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arXiv:cond-mat/9904147 · Physical Review A

Josephson tunneling between weakly interacting Bose-Einstein condensates

F. Meier, W. Zwerger

Published Aug 17, 2001• 88 citations
arXivHTMLPDF

Abstract

Based on a tunneling Hamiltonian description, we calculate the Josephson, normal, and interference currents between two Bose-Einstein condensates described by the Bogoliubov theory. The dominant Josephson term is of first order in the tunneling with a critical current density proportional to the ground state pressure. In contrast to superconductors, the normal current remains finite at zero temperature. We discuss the dynamics of the relative phase in a semiclassical approximation derived from an exact functional integral approach, which includes the interaction effects at fixed total particle number. It is shown that the normal current leads to a damping of the Josephson oscillations and, at long times, eliminates the macroscopic quantum self-trapping predicted by Smerzi et al. Finally, we give estimates for an experimental realization of Josephson tunneling in cold atomic gases, which indicate that coherent transfer of atoms might be realized with a ${}^{23}\mathrm{Na}$ condensate.

DOI
10.1103/physreva.64.033610

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Based on a tunneling Hamiltonian description, we calculate the Josephson, normal, and interference currents between two Bose-Einstein condensates described by the Bogoliubov theory. The dominant Josephson term is of first order in the tunneling with a critical current density proportional to the ground state pressure. In contrast to superconductors, the normal current remains finite at zero temperature. We discuss the dynamics of the relative phase in a semiclassical approximation derived from an exact functional integral approach, which includes the interaction effects at fixed total particle number. It is shown that the normal current leads to a damping of the Josephson oscillations and, at long times, eliminates the macroscopic quantum self-trapping predicted by Smerzi et al. Finally, we give estimates for an experimental realization of Josephson tunneling in cold atomic gases, which indicate that coherent transfer of atoms might be realized with a 23Na{}^{23}\mathrm{Na}23Na condensate.
Atomtronics experiments with ultracold atomic gases allow us to explore quantum transport phenomena of a weakly-interacting Bose-Einstein condensate (BEC). Here, we focus on two-terminal transport of such a BEC in the vicinity of zero temperature. By using the tunnel Hamiltonian and Bogoliubov theory, we obtain a DC Josephson current expression in the BEC and apply it to experimentally relevant situations such as quantum point contact and planar junction. Due to the absence of Andreev bound states but the presence of couplings associated with condensation elements, a current-phase relation in the BEC is found to be different from one in an sss-wave superconductor. In addition, it turns out that the DC Josephson current in the BEC depends on the sign of tunneling elements, which allows to realize the so-called $\ensuremath{\pi}$ junction by using techniques of artificial gauge fields.
Tunneling of atoms between two trapped Bose-condensed gases at finite temperatures is explored using a many-body linear-response tunneling formalism similar to that used in superconductors. To lowest order, the tunneling currents can be expressed quite generally in terms of the single-particle Green's functions of isolated Bose gases. A coherent first-order tunneling Josephson current between two atomic Bose-Einstein condensates is found, in addition to coherent and dissipative contributions from second-order condensate-noncondensate and noncondensate-noncondensate tunneling. Our work is a generalization of Meier and Zwerger, who recently treated tunneling between uniform atomic Bose gases. We apply our formalism to the analysis of an out-coupling experiment induced by light wave fields, using a simple Bogoliubov-Popov quasiparticle approximation for the trapped Bose gas. For tunneling into the vacuum, we recover the results of Japha, Choi, Burnett, and Band, who recently pointed out the usefulness of studying the spectrum of out-coupled atoms. In particular, we show that the small tunneling current of noncondensate atoms from a trapped Bose gas has a broad spectrum of energies, with a characteristic structure associated with the Bogoliubov quasiparticle u2{u}^{2}u2 and v2{v}^{2}v2 amplitudes.
We study the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BEC) confined in a double-well magnetic trap. Two Gross-Pitaevskii equations for the self-interacting BEC amplitudes, coupled by a transfer matrix element, describe the dynamics in terms of the interwell phase difference and population imbalance. In addition to the anharmonic generalization of the familiar ac Josephson effect and plasma oscillations occurring in superconductor junctions, the nonlinear BEC tunneling dynamics sustains a self-maintained population imbalance: a novel ``macroscopic quantum self-trapping'' effect.
In this thesis, I present the first experimental implementation of a Josephson junction for Bose-Einstein condensates. The weak link between two BECs constitutes the nonlinear generalization of the well known Josephson junction of weakly-coupled superconductors that are separated by a thin insulating barrier. In our experiment, the required overlap of two macroscopic wavefunctions is provided by loading a BEC into an optical double well potential. It is realized by a superposition of a one-dimensional optical lattice with a focused laser beam optical dipole trap. The tunneling dynamics between the two potential wells exhibits two distinct dynamical regimes. For small initial population imbalances of the two wells we observe nearly sinusoidal Josephson tunneling oscillations, which are characterized by an oscillating population and relative phase. The situation changes drastically, if the initial population imbalance is chosen above a critical value. In this case, resonant tunneling between the two wells is prohibited because the difference between the on-site particle interaction energies in the two wells exceeds the tunneling energy splitting. As a consequence, the atomic distribution becomes self-locked and the relative phase evolves unbound in time. This regime of prohibited tunneling, which has no analogon in superconducting Josephson junctions, is called “macroscopic quantum self-trapping”.
We report on the first realization of a single bosonic Josephson junction, implemented by two weakly linked Bose-Einstein condensates in a double-well potential. In order to fully investigate the nonlinear tunneling dynamics we measure the density distribution in situ and deduce the evolution of the relative phase between the two condensates from interference fringes. Our results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions. Additionally, we confirm a novel nonlinear effect known as macroscopic quantum self-trapping, which leads to the inhibition of large amplitude tunneling oscillations.