Realization of balanced gain and loss in a time-dependent four-mode Bose-Hubbard model

A quantum system exhibiting $\mathcal{PT}$ 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.

Figure 1

(a) Open non-Hermitian two-mode system and (b) closed Hermitian four-mode system. The currents between the outer reservoir wells and the inner wells account for the in- and out-coupling of particles in the open quantum system. To reproduce the properties of an open system the additional parameters $J_{01}\left(t\right)$, $J_{23}\left(t\right)$, ${\mu }_0\left(t\right)$, and ${\mu }_3\left(t\right)$ have to be varied time dependently.

Figure 2

Observables and parameters for a weakly interacting BEC with $g=0.1$ described by the BH model. The coupling strength is fixed to $\gamma =0.5$ and the occupation ratio between the embedded system and the reservoir is 10 for a total number of $N=110$ particles. In comparison with the MF limit (dashed lines), the many-body system (solid lines) shows no quasistationary behavior, which is clearly visible by looking at (a) the occupation numbers $n_1$ and $n_2$, and the reduced current (b) $\tilde{j}{}_{12}$ and (d) $c_{12}$. The many-body calculations also show an earlier breakdown of the system (dotted vertical line) due to (c) $\tilde{j}{}_{23}$ dropping to zero, leading to (e) divergent tunneling rates and (f) on-site energies. Note that panels (a) and (d) have two separate vertical axes due to the different ranges of their observables.

Figure 3

Equations (30) in the vicinity of pure states. Since both lines are almost parallel, the unique solution cannot be calculated accurately. By projection of the parameters (29) onto the imaginary part yielding $({\tilde{\mu }}_0,{\tilde{\mu }}_3)$, a constant particle flow between the inner wells is achieved. While in principle all points on the solid curve representing Eq. (30b) are valid, our approach stays as close as possible to the former approach adapted from the MF.

Figure 4

Same as Fig. 2 but with modified on-site energies as shown in Fig. 3. The BH dynamics (solid lines) show (a) stationary occupation numbers in the embedded wells and (b) stationary currents between neighboring wells, which we call the BGL state. There is still an earlier breakdown in comparison with the MF dynamics (dashed lines), now caused by $\tilde{j}{}_{01}$ dropping to zero. (f) Due to the modifications (32), some parameters are no longer monotonic. Note that panels (a) and (d) have two separate vertical axes due to the different ranges of their observables.

Figure 5

Comparison of the first approach from Fig. 2 (dashed lines) with the second approach from Fig. 4 (solid lines) for a total number of 1100 particles with the same occupation ratio as before. Many-body calculations were performed within the BBR approximation. (a)–(d) Although the time of the breakdown (dotted vertical line) almost matches that of the MF limit in Figs. 2 and 4 (note that the calculation ends slightly before $t=10$), there are still deviations from the MF dynamics; cf. Figs. 2 and 4. For both approaches the time-dependent parameters are in good agreement and only show different behavior close to the breakdown. (f) The same holds for the purities of the four-mode system $P_4$ and the embedded two-mode system $P_2$. (g), (h) The standard deviations for the embedded observables are shown for the BGL state. Note that panels (a), (d), and (e) have two separate vertical axes due to the different ranges of their observables.