Hermitian four-well potential as a realization of a $\mathcal{PT}$-symmetric system

A $\mathcal{PT}$-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential; however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a $\mathcal{PT}$-symmetric system, where the $\mathcal{PT}$-symmetric currents of a two-well system are implemented by coupling two additional wells to the system, which act as particle reservoirs. In terms of a simple four-mode model we derive conditions under which the two middle wells of the Hermitian four-well system behave exactly as the two wells of the $\mathcal{PT}$-symmetric system. We apply these conditions to calculate stationary solutions and oscillatory dynamics. By means of frozen Gaussian wave packets we relate the Gross-Pitaevskii equation to the four-mode model and give parameters required for the external potential, which provides approximate conditions for a realistic experimental setup.

Figure 1

(a) $\mathcal{PT}$-symmetric two-well system with tunneling rate $J$ and complex potential $\pm i\Gamma$, which describes coherent in- and outcoupling of atoms. (b) Alternatively, two additional wells serve as particle reservoirs and induce particle currents that—under application of specific system parameters to be determined—are equivalent to the $\mathcal{PT}$-symmetric currents of system (a).

Figure 2

(a) Populations $n_1=n_2=1/2$ of the middle wells, which are both equal and constant in time. (b) Populations $n_0$ (solid) and $n_3$ (dashed) of the outer wells, which depend linearly on the time, i.e., ${\dot{n}}_0=-\Gamma$ and ${\dot{n}}_3=\Gamma$. (c) Time-dependent on-site energies $E_0$ (solid) and $E_3$ (dashed). (d) Tunneling amplitudes $J_{01}$ (solid) and $J_{23}$ (dashed). Time is given in units of $t_0=\hslash /J_{12}$.

Figure 3

Same as Fig. 2, but for oscillatory dynamics. Additionally, the total number of particles in wells 1 and 2 are plotted as black crosses in (a).

Figure 4

(a) Adiabatic ramp of the current from $\Gamma =0$ to a target value of ${\Gamma }_{\mathrm{f}}$ as a function of the time $t$. Time dependences of (b) the potential depths $V_0$ (solid) and $V_3$ (dashed) and (c) the displacements of the outer wells ${\delta }_0$ (solid) and ${\delta }_3$ (dashed). (d) and (e): Number of particles in each well.