$\mathcal{PT}$-symmetric gain and loss in a rotating Bose-Einstein condensate

$\mathcal{PT}$-symmetric quantum mechanics allows finding stationary states in mean-field systems with balanced gain and loss of particles. In this work we apply this method to rotating Bose-Einstein condensates with contact interaction which are known to support ground states with vortices. Due to the particle exchange with the environment transport phenomena through ultracold gases with vortices can be studied. We find that even strongly interacting rotating systems support stable $\mathcal{PT}$-symmetric ground states, sustaining a current parallel and perpendicular to the vortex cores. The vortices move through the nonuniform particle density and leave or enter the condensate through its borders creating the required net current.

Figure 1

Ground states of the rotating isotropic harmonic oscillator without gain and loss ($\gamma =0$) for $\mathrm{\Omega }=0.85$ (a), $\mathrm{\Omega }=0.88$ (b, c), $\mathrm{\Omega }=0.91$ (d), $\mathrm{\Omega }=0.94$ (e, f) in all possible $y$-symmetric configurations. Note that the states with two, three, and four vortices exhibit a two-, three-, and fourfold symmetry instead of full rotational symmetry. The interaction strength is fixed to $Na=1$.

Figure 2

Mean-field energy of the four ground states $v_1$ at $\mathrm{\Omega }=0.85$ (a), $v_2$ at $\mathrm{\Omega }=0.88$ (b, c), $v_3$ at $\mathrm{\Omega }=0.91$ (d), and $v_4$ at $\mathrm{\Omega }=0.94$ (e, f) for $Na=1$ as a function of the in- and out-coupling strength $\gamma$. The orientations are chosen in the same way as presented in Fig. 1. While both ground states with two vortices on a central axis parallel to the current are unstable for any $\gamma \ne 0$ (b, f), all other configurations show stable stationary ground states in some parameter regimes. All spectra with the exception of the state $v_3$ (d) are even functions of $\gamma$. Inside the spectra of the state $v_3$ (d), the two tangent bifurcations and the involved states at the parameter $\gamma =0$ are marked as a reference for Fig. 3.

Figure 3

Wave functions marked in the spectra of the state $v_3$ for $\mathrm{\Omega }=0.91$ and $Na=1$ shown in Fig. 2. The two states at the bifurcations for negative (a) and positive (b) values of $\gamma$ are shown above the respective bifurcation partners (c) and (d) taken at $\gamma =0$. The particle density is shown as a color map, in which darker regions correspond to higher densities, while the currents are depicted by bright blue arrows. Their lengths are proportional to the current strength.

Figure 4

The particle density of the four ground states in three dimensions with one to four vortices is shown as a color map, in which darker regions correspond to higher densities, and the currents are depicted by bright blue-headed arrows. The vortex centers are highlighted by white lines. For $\mathrm{\Omega }=0.85$ (a) one central vortex exists, while for $\mathrm{\Omega }=0.87$ (b), $\mathrm{\Omega }=0.9$ (c), and $\mathrm{\Omega }=0.94$ (d) all vortices are located off center.

Figure 5

Mean-field energy of the ground states with zero to four vortices. The in- and out-coupling parameter $\gamma$ is increased until the mean-field energy undergoes a tangent bifurcation, at which the states vanish. The bifurcation partners are not shown and correspond to the same states with an additional excitation in the $z$ direction.

Figure 6

The four ground states with one to four vortices after evolving to their maximum $\gamma$. The particle density is shown as a color map, in which darker regions correspond to higher densities, and the currents are depicted by bright blue-headed arrows. The vortex centers are highlighted by white lines. Additional vortices are added to the ground state at $\mathrm{\Omega }=0.85$ and $\gamma =0.52$ (a), $\mathrm{\Omega }=0.87,\gamma =0.5$ (b), and $\mathrm{\Omega }=0.94,\gamma =0.54$ (d). The three vortex state $\mathrm{\Omega }=0.9,\gamma =0.51$ (c) is mainly unchanged.

Figure 7

The screwing strength given as ${d\varphi /dz|}_{z=0}$ of the parametrized vortices, as a function of $\gamma$ and for different rotation frequencies $\mathrm{\Omega }$. The original vortices existing from $\gamma =0$ upward are shown as lines, and the new vortices arising for larger parameters $\gamma$ are depicted as different points. Note that all vortices of the same type and from the same wave function are shifted equally.