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

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.

Figure 1

(a) Time evolution of the purity $P$ for four different values of the gain-loss parameter $\gamma$. With increasing values of $\gamma$ the oscillations become stronger and their frequencies become smaller. The comparison of the purity with the total particle number $\langle n\rangle$ for (b) $\gamma =0.5$ and (c) $\gamma =1.5$ shows the similar behavior of the two oscillation frequencies. In all calculations the pure initial state $c_{1/2}=0.5\pm 0.5i$ and the parameters $J=1, g=0.5$ and $N_0=100$ were used and it was averaged over 500 trajectories.

Figure 2

(a) The purity of the stationary ground and excited states of the $\mathcal{PT}$-symmetric Gross-Pitaevskii equation does not show oscillations but decays slowly, and in contrast to an oscillating state with $c_{1/2}=0.5\pm 0.5i$ does not reach small purities. The results were obtained using $g=0.5$. (b) Increasing the nonlinearity parameter $g$ results in a faster overall decay of the purity but changes only slightly the frequency of the oscillations. The initial state is pure with $c_{1/2}=0.5\pm 0.5i$. In all calculations the parameters $J=1, \gamma =0.5$, and $N_0=100$ were used and it was averaged over 500 trajectories.

Figure 3

(a) The contrast $\nu$ for three different values of $\gamma$ shows oscillations. Every second peak is smaller and narrower. For panel (b) $\gamma =0.5$ the overall behavior of the contrast is dominated by the imbalance $I$ and for panel (c) $\gamma =1.5$ by the purity $P$. (d) The contrasts of the stationary ground and excited states of the $\mathcal{PT}$-symmetric Gross-Pitaevskii equation lie nearly perfectly on top of each other. They do not oscillate and stay high compared to an oscillating state. (e) The main effect of the nonlinearity $g$ is that it dampens the oscillations of the contrast. For all calculations except the stationary states the pure initial state $c_{1/2}=0.5\pm 0.5i$ was used. If not stated otherwise in the legend the parameters are $J=1, \gamma =0.5, g=0.5$, and $N_0=100$ and it was averaged over 500 trajectories.