wavefunctions ul(r) with boundary conditions ul(0) = 0 and continuity at r = R one finds the phase shift for the swave (l=0) to be δ0(k) = −kR + tan−1[ k k′ tan(k′R)] k and k′ are the wavevectors outside and inside the well respectively. Thus from Eq. 5.9 the scattering length can be shown to be a = R[1 − tan(λR) λR ] (5.11) where λ = √ mV0/~2. Figure 5.3 plots the scattering length as a function of λ. As can be seen, increasing λ which is equivalent to increasing the well depth V0 leads to periodic divergences of a. 71 Figure 5.3: Variation of the scattering length a as a function of λ = √ mV0/~2. As the well depth V0 increases ((a) to (c)) a bound state is formed (dashed line) and the scattering length passes through a divergence and changes sign. 72 The negative scattering length for an attractive squarewell potential begins to decrease as the welldepth increases, diverges when the well can hold a boundstate and becomes positive for a weakly bound state. The dependence of the energy of the last bound state, Eb, on the scattering length is given by Eb = −~2/ma2 for a >> R. The relationship between the scattering length and the position of the last bound state is at the heart of the concept of Feshbach resonances. The idea of inducing such a resonance involves using an external field to change the internal states of the colliding atoms to couple them to a quasibound state of another interatomic potential. Using a magnetic field for instance, two atoms colliding in an open channel V0 in a hyperfine asymptote can be Zeeman shifted into resonance with a bound state in another hyperfine state where they can stay temporarily bind. The upper potential with a higher threshold energy for the low energy atoms is usually called a closed channel, since energy conservation prohibits the escape of atoms from this potential. The tunability of the scattering length by a magnetic field is expressed as a(B) = abg[1 − Δ/(B − B0)] where abg is the background scattering length far away from resonance, B0 is the field on resonance, and Δ is the width of the resonance. Such a magnetic Feshbach resonance [109] was first observed in a 23Na BEC [110]. 5.3.1 Optical Feshbach Resonance Fedichev et al. [111] proposed the use of optical fields near a photoassociation resonance as a means of modifying the scattering length. An intense field of light tuned near such a resonance can be used to couple a pair of atoms colliding in a ground state potential to a bound excited state of a 1/R3 potential and thereby change the scattering properties of the pair of atoms. Bohn and Julienne [112] introduced a complex phase shift δ = λ + iμ, to include the presence of inelastic loss near a PA resonance. This leads to a complex scattering length, the imaginary part of which accounts for the spontaneous loss processes from the excited state. The expressions 73 Figure 5.4: A Feshbach resonance occurs when an excited state has a bound state close to the collisional threshold. Changing the detuning Δ by an external field can couple the collisional to the bound state and change the scattering length. 74 for the scattering length, a and the inelastic loss rate coefficient, Kinel were given as a = abg + 1 2ki ΓstimΔ Δ2 + (Γspon/2)2 (5.12) Kinel = 2π~ m 1 ki ΓstimΓspon Δ2 + (Γspon/2)2 (5.13) where abg is the background scattering length in the absence of the light, ki is the wavenumber of the condensate atoms, Δ is the detuning from a PA resonance and Γstim and Γspon are the stimulated transition rate and spontaneous decay rate constants respectively. In arriving at these expressions the assumption Γspon >> Γstim has been made [114]. This is usually true with the measured Γstim/2π being of the order of a few kHz whereas Γspon/2π is around 20 MHz. From Eq. 5.12 it can be seen that there are two parameters that can tune the scattering length, the detuning Δ, and the intensity I, contained in the stimulated transition rate Γstim (Eq. 5.1). To minimise inelastic loss processes and still have a significant change in the scattering length, it becomes necessary to choose a light source far detuned from a PA resonance (Eq. 5.13) and with a high intensity (large Γstim, Eq. 5.12). Such an optically induced Feshbach resonance was first achieved in a 87Rb BEC in 2004 [114]. Another way of viewing the Optical Feshbach Resonance process is in the dressedstate picture. The relevant states are the collisional ground state dressed in n photons from the laser g, n⟩ and the excited state dressed in n−1 photons e, n−1⟩. Altering the frequency of the light enables one to change the position of the bound state in the excited state relative to the ground state potential. 5.4 Experiment and Results The photoassociation light was derived from a grating stabilized master laser (Toptica Photonics DL 100). The output light was monitored with a wavemeter (Coherent Wavemaster, resolution ∼500 MHz). To set the laser’s initial frequency to a value 75 near a known molecular level, its grating angle, current, and temperature were appropriately adjusted. It was then offsetlocked to the MOT laser using a scanning FabryPerot interferometer cavity (Coherent, 300 MHz free spectral range). The majority of the light was used to injection lock a home built diode laser (Sanyo diode, 120 mW). The output of this slave laser was coupled through an optical fiber to the table for the BEC experiment. A small amount of the slave light was sent into the scanning cavity through a flipper mirror (Fig. 5.5). At the right current, transmission peaks of the slave and master overlap on the photodiode signal, indicating that the slave was following the master. The flipper mirror was then lowered and the path kept open for the reference beam to be input into the cavity. The master PA laser was locked to the reference light using home built electronics. The basic operating principle for the locking is shown in Fig. 5.6. Around 1 mW each from the master PA laser and reference light (from MOT or Repump laser) were coupled to the scanning cavity through a polarizing beamsplitter cube. The cavity was scanned with a free spectral range of 300 MHz with the help of a piezoelectric transducer attached to one of the mirrors. The frequency difference (modulo FSR) between the transmission peaks of the beams from the photodiode was converted to a ‘separation’ voltage, Vsep [115,116]. The difference between this and a setpoint voltage Vset from a National Instruments analog card gave the error signal, Verror (Fig. 5.6). This error signal was sent to a PID controller, built inhouse. The correction output from the PID was combined with the scan control voltage from the DL 100 unit to control the PA master laser’s grating angle. Changing Vset during a photoassociation experiment enabled control over the frequency of the PA light with an accuracy of around 5 MHz. The power in the PA beam, before it had entered the BEC chamber, was measured at 33 mW. It had a waist radius of 70 μm on the condensate. Higher intensities, which are especially vital for Optical Feshbach Resonances, are possible by reducing 76 Figure 5.5: Schematic of the optical setup for the photoassociation light. 77 Figure 5.6: Flowchart of the locking for the photoassociation master laser. 78 the beam waist size. However, this makes the dipole force of the PA beam strong enough to displace the condensate from the FORT making it increasingly difficult to perform any photoassociation experiment. Figure 5.8 shows the photoassociation spectrum of the 1g(P3/2), v = 152 molecular state. This spectrum was obtained on an ultracold sample (kept in the FORT and just before the BEC transition) with a 1 mW beam pulsed on for 100 ms. The state has a binding energy of 24.1 cm−1 below the 87Rb D2 line and asymptotically connects to 52S1/2+52P3/2 at large internuclear distances [105]. Five molecular lines of the hyperfine structure can be seen. These have been categorized in Ref. [118] according to Hund’s rule (c). The good quantum numbers for these states are the total molecular angular momentum F, total nuclear spin I and their projections Fz and Iz on the internuclear axis. The 0− g (∼ S1/2 + P3/2), v = 1 photoassociation spectrum of a BEC (∼ 25,000 atoms 87Rb) was observed with a 5 ms square pulse with a power of 33 mW. Two rotational lines J=0 and J=2 separated by 200 MHz can be seen in Fig. 5.9. This state located 26.8 cm−1 below the 87Rb D2 line is a ‘pure long range’ state with internuclear distances > 20a0 [117], much greater than ordinary chemical bonds (a0 = 0.53˚A is the Bohr radius). Single atom spontaneous emission losses in the presence of the PA light is proportional to (Ω/δ)2, where Ω is the Rabi frequency and δ the light detuning from an atomic resonance. With the high detuning (800 GHz) of the 0− g state from the atomic resonance such losses are therefore small. Moreover, a strong FranckCondon overlap with the incident collisional state [117] and absence of many neighboring molecular states makes the 0− g (P3/2), v = 1 particularly suitable for studying Optical Feshbach resonances near it. Bragg spectroscopy [119] was utilised in Ref. [114] as a technique to measure the scattering length as a function of the photoassociation detuning. Nth order Bragg diffraction is a process where atoms exposed to a standing wave of offresonant 79 light undergo absorption of N photons from one beam, followed by emission into the other. The internal state of an atom is left unchanged but it ends up acquiring 2N photon recoils of momentum (2N~k, where k = 2π/λ is the wavevector of light). For a moving standing wave created by two beams with a frequency difference ν, the energy and momentum conservation condition for noninteracting atoms is satisfied when hν0 = (2~k)2/2m+ 2~k.pi/m [119]. pi is the initial momentum of the atoms. In a weakly interacting condensate of uniform density n, the Bogoliubov dispersion relation is ν = √ ν2 0 + 2ν0nU/h where nU = n4π~2a/m is the chemical potential. For hν ≫ nU ν = ν0 + nU/h (5.14) The free particle resonance frequency has thus been shifted by the mean field energy [119]. In the presence of a photoassociation light tuned near a molecular state, any optically induced change in the scattering length should therefore be apparent in a shift of the Bragg resonance frequency. Therefore for a known condensate density, such a shift in the resonance frequency would allow measurement of the scattering length a. Light 6.8 GHz red detuned from the 87Rb D2 line was used to create a standing wave. As shown in Fig. 5.7 the Bragg (kicking) beams were aligned such that the wavevector of the standing wave was along the long axis of the FORT. Each of the beams were passed through two acoustooptic modulators. The radiofrequency driving one of these modulators was changed to vary the frequency of the beam passing through it. This created a moving standing wave required to perform a Bragg scan around the resonance frequency. Figure 5.10(a) is an image of a Bragg scan performed by changing the relative frequency between the two beams creating the standing wave. A 90 μs long Bragg pulse transferred nearly 40% atoms on resonance 80 to the −2~k momentum state. Figure 5.10(b) shows the Bragg resonance curves for photoassociation light detunings of Δ/2π =+10 MHz and 10 MHz with respect to the 0− g (P3/2), v = 1, J = 2 molecular state. No shift of the resonance is observed. An alternate approach can be adopted to extract the scattering length values from the loss spectrum [114,118]. The twobody loss rate equation for the condensate density n(r, t) is ∂n(r, t) ∂t = −2Kineln(r, t)2 (5.15) where Kinel is the inelastic rate coefficient. Assuming local density changes only due to photoassociation loss, McKenzie et. al. [120] give a relationship between the fraction of remaining atoms f(η) and Kinel f(η) = 15 2 η−5/2{η1/2 + 1 3 η3/2 − (1 + η)1/2 tanh −1[ √ η/(1 + η)]} (5.16) where η = 2Kineln0t and t is the PA pulse length. The Kinel values calculated from the loss spectrum of the 0− g (P3/2), v = 1, J = 2 state using Eq. 5.16 is shown in Fig. 5.11. A lorentzian fit to this yields a spontaneous decay rate, Γspon, of 8 MHz(Eq. 5.13). In addition, the amplitude A0 of the fit can be related to the stimulated rate constant Γstim as Γstim/ki = A0(mΓspon/8π~) from Eq.5.13. For the data in Fig. 5.11, Γstim/ki = 2.3 × 10−3ms−1. These values of Γstim and Γspon can be used in Eq. 5.12 to calculate the scattering length and the result is plotted in Fig. 5.12. A maximum change of 6a0 can be seen. This corresponds to a variation of 5 Hz in the Bragg resonance frequency, quite a small value to be detected using Bragg analysis. To achieve larger variations in the scattering length, a higher intensity of the PA light is necessary as can be seen from Eq. 5.12. Moreover any alteration in a can be amplified in the meanfield induced Bragg shift by using a higher condensate density (Eq. 5.14). Thus an improvement in these two parameters is crucial for the direct observation of an optically induced Feshbach resonance. 81 5.5 Conclusion and Outlook Photoassociation spectroscopy performed on a 87Rb BEC was able to resolve five hyperfine levels of the 1g long range molecular state and two rotational levels of the 0− g ‘pure long range’ state, matching the available data in the literature. Bragg spectroscopy done near the 0− g (P3/2), v = 1, J = 2 state did not yield a detectable change in the swave scattering length a. Analysis of the loss spectrum of this state as a function of the photoassociation detuning revealed a change of 6 Bohr radius in the scattering length. To achieve a larger and directly observable change, an improvement in the condensate density and the photoassociation light intensity is necessary and is currently being pursued. Another approach could be to more easily detect small changes in ‘a’ using a different measurement technique. Such a possibility was investigated using a method based on the interference between two copies of the condensate, made by standing wave pulses separated by Δt as shown in Fig. 5.13(a) [121]. The final interference amplitude depends on the difference in phase between the two copies accumulated during Δt, ϕ = [ ~G2 2m + 2μ 7π~]Δt where the first term arises from the recoil energy of the initial copy and the second term is due to a mean field effect of the condensate Fig. 5.13(a). Here it is assumed that one of the condensates has few atoms and consequently negligible mean field energy. The measured signal is then the number of atoms diffracted by the two pulses divided by the maximum number of atoms diffracted (i.e., without the second pulse). It can be seen in Fig. 5.13(c) that the signal oscillates faster for a larger mean field of a condensate with a higher number of atoms. Such a change in the oscillation frequency can be expected to occur in the presence of Optically Induced scattering length changes. Compared to the Bragg technique, this method yields a higher frequency resolution of mean field shifts simply with an increase in the measurement time Δt (Fig. 5.13(d)). Thalhammer et al. recently induced an optical Feshbach resonance via a coherent 82 Raman transition [122]. This stimulated Raman scheme offers a more stable control of the photoassociation light frequency and has a slightly less severe restriction on the light intensity. It can be achieved in our existing experiment by adding a second laser with a suitable frequency in order to couple the colliding atoms to a bound state in the ground molecular potential. 83 Figure 5.7: Experimental configuration for photoassociation. Shown are the CO2 laser FORT and the photoassociation beams. The Bragg (kicking) beams were aligned such that a horizontal standing wave was created along the long axis of the FORT. 84 0 500 1000 1500 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 Laser detuning(D/2p) in MHz Number of atoms a b c d e Figure 5.8: Photoassociation spectrum of the 1g(P3/2), v = 152 state. 0 MHz on this scale corresponds to a point 713 GHz below the 87Rb D2 line. The states a through e correspond to 2,−2, 3,−3⟩, 3,−2, 3,−3⟩, 1,−1, 3,−2⟩, 2,−1, 3,−2⟩, 1, 0, 1,−1⟩ of Ref. [118] labeled by the F, f, I, i⟩ quantum numbers. 85 −350 −300 −250 −200 −150 −100 −50 0 50 100 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 Laser detuning in MHz Number of atoms J=0 J=2 Figure 5.9: Photoassociation of the 0− g (∼ S1/2 + P3/2), v = 1 state showing the J=0 and J=2 rotational levels. Each point is separated by 5 MHz. 86 Figure 5.10: (a) First order Bragg diffraction as a function of the frequency difference between the beams used to create the standing wave. (b) Percentage of Bragg diffracted atoms for two photoassociation light detunings, Δ/2π =+10MHz (red) and 10MHz (blue). 87 −20 −15 −10 −5 0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 x 10−12 Kinel(cm3/s) detuning (MHz) Figure 5.11: Measured inelastic collision rate coefficients for the 0− g (P3/2), v = 1, J = 2 state. Also shown is a lorentzian fit to the data from which values of Γspon and Γstim are obtained. 88 −20 −15 −10 −5 0 5 10 15 20 25 97 98 99 100 101 102 103 scattering length (a 0 ) detuning (MHz) Figure 5.12: Plot of the scattering length a in units of a0 = 0.53˚A , the Bohr radius, calculated from the data in Fig. 5.11. The dashed line is the background scattering length of 100a0 and the detuning is with respect to the 0− g (P3/2), v = 1, J = 2 state. 89 Figure 5.13: Interference between condensates as a measure of mean field energy. Please refer to text for details. 90 CHAPTER 6 CONCLUSIONS 6.0.1 Summary The work covered in this study explored two features of a periodically kicked quantum system  transport and sensitivity to deviations from resonance. To prepare the required initial momentum state for these experiments, a BEC was created in an alloptical trap. Exposing it to short periodic kicks from an offresonant standing wave, with the correct choice of parameters, enabled investigation of momentum transport. The first system was a quantum δkicked accelerator. Specifically three higher order resonances of an accelerator mode were observed and their parameter dependence characterized. Two theoretical approaches were used to analyze the modes  one based on rephasing of momentum states separated by the order of the resonance. It was also seen that the dynamics could be explained as contributions from a family of rays of a fictitious classical system, obtained close to the resonance (ϵ → 0). A quantum ratchet was realized at a primary resonance (T1/2). The dependence of the directed ratchet current on the momentum of the initial state, or quasimo mentum, was investigated and explained by theory. A result of this quasimomentum dependence is that the finite momentum width of a BEC was found to suppress the ratchet current after some kicks. Measurement of the fidelity or overlap of a resonant QDKR state with an offresonant state was performed. The widths of the resultant peaks were found to scale as the inverse cube of the measurement time, in units of kicks. A theoretical analysis revealed a similar scaling behavior in the presence of acceleration and was validated 91 by experiment. Furthermore, velocity selection in the measurement process led to a reduced momentum width of the final zeroth order state compared to the initial state. Finally, photoassociation spectroscopy was performed on a 87Rb BoseEinstein condensate for the 1g and 0− g long range molecular states to realize an Optical Feshbach resonance. The observed loss spectrum was used to analyze the change in the elastic scattering length. 6.0.2 Future work The results described in this work lays out the groundwork for further refinements to the experiments and future research. It became clear during the search for Optical Feshbach resonances that extracting the scattering lengths near a photoassociation resonance is more involved than with a Magnetic Feshbach resonance. Bragg spectroscopy as a measure of the scattering length is time consuming, with the photoassociation losses adding to the difficulties. Recently it was shown that the benefits of the two could instead be combined: a robust magnetic Feshbach resonance in the presence of a boundbound photoassociation light provides the flexibility of the optical method with a minimized atom loss [123]. Some changes to our existing experimental setup are currently being undertaken to implement this technique. Once tunability of interatomic interactions is achieved, its effect on the resonances of the QDKR and QDKA can be probed. Complex optical potentials can be produced by overlapping standing waves of on resonant and off resonant light. Presence of a Feshbach resonances in the light will allow a new degree of control and absorptive and refractive indices of atoms diffracted by these potentials can be studied. Another model popular with quantum chaologists is the δkicked harmonic oscillator. Its phase space displays a stochastic web, the thickness of which is determined by the amount of underlying classical chaos. Identifying such a web in the quantum 92 version and exploring its properties can yield answers to questions on quantum chaos. Such a system can be implemented for instance by kicking the condensate while it is still in the trap. 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Raizen, “Atom optics realization of the quantum deltakicked rotor,” Phys. Rev. Lett. 75, 4598 (1995). [7] M. G. Raizen, F. L. Moore, J. C. Robinson, C. F. Bharucha, and Bala Sundaram, “An Experimental Realization of the Quantum deltaKicked Rotor,” Quantum and Semiclass. Optics 8, 687 (1996). [8] E. J. Galvez, B. E. Sauer, L. Moorman, P. M. Koch, and D. Richards, “Microwave Ionization of H Atoms: Breakdown of Classical Dynamics for High Frequencies,” Phys. Rev. Lett. 61, 2011 (1988). 94 [9] J. E. Bayfield, G. Casati, I. Guarneri, and D. W. Sokol, “Localization of Classically Chaotic Diffusion for Hydrogen Atoms in Microwave Fields,” Phys. Rev. Lett. 63, 364 (1989). [10] R. Bl¨umel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, “Dynamical localization in the microwave interaction of Rydberg atoms: The influence of noise,” Phys. Rev. A 44, 4521 (1991). [11] F. L. Moore, J. C. Robinson, C. F. Bharucha, P. E. Williams, and M. G. 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Title  Transport and Resonances in Kicked Boseeinstein Condensates 
Date  20101201 
Author  Talukdar, Ishan 
Keywords  Atom optics, BoseEinstein condensation, Precision measurements, Quantum chaos, Quantum ratchets, Quantum transport 
Department  Physics 
Document Type  
Full Text Type  Open Access 
Abstract  A purpose of this research was to study the scaling behavior of resonances of a periodically kicked quantum system. Specifically, a model known as a quantum δkicked rotor (QDKR) was implemented with a 87Rb BoseEinstein Condensate. Two variants of this system, a quantum δkicked accelerator (QDKA) and a quantum ratchet, were also realized experimentally and used to study quantum transport. Furthermore, experiments were undertaken to introduce meanfield interactions in these systems through an Optical Feshbach resonance. An overlap or fidelity measurement between a resonant and an offresonant state of the QDKR yielded a scaling that was dependent on the cube of the measurement time, in units of kicks. Such a subFourier behavior was also calculated to appear in the presence of an acceleration, and was verified experimentally. Quantum accelerator modes were observed in the QDKA and a detailed study of the parameter dependencies of three higher order resonances was done. The modes were explained with two models: a wavefront rephasing model and a quasi classical analysis. A quantum ratchet was realized at a QDKR resonance and the dependence of the directed momentum current on the momentum of the initial state investigated. Finally, photoassociation spectroscopy was performed on a <super>87</super>Rb BoseEinstein condensate for the 1g and 0g long range molecular states to realize an Optical Feshbach resonance. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  TRANSPORT AND RESONANCES IN KICKED BOSEEINSTEIN CONDENSATES By ISHAN TALUKDAR Master of Science University of Delhi Delhi, India 2003 Master of Science Oklahoma State University Stillwater, Oklahoma, USA 2006 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, 2010 COPYRIGHT c⃝ By ISHAN TALUKDAR December, 2010 TRANSPORT AND RESONANCES IN KICKED BOSEEINSTEIN CONDENSATES Dissertation Approved: Dr. Gil S. Summy Dissertation Advisor Dr. Girish S. Agarwal Dr. John W. Mintmire Dr. Nicholas F. Materer Dr. Mark E. Payton Dean of the Graduate College iii ACKNOWLEDGMENTS I would like to begin by expressing my gratitude to my advisor Gil Summy. Gil has taught me much of what I know of this field today, patiently answering my questions and showing me new, creative ways of looking at scientific problems. His enthusiasm and words of encouragement guided me during the difficult times and will remain invaluable in the years to come. In the lab it has always amazed and inspired me how Gil finds a way to simply make things work. I would like to thank my committee members, Prof. G. Agarwal, Prof. J. Mintmire, and Prof. N. Materer for their time, help and advice. Without the love and continued support from my parents Kanteswar and Mohini Talukdar, I would not be where I am today. They showed me the importance of aiming high and working hard. I wish to thank them and my sister Shruti for her cheer and motivation. I have had the good fortune of meeting some incredible people during my stay at OSU. Peyman had defended his thesis when I joined the group. An exceptional physicist and a great individual, he has remained a good friend and mentor throughout these years. Ghazal cheerfully introduced me to the experiments and helped me adjust to my new lab. Vijay, who was my roommate as well, explained the nitty gritties of the experiments and gladly fielded my incessant questions. It has been a nice experience working with Raj who joined our lab later and brought a refreshing way of doing things. Part of my physics family were our department staff members, Susan, Cindy, Stephanie, Danyelle, and recently Tamara. Besides their helpful and supportive attiiv tude, they are among the nicest people I have met in my life. Melissa has been a kind and understanding labcoordinator. Thanks also to Warren, our technical support. He is a one man army, handling all kinds of requests from the students. Mike and his staff at the machine shop have the special ability to bring any model design to life, only better. I would like to thank the physics faculty for providing me with the knowledge and guidance that has helped me pursue my research. Especially, I wish to thank Prof.Paul Westhaus, our former graduate coordinator and Prof. R. Hauenstein, our current graduate coordinator for their support during my years at the physics department. Thanks are also due to my former professors at the University of Delhi, Prof. Seshadri and Prof. Annapoorni for having shown me the path of research. Finally, I would like to express my thanks to all my friends at Stillwater, especially DJ (Deok Jin Yu), Prem and Amit. v TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 DeltaKicked Rotor : Theory and Experiments 5 2.1 Classical kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Atom optics δkicked rotor . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Quantum resonances and antiresonances . . . . . . . . . . . . 9 2.2.2 Dynamical Localization . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 BoseEinstein Condensation . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . 15 2.3.2 Limits of laser cooling . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . 19 3 Quantum transport with a kicked BEC 27 3.1 Highorder resonances of a quantum accelerator mode . . . . . . . . . 27 3.1.1 Rephasing theory . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.2 ϵclassical theory . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 A quantum ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 vi 4 SubFourier resonances of the kicked rotor 48 4.1 A fidelity measurement on the QDKR . . . . . . . . . . . . . . . . . . 49 4.1.1 Effect of ‘gravity’ . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Experimental Configuration and Results . . . . . . . . . . . . . . . . 53 5 Photoassociation of a 87Rb BEC 64 5.1 Photoassociative spectroscopy . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Ultracold Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 Scattering length and Feshbach Resonances . . . . . . . . . . . . . . . 69 5.3.1 Optical Feshbach Resonance . . . . . . . . . . . . . . . . . . . 73 5.4 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 CONCLUSIONS 91 6.0.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.0.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 BIBLIOGRAPHY 94 A Publications 108 vii LIST OF FIGURES Figure Page 2.1 Classical δkicked rotor phase space for different kick strengths K. Onset of stochastic regions can be seen in (c) while (d) is predominantly chaotic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 (a) and (b), A quadratic growth in the mean energy at a quantum resonance at the Talbot time, and (c) and (d), an oscillatory mean energy at an antiresonance at T = T1/2. . . . . . . . . . . . . . . . . 11 2.3 Quantum suppression of classical chaos. Simulation of the QDKR near the Talbot time with ϕd=3.0 shows the onset of dynamical localization after six kicks. Also shown is the classical momentum diffusion (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Evolution of thermodynamic quantities as a function of the trap truncation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 A schematic of the rubidium D2 transition. . . . . . . . . . . . . . . . 22 2.6 Setup of the optical table for the BEC experiments. Not shown is a final 80 MHz AOM for both the MOT Slave beams before the optical fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Alignment of beams for the MagnetoOptic Trap and the FORT inside the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 viii 3.1 Phase space of quantum accelerator modes generated by the map of (3.22) for τ = 5.744 and ϕd = 1.4. Mode (a) with (p, j) = (1, 0) for η = 2.1459. is a primary QAM. Higher order modes are seen in (b) with (p, j) = (2, 1), η = 2.766, and (c) (p, j) = (5, 1), η = 4.1801. . . . . 32 3.2 Setup for the kicking experiments. Two counterpropagating beams formed the standing wave oriented at an angle of 52◦ to the vertical. AOM2 was driven by an RF signal with a fixed frequency, ω/2π = 40MHz. 34 3.3 Quantum accelerator modes at (a) T=22.68 μs close to (2/3)T1/2, (b) T=17.1 μs which is close to (1/2)T1/2, and (c) T=72.4 μs close to 2T1/2. a value of g′=6 ms−2 was used in these scans. The arrows in (a) and (b) show orders separated by b~G which participate in the QAMs. Dashed lines correspond to the ϵclassical theory of Eq. (3.27) . . . . 36 3.4 Horizontally stacked momentum distributions across (a) (1/2)T1/2 for 40 kicks, ϕd = 1.4 and effective acceleration g′ = 6 ms−2; and (b) (1/3)T1/2 for 30 kicks, ϕd = 1.8, and g′=4.5 ms−2. The dashed curve is a fit to the theory in Eq. (3.27) . . . . . . . . . . . . . . . . . . . . 38 3.5 Initial momentum scans for QAMs near (a) (1/2)T1/2 (T=17.1 μs) for 30 kicks and g′=6 ms−2; and (b) (2/3)T1/2 (T=22.53 μs) with 40 kicks and g′=4.5 ms−2. The dashes indicate QAMs at the resonant β. . . . 39 3.6 Dependence of the mean momentum of the quantum ratchet on the offset angle γ for 5 kicks and β = 0.5. The dashed and solid lines represent Eqs. (3.33) and (3.36) respectively. The inset shows the offset γ created between the symmetry centers of the initial distribution (blue curve) and the kicking potential V (red curve). . . . . . . . . . . . . . 43 ix 3.7 The ratchet effect. This time of flight image shows growth of mean momentum with each standing wave pulse applied with a period of T1/2 and a maximum offset (γ = π/2) between the standing wave and the initial state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.8 Mean momentum as a function of kicks. The data and error bars are from experiments with ϕd = 1.4, γ = π/s and β = 0.5. The solid line is Eq. (3.33) while the dashed line corresponds to Eq. (3.36). . . . . 45 3.9 Change in mean momentum vs the quasimomentum β for ϕd = 1.4 and (a) γ = −π/2, (b) γ = π/2. Shown are the fits of Eqs. (3.33), dashed line and (3.36), solid line respectively to the experimental data (filled circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Momentum distributions of a sequence of 8 kicks of strength ϕd followed by a final πphase reversed kick of strength Nϕd, with a time period equal to the Talbot time 106.5μs. . . . . . . . . . . . . . . . . . . . . 54 4.2 (a) Horizontally stacked timeofflight images of a fidelity scan around the Talbot time. Each TOF image was the result of 5 kicks with ϕd = 0.8 followed by a πphase shifted kick at 5ϕd. (b) Mean energy distribution of the 5 kick rotor with the same ϕd. (c) The measured fidelity distribution (circles) from (a). The mean energy of the scan in (b) is shown by the triangles. Numerical simulations of the experiment for a condensate with momentum width 0.06 ~G are also plotted for fidelity (bluedashed line) and mean energy (redsolid line). The amplitude and offset of the simulated fidelity were adjusted to account for the experimentally imperfect reversal phase. . . . . . . . . . . . . . . 56 x 4.3 Experimentally measured fidelity (circles) and mean energy (triangles) widths (FWHM) as a function of (a) the number of pulses, and (b) the kicking strength ϕ˜d scaled to ϕd of the first data point. In (a), the data are for 4 to 9 kicks in units normalized to the 4th kick. Error bars in (a) are over three sets of experiments and in (b) 1σ of a Gaussian fit to the distributions. Dashed lines are linear fits to the data. Stars are numerical simulations for an initial state with a momentum width of 0.06~G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Variation of the fidelity peak width around β=0 as a function of kick number N(N +1)s = N(N +1)/20 scaled to the 4th kick. The straight line is a linear fit to the data with a slope of −0.92 ± 0.06. Error bars as in Fig. 2(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.5 (a) Momentum width of the reversed zeroth order state as a function of kick number. Error bars are an average over three experiments. (b) Optical density plots for the initial state (red,solid) and kick numbers 2 (magenta,dotdashed),4 (black,dotted), and 6 (blue,dashed) after summation of the timeofflight image along the axis perpendicular to the standing wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.6 Dependence of the acceleration resonance peak width as a function of the kick number in units scaled to the 4th kick. Error bars are over three sets of experiments. . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1 Schematic of a photoassociation process. Two atoms colliding along the ground state potential (S+S) absorb a photon and get excited to the (S+P) molecular potential. The excited molecule can subsequently decay to free atoms or a ground state molecule. . . . . . . . . . . . . 66 xi 5.2 Centrifugal energy term ~2l(l + 1)/2μr2 of the Hamiltonian for three partial waves, l=0,1,2. For low energy scattering all partial waves l >0 are blocked by the centrifugal barrier. . . . . . . . . . . . . . . . . . . 70 5.3 Variation of the scattering length a as a function of λ = √ mV0/~2. As the well depth V0 increases ((a) to (c)) a bound state is formed (dashed line) and the scattering length passes through a divergence and changes sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4 A Feshbach resonance occurs when an excited state has a bound state close to the collisional threshold. Changing the detuning Δ by an external field can couple the collisional to the bound state and change the scattering length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.5 Schematic of the optical setup for the photoassociation light. . . . . 77 5.6 Flowchart of the locking for the photoassociation master laser. . . . . 78 5.7 Experimental configuration for photoassociation. Shown are the CO2 laser FORT and the photoassociation beams. The Bragg (kicking) beams were aligned such that a horizontal standing wave was created along the long axis of the FORT. . . . . . . . . . . . . . . . . . . . . 84 5.8 Photoassociation spectrum of the 1g(P3/2), v = 152 state. 0 MHz on this scale corresponds to a point 713 GHz below the 87Rb D2 line. The states a through e correspond to 2,−2, 3,−3⟩, 3,−2, 3,−3⟩, 1,−1, 3,−2⟩, 2,−1, 3,−2⟩, 1, 0, 1,−1⟩ of Ref. [118] labeled by the F, f, I, i⟩ quantum numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.9 Photoassociation of the 0− g (∼ S1/2+P3/2), v = 1 state showing the J=0 and J=2 rotational levels. Each point is separated by 5 MHz. . . . . 86 xii 5.10 (a) First order Bragg diffraction as a function of the frequency difference between the beams used to create the standing wave. (b) Percentage of Bragg diffracted atoms for two photoassociation light detunings, Δ/2π =+10MHz (red) and 10MHz (blue). . . . . . . . . . . . . . . . 87 5.11 Measured inelastic collision rate coefficients for the 0− g (P3/2), v = 1, J = 2 state. Also shown is a lorentzian fit to the data from which values of Γspon and Γstim are obtained. . . . . . . . . . . . . . . . . . . . . . . 88 5.12 Plot of the scattering length a in units of a0 = 0.53˚A , the Bohr radius, calculated from the data in Fig. 5.11. The dashed line is the background scattering length of 100a0 and the detuning is with respect to the 0− g (P3/2), v = 1, J = 2 state. . . . . . . . . . . . . . . . . . . . . . 89 5.13 Interference between condensates as a measure of mean field energy. Please refer to text for details. . . . . . . . . . . . . . . . . . . . . . 90 xiii CHAPTER 1 INTRODUCTION Since its birth, quantum mechanics has stood the test of experiment. It has made remarkable progresses from theory to practice in the form of devices like atom interferometers, scanning tunneling microscopes and atomic clocks, to name a few. Despite the success of quantum theory, subtle issues still remain in our understanding. According to Bohr’s Correspondence Principle quantum mechanics reduces to Newtonian dynamics when the classical unit of action S becomes larger than the Planck’s constant ~. However, the exact nature and causes for this transition is an open subject for discussion. This is perhaps best seen in the problem of quantizing chaos, noticed by none other than Einstein [1]. The existence of classical chaos was first shown by the French mathematicianphysicist Henri Poincar´e. Attempting to find an analytical solution to the dynamics of three gravitationally interacting bodies, he discovered the possibility of irregular motion, where slight changes in initial conditions could lead to vastly different trajectories. This was chaos in a deterministic system, without any random parameters. The problem was revisited in the midtwentieth century. The result was the KolmogorovArnoldMoser (KAM) theorem [2]. It quantified the amount of perturbation necessary for an integrable system to develop chaotic motion. Carrying this concept of chaos over to the quantum realm runs into difficulties however. The overlap integral of two initially close wave packets undergoes unitary evolution to remain preserved over time. A better way of finding the quantum mechanical equivalent of chaos might be to look at the overlap of two similar states 1 evolving under slightly different Hamiltonians, as suggested by Peres [3]. These strategies for governing the evolution in phase space necessitate a deeper understanding of quantum transport in a classically chaotic system. One of the widely researched systems of quantum transport in momentum or energy space is the multiphoton ionization of Rydberg atoms under periodic microwave driving [4, 5]. This was the first demonstration of the δkicked rotor, a paradigm for studying classical and quantum chaos. Raizen’s group at Austin demonstrated a much more experimentally manageable version with atom optics. Here, cold atoms were subjected to periodic flashes from an optical standing wave [6, 7]. In general, onedimensional autonomous Hamiltonian systems are integrable as a result of the conservation of energy. There is therefore no chaos and the dynamics is regular. However, the energy periodically pumped into the kicked rotor breaks the timeinvariance of the Hamiltonian and makes chaotic dynamics possible in this one dimensional classical system. Surprisingly, the quantum kicked rotor, despite the inherent nonintegrability, was found to suppress classical chaotic motion. This was termed dynamical localization [6, 8–11], since it was reminiscent of Anderson localization in disordered solids. Decoherence introduced into the system for instance via spontaneous emissions or noise added to the kicking strength was found to destroy dynamical and restore classical diffusion [12–15]. Dynamical localization to this day remains an invaluable resource in investigating quantumclassical correspondence. Such atom optical systems have also proved their versatility in modeling other physical phenomena in the spirit of Feynman’s quantum simulator [16]. Cold atoms in accelerating optical lattices were found to exhibit Bloch oscillations, Wannier Stark ladders and tunneling, transport behavior normally associated with solid state physics [17–22]. Variants of the kicked rotor have enabled the demonstrations of a quantum accelerator and quantum ratchet [23–27]. The sensitivity of momentum transport to the underlying classical chaos in this system also offers possibilities for 2 precision measurements. A group in France observed subFourier resonances in a kicked rotor subject to two kicking frequencies [28]. Another study looked at the prospect of exploiting quantum rotor resonances to improve atom interferometric measurements of physical constants like the photon recoil frequency [29, 30]. The creation of a BoseEinstein condensate by Cornell and Wieman in 1995 heralded a new era of experiments [31, 32]. Unprecedented control over quantum state preparation and manipulation was now possible. A BEC has a momentum width less than that of a photon recoil. In the context of the quantum kicked rotor and accelerator, it allowed a close look at their phase spaces. The first BEC was achieved by evaporative cooling of already cold atoms in a magnetic trap. Condensates in our lab are produced in an optical trap, which has proved to be a simple yet robust method [33]. 1.1 Organization Two important aspects of the atom optics kicked rotor are investigated in this thesis  transport and resonance width scaling. Experiments were also conducted to observe an Optical Feshbach resonance in a 87Rb BoseEinstein Condensate, which could be useful for further kicked rotor research. To provide a theoretical and experimental backdrop for these studies, we begin with Chapter 2 which introduces the quantum deltakicked rotor and discusses properties exhibited by it and its variants. The laboratory realization of the quantum rotor is in the form of a BoseEinstein Condensate subjected to a periodically pulsed standing wave of light detuned far from any atomic resonances. An account of the theoretical background, from laser cooling techniques to evaporative cooling, necessary in achieving a phase space transition to a BEC is given. This is followed by a description of the experimental techniques which we employ in order to create such a condensate and perform experiments on it. Chapter 3 reviews two kinds of quantum transport behavior made possible by this 3 system. The first one details the observation and properties of quantum accelerator modes, specifically higher order resonances of such a mode. Two models, an interference model based on the rephasing of wavefronts, and a classicallike model can explain the occurrences of such modes and are discussed. The momentum evolution predicted by the two theories is then compared with experiment. Next we see how preparation of a special initial state of the rotor can establish a momentum current even in the absence of a net bias force. The directed current appears in the absence of dissipation and is an example of a quantum ratchet. Its realization will be described next. The effect of the finite momentum width of a condensate on the ratchet current is also investigated. The relatively nascent topic of applying the quantum kicked rotor towards precision measurements is the subject of Chapter 4. This system exhibits resonances in the mean energy at specific combinations of the kicking period and initial momentum. It is shown that the fidelity or quantum mechanical overlap of an offresonant state with one that is on resonance scales at a subFourier rate, with respect to the measurement time. The existence of resonances at particular values of the acceleration of the standing wave is also analyzed and verified by experiment. These resonances are also shown to have subFourier widths. Interactions among atoms in a BoseEinstein condensate can be strongly tuned near a Feshbach resonance. Presence of such interactions can introduce nonlinearity into the kicked rotor, thereby enabling a probe at quantum chaos. Chapter 5 details photoassociation spectroscopy performed on a BoseEinstein condensate for the 1g and 0− g long range molecular states. Bragg spectroscopy was employed to detect optically induced changes in the elastic scattering length near one of these molecular states, the sign of an Optical Feshbach resonance. Finally, the conclusions are laid out in chapter 6 including the outlook for future research. 4 CHAPTER 2 DeltaKicked Rotor : Theory and Experiments For more than three decades the delta kicked rotor has been at the centerstage in the study of quantum chaos. The relative ease with which it can be handled analytically has drawn many researchers to it. It is the basic model for studying dynamical chaos [34] and is described classically by the Standard Map. In this chapter we review this model and its quantum analogue. Along the way we examine some of the rich features this simple system has revealed so far. Finally, we look at the implementation of this model in our lab in an atom optics setting. 2.1 Classical kicked rotor The kicked rotor is a pendulum exposed to a periodic constant force like gravity. Its Hamiltonian is H = p2 2I + V0 cos(θ) Σ N δ(t − NT) (2.1) p is the angular momentum, θ the angular displacement, I the moment of inertia and V0 the kick strength. t is the time and N counts the kicks which are switched on at time intervals of T. Choosing I = 1, we can write Hamilton’s equation’s of motion as ∂H ∂θ = −p˙ = −V0 sin(θ)δ(t − NT) (2.2) ∂H ∂p = θ˙ = p Integrating Eqs. (2.2) over one period between t = N and t = N + 1, pN+1 = pN + V0 sin(θN) (2.3) θN+1 = θN + pN+1T 5 Using the rescaled variables ρ = pT and K = V0T, (2.3) can be written as ρN+1 = ρN + K sin(θN) (2.4) θN+1 = θN + ρN+1 This is known as the Standard or TaylorChirikov Map where the dynamics is completely determined by the ‘stochasticity parameter’ K [35]. As seen in the ρ − θ phase space (modulo 2π) of Fig. 2.1, the islands of stability at small K give way to chaotic regions with increasing K. Numerical analysis showed that the transition to global stochasticity occurs at a value of K ≈ 0.9716 [36]. At the end of ‘t’ kicks, ⟨(ρt − ρ0)2⟩ = K2 Σt−1 q,r (sin θq sin θr) (2.5) Ignoring the correlation terms for large K, we get ⟨ρ2t ⟩ ∼= 1 2 K2t (2.6) Thus the chaotic domain of the classical δkicked rotor is characterized by a diffusive growth of the mean energy. 2.2 Atom optics δkicked rotor Study of the quantum version of the δkicked rotor was made possible by remarkable advances in atom optics. Its experimental realization can be described in terms of the center of mass Hamiltonian of an atom of mass M exposed to short periodic pulses of far detuned light [6, 7, 37], ˆH = ˆ P2 2M + ~ϕd cos(G ˆX ) Σ∞ N=−∞ δ(t′ − NT) (2.7) G = 4π/λ is the grating vector of the standing wave formed from a laser of wavelength λ. It is convenient to convert this Hamiltonian to dimensionless units. This can be 6 −pi 0 pi −pi 0 pi K=0.5 q r −pi 0 pi −pi 0 pi K=0.9 q r −pi 0 pi −pi 0 pi K=1.5 q r −pi 0 pi −pi 0 pi K=5 q r (a) (c) (b) (d) Figure 2.1: Classical δkicked rotor phase space for different kick strengths K. Onset of stochastic regions can be seen in (c) while (d) is predominantly chaotic. 7 done by expressing position in units of G−1, ˆx = G ˆX and momentum in units of ~G, ˆp = ˆ P/~G. Hˆ = ˆp2 2 + ϕd cos(ˆx) Σ N δ(t − Nτ ) (2.8) The scaled period is τ = 2πT/T1/2, time is measured in t = 2πt′/T1/2 and Hˆ = (M/(~G)2) ˆH . T1/2 = 2πM/~G2 is known as the halfTalbot time, the physical significance of which we shall soon see. The periodicity of the potential provides the connection between the particle propagating along it and the kicked rotor. The position of the particle can be folded into an angular coordinate θ = x mod(2π). From a quantum mechanical perspective, the periodicity of the potential allows us to use Bloch’s theorem on the atomic deBroglie wave. The solutions are then invariant under translations by one period of the potential [38]. A result of this invariance is the conservation of the particle quasimomentum. In terms of photon exchange, each atom absorbs a photon from one of the standing wave beams followed by its subsequent stimulated emission into the other beam. The net result is that the atom momentum changes by two photon recoils (2~k or ~G) while leaving the fractional part of its momentum (or the quasimomentum, in units of ~G) unchanged. The light shift of the atomic ground state in the presence of this optical potential is ΔEg = ~Ω2 4δ cos2(x/2) (2.9) where Ω ≡ −eE0 ~ ⟨erg⟩ is the Rabi frequency between the excited and ground states. E0 is the amplitude of the potential and it is detuned δ from the atomic transition. This sinusoidal energy shift leads to the standing wave acting as a phase grating on the atomic wave function. The strength of the kicks is therefore given by the phase modulation depth parameter, ϕd = Ω2Δt/8δ. Change in the kinetic energy of the atoms is negligible during the short time the potential is on (RamanNath regime). In this limit, the imprint of the thin phasegrating on an incident plane wave 0⟩ 8 results in a final state ψ(Δt)⟩ = e−iϕd cos(ˆx)0⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)e−inˆx0⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)n⟩ (2.10) where the second line derives from the JacobiAnger expansion, Jn is the nth order Bessel function of the first kind, and n⟩ is a momentum order along the grating in units of ~G. After a free evolution for a period T, the wave function is ψ(t)⟩ = Σ n (−i)nJn(ϕd)e−i ^p2 2 τ n⟩ (2.11) 2.2.1 Quantum resonances and antiresonances We can now write the one period evolution operator as ˆU = e−iϕd cos(ˆx)e−i ^p2 2 τ . (2.12) When the time period τ is an integer multiple of 4π, the free evolution factor is unity. N consecutive kicks of strength ϕd are then equivalent to one kick of strength Nϕd. That is, ˆU N = e−iNϕd cos(ˆx). (2.13) From the initial state 0⟩, we then have a momentum distribution pn = ⟨nUN0⟩2 = J2 n(Nϕd) (2.14) The mean energy at the end of N kicks is then ⟨E⟩ = Σ∞ n=−∞ n2pn = Σ n n2J2 n(Nϕd) = 1 2 N2ϕ2 d (2.15) 9 This uniquely quantum effect, characterized by the quadratic growth of energy, is termed a quantum rotor resonance. It is the temporal equivalent of the Talbot effect in optics and the resonant time period is therefore called the Talbot time, TT = 4πM/~G2 [6, 39]. Next we look at a zero initial momentum state, exposed to a kick, and allowed to evolve for half of the Talbot time (T1/2). Now, τ = 2π (or any odd integer multiple of it), and the phases acquired by the momentum orders are +1 or 1 depending on whether the order is even or odd. The state at the end of the halfTalbot time is, ψ(t = T1/2)⟩ = Σ∞ n=−∞ (−i)nJn(ϕd)e−iπn2 e−inˆxψ0⟩ = eiϕd cos(ˆx)ψ0⟩ (2.16) where we have used the property, e−iπn2 = e−iπn. A kick applied at this point will cancel out the spatial variation due to the first kick recreating the original state. This phenomenon is known as a quantum antiresonance. A numerical simulation of these two effects can be performed using a technique shown in Ref. [39]. Beginning with a zero momentum initial state (convoluted by a finite width Gaussian to account for the BEC momentum spread), the quantum deltakicked rotor is realized by repeated application of the one period evolution operator of (2.12). Figure 2.2 shows a simulated quantum resonance and antiresonance. 2.2.2 Dynamical Localization We now turn our attention to the generic behavior of the quantum delta kicked rotor, away from these resonances, that is when τ/2π is an irrational number. In Section 2.1 we saw how strong chaos dictates a diffusive energy growth of the classical kicked rotor. In order to investigate the behavior of its quantum counterpart, Casati, Chirikov, Ford and Izrailev [40, 41] in 1979 simulated the quantum kicked rotor. Contrary to expectations, however, they were surprised when the quantum rotor showed corre 10 0 5 10 −30 −20 −10 0 10 20 30 kicks Momentum Resonance 0 5 10 0 20 40 60 80 100 kicks Mean Energy 0 5 10 −30 −20 −10 0 10 20 30 kicks Momentum Anti−resonance 0 5 10 0 0.2 0.4 0.6 0.8 1 kicks Mean Energy (a) (b) (c) (d) Figure 2.2: (a) and (b), A quadratic growth in the mean energy at a quantum resonance at the Talbot time, and (c) and (d), an oscillatory mean energy at an antiresonance at T = T1/2. 11 spondence with the classical case only upto a certain time even in the deeply chaotic domain (ϕd ≫1). After this, quantum interference was found to suppress any further growth of momentum [42]. This phenomenon came to be known as dynamical localization, and was shown to be analogous to Anderson localization in disordered solids [34, 43]. In the rotor, destructive interference between momentum orders separated by irrational multiples of 2π leads to a final eigenstate exponentially localized in momentum. 2.2.3 Quantum Transport While the mean energy of the QDKR increases ballistically with kicks at a quantum resonance, the mean momentum ⟨p⟩ remains fixed. Presence of a linear potential like gravity breaks the symmetry of the system. The momentum acquired by the qth order at the end of N kicks is mvi +q~G+mgNT. The phase acquired by the momentum state q⟩ therefore has a gravity dependent term, ϕq = viGTq + ~G2 2m Tq2 + gGT2Nq. When the phase difference between q⟩ and q − 1⟩, ϕq − ϕq−1 = ~G2 2m T(2q − 1) + viGT + gGT2N (2.17) is a multiple of 2π, order q − 1⟩ can be perfectly coupled to q⟩ by the next kick. This imposes two conditions on rephasing, one which depends on the kick number and another which does not. ~G2 2m T2q + gGT2N = 2πql′ (2.18) viGT − ~G2 2m T = 2πl (2.19) Equation (2.18) can be solved to find the momentum at the end of N kicks, q = N γ α2 l′ − α (2.20) where α = T/T1/2 and γ = ~2G3/2πm2g. Therefore, near a resonance, a fixed momentum can be imparted to a section of the atoms by each kick. This is a quan tum accelerator mode created with the quantum delta kicked accelerator (QDKA). 12 0 2 4 6 8 10 0 5 10 15 20 25 30 35 40 45 kicks Mean energy (Er) Figure 2.3: Quantum suppression of classical chaos. Simulation of the QDKR near the Talbot time with ϕd=3.0 shows the onset of dynamical localization after six kicks. Also shown is the classical momentum diffusion (solid line). 13 Among many possibilities, its use as a coherent beam splitter for an interferometer has brought it a great deal of attention among researchers since its discovery almost a decade ago in Oxford [44]. Another topic of recent focus in rectified transport is a ratchet, where a directed current of particles along a periodic potential can be established even in the absence of a biased force. A pure quantum ratchet was first demonstrated at a QDKR resonance by Sadgrove et al. [45]. It was realized with an initial state prepared in a superposition of a zeroth and first order momentum states (in units of ~G), ψ⟩i = √1 2 [0⟩ + 1⟩]. (2.21) After a time ‘t’ of free evolution, the first order accumulates a phase θ, ψ(t)⟩ = √1 2 [0⟩ + eiθ1⟩]. (2.22) The state after application of a kick become, ψ(t+)⟩ = e−iϕd cos(ˆx)ψ⟩i = √1 2 Σ q (−i)q[Jq(ϕd) + eiθJq−1(ϕd)]q⟩ (2.23) with a distribution, pq = 1 2 [Jq(ϕd)2 + Jq−1(ϕd)2 + 2 cos θJq(ϕd)Jq−1(ϕd)] (2.24) Recalling the Bessel function property, J−n(x) = (−1)nJn(x), we notice that the momentum distribution is asymmetric whether q is odd or even. This is a ratchet current, the strength of which can be tuned by the relative phase θ. Experiments on these and other exotic properties of the δkicked rotor depend on an initial momentum state with a spread that is less than one photon recoil. The ideal atomic physics system for this is a BoseEinstein condensate. The next sections outline the stages in reaching this quantum state of matter and its realization in our lab. 14 2.3 BoseEinstein Condensation In 1924, Satyendranath Bose proposed a statistical technique to evaluate the photon black body spectrum [46]. Einstein extended the theory to the general case of identical particles leading to the birth of BoseEinstein statistics [47]. The distribution function for particles obeying this statistics is N(E) = 1 eβ(E−μ) − 1 (2.25) where β = 1/kBT and μ is the chemical potential. Einstein noticed a peculiarity, that below a certain critical temperature, these bosonic atoms would accumulate in the lowest energy quantum state: the onset of the BoseEinstein Condensate (BEC) phase. In terms of the phase space density, ρ = nλ3 dB, where n is the particle number density, a BEC phase transition happens when ρ = ζ( 3 2 ) = 2.612. In other words once a particle’s thermal deBroglie wavelength λdB = ( 2π~2 mkBT )1/2 becomes greater than the interparticle separation, λdB > n−1 3 , this macroscopic quantum state starts appearing. A subject of academic curiosity for more than half a century, achievement of a BEC was given serious thought when laser cooling of atoms was realized. This cooling scheme is based on using optical forces to reduce the thermal velocity distribution of atoms. An indepth discussion of BEC in the Summy lab at OSU can be found in the theses of Ahmadi, Timmons, and Behin Aein [48–50]. I shall therefore only outline the general principles involved. 2.3.1 Laser Cooling and Trapping The light force on an atom in general comprises of two parts, F = Fdip + Fsc, a conservative dipole force, Fdip and a dissipative scattering force Fsc. The science of laser cooling was built primarily on the second kind, which is a result of absorption 15 of photons by atoms followed by spontaneous emission. This force is expressed as Fsc = ~kγρee (2.26) where ~k is the momentum transferred by a photon, γ is the rate of decay of the excited state of the atom and ρee is the probability for the atoms to be in the excited state. Evaluating ρee using the optical Bloch equations, one can show that, Fsc = ~k s0γ/2 1 + s0 + (2δ/γ)2 . (2.27) Here, s0 = 2Ω2/γ2 is the onresonance saturation parameter and δ = ωl − ωa is the detuning of the light from the atomic transition. We now consider the case of an atom with velocity ⃗v placed in the light field of two beams counterpropagating along the zaxis. In addition we arrange a linearly inhomogeneous magnetic field B = Az formed by a magnetic quadrupole field. In the limit of low light intensity, the total force on the atom due to the two beams is, ⃗F = ⃗F+ + ⃗F−, where ⃗F± = ~⃗k s0γ/2 1 + s0 + (2δ±/γ)2 . (2.28) The detuning is δ± = δ ∓ ωD ± ωZ, where we now have to include the Doppler shift ωD = ⃗k.⃗v and the Zeeman shift ωZ = μ′B/~, μ′ being an effective magnetic moment [51]. In the limit of Doppler and Zeeman shifts that are small compared to δ, we arrive at the total force, ⃗F = −β⃗v − μ′A ~k β⃗r (2.29) which is the motion of a damped harmonic oscillator with the damping constant β = − 8~k2δs0 γ{1+s0+(2δ/γ)2}2 . Thus with a configuration of three retroreflecting beams tuned below the atomic resonance, the Zeeman shift provides a confining potential for the atoms creating what is known as the MagnetoOptic Trap (MOT). The presence of a viscous damping force due to the Doppler effect leads to a significant reduction in the velocity distribution of the atoms forming an “Optical Molasses” [52, 53]. 16 2.3.2 Limits of laser cooling There is a limit to the temperatures that can be obtained in an optical molasses due to recoil heating. This effect results from a diffusion of the atoms in momentum space set off by the random nature of the photon scattering events. A steady state is reached when molasses cooling equals recoil heating which determines the limiting Doppler temperature, TD, TD = ~γ 2kB . (2.30) For 87Rb the Doppler temperature is 146 μK. Surprisingly, in one of the early experiments with Na atoms, temperatures 10 times lower than the Doppler temperature were observed [54]. Later, a theory which included the multilevel structure of the atomic states and the optical pumping among these sublevels was able to explain these subDoppler temperatures [55, 56]. This process became known as polarization gradient cooling. The next limit to the laser cooling temperature is set by the energy associated with a photon recoil, Er = ~2k2/2m. The recoil limit temperature, Tr = ~2k2 mkB (2.31) has a value of 360 nK for 87Rb. 2.3.3 Evaporative cooling The maximum phase space density possible with laser cooling is 10−5 − 10−4. Cold atom physicists soon realized that in order to increase it any further, evaporative cooling would be the way to go. Originally proposed by Hess [57] for atomic hydrogen, the method is based on the preferential removal of high energy atoms from a confined sample, followed by rethermalization of the remaining atoms by elastic collisions. The simultaneous decrease in the temperature and the volume leads to an increase in the phase space density. 17 Several models have been developed to explain the process of evaporative cooling. These include analytical and numerical treatments by Doyle and coworkers [58], Luiten et al., and Wu and Foot [60] among a few. We shall here focus on a simple yet highly instructive analytical model due to Davis et al. [59]. In this model the trap depth is lowered in one single step to a finite value ηkBT and the effect of removal of high energy atoms on the thermodynamical quantities is calculated. The remaining fraction of atoms is ν = N′/N. The decrease in temperature caused by the release of the hot atoms can be defined by the quantity, γ = log(T′/T ) log ν . (2.32) In a ddimensional potential, U(r) ∝ rd/ξ and the volume V ∝ Tξ [61]. The value of ξ describes the type of the potential. For a linear potential like a spherical quadrupole trap, ξ = 3 and for a harmonic potential as in an optical trap, ξ = 3/2. We thus have the scaling of the important thermodynamic quantities, N′ = Nν, T′ = Tνγ, and V ′ = V νγξ. The phase space density ρ = nλ3 dB scales as ρ′ = ρν1−γ(ξ+3/2). With the knowledge of ξ, ν(η), and γ(η) one can track the evolution of these quantities with the lowering of the trap depth. The density of states for atoms in a trapping potential U(x, y, z) is D(E) = 2π(2M)3/2 ~3 ∫ V √ E − U(x, y, z)d3r (2.33) The fraction of atoms which remain in the trap after its depth has been decreased to ηkBT is ν(η) = 1 N ∫ ηkBT 0 D(E)e−(E−μ)/kBTdE. (2.34) The occupation number is given by the MaxwellBoltzmann distribution e−(E−μ)/kBT since the effects of quantum statistics can be neglected for a dilute gas. μ is the chemical potential of the gas. We can write Eq.(2.34) as, ν(η) = ∫ η 0 Δ(ϵ)e−ϵdϵ (2.35) 18 where ϵ = E/kBT is the reduced energy and the reduced density of states is Δ(ϵ) = ϵ1/2+ξ Γ(3/2 + ξ) (2.36) After truncation, the total energy of the atoms is α(η)NkBT where α(η) = ∫ η 0 ϵ Δ(ϵ)e−ϵdϵ. (2.37) Therefore the average total energy per atom(in units of kBT) is α(η)/ν(η). Before truncation (η → ∞) this quantity is α(∞)/ν(∞) = (3/2 + ξ)/1. The decrease in temperature is thus T′ T = α(η)/ν(η) α(∞)/ν(∞) . (2.38) Using Eq.(2.32) we now have, γ(η) = log( α(η) ν(η)α(∞) ) log[ν(η)] . (2.39) Solving Eqs. (2.35), (2.36), and (2.37) for a specific form of potential, one can determine γ(η). For ξ = 3/2, ν(η) = 1 − 2+2η+η2 2eη and α(η) = 3 − 6+6η+3η2+η3 2eη . Figure 2.4 shows the dependence of the number of atoms, temperature, density and phase space density on the normalized truncation parameter ˜η = η 3/2+ξ for ξ = 3/2 and ξ = 3. It can be seen that for the same truncation, ˜η, a higher phase space density is achieved with a larger ξ due to a faster shrinking of volume with decreasing temperature (V ∝ Tξ). 2.4 Experimental Configuration Ever since the first BoseEinstein condensate [31], rubidium 87 has been an atom of choice, due to its large elastic cross sections and the convenience of trapping it with inexpensive diode lasers. The light needed to trap and cool these atoms is tuned close to the Rb87 F=2 → F′=3 transition shown in the D2 hyperfine structure in Fig. 2.5. In our experiment this light was derived from a grating stabilized Toptica 19 0 1 2 3 0 2 4 6 8 10 ˜´ phase space density linear harmonic 0 1 2 3 0 0.5 1 1.5 2 2.5 ˜´ density (n¢/n) 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ˜´ number of atoms (N¢/N) 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ˜´ temperature (T¢/T) Figure 2.4: Evolution of thermodynamic quantities as a function of the trap truncation parameter. 20 DL100 diode laser with a 1 MHz linewidth and 15 mW of output power. We call this our master laser. Using saturated absorption spectroscopy it was locked to the F=2 → F′=2 and F′=3 crossover line. This light was injected into a homemade temperature stabilized slave laser with an output of 110 mW as shown in Fig. 2.6. Three frequency detunings of the light are needed in the experiment: 20 MHz for the MOT, 90 MHz for molasses cooling, and 0 MHz for imaging. To enable fast switching between these, the output from this main slave was sent through an AcoustoOptic Modulator (AOM) in a doublepass configuration. The frequency adjusted light was finally injected into a series of two slave lasers used for the actual trapping and cooling of atoms. Due to nonresonant excitation to the F′=2 state, some of the atoms can fall down to the F=1 ground state and get out of the trapping cycle. To bring them back into the MOT, a repumping laser tuned to the F=1→F′=2 state is used. Control over the intensity of this light is also crucial during the FORT loading stage and is adjusted by an AOM. All of these optics are placed in an isolated table as seen in Fig. 2.6. The trapping and repump beams are coupled through two fibers to the table containing the vacuum chamber. The vacuum chamber consisted of an octagonal multiport chamber attached to a flange of a sixway cross. An ion pump attached to another flange kept the vacuum pressure at 10−11 Torr. After exiting the fiber on the second table, the MOT light is divided into three beams, circularly polarized and diameters expanded to 2.2 cm. These are then sent into the vacuum chamber from three orthogonal directions and then retroreflected with opposite circular polarization. The MOT is formed at the intersection of these six beams. The inhomogeneous magnetic field required to create the confining potential of the MOT was formed using two sets of water cooled coils arranged in an anti Helmholtz configuration. A magnetic field gradient of 10 Gauss/cm along the axial direction was obtained with the coil geometry of our experiment. Around 10 million 21 Figure 2.5: A schematic of the rubidium D2 transition. 22 Figure 2.6: Setup of the optical table for the BEC experiments. Not shown is a final 80 MHz AOM for both the MOT Slave beams before the optical fibers. 23 atoms were trapped in this MagnetoOptic trap. Finally, stray magnetic fields in the region of the MOT were removed with three pairs of Helmholtz coils, commonly known as the nulling coils. A 50W CO2 laser formed the Far OffResonant Trap (FORT) for the evaporative cooling stage. With a wavelength of 10.6μm, the laser is far detuned from the atomic resonance and its effect on the atoms can be considered as that of a static electric field. An atom with a ground state polarizability αg placed in the electric field E(x, y, z) of the Gaussian beam given by E2 = E2 0 exp[ −2(x2+y2) w2 0(1+( z zR )2) ] 1 + ( z zR )2 . (2.40) experiences a potential U = 1 2αgE2. Here w0 is the beam waist and zR = πw2 0/λ is the Rayleigh length. The focus of the beam where z << zR, serves as a harmonic trapping potential U = −1 2 αgE2 0(1 − 2x2 w2 0 − 2y2 w2 0 − z2 z2R ). (2.41) The output of the CO2 laser passed through a 40 MHz RFdriven AOM. The 35W first order from the AOM overlapped the MOT for a loading time of typically 20 seconds. The MOT light detuning was then increased to 90 MHz for optical molasses cooling. This was followed by reduction of the repump intensity to produce a temporal dark MOT. During this crucial step, atoms start entering into a state that is “dark” to the cooling light. The resultant decrease in the recoil heating and excited state collisions leads to an increase in the phase space density. After around 100ms, the MOT and repump beams were extinguised and the magnetic coil current switched off. The FORT beam was transported into the vacuum chamber by a three lens assembly: the first two lenses formed a 2x beam expander followed by a third focussing lens mounted inside the chamber. The final spot size at the center of the MOT was w0 = λf/(πR), where f = 38.1mm is the focal length of the third lens and R is the 24 radius of the beam incident on it. The second lens of the beam expander was mounted on a translation stage (Fig. 2.7). Moving this by 15 mm in 1 s increased the beam size R thereby compressing the FORT. A tightly focussed trap was essential to increase the elastic collision rate and enhance the evaporative cooling. The cloud of around a million ultracold atoms was then subjected to forced evaporative cooling, where the trap depth was lowered at an exponential rate by reducing the FORT beam intensity. This was done by decreasing the RF power driving the CO2 AOM. After 5 s and at a final laser power of 40 mW, a pure condensate of around 30,000 atoms in the 5S1/2, F=1 state was obtained. To image the BEC, it was released from the FORT and pulsed with the repump beam at full intensity to pump the atoms to the F=2 state. After 8 ms of expansion, a 100ns pulse of light tuned to the resonant F=2 → F′=3 transition cast a shadow of the falling condensate on a CCD camera. Subtracting a reference image (without the condensate) from this gave the final distribution of the BEC. Before we conclude it must be mentioned that aspects of the experimental setup for kicking and photoassociation are explained in their respective chapters. 25 Figure 2.7: Alignment of beams for the MagnetoOptic Trap and the FORT inside the vacuum chamber. 26 CHAPTER 3 Quantum transport with a kicked BEC Unraveling the details of transport in solid state systems is important for fundamental physics and is becoming particularly relevant for new nanoscopic devices [62, 63]. Atom optic systems offer an easily configurable and ‘clean’ system to study many aspects of it. In this chapter we investigate two models of directed transport in a quantum Hamiltonian. Section 3.1 is on higher order resonances of a quantum accelerator mode, which was discussed in the previous chapter. First, two theoretical approaches of explaining these resonances are described. These are then used to analyze experimental results. In section 3.2, the concept of a quantumresonance ratchet is discussed. An expression of generalized momentum current is derived. Our realization of such a ratchet is then compared to this theory and the effect of the various experimental parameters studied. 3.1 Highorder resonances of a quantum accelerator mode 3.1.1 Rephasing theory Presence of a linear potential like gravity along the grating leads to the quantum kicked accelerator. In dimensionless units, ˆH = ˆp2 2 − η τ ˆx + ϕd cos(ˆx) Σ∞ Np=0 δ(t − Npτ ). (3.1) Here, η = mg′T/~G is the unitless ‘gravity’ g′. As seen in the previous chapter, near multiples of the halfTalbot time T/T1/2, a group of atoms can acquire a fixed momentum with each kick. In general, such accelerator modes can exist near rational 27 fractions of the Talbot time, that is when τ = 2π(a/b). As in section 2.3, and using the definitions of τ and η, the phase acquired by the qth order is ϕq = τ q(pi + ηNp + 1 2 q) (3.2) where pi = Pi/~G is the scaled initial velocity. For these higher order resonances, rephasing occurs between momentum orders separated by b~G [64]. We can again divide the phase difference between these orders, ϕq − ϕq−b = τ b[pi + ηNp + 1 2 (2q − b)] (3.3) into two parts, τ b(q + ηNp) = 2πqa (3.4) τ b(pi − b/2) = 2πl (3.5) the first of which is dependent on the pulse number. Solution of Eq. (3.4) gives the momentum acquired by the accelerator mode, q = − ητ τ − 2πa b Np (3.6) For a primary resonance, (a = l′b), the above expression reduces to Eq. (2.20) as expected. Equation (3.5) also imposes a condition on the initial momentum of the atoms in the mode, pi = 2πl τ b + b 2 (3.7) 3.1.2 ϵclassical theory Fishman, Guarneri, and Rebuzzini (FGR) in 2002 proposed a theory where the detuning, ϵ, of the pulse period from resonance plays the role of Planck’s constant. The map obtained in the classical (or correctly, ϵclassical, ϵ → 0) limit of this system was able to successfully explain quantum accelerator modes. It further predicted the existence of higher order modes and spurred the experimental search for them [68]. 28 We review this theory here. The one period evolution operator or the Floquet operator for the Hamiltonian (3.1) can be written as ˆU = ˆK ˆ F = e−iϕd cos(ˆx) ˆ F (3.8) The matrix elements of this operator are evaluated in a basis of eigenfunctions of a particle falling under gravity, uE(p) = ⟨pE⟩, uE(p) = ( τ 2πη )1/2 ei τ η (Ep−p3 6 ) (3.9) The matrix elements of the free evolution operator are, ⟨p′ ˆ Fp′′⟩ = ∫ dEe−EτuE(p′)u∗ E(p′′) = δ(p′ − p′′ − η)e−i τ 2 (p′−η 2 )2 (3.10) The kick operator ˆK reads, ⟨pe−iϕd cos(ˆx)p′⟩ = Σ∞ n=−∞ (−i)nJnδ(p − p′ − n) (3.11) Thus we have the propagator, (ˆU ψ)(p) = ∫ dp′⟨pˆU p′⟩⟨p′ψ⟩ = Σ∞ n=−∞ (−i)nJne−i τ 2 (p−n−η 2 )2 ψ(p − n − η) (3.12) The presence of the linear term in the Hamiltonian (3.1) breaks the kicking potential’s periodicity, and quasimomentum is no longer conserved. By means of a gauge transformation, spatial periodicity and therefore conservation of the particle quasimomentum, β, can be restored, ˆH g(t) = 1 2 ( ˆp + η τ t )2 + ϕd cos(ˆx) Σ∞ Np=−∞ δ(t − Npτ ). (3.13) 29 This amounts to writing the state in a momentum basis falling freely with the particle, ψg(p,Np) = ⟨p + ηNpˆU Np ψ⟩ (3.14) where ηNp is the momentum gained due to gravity. The propagator (3.12) in this frame reads, ˆU g(Np) = e−iϕd cos(ˆx)e−i τ 2 (ˆp+ηNp+η 2 )2 (3.15) The kickedrotor We can write the Bloch eigenstates for the above periodic potential as eiβxψβ(x), where ψβ(x) is 2π periodic in x. Introducing an angular coordinate θ = x mod(2π), ψβ(θ) can be considered a fictitious rotor (also called a βrotor). The angular momentum representation is related to the kicked particle by, ⟨nψβ⟩ = ⟨n + βψ⟩ (3.16) In the θrepresentation, ⟨θψβ⟩ = √1 2π Σ n ⟨n + βψ⟩einθ (3.17) The one kick evolution operator for ψβ⟩ is ˆU β = e−iϕd cos(ˆθ)e−i τ 2 (Nˆ+β+ηNp+η 2 )2 (3.18) where the angular momentum operator Nˆ = −i d dθ is related to the particle momentum by pˆ = Nˆ + β. Dynamics near a Resonance: ϵClassical treatment Let us now investigate the rotor dynamics near a resonance and define ϵ = τ − 2πl (ϵ << 1). With ˜k = ϵϕd, and using e−iπln2 = e−iπln, the evolution operator (3.18) becomes, ˆU β(Np) = e −i ~k ϵ cos ˆθe − i ϵ Hˆβ(ˆI,Np). (3.19) 30 Here we have defined Hˆβ(ˆI, t) = 1 2 ϵ ϵ (ϵ)ˆI2+ ˆI[πl+τ (β+Npη+ η 2 )] with Iˆ= ϵNˆ = −iϵ d dθ (3.20) If ϵ in Eq. (3.19) is treated as a Planck’s constant, (3.19) would follow from quantization of the following classical (timedependent) map, INp+1 = INp + ˜k sin(θNp+1) θNp+1 = θNp + ϵ ϵINp + πl + τ (β + Npη + η 2 ) (3.21) In other words, as ϵ → 0, the quantum rotor dynamics can be described in terms of ‘classical rays’ along trajectories of (3.21). This limit has been dubbed “ϵclassical” to distinguish it from the actual ~ → 0 classical limit. We can define JNp = INp + ϵ ϵ [πl + τ (β + Npη + η 2 )] and remove the explicit time dependence, JNp+1 = JNp + ˜k sin(θNp+1) + ϵ ϵτη θNp+1 = θNp + ϵ ϵJNp (3.22) These describe stable periodic orbits on a 2torus parametrized by J and θ (mod2π). If the orbit has periodicity p (measured in kicks), then Jp = J0 + 2πj, θp = θ0 + 2πn (3.23) j, called the jumping index, gives the number of times the orbit has wound around itself in the J direction. Figure shows phase space maps of Eq. (3.23) for a primary and two higher order modes. After Np cycles, the physical momentum qNp = I/ϵ along this orbit is [67] qNp ≃ n0 − ητ ϵ Np + 2π j pϵNp (3.24) where we have used Eq. (3.23) and the definition of J. These are trajectories surrounding a ‘stable fixed point’. Together they form an island of stability in the phase space. A wave packet starting with a sizable overlap with such an island grows linearly in momentum with the pulse number (3.24), and forms an ‘accelerator mode’. The mode is characterised by the parameters (p, j). 31 Figure 3.1: Phase space of quantum accelerator modes generated by the map of (3.22) for τ = 5.744 and ϕd = 1.4. Mode (a) with (p, j) = (1, 0) for η = 2.1459. is a primary QAM. Higher order modes are seen in (b) with (p, j) = (2, 1), η = 2.766, and (c) (p, j) = (5, 1), η = 4.1801. 32 ϵclassical theory of higher order resonances Higher order resonances appear near τ = 2πa/b where a, b are coprime integers. Near a higher order resonance, we define ϵ = τ −2πa/b. However, in general, no ϵclassical limit exists near these higher order times. Nevertheless, it was shown in Ref. [65] that the QAM’s can still be generated, not by a single classical map, but by bundles of “classical rays” which follow the trajectories, θNp+1 = θNp + INp + τ (δβ + Npη + η 2 ) + 2πSNp/b INp+1 = INp − ˜k sin(θNp+1 + 1) (3.25) where δβ = β − βr gives the closeness to resonant βr. The integers SNp can take integer values between 1 and b and are arbitrary. The average physical momentum is then, qNp ≃ −ητ ϵ Np + ( 2π j p − TΣ′−1 r=0 ΔSr ) Np T′ϵ (3.26) Here, ΔSr = 2π(Sr+1 − Sr)/b and T′ satisfies ΔSr+T′ = ΔSr. For a primary QAM which we investigate experimentally, j = 0. Also with the simple case of ΔSr = 0, Eq. (3.26) reduces to, qNp ≃ −ητ ϵ Np (3.27) identical to Eq. (3.6) obtained from the interference model. 3.1.3 Experiment To experimentally realize such higher order resonances, an initial state was produced in the form of a BoseEinstein condensate of around 30,000 atoms of Rb87 in the F = 1, 5S1/2 state. The CO2 beam forming the optical trap was then extinguished and the condensate released from the trap. The kicking beams were derived from a slave laser injection locked to the F = 2 → F′ = 3 transition frequency of the master laser. It was thus 6.8 GHz reddetuned with respect to the condensate atoms 33 AOM2: Asin(wt) AOM1: Asin[(w+w )t] D Kicking beams Figure 3.2: Setup for the kicking experiments. Two counterpropagating beams formed the standing wave oriented at an angle of 52◦ to the vertical. AOM2 was driven by an RF signal with a fixed frequency, ω/2π = 40MHz. 34 (Fig. 2.5). An optical fiber transported this beam from the optical table to the BEC table where it was divided into two by a 5050 beam splitter cube. Each of these beams had 25 mW of power and was directed into an acoustooptic modulater (Isomet model 40N). The frequency of the first order beam diffracted by the AOM is Doppler shifted by an amount equal to the frequency of the acoustic wave in the AOM. The acoustic wave was generated by an RF electrical signal supplied by an arbitrary waveform generator HP8770A after amplification. One of the AOMs was driven at a fixed 40 MHz while the other had a variable frequency input from a second HP8770A which was phaselocked to the first generator. By adjusting the variable part of this frequency, ωD = 2π T1/2 β + 1 2Gat, the initial momentum β, and acceleration a, of the standing wave relative to the condensate could be changed. A schematic of the kicking setup is shown in Fig. 3.2. After the application of the kicking sequence, the condensate was allowed to expand for 8 ms to allow the diffracted momentum orders to separate before imaging. Pulse lengths of 1.8 μs were used for the kicks. The probability of the nth momentum order getting populated is Jn(ϕd)2. Thus, by examining a distribution after one kick, the strength ϕd of a kick was inferred to be ≈ 1.5. Figure 3.3 depicts the evolution of a QAM as a function of kicks. The plots show horizontally stacked distributions with increasing pulse number near a resonance time. Figure 3.3(a) is a kick scan performed at T = 22.68 μs, close to the resonance at (2/3)T1/2 [70]. The linear growth of a QAM with orders separated by 3~G can be seen. This behavior where states separated by b~G rephase to form the bth order resonance is expected from the fractional Talbot effect [71]. The effect is clearer in Fig. 3.3(b) which is near the resonance at (1/2)T1/2 with a pulse separation of T = 17.1 μs. Figure 3.3(c) is at the Talbot time (2T1/2), where the QAM can be seen to comprise of neighboring momentum states. Next, time scans across two highorder resonances at (1/2)T1/2 and (1/3)T1/2 are 35 20 Figure 3.3: Quantum accelerator modes at (a) T=22.68 μs close to (2/3)T1/2, (b) T=17.1 μs which is close to (1/2)T1/2, and (c) T=72.4 μs close to 2T1/2. a value of g′=6 ms−2 was used in these scans. The arrows in (a) and (b) show orders separated by b~G which participate in the QAMs. Dashed lines correspond to the ϵclassical theory of Eq. (3.27) 36 shown in Figs. 3.4(a) and (b) respectively. The dashed curves are Eq. (3) demonstrating a good fit with the ϵclassical theory. The relatively weak experimental signal for the resonance at (1/3)T1/2 is also noticed in Fig. 3.4. This difficulty in creating a highorder QAM stems from the high kicking strength ϕd needed to populate states comparable to b~G. It can be seen from the fact that the population of the state b~G is proportional to Jb(ϕd)2 with a maximum at b ∼ ϕd. This high value of ϕd has the unwanted effect of increasing the distribution of states not participating in the QAM, thereby masking the presence of a QAM. Finally, the dependence of QAM on the initial momentum of the BEC was investigated by moving the standing wave using the variable frequency of a kicking AOM. The initial momentum scans performed near resonances at (1/2)T1/2 and (2/3)T1/2 are shown in Fig. 3.5(a) and (b) respectively. The QAM appears once every 1~G of initial momentum at (1/2)T1/2 and twice at (2/3)T1/2. To understand this, we can use Eq. (3.7) to obtain the separation in initial momenta for a QAM to be Δpi = 2π/τb ≈ 1/a. Thus our experimental results bear out the predictions of the theory of highorder resonances. 37 Figure 3.4: Horizontally stacked momentum distributions across (a) (1/2)T1/2 for 40 kicks, ϕd = 1.4 and effective acceleration g′ = 6 ms−2; and (b) (1/3)T1/2 for 30 kicks, ϕd = 1.8, and g′=4.5 ms−2. The dashed curve is a fit to the theory in Eq. (3.27) 38 Figure 3.5: Initial momentum scans for QAMs near (a) (1/2)T1/2 (T=17.1 μs) for 30 kicks and g′=6 ms−2; and (b) (2/3)T1/2 (T=22.53 μs) with 40 kicks and g′=4.5 ms−2. The dashes indicate QAMs at the resonant β. 39 3.2 A quantum ratchet Feynmann in his classic Lectures on Physics [72] discussed the concept of a ratchet, and demonstrated how directed motion could not be extracted from a spatially asymmetric system kept in thermal equilibrium. However away from equilibrium, a ratchet current, defined as directed transport in a spatially periodic system with no bias field, becomes a possibility [73]. The best examples of this are seen in nature in the form of biological motors which can use thermal fluctuations to establish a particulate current. Hamiltonian ratchets, where deterministic chaos replaces dissipation to drive motion, have recently gathered theoretical and experimental interest. For instance, asymmetric dynamical localization led to a unidirectional current in a kicked rotor with spatiotemporal asymmetry [75]. The possibility of using a resonance of a kicked rotor with a broken spatial symmetry was shown in [74] and subsequently observed experimentally [26]. We realize such a quantum ratchet at arbitrary initial momenta (quasimomentum) and investigate its various parameter dependencies. The initial state is prepared as a superposition of the momentum states ψ0⟩ = √1 2 [0⟩ + 1⟩] (3.28) With Eq. (3.17), we can express this as a rotor state, ψ0(θ) = √1 4π [1 + eiθ] (3.29) After a time T, the 1st order picks up a phase γ = (~G)2 2m~ T. Equivalently, the free evolution can be substituted by a translation of the grating from cos(θ) → cos(θ − γ). We shall adopt this latter approach in the analysis below, and the experiments thereafter, where an offset kicking potential V (θ) = cos(θ − γ) is applied on the initial state. The one period evolution operator for the Hamiltonian ˆH = ˆp2 2 + ϕd cos(ˆx) Σ∞ Np=0 δ(t − Npτ ) is given by (Eq. 3.18 with η = 0), ˆU β = e−iϕdV (ˆθ)e−iτ(Nˆ+β)2/2 40 = e−iϕd cos(θˆ−γ)e−iNˆ τβ (3.30) Here we have ignored an irrelevant phase factor and τβ = πl0(2β + 1) with l0 an integer. After Np kicks, ψNp(θ) = ˆU Np β ψ0(θ) = ˆU Np−1 β e−iϕd cos(ˆθ−γ)ψ0(θ − τβ) = ˆU Np−2 β e−iϕd cos(ˆθ−γ)e−iϕd cos(ˆθ−γ−τβ)ψ0(θ − 2τβ) = e−iϕd ΣNp−1 s=0 cos(ˆθ−γ−sτβ)ψ0(θ − Npτβ) = e [−iϕd sin(τβNp/2) sin(τβ/2) cos{θ−(Np−1) τβ 2 −γ}] ψ0(θ − Npτβ) (3.31) where we have used the relation Σt−1 s=0 eas = (1 − eat)/(1 − ea) in the last step. The momentum current at time Np is, ⟨Nˆ ⟩Np ≡ ⟨ψNp Nˆ ψNp ⟩ = −i ∫ 2π 0 dθψ∗ Np(θ) dψNp(θ) dθ = −i ∫ 2π 0 dθ [iϕd sin(τβNp/2) sin(τβ/2) sin{θ − (Np − 1) τβ 2 − γ}ψ0(θ − Npτβ)2 +ψ∗ 0(θ − Npτβ)ψ′ 0(θ − Npτβ)] (3.32) Here the prime denotes a derivative with respect to θ. Using the initial state (3.29) and identifying the second term as the mean momentum of the initial state, the change in the momentum current is [76], Δ⟨ˆp⟩Np = ⟨Nˆ ⟩Np − ⟨Nˆ ⟩0 = ϕd 2 sin(τβNp/2) sin(τβ/2) sin[(Np + 1)τβ/2 − γ] (3.33) When τβ → 2rπ, where r is an integer, we obtain a linear growth in the current, a ratchet acceleration: Δ⟨ˆp⟩Np,r = −ϕd 2 sin(γ)Np. (3.34) 41 At this point we would like to note that for a plane wave initial state, the integral of the first term in (3.32) vanishes and there is no change in the momentum current from its initial value. This explains our choice of the special superposition initial state (3.29) to create a ratchet. Of course one has to account for the finite initial momentum width of the condensate in an experiment. Towards this end we consider a condensate with a Gaussian distribution, √ 1 2π(Δβ)2 exp−(β′−β)2 2(Δβ)2 , where β and Δβ are the average and standard deviation respectively. The average of the momentum current (3.33) over β = β′ is [27, 77], ⟨Δ⟨ˆp⟩Np ⟩Δβ = ϕd 2 ΣNp s=1 sin(τβs − γ) exp[−2(πl0Δβs)2] (3.35) On resonance, that is when τβ = 2rπ, Eq. (3.35) reduces to ⟨Δ⟨ˆp⟩Np ⟩Δβ = −ϕd 2 sin(γ) ΣNp s=1 exp[−2(πl0Δβs)2] (3.36) 3.2.1 Experiment The experimental configuration was similar to that used to observe quantum accelerator modes but with η = 0. The initial state was prepared by applying a weak but long pulse (duration of 38 μs) to the released condensate. This resulted in a Bragg diffraction of the atoms into an equal superposition of the 0~G⟩ and 1~G⟩ states. This was followed by short (2 μs) kicks of the rotor, each with a strength ϕd ∼ 1.4. To produce an offset γ, the standing wave was shifted by changing the phase of the variable kicking AOM. A systematic study was done to experimentally determine the parameter dependence of a quantum ratchet. We worked at the first quantum resonance where τβ = 2π, corresponding to the halfTalbot time, T1/2. This gives a resonant β of 0.5 (for l0 = 1). Figure 3.6 shows the mean momentum as a function of the offset phase, γ, after 5 kicks on the initial state. The largest change in momentum was at γ = π/2, 3π/2 etc. This can be seen classically as a maximum gradient of the optical potential acting on 42 Figure 3.6: Dependence of the mean momentum of the quantum ratchet on the offset angle γ for 5 kicks and β = 0.5. The dashed and solid lines represent Eqs. (3.33) and (3.36) respectively. The inset shows the offset γ created between the symmetry centers of the initial distribution (blue curve) and the kicking potential V (red curve). 43 Figure 3.7: The ratchet effect. This time of flight image shows growth of mean momentum with each standing wave pulse applied with a period of T1/2 and a maximum offset (γ = π/2) between the standing wave and the initial state. 44 Figure 3.8: Mean momentum as a function of kicks. The data and error bars are from experiments with ϕd = 1.4, γ = π/s and β = 0.5. The solid line is Eq. (3.33) while the dashed line corresponds to Eq. (3.36). 45 the atomic distribution when it is displaced by an integer multiple of γ = π/2 (Fig. 3.6). The ratchet effect can be seen clearly in the time of flight image of Fig. 3.7 at γ = π/2. The mean momentum as a function of kicks is plotted in Fig. 3.8, and exhibits a saturation of the ratchet acceleration. A fit of the mean momentum in Figs. 3.6 and 3.8 to Eq. (3.36) yields a value of Δβ = 0.056. This value is consistent with a condensate momentum width measured using a timeofflight technique. Finally, with γ = ±π/2, the dependence of the mean momentum on β is depicted in Fig. 3.9. As expected, a pronounced ratchet effect is seen at the resonant β = 0.5 but is suppressed overall due to the finite Δβ. Thus, we realized a quantumresonance ratchet by applying periodic kicks from a symmetric optical potential on a superposition state. An asymmetry introduced between the centers of the initial state and the kicking potential created a ratchet current (Fig. 3.6), with a maximum magnitude at γ = ±π/2. A linear growth in momentum was observed at the resonant β = 0.5, but was found to be suppressed by the finite momentum width of the BEC. It is therefore desirable to reduce Δβ to obtain a higher ratchet current. 46 0 0.2 0.4 0.6 0.8 1 −4 −2 0 Quasimomentum (b) Mean−momentum change (2 photon recoils) 0 2 4 (a) (b) Figure 3.9: Change in mean momentum vs the quasimomentum β for ϕd = 1.4 and (a) γ = −π/2, (b) γ = π/2. Shown are the fits of Eqs. (3.33), dashed line and (3.36), solid line respectively to the experimental data (filled circles). 47 CHAPTER 4 SubFourier resonances of the kicked rotor A discrete Fourier transform of input test signals was recently performed with a BEC in an optical lattice [78]. The resultant frequency spectra were resolved more accurately than possible by classical Fourier analysis. Such examples exploit the nonlinear response of the quantum system to an excitation. For instance, the linewidth of the nth harmonic of a multiphoton Raman resonance was shown to be nfold narrower than the Fourier transform linewidth of the driving optical pulse [79]. In another demonstration, dynamical localization in the classically chaotic regime of the quantum kicked rotor was utilized to discriminate between two driving frequencies with a sub Fourier resolution [28]. At a primary quantum resonance at the Talbot time, adjacent momentum orders evolve in phases which are integer multiples of 2π. As we saw in chapter 2, this leads to a quadratic growth in mean energy, ⟨E⟩ = 2πErϕ2 dN2 where Er = ~2G2/8M is the photon recoil energy. The width of mean energy distribution around the resonance time was found to decrease with kicks and kicking strength as 1/(N2ϕd) [91]. This subFourier behavior was attributed to the nonlinear nature of the quantum delta kicked rotor and explained using the ϵclassical theory [80]. Highprecision measurements using quantum mechanical principles have been carried with atom interferometers for many years [81]. Such devices were used to determine the Earth’s gravitational acceleration [82–84], fine structure constant α [86–88], and the Newtonian constant of gravity [89]. The promise of the QDKR as a candidate for making these challenging measurements has begun to be realized [29]. Recently a scheme was proposed for measuring the overlap or fidelity between a nearresonant 48 δkicked rotor state and a resonant state via application of a tailored pulse at the end of a rotor pulse sequence [90]. It predicted a 1/N3 scaling of the temporal width of the fidelity peak. In this chapter we describe our observation of such fidelity resonance peaks and their subFourier nature. In section 4.1 we discuss the theory. We also calculate the effect of gravity on such a measurement. Section 5.7 details the experimental configuration and the results. 4.1 A fidelity measurement on the QDKR We begin with the evolution of a state ψ0⟩ due to N kicks at a period near the Talbot time, i.e., τ = 4π + ϵ, ψf ⟩ = ˆU Nψ0⟩ = Σ n cnn⟩ (4.1) where the n⟩ are momentum eigenstates in units of ~G. The effect of the deviation from resonance (to the first order in ϵ), results in a change of the phase of the complex coefficients cn = An(ϵ) exp{iθn(ϵ)}, cn(ϵ) ≃ cn(0) exp ( i ∂θn ∂ϵ ϵ=0ϵ ) = (−i)nJn(ϕd) exp ( i ∂θn ∂ϵ ϵ=0ϵ ) (4.2) cn(0) are the resonant coefficients as seen in Eq. (2.10). We notice that a measurement of the mean energy ⟨E⟩ = 1 2 Σ n n2pn for the distribution pn = ⟨nψ⟩f 2 will not detect these phases. To probe the effect of these phase changes, Ref. [90] proposed a measurement of the projection of this final offresonant state onto the resonant state. To this end one can define a “fidelity”, F = ⟨ψ0ˆU rˆU Nψ0⟩2 (4.3) 49 where ˆU = exp(−i τ 2 ˆp2) exp[−iϕd cos(ˆx)] and ˆU r = exp[iNϕd cos(ˆx)]. Experimentally, ˆU r can be implemented with a kick changed in phase by π and carrying a strength of Nϕd. Note that on resonance, exp(−i τ 2 ˆp2) = 1 and the final state is the same as the initial. F therefore measures the probability of revival of the initial state as a function of these deviations. Using Eqs. (4.1) and (4.2), the fidelity can be expressed as, F ≃  Σ n J2 n(Nϕd) exp(iΘn)2 (4.4) Here we have introduced Θn = ∂θn ∂ϵ ϵ=β=η=0ϵ + ∂θn ∂β ϵ=β=η=0β + ∂θn ∂η ϵ=β=η=0η (4.5) to account for the general case of phases accruing from deviations in resonant time (ϵ), initial momentum (β) or a nonzero gravity (η). This perturbative treatment was used to show that near the Talbot time, τ = 4π, the fidelity is [90] F(ϵ, β = 0, η = 0) ≃ J2 0 ( 1 12 N3ϕ2 dϵ ) (4.6) The width of such a peak in ϵ therefore changes as 1/(N3ϕ3 d), displaying a subFourier dependence on the measurement time, expressed here in units of kicks, N. We now investigate the behavior of the fidelity in the presence of a linear potential like gravity. 4.1.1 Effect of ‘gravity’ For a perturbation only due to gravity, Θn = ∂θn ∂η ϵ=β=η=0η (4.7) Equation (4.2) can be rearranged as, ∂θn ∂η (ϵ=β=η=0) = ∂cn ∂η (ϵ=β=η=0) icn(0, 0, 0) (4.8) 50 Coefficients cn are cn(ϵ, β, η) = ⟨n + βUˆgN ...Uˆg2 Uˆg1 β⟩ (4.9) Uˆgt = exp[−i τ 2 (ˆp + tη + η 2 )2] exp[−iϕd cos(ˆx)] is the tth kick evolution operator in the freely falling frame which was introduced in section (3.1.2) obtained after a gauge transformation of the QDKA Hamiltonian. After 3 kicks on the initial zero momentum eigenstate, ∂cn ∂η (ϵ=β=η=0) = ⟨n∂ ˆ Ug3 ∂η Uˆg2 Uˆg1 + Uˆg3 ∂Uˆg2 ∂η Uˆg1 + Uˆg3 Uˆg2 ∂Uˆg1 ∂η 0⟩(ϵ=β=η=0) = −2iτ 2 ⟨n(ˆp + 3η + η 2 )(3 + 1 2 ) exp(−iϕd cos xˆ)Uˆg2 Uˆg1 + Uˆg3(ˆp + 2η + η 2 )(2 + 1 2 ) exp(−iϕd cos xˆ)Uˆg1 + Uˆg3 Uˆg2(ˆp + 1η + η 2 )(1 + 1 2 ) exp(−iϕd cos ˆx)0⟩(ϵ=β=η=0) = −iτ ⟨ne−iϕd(1−1) cos ˆx ˆp (3 + 1 2 )e−iϕd(3+1−1) cos ˆx + e−iϕd(2−1) cos ˆx ˆp (2 + 1 2 )e−iϕd(3+1−2) cos ˆx + e−iϕd(3−1) cos ˆx ˆp (1 + 1 2 )e−iϕd(3+1−3) cos ˆx0⟩ Therefore for N kicks, ∂cn ∂η (ϵ=β=η=0) = −iτ ΣN r=1 bn,r (4.10) where bn,r = ⟨ne−iϕd(r−1) cos ˆx ˆp (N − r + 3 2 )e−iϕd(N+1−r) cos ˆx0⟩ (4.11) We make an observation here. Equation (4.11) can be expanded as bn,r = (N − r + 3 2 ) Σ∞ m=−∞m2Jm[(N + 1 − r)ϕd]Jm−n[(r − 1)ϕd] (−1)m (−i)n . This expression and Eq. (4.10) show that, a first order perturbation in η can be viewed as a rotation in the complex plane (real→imaginary and vice versa). Similar results hold for deviations from resonant ϵ and β and explain the basis of assumption (4.2). We now return back to evaluate (4.10) with (4.11), ∂cn ∂η (ϵ=β=η=0) = −iτϕd ΣN r=1 (N − r + 3 2 )(N + 1 − r)⟨ne−iϕd(r−1) cos ˆx sin ˆx e−iϕd(N−1+r) cos ˆx0⟩ 51 = −iτϕd⟨n sin ˆx e−iNϕd cos ˆx0⟩ ΣN r=1 (N − r + 3 2 )(N + 1 − r) = iτ 2i ϕd⟨n(eiˆx − e−iˆx) e−iNϕd cos ˆx0⟩ ΣN r=1 (N − r + 3 2 )(N + 1 − r) = τ 2 ϕd(⟨n + 1 − ⟨n − 1){ Σ l (−i)lJl(Nϕd)l⟩} ΣN r=1 (N − r + 3 2 )(N + 1 − r) = τ 2 ϕd[(−i)n+1Jn+1(Nϕd) − (−i)n−1Jn−1(Nϕd)] ΣN r=1 (N − r + 3 2 )(N + 1 − r) = −iτn N (−i)nJn(Nϕd) ΣN r=1 (N − r + 3 2 )(N + 1 − r) (4.12) using Jn+1(x) = 2n x Jn(x) − Jn−1(x). This gives us, ∂θn ∂η (ϵ=β=η=0) = ∂cn ∂η (ϵ=β=η=0) icn(0, 0, 0) = −τn[N2 + 5 2 N + 3 2 + (N + 1) 2 {2N + 1 3 − (2N + 5 2 )}] Keeping only terms in N2, ∂θn ∂η (ϵ=β=η=0) = −τn N2 3 = −4π 3 nN2 (4.13) Therefore, the fidelity is F(η, ϵ = β = 0) ≃  Σ n J2 n(Nϕd) exp(i ∂θn ∂η η)2 = Σ n J−n(Nϕd)Jn(Nϕd) exp{i(−Δ + π)n}2 where we have used result (4.13) to define Δ = 4π 3 N2η and used the Bessel function property, J−m(x) = (−1)mJm(x). Finally with the help of Graf’s identity, Σ l J−l+m(A)Jl(B) eilϕ = A+Beiϕ A+Be−iϕ Jm[(A2 + B2 + 2AB cos ϕ)1/2], we can simplify this 52 to F(η, ϵ = β = 0) = Nϕd(1 + e−iΔ) Nϕd(1 + eiΔ) J0[Nϕd(2 − 2 cos Δ)1/2] 2 = J2 0 (NϕdΔ) = J2 0 ( 4π 3 N3ϕdη) (4.14) keeping the lowest terms in η. Thus the width of a fidelity peak centered at the resonant zero acceleration should drop as 1/N3. A similar procedure gives the fidelity near the resonant(β = 0) initial momentum [90], F(β, ϵ = η = 0) = J2 0 (2πϕdN(N + 1)β) (4.15) 4.2 Experimental Configuration and Results These experiments were performed with a BoseEinstein condensate of around 20,000 87Rb atoms in the 5S1/2, F = 1,mF = 0 level. The selection of the mF = 0 Zeeman sublevel was done by keeping the MOT coils on during the evaporative cooling. Only the mF = 0 atoms which are insensitive to the magnetic field gradient (10G/cm) of the MOT coils could then undergo a stable evaporation [94], producing a pure mF = 0 condensate. After being released from the trap, the condensate was exposed to a horizontal optical grating created by two beams of wavelength λ = 780 nm light detuned 6.8 GHz to the red of the atomic transition. The wave vector of each beam was aligned θ = 52◦ to the vertical. This formed a horizontal standing wave at the point of intersection with a wavelength of λG = λ/2 sin θ. With this grating vector G = 2π/λG the resonant Talbot time is TT = 4πM/~G2 = 106.5 μs. The kicking pulse length was Δt = 0.8 μs with a measured ϕd ≈ 0.6. To create the reversal operator ˆU r in Eq. (4.3), a final kick of strength Nϕd was applied at the end of the N rotor kicks. We did this by increasing the intensity rather than the pulse length in order to keep this 53 Figure 4.1: Momentum distributions of a sequence of 8 kicks of strength ϕd followed by a final πphase reversed kick of strength Nϕd, with a time period equal to the Talbot time 106.5μs. 54 final kick within the RamanNath regime. Moreover, the standing wave for this kick was shifted by half a wavelength by changing the phase of one of the kicking AOM’s rf driving signal. A timeofflight image of a fidelity kicking sequence on resonance is depicted in Fig. 4.1. The sensitivity of the phases in this experiment to external perturbations made it necessary to identify and minimize such sources. Calculating the photon scattering rate γp = Ω2 δ2 γ for our kicking strength ϕd(= Ω2Δt 8δ ), we find that decoherence due to spontaneous emission is expected to be important only beyond ∼ 35 kicks. Dephasing primarily due to vibrations made the reversal process inconsistent for N > 6. To reduce this, the standing wave at each kick was shifted by half a wavelength with respect to the previous kick. This had the effect of shifting the Talbot time resonance to T1/2. Consequently the reduced experimental time led to much improved results. After 8 ms of expansion, the separated momentum orders were absorption imaged. In the experiment fidelity is measured by the fraction of atoms which have returned to the initial zero momentum state. That is we measure F = P0/ Σ n Pn where Pn is the number of atoms in the nth momentum order obtained from the timeofflight images. To facilitate the analysis of the data, all of the resonance widths (δϵ) were scaled to that at a reference kick number of N = 4. That is we define a scaled fidelity width Δϵ = δϵ/δϵN=4 for each scaled kick number Ns = N/4 and recover logΔϵ = −3 logNs using Eq. (4.6). For each kick, a scan is performed around the resonance time. To ensure the best possible fit of the central peak of the fidelity spectrum to a gaussian, the time is scanned between values which make the argument of J2 0 of Eq. (4.6) ≈ 2.4 so that the first side lobes are only just beginning to appear. Figure 4.3(a) plots the logarithm of the FWHM for 4 to 9 kicks scaled to the fourth kick. A linear fit to the data gives a slope of −2.73±0.19 giving a reasonable agreement with the predicted value of 3 within the experimental error [93]. As seen in the same figure, the results are close to the numerical simulations which take into account the 55 Figure 4.2: (a) Horizontally stacked timeofflight images of a fidelity scan around the Talbot time. Each TOF image was the result of 5 kicks with ϕd = 0.8 followed by a πphase shifted kick at 5ϕd. (b) Mean energy distribution of the 5 kick rotor with the same ϕd. (c) The measured fidelity distribution (circles) from (a). The mean energy of the scan in (b) is shown by the triangles. Numerical simulations of the experiment for a condensate with momentum width 0.06 ~G are also plotted for fidelity (bluedashed line) and mean energy (redsolid line). The amplitude and offset of the simulated fidelity were adjusted to account for the experimentally imperfect reversal phase. 56 finite width of the initial state of 0.06 ~G [27]. We also compared the resonance widths of the kickedrotor mean energy ⟨E⟩ to that of the fidelity widths. As in the fidelity, the plotted values Δ⟨E⟩ have been normalized to that of the fourth kick. On the log scale, the width of each peak gets narrower with the kick number with a slope of −1.93±0.21 (Fig. 4.3(a)) in agreement with previous results [92, 95]. The relative scalings of the fidelity and mean energy distributions can be seen clearly in Fig. 4.2. It can be seen that even for relatively few kicks the fidelity peak can be significantly narrower. As a further test of Eq. (4.6), the variation in the widths of the fidelity and mean energy peaks were studied as a function of ϕd. Figure 4.3(b) shows the fidelity width changing with a slope of −1.96 ± 0.30, close to the predicted value of 2. This is again a faster scaling compared to the mean energy width which decreases with a slope of −0.88 ± 0.24 (the theoretical value being 1). The resonances studied here appear for pulses separated by the Talbot time and an initial momentum state of β = 0. As seen in Eq. (4.15), the peak width in β space is expected to change as 1/[N(N + 1)] around this resonant, as against a 1/N scaling of the mean energy width [92]. To verify this, the initial momentum of the condensate with respect to the standing wave was varied and the kicking sequence applied. The experimentally measured widths Δβ = δβ/δβN=4 in Fig. 4.4 display a scaling of Δβ _ [N(N + 1)]−0.92 close to the theoretical value. For an initial state β+n⟩, the wave function acquires a nonzero phase during the free evolution even at the Talbot time. Therefore the final kick performs a velocity selective reversal, preferentially bringing back atoms closer to an initial momentum of β = 0. This is similar to the timereversed Loschmidt cooling process proposed in Refs. [96, 97], although in that technique a forward and reverse path situated on either side of the resonant time was used in order to benefit from the chaotic dynamics. To observe this effect the current scheme offers experimental advantage in terms of 57 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −2 −1.5 −1 −0.5 0 (a) log ¢², log ¢hEi log Ns 0 0.1 0.2 0.3 0.4 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (b) log ¢², log ¢hEi log ˜ Ád Figure 4.3: Experimentally measured fidelity (circles) and mean energy (triangles) widths (FWHM) as a function of (a) the number of pulses, and (b) the kicking strength ϕ˜d scaled to ϕd of the first data point. In (a), the data are for 4 to 9 kicks in units normalized to the 4th kick. Error bars in (a) are over three sets of experiments and in (b) 1σ of a Gaussian fit to the distributions. Dashed lines are linear fits to the data. Stars are numerical simulations for an initial state with a momentum width of 0.06~G. 58 0 0.5 1 1.5 −1.5 −1 −0.5 0 log ¢¯ log N(N +1)s Figure 4.4: Variation of the fidelity peak width around β=0 as a function of kick number N(N +1)s = N(N +1)/20 scaled to the 4th kick. The straight line is a linear fit to the data with a slope of −0.92 ± 0.06. Error bars as in Fig. 2(b). 59 0 1 2 3 4 5 6 7 8 9 0.12 0.13 0.14 0.15 0.16 (a) momentum width Kicks (¯h G) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 (¯h G) Optical density (b) momentum N=0 N=2 N=4 N=6 Figure 4.5: (a) Momentum width of the reversed zeroth order state as a function of kick number. Error bars are an average over three experiments. (b) Optical density plots for the initial state (red,solid) and kick numbers 2 (magenta,dotdashed),4 (black,dotted), and 6 (blue,dashed) after summation of the timeofflight image along the axis perpendicular to the standing wave. 60 stability due to the reduced length of the pulse sequence. Here, only a single pulse performs the velocity selection at the end, whereas in the Loschmidt technique N phase reversed kicks separated by a finite time are used. Figure 4.5 demonstrates the reduction of the momentum distribution width. Accompanying this decrease is a drop in the peak height. Our simulations and the results of Ref. [97] predict that for the case of a noninteracting condensate this should remain constant. In addition to interactions we expect experimental imperfections in the fidelity sequence to play a role in the smaller peak densities with increasing kick numbers. We performed the same experiment 4.5 ms after the BEC was released from the trap when the mean field energy had mostly been transformed to kinetic energy in the expanding condensate. A similar reduction in the momentum width of the reversed state alongwith a decrease in the peak density was observed. Finally to investigate the sensitivity of the fidelity resonance to gravity, the standing wave was accelerated during the application of the pulses. This acceleration was scanned across the resonant zero value and readings of the fidelity collected. Since a typical value of the halfwidth at half maximum is η = 0.05 for N=4 (corresponding to an acceleration of 4m/s2), the perturbative treatment of acceleration on fidelity used above is justified. Figure 4.6 plots the experimental data for 4 to 9 kicks, where the widths of the peaks decrease with a slope of −3.00 ± 0.23 in excellent agreement with the theory. In conclusion, we performed experimental measurements of the fidelity widths of a δkicked rotor state near a quantum resonance. The width of these peaks centered at the Talbot time decreased at a rate of N−2.73 comparable to the predicted exponent of −3. By comparison, the mean energy widths was found to reduce only as N−1.93. Furthermore, the fidelity peaks in momentum space changed as (N(N +1))−0.92, also consistent with theory. The reversal process used in the fidelity experiments led to a decrease in the momentum distribution of the final zeroth order state by ∼ 25% (for N=9) from the initial width. The subFourier dependencies of the 61 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −2.5 −2 −1.5 −1 −0.5 0 log ¢g log Ns Figure 4.6: Dependence of the acceleration resonance peak width as a function of the kick number in units scaled to the 4th kick. Error bars are over three sets of experiments. 62 mean energy and fidelity observed here are characteristic of the dynamical quantum system that is the QDKR [90]. The narrower resonances of the fidelity scheme could be exploited in locating the resonance frequency with a resolution below the limit imposed by the fourier relation. This can help determine the photon recoil frequency (ωr = Er/~) which together with the photon wavelength enables measurement of the fine structure constant with a high degree of precision [29, 86–88]. We also demonstrated a N−3 dependence of the resonance width in acceleration space in accordance with the extended theory. The sensitivity of an atom interferometer based gravimeter scales as the square of the loop time, hence the pursuit of large area interferometers to improve accuracy [82–84]. By comparison, the fidelity is responsive to the gravitational acceleration g with the cube of the ‘time’ N, leading to the possibility of higher precision measurements. One could perform a fidelity measurement on a freely falling condensate exposed to kicks accelerating at the local value of g (to realize η ≪ 1). Variation in g would then manifest itself as a shift of the resonant acceleration. A parts per billion precision [85] would require a judicious selection of the parameters (N, ϕd, T), for instance (150, 10, 16T1/2). Such a resolution, though not feasible in the current setup without addressing stability related issues, could be possible with future refinements, for instance through active stabilization measures. 63 CHAPTER 5 Photoassociation of a 87Rb BEC Our studies on kicked atoms have so far employed the linear kicked rotor model. A nonlinearity can be introduced via interactions among the condensate atoms. Such interactioninduced nonlinearity in the kicked rotor was shown to lead to a shift in the resonance and a striking cutoff at the maximum, due to coupling between phonon modes [98]. Another system studied in this context is the quantum δkicked harmonic oscillator. It is the realized with a kicked rotor in a harmonic trap and exhibits resonances like the rotor. Dephasing and destruction of the resonant motion were found to occur in the presence of interactions [99]. Furthermore, the nonlinear version remains an excellent candidate to study quantumclassical correspondence [100]. To pave the way for such studies, it is necessary to tune the interactions on demand. Magnetic Feshbach resonances, and more recently Optical Feshbach resonances, have enabled such control over the condensate interaction. In this chapter, we describe our experiments on photoassociation spectroscopy performed near long range molecular states of a 87Rb BEC to realize an Optical Feshbach resonance. In section 5.1 we discuss such spectroscopy. This is followed by a review of the theory ultracold collisions and Feshbach resonances in sections 5.2 and 5.3 respectively. We then discuss the experimental configuration and results in section 5.4, and the conclusion in section 5.5. 64 5.1 Photoassociative spectroscopy Photoassociation (PA) is a process in which two atoms colliding in the presence of a light field absorb a photon to form a bound, excited molecule. This effect was first observed for the H2 molecule and rare gas halide molecules [101,102]. With the advent of laser cooling, photoassociation of samples of cold atoms became a high resolution spectroscopic technique as a result of the very low energy spread of the atoms [103]. It has since been used to precisely measure atomic lifetimes, map ground state collisional wavefunctions, probe long range and ‘pure long range’ molecular states and to produce translationally cold molecules from cold atoms [104–107]. Furthermore, it was discovered that the collisional state scattering length can be altered near a photoassociation resonance [111]. This effect has been termed as an optically induced Feshbach resonance and has interesting implications for the dynamics of BoseEinstein condensates [112–114]. Experimental results of photoassociation of a 87 BoseEinstein condensate and the effect on the scattering length will be outlined in this chapter. Figure 1 depicts a typical photoassociative process. The interaction of the groundstate atoms (S+S) occurs along a Vander Waals potential Vg(R) = −C6/R6 at long ranges. For short internuclear distances the interaction is described by a 1/R12 strong electron exchange repulsion. Atoms colliding in the presence of light tuned close to the SP transition can absorb a photon and form an electronically excited molecule. The interaction between the two atoms in the excited molecule is determined by a resonant dipoledipole interaction, Ve = −C3/R3, at long ranges. Many of the bound states that are formed have energies close to the threshold and are therefore easier to access via freebound photoassociation compared to traditional boundbound spectroscopy [104]. The absorption takes place at an internuclear separation called the Condon point RC where the laser photon energy is equal to the difference between the ground the excited state potentials. The stimulated rate of transition from the 65 Figure 5.1: Schematic of a photoassociation process. Two atoms colliding along the ground state potential (S+S) absorb a photon and get excited to the (S+P) molecular potential. The excited molecule can subsequently decay to free atoms or a ground state molecule. 66 free to bound state is given by Γstim = 4π2Id2 M c ⟨eg⟩2, (5.1) where I is the laser intensity, dM is the molecular dipole transition moment, g⟩ is the ground state collisional wavefunction and e⟩ the bound, excited state [112]. The decay of the excited state molecule to either ground state molecules or to hot atoms leads to loss from the trap containing the molecules. Monitoring this loss as a function of the PA laser frequency results in a spectrum of the freebound transitions. This technique is generally known as photoassociation spectroscopy. This is a very high resolution technique resulting from the very low energy spread of ultracold collisions (a few MHz at μK temperatures, which is comparable to the natural molecular linewidths) [104]. As can be seen from Eqn. (5.1), the transition rate depends on the FranckCondon overlap between the ground and excited state wavefunctions. With larger collisional wavefunctions at greater internuclear distances, ultracold photoassociation is ideal for probing longrange excited molecular states [104]. The molecular states investigated using photoassociation in this study were the 1g and the 0− g pure long range state. These are labeled by Hund’s case (c). The total electron orbital angular momentum ⃗l couples strongly with the total spin ⃗s to form ⃗j = ⃗l + ⃗s. The projection of ⃗j along the internuclear axis Ω is conserved due to cylindrical symmetry of the molecule. The molecular states are labeled as Ωg/u. The g/u denotes the inversion symmetry of the electronic wavefunction. The 1g state has Ω=1 and is symmetric with respect to a reflection about the center of the internuclear axis(a gerade state, g). The 0− g state has an additional reflection symmetry and changes sign upon reflection in a plane containing the internuclear axis (the negative sign) [106]. 67 5.2 Ultracold Collisions Elastic twobody collisions are central to the description of BoseEinstein condensates as they determine the behavior as well as the stability of the condensates. The following discussion reviews basic elastic collision theory. One begins by looking for solutions to the timeindependent Schr¨odinger equation [ ˆp2 2μ + V (r)]ψ(r) = Eψ(r) (5.2) which describes the motion of two atoms with relative momentum p = ~k colliding in the potential V (r). μ is the reduced mass of the particles and E = ~2k2/2μ is the collision energy. At large r, where the potential is negligible, the scattering wave function satisfies the freeparticle Schr¨odinger equation and the solution can be written as ψk(r) −−→ r→∞ A[exp(ik.r) + f(k, θ, ϕ) exp(ikr) r ] (5.3) That is, the scattering wave function at large distances is a superposition of an incident plane wave and an outgoing spherical wave. A is a normalization constant. To calculate the scattering amplitude f, we consider the case of a central potential V (r). For such a potential the Hamiltonian commutes with the total angular momentum, L2 and its projection Lz. Also the scattering is symmetric about the incident direction and therefore independent of ϕ. One can thus separate the solutions in terms of radial components Rl(k, r) and Legendre polynomials Pl(cosθ) with contributions of l different partial waves. ψk(k, r, θ) = Σ∞ l=0 Rl(k, r)Pl(cosθ) (5.4) The Schr¨odinger equation for the radial part of the wavefunction becomes [ d2 dr2 − l(l + 1) r2 − U(r) + k2]ul(k, r) = 0 (5.5) 68 where ul(k, r) = rRl(k, r) and U(r) = 2μV (r)/~2. The total scattering cross section is defined as σ = ∫ f2dΩ, where the integration is over solid angle, can be shown to be σ = 4π k2 Σ l (2l + 1)sin2δl (5.6) where δl is the phase shift of the lth partial wave as a result of the collision. For identical bosons, only even partial waves contribute, σ = 8π k2 Σ l even (2l + 1)sin2δl (5.7) In the ultracold regime the collisional energy is too small to overcome the centrifugal barrier ~2l(l + 1)/2μr2. As can be seen from Fig. 5.2, any partial wave with l>0 cannot enter inside the potential. Therefore only l=0 (i.e. swaves) need to be considered for ultracold collisions. From Eq. 5.7 the scattering crosssection for the l=0 wave is σl=0 → 8πa2 (5.8) where the scattering length a has been defined as a = − lim k→0 tanδ0 k (5.9) Physically, the scattering length can be understood as the intercept of the unperturbed collisional wavefunction with the internuclear axis. 5.3 Scattering length and Feshbach Resonances The dynamics of an interacting BoseEinstein condensate are described by the Gross Pitaevskii equation [− ~2 2m ∇2 + U(r) + gψ(r)2]ψ(r) = −i~ ∂ ∂t ψ(r) (5.10) 69 Figure 5.2: Centrifugal energy term ~2l(l + 1)/2μr2 of the Hamiltonian for three partial waves, l=0,1,2. For low energy scattering all partial waves l >0 are blocked by the centrifugal barrier. 70 ψ(r) is the condensate wave function. The first two terms in the Hamiltonian are the kinetic energy and the trapping potential U(r). It also contains a nonlinear mean field energy term gψ(r)2 which describes the interation energy of an atom in the meanfield produced by the other bosons. It is proportional to the condensate density n = ψ(r)2 and the interaction coefficient g = 4π~2a/m for twobody elastic collisions between the bosons. The elastic scattering length, a, thus determines the interactions. For a > 0 the interactions are repulsive, and a < 0 leads to a condensate with attractive interactions stable only below a certain critical density [108]. Changing the scattering length has therefore received much attention and has been made possible by the use of Feshbach resonances. The scattering length depends strongly on the nature of the interatomic potential. This is best illustrated by the textbook problem of scattering by a squarewell potential: V (r) = −V0 if r < R = 0 if r > R Upon solving the radial wavefunctions ul(r) with boundary conditions ul(0) = 0 and continuity at r = R one finds the phase shift for the swave (l=0) to be δ0(k) = −kR + tan−1[ k k′ tan(k′R)] k and k′ are the wavevectors outside and inside the well respectively. Thus from Eq. 5.9 the scattering length can be shown to be a = R[1 − tan(λR) λR ] (5.11) where λ = √ mV0/~2. Figure 5.3 plots the scattering length as a function of λ. As can be seen, increasing λ which is equivalent to increasing the well depth V0 leads to periodic divergences of a. 71 Figure 5.3: Variation of the scattering length a as a function of λ = √ mV0/~2. As the well depth V0 increases ((a) to (c)) a bound state is formed (dashed line) and the scattering length passes through a divergence and changes sign. 72 The negative scattering length for an attractive squarewell potential begins to decrease as the welldepth increases, diverges when the well can hold a boundstate and becomes positive for a weakly bound state. The dependence of the energy of the last bound state, Eb, on the scattering length is given by Eb = −~2/ma2 for a >> R. The relationship between the scattering length and the position of the last bound state is at the heart of the concept of Feshbach resonances. The idea of inducing such a resonance involves using an external field to change the internal states of the colliding atoms to couple them to a quasibound state of another interatomic potential. Using a magnetic field for instance, two atoms colliding in an open channel V0 in a hyperfine asymptote can be Zeeman shifted into resonance with a bound state in another hyperfine state where they can stay temporarily bind. The upper potential with a higher threshold energy for the low energy atoms is usually called a closed channel, since energy conservation prohibits the escape of atoms from this potential. The tunability of the scattering length by a magnetic field is expressed as a(B) = abg[1 − Δ/(B − B0)] where abg is the background scattering length far away from resonance, B0 is the field on resonance, and Δ is the width of the resonance. Such a magnetic Feshbach resonance [109] was first observed in a 23Na BEC [110]. 5.3.1 Optical Feshbach Resonance Fedichev et al. [111] proposed the use of optical fields near a photoassociation resonance as a means of modifying the scattering length. An intense field of light tuned near such a resonance can be used to couple a pair of atoms colliding in a ground state potential to a bound excited state of a 1/R3 potential and thereby change the scattering properties of the pair of atoms. Bohn and Julienne [112] introduced a complex phase shift δ = λ + iμ, to include the presence of inelastic loss near a PA resonance. This leads to a complex scattering length, the imaginary part of which accounts for the spontaneous loss processes from the excited state. The expressions 73 Figure 5.4: A Feshbach resonance occurs when an excited state has a bound state close to the collisional threshold. Changing the detuning Δ by an external field can couple the collisional to the bound state and change the scattering length. 74 for the scattering length, a and the inelastic loss rate coefficient, Kinel were given as a = abg + 1 2ki ΓstimΔ Δ2 + (Γspon/2)2 (5.12) Kinel = 2π~ m 1 ki ΓstimΓspon Δ2 + (Γspon/2)2 (5.13) where abg is the background scattering length in the absence of the light, ki is the wavenumber of the condensate atoms, Δ is the detuning from a PA resonance and Γstim and Γspon are the stimulated transition rate and spontaneous decay rate constants respectively. In arriving at these expressions the assumption Γspon >> Γstim has been made [114]. This is usually true with the measured Γstim/2π being of the order of a few kHz whereas Γspon/2π is around 20 MHz. From Eq. 5.12 it can be seen that there are two parameters that can tune the scattering length, the detuning Δ, and the intensity I, contained in the stimulated transition rate Γstim (Eq. 5.1). To minimise inelastic loss processes and still have a significant change in the scattering length, it becomes necessary to choose a light source far detuned from a PA resonance (Eq. 5.13) and with a high intensity (large Γstim, Eq. 5.12). Such an optically induced Feshbach resonance was first achieved in a 87Rb BEC in 2004 [114]. Another way of viewing the Optical Feshbach Resonance process is in the dressedstate picture. The relevant states are the collisional ground state dressed in n photons from the laser g, n⟩ and the excited state dressed in n−1 photons e, n−1⟩. Altering the frequency of the light enables one to change the position of the bound state in the excited state relative to the ground state potential. 5.4 Experiment and Results The photoassociation light was derived from a grating stabilized master laser (Toptica Photonics DL 100). The output light was monitored with a wavemeter (Coherent Wavemaster, resolution ∼500 MHz). To set the laser’s initial frequency to a value 75 near a known molecular level, its grating angle, current, and temperature were appropriately adjusted. It was then offsetlocked to the MOT laser using a scanning FabryPerot interferometer cavity (Coherent, 300 MHz free spectral range). The majority of the light was used to injection lock a home built diode laser (Sanyo diode, 120 mW). The output of this slave laser was coupled through an optical fiber to the table for the BEC experiment. A small amount of the slave light was sent into the scanning cavity through a flipper mirror (Fig. 5.5). At the right current, transmission peaks of the slave and master overlap on the photodiode signal, indicating that the slave was following the master. The flipper mirror was then lowered and the path kept open for the reference beam to be input into the cavity. The master PA laser was locked to the reference light using home built electronics. The basic operating principle for the locking is shown in Fig. 5.6. Around 1 mW each from the master PA laser and reference light (from MOT or Repump laser) were coupled to the scanning cavity through a polarizing beamsplitter cube. The cavity was scanned with a free spectral range of 300 MHz with the help of a piezoelectric transducer attached to one of the mirrors. The frequency difference (modulo FSR) between the transmission peaks of the beams from the photodiode was converted to a ‘separation’ voltage, Vsep [115,116]. The difference between this and a setpoint voltage Vset from a National Instruments analog card gave the error signal, Verror (Fig. 5.6). This error signal was sent to a PID controller, built inhouse. The correction output from the PID was combined with the scan control voltage from the DL 100 unit to control the PA master laser’s grating angle. Changing Vset during a photoassociation experiment enabled control over the frequency of the PA light with an accuracy of around 5 MHz. The power in the PA beam, before it had entered the BEC chamber, was measured at 33 mW. It had a waist radius of 70 μm on the condensate. Higher intensities, which are especially vital for Optical Feshbach Resonances, are possible by reducing 76 Figure 5.5: Schematic of the optical setup for the photoassociation light. 77 Figure 5.6: Flowchart of the locking for the photoassociation master laser. 78 the beam waist size. However, this makes the dipole force of the PA beam strong enough to displace the condensate from the FORT making it increasingly difficult to perform any photoassociation experiment. Figure 5.8 shows the photoassociation spectrum of the 1g(P3/2), v = 152 molecular state. This spectrum was obtained on an ultracold sample (kept in the FORT and just before the BEC transition) with a 1 mW beam pulsed on for 100 ms. The state has a binding energy of 24.1 cm−1 below the 87Rb D2 line and asymptotically connects to 52S1/2+52P3/2 at large internuclear distances [105]. Five molecular lines of the hyperfine structure can be seen. These have been categorized in Ref. [118] according to Hund’s rule (c). The good quantum numbers for these states are the total molecular angular momentum F, total nuclear spin I and their projections Fz and Iz on the internuclear axis. The 0− g (∼ S1/2 + P3/2), v = 1 photoassociation spectrum of a BEC (∼ 25,000 atoms 87Rb) was observed with a 5 ms square pulse with a power of 33 mW. Two rotational lines J=0 and J=2 separated by 200 MHz can be seen in Fig. 5.9. This state located 26.8 cm−1 below the 87Rb D2 line is a ‘pure long range’ state with internuclear distances > 20a0 [117], much greater than ordinary chemical bonds (a0 = 0.53˚A is the Bohr radius). Single atom spontaneous emission losses in the presence of the PA light is proportional to (Ω/δ)2, where Ω is the Rabi frequency and δ the light detuning from an atomic resonance. With the high detuning (800 GHz) of the 0− g state from the atomic resonance such losses are therefore small. Moreover, a strong FranckCondon overlap with the incident collisional state [117] and absence of many neighboring molecular states makes the 0− g (P3/2), v = 1 particularly suitable for studying Optical Feshbach resonances near it. Bragg spectroscopy [119] was utilised in Ref. [114] as a technique to measure the scattering length as a function of the photoassociation detuning. Nth order Bragg diffraction is a process where atoms exposed to a standing wave of offresonant 79 light undergo absorption of N photons from one beam, followed by emission into the other. The internal state of an atom is left unchanged but it ends up acquiring 2N photon recoils of momentum (2N~k, where k = 2π/λ is the wavevector of light). For a moving standing wave created by two beams with a frequency difference ν, the energy and momentum conservation condition for noninteracting atoms is satisfied when hν0 = (2~k)2/2m+ 2~k.pi/m [119]. pi is the initial momentum of the atoms. In a weakly interacting condensate of uniform density n, the Bogoliubov dispersion relation is ν = √ ν2 0 + 2ν0nU/h where nU = n4π~2a/m is the chemical potential. For hν ≫ nU ν = ν0 + nU/h (5.14) The free particle resonance frequency has thus been shifted by the mean field energy [119]. In the presence of a photoassociation light tuned near a molecular state, any optically induced change in the scattering length should therefore be apparent in a shift of the Bragg resonance frequency. Therefore for a known condensate density, such a shift in the resonance frequency would allow measurement of the scattering length a. Light 6.8 GHz red detuned from the 87Rb D2 line was used to create a standing wave. As shown in Fig. 5.7 the Bragg (kicking) beams were aligned such that the wavevector of the standing wave was along the long axis of the FORT. Each of the beams were passed through two acoustooptic modulators. The radiofrequency driving one of these modulators was changed to vary the frequency of the beam passing through it. This created a moving standing wave required to perform a Bragg scan around the resonance frequency. Figure 5.10(a) is an image of a Bragg scan performed by changing the relative frequency between the two beams creating the standing wave. A 90 μs long Bragg pulse transferred nearly 40% atoms on resonance 80 to the −2~k momentum state. Figure 5.10(b) shows the Bragg resonance curves for photoassociation light detunings of Δ/2π =+10 MHz and 10 MHz with respect to the 0− g (P3/2), v = 1, J = 2 molecular state. No shift of the resonance is observed. An alternate approach can be adopted to extract the scattering length values from the loss spectrum [114,118]. The twobody loss rate equation for the condensate density n(r, t) is ∂n(r, t) ∂t = −2Kineln(r, t)2 (5.15) where Kinel is the inelastic rate coefficient. Assuming local density changes only due to photoassociation loss, McKenzie et. al. [120] give a relationship between the fraction of remaining atoms f(η) and Kinel f(η) = 15 2 η−5/2{η1/2 + 1 3 η3/2 − (1 + η)1/2 tanh −1[ √ η/(1 + η)]} (5.16) where η = 2Kineln0t and t is the PA pulse length. The Kinel values calculated from the loss spectrum of the 0− g (P3/2), v = 1, J = 2 state using Eq. 5.16 is shown in Fig. 5.11. A lorentzian fit to this yields a spontaneous decay rate, Γspon, of 8 MHz(Eq. 5.13). In addition, the amplitude A0 of the fit can be related to the stimulated rate constant Γstim as Γstim/ki = A0(mΓspon/8π~) from Eq.5.13. For the data in Fig. 5.11, Γstim/ki = 2.3 × 10−3ms−1. These values of Γstim and Γspon can be used in Eq. 5.12 to calculate the scattering length and the result is plotted in Fig. 5.12. A maximum change of 6a0 can be seen. This corresponds to a variation of 5 Hz in the Bragg resonance frequency, quite a small value to be detected using Bragg analysis. To achieve larger variations in the scattering length, a higher intensity of the PA light is necessary as can be seen from Eq. 5.12. Moreover any alteration in a can be amplified in the meanfield induced Bragg shift by using a higher condensate density (Eq. 5.14). Thus an improvement in these two parameters is crucial for the direct observation of an optically induced Feshbach resonance. 81 5.5 Conclusion and Outlook Photoassociation spectroscopy performed on a 87Rb BEC was able to resolve five hyperfine levels of the 1g long range molecular state and two rotational levels of the 0− g ‘pure long range’ state, matching the available data in the literature. Bragg spectroscopy done near the 0− g (P3/2), v = 1, J = 2 state did not yield a detectable change in the swave scattering length a. Analysis of the loss spectrum of this state as a function of the photoassociation detuning revealed a change of 6 Bohr radius in the scattering length. To achieve a larger and directly observable change, an improvement in the condensate density and the photoassociation light intensity is necessary and is currently being pursued. Another approach could be to more easily detect small changes in ‘a’ using a different measurement technique. Such a possibility was investigated using a method based on the interference between two copies of the condensate, made by standing wave pulses separated by Δt as shown in Fig. 5.13(a) [121]. The final interference amplitude depends on the difference in phase between the two copies accumulated during Δt, ϕ = [ ~G2 2m + 2μ 7π~]Δt where the first term arises from the recoil energy of the initial copy and the second term is due to a mean field effect of the condensate Fig. 5.13(a). Here it is assumed that one of the condensates has few atoms and consequently negligible mean field energy. The measured signal is then the number of atoms diffracted by the two pulses divided by the maximum number of atoms diffracted (i.e., without the second pulse). It can be seen in Fig. 5.13(c) that the signal oscillates faster for a larger mean field of a condensate with a higher number of atoms. Such a change in the oscillation frequency can be expected to occur in the presence of Optically Induced scattering length changes. Compared to the Bragg technique, this method yields a higher frequency resolution of mean field shifts simply with an increase in the measurement time Δt (Fig. 5.13(d)). Thalhammer et al. recently induced an optical Feshbach resonance via a coherent 82 Raman transition [122]. This stimulated Raman scheme offers a more stable control of the photoassociation light frequency and has a slightly less severe restriction on the light intensity. It can be achieved in our existing experiment by adding a second laser with a suitable frequency in order to couple the colliding atoms to a bound state in the ground molecular potential. 83 Figure 5.7: Experimental configuration for photoassociation. Shown are the CO2 laser FORT and the photoassociation beams. The Bragg (kicking) beams were aligned such that a horizontal standing wave was created along the long axis of the FORT. 84 0 500 1000 1500 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 Laser detuning(D/2p) in MHz Number of atoms a b c d e Figure 5.8: Photoassociation spectrum of the 1g(P3/2), v = 152 state. 0 MHz on this scale corresponds to a point 713 GHz below the 87Rb D2 line. The states a through e correspond to 2,−2, 3,−3⟩, 3,−2, 3,−3⟩, 1,−1, 3,−2⟩, 2,−1, 3,−2⟩, 1, 0, 1,−1⟩ of Ref. [118] labeled by the F, f, I, i⟩ quantum numbers. 85 −350 −300 −250 −200 −150 −100 −50 0 50 100 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 Laser detuning in MHz Number of atoms J=0 J=2 Figure 5.9: Photoassociation of the 0− g (∼ S1/2 + P3/2), v = 1 state showing the J=0 and J=2 rotational levels. Each point is separated by 5 MHz. 86 Figure 5.10: (a) First order Bragg diffraction as a function of the frequency difference between the beams used to create the standing wave. (b) Percentage of Bragg diffracted atoms for two photoassociation light detunings, Δ/2π =+10MHz (red) and 10MHz (blue). 87 −20 −15 −10 −5 0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 x 10−12 Kinel(cm3/s) detuning (MHz) Figure 5.11: Measured inelastic collision rate coefficients for the 0− g (P3/2), v = 1, J = 2 state. Also shown is a lorentzian fit to the data from which values of Γspon and Γstim are obtained. 88 −20 −15 −10 −5 0 5 10 15 20 25 97 98 99 100 101 102 103 scattering length (a 0 ) detuning (MHz) Figure 5.12: Plot of the scattering length a in units of a0 = 0.53˚A , the Bohr radius, calculated from the data in Fig. 5.11. The dashed line is the background scattering length of 100a0 and the detuning is with respect to the 0− g (P3/2), v = 1, J = 2 state. 89 Figure 5.13: Interference between condensates as a measure of mean field energy. Please refer to text for details. 90 CHAPTER 6 CONCLUSIONS 6.0.1 Summary The work covered in this study explored two features of a periodically kicked quantum system  transport and sensitivity to deviations from resonance. To prepare the required initial momentum state for these experiments, a BEC was created in an alloptical trap. Exposing it to short periodic kicks from an offresonant standing wave, with the correct choice of parameters, enabled investigation of momentum transport. The first system was a quantum δkicked accelerator. Specifically three higher order resonances of an accelerator mode were observed and their parameter dependence characterized. Two theoretical approaches were used to analyze the modes  one based on rephasing of momentum states separated by the order of the resonance. It was also seen that the dynamics could be explained as contributions from a family of rays of a fictitious classical system, obtained close to the resonance (ϵ → 0). A quantum ratchet was realized at a primary resonance (T1/2). The dependence of the directed ratchet current on the momentum of the initial state, or quasimo mentum, was investigated and explained by theory. A result of this quasimomentum dependence is that the finite momentum width of a BEC was found to suppress the ratchet current after some kicks. Measurement of the fidelity or overlap of a resonant QDKR state with an offresonant state was performed. The widths of the resultant peaks were found to scale as the inverse cube of the measurement time, in units of kicks. A theoretical analysis revealed a similar scaling behavior in the presence of acceleration and was validated 91 by experiment. Furthermore, velocity selection in the measurement process led to a reduced momentum width of the final zeroth order state compared to the initial state. Finally, photoassociation spectroscopy was performed on a 87Rb BoseEinstein condensate for the 1g and 0− g long range molecular states to realize an Optical Feshbach resonance. The observed loss spectrum was used to analyze the change in the elastic scattering length. 6.0.2 Future work The results described in this work lays out the groundwork for further refinements to the experiments and future research. It became clear during the search for Optical Feshbach resonances that extracting the scattering lengths near a photoassociation resonance is more involved than with a Magnetic Feshbach resonance. Bragg spectroscopy as a measure of the scattering length is time consuming, with the photoassociation losses adding to the difficulties. Recently it was shown that the benefits of the two could instead be combined: a robust magnetic Feshbach resonance in the presence of a boundbound photoassociation light provides the flexibility of the optical method with a minimized atom loss [123]. Some changes to our existing experimental setup are currently being undertaken to implement this technique. Once tunability of interatomic interactions is achieved, its effect on the resonances of the QDKR and QDKA can be probed. Complex optical potentials can be produced by overlapping standing waves of on resonant and off resonant light. Presence of a Feshbach resonances in the light will allow a new degree of control and absorptive and refractive indices of atoms diffracted by these potentials can be studied. Another model popular with quantum chaologists is the δkicked harmonic oscillator. Its phase space displays a stochastic web, the thickness of which is determined by the amount of underlying classical chaos. Identifying such a web in the quantum 92 version and exploring its properties can yield answers to questions on quantum chaos. Such a system can be implemented for instance by kicking the condensate while it is still in the trap. 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