Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects,πoscillations, and macroscopic quantum self-trapping

Abstract

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to π (π − phase oscillations); 2) the average population imbalance is nonzero, in states with "macroscopic quantum self-trapping" (MQST). The (non-sinusoidal) gener-alization of the SJJ 'ac' and 'plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations.Analogies with Josephson oscillations between two weakly coupled reservoirs of 3 He-B and the internal Josephson effect in 3 He-A are also discussed.

Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons

“…The solution of equations (2) with the parameters of the experiment reproduces both the time evolution of E * L − E * R and the anharmonicity of the oscillations, as shown in Fig. 2c,g. In the high-density regime, if the initial population imbalance is prepared above the threshold 4 , z(0) > (4J /UN T )((UN T /2J )−1) 1/2 , a strongly asymmetric renormalization of the energy takes place between the two sites (Fig. 3a).…”

“…This is the main principle of the bosonic Josephson effect, which manifests in an ensemble of oscillatory regimes. In the absence of interactions, sinusoidal oscillations take place 4,7 with a frequencȳ Josephson physics shows the most spectacular phenomena in the nonlinear regime, when the interaction energy (U |ψ| 2 ) is greater than the coupling J . The transfer of particles from one site to the other gives rise to a dynamical renormalization of the energy in each site, resulting in anharmonic oscillations.…”

“…The Madelung transformation ψ L,R (t ) = √ N L,R (t )e iθL,R(t ) allows us to rewrite equations (1) in their dynamical form 4 :…”

“…The coupling of two macroscopic quantum states through a tunnel barrier gives rise to Josephson phenomena 1 such as Rabi oscillations 2 , the a.c. and d.c. effects 3 , or macroscopic self-trapping, depending on whether tunnelling or interactions dominate 4 . Nonlinear Josephson physics was first observed in superfluid helium 5 and atomic condensates 6,7 , but it has remained inaccessible in photonic systems because it requires large photon-photon interactions.…”

Dipolar Bose–Einstein condensates in triple-well potentials

“…The second is the symmetry-broken phase (macroscopic quantum self-trapping, MQST), in which the majority of the particles populates one well, and the third is the unstable phase, where the condensate wave function collapses. It has been pointed out in [12] and [13] that MQST is a dynamical effect arising from the interaction of the particles which reflects itself in the nonlinearity of the GPE.…”