Theory of quantum dynamics in fermionic environment: An influence functional approach

Abstract

Quantum dynamics of a particle coupled to a fermionic environment is considered, with particular emphasis on the formulation of macroscopic quantum phenomena. The framework is based on a path integral formalism for the real-time density matrix. After integrating out of the fermion variables of the environment, we embed the whole environmental effects on the particle into the so-called influence functional in analogy to Feynman and Vernon's initial work. We then show that to the second order of the coupling constant, the exponent of the influence functional is in exact agreement with that due to a linear dissipative environment (boson bath). Having obtained this, we turn to a specific model in which the influence functional can be exactly evaluated in a long-time limit (long compared to the inverse of the cutoff frequency of the environmental spectrum). In this circumstance, we mainly address our attention to the quantum mechanical representation of the system-plus-environment from the known classical properties of the particle. It is shown that, in particular, the equivalence between the fermion bath and the boson bath is generally correct for a singlechannel coupling provided we make a simple mapping between the nonlinear interaction functions of the baths. Finally, generalizations of the model to more complicated situations are discussed and significant applications and connections to certain practically interesting problems are mentioned.