Abstract
We study the reduced time-evolution of general open quantum systems by combining insights from quantum-information and statistical field theory. Inspired by prior work [Eur. Phys. Lett.~102, 60001 (2013) and Phys. Rev. Lett.~111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling.This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage naturally defines the dissipative timescales of the system -before having integrated out the environment completely- the second stage sums up elementary physical processes, each described by a CP superoperator. This allows us to establish the highly nontrivial functional relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the reduced density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. We illustrate the physically different roles of the two emerging coupling-expansion parameters for a simple solvable model. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit.Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by `cutting' a group of Keldysh real-time diagrams `in half'. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schroedinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion.
Highlights
The experimental progress in nanoscale and mesoscopic devices has continued to drive the development of theoretical methods to tackle models of nonequilibrium quantum systems that interact with their environment [1]
An advantage of the diagrammatic approach of statistical field theory is that it provides a uniform treatment of all models of interest: The primary quantities of importance are the environment modes (6) with which the system has interacted. This is precisely what is needed to connect to quantum information theory where the operational formulation of the complete positivity (CP)-TP dynamics hides all details except for the outcomes of measurements performed on the environment
Since the approach we develop here is centered around the time-evolution of Kraus operators, it provides a systematic way for going beyond the limited GKSL approach without giving up CP
Summary
The experimental progress in nanoscale and mesoscopic devices has continued to drive the development of theoretical methods to tackle models of nonequilibrium quantum systems that interact with their environment [1]. The experimental importance of effects due to strong coupling and non-Markovianity– which are tied together [9]– has spurred progress in a variety of other approaches: Inclusion of parts of the environment into the system [10], time-convolutionless master equations [11, 12], stochastic descriptions such as quantum trajectories [13], pathintegral [14], quantum Monte Carlo (QMC) [15], and hierarchical methods [16, 17], multilayer multiconfiguration time-dependent Hartree method [18], perturbative expansions [19,20,21,22,23,24,25], resummation [26,27,28] and related techniques [29,30,31], projection techniques [32,33] and real-time renormalization-group methods [34,35,36,37,38,39,40,41] These are applicable to more general reduced dynamics derived from unitary evolution U(t) of initially uncorrelated states ρ(0) of the system (S) and ρE of the environment (E): ρ(t) := Π(t)ρ(0) = Tr U(t) ρ(0) ⊗ ρE U†(t).
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