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

A strong decrease of the resistance of a superconducting film measured in magnetic field has been observed when a metallic gate is placed above the film.~\footnote{N. Mason, and A. Kapitulnik, Phys. Rev. B. 65, 220505(R) (2002).} We study how the magnetic coupling between vortices in a superconducting film and electrons in a remote unbiased gate affects the tunneling rate of the vortices. We examine two approaches to analyze tunneling in the presence of low-energy degrees of freedom: (i) the response of the electrons inside the metallic gate to a change in the vortex position is described as ``tunneling with dissipation'' (ii) the scattering of the electrons by the magnetic flux of the vortex leads to the Orthogonality Catastrophe that opposes the vortex tunneling. We show that the gate can cause a dramatic suppression of the vortex tunneling restoring the superconducting property in accord with the experiment.