Relation between non-equilibrium Green's function (NEGF) and density matrix dynamics.

The study of non-equilibrium phenomena is becoming increasingly important in recent years due to advancements in laser technology that allow to probe excitations in materials on a very short time scale. Pump-probe experiments are routinely employed to study relaxation and decoherence effects, which are important in a wide range of fields, from quantum information science to solar energy conversion.

Various theory based on non-equlibrium Green's function (NEGF) or open quantum dynamics with ab-initio density-matrix formalism [1] have been developed in recent years; however, their underlying connection remains to be studied. More importantly, the complete ab-initio theoretical framework, which takes into account an accurate treatment of decoherence due to electron correlation and interactions with environment, still needs to be developed.

We present a theory derived from the Keldysh NEGF formalism for real-time density matrix dynamics. We employ the non-equilibrium GW and Fan-Migdal self-energies to express electron-electron and electron-phonon contributions to decoherence, respectively. We obtain a non-Markovian equation of motion and analyze two different strategies to perform the Markov limit. One is based on the completed collision limit, where all the retardations of the density matrix are ignored. The other is based on a time symmetrization between microscopic and macroscopic scales, which allows us to obtain a Lindblad dynamics that preserves the positive definiteness of the density matrix. We then test the different levels on approximations we propose on the Holstein-Hubbard model.

[1] J. Xu, A. Habib, R. Sundararaman, and Y. Ping, Phys. Rev. B 104, 184418 (2021), URL https://link.aps.org/doi/10.1103/PhysRevB.104.184418.