Coupling approach for the realization of a $\mathcal{PT}$-symmetric potential for a Bose-Einstein condensate in a double well

We show how non-Hermitian potentials used to describe probability gain and loss in effective theories of open quantum systems can be achieved by a coupling of the system to an environment. We do this by coupling a Bose-Einstein condensate (BEC) trapped in an attractive double-$\delta$ potential to a condensate fraction outside the double well. We investigate which requirements have to be imposed on possible environments with a linear coupling to the system. This information is used to determine an environment required for stationary states of the BEC. To investigate the stability of the system we use fully numerical simulations of the dynamics. It turns out that the approach is viable and possible setups for the realization of a $\mathcal{PT}$-symmetric potential for a BEC are accessible. Vulnerabilities of the whole system to small perturbations can be attributed to the singular character of the simplified $\delta$-shaped potential used in our model.

Figure 1

Eigenvalues $\mu$ of the double-$\delta$ potential in dependence of the non-Hermiticity parameter $\Gamma$. The spectrum is shown for $g=1.0$ (nonlinear case, strong and colored lines) and $g=0.0$ (linear case, light gray solid lines). Other parameters are fixed to $a=1.1$ and $V=-1.0$. The spectrum consists in both cases of a ground state ${\mu }_{\text{gr}}$ and an excited state ${\mu }_{\text{ex}}$ which merge in an exceptional point at ${\Gamma }_c$. Above a critical value ${\Gamma }_{c^{*}}$, where ${\Gamma }_{c^{*}}<{\Gamma }_c$ for $g\ne 0$, two $\mathcal{PT}$-broken solutions with complex and complex-conjugate chemical potentials ${\mu }_{\text{br}}$ appear, of which only the real part is shown in the figure. Ground-state solutions between both critical points ${\Gamma }_c$ and ${\Gamma }_{c^{*}}$ are unstable (${\mu }_{\text{gr}\text{unstab}}$). All values are given in the dimensionless units introduced in Ref. [27].

Figure 2

Parameter space of the double-$\delta$ potential for fixed $a=1.1$ and $V=-1.0$. The dimensionless units introduced in Ref. [27] are used for all values. In the shaded areas stationary states with real eigenvalues exist. The lines separating the different areas show the location of the two critical points ${\Gamma }_c$ and ${\Gamma }_{c^{*}}$. Between these critical points the ground-state solutions are unstable. The figure also shows where stationary states that satisfy relation (15) in Sec. 5b can be found (${\psi }_c$). Many of these solutions are in the unstable regime (upper shaded area).

Figure 3

Sketch of the geometric alignment of two one-dimensional stationary states ${\psi }_1\left(x\right)$ and ${\psi }_2\left(x\right)$. Both states intersect at $x_0$ in their coordinate system and interact at this point.

Figure 4

Both figures show an environment supporting a stationary state ${\psi }_1$ of the double-$\delta$ potential. Particle currents $j$ are represented by solid blue arrows. Dashed red arrows represent a particle exchange between two wave functions. The free parameters for creating the environments are indicated as green dots. In (a) the environment consists of two feeders, particles are incoming via ${\psi }_2\left(x\right)$ and leaving via ${\psi }_3\left(x\right)$. In (b) the environment consists of only one wave function. The current it is carrying is partially redirected through ${\psi }_1\left(x\right)$. Figure (a) features a sketch of how ${\psi }_3\left(x\right)$ is constructed with use of the soliton solution (11) (black dotted line). Its maximum $\tilde{x}$ is positioned such that the correct function value at the coupling point is achieved.

Figure 5

Squared moduli of a stationary ground state of the double-$\delta$ potential ${\psi }_1\left(x\right)$ and the two feeders ${\psi }_2\left(x\right)$ and ${\psi }_3\left(x\right)$. The numerical data were created with the parameters $a=1.1$, $g=1.0$, $\Gamma =0.1$, and ${\psi }_{2/3}(\pm a)=0.7$, which are given as all values in the figure in the dimensionless units introduced in Ref. [27].

Figure 6

(a) Modulus squared of a stationary state of the double $\delta$ potential ${\psi }_1\left(x\right)$ and its feeder ${\psi }_2\left(x\right)$. (b) Simulated time evolution of both states. The system parameters in the dimensionless units from Ref. [27] used in the figure are $a=1.1$, $V=-1.0$, $\Gamma =0.1$, and $g=1.0$. Numerical perturbations cause both states to develop fluctuations, thus destroying the dynamics. They lose the stationary property at $t\approx 45$. Bright colors represent high probability amplitudes; compare with (a) for absolute values.