SciPost Phys. 7, 012 (2019) ·
published 25 July 2019
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We study the reduced time-evolution of open quantum systems by combining
quantum-information and statistical field theory. Inspired by prior work [EPL
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 defines the dissipative timescales of the system
--before having integrated out the environment completely-- the second stage
sums up elementary physical processes described by CP superoperators. This
allows us to establish the nontrivial relation between the (Nakajima-Zwanzig)
memory-kernel superoperator for the density operator and novel memory-kernel
operators that generate the Kraus operators of an operator-sum. 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 Schr\"odinger 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.
