Quantum impurity models coupled to Markovian and non-Markovian baths

Abstract

We develop a method to study quantum impurity models, small interacting quantum systems bilinearly coupled to an environment, in presence of an additional Markovian quantum bath, with a generic non-linear coupling to the impurity. We aim at computing the evolution operator of the reduced density matrix of the impurity, obtained after tracing out all the environmental degrees of freedom. First, we derive an exact real-time hybridization expansion for this quantity, which generalizes the result obtained in absence of the additional Markovian dissipation, and which could be amenable to stochastic sampling through diagrammatic Monte Carlo. Then, we obtain a Dyson equation for this quantity and we evaluate its self-energy with a resummation technique known as the Non-Crossing Approximation. We apply this novel approach to a simple fermionic impurity coupled to a zero temperature fermionic bath and in presence of Markovian pump, losses and dephasing.

Self-consistent dynamical maps for open quantum systems

“…1. The NCA approach is a non-linear, non-Markovian quantum dynamical map which can be shown to preserve the trace and hermiticity of the density matrix [59]. While it is not guaranteed to preserve complete positivity, in our application to the spin-boson model we find that this approximation is well behaved way beyond perturbative master equations.…”

“…Recent efforts have focused on deriving a Lindblad master equations without a rotating-wave approximation [45][46][47][48][49][50] and on including non-Markovian effects [51], which is often achieved via phenomenological master equations [52][53][54]. Microscopic approaches beyond Born-Markov master equations exist, including diagrammatic techniques [36,[55][56][57][58][59][60], real-time Quantum Monte Carlo methods [61][62][63][64][65][66][67], an exact hierarchy of equations of motion [68][69][70] or similar stochastic approaches [71][72][73][74][75] and a matrix product states ansatz [76]. The latter techniques are nevertheless numerically heavy, limiting their applicability in many practical cases.…”

“…Noncrossing approximations have a long tradition in condensed matter physics: they have been used for disordered systems [77], for quantum impurity models both in and out of equilibrium [78][79][80][81][82][83][84][85][86] and for quantum transport [87][88][89]. In [59,90] we introduced a novel superoperator NCA in real time to deal simultaneously with Markovian and non-Markovian environments, which we adapt here to a conventional open quantum system setting, a small system coupled to a bath.…”

“…The system-bath coupling is written in the general form H SB = α X α ⊗ B α , where X † α = X α are system operators and B † α = B α are linear combination of bath creation and annihilation operators. Differently from master equations for the reduced density matrix of the system, we focus here on its evolution superoperator ρ(t) = tr B ρ tot (t) = V(t)ρ(0), which defines a quantum dynamical map for the reduced system dynamics, and consider its full series expansion in the system-bath coupling [59]. Taking advantage of the non-interacting nature of the bath, a real-time diagrammatics [55] can then be defined such that one-particle irreducible (1PI) diagrams are formally resummed into a "self-energy" Ŝ and the evolution superoperator is found to obey the Dyson-like equation [59]…”

Dynamics of a nonlinear quantum oscillator under non-Markovian pumping

“…If a reservoir cannot be fully separated from a considered quantum state, dynamics becomes essentially non-Markovian. Non-Markovian open quantum systems are basically governed by evolution equations which are non-local in time and contain integrals with memory kernels [22][23][24][25][26][27]. Inclusion of memory effects can remarkably enrich physical properties of such systems (see [28][29][30] for particular illustrations).…”