Classical Floquet Drives from Quantized Cavity Fields: Error Budgets, Geometric Phases, Gauge Consistency, and Entanglement Bounds at Large Photon Number
Floquet engineering treats periodically driven quantum systems with classical time-periodic Hamiltonians, while cavity quantum electrodynamics (QED) treats the drive itself as a quantized field. The precise connection between the two descriptions beyond weak coupling has remained subtle. Building on the program of [1], in which the cavity field is represented in a large-photon-number phase basis so that the time-dependent Schrödinger equation of a classically driven system emerges as a controlled limit of the fully quantized dynamics, we supply a quantitative bridge with three components. First, we derive the effective classical drive amplitude Ω = 2g√N and its finite-photon-number corrections, showing that the classical limit is approached as a power law in the mean photon number n̄, not exponentially: the relative amplitude correction scales as 1/(2√n̄), bounded conservatively by 1/√n̄. Second, we quantify the gauge-inconsistency phase error per drive cycle incurred by ignoring photon-number quantization, obtaining π/√n̄ to 2π/√n̄ radians, and connect the resulting geometric phase to the gerbe-level structure of adiabatic Floquet theory [4], with the mixed-state operational phase of [6] as the natural observable when fluctuations are retained. Third, we derive entanglement-depletion bounds tied to the Schmidt-gap phenomenology of pulsed Dicke dynamics [7]. For a representative cavity at n̄ = 10⁴ we compute: an effective drive frequency of 10 ± 1.1 GHz for strong-coupling parameters (g/ω_c = 0.01, ω_c/2π = 5 GHz); a 0.5% relative amplitude correction; a per-cycle gauge phase error of 0.031–0.063 radians; and a classical-regime crossover at n̄ ≈ 3.9 × 10⁵ photons for 0.01-radian phase resolution. These numbers convert the formal claims of [1] into falsifiable experimental targets. ## 1. Introduction Periodic driving is a cornerstone of modern quantum control, enabling synthetic Hamiltonians, topological band structures, and dynamical stabilization. In the Floquet Full text and updates: papers.qnfo.org/papers/classical-floquet-drives-from-quantized-cavity-fields-error-budgets-geometric-ph/
Floquet engineering treats periodically driven quantum systems with classical time-periodic Hamiltonians, while cavity quantum electrodynamics (QED) treats the drive itself as a quantized field. The precise connection between the two descriptions beyond weak coupling has remained subtle. Building on the program of [1], in which the cavity field is represented in a large-photon-number phase basis so that the time-dependent Schrödinger equation of a classically driven system emerges as a controlled limit of the fully quantized dynamics, we supply a quantitative bridge with three components. First, we derive the effective classical drive amplitude Ω = 2g√N and its finite-photon-number corrections, showing that the classical limit is approached as a power law in the mean photon number n̄, not exponentially: the relative amplitude correction scales as 1/(2√n̄), bounded conservatively by 1/√n̄. Second, we quantify the gauge-inconsistency phase error per drive cycle incurred by ignoring photon-number quantization, obtaining π/√n̄ to 2π/√n̄ radians, and connect the resulting geometric phase to the gerbe-level structure of adiabatic Floquet theory [4], with the mixed-state operational phase of [6] as the natural observable when fluctuations are retained. Third, we derive entanglement-depletion bounds tied to the Schmidt-gap phenomenology of pulsed Dicke dynamics [7]. For a representative cavity at n̄ = 10⁴ we compute: an effective drive frequency of 10 ± 1.1 GHz for strong-coupling parameters (g/ω_c = 0.01, ω_c/2π = 5 GHz); a 0.5% relative amplitude correction; a per-cycle gauge phase error of 0.031–0.063 radians; and a classical-regime crossover at n̄ ≈ 3.9 × 10⁵ photons for 0.01-radian phase resolution. These numbers convert the formal claims of [1] into falsifiable experimental targets. ## 1. Introduction Periodic driving is a cornerstone of modern quantum control, enabling synthetic Hamiltonians, topological band structures, and dynamical stabilization. In the Floquet Full text and updates: papers.qnfo.org/papers/classical-floquet-drives-from-quantized-cavity-fields-error-budgets-geometric-ph/
Floquet engineering and cavity quantum electrodynamics represent two complementary descriptions of light-matter interaction, yet their precise connection beyond the small-coupling regime remains subtle. Here, by representing the cavity field in a large-photon-number phase basis, we show that the time-dependent Schrödinger equation of a classically driven system emerges directly from the fully quantized problem, without replacing the field by a classical trajectory or assuming a mean-field decoupling. This establishes a one-to-one correspondence between quasienergies and photon-sector-shifted quantum energies, while providing a direct quantum interpretation of the Floquet mean energy and Anandan phase. Because gauge invariance is exactly preserved at the level of the quantum Hamiltonian, the quantum-to-classical correspondence remains valid beyond perturbative and small-coupling regimes. We further show that light--matter entanglement is encoded in matter--harmonic correlations of the corresponding Floquet mode, thus identifying a semiclassical limit in which Floquet physics emerges without requiring the quantum field itself to approach a coherent classical state.
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.
Periodically driven quantum systems have emerged as one of the most versatile tools for quantum simulation, enabling the engineering of effective Hamiltonians, topological phases, and inherently nonequilibrium phenomena such as discrete time crystals. Yet fundamental challenges persist: from a spectral ambiguity that precludes a unique Floquet ground state to an inability to manipulate Floquet eigenstates beyond the adiabatic regime. This thesis demonstrates that counterdiabatic driving—a technique from adiabatic quantum control—provides a unifying framework to address these challenges. We first show that the periodic lab-frame Hamiltonian generates counterdiabatic driving for the Floquet eigenstates, revealing Floquet theory as an inverse counterdiabatic driving problem. Exploiting this equivalence, we develop geometric Floquet theory: a reformulation based on the parallel-transport gauge, which uniquely decomposes Floquet dynamics into a geometric contribution, encoded in Berry phases, and a dynamical contribution, generated by the average-energy operator. The average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifies a unique Floquet ground state at all drive frequencies, and serves as a sensitive diagnostic for heating and symmetry-breaking transitions. Inherently nonequilibrium phenomena—the 𝜋-quasienergy splitting of discrete time crystals and the anomalous edge modes of Floquet topological insulators—are traced to geometric phases. On the practical side, we exploit the intrinsic robustness of prethermal discrete time crystals for a highly frequency-selective DTC-based AC magnetic field sensing scheme and demonstrate it experimentally on a nuclear spin quantum simulator. We then generalize counterdiabatic driving to periodically driven systems. In particular, we derive a non-perturbative variational principle for the Floquet adiabatic gauge potential; this yields local, experimentally feasible control protocols for transitionless manipulation of Floquet states. We demonstrate this formalism on Floquet state preparation in an ultracold atom quantum simulator and many-body state transfer across a quantum phase transition, with significant improvements over unassisted protocols in both cases.
Periodically driven systems have emerged as a useful technique to engineer the properties of quantum systems, and are in the process of being developed into a standard toolbox for quantum simulation. An outstanding challenge that leaves this toolbox incomplete is the manipulation of the states dressed by strong periodic drives. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. To achieve fast control of nonequilibrium quantum matter, we generalize the notion of variational counterdiabatic driving away from equilibrium focusing on Floquet systems. We derive a nonperturbative variational principle to find local approximations to the adiabatic gauge potential for the effective Floquet Hamiltonian. It enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. We discuss applications to two-level, Floquet band, and interacting periodically driven models. The developed technique allows us to capture nonperturbative photon resonances and obtain high-fidelity protocols that respect experimental limitations like the locality of the accessible control terms.
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.
Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024