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

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 $\mathcal{PT}$-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 $\mathcal{PT}$-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.

Figure 1

The nonoscillatory solutions (solid lines) and the stationary $\mathcal{PT}$-symmetric solutions (dotted lines) as a function of the gain-loss parameter $\gamma$ for $N_0=100$ and $U=0$. The nonoscillatory states coalesce and vanish slightly before the $\mathcal{PT}$-symmetric solutions. Both types of states vanish before the critical point ${\gamma }_{+}=2J$ (i.e., $\gamma =2J\frac{N_0+2}{N_0+1}$) indicated by the vertical black line, at which the oscillatory regime ends.

Figure 2

Time evolution of the reduced quantities [Eq. (26)] in the noninteracting limit $U=0$. All trajectories encircle the two nonoscillatory states (blue lines), thus motivating their interpretation as the many-particle equivalent of the $\mathcal{PT}$-symmetric solutions (green lines in the right panel). The right panel shows a single cone with $s_x^{\prime }>0$ in more detail. The initial particle number is $N_0=100$ and the gain-loss parameter is $\gamma =1.5$.

Figure 3

The probabilities of finding $j$ particles at site 1, at site 2, and at both sites obtained via the iterative approach (data points) are in very good agreement with the probabilities of the analytically solvable single-mode approach (solid lines) given by Eq. (35) and Eq. (37), respectively. The parameters used are $N_0=5, g=0.5$, and $\gamma =0.5$.

Figure 4

The first-order moments $s_{x,y,z}$ and $n$ of the nonequilibrium steady state for the initial particle number $N_0=100$ and different values of the interaction strength $g$. The results are obtained by a root search of the equations of motion of the Bogoliubov back-reaction method. (d) The uncertainty of the particle number defined in Eq. (39) is shown along with the expectation value.

Figure 5

(a) The purity of the steady state as a function of the gain-loss parameter $\gamma$ for different values of the interaction strength $g$. (b) For the parameter values in the colored area a steady state exists. Both the critical value of $\gamma$ where the steady state vanishes and its purity decrease rapidly for stronger interactions $g$. (c) Same as (a) but the macroscopic interaction strength $g$ is held constant using the replacement (40). (d) With $g$ held constant the critical value of $\gamma$ at which the steady state vanishes decreases much slower as compared with (b) and, as a result, pure steady states are achieved in the presence of interaction. For all calculations $N_0=100$ was used.