Tunneling of an extended object in a dissipative environment: Suppression of tunneling of superconducting vortices caused by a remote gate

We discuss a recent experiment in which the resistance of a superconducting film has been measured in magnetic field. A strong decrease of the superconducting film resistance has been observed when a metallic gate is placed above the film. We study how the magnetic coupling between vortices in a thin superconducting film and electrons in a remote unbiased gate affects the tunneling rate of the vortices. We show that the response of electrons inside the gate to a change in the vortex position can lead to a dramatic suppression of the vortex tunneling, restoring the superconducting property in accord with the experiment. We interpret the change in the resistance of the film as a transition from “metallic” to “insulating” phases in the system of tunneling vortices caused by the gate. We examine two general approaches to analyze tunneling in the presence of slow low-energy degrees of freedom: the functional-integral and scattering formalisms. In the first one, the response of the electrons inside the metallic gate to a change in the vortex position is described by the “tunneling with dissipation”. We consider the Eddy current induced in the gate by the magnetic flux of the vortex as a result of tunneling. In the second approach, the response is given in terms of scattering of the electrons by the magnetic flux of the vortex in a way similar to the Aharonov-Bohm scattering. The response of the electrons to a sudden change in the vortex position leads to the Orthogonality Catastrophe that opposes the vortex tunneling. We conclude that adding a gate to superconducting films can be an effective tool for studying the dynamics of vortices at low temperatures.