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arXiv:1305.1828 · Physical Review E

Dynamical tunneling of a Bose-Einstein condensate in periodically driven systems

Rajendra K. Shrestha, Jiating Ni, Wa Kun Lam, G. S. Summy, Sandro Wimberger

Published Sep 6, 2013• 25 citations
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Abstract

We report measurements of dynamical tunneling rates of a Bose-Einstein condensate across a barrier in classical phase space. The atoms are initially prepared in quantum states that extend over a classically regular island region. We focus on the specific system of quantum accelerator modes of the kicked rotor in the presence of gravity. Our experimental data is supported by numerical simulations taking into account imperfections mainly from spontaneous emission. Furthermore, we predict experimentally accessible parameter ranges over which direct tunneling could be readily observed if spontaneous emission was further suppressed. Altogether, we provide a proof-of-principle for the experimental accessibility of dynamical tunneling rates in periodically driven systems.

DOI
10.1103/physreve.88.034901

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We report measurements of dynamical tunneling rates of a Bose-Einstein condensate across a barrier in classical phase space. The atoms are initially prepared in quantum states that extend over a classically regular island region. We focus on the specific system of quantum accelerator modes of the kicked rotor in the presence of gravity. Our experimental data is supported by numerical simulations taking into account imperfections mainly from spontaneous emission. Furthermore, we predict experimentally accessible parameter ranges over which direct tunneling could be readily observed if spontaneous emission was further suppressed. Altogether, we provide a proof-of-principle for the experimental accessibility of dynamical tunneling rates in periodically driven systems.
The study of dynamical tunneling in a periodically driven anharmonic potential probes the quantum-classical transition via the experimental control of the effective Planck's constant for the system. In this paper we consider the prospects for observing dynamical tunneling with ultracold atoms in magnetic microtraps on atom chips. We outline the driven anharmonic potentials that are possible using standard magnetic traps and find the Floquet spectrum for one of these as a function of the potential strength, modulation, and effective Planck's constant. We develop an integrable approximation to the nonintegrable Hamiltonian and find that it can explain the behavior of the tunneling rate as a function of the effective Planck's constant in the regular region of parameter space. In the chaotic region we compare our results with the predictions of models that describe chaos-assisted tunneling. Finally, we examine the practicality of performing these experiments in the laboratory with Bose-Einstein condensates.
The simplicity and versatility of ultra-cold atoms make it an ideal system for studying Quantum Mechanical phenomenon like tunneling. This thesis describes progress towards experimentally measuring the time spent by a tunneling particle in the classically forbidden region. A detailed analysis of the experimental requirements to measure the tunneling time is presented along with the experimental tools developed to meet these requirements. These include a thin \SI{1.3}{\micro m} optical tunnel barrier, a smooth atomic waveguide, and a Larmor clock to measure the tunneling time. Delta-kick cooling is used to achieve the extremely low temperatures required for the tunneling time experiments. A 2.4x reduction in the atomic velocity spread is demonstrated, reducing the rms velocity spread to \SI{0.46(5)}{mm/s}. This corresponds to cooling to an effective temperature of \SI{2.0(4)}{nK}. The thin optical barrier opens up the possibility to study the decay of quasi-bound condensates via tunneling. We develop a novel trapping configuration for this purpose, in which the barrier acts as one of the walls of the trap. Inter-atomic interactions strongly dictate the escape dynamics of the condensate out of this trap, giving rise to three distinct regimes--- classical over the barrier spilling, quantum tunneling driven by interactions, and decay dominated by background losses. We show that in the tunneling regime, the decay rate depends exponentially with the chemical potential of the condensate. Experimental results show good agreement with numerical solutions of the 3D Gross-Pitaevskii equation.
Dynamical tunneling is a phenomenon that can be observed in certain dynamical systems at the borderline of quantum and classical dynamics. Rather than tunneling through an energy barrier (as in the case of “conventional” tunneling), a different constant of motion forbids this classically forbidden process. It is mostly observed in dynamical systems that exhibit a mixed classical phase space. In a mixed phase space, the dynamics of a particle can be either chaotic or regular depending on the initial position and momentum of the particle.
In the two-mode approximation Bose-Einstein condensates (BEC) in a double-well potential can be described by a many particle Hamiltonian of Bose-Hubbard type. We focus on such a BEC whose interatomic interaction strength is modulated periodically with delta\\deltadelta-kicks. This System in fact represents a model of a kicked top. The mean-field dynamics provide a rich mixed phase space with regular and chaotic regions. By increasing the kick-strength a bifurcation leads to the appearance of self-trapping states localized in regular islands. This self-trapping is also found for the many particle system, however in general suppressed by tunneling oscillations. We show that the tunneling time can be calculated from the quasi-energy splitting of the corresponding Floquet states. By varying the kick-strength these quasi-energy levels undergo both avoided and even real crossings. Therefore stimulation or actually complete destruction of tunneling is observed for this many particle system.
In a two-mode approximation, Bose–Einstein condensates (BEC) in a double-well potential can be described by a many particle Hamiltonian of Bose–Hubbard type. We focus on such a BEC whose interatomic interaction strength is modulated periodically by δ-kicks which represents a realization of a kicked top. In the (classical) mean-field approximation it provides a rich mixed phase space dynamics with regular and chaotic regions. By increasing the kick-strength a bifurcation leads to the appearance of self-trapping states localized on regular islands. This self-trapping is also found for the many particle system, however, in general suppressed by coherent many particle tunneling oscillations. The tunneling time can be calculated from the quasi-energy splitting of the corresponding Floquet states. By varying the kick-strength these quasi-energy levels undergo both avoided and even actual crossings. Therefore, stimulation or complete destruction of tunneling can be observed for this many particle system.
A Bose-Einstein condensate in a modulated, one-dimensional, anharmonic potential can exhibit dynamical tunneling between islands of regular motion in phase space. With increasingly repulsive atomic interactions, dynamical tunneling is predicted to cease due to self-trapping [S. Wüster ]. This suppression of tunneling oscillations is related to the same phenomenon that occurs in the two-mode dynamics of a repulsively interacting Bose-Einstein condensate in a double-well potential. Here we present a two-mode model for dynamical tunneling based on nonlinear Floquet states and examine the range of validity of the approximation. We characterize nonlinear dynamical tunneling for different trap strengths, modulation amplitudes, and effective Planck constants. Using the linear Floquet states we derive an expression for the critical nonlinearity beyond which tunneling ceases. Finally we demonstrate the dynamical instability of selected nonlinear Floquet states and show how to initialize some Floquet states in experiments. Our detailed survey will enable experiments to target accessible parameter regimes for the study of nonlinear dynamical tunneling.