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arXiv:2407.17732 · Physical Review A

Photon-resolved Floquet theory. I. Full counting statistics of the driving field in Floquet systems

Georg Engelhardt, JunYan Luo, V. M. Bastidas, Gloria Platero

Published Dec 12, 2024• 4 citations
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Abstract

Floquet theory and other established semiclassical approaches are widely used methods to predict the state of externally driven quantum systems, yet they do not allow the prediction of the state of the photonic driving field. To overcome this shortcoming, the photon-resolved Floquet theory (PRFT) was developed recently [], which deploys concepts from full-counting statistics to predict the statistics of the photon flux between several coherent driving modes. In this paper, we study in detail the scaling properties of the PRFT in the semi-classical regime. We find that there is an ambiguity in the definition of the moment-generating function, such that different versions of this function produce the same photonic probability distribution in the semiclassical limit, and generate the same leading-order terms of the moments and cumulants. Using this ambiguity, we establish a simple expression for the Kraus operators, which describe the decoherence dynamics of the driven quantum system appearing as a consequence of the light-matter interaction. The PRFT will pave the way for improved quantum sensing methods, e.g., for spectroscopic quantum sensing protocols, reflectometry in semiconductor nanostructures, and other applications, where the detailed knowledge of the photonic probability distribution is necessary. Published by the American Physical Society 2024

DOI
10.1103/physreva.110.063707

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Floquet theory and other established semiclassical approaches are widely used methods to predict the state of externally driven quantum systems, yet they do not allow the prediction of the state of the photonic driving field. To overcome this shortcoming, the photon-resolved Floquet theory (PRFT) was developed recently [], which deploys concepts from full-counting statistics to predict the statistics of the photon flux between several coherent driving modes. In this paper, we study in detail the scaling properties of the PRFT in the semi-classical regime. We find that there is an ambiguity in the definition of the moment-generating function, such that different versions of this function produce the same photonic probability distribution in the semiclassical limit, and generate the same leading-order terms of the moments and cumulants. Using this ambiguity, we establish a simple expression for the Kraus operators, which describe the decoherence dynamics of the driven quantum system appearing as a consequence of the light-matter interaction. The PRFT will pave the way for improved quantum sensing methods, e.g., for spectroscopic quantum sensing protocols, reflectometry in semiconductor nanostructures, and other applications, where the detailed knowledge of the photonic probability distribution is necessary. Published by the American Physical Society 2024
Photon-resolved Floquet theory keeps track of the photon exchange of a quantum system with a coherent driving field. It thus complements the standard full-counting statistics that counts the number of photons exchanged with incoherent photon modes giving rise to dissipation. In this paper, we introduce a unifying framework describing both situations. We develop methods suitable for an analytical evaluation of low-order cumulants of photonic probability distributions. Within this framework we analyze the two-mode Jaynes-Cummings model to demonstrate that the photon-resolved Floquet theory and the standard full-counting statistics make consistent statistical predictions. Interestingly, we find that the photon-flux fluctuations diverge for vanishing dissipation, which can be related to an entanglement effect between the driven matter system and the driving field. To substantiate our results, we use our framework to describe efficient photon up-conversion in an ac-driven lambda system, that is characterized by a high signal-to-noise ratio. Because the framework is nonperturbative and predicts fluctuations, it paves the way toward nonperturbative spectroscopy, which will assist to improve metrological methods. Published by the American Physical Society 2024
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 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.
Periodically driven quantum systems provide a powerful route to engineer novel states of matter by controlling their effective Hamiltonians through external fields. However, most treatments are usually simplified by considering the high-frequency limit, and do not account for the processes taking place in the transient regime during the onset of the drive. In this work, we introduce a time and momentum resolved tunneling spectroscopy protocol to probe the instantaneous energy spectrum of Floquet-driven systems beyond the high-frequency regime, capturing both emergent effects and non-adiabatic phenomena without requiring explicit reconstruction of the full time-dependent Green's function. We benchmark the method on driven non-interacting fermionic models, and then generalize the approach to strongly correlated systems with the aid of time-dependent density matrix renormalization group techniques. We also provide a microscopic description to the emergence of in-gap states under resonant driving, and briefly explore its finite temperature analog.
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.
Periodically driven quantum systems known as Floquet insulators can host topologically protected bound states known as “ π modes” that exhibit response at half the frequency of the drive. Such states can also appear in undriven lattice field theories when time is discretized as a result of fermion doubling, raising the question of whether these two phenomena could be connected. Recently we demonstrated such a connection at the level of an explicit mapping between the spectra of a continuous-time Floquet model and a discrete-time undriven lattice fermion model. However, this mapping relied on a symmetry of the single-particle spectrum that is not present for generic drive parameters. Inspired by the example of the temporal Wilson term in lattice field theory, in this paper we extend this mapping to the full drive parameter space by allowing the parameters of the discrete-time model to be frequency-dependent. The spectra of the resulting lattice fermion models exactly match the quasienergy spectrum of the Floquet model in the thermodynamic limit. Our results demonstrate that spectral features characteristic of beyond-equilibrium physics in Floquet systems can be replicated in static systems with appropriate time discretization. Published by the American Physical Society 2024