Describing non-Hermitian dynamics using a Generalized Three-Time NEGF for a Partition-free Molecular Junction with Electron-Phonon Coupling

Abstract

In this paper we develop the Non-Equilibrium Green's Function (NEGF) formalism for a dissipative molecular junction that consists of a central molecular system with one-dimensional electronic transport coupled to a phonon environment and attached to multiple electronic leads. Our approach is partitionless - initial preparation of the system places the whole system in the correct canonical equilibrium state - and is valid for an external bias with arbitrary time dependence. Using path integrals as an intermediary tool, we apply a two-time Hubbard-Stratonovich transformation to the phonon influence functional with mixed real and imaginary times to obtain an exact expression for the electronic density matrix at the expense of introducing coloured Gaussian noises whose properties are rigorously derived from the environment action. This results in a unique stochastic Hamiltonian on each branch of the Konstantinov-Perel' contour (upper, lower, vertical) such that the time evolution operators in the Liouville equation no longer form a Hermitian conjugate pair, thus corresponding to non-Hermitian dynamics. To account for this we develop a generalized three-time NEGF which is sensitive to all branches of the contour, and relate it to the standard NEGF in the absence of phonons via a perturbative expansion of the noises. This approach is exact and fully general, describing the non-equilibrium driven dynamics from an initial thermal state while subject to inelastic scattering, and can be applied to non-Hermitian dynamics in general.