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Materials Today

Cold atoms in optical lattices

Robin George Scott

Published Apr 28, 2010• 1 citation
Publisher

Abstract

This thesis describes the behaviour of cold atoms in optical lattices. In particular, it explores how transport through the energy bands of the optical lattice can be used to study quantum chaos and Bose-Einstein condensation.\n\nFirstly, this study examines the dynamics of ultra-cold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap, using both semi-classical and quantum-mechanical analyses. The atoms show mixed stable-chaotic classical dynamics, which originate from the intrinsically quantum-mechanical nature of the energy band. \nThe quantised energy levels exhibit Gutzwiller fluctuations, and the wavefunctions are scarred by an unstable periodic orbit. Distinct types of wavefunction are identified and related directly to particular parts of the classical phase space via a Wigner function analysis.\n\nSecondly, this report studies the dynamics of a rubidium Bose-Einstein condensate in a one-dimensional optical lattice and three-dimensional harmonic trap. The condensates are set in motion by displacing the trap and initially follow simple semi-classical paths, shaped by the lowest energy band. Above a critical displacement, the condensate undergoes Bragg reflection, and performs Bloch oscillations. After multiple Bragg reflections, solitons and vortices form which damp the centre-of-mass motion.\n\nFinally, the dynamics of Bose-Einstein condensates in optical lattices are investigated for different parameter regimes, as realised in recent experiments. The results reveal how the experiments can be understood, and identify regimes in which vortices trigger explosive expansion of the condensate.

DOI
10.1016/s1369-7021(10)70086-9

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This thesis describes the behaviour of cold atoms in optical lattices. In particular, it explores how transport through the energy bands of the optical lattice can be used to study quantum chaos and Bose-Einstein condensation.\n\nFirstly, this study examines the dynamics of ultra-cold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap, using both semi-classical and quantum-mechanical analyses. The atoms show mixed stable-chaotic classical dynamics, which originate from the intrinsically quantum-mechanical nature of the energy band. \nThe quantised energy levels exhibit Gutzwiller fluctuations, and the wavefunctions are scarred by an unstable periodic orbit. Distinct types of wavefunction are identified and related directly to particular parts of the classical phase space via a Wigner function analysis.\n\nSecondly, this report studies the dynamics of a rubidium Bose-Einstein condensate in a one-dimensional optical lattice and three-dimensional harmonic trap. The condensates are set in motion by displacing the trap and initially follow simple semi-classical paths, shaped by the lowest energy band. Above a critical displacement, the condensate undergoes Bragg reflection, and performs Bloch oscillations. After multiple Bragg reflections, solitons and vortices form which damp the centre-of-mass motion.\n\nFinally, the dynamics of Bose-Einstein condensates in optical lattices are investigated for different parameter regimes, as realised in recent experiments. The results reveal how the experiments can be understood, and identify regimes in which vortices trigger explosive expansion of the condensate.
We investigate the classical and quantum properties of ultracold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap. The energy versus crystal momentum dispersion relation for the lowest energy band of the optical lattice, together with the harmonic potential, define an effective Hamiltonian that we use to calculate classical atom paths. When one of the symmetry axes of the harmonic trap is aligned with the optical lattice, the atoms follow stable trajectories. But tilting this symmetry axis away from the optical lattice direction creates an unusual type of mixed stable-chaotic classical dynamics, which originates from the intrinsically quantum-mechanical nature of energy bands. In this regime, the density of quantized energy levels for the system exhibits periodic fluctuations associated with both stable and unstable periodic classical orbits. One of the unstable orbits also produces well-defined scar patterns in a subset of eigenfunctions. Wave functions with distinct spatial forms are identified and related directly to particular parts of the classical phase space using a Wigner function analysis.
We present theoretical and experimental studies of the dynamics of Bose-Einstein condensates in a one-dimensional optical lattice and a three-dimensional harmonic trap. For low atom densities and inertial forces, the condensate performs regular Bloch oscillations, and its center-of-mass motion closely follows semiclassical single-particle trajectories, shaped by the lowest-energy band. But in other regimes, the center-of-mass motion disrupts the internal structure of the condensate by generating solitons and vortex rings, which can trigger explosive expansion of the atom cloud. We use images of the atom cloud to provide experimental evidence for this internal disruption, and find that the process occurs most readily in high-density condensates undergoing slow Bloch oscillations.
Summary form only given. Since optical lattices can be used to investigate both classical energy band transport and the corresponding quantized states, they are ideal for studying the classical-quantum correspondence. Here, we investigate this correspondence for cold sodium atoms in a one-dimensional optical lattice along the x-direction and a three-dimensional harmonic trap. The energy versus crystal momentum dispersion relation for the lowest energy band of the optical lattice, together with the harmonic potential, define an effective Hamiltonian that we use to calculate classical atom paths. When a symmetry axis of the harmonic trap is aligned with the optical lattice, the atoms follow stable trajectories similar to Bloch oscillations But tilting this symmetry axis away from the optical lattice direction induces a transition to strongly chaotic dynamics.
We study the dynamics of Bose-Einstein condensates in an optical lattice and harmonic trap. The condensates are set in motion by displacing the trap and initially follow simple semiclassical paths, shaped by the lowest energy band. Above a critical displacement, the condensate undergoes Bragg reflection. For high atom densities, the first Bragg reflection generates a train of solitons and vortices, which destabilize the condensate and trigger explosive expansion. At lower densities, soliton and vortex formation requires multiple Bragg reflections, and damps the center-of-mass motion.
We show how ultracold (sub-µK temperature) atoms in a one-dimensional periodic optical potential with a harmonic trap can provide a new quantum chaotic system that is accessible to experiment. Provided that the harmonic potential varies slowly over a lattice period, the optical lattice has well defined energy bands. The energy-wavevector dispersion curves define an effective Hamiltonian that can be used to calculate classical orbits for atoms confined to a single band. We show that tilting the harmonic trap relative to the optical lattice induces a controllable transition from stable regular motion to classical chaos. The onset of chaos strongly delocalizes the atom orbits and should be manifest in both the classical and quantum properties of the trapped atoms. We also consider how the transition to chaos might affect the collective time-dependent dynamics of interacting atoms in Bose-Einstein condensates.
Experiments on atomic Bose-Einstein condensates inside quasi-one-dimensional optical lattices and related developments in the realization of atom lasers are currently at the frontiers in atomic physics. We give a short review of theoretical progress in evaluating the coherent transport of matter in such configurations, which has been based on an adaptation of the Wannier function representation of quasi-particle states in periodic potentials and on the use of advanced numerical methods for the solution of the time-dependent Gross-Pitaevskii equation. We present various methods by which the band structure of the elementary excitations of a periodic condensate may be probed and describe in terms of Bloch oscillations the coherent emission of matter pulses from a condensate inside an optical lattice under the force of gravity. A harmonic force can be applied to a condensate inside a magnetic trap by a rapid displacement of the center of the trap and a transition of the condensate from superfluid to dissipative behavior is driven by superposing an optical lattice. Finally, in the superfluid regime the combination of a constant force and a harmonic force in the presence of an optical lattice drives Josephson-type oscillations of the condensate, leading to observable resonances and multimode behavior.