Quantum Tunneling of a Vortex between Two Pinning Potentials

A vortex can tunnel between two pinning potentials in an atomic Bose-Einstein condensate on a time scale of the order of $1s$ under typical experimental conditions. This makes it possible to detect the tunneling experimentally. We calculate the tunneling rate by phenomenologically treating vortices as charged particles moving in an inhomogeneous magnetic field. The obtained results are in close agreement with numerical simulations based on the stochastic $c$-field theory.

Figure 1

Equilibrium positions $x_{\min }$ of a vortex in the presence of two pinning potentials separated by the distance $d_0=2r_0$. Solid line: analytical result based on Eq. (1). Open circles: results of numerical simulations based on the solutions of Eq. (2). Inset: Number of particles inside a circle with the radius equal to a pinned position of the vortex. Dashed line: result of the numerical simulation. Solid line: analytical result based of the assumption that the vortex is structureless.

Figure 2

Effective magnetic field $B$ along the axis through the trap center (solid line) and density $n$ (dashed line). The two pinning potentials are separated by the distance $d_0=2\xi$. $B_0=n_{0}h/q$, where $n_0$ is the bulk density at the trap center. In this configuration the vortex is pinned classically (cf. Fig. 1) and a double-well potential appears within the effective theory.

Figure 3

The period of quantum oscillations of a vortex between two pining potentials as a function of distance between them. The open circles represent the result of the truncated Wigner approximation. The open squares are the result of the effective theory (see text). Inset: One of the trajectories of a vortex. Dashed lines indicate the $x_{\min }\approx \pm 0.6\xi$ values corresponding to $d_0\approx 1.8\xi$ in Fig. 1.