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Journal of Optics B Quantum and Semiclassical Optics

Quantum chaos for cold atoms in an optical lattice with a tilted harmonic trap

T. M. Fromhold, Christopher R. Tench, Sylwia Bujkiewicz, Paul Bryan Wilkinson, F. W. Sheard

Published Oct 1, 2000• 11 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/2/5/310

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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.
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.
Summary form only given. We consider the semiclassical orbits of cold atoms in an optical lattice with a magnetically-induced harmonic trap. When the laser beams are aligned with a principal axis of the trap, the atoms follow regular classical paths. But when the laser beams are tilted at an angle to the axis, the atoms undergo a controllable transition from regular motion to strong classical chaos. Our work introduces atom traps in a tilted laser field as a new type of quantum chaotic system that is accessible to experiment.
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.
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.
Using the direct perturbation technique,a general unstable chaotic solution was obtained of the Bose-Einstein condensates trapped in one-dimensional tilted optical lattice potential.The dynamical phase dia-gram of this system was studied.Theoretical analysis results and the corresponding numerical results showed the chaotic behavior of the system.It was found that Bose-Einstein condensate chaos could be controlled by chang-ing the wave length of the incident laser light or adjusting the system parameters.
Abstract Experiments with ultracold atoms in optical lattices usually involve a weak parabolic trapping potential which merely serves to confine the atoms, but otherwise remains negligible. In contrast, we suggest a different class of experiments in which the presence of a stronger trap is an essential part of the set-up. Because the trap-modified on-site energies exhibit a slowly varying level spacing, similar to that of an anharmonic oscillator, an additional time-periodic trap modulation with judiciously chosen parameters creates nonlinear resonances which enable efficient Floquet engineering. We employ a Mathieu approximation for constructing the near-resonant Floquet states in an accurate manner and demonstrate the emergence of effective ground states from the resonant trap eigenstates. Moreover, we show that the population of the Floquet states is strongly affected by the phase of a sudden turn-on of the trap modulation, which leads to significantly modified and rich dynamics. As a guideline for further studies, we argue that the deliberate population of only the resonance-induced effective ground states will allow one to realize Floquet condensates which follow classical periodic orbits, thus providing challenging future perspectives for the investigation of the quantum–classical correspondence.