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arXiv:quant-ph/9901001

Quantum slow motion

M. Hug, Gerard J Milburn

Published Jan 1, 1999• 0 citations
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Abstract

We simulate the center of mass motion of cold atoms in a standing, amplitude modulated, laser field as an example of a system that has a classical mixed phase-space. We show a simple model to explain the momentum distribution of the atoms taken after any distinct number of modulation cycles. The peaks corresponding to a classical resonance move towards smaller velocities in comparison to the velocities of the classical resonances. We explain this by showing that, for a wave packet on the classical resonances, we can replace the complicated dynamics in the quantum Liouville equation in phase-space by the classical dynamics in a modified potential. Therefore we can describe the quantum mechanical motion of a wave packet on a classical resonance by a purely classical motion.

DOI
10.48550/arxiv.quant-ph/9901001

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We simulate the center of mass motion of cold atoms in a standing, amplitude modulated, laser field as an example of a system that has a classical mixed phase-space. We show a simple model to explain the momentum distribution of the atoms taken after any distinct number of modulation cycles. The peaks corresponding to a classical resonance move towards smaller velocities in comparison to the velocities of the classical resonances. We explain this by showing that, for a wave packet on the classical resonances, we can replace the complicated dynamics in the quantum Liouville equation in phase-space by the classical dynamics in a modified potential. Therefore we can describe the quantum mechanical motion of a wave packet on a classical resonance by a purely classical motion.
We investigate the center-of-mass motion of cold atoms in a standing amplitude modulated laser field. We use a simple model to explain the momentum distribution of the atoms after any distinct number of modulation cycles. The atoms starting near a classical phase-space resonance move slower than we would expect classically. We explain this by showing that for a wave packet on the classical resonances we can replace the complicated dynamics in the quantum Liouville equation in phase space by its classical dynamics with a modified potential.
Summary form only given. We investigate the nonlinear motion of two-level atoms in an amplitude modulated standing laser field in the limit of large detuning. In this case the center-of-mass-motion is described by the Hamiltonian H(p,q,t)=p/sup 2//2/spl middot//spl kappa/(1/spl middot/2/spl epsiv/sint)cosq, where p and q denote scaled dimensionless momentum and position, t time, and /spl kappa/ and /spl epsiv/ are the parameters defining the depth of the standing wave and the strength of the amplitude modulation, respectively. This system is prominent for proposing the appearance of coherent tunneling between nonlinear resonances of the corresponding classical stroboscopic phase-space map. That means a wave packet initially confined near one such resonance will stay localized and, depending on the modulation parameters will coherently tunnel between delocalized resonances.
We present experimental results for the dynamics of cold atoms in a far detuned amplitude-modulated optical standing wave. Phase-space resonances constitute distinct peaks in the atomic momentum distribution containing up to 65% of all atoms resulting from a mixed quantum chaotic phase space. We characterize the atomic behavior in classical and quantum regimes and we present the applicable quantum and classical theory, which we have developed and refined. We show experimental proof that the size and the position of the resonances in phase space can be controlled by varying several parameters, such as the modulation frequency, the scaled well depth, the modulation amplitude, and the scaled Planck's constant of the system. We have found a surprising stability against amplitude noise. We present methods to accurately control the momentum of an ensemble of atoms using these phase-space resonances which could be used for efficient phase-space state preparation.
We present a nonperturbative result for the spectrum of spontaneous photons emitted by an atom coupled on resonance to a standing-wave cavity mode. Motion of the atom through the standing-wave mode function is included in a classical way. We discuss the form of the spectrum as a function of the velocity of the atom in the limit of strong dipole coupling. For slow atoms the normal-mode doublet (``vacuum'' Rabi spectrum) survives with small changes. The motion of a fast atom through the standing wave averages the coupling constant to zero, and the spectrum reverts to a single peak. At intermediate velocities the spectrum has a complicated multipeaked structure that can be understood in terms of interfering probability amplitudes for different quantum-mechanical emission paths or as a frequency-modulation spectrum for radiating coupled oscillators.
Cold atoms in optical potentials provide an ideal test bed to explore quantum nonlinear dynamics. Atoms are prepared in a magneto-optic trap or as a dilute Bose–Einstein condensate and subjected to a far detuned optical standing wave that is modulated. They exhibit a wide range of dynamics, some of which can be explained by classical theory while other aspects show the underlying quantum nature of the system. The atoms have a mixed phase space containing regions of regular motion which appear as distinct peaks in the atomic momentum distribution embedded in a sea of chaos. The action of the atoms is of the order of Planck’s constant, making quantum effects significant. This tutorial presents a detailed description of experiments measuring the evolution of atoms in time-dependent optical potentials. Experimental methods are developed providing means for the observation and selective loading of regions of regular motion. The dependence of the atomic dynamics on the system parameters is explored and distinct changes in the atomic momentum distribution are observed which are explained by the applicable quantum and classical theory. The observation of a bifurcation sequence is reported and explained using classical perturbation theory. Experimental methods for the accurate control of the momentum of an ensemble of atoms are developed. They use phase space resonances and chaotic transients providing novel ensemble atomic beamsplitters. The divergence between quantum and classical nonlinear dynamics is manifest in the experimental observation of dynamical tunnelling. It involves no potential barrier. However a constant of motion other than energy still forbids classically this quantum allowed motion. Atoms coherently tunnel back and forth between their initial state of oscillatory motion and the state 180 ° out of phase with the initial state.
We study the motion of a cold atom in a frequency-modulated standing laser wave. If the detuning between the atomic electronic transition and the field is large, the atom moves in a modulated optical potential demonstrating known classical nonlinear effects such as chaos and nonlinear resonances. If the atom-field detuning is small, then two optical potentials emerge in the system, and the atom performs Landau-Zener (LZ) tunnelings between them. It is a radically non-classical behavior. However, we show that classical nonlinear structures in system's phase space (KAM-tori and chaotic layers) survive. Quantum effect of LZ tunnelings only induces small random jumps of trajectories between these structures (dynamical tunneling).