A new solver, named FLiMESolve, is exhibited which is well suited for time-periodic (Floquet-type) Hamiltonians. This solver maintains the benefits of other solvers presently considered, while superceding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular approximation inherent to the Lindblad Equation.
A new solver, named FLiMESolve, is exhibited which is well suited for time-periodic (Floquet-type) Hamiltonians. This solver maintains the benefits of other solvers presently considered, while superceding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular approximation inherent to the Lindblad Equation.
Computational physics has been a backbone of experimental and theoretical physics since the 1940’s, when the first nuclear bomb and ballistic simulations were per- formed at Los Alamos Laboratory. Nowadays, however, simulation of open quan- tum systems has been formalized into coding packages and user-friendly functions, such as QuTiP or Qiskit, which allow users to simulate these systems without hav- ing the in-depth knowledge required to build the computational backbone from the ground-up. With the shift in modern quantum physics towards the realization of quantum computation, a strong computational background is still necessary to lend validity to theoretical results and to help explain experimental results. Here, a new solver is considered which fits well into the regime of many modern quantum com- putational experiments, i.e. one which works well for systems which have Hamil- tonians that are repeating in time. This solver, named FLiMESolve (for Floquet- Lindblad Master Equation Solver), is derived from a Floquet interpretation of the Lindblad master equation, is well suited for these time-periodic Hamiltonians, and will be integrated into QuTiP. This solver maintains the benefits of other solvers used in QuTiP, while superseding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular ap- proximation inherent to the Lindblad Equation.
The dynamics of open quantum systems is commonly described by quantum master equations derived under the assumption of weak system-bath coupling and a separation of timescales between system and bath. When the system is additionally subjected to a periodic driving, the validity of the resulting Floquet master equations is further restricted to regimes of weak or high-frequency driving. Here, we benchmark a set of commonly used Floquet master equations for a model of two locally driven spins coupled to a shared Ohmic reservoir at finite temperature. We systematically probe the accuracy of the equations as a function of the driving parameters, thus identifying limits of their applicability. Dynamical maps predicted by each master equation are compared against numerically exact non-Markovian simulations, tracking the full relaxation dynamics. We find that the accuracy of each master equation closely reflects the assumptions underlying its derivation. For the Floquet-Lindblad equation, errors can be strongly amplified near resonances where the secular approximation breaks down, while approaches that avoid the secular approximation perform better and exhibit a more systematic dependence of the error on driving frequency and amplitude.
For a closed quantum system described by a time-periodic Hamiltonian, Floquet theorem says that the time evolution operator can be written as $U(t,0)\ensuremath{\equiv}P(t){e}^{\frac{\ensuremath{-}i}{\ensuremath{\hbar}}{H}_{F}t}$ with P(t+T)=P(t), and HF is Hermitian and time independent and called the Floquet Hamiltonian. In this work, we extend the Floquet theorem from closed to open systems described by a time-periodic Lindblad master equation with period $T=\frac{2\ensuremath{\pi}}{\ensuremath{\omega}}$. We derive an expansion for the Lindbladian in powers of $\frac{1}{\ensuremath{\omega}}$. Two examples are presented to illustrate the theorem. We find that the results given by the Lindbladian expansion based on the open system Floquet theorem agree well with the exact dynamics in the high-frequency limit. It may find applications in quantum engineering with systems subject to decoherence.
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.