Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0312016 · Physical Review A

Quantum phase-space picture of Bose-Einstein condensates in a double well

Khan W. Mahmud, Heidi Perry, William P. Reinhardt

Published Feb 28, 2005• 154 citations
arXivHTMLPDF

Abstract

We present a quantum phase-space model of the Bose-Einstein condensate (BEC) in a double-well potential. In a quantum two-mode approximation we examine the eigenvectors and eigenvalues and find that the energy correlation diagram indicates a transition from a delocalized to a fragmented regime. Phase-space information is extracted from the stationary quantum states using the Husimi distribution function. We show that the mean-field phase-space characteristics of a nonrigid physical pendulum arises from the exact quantum states, and that only 4--8 particles per well are needed to reach the semiclassical limit. For a driven double-well BEC, we show that the classical chaotic dynamics is manifest in the dynamics of the quantum states. Phase-space analogy also suggests that a $\ensuremath{\pi}$ phase-displaced wave packet put on the unstable fixed point on a separatrix bifurcates to create a superposition of two pendulum rotor states---a macroscopic superposition state of BEC. We show that the choice of initial barrier height and ramping, following a $\ensuremath{\pi}$ phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.

DOI
10.1103/physreva.71.023615

Related papers

Phase engineering of controlled entangled number states in a single component Bose–Einstein condensate in a double well

Khan W. Mahmud, Heidi Perry, William P. Reinhardt

We propose a model for the creation of entangled number states (Schrödinger cat states) of a Bose–Einstein condensate in a double well through simple phase engineering. We show that a π-phase imprinted condensate in a double well evolves, with a simultaneous change of barrier height, to number states with well defined and controlled entanglement. The cat state generation is understood in terms of the underlying classical phase space dynamics of a π-phase displaced coherent state put at the hyperbolic fixed point of the separatrix of a physical pendulum. The extremity and sharpness of the final cat state is determined by the initial barrier height and the rate at which it is ramped during the evolution.

arXiv:cond-mat/0307453•Journal of Physics B Atomic Molecular and Optical Physics•Aug 13, 2003•43 citations

Dynamics of Bose-Einstein condensates in double-well potentials

M. Jääskeläinen, P. Meystre

We study the dynamics of Bose-Einstein condensates in symmetric double-well potentials following a sudden change of the potential from the Mott-insulator to the superfluid regime. We introduce a continuum approximation that maps that problem onto the wave-packet dynamics of a particle in an anharmonic effective potential. For repulsive two-body interactions the visibility of interference fringes that result from the superposition of the two condensates following a stage of ballistic expansion exhibits a collapse of coherent oscillations onto a background value whose magnitude depends on the amount of squeezing of the initial state. Strong attractive interactions are found to stabilize the relative number dynamics. We visualize the dynamics of the system in phase space using a quasiprobability distribution that allows for an intuitive interpretation of the various types of dynamics.

Physical Review A•Apr 11, 2005•47 citations

Fock-space WKB method for the boson Josephson model describing a Bose-Einstein condensate trapped in a double-well potential

V. S. Shchesnovich, Marek Trippenbach

We develop a Fock-space WKB method for a Bose-Einstein condensate (BEC) in a double-well trap, using an analogy with a single particle of either positive or negative mass in a quantum potential. The usual mean-field approach is the classical limit, and the inverse number of atoms plays the role of Planck constant. The ground state of positive-mass particles corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BECs above the threshold for symmetry breaking, the ground state is the Schr\"odinger cat state and we relate this to the double-well shape of the potential in Fock space. In the repulsive case, we identify the quantum states corresponding to the mean-field macroscopic self-trapping phenomenon. In particular, the phase-locked ($\ensuremath{\pi}$ phase) macroscopic quantum self-trapping of BECs is related to the double-well shape of the potential in Fock space. The running-phase macroscopic quantum self-trapping state is found to be subject to quantum collapses and revivals. The phase dispersion grows exponentially, reaching the absolute maximum just before the first collapse of the running phase, which may explain the growth of the phase fluctuations seen in the experiment on macroscopic quantum self-trapping.

Physical Review A•Aug 6, 2008•61 citations

Nonlinear Dynamics of Attractive Bose-Einstein Condensates in Double-Well Potential with Tunable Gaussian Barrier

Yubo Liu, Chen Zheng, Yue Zhou +5 more

Abstract Bose-Einstein Condensate (BEC) is a paradigmatic representative of macroscopic quantum states. 39 K atoms serve as an ideal system for realizing a nearly ideal BEC owing to their ultra-small background scattering length. This study further investigates the nonlinear dynamics of 39 K BEC confined in a double-well potential under attractive interactions. In this work, a Gaussian barrier is integrated with a double-well potential, and we innovatively adopt a transient dynamics approach to construct a highly non-equilibrium initial state where all atoms are confined to the left well. We track the full evolutionary process of the system from non-equilibrium excitation to stable self-trapping, elucidating wave function contraction and energy conversion. Under high barrier height conditions, tunneling is substantially suppressed: atoms localize in a single well, the longitudinal position expectation value remains constant, and the phase difference is locked, leading to the formation of a macroscopic quantum self-trapping state. Radial oscillations exhibit reduced amplitude and a slower evolution rate owing to density homogenization. A smooth, fluctuation-free momentum density profile corresponds to the static localized regime of self-trapping. Coherence transitions from dynamic oscillation to static locking, and weak attractive interactions in 39 K atoms further prolong the coherence time. This study elucidates key microscopic mechanisms, including the nonlinear coupling mechanism of attractive interactions, the dynamic-static transition rules of coherence, and momentum density acting as a microscopic probe for tunneling and soliton dynamics, thereby providing a quantitative basis for the preparation of tunable quantum states.

Chinese Physics B•Mar 16, 2026•0 citations

Reproducible mesoscopic superpositions of Bose-Einstein condensates and mean-field chaos

Bettina Gertjerenken, Stephan Arlinghaus, Niklas Teichmann +1 more

In a parameter regime for which the mean-field (Gross-Pitaevskii) dynamics becomes chaotic, mesoscopic quantum superpositions in phase space can occur in a double-well potential, which is shaken periodically. For experimentally realistic initial states, such as the ground state of some 100 atoms, the emergence of mesoscopic quantum superpositions in phase space is investigated numerically. It is shown to be reproducible, even if the initial conditions change slightly. Although the final state is not a perfect superposition of two distinct phase states, the superposition is reached an order of magnitude faster than in the case of the collapse-and-revival phenomenon. Furthermore, a generator of entanglement is identified.

arXiv:1008.0314•Physical Review A•Aug 30, 2010•14 citations

Dynamics of a spin-1 BEC in the regime of a quantum inverted pendulum

Corey Gerving

The primary study of this thesis is the experimental realization of the non-equilibrium dynamics of a quantum inverted pendulum as examined in the collective spin dynamics of a spin-1 Bose-Einstein condensate. In order to compare experimental results with the simulation past the low depletion limit, current simulation techniques needed to be extended to model atomic loss. These extensions show that traditional measurements of the system evolution (e.g. measuring the mean and standard deviation of the evolving quantity) were insufficient in capturing the quantum nature of the evolution. It became necessary to look at higher order moments and cumulants of the distributions in order to capture the quantum fluctuations. Extending the implications of the loss model further, it is possible that the system evolves in a way previously unpredicted. Spin-mixing from a hyperbolic fixed point in the phase space and low noise atom counting form the core of the experiment to measure the evolution of the distributions of the spin populations. The evolution of the system is also compared to its classical analogue, the momentum-shortened inverted pendulum. \n \nThe other experimental study in this thesis is mapping the mean-field phase space. The mean-field phase space consists of different energy contours that are divided into both phase-winding trajectories and closed orbits. These two regions are divided by a separatrix whose orbit has infinite period. Coherent states can be created fairly accurately within the phase space and allowed to evolve freely. The nature of their subsequent evolution provides the shape of the phase space orbit at that initial condition. From this analysis a prediction of the nature of the entire phase space is possible.

•Apr 3, 2013•1 citation
We present a quantum phase-space model of the Bose-Einstein condensate (BEC) in a double-well potential. In a quantum two-mode approximation we examine the eigenvectors and eigenvalues and find that the energy correlation diagram indicates a transition from a delocalized to a fragmented regime. Phase-space information is extracted from the stationary quantum states using the Husimi distribution function. We show that the mean-field phase-space characteristics of a nonrigid physical pendulum arises from the exact quantum states, and that only 4--8 particles per well are needed to reach the semiclassical limit. For a driven double-well BEC, we show that the classical chaotic dynamics is manifest in the dynamics of the quantum states. Phase-space analogy also suggests that a \ensuremathπ\ensuremath{\pi}\ensuremathπ phase-displaced wave packet put on the unstable fixed point on a separatrix bifurcates to create a superposition of two pendulum rotor states---a macroscopic superposition state of BEC. We show that the choice of initial barrier height and ramping, following a \ensuremathπ\ensuremath{\pi}\ensuremathπ phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.
We propose a model for the creation of entangled number states (Schrödinger cat states) of a Bose–Einstein condensate in a double well through simple phase engineering. We show that a π-phase imprinted condensate in a double well evolves, with a simultaneous change of barrier height, to number states with well defined and controlled entanglement. The cat state generation is understood in terms of the underlying classical phase space dynamics of a π-phase displaced coherent state put at the hyperbolic fixed point of the separatrix of a physical pendulum. The extremity and sharpness of the final cat state is determined by the initial barrier height and the rate at which it is ramped during the evolution.
We study the dynamics of Bose-Einstein condensates in symmetric double-well potentials following a sudden change of the potential from the Mott-insulator to the superfluid regime. We introduce a continuum approximation that maps that problem onto the wave-packet dynamics of a particle in an anharmonic effective potential. For repulsive two-body interactions the visibility of interference fringes that result from the superposition of the two condensates following a stage of ballistic expansion exhibits a collapse of coherent oscillations onto a background value whose magnitude depends on the amount of squeezing of the initial state. Strong attractive interactions are found to stabilize the relative number dynamics. We visualize the dynamics of the system in phase space using a quasiprobability distribution that allows for an intuitive interpretation of the various types of dynamics.
We develop a Fock-space WKB method for a Bose-Einstein condensate (BEC) in a double-well trap, using an analogy with a single particle of either positive or negative mass in a quantum potential. The usual mean-field approach is the classical limit, and the inverse number of atoms plays the role of Planck constant. The ground state of positive-mass particles corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BECs above the threshold for symmetry breaking, the ground state is the Schr\"odinger cat state and we relate this to the double-well shape of the potential in Fock space. In the repulsive case, we identify the quantum states corresponding to the mean-field macroscopic self-trapping phenomenon. In particular, the phase-locked ($\ensuremath{\pi}$ phase) macroscopic quantum self-trapping of BECs is related to the double-well shape of the potential in Fock space. The running-phase macroscopic quantum self-trapping state is found to be subject to quantum collapses and revivals. The phase dispersion grows exponentially, reaching the absolute maximum just before the first collapse of the running phase, which may explain the growth of the phase fluctuations seen in the experiment on macroscopic quantum self-trapping.
Abstract Bose-Einstein Condensate (BEC) is a paradigmatic representative of macroscopic quantum states. 39 K atoms serve as an ideal system for realizing a nearly ideal BEC owing to their ultra-small background scattering length. This study further investigates the nonlinear dynamics of 39 K BEC confined in a double-well potential under attractive interactions. In this work, a Gaussian barrier is integrated with a double-well potential, and we innovatively adopt a transient dynamics approach to construct a highly non-equilibrium initial state where all atoms are confined to the left well. We track the full evolutionary process of the system from non-equilibrium excitation to stable self-trapping, elucidating wave function contraction and energy conversion. Under high barrier height conditions, tunneling is substantially suppressed: atoms localize in a single well, the longitudinal position expectation value remains constant, and the phase difference is locked, leading to the formation of a macroscopic quantum self-trapping state. Radial oscillations exhibit reduced amplitude and a slower evolution rate owing to density homogenization. A smooth, fluctuation-free momentum density profile corresponds to the static localized regime of self-trapping. Coherence transitions from dynamic oscillation to static locking, and weak attractive interactions in 39 K atoms further prolong the coherence time. This study elucidates key microscopic mechanisms, including the nonlinear coupling mechanism of attractive interactions, the dynamic-static transition rules of coherence, and momentum density acting as a microscopic probe for tunneling and soliton dynamics, thereby providing a quantitative basis for the preparation of tunable quantum states.
In a parameter regime for which the mean-field (Gross-Pitaevskii) dynamics becomes chaotic, mesoscopic quantum superpositions in phase space can occur in a double-well potential, which is shaken periodically. For experimentally realistic initial states, such as the ground state of some 100 atoms, the emergence of mesoscopic quantum superpositions in phase space is investigated numerically. It is shown to be reproducible, even if the initial conditions change slightly. Although the final state is not a perfect superposition of two distinct phase states, the superposition is reached an order of magnitude faster than in the case of the collapse-and-revival phenomenon. Furthermore, a generator of entanglement is identified.
The primary study of this thesis is the experimental realization of the non-equilibrium dynamics of a quantum inverted pendulum as examined in the collective spin dynamics of a spin-1 Bose-Einstein condensate. In order to compare experimental results with the simulation past the low depletion limit, current simulation techniques needed to be extended to model atomic loss. These extensions show that traditional measurements of the system evolution (e.g. measuring the mean and standard deviation of the evolving quantity) were insufficient in capturing the quantum nature of the evolution. It became necessary to look at higher order moments and cumulants of the distributions in order to capture the quantum fluctuations. Extending the implications of the loss model further, it is possible that the system evolves in a way previously unpredicted. Spin-mixing from a hyperbolic fixed point in the phase space and low noise atom counting form the core of the experiment to measure the evolution of the distributions of the spin populations. The evolution of the system is also compared to its classical analogue, the momentum-shortened inverted pendulum. \n \nThe other experimental study in this thesis is mapping the mean-field phase space. The mean-field phase space consists of different energy contours that are divided into both phase-winding trajectories and closed orbits. These two regions are divided by a separatrix whose orbit has infinite period. Coherent states can be created fairly accurately within the phase space and allowed to evolve freely. The nature of their subsequent evolution provides the shape of the phase space orbit at that initial condition. From this analysis a prediction of the nature of the entire phase space is possible.