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arXiv:1706.03489 · Physical Review A

Balanced gain and loss in Bose-Einstein condensates without PT symmetry

Philipp Lunt, Daniel Haag, Dennis Dast, Holger Cartarius, Günter Wunner

Published Aug 16, 2017• 13 citations
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Abstract

Balanced gain and loss renders the mean-field description of Bose-Einstein condensates $\mathcal{PT}$ symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the $\mathcal{PT}$ symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.

DOI
10.1103/physreva.96.023614

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•Jan 1, 2017•0 citations
Balanced gain and loss renders the mean-field description of Bose-Einstein condensates PT\mathcal{PT}PT symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the PT\mathcal{PT}PT symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.
Bose-Einstein condensates with balanced gain and loss can support stationary states despite the exchange of particles with the environment. In the mean-field approximation this is described by the PT\mathcal{PT}PT-symmetric Gross-Pitaevskii equation with real eigenvalues. In this work we study the role of stationary states in the appropriate many-particle description. It is shown that without particle interaction there exist two nonoscillating trajectories which can be interpreted as the many-particle equivalent of the stationary PT\mathcal{PT}PT-symmetric mean-field states. Furthermore, the system has a nonequilibrium steady state which acts as an attractor in the oscillating regime. This steady state is a pure condensate for strong gain and loss contributions if the interaction between the particles is sufficiently weak.
PT\mathcal{PT}PT-symmetric quantum mechanics allows finding stationary states in mean-field systems with balanced gain and loss of particles. In this work we apply this method to rotating Bose-Einstein condensates with contact interaction which are known to support ground states with vortices. Due to the particle exchange with the environment transport phenomena through ultracold gases with vortices can be studied. We find that even strongly interacting rotating systems support stable PT\mathcal{PT}PT-symmetric ground states, sustaining a current parallel and perpendicular to the vortex cores. The vortices move through the nonuniform particle density and leave or enter the condensate through its borders creating the required net current.
We present a quantum master equation describing a Bose-Einstein condensate with particle loss on one lattice site and particle gain on the other lattice site whose mean-field limit is a non-Hermitian PT\mathcal{PT}PT-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of PT\mathcal{PT}PT-symmetric systems, such as the existence of stationary states and the phase shift of pulses between two lattice sites, are also found in the many-particle system. Visualizing the dynamics on a Bloch sphere allows us to compare the complete dynamics of the master equation with that of the Gross-Pitaevskii equation. We find that even for a relatively small number of particles the dynamics are in excellent agreement and the master equation with balanced gain and loss is indeed an appropriate many-particle description of a PT\mathcal{PT}PT-symmetric Bose-Einstein condensate.
In this work we present a generic feature of PT\mathcal{PT}PT-symmetric Bose-Einstein condensates by studying the many-particle description of a two-mode condensate with balanced gain and loss. This is achieved using a master equation in Lindblad form whose mean-field limit is a PT\mathcal{PT}PT-symmetric Gross-Pitaevskii equation. It is shown that the purity of the condensate periodically drops to small values but then is nearly completely restored. This has a direct impact on the average contrast in interference experiments which cannot be covered by the mean-field approximation, in which a completely pure condensate is assumed.
Most of the work done in the field of Bose-Einstein condensates with balanced gain and loss has been performed in the mean-field approximation using the PT\mathcal{P}TPT-symmetric Gross-Pitaevskii equation. In this work we study the many-particle dynamics of a two-mode condensate with balanced gain and loss described by a master equation in Lindblad form whose purity periodically drops to small values but then is nearly completely restored. This effect cannot be covered by the mean-field approximation, in which a completely pure condensate is assumed. We present analytic solutions for the dynamics in the noninteracting limit and use the Bogoliubov backreaction method to discuss the influence of the on-site interaction. Our main result is that the strength of the purity revivals is almost exclusively determined by the strength of the gain and loss and is independent of the amount of particles in the system and the interaction strength. For larger particle numbers, however, strong revivals are shifted towards longer times, but by increasing the interaction strength these strong revivals again occur earlier.
PT symmetry, alongside its applications and possible realizations, has established itself as a hot topic in recent years. Next to its possible impact on the fundamental nature of quantum physics, it has proved to be useful in the research on open systems of all kind. While PT physics has already been very successful in the field of optics, this thesis presents numerical calculations that show how PT symmetry can be applied to the mean-field theory of ultra-cold gases, where particles are added and removed coherently from a Bose-Einstein condensate. As long as the eigenvalues of a system are continuous functions with respect to such an exchange rate of particles, PT symmetry is preserved for weak perturbations. Therefore, a prediction of these changes for high excitation numbers is incredibly valuable to ensure stable stationary realizations. Not only does the present work test such rigorous estimates with regard to their validity, it also goes beyond analytically accessible regimes and presents numerical estimations for the nonlinear description of interacting particles. With this in mind, only low excited states have to be considered when studying non-trivial modifications to the PT-symmetric double-well potential. Not only do the results at hand show that this simplest implementation of PT-symmetric concepts is extremely stable with respect to experimental inaccuracies, but such modifications give rise to possible applications. Weakening the barrier inside the double well by introducing another particle channel from the gain to the loss well has a non-trivial effect on the net transport: In general, the net current does not profit from this modification, specific configurations even allow its complete suppression. Further, not only static systems, but rotating condensates are considered. It is shown, that the net particle transport is not only possible in the presence of the rotation; the vortices in the rotating wave functions even generate the necessary current by moving in such a way that their circular current adds to the net current.