Non-equilibrium quantum dynamics based on density-matrix formalism for strong electron-electron and electron-phonon interactions

Prediction of non-equilibrium quantum phenomena is one of the most important theoretical challenges in the past decades. Having a self-consistent formalism to describe the non-adiabatic propagation of electrons and phonons with many-body effects is extremely important for the study of ultrafast phenomena.

We present a theory based on real-time density matrix formalism that includes electron-electron and electron-phonon interactions through non-equilibrium GW, Fan-Migdal, and Ehrenfest self-energies. We obtain a set of five differential equations that describe the time evolution of the reduced density matrix, two particle correlator, electron-phonon correlator, coherent phonon and phonon propagator. These equations are coupled with an external field that drives the system out of equilibrium. By considering a small external field perturbation, we recover the limit of linear-response theory, but our formalism is general and capable of describing interactions with time-dependent strong external fields beyond linear-response.

We then test our equations and different approximations with the exact result for the Holstein-Hubbard model. We show that coherent phonons are responsible for the creation of polaronic feature from the spectrum function, and we can describe them with good accuracy for both weak and strong electron-phonon coupling. Meanwhile, with the e-e interaction, we are able to recover both singlet and triplet excitations, same as the exact solution.