Reversal of the circulation of a vortex by quantum tunneling in trapped Bose systems

We study the quantum dynamics of a model for a vortex in a Bose gas with repulsive interactions in an anisotropic, harmonic trap. By solving the Schrödinger equation numerically, we show that the circulation of the vortex can undergo periodic reversals by quantum-mechanical tunneling. With increasing interaction strength or particle number, vortices become increasingly stable, and the period for reversals increases. Tunneling between vortex and antivortex states is shown to be described to a good approximation by a superposition of vortex and antivortex states (Schrödinger cat state), rather than the mean-field state, and we derive an analytical expression for the oscillation period. The problem is shown to be equivalent to that of the two-site Bose-Hubbard model with attractive interactions.

Figure 1

(Color online) Landscape of the energy per particle $\mathcal{E}$ for $\Gamma =0$ (a), 0.8 (b), 4.0 (c), and 8.0 (d). Purple regions (the darker area for $\Delta n\lesssim 0$) correspond to lower $\mathcal{E}$ and red ones (the darker area for $\Delta n\simeq 1$) to higher $\mathcal{E}$. For a given panel, the contour lines are equally spaced in $\mathcal{E}$, but the spacing varies from panel to panel.

Figure 2

(Color online) Time evolution of $\overline{l}$ for $\Gamma =4.0$ and $N=2$ (a), 3 (b), and 5 (c). Panel (d) is the corresponding energy landscape showing that the vortex and antivortex states lie on closed orbits, and the arrow indicates a possible trajectory for tunneling between these states.

Figure 3

(Color online) Oscillation period $T$ as a function of $N$ at fixed interaction strength $\Gamma$ (a) and as a function of $\Gamma$ at fixed $N$ (b). The thick solid lines for $\Gamma ⩾2$ are calculated from the analytic formula (8). The lines connecting the data points in (b) are to guide the eye. The period $T$ for the data points at $\Gamma =1.6$ in (b) is hard to define and has an uncertainty of order $0.1T_0$.

Figure 4

(Color online) Overlap $\mathbf{\mid }⟨\Psi \mid \Delta {\varphi }_{\mathrm{opt}},0⟩\mathbf{\mid }$ of the wave function with the optimized phase state for $N=5$ and $\Gamma =0$ (a), 1.6 (b), 2.4 (c), and 8.0 (d).

Figure 5

(Color online) Overlap $\mathbf{\mid }⟨\Psi \mid \text{Cat};{\theta }_{\mathrm{opt}}⟩\mathbf{\mid }$ of the numerical solution of the wave function with the cat state for $N=5$ and $\Gamma =8.0$.