Self-consistent projection operator theory for quantum many-body systems

Abstract

We derive an exact equation of motion for the reduced density matrices of individual subsystems of quantum many-body systems of any lattice dimension and arbitrary system size. Our projection operator based theory yields a highly efficient analytical and numerical approach. Besides its practical use it provides a novel interpretation and systematic extension of mean-field approaches and an adaption of open quantum systems theory to settings where a dynamically evolving environment has to be taken into account. We show its high accuracy for two significant classes of complex quantum many-body dynamics, unitary evolutions of non-equilibrium states in closed and stationary states in driven-dissipative systems.

Absorbing State Phase Transition with Competing Quantum and Classical Fluctuations

“…To generate the incoherent branching/coagulation the atoms are coupled (with coupling g) to a second equallydetuned light field with strong phase noise (dephasing rate λ g) [63]; for a correlation length shorter than the interatomic distance, the bath is modeled as independent bosonic modes b k , b † k acting on each lattice site. The effective equation of motion for the atoms is obtained by performing second order perturbation theory in the small parameter g/λ [44,60,64]. The resulting master equation for the reduced atomic density matrix ρ iṡ…”

Self-consistent projection operator theory in nonlinear quantum optical systems: A case study on degenerate optical parametric oscillators

“…The term "self-consistent" is chosen because the state of the pump in P p t is not a time-independent reference state but is rather obtained consistently from the time-evolving state ρ(t) of the full dynamics. Using this projector, we derive a generalized Nakajima-Zwanzig equation which is an exact equation for the reduced state of the signal mode [18]. The effective Liouvillian describing such a Nakajima-Zwanzig equation will depend on the state of the pump ρ p (t).…”

“…To explain the approach employed in our calculations, we will first recapitulate some basic ideas of the selfconsistent projection operator theory [18]. The first step is to divide the entire system into subsystems.…”

“…Such a step is important as our theory expands in powers of the interaction Liouvillian L I in order to solve the Nakajima-Zwanzig equation. As in reference [18] we will expand to second order in the system-environment interaction. This approximation is known as the Born approximation [24].…”

“…For example, adiabatic elimination techniques can be exploited to derive effective equations of motion [16,17]. In this work, we apply the recently introduced self-consistent projection operator theory [18] to the degenerate optical parametric oscillator, and exemplify how it generalizes adiabatic elimination approaches. This theory takes dynamical back-action between the degrees of freedom into account and therefore does not require any time-scale separation.…”