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Analysis of dynamical tunnelling experiments with a Bose-Einstein condensate

W. K. Hensinger, A Mouchet (13669555), P S Julienne (16105424), D Delande (16105427), N R Heckenberg (16060925), Halina H. Rubinsztein-Dunlop

Published Jul 15, 2004• 0 citations
OpenAlex

Abstract

Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohibited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime

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Analysis of dynamical tunneling experiments with a Bose-Einstein condensate

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Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohibited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime.

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Dynamical Tunneling with a Bose–Einstein Condensate

W. K. Hensinger

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.

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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.

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The transmission of an interacting Bose-Einstein condensate incident on a repulsive Gaussian barrier is investigated through numerical simulation. The dynamics associated with interatomic interactions are studied across a broad parameter range not previously explored. Effective one-dimensional Gross-Pitaevskii equation (GPE) simulations are compared to classical Boltzmann-Vlasov equation (BVE) simulations in order to isolate purely coherent matter-wave effects. Quantum tunneling is then defined as the portion of the GPE transmission not described by the classical BVE. An exponential dependence of transmission on barrier height is observed in the classical simulation, suggesting that observing such an exponential dependence is not a sufficient condition for quantum tunneling. Furthermore, the transmission is found to be predominately described by classical effects, although interatomic interactions are shown to modify the magnitude of the quantum tunneling. Interactions are also seen to affect the amount of classical transmission, producing transmission in regions where the noninteracting equivalent has none. This theoretical investigation clarifies the contribution quantum tunneling makes to overall transmission in many-particle interacting systems, potentially informing future tunneling experiments with ultracold atoms.

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One-dimensional quantum tunneling dynamics between two-component Bose-Einstein condensates confined in a double-well magnetic trap is investigated. One-dimensional Gross-Pitaevskii equations for two-component Bose-Einstein condensates are derived from the three-dimensional ones. We derive Feynman equations from one-dimensional Gross-Pitaevskii equations. To study tunneling dynamics we solve Feynman equations in terms of a completely numerical procedure. In contrast to single-component condensates between two-component condensates, we find that this system can take on abundant tunneling results, the full tunneling dynamical behavior is summarized in phase portrait with constant energy lines. It is found that this system can achieve self-trapping when increase interatomic interactions exceed a critical value. We give the analytical critical expressions of interatomic interactions from the system Hamiltonian.

Acta Physica Sinica•Jan 1, 2010•5 citations
Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohibited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime
Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohibited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime.
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.
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.
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.
The transmission of an interacting Bose-Einstein condensate incident on a repulsive Gaussian barrier is investigated through numerical simulation. The dynamics associated with interatomic interactions are studied across a broad parameter range not previously explored. Effective one-dimensional Gross-Pitaevskii equation (GPE) simulations are compared to classical Boltzmann-Vlasov equation (BVE) simulations in order to isolate purely coherent matter-wave effects. Quantum tunneling is then defined as the portion of the GPE transmission not described by the classical BVE. An exponential dependence of transmission on barrier height is observed in the classical simulation, suggesting that observing such an exponential dependence is not a sufficient condition for quantum tunneling. Furthermore, the transmission is found to be predominately described by classical effects, although interatomic interactions are shown to modify the magnitude of the quantum tunneling. Interactions are also seen to affect the amount of classical transmission, producing transmission in regions where the noninteracting equivalent has none. This theoretical investigation clarifies the contribution quantum tunneling makes to overall transmission in many-particle interacting systems, potentially informing future tunneling experiments with ultracold atoms.
One-dimensional quantum tunneling dynamics between two-component Bose-Einstein condensates confined in a double-well magnetic trap is investigated. One-dimensional Gross-Pitaevskii equations for two-component Bose-Einstein condensates are derived from the three-dimensional ones. We derive Feynman equations from one-dimensional Gross-Pitaevskii equations. To study tunneling dynamics we solve Feynman equations in terms of a completely numerical procedure. In contrast to single-component condensates between two-component condensates, we find that this system can take on abundant tunneling results, the full tunneling dynamical behavior is summarized in phase portrait with constant energy lines. It is found that this system can achieve self-trapping when increase interatomic interactions exceed a critical value. We give the analytical critical expressions of interatomic interactions from the system Hamiltonian.