Balanced gain and loss in Bose-Einstein condensates without $\mathcal{PT}$ symmetry

Balanced gain and loss renders the mean-field description of Bose-Einstein condensates $\mathcal{PT}$ symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the $\mathcal{PT}$ symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.

Figure 1

(a) Imaginary part of the nonlinear spectrum of the asymmetric double well of Eq. (1) with $a_I=-0.2$. For stronger nonlinearity parameters $U$ the upper branches intersect the axis $Im\mu =0$ at lower parameters $\gamma$. (b) The imaginary part of the Bogoliubov–de Gennes eigenvalues $\omega$ are given for the upper branch, which contains the stationary state. In the nonlinear case this state is unstable. (c) For the system described by Eq. (4) with $a_R=-0.15$, i.e., the case of asymmetric on-site energies, this behavior changes. Stronger nonlinearity parameters $U$ now lead to upper branches that intersect the axis $Im\mu =0$ at larger parameters $\gamma$. This holds up to $U\approx 1.5$. For even stronger nonlinearities, the intersection with the axis again moves to lower values of $\gamma$. (d) The appropriate stability eigenvalues of the Bogoliubov–de Gennes equations are now negative imaginary numbers, which shows that the stationary state is a dynamical attractor.

Figure 2

Bloch sphere for the parameters $U=1$ and $\gamma =0.7$. Since the radius of each state equals its norm, the transparent sphere at radius one represents all normalized states. Large $z$ values, which reside on the left side of the figure, correspond to states mainly residing in well one, i.e. the gain well, while the right side contains all states favoring the loss well. The $\mathcal{PT}$-symmetric sphere in (a) shows stable oscillations (thin blue closed lines), while diverging trajectories (thick red open lines) start in the loss well for $t\to -\infty$ and end up in the gain well for $t\to \infty$. All trajectories are symmetric with respect to the $x-y$ plane. In the $\mathcal{PT}$-broken case in (b) this symmetry is lost. While the upper two diverging trajectories (thick red lines) still run from the right to the left side, the converging trajectories (thin blue converging lines) are no longer closed.

Figure 3

Intersection between the separatrix and spheres with given radii in the three-dimensional Bloch space for $U=1$ and $\gamma =0.7$. The upper panel shows the $\mathcal{PT}$-symmetric case with $a_R=0$ and $a_I=0$ where the separatrix divides the space between stable oscillations and diverging wave functions. The lower panel presents the asymmetric case with $a_R=-0.15$ and $a_I=-0.2$; due to the attractor, no oscillating trajectories can be found and all wave functions are either divergent or convergent. In both cases, the separatrix reaches the lowest radii for $\theta \approx 0.5\pi$ and $\varphi \approx 1.25\pi$. For higher radii, more wave functions become divergent, until only a small region around $\theta \approx 0.5\pi$ and $\varphi \approx 0.25\pi$ remains stable. The two cases differ in two aspects: The asymmetric case supports convergent wave functions with higher norms and is not symmetric to the axis $\theta =0.5\pi$ like the $\mathcal{PT}$-symmetric situation.

Figure 4

(a) Real part and (b) imaginary part of the spatially extended potential [Eqs. (11) and (10)] for various asymmetry parameters and $\gamma =0.05$.

Figure 5

(a) Norm of the stationary states and (b) their four smallest Bogoliubov–de Gennes eigenvalues over the value of the in- and out-coupling parameter $\gamma$ for $g=0.1, a_R=-0.01$, and $a_I=-0.08$. As long as larger values of $\gamma$ support a higher norm of the stationary state (solid line), it is stable. Therefore, the upper branch (dotted line) is unstable for all parameters $\gamma$.

Figure 6

Time evolution of the Bloch coordinates of the two stationary states ${\psi }_i$ for ${\gamma }_1\approx 0.0277$ (green dotted line) and ${\gamma }_2\approx 0.0366$ (magenta solid line). In the left panel a system with $g=0.1, a_R=-0.01, a_I=-0.08$, and $\gamma \approx 0.0277$ is used. Hence, ${\psi }_1$ is stationary while ${\psi }_2$ converges to ${\psi }_1$ within approximately four oscillations. In the right panel the in- and out-coupling parameter is changed to $\gamma \approx 0.0366$, and the two states exchange their roles.