Escape of trapped electrons from a helium surface: a dynamical theory

The system of electrons in a metastable well near a helium surface is an ideal system to test our understanding of the escape process from a metastable well with many body correlations due to Coulomb interaction. A number of experiments have been performed on the escaping of electrons from a helium surface. Recent theoretical studies have mainly concentrated on how the static correlations affect the escape rate. In the present paper we describe a dynamical theory for the escape of an electron from a helium surface, which accounts for the effects of both static and dynamical correlations as well as of magnetic fields. We consider the experimentally relevant situation in which the lifetime of the metastable state of an electron is much longer than its relaxation time in the metastable well and the density of 2-d electrons is low such that the Fermi temperature is the smallest energy scale in the problem. The escaping electrons are then statistically independent of each other and the exchange effect of an escaping electron with 2-d electrons can be ignored. A separation between the escaping electron and the remaining 2-d electrons for each escape event can be made. Firstly we find the effective Hamiltonian which describes the escape process in the high temperature limit. Starting from it we use an imaginary time path integral method to calculate the tunneling rate at low temperatures. We shall ignore the weaker interactions between the escaping electron and the surface waves of liquid helium and the helium vapor atoms, which have been discussed elsewhere. The key element in the present theory is to treat properly the dynamics of the 2-d electrons in response to the motion of the escaping electron, which presents an induced electric force in the equation of motion for the escaping electron. The 2-d electron fluid is described by a set of hydrodynamical equations: the continuity equation and the Euler’s equation. In the small density deviation and nonrelativistic limit, we linearize the hydrodynamical equations. We solve for the density deviation, which is determined by the motion of plasma modes. Then using the Poisson equation we can calculate the induced electric field. At this point we find that we are facing a problem similar to the one in the discussion of the macroscopic quantum effect, where the total Hamiltonian has three parts, a dissipative bath consisting of harmonic oscillators, a system of interest, and the coupling between the system and the bath. Using this analogy, we obtain the effective Hamiltonian to describe the motion of the escaping electron: