Quantum master equation with balanced gain and loss

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 $\mathcal{PT}$-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric Bose-Einstein condensate.

Figure 1

Real and imaginary parts of the eigenvalue spectrum of the $\mathcal{PT}$-symmetric discrete Gross-Pitaevskii equation (9). For $\left|\gamma \right|\le 2$ two $\mathcal{PT}$-symmetric solutions with real eigenvalues exist. Two $\mathcal{PT}$-broken solutions with complex eigenvalues emerge at $\left|\gamma \right|=\sqrt{4-g^2}$.

Figure 2

The stationary solutions of the Gross-Pitaevskii equation (9) are transformed to many-particle states and the time evolution is calculated using the master equation (3) for (a) the ground state and (b) the excited state. The expectation value of the particle number divided by the total initial particle number at the loss site $\langle n_1\rangle$ and at the gain site $\langle n_2\rangle$ stay constant. The parameters $g=0.5$, $\gamma =0.5$, and $N_0=200$ were used and it was averaged over 2000 trajectories.

Figure 3

The expectation value of the particle number at the loss site $\langle n_1\rangle$, the gain site $\langle n_2\rangle$, and at both sites divided by the initial number of particles in the system $N_0=100$ is shown for (a) $\gamma =0$, (b) $\gamma =0.5$, (c) $\gamma =1$, and (d) $\gamma =1.5$. The initial wave functions are superpositions of the stationary states (11) with $\theta =0.2$. The strength of the on-site interaction is $g=0.5$ and it was averaged over 500 trajectories. The oscillations at the two lattice sites become more and more in phase as $\gamma$ is increased. The calculations using the master equation (solid lines) are in excellent agreement with the results of the $\mathcal{PT}$-symmetric Gross-Pitaevskii equation (dashed lines). The dashed lines are exactly on top of the solid lines in (a) and (b). Small deviations can only be seen in (c) and (d) for large times.

Figure 4

The expectation value of the particle number for two different initial wave functions. The initial wave functions are superpositions of the stationary states (11) with (a) $\theta =1.4$ and (b) $\theta =0.2$. The parameters $g=1$, $\gamma =1$, $N_0=100$ are used and the expectation values were averaged over 500 trajectories. Depending on the initial superposition the number of particles (a) diverges or (b) oscillates. Again the results of the master equation (solid lines) and the Gross-Pitaevskii equation (dashed lines) are in excellent agreement. In (b) the dashed lines are not even visible since they lie exactly on top of the solid lines.

Figure 5

The coordinate system used for the Bloch vector $b_{\alpha }=\langle {\Sigma }_{\alpha }\rangle$, $\alpha =x,y,z$. The north pole corresponds to the stationary excited state $|{\psi }_e\rangle$ of the system in the mean-field limit and the south pole is the state orthogonal to $|{\psi }_e\rangle$ in the two-dimensional space spanned by $|{\psi }_e\rangle$ and the stationary ground state. In the Hermitian case the south pole represents exactly the ground state. All initial states reside on the great circle in the $xz$ plane.

Figure 6

Dynamics on a Bloch sphere described by the master equation with balanced gain and loss (upper panels) and the $\mathcal{PT}$-symmetric Gross-Pitaevskii equation (lower panels), respectively. In all graphs the coordinate system introduced in Fig. 5 was used and all spheres are aligned appropriately. The gain-loss parameter is $\gamma =0.1$ in the left panels, $\gamma =0.7$ in the middle panels, and $\gamma =1.3$ in the right panels. The parameters $g=0.5$, $N_0=50$ [(a), (b)], $N_0=100$ (c) were used and it was averaged over 500 trajectories. The elliptic fixed point on the north pole is the excited state of the system. The ground state of the system is the second fixed point which for $\gamma =0$ resides on the south pole and wanders towards the north pole as $\gamma$ increases. The many-particle calculations and the mean-field calculations are in excellent agreement.