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arXiv:2506.17034 · Physical Review A

Quantum-corrected Floquet dynamics: Bridging fully quantum and semiclassical regimes

E. K. Twyeffort, A. D. Armour

Published Sep 17, 2026• 0 citations
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Abstract

Semiclassical descriptions of a few-level system coupled to an electromagnetic field mode reduce the field to a time-dependent driving term. Although such methods are widely used, the underlying quantum character of the field generates corrections to the dynamics that can become significant. Here, we develop a general approach for systematically calculating quantum corrections to the time-dependent Floquet dynamics that emerges in the semiclassical description. Taking the Rabi model of a spin-field system as an example, we show how approximate analytical expressions for short-time quantum corrections to the semiclassical dynamics can be obtained. This framework helps explain the emergence of semiclassical behavior, sheds light on collapse and revival dynamics in the Rabi model, and provides a tool for assessing corrections to semiclassical control techniques.

DOI
10.1103/y5yj-2p6b

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The semiclassical interaction of a two-level atom with a singlemode field, taking into account the atomic reaction onto the field, is considered. Analytical expressions for the field and atomic dynamics are found. An expansion of this solution in terms of the inverse of the field intensity is performed. To zeroth order, the classical Rabi solution is recovered, while first-order corrections give rise to a slow modulation of the atomic dipole, in agreement with a fully quantum model. The evolution of the atomic dipole is studied in the framework of the reduced Q function, both in the semiclassical and quantum regimes. PACS 42.50.Dv — 03.65.-w.
We investigate the emergence of semiclassical dynamics in the quantum Rabi model using a recently developed limiting procedure that formally establishes correspondence with the semiclassical Rabi Hamiltonian [E. K. Twyeffort Irish and A. D. Armour, Phys. Rev. Lett. 129, 183603 (2022)]. While the limit itself is defined at the Hamiltonian level, how it is reached depends on the choice of quantum states. Defining a set of quantitative measures that capture the differences between quantum and semiclassical dynamics, we examine convergence to the semiclassical limit when the field is prepared in a displaced number state. These states, which interpolate to Fock states for zero displacement, are more general than the set of coherent states usually employed when considering the emergence of semiclassical behavior. Numerical computations of these measures consistently demonstrate the progressive emergence of semiclassical behavior as the joint limit of vanishing coupling and infinite displacement is approached. Complementing the numerical results, analytical approximations are developed that reproduce the behavior in the vicinity of the semiclassical limit with a high degree of fidelity and allow scaling relations to be derived. Although any initial displaced number state will eventually converge to the corresponding semiclassical dynamics as the limit is taken, the rate of convergence depends on the Fock number nnn of the state. States with larger values of nnn, which behave less classically than coherent states, converge more slowly to the limit.
It is shown that the dynamics of an N-level quantum system driven by a classical radiation field can be derived rigorously from an associated fully quantum-mechanical model with the help of an asymptotic limit. By formally expanding the full quantum dynamics around the semiclassical limit, quantum corrections to the semiclassical description are systematically constructed. The observables describing these corrections are shown to exist in any order, and explicit expressions for their time dependence are given.
Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous in various fields of physics. Though they have been exactly solved, the physical pictures in these exact solutions are not clear. Recently, approximate analytical solutions with neat physics have been obtained by using the counter-rotating hybridized rotating wave (CHRW) method, which has been proven to be effective over a wider range of parameters than the previous analytical solutions. However, the CHRW depends on a parameter $\ensuremath{\xi}$, which has no solution in some regimes. Here, we combine the double-unitary-transformation approach with the generalized Van Vleck nearly degenerate perturbation theory, and present approximate analytical results with clear physics for almost all parameter regimes, which agree well with the numerical solutions and the previous experimental results. Moreover, the dynamic frequencies of the Rabi model are regular, and the frequency with the highest Fourier amplitude changes from the Rabi frequency to $2n\ensuremath{\omega}$ with driving frequency $\ensuremath{\omega}$ and integer nnn, as the driving intensity increases from weak to deep-strong. In addition, we further explore the Floquet dynamics of the dissipative open Rabi model. Remarkably, the dissipations are tunable in the rotating frame, and the approximate analytical results obtained by our method are in good agreement with the numerical results in the strong driving regime. These results pave the way to quantum control using strong and deep-strong driving with applications in quantum technologies.
The crossover from quantum to semiclassical behavior in the seminal Rabi model of light-matter interaction still, surprisingly, lacks a complete and rigorous understanding. A formalism for deriving the semiclassical model directly from the quantum Hamiltonian is developed here. Working in a displaced Fock-state basis |α,n⟩, the semiclassical limit is obtained by taking |α|→∞ and the coupling to zero. This resolves the discrepancy between coherent-state dynamics and semiclassical Rabi oscillations in both standard and ultrastrong coupling and driving regimes. Furthermore, it provides a framework for studying the quantum-to-semiclassical transition, with potential applications in quantum technologies.
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.