# Stationary states in the many-particle description of Bose-Einstein condensates with balanced gain and loss

## TL;DR

This paper investigates the existence and properties of stationary states in many-particle Bose-Einstein condensates with balanced gain and loss, extending mean-field PT-symmetric models to a full quantum description.

## Contribution

It demonstrates the presence of non-oscillating trajectories and a non-equilibrium steady state in the many-particle framework, highlighting the effects of particle interactions.

## Key findings

- Existence of two non-oscillating trajectories as many-particle counterparts of mean-field states.
- Identification of a non-equilibrium steady state acting as an attractor.
- Steady state remains a pure condensate under weak interactions with strong gain and loss.

## Abstract

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-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 non-oscillating trajectories which can be interpreted as the many-particle equivalent of the stationary PT-symmetric mean-field states. Furthermore the system has a non-equilibrium 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.