How Quantum Evolution with Memory is Generated in a Time-Local Way

Abstract

Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel K, whereas the second achieves the same using a time-local generator G. Here we show that the two are connected by a simple yet general fixed-point relation:. This allows one to extract nontrivial relations between the two completely different ways of computing the time-evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient / Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces non-semigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.

Fermionic duality: General symmetry of open systems with strong dissipation and memory

“…Finally, in Sec. 5 we combine these approaches to gain deeper insight into a generally applicable nonperturbative semigroup approximation [5,23] and its correction by an "initial slip". Here we combine the fermionic duality with a recently found exact functional relation between the time-local generator G and the time-nonlocal memory kernel K [5].…”

“…one pair of contributions is of particular interest. The right eigenvector to eigenvalue π 0 = 1 is a time-dependent fixed point 5 , Π(t) π 0 (t) = π 0 (t) , which is guaranteed to exist by the evolution's TP property, π 0 Π(t) = π 0 writing π 0 = Tr. Often the operator π 0 (t) is unique and can then be scaled to a positive, tracenormalized physical state 6 .…”

“…However, when considering an open system with evolving density operator ρ(t) = Π(t)ρ(0) the above mentioned shortcut completely breaks down. Although it turns out that non-unitary open-system evolutions can still be generated time-locally [2][3][4][5] as ∂ t ρ(t) = −iG(t)ρ(t), new problems arise because the generator is a time-dependent, non-Hermitian superoperator G(t) = [G(t)] † even though the total system evolution is generated by a timeconstant, Hermitian Hamiltonian operator. Physically, these new complications derive from memory (retardation) and dissipation effects, hallmarks of open-system dynamics.…”

“…In practice, time-local QMEs also arise naturally from the time-nonlocal QMEs of approach (ii) when consistently accounting for frequency dependence of the memory kernel in decay problems [37,38]-recently generalized in Ref. [5]-or in adiabatic expansions for situations with external driving [9,[39][40][41]. The microscopic calculation of G(t) is, however, very challenging making additional analytic insights valuable [34,42,43].…”

Memory Effects in Quantum Dynamics Modelled by Quantum Renewal Processes

“…The superoperators L(t) and K(t) are generally related [70][71][72][73], though in a highly non-trivial way. Moreover general conditions on their expression warranting CPTP are not known, except for special cases.…”