Floquet Analysis of Atom Optics Tunneling Experiments

Dynamical tunneling has been observed in atom optics experiments by two groups. We show that the experimental results are extremely well described by time-periodic Hamiltonians with momentum quantized in units of the atomic recoil. The observed tunneling has a well defined period when only two Floquet states dominate the dynamics. Beat frequencies are observed when three Floquet states dominate. We find frequencies which match those observed in both experiments. The dynamical origin of the dominant Floquet states is identified.Comment: Accepted in Physical Review

Feedback cooling of atomic motion in cavity QED

We consider the problem of controlling the motion of an atom trapped in an optical cavity using continuous feedback. In order to realize such a scheme experimentally, one must be able to perform state estimation of the atomic motion in real time. While in theory this estimate may be provided by a stochastic master equation describing the full dynamics of the observed system, integrating this equation in real time is impractical. Here we derive an approximate estimation equation for this purpose, and use it as a drive in a feedback algorithm designed to cool the motion of the atom. We examine the effectiveness of such a procedure using full simulations of the cavity QED system, including the quantized motion of the atom in one dimension.Comment: 22 pages, 17 figure

Quantum feedback control of atomic motion in an optical cavity

We study quantum feedback cooling of atomic motion in an optical cavity. We design a feedback algorithm that can cool the atom to the ground state of the optical potential with high efficiency despite the nonlinear nature of this problem. An important ingredient is a simplified state-estimation algorithm, necessary for a real-time implementation of the feedback loop. We also describe the critical role of parity dynamics in the cooling process and present a simple theory that predicts the achievable steady-state atomic energies

Probability Bifurcations of L\'evy Bridges

A L\'evy bridge--a stable L\'evy stochastic process conditioned to arrive at some state at some later time--can exhibit behavior differing dramatically from the more widely studied case of conditioned Brownian (Gaussian) processes. This difference stems from a structural change in the conditioned probability density at intermediate times as the arrival position varies. This structural shift gives rise to a distinction between "short" and "long" jumps. We explore the consequences of this idea for the statistics of L\'evy vs. Brownian bridges, with applications to the analysis of the boundary-crossing problem and a computationally useful representation of L\'evy bridges that does not carry over directly from the Gaussian case.Comment: 5 pages, 7 figure

Engineering Quantum States, Nonlinear Measurements, and Anomalous Diffusion by Imaging

We show that well-separated quantum superposition states, measurements of strongly nonlinear observables, and quantum dynamics driven by anomalous diffusion can all be achieved for single atoms or molecules by imaging spontaneous photons that they emit via resonance florescence. To generate anomalous diffusion we introduce continuous measurements driven by L\'evy processes, and prove a number of results regarding their properties. In particular we present strong evidence that the only stable L\'evy density that can realize a strictly continuous measurement is the Gaussian.Comment: revtex4-1, 17 pages, 7 eps figure

Fractal templates in the escape dynamics of trapped ultracold atoms

We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in the escape data. This data would be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature--a direct result of the extreme phase space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.Comment: 11 pages with 5 figure