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Quantum Electronics and Laser Science Conference

Cold atoms and condensates in a harmonic trap with a tilted laser field: A new system for the study of quantum chaos

Christopher R. Tench, Paul Bryan Wilkinson, F. W. Sheard, L. Eaves

Published May 12, 2000• 0 citations
OpenAlex

Abstract

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.

Related papers

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

T. M. Fromhold, Christopher R. Tench, Sylwia Bujkiewicz +2 more

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.

Journal of Optics B Quantum and Semiclassical Optics•Oct 1, 2000•11 citations

Effects of chaotic energy band transport on the quantum properties of cold atoms and condensates in an optical lattice with a tilted harmonic trap

T. M. Fromhold, R. G. Scott, Sylwia Bujkiewicz +2 more

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.

Quantum Electronics and Laser Science Conference•Jun 25, 2003•0 citations

Cold atoms in optical lattices

Robin George Scott

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.

Materials Today•Apr 28, 2010•1 citation

Quantum–classical correspondence in chaotic dynamics of laser-driven atoms

Sergei V Prants

Abstract This paper is a review article on some aspects of quantum–classical correspondence in chaotic dynamics of cold atoms interacting with a standing-wave laser field forming an optical lattice. The problem is treated from both (semi)classical and quantum points of view. In both approaches, the interaction of an atomic electic dipole with the laser field is treated quantum mechanically. Translational motion is described, at first, classically (atoms are considered to be point-like objects) and then quantum mechanically as a propagation of matter waves. Semiclassical equations of motion are shown to be chaotic in the sense of classical dynamical chaos. Point-like atoms in an absolutely deterministic and rigid optical lattice can move in a random-like manner demonstrating a chaotic walking with typical features of classical chaos. This behavior is explained by random-like ‘jumps’ of one of the atomic internal variable when atoms cross nodes of the standing wave and occurs in a specific range of the atom-field detuning. When treating atoms as matter waves, we show that they can make nonadiabatic transitions when crossing the standing-wave nodes. The point is that atomic wave packets split at each node in the same range of the atom-field detuning where the classical chaos occurs. The key point is that the squared amplitude of those semiclassical ‘jumps’ equal to the quantum Landau–Zener parameter which defines the probability of nonadiabatic transitions at the nodes. Nonadiabatic atomic wave packets are much more complicated compared to adiabatic ones and may be called chaotic in this sense. A few possible experiments to observe some manifestations of classical and quantum chaos with cold atoms in horizontal and vertical optical lattices are proposed and discussed.

Physica Scripta•Mar 6, 2017•9 citations

Chaotic Walking of Cold Atoms in a 2D Optical Lattice

Sergey V. Prants

Abstract Coherent dynamics of cold atoms in a 2D optical lattice with interfering laser beams is studied with account for internal and external degrees of freedom of an atom. A system of differential equations for coupled degrees of freedom obtained in the semiclassical approximation has regular and chaotic solutions depending on the atomic-field detuning from resonance. The Hamilton chaos is manifested in the form of chaotic Rabi oscillations and random walks of cold atoms in the lattice for relatively small resonance detunings. It is shown that the deterministic chaos appears as a result of jumps in the value of the electric dipole moment of an atom approaching the nodes of a 2D standing wave. This in turn causes a pseudorandom behavior of momenta of atoms and, as a consequence, their random walks in the absolutely rigid 2D optical lattice without any external modulation of its parameters. It is shown in numerical experiments with 106 atoms that their distributions over the lattice for different resonance detunings differ significantly. This fact can be used for detecting the effect of random walks of cold atoms in a real experiment by the absorption image method.

Journal of Experimental and Theoretical Physics•Sep 1, 2020•4 citations

On the possibility of observing chaotic motion of cold atoms in rigid optical lattices

Sergei V Prants, Leonid E. Kon’kov

The possibility of experimental observation of chaotic motion of cold atoms in a one-dimensional rigid optical lattice without any modulation of its parameters and additional impact is demonstrated theoretically and numerically. This effect of deterministic chaos arises near the optical resonance due to the nonlinear coupling of electronic and mechanical degrees of freedom in the atom. In a certain range of resonance detunings, at each crossing of the lattice-forming standing wave node by the atom, the atomic dipole moment changes pseudorandomly in a step-like manner, accompanied by an appropriate change in the atom momentum. This leads to the effect of chaotic wandering of atoms, possessing the properties of the classical deterministic chaos, namely, to the exponential sensitivity of atomic trajectories to small variations of the initial conditions and/or the control parameters. With time, part of atoms from the initial cloud changes the direction of their motion to the opposite one, which can be detected in a real experiment. Numerical experiments with lithium atoms with the spontaneous emission taken into account show that by varying the laser radiation intensity (normalised detuning of resonance) in the rigid optical lattice, one can implement the transition from the regular regime of the atomic motion to the chaotic wandering in a real experiment.

Quantum Electronics•May 9, 2017•3 citations
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 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.
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.
Abstract This paper is a review article on some aspects of quantum–classical correspondence in chaotic dynamics of cold atoms interacting with a standing-wave laser field forming an optical lattice. The problem is treated from both (semi)classical and quantum points of view. In both approaches, the interaction of an atomic electic dipole with the laser field is treated quantum mechanically. Translational motion is described, at first, classically (atoms are considered to be point-like objects) and then quantum mechanically as a propagation of matter waves. Semiclassical equations of motion are shown to be chaotic in the sense of classical dynamical chaos. Point-like atoms in an absolutely deterministic and rigid optical lattice can move in a random-like manner demonstrating a chaotic walking with typical features of classical chaos. This behavior is explained by random-like ‘jumps’ of one of the atomic internal variable when atoms cross nodes of the standing wave and occurs in a specific range of the atom-field detuning. When treating atoms as matter waves, we show that they can make nonadiabatic transitions when crossing the standing-wave nodes. The point is that atomic wave packets split at each node in the same range of the atom-field detuning where the classical chaos occurs. The key point is that the squared amplitude of those semiclassical ‘jumps’ equal to the quantum Landau–Zener parameter which defines the probability of nonadiabatic transitions at the nodes. Nonadiabatic atomic wave packets are much more complicated compared to adiabatic ones and may be called chaotic in this sense. A few possible experiments to observe some manifestations of classical and quantum chaos with cold atoms in horizontal and vertical optical lattices are proposed and discussed.
Abstract Coherent dynamics of cold atoms in a 2D optical lattice with interfering laser beams is studied with account for internal and external degrees of freedom of an atom. A system of differential equations for coupled degrees of freedom obtained in the semiclassical approximation has regular and chaotic solutions depending on the atomic-field detuning from resonance. The Hamilton chaos is manifested in the form of chaotic Rabi oscillations and random walks of cold atoms in the lattice for relatively small resonance detunings. It is shown that the deterministic chaos appears as a result of jumps in the value of the electric dipole moment of an atom approaching the nodes of a 2D standing wave. This in turn causes a pseudorandom behavior of momenta of atoms and, as a consequence, their random walks in the absolutely rigid 2D optical lattice without any external modulation of its parameters. It is shown in numerical experiments with 106 atoms that their distributions over the lattice for different resonance detunings differ significantly. This fact can be used for detecting the effect of random walks of cold atoms in a real experiment by the absorption image method.
The possibility of experimental observation of chaotic motion of cold atoms in a one-dimensional rigid optical lattice without any modulation of its parameters and additional impact is demonstrated theoretically and numerically. This effect of deterministic chaos arises near the optical resonance due to the nonlinear coupling of electronic and mechanical degrees of freedom in the atom. In a certain range of resonance detunings, at each crossing of the lattice-forming standing wave node by the atom, the atomic dipole moment changes pseudorandomly in a step-like manner, accompanied by an appropriate change in the atom momentum. This leads to the effect of chaotic wandering of atoms, possessing the properties of the classical deterministic chaos, namely, to the exponential sensitivity of atomic trajectories to small variations of the initial conditions and/or the control parameters. With time, part of atoms from the initial cloud changes the direction of their motion to the opposite one, which can be detected in a real experiment. Numerical experiments with lithium atoms with the spontaneous emission taken into account show that by varying the laser radiation intensity (normalised detuning of resonance) in the rigid optical lattice, one can implement the transition from the regular regime of the atomic motion to the chaotic wandering in a real experiment.