Five approaches to exact open-system dynamics: Complete positivity, divisibility, and time-dependent observables

Abstract

To extend the classical concept of Markovianity to an open quantum system, different notions of the divisibility of its dynamics have been introduced. Here we analyze this issue by five complementary approaches: equations of motion, real-time diagrammatics, Kraus-operator sums, as well as time-local (TCL) and nonlocal (Nakajima-Zwanzig) quantum master equations. As a case study featuring several types of divisible dynamics, we examine in detail an exactly solvable noninteracting fermionic resonant level coupled arbitrarily strongly to a fermionic bath at arbitrary temperature in the wideband limit. In particular, the impact of divisibility on the time-dependence of the observable level occupation is investigated and compared with typical Markovian approximations. We find that the loss of semigroup-divisibility is accompanied by a prominent reentrant behavior: Counter to intuition, the level occupation may temporarily increase significantly in order to reach a stationary state with smaller occupation, implying a reversal of the measurable transport current. In contrast, the loss of the so-called completely-positive divisibility is more subtly signaled by the prohibition of such current reversals in specific time-intervals. Experimentally, it can be detected in the family of transient currents obtained by varying the initial occupation. To quantify the nonzero footprint left by the system in its effective environment, we determine the exact time-dependent state of the latter as well as related information measures such as entropy, exchange entropy and coherent information.

Relaxation dynamics in a Hubbard dimer coupled to fermionic baths: Phenomenological description and its microscopic foundation

“…Recalling Eq. (28), which describes differences between the secular (L S ) and the coherent (L C ) approximation, we find for the particle-hole symmetric case f α (ε + U ) = 1 − f α (ε) (shown in Fig. 1), that their difference scales with f α (ε)f α (ε + U ) = [2 cosh(βU/4)] −1 .…”

Density-operator evolution: Complete positivity and the Keldysh real-time expansion

“…Obtaining an explicit Kraus form in such settings is usually restricted to simple models which can either be solved exactly or treated in the simplifying GKSL limit. Besides guaranteeing the CP property, access to the individual Kraus operators also allows the calculation of additional measures of information yielding interesting insights, even for well-known solvable models [48].…”

“…Here we are emphatically not interested in such situations where one takes the 'easy way out' by relying on one of several simplifications: (i) Models described by GKSL equations where CP and TP are encoded into the simple form of the generator. (ii) Models with exactly solvable non-GKSL dynamics where CP and TP can be seen explicitly [48]. (iii) Approximations that reduce 4 models to these cases.…”

“…In this context the Keldysh approach has proven useful by its extension to 'parallel-worlds' [25] to compute Renyi entropies. Our operator-sum formulation of the Keldysh approach identifies a variety of approximations that give meaningful entropic quantities, including also the exchange entropy [67,68] associated with the environment [48] which requires the evolution Π(t) to be completely positive.…”

“…Although we could have derived [48] a GKSL master equation ∂ t ρ(t) = Lρ(t) -which is exact for this model-to more easily obtain the result (52), all steps and physical arguments that we discussed can also be applied for general evolutions -for which no GKSL equation can be derived. Clearly, in the latter case the evaluation steps will be technically more difficult.…”