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Quantum Electronics and Laser Science Conference

Nonlinear motion of cold atoms in an amplitude modulated standing laser field

M. Hug, Gerard J Milburn

Published Jan 20, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1109/qels.1999.807607

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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 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.
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.
Summary form only given. In our experiments ultracold rubidium atoms are loaded into an optical standing wave. The amplitude of the standing wave is then modulated, creating a time dependent potential given by V(x,t)=/spl kappa/(1-2/spl epsiv/sin(2/spl pi/f/sub m/t))cos(2kx) where /spl kappa/ is the strength of the potential, /spl epsiv/ is the amplitude of modulation and f/sub m/ is the frequency of modulation. The dynamics of atoms in such a potential is chaotic except for small islands of stability which exist for certain values of /spl kappa/, /spl epsiv/ and f/sub m/. These islands are phase space resonances and rotate in phase space at the modulation frequency. By viewing the dynamics of the interaction stroboscopically, at intervals of the modulation frequency it is possible to observe these resonances in the final atomic velocity distribution.
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).
The wave function of a moderately cold atom in a stationary near-resonant standing light wave delocalizes very fast due to wave packet splitting. However, we show that frequency modulation of the field can suppress packet splitting for some atoms whose specific velocities are in a narrow range. These atoms remain localized in a small space for a long time. We demonstrate and explain this effect numerically and analytically. We also demonstrate that the modulated field can not only trap but also cool the atoms. We perform a numerical experiment with a large atomic ensemble having wide initial velocity and energy distributions. During the experiment, most of atoms leave the wave while the trapped atoms have a narrow energy distribution.
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.