GAUGE-INVARIANT DESCRIPTION OF NONLINEAR QUANTUM TRANSPORT OF WEAKLY COUPLED ELECTRON-PHONON SYSTEMS IN UNIFORM ELECTRIC-FIELDS

We present a gauge-invariant formulation of the eigenfunction representation of the Baym-Kadanoff-Keldysh theory of non-equilibrium quantum statistical mechanics. The eigenfunction representation, which for uniform fields consist of Airy functions, allows a non-perturbative treatment of the accelerating electric field. For the special case of a momentum-independent electron-phonon coupling, treated in the first Born approximation, we are able to carry through the complete non-equilibrium calculation, which in our case consists of the following steps: (i) determination of spectral densities, which are consistent with the frequency sum rule, and include collisional broadening and intra-collisional field effect on equal footing; (ii) derivation of a closed equation for a generalized distribution function, which contains information about the non-equilibrium occupation of energy states (in our special case the central 'kinetic' equation is a homogenous integral equation for a function of only one variable, well suited for a numerical solution); (iii) relation of the function determined in step (ii) to experimentally observable quantities, such as number densities or non-linear conductivities; (iv) checking of the consistency of our results with the linearized Boltzmann equation in the appropriate limit; and (v) finding a numerical solution of the derived transport equation.