We formulate Floquet Green's functions for noninteracting single-band lattice electrons driven by a reservoir-stabilized single-mode Gaussian quantum light source. The source is prescribed externally and is not updated by the many-electron polarization, while an active electronic probe still conditions the source evolution through the Peierls coupling. The two time arguments of a Green's function share one source history: the lesser and greater components are obtained by convolving a shared-history four-endpoint kernel with the continuous bath kernels before the final source trace, while the bath canonical anticommutation relation yields an equal-time covariance that seeds the retarded and advanced one-leg propagations. The Peierls coupling is treated nonperturbatively within the prescribed-source model, and classical Floquet theory is recovered in the appropriate limit. Numerical calculations on a minimal one-dimensional model show finite-coupling quantum-source corrections beyond a prescribed classical drive, together with spectral reconstruction and occupancy redistribution for squeezed vacuum and squeezed coherent sources. The squeezing parameter and phase provide additional control knobs, beyond classical amplitude modulation, for both sideband structure and occupied weight. This work provides a theoretical framework for quantum Floquet engineering of condensed matter with an externally prescribed quantum light source.
We formulate Floquet Green's functions for noninteracting single-band lattice electrons driven by a reservoir-stabilized single-mode Gaussian quantum light source. The source is prescribed externally and is not updated by the many-electron polarization, while an active electronic probe still conditions the source evolution through the Peierls coupling. The two time arguments of a Green's function share one source history: the lesser and greater components are obtained by convolving a shared-history four-endpoint kernel with the continuous bath kernels before the final source trace, while the bath canonical anticommutation relation yields an equal-time covariance that seeds the retarded and advanced one-leg propagations. The Peierls coupling is treated nonperturbatively within the prescribed-source model, and classical Floquet theory is recovered in the appropriate limit. Numerical calculations on a minimal one-dimensional model show finite-coupling quantum-source corrections beyond a prescribed classical drive, together with spectral reconstruction and occupancy redistribution for squeezed vacuum and squeezed coherent sources. The squeezing parameter and phase provide additional control knobs, beyond classical amplitude modulation, for both sideband structure and occupied weight. This work provides a theoretical framework for quantum Floquet engineering of condensed matter with an externally prescribed quantum light source.
Abstract Recent experimental advances enable the manipulation of quantum matter by exploiting the quantum nature of light. However, paradigmatic exactly solvable models, such as the Dicke, Rabi or Jaynes-Cummings models for quantum-optical systems, are scarce in the corresponding solid-state, quantum materials context. Focusing on the long-wavelength limit for the light, here, we provide such an exactly solvable model given by a tight-binding chain coupled to a single cavity mode via a quantized version of the Peierls substitution. We show that perturbative expansions in the light-matter coupling have to be taken with care and can easily lead to a false superradiant phase. Furthermore, we provide an analytical expression for the groundstate in the thermodynamic limit, in which the cavity photons are squeezed by the light-matter coupling. In addition, we derive analytical expressions for the electronic single-particle spectral function and optical conductivity. We unveil quantum Floquet engineering signatures in these dynamical response functions, such as analogs to dynamical localization and replica side bands, complementing paradigmatic classical Floquet engineering results. Strikingly, the Drude weight in the optical conductivity of the electrons is partially suppressed by the presence of a single cavity mode through an induced electron-electron interaction.
Strong periodic driving of cavity-quantum electrodynamics (QED) in the ultrastrong-coupling regime creates nonequilibrium states whose dissipation is governed by Floquet quasienergies rather than undriven dressed resonances. However, modeling such a regime is a significant theoretical challenge, including a number of subtle problems such as the need to ensure gauge invariance for truncated matter-cavity systems with time-dependent driving. To fill this theoretical gap, we introduce a nonsecular Floquet generalized master equation framework for strongly driven open cavity-QED systems, formulated in the dressed basis of the quantum Rabi model and applicable to structured reservoirs without rotating-wave approximations. Our theory can thus model Floquet-driven dynamics in open ultrastrong-coupling cavity-QED, and demonstrates a wide range of quantum state control. Using strong optical pumping and parametric mechanical modulation, we compute long-time populations, fluorescence spectra, and the Floquet-Liouville eigenspectra, resolving observable resonances into hybridized quasienergy channels and decay rates. By systematically comparing with conventional time-independent dressed-basis generalized master equations, we show that static approaches only reproduce steady-state populations in restricted excitation regimes, and fail for frequency-resolved observables and break down under appropriate Floquet engineering, surprisingly, even for spectrally flat baths. Structured environments, such as Lorentzian-Ohmic reservoirs, further amplify these discrepancies through sideband-selective decay. Our results demonstrate that dissipation in driven ultrastrong cavity-QED is intrinsically quasienergy resolved and we establish Floquet-dissipative theory as an accurate and powerful framework for predicting spectra, controlling decay pathways, and engineering nonequilibrium quantum states and reservoirs.
We derive an analytical expression for the steady-state quantum statistics of periodically driven quantum systems coupled to a bath using the nonequilibrium Green's function (NEGF) formalism. By embedding Floquet theory into NEGF, we obtain closed expressions for the retarded, advanced, and lesser Green's functions in the Floquet representation, yielding the Floquet Fermi distribution in which the steady-state occupation is expressed as a weighted sum of Fermi functions shifted by integer multiples of the driving frequency. The weights are determined solely by the Fourier components of the micromotion operator, providing a transparent interpretation of Floquet sideband occupations. Our analysis extends beyond the diagonal commuting Hamiltonians treated in earlier work, and further shows that the robust Floquet distribution remains valid for a broad class of weakly coupled bath spectral functions beyond the ideal featureless-bath approximation. Finally, we establish a Floquet version of the Landauer formula for the DC part of the current, in which the equilibrium Fermi functions are replaced by their Floquet-modified counterparts. Together, these results provide a coherent description of Floquet quantum statistics and transport in periodically driven open quantum systems.
As many-body Floquet theory becomes more popular, it is important to find ways to connect theory with experiment. Theoretical calculations can have a periodic driving field that is always on, but experiment cannot. Hence, we need to know how long a driving field is needed before the system starts to look like the periodically driven Floquet system. We answer this question here for noninteracting band electrons in the infinite-dimensional limit by studying the properties of the system under pulsed driving fields and illustrating how they approach the Floquet limit. Our focus is on determining the minimal pulse lengths needed to recover the qualitative and semiquantitative Floquet theory results.
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.
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.