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 method for the direct computation of the Wigner function by solving a coupled system of linear partial differential equations. Our procedure is applicable to arbitrary binding potentials. We introduce a modified spectral tau method that uses Chebyshev polynomials as shape functions to approximate the solution.Since two differential equations are solved simultaneously, the resulting linear equation system is overdetermined. We approximate its solution by a least-squares method. We prove the stability and convergence of our scheme. As an application, we compute numerically the Wigner function for the harmonic oscillator. Our calculations show excellent agreement with known analytic results. ͓S1050-2947͑98͒04704-0͔ PACS number͑s͒: 03.65.Bz, 02.70.Hm 2ប ͵ Ϫϱ ϱ dy e ipy/ប Ẽ * ͩ qϪ y 2 ͪ Ẽͩ qϩ y 2 ͪ

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. To have an intuitive picture of the quantum-mechanical dynamics of a wave packet we are usually confined to the semiclassical regime, that is, to orbits with action large compared to Planck's constant ͓1,2͔, or to special systems like the harmonic oscillator, where the quantum evolution equations in phase space are identical to the classical ones ͓3͔. In this paper we analyze the center-of-mass motion of cold atoms in an amplitude-modulated standing laser field in the limit of large detuning. In this limit we can describe the dynamics by a sinusoidally modulated cosine potential.In terms of this physical system we propose a scheme which enables us to describe a wave packet, localized near a resonance of a classical mixed phase space, by classical dynamics in a modified potential. We apply the theory of Henriksen et al. ͓4͔ to replace the potential in the high-order quantum Liouville equation by an effective potential in such a way that we obtain a classical Liouville equation. Hence we describe the quantum motion as a classical motion in this modified potential. We are then able to characterize the quantum effect by comparing the modified dynamics with the dynamics in the original potential. This method is applicable well beyond the semiclassical regime for many different potentials.Usually quantum effects on wave packets express themselves in the revival and fractional revival properties ͓5͔ or in the occurrence of tunneling phenomena ͓6͔. Both take place on a comparatively long-time scale so that we intuitively do not expect quantum effects to be visible on a short-time scale. We disprove this intuitive assumption in our model where we use the center-of-mass motion of cold atoms in a standing amplitude-modulated laser field. Here we demonstrate that the momentum distribution after each cycle of the modulation is peaked at smaller momenta than we would expect classically. This shows that the atoms are traveling slower than we would expect from classical simulations and we can give a very simple explanation of this ''quantum slow motion'' phenomenon.We investigate a cloud of two-level atoms situated in a standing laser field with a periodic modulated amplitude. This system has been the subject of several experiments ͓7,8͔. The Hamiltonian of the center-of-mass motion in the limit of large detuning is ͓9͔ H͑t ͒ϭ p 2 2 Ϫ͑1Ϫ2⑀ cos t ͒cos q, ͑1͒where p and q denote scaled dimensionless momentum and position, t time, and and ⑀ are the parameters defining the depth of the standing w...

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