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arXiv:cond-mat/0204086 · Physical Review A

Josephson spectroscopy of a dilute Bose-Einstein condensate in a double-well potential

E. Sakellari, M. L. Leadbeater, Niels J. Kylstra, Charles S. Adams

Published Sep 23, 2002• 20 citations
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Abstract

The dc and ac Josephson dynamics of a Bose-Einstein condensate in a double-well potential is analyzed in terms of transitions between energy eigenstates. By solving the time-independent and time-dependent Gross-Pitaevskii equations in one dimension, we identify population transfer associated with level crossings, and determine the critical velocity that characterizes the transition. We test the validity of a nonlinear two-state model, and show that for the experimentally interesting case, where the critical velocity is large, the influence of higher-lying states is important.

DOI
10.1103/physreva.66.033612

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The dc and ac Josephson dynamics of a Bose-Einstein condensate in a double-well potential is analyzed in terms of transitions between energy eigenstates. By solving the time-independent and time-dependent Gross-Pitaevskii equations in one dimension, we identify population transfer associated with level crossings, and determine the critical velocity that characterizes the transition. We test the validity of a nonlinear two-state model, and show that for the experimentally interesting case, where the critical velocity is large, the influence of higher-lying states is important.
We propose an experiment that would demonstrate the dc and ac Josephson effects in two weakly linked Bose-Einstein condensates. We consider a time-dependent barrier, moving adiabatically across the trapping potential. The phase dynamics are governed by a "driven-pendulum" equation, as in current-driven superconducting Josephson junctions. At a critical velocity of the barrier (proportional to the critical tunneling current), there is a sharp transition between the dc and ac regimes. The signature is a sudden jump of a large fraction of the relative condensate population. Analytical results are compared with a numerical integration of the Gross-Pitaevskii equation, in an experimentally realistic situation.
The dynamics of an F=1F=1F=1 spinor condensate in a two-well potential is studied within the framework of the Gross-Pitaevskii equation. We derive two-mode equations relating the population imbalances, the phase differences among the condensates at each side of the barrier and the time evolution of the different Zeeman populations for the case of small population imbalances. The case of zero total magnetization is scrutinized in this limit demonstrating the ability of a two-mode analysis to describe to a large extent the dynamics observed in the Gross-Pitaevskii equations. It is also demonstrated that the time evolution of the different total populations fully decouples from the Josephson tunneling phenomena. All the relevant time scales are clearly identified with microscopic properties of the atom-atom interactions.
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Two aspects of the transport of the repulsive Bose-Einstein condensate (BEC) in a double-well trap are inspected: impact of the interatomic interaction and analogy to the Josephson effect. The analysis employs a numerical solution of 3D time-dependent Gross-Pitaevskii equation for a total order parameter covering all the trap. The population transfer is driven by a time-dependent shift of a barrier separating the left and right wells. Sharp and soft profiles of the barrier velocity are tested. Evolution of the relevant characteristics, involving phase differences and currents, is inspected. It is shown that the repulsive interaction substantially supports the transfer making it possible i) in a wide velocity interval and ii) three orders of magnitude faster than in the ideal BEC. The transport can be approximately treated as the d.c. Josephson effect. A dual origin of the critical barrier velocity (break of adiabatic following and d.c.-a.c. transition) is discussed. Following the calculations, robustness of the transport (d.c.) crucially depends on the interaction and barrier velocity profile. Only soft profiles which minimize undesirable dipole oscillations are acceptable.
We investigate the stability of the first excited state, the so-called “π-state,” of Bose-Einstein condensates in a double-well potential. From the condition of complex excitation energies, we determine the critical barrier height, above which the π-state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the π-state by solving the time-dependent Gross-Pitaevskii equation. Our simulation results show that the π-state in the dynamically unstable region exhibits distinctively different behavior from that in the dynamically stable region.