Vortices in Bose-Einstein condensates with $\mathcal{PT}$-symmetric gain and loss

We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex $\mathcal{PT}$-symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to nonvortex states. In addition, we study the influences of $\mathcal{PT}$ symmetry on the dynamics of nonstationary vortex states placed at off-center positions.

Figure 1

The three example potentials from Eq. (9) used in this work. For potential (c) the parameter is $d=3$.

Figure 2

Spectra as a function of the gain-loss strength $\mathrm{\Gamma }$ with the imaginary potentials (9a) (left) and (9b) (right) of the lowest states for $g=1$. The lower figures are extracts of the upper figures in the range of the bifurcation between the vortex and the excited state. The two degenerate vortex states are shown as a single state.

Figure 3

Chemical potential (top) and the individual contributions $E_{\mathrm{kin}}$ and $E_{\mathrm{pot}}$ (bottom) of the excited states as a function of the gain-loss peaks position $d$ from the imaginary potential (9c). The parameters are $g=1$ and $\mathrm{\Gamma }=2$.

Figure 4

Modulus squared of the wave functions of the excited states in potential (9c) (upper two rows) and the corresponding potentials (lower row) for different values of the gain-loss peaks position $d$. The parameters are $g=1$ and $\mathrm{\Gamma }=2$.

Figure 5

Modulus squared of the wave functions with arrows representing the current density (top) and phase (bottom) for the imaginary potential (9a) (left) and (9b) (right) for different values of the gain-loss strength $\mathrm{\Gamma }$. The nonlinearity is $g=1$.

Figure 6

Integrated projection $J_{\phi }$ of the current density on the azimuthal direction for the vortex from Fig. 5. It starts with a nonvanishing value for the vortex clearly distinguishing it from the excited $x$ state. Close to the bifurcation it rapidly decreases to zero.

Figure 7

Spectra dependent on the gain-loss strength $\mathrm{\Gamma }$ of the lowest states for potentials (9a) (left) and (9b) (right) for different values of the nonlinearity $g$. For potential (9a), the spectrum stays qualitatively the same independent of $g$. The spectrum for potential (9c) changes qualitatively to the spectrum for potential (9a) for increasing $g$.

Figure 8

Positive imaginary parts of the Bogoliubov-de Gennes eigenvalues $\omega$ for potentials (9a) (top) and (9b) (bottom). The nonlinearity is $g=1$. The ground state is always stable and has no imaginary contribution.

Figure 9

Time evolution for 1000 time steps of the stationary vortex state of potentials (9a) (left) and (9b) (right) for $g=1$ and different values of $\mathrm{\Gamma }$. The initial state is distorted with random noise on the order of ${10}^{-2}$ and propagated with the split-operator method with a step size of $dt={10}^{-3}$.

Figure 10

Vortex dynamics of the nonstationary state (21) with $x_0=y_0=0.2$. The first three plots are results for potential (9c) with $d=1$ and $\mathrm{\Gamma }=0, \mathrm{\Gamma }=0.5$, and $\mathrm{\Gamma }=0.8$. The fourth plot is the result for the $\mathcal{PT}$-broken potential (22) with $\mathrm{\Gamma }=0.5$. In each plot the vortex center trajectory is shown as a function of time. The time evolution was performed with the split-operator method with a step size of $t={10}^{-3}$.