We show that Floquet states can be constructed as strong-field limits of cavity-dressed states. The interaction between laser beams propagating outside the cavity with atoms or molecules are described by Floquet states, constructed from dressed states of average photon number quantized in a cavity of volume V, by taking the limit , , while keeping the photon density constant. Thus, Floquet theory can be seen as a fully quantum-mechanical model in the sense that it describes the photon exchanges between matter and radiation containing a large amount of photons. We discuss in this context adiabatic Floquet theory to treat a slow time dependence of the laser amplitude, to describe pulses, and of its frequency (chirping).
We show that Floquet states can be constructed as strong-field limits of cavity-dressed states. The interaction between laser beams propagating outside the cavity with atoms or molecules are described by Floquet states, constructed from dressed states of average photon number quantized in a cavity of volume V, by taking the limit , , while keeping the photon density constant. Thus, Floquet theory can be seen as a fully quantum-mechanical model in the sense that it describes the photon exchanges between matter and radiation containing a large amount of photons. We discuss in this context adiabatic Floquet theory to treat a slow time dependence of the laser amplitude, to describe pulses, and of its frequency (chirping).
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
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.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
We investigate QED field-dependent effects in radiative shifts of atomic levels driven by a strong laser field. Our focus is on indirect laser-induced corrections to the Lamb shift of an atomic reference state due to a strong laser field which near-resonantly couples two other states, but not the reference state. Our aim is to show how to generalize previous treatments of related scenarios to the case of an induced indirect coupling. We also derive an expression of self-energy corrections in the Floquet picture and analyse various coupling schemes with respect to the shift of Floquet quasi-energies. The semi-classical Floquet picture and the fully quantized dressed-state approach are contrasted.
Periodically driven quantum systems can exhibit a plethora of intriguing nonequilibrium phenomena that can be analyzed using Floquet theory. Naturally, Floquet theory is employed to describe the dynamics of atoms interacting with intense laser fields. However, this semiclassical analysis cannot account for quantum-optical phenomena that rely on the quantized nature of light. In this paper, we take a significant step to go beyond the semiclassical description of atom-photon coupled systems by unifying Floquet theory with quantum optics using the framework of full-counting statistics. This is achieved by introducing counting fields that keep track of the photonic dynamics. This formalism, which we dub “photon-resolved Floquet theory” (PRFT), is based on two-point tomographic measurements, instead of the two-point projective measurements used in standard full-counting statistics. Strikingly, the PRFT predicts the generation of macroscopic light-matter entanglement when atoms interact with multimode electromagnetic fields, thereby leading to complete decoherence of the atomic subsystem in the basis of the Floquet states. This decoherence occurs rapidly in the optical-frequency regime, but is negligible in the radio-frequency regime. Our results thus pave the way for the design of efficient quantum memories and quantum operations. Finally, employing the PRFT, we propose a quantum-communication protocol that can significantly outperform the state-of-art few-photon protocols by two orders of magnitude or better. The PRFT potentially leads to insights in various Floquet settings including spectroscopy, thermodynamics, quantum metrology, and quantum simulations. Published by the American Physical Society 2024
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/