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 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 an 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.
REFERENCES
Representations with mixed effective environments may exist, but only if certain additional conditions hold, see, e.g., the appendix “Mixed-state measurement models and extreme operations” in Ref. 153.
One can in general extend U′(t) → ∑mKm(t) ⊗ |m⟩⟨0| + X(t) to be unitary.73,154 However, the additional term obeys the property X(t)|0⟩ = 0 and thus drops out of all physical quantities because the effective environment starts out in a pure state |0⟩.
It is often not realized that fermionic particle statistics only requires fields acting on the same mode to anticommute, see paragraph 65 of Ref. 156. Fields acting on different modes can be chosen to either commute or anticommute, see Ref. 15 for a detailed discussion. We stress that this is not specific to noninteracting models and allows for drastic simplifications, especially in open-system dynamics.
Within the spin formulation (6), the lack of parity-superselection complicates the structure of both system and effective environment density matrices but leaves their positivity properties unaffected, see Appendix E 4 for a detailed discussion.
Since the effective environment has finite dimension 4, it can always be considered as consisting of two qubits.73 However, because the joint state is block-diagonal by the superselection property of the dynamics, it is consistent to consider the environment as two fermions.
The time-local generator (51a) is obtained either by taking the time-derivative of the exponential form (41b), by superfermion considerations (Appendix D 3), or by explicitly inverting the EOM result (24) to calculate (Appendix B 5).
In the “initial slip” approach,155,156 one also replaces the time-local QME by another time-constant QME which has the same stationary state as the original equation. To further improve this approximation, one additionally modifies the initial condition, see Ref. 119. Importantly, a constant initial slip will never recover the interesting reentrant behavior discussed in Sec. VIII because the dynamical map remains a semigroup.
