We present a general method to calculate the periodic steady state of a driven-dissipative system coupled to a transmission line (and more generally, to a reservoir) under periodic modulation of its parameters. Using Floquet's theorem, we formulate the differential equation for the system's density operator which has to be solved for a single period of modulation. On this basis we also provide systematic expansions in both the adiabatic and high-frequency regime. Applying our method to three different systems---two- and three-level models as well as the driven nonlinear cavity---we propose periodic modulation protocols of parameters leading to a temporary suppression of effective dissipation rates, and study the arising nonadiabatic features in the response of these systems.
We present a general method to calculate the periodic steady state of a driven-dissipative system coupled to a transmission line (and more generally, to a reservoir) under periodic modulation of its parameters. Using Floquet's theorem, we formulate the differential equation for the system's density operator which has to be solved for a single period of modulation. On this basis we also provide systematic expansions in both the adiabatic and high-frequency regime. Applying our method to three different systems---two- and three-level models as well as the driven nonlinear cavity---we propose periodic modulation protocols of parameters leading to a temporary suppression of effective dissipation rates, and study the arising nonadiabatic features in the response of these systems.
In this paper, we investigate periodically driven open quantum systems within the framework of Floquet-Lindblad master equations. Specifically, we discuss Lindblad master equations in the presence of a coherent, time-periodic driving and establish their general spectral features. We also clarify the notions of transient and nondecaying solutions from this spectral perspective, and then prove that any physical system described by a Floquet-Lindblad equation must have at least one physical nonequilibrium steady state (NESS), corresponding to an eigenoperator of the Floquet-Lindblad evolution superoperator UF with unit eigenvalue. Since the Floquet-Lindblad formalism encapsulates the entire information regarding the NESS, it in principle enables us to obtain nonlinear effects to all orders at once. The Floquet-Lindblad formalism thus provides a powerful tool for studying driven-dissipative solid-state systems, which we illustrate by deriving the nonlinear optical response of a simple two-band model of an insulating solid and comparing it with prior results established through Keldysh techniques.
Using the density matrix formalism, we prove the existence of the periodic steady state for an arbitrary periodically driven system described by linear dynamic equations. This state has the same period as the modulated external influence, and it is realized as an asymptotic solution $(t\ensuremath{\rightarrow}+\ensuremath{\infty})$ due to relaxation processes. The presented derivation simultaneously contains a simple and effective computational algorithm (without using either the Floquet or Fourier formalisms), which automatically guarantees a full account of all frequency components. As a particular example, for three-level $\mathrm{\ensuremath{\Lambda}}$ system we calculate the line shape and field-induced shift of the dark resonance formed by the field with a periodically modulated phase. Also we have analytically solved a basic theoretical problem of the direct frequency comb spectroscopy, when the two-level system is driven by the periodic sequence of rectangular pulses. In this case, the radical dependence of the spectroscopy line shape on pulse area is found. Moreover, the existence of quasiforbidden spectroscopic zones, in which the Ramsey fringes are significantly reduced, is predicted. Our results have a wide area of applications in laser physics, spectroscopy, atomic clocks, and magnetometry. Also they can be useful for any area of quantum physics where periodically driven systems are considered.
We investigate a long-time asymptotic state of periodically driven open quantum systems analytically. The model we consider in this paper is a free fermionic system coupled to an energy and particle reservoir. We clarify some generic properties of the system which are independent of the details of the reservoir in the high-frequency regime of the external driving. When the frequency of the external driving is much larger than the energy cutoff of the system-reservoir coupling, the low-energy properties of the system are equivalent to those of the Gibbs distribution of a Floquet effective Hamiltonian. Furthermore, we investigate the effect of finite dissipation on the system, and elucidate that we cannot suppress excitations by merely increasing the system-reservoir coupling because the excitations are created through the reservoir by the external driving when the frequency is smaller than the energy cutoff of the system-reservoir coupling.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.
Periodically driven open quantum systems are central to quantum thermodynamics and quantum control. These systems are typically described using Floquet-Born-Markov master equations, derived with the use of a full secular approximation, and whose thermodynamic implications are often overlooked. In this context, we show that such a strong secular approximation may lead to unphysical predictions for steady state energy currents. We then demonstrate that a coarse-grained formulation of the master equation can regularize these issues while yielding completely positive dynamics and consistent energy currents. The coarse-graining time has a clear physical interpretation, as it defines the temporal resolution at which a Markovian master equation can describe the evolution of the periodically driven system. We show the consistency and validity of our approach by comparing to an exact non-Markovian simulation in paradigmatic examples: a driven two-level system and a three-level maser coupled to hot and cold thermal reservoirs. Our work provides a practical framework for correctly applying the secular approximation in periodically driven-dissipative systems and for assessing the accuracy of master equations of the GKSL form.
Periodically driven quantum systems are commonly modeled using master equations constructed in the eigenbasis of an undriven Hamiltonian, implicitly assuming that environmental dissipation couples to static energy transitions even under strong time-periodic driving. The validity of this approximation beyond weak or near-resonant driving remains poorly understood. To address the need for a more self-consistent quantum theory approach, we formulate a nonsecular Floquet--Markov generalized master equation (F-GME) in the quasienergy basis, treating interaction-induced (internal) and drive-induced (external) nonperturbative dressing on an equal footing. We subsequently investigate dissipation in two minimal driven open quantum systems---a harmonically driven two-level system and a harmonically driven coupled-two-level-system---each weakly coupled to a Markovian bath. Comparing the F-GME to a time-independent dressed-basis master equation, we show that even for a flat-bath spectral density and weak dissipation, the two approaches can yield qualitatively different steady-state populations and emission spectra. We resolve dissipation into drive-assisted sideband processes decaying via Floquet extended-space quasienergy channels, and show these channels can hybridize through nonsecular couplings into collective Floquet--Liouville modes governing observable spectral resonances. This analysis demonstrates that time-independent dissipative descriptions can incorrectly weight multiphoton Floquet transitions by collapsing quasienergy-resolved decay pathways into static energy gaps. The F-GME framework provides a systematic diagnostic for identifying regimes where Floquet-consistent dissipation is essential and clarifies the physical origin of discrepancies between commonly used master-equation approaches.