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arXiv:1509.05656 · Physical Review A

Purity oscillations in Bose-Einstein condensates with balanced gain and loss

Dennis Dast, Daniel Haag, Holger Cartarius, Günter Wunner

Published Mar 9, 2016• 24 citations
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Abstract

In this work we present a generic feature of $\mathcal{PT}$-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 $\mathcal{PT}$-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.

DOI
10.1103/physreva.93.033617

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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.
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.
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.
Distinct oscillations of the purity of the single-particle density matrix for many-body open quantum systems have been shown to exist [D. Dast, D. Haag, H. Cartarius, and G. Wunner, Phys. Rev. A 93, 033617 (2016)]. They are found in PT\mathcal{PT}PT-symmetric Bose-Einstein condensates, in which the coherence of the condensate drops and is almost completely restored periodically. For this effect the presence of a gain and loss of particles turned out to be essential. We demonstrate that it can also be found in closed quantum systems whose subsystems experience a gain and loss of particles. This is shown with two different lattice setups for cold atoms, viz., a ring of six lattice sites with periodic boundary conditions and a linear chain of four lattice wells. In both cases pronounced purity oscillations are found, and it is shown that they can be made experimentally accessible via the average contrast in interference experiments. This shows that it is possible to identify this characteristic effect in closed quantum systems which have been proposed to realize PT\mathcal{PT}PT-symmetric quantum mechanics via separation into subsystems.
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.
A quantum system exhibiting PT\mathcal{PT}PT symmetry is a Bose-Einstein condensate in a double-well potential with balanced particle gain and loss, which is described in the mean-field limit by a Gross-Pitaevskii equation with a complex potential. A possible experimental realization of such a system by embedding it into a Hermitian time-dependent four-mode optical lattice was proposed by Kreibich et al. [Phys. Rev. A 87, 051601(R) (2013)], where additional potential wells act as reservoirs and particle exchange happens via tunneling. Since particle influx and outflux have to be controlled explicitly, a set of conditions on the potential parameters had to be derived. In contrast to previous work, our focus lies on a full many-body description beyond the mean-field approximation using a Bose-Hubbard model with time-dependent potentials. This gives rise to additional quantum effects, such that the differences between mean-field and many-body dynamics are of special interest. We further present stationary analytical solutions for the embedded wells in the mean-field limit, different approaches for the embedding into a many-body system, and a very efficient method for the evaluation of hopping terms to calculate exact Bose-Hubbard dynamics.