Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be nontrivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet generator (describing the stroboscopic evolution) is either again Markovian and of Lindblad type or not. On the other hand, even though in the high-frequency regime a Lindbladian Floquet generator (Floquet Lindbladian) is numerically found to exist, this behavior is, curiously, not correctly reproduced within analytical high-frequency expansions. Here, we demonstrate that a proper Floquet Lindbladian can still be obtained from a high-frequency expansion, when treating the problem in a suitably chosen rotating frame. Within this approach, we can then also describe the transition to a phase at lower driving frequencies, where no Floquet Lindbladian exists, and show that the emerging non-Markovianity of the Floquet generator can entirely be attributed to the micromotion of the open driven system.
We study driven and open quantum systems described by a time-periodic Lindblad master equation. In closed systems, the stroboscopic dynamics can always be described by an effective time-independent Floquet Hamiltonian; this idea is the basis of Floquet engineering. However, in the presence of dissipation, the existence of an effective time-independent Floquet Lindbladian is not guaranteed due to the non-unitary nature of the evolution. Using Floquet theory, we construct a well-defined time-independent Floquet Lindbladian in an extended Sambe-Liouville space, transforming the initial time-dependent problem to a static and non-Hermitian eigenvalue problem. For harmonic driving, we introduce a matrix continued fraction method to nonperturbatively resum multiphoton processes and construct an effective Floquet Lindbladian acting only on the physical Liouville space. Compared to other high-frequency expansions, this method has the advantage of providing the whole infinite series expansion at once. Using a resolvent formalism, we show how to obtain a spectral Floquet representation of correlation functions of an open quantum system. As an application, we consider a dissipating two-level system in a linearly polarized field and calculate its resonance fluorescence spectrum. Furthermore, we consider a parametrically driven quantum dot with pump and loss for which we calculate its spectral function and current-voltage characteristics.
The stroboscopic evolution of a time-periodically driven isolated quantum system can always be described by an effective time-independent Hamiltonian. Whether this concept can be generalized to open Floquet systems, described by a Markovian Lindblad master equation with a time-periodic generator (Lindbladian), remains an open question. By using a two-level system as a model, we explicitly show the existence of two well-defined parameter regions. In one region the stroboscopic evolution can be described by a Markovian master equation with a time-independent Floquet Lindbladian. In the other it cannot; but here the one-cycle evolution operator can be reproduced with an effective non-Markovian master equation that is homogeneous but nonlocal in time. Interestingly, we find that the boundary between the phases depends on when during the period the evolution is stroboscopically monitored. This reveals the nontrivial role played by the micromotion in the dynamics of open Floquet systems.
We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian; explicit finite-time error bounds that quantify leakage from the unperturbed center sector are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.
Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely positive trace-preserving maps, while time-continuous evolution is often specified with superoperators referred to as "Lindbladians." Here, we address the following question: Being given a quantum map, can we find a Lindbladian that generates an evolution identical-when monitored at discrete instances of time-to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension N of the Hilbert space, the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of machine learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentarily, we use the performance of different ML methods as a tool to validate a hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME)
dtdρt=Lρt, where
L is a linear (super-)operator and
ρt is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state
ρ∞ (such that
Lρ∞=0), the first few excited states, and trajectories
t↦ρt. In finite dimension,
ρt is an
n×n matrix,
L thus typically costs
n4 to store explicitly as a dense matrix, and
O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However,
L usually costs only
O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution
S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map
Φ whose fixed point is directly related to
ρ∞, all the other eigenvalues having smaller magnitude. The map
Φ is thus well suited to iterative methods, and
ρ∞ can be found in a few Arnoldi iterations. Using the same inverse map
S−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like
O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.