Realizing $\mathcal{PT}$-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials

We discuss the possibility of realizing a non-Hermitian, i.e., an open two-well system of ultracold atoms by enclosing it within additional time-dependent wells that serve as particle reservoirs. With the appropriate design of the additional wells, $\mathcal{PT}$-symmetric currents can be induced to and from the inner wells, which support stable solutions. We show that interaction in the mean-field limit does not destroy this property. As a first method we use a simplified variational ansatz leading to a discrete nonlinear Schrödinger equation. A more accurate and more general variational ansatz is then used to confirm the results.

Figure 1

(a) Two-mode model with an imaginary potential $V$, where—in this case—particles are removed from the left well and injected into the right. (b) Such a system could be realized by coupling two additional wells to the system, where the tunneling currents to and from the outer wells are used as an implementation of the imaginary potential. The outer wells act as (limited) particle reservoirs and have to be chosen as time dependent.

Figure 2

Realization of a stationary $\mathcal{PT}$-symmetric solution within the four-mode model. (a) Particle numbers in well 1 (solid) and well 2 (dashed) both equal to $1/2$. (b) Particle numbers in well 0 (solid) and well 3 (dashed). (c) Particle currents $j_{01}$ (solid), $j_{12}$ (dashed), and $j_{23}$ (dash-dotted) in units of $J_{12}/\hslash$. (d) Tunneling elements $J_{01}$ (solid), $J_{23}$ (dashed) and on-site energies $E_0$ (solid), $E_3$ (dashed), in units of $J_{12}$.

Figure 3

As Fig. 2, but for oscillatory dynamics. Additionally, the total number of particles $n_1+n_2$ is plotted as crosses in (a). As the particle number in the reservoir well 0 decreases and gets close to zero, the conditions cannot be fulfilled anymore and the simulation breaks down.

Figure 4

As Fig. 2, but for the interacting case ($c_i=-1$), which leads to a collapsing dynamics, indicating a change of stability. The reservoir empties in a short time domain such that the time available for $\mathcal{PT}$ symmetry is quite short.

Figure 5

As Fig. 2, but for an adiabatic current ramp (${\Gamma }_f/J_{12}=0.5$, $t_f/t_0=60$) for realistic parameters of the external potential. Instead of the matrix elements, in (d) we show the parameters of the potential; ${\delta }^k$ marks the displacement from equidistant potential wells. There is a total number of particles of $N={10}^4$. See the text for units.

Figure 6

External potential $V_{\mathrm{ext}}$; the darker, the deeper the potential. The wells are separated into regions by walls at positions right in the middle between two minima. The integral over ${\left|\psi \right|}^2$ in a region $k$ is the particle number $n_k$; the current through a wall is denoted by $j_{k,k+1}$. See the text for exact definitions. With this separation we obtain a formulation analogous to the four-mode model (Sec. 3).

Figure 7

Adiabatic current ramp calculated with variational ansatz (lines) and four-mode model (crosses) for the same system. Shown are (a) particle numbers in the middle wells, (b) particle numbers in the outer wells, (c) potential depths of the outer wells in units of $E_0$, and (d) particle current $j_{12}$. There are small deviations, but overall the four-mode model is a valid approximation compared to the variational ansatz.

Figure 8

Modulus squared of the wave function ${\left|\psi \right|}^2$ at $x,y=0$ for different times $t$. The amplitudes of the outer parts increase or decrease, whereas the amplitudes in the middle wells stay constant, a clear sign of $\mathcal{PT}$ symmetry, as also in an experiment.