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We discuss a recent experiment in which the resistance of a superconductingfilm has been measured in magnetic field. A strong decrease of thesuperconducting film resistance has been observed when a metallic gate isplaced above the film. We study how the magnetic coupling between vortices in athin superconducting film and electrons in a remote unbiased gate suppressesthe tunneling rate of the vortices. We examine two general approaches toanalyze tunneling in the presence of slow low-energy degrees of freedom: thefunctional-integral and scattering formalisms. In the first one, the responseof the electrons inside the metallic gate to a change in the vortex position isdescribed by the "tunneling with dissipation". We consider the Eddy currentinduced in the gate by the magnetic flux of the vortex as a result oftunneling. In the second approach, the response is given in terms of scatteringof the electrons by the magnetic flux of the vortex in a way similar to theAharonov-Bohm scattering. A sudden change in the vortex position leads to theOrthogonality Catastrophe that opposes the vortex tunneling. We show that themagnetic coupling between the vortices and the electrons inside the gate canlead to a dramatic suppression of the vortex tunneling, restoring thesuperconducting property in accord with the experiment.

Suppression of Tunneling of Superconducting Vortices Caused by a Remote Gate: Example of an Extended Object Tunneling