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arXiv:2607.22340

Fluctuational Quantum Electrodynamics of Dispersive Time-Varying Media

Jaime E. Sustaeta-Osuna, Thomas F. Allard, Francisco J. García-Vidal, Paloma A. Huidobro

physics.opticsquant-phSubmitted Jul 24, 2026• Announced Jul 27, 2026
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Abstract

We present the theoretical framework of fluctuational quantum electrodynamics in frequency-dispersive and dissipative time-varying media. Our theory accounts for dispersion and losses in the temporal modulation, which is treated in an exact manner, without relying on perturbative methods. Thus, our work constitutes the first consistent quantization of the electromagnetic field in time-modulated material bodies. We derive a Fermi Golden Rule for time-varying media and use it to define the local density of states for these time-dependent systems, which includes both loss and gain contributions. Additionally, we prove the equivalence between the quantum Fermi Golden Rule and the power emitted by a classical harmonic point dipole. Moreover, we show that neglecting the dispersive and dissipative nature of the time modulation leads to erroneous predictions for both slow and fast modulations. Furthermore, we analyze the thermal radiation emitted by a time-varying material body, revealing new features in the enhancement of thermal emission in time-varying media. Finally, we study the dynamical Casimir effect, showing how the time modulation amplifies vacuum fluctuations and generates entangled pairs of polaritons exhibiting non-local spatial correlations.

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We present the theoretical framework of fluctuational quantum electrodynamics in frequency-dispersive and dissipative time-varying media. Our theory accounts for dispersion and losses in the temporal modulation, which is treated in an exact manner, without relying on perturbative methods. Thus, our work constitutes the first consistent quantization of the electromagnetic field in time-modulated material bodies. We derive a Fermi Golden Rule for time-varying media and use it to define the local density of states for these time-dependent systems, which includes both loss and gain contributions. Additionally, we prove the equivalence between the quantum Fermi Golden Rule and the power emitted by a classical harmonic point dipole. Moreover, we show that neglecting the dispersive and dissipative nature of the time modulation leads to erroneous predictions for both slow and fast modulations. Furthermore, we analyze the thermal radiation emitted by a time-varying material body, revealing new features in the enhancement of thermal emission in time-varying media. Finally, we study the dynamical Casimir effect, showing how the time modulation amplifies vacuum fluctuations and generates entangled pairs of polaritons exhibiting non-local spatial correlations.
The recent first-principles derivation of Langevin-Bloch equations of motion for near-resonant atoms embedded in a linear dielectric host leads to a clear, direct physical reinterpretation of fundamental physical principles as applied to one of the most basic problems of quantum mechanics: the renormalization of the spontaneous emission rate of an atom in a dielectric host. Here we show the role of the Lorentz local field, the fluctuation-dissipation theorem, the wave number and wave vector, the density of states, Fermi's golden rule, the electromagnetic energy density, canonical quantization, and the classical constitutive relations in this context.
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We discuss the quantum fluctuations of the fields associated with a broad class of optical scattering and transmission problems by developing the quantum electrodynamics of an idealized linear, but nonuniform, dielectric medium. We present and compare two quantization schemes for this purpose. The first is based on the expansion of the field in terms of a set of single-frequency solutions of the Maxwell equations. The second involves expanding the field in the set of plane-wave solutions of the Maxwell equations in the vacuum. The relation between the two quantization schemes is discussed in the framework of the scattering theory that connects them. The methods presented are used to show that various field components within a dielectric medium may be either superfluctuant or subfluctuant relative to their fundamental uncertainties in the vacuum. These alterations of the fluctuation properties of the fields are shown to lead to changes in the spontaneous emission rates for both electric and magnetic dipole transitions of excited atoms within or near dielectric media. We also analyze the quantum properties of the transition radiation emitted by a fast charged particle in passing from one dielectric medium to another.
Abstract This book is an introduction to quantum optics for students who have studied electromagnetism and quantum mechanics at an advanced undergraduate or graduate level. It provides detailed expositions of theory with emphasis on general physical principles. Foundational topics in classical and quantum electrodynamics, including the semiclassical theory of atom-field interactions, the quantization of the electromagnetic field in dispersive and dissipative media, uncertainty relations, and spontaneous emission, are addressed in the first half of the book. The second half begins with a chapter on the Jaynes-Cummings model, dressed states, and some distinctly quantum-mechanical features of atom-field interactions, and includes discussion of entanglement, the no-cloning theorem, von Neumann’s proof concerning hidden variable theories, Bell’s theorem, and tests of Bell inequalities. The last two chapters focus on quantum fluctuations and fluctuation-dissipation relations, beginning with Brownian motion, the Fokker-Planck equation, and classical and quantum Langevin equations. Detailed calculations are presented for the laser linewidth, spontaneous emission noise, photon statistics of linear amplifiers and attenuators, and other phenomena. Van der Waals interactions, Casimir forces, the Lifshitz theory of molecular forces between macroscopic media, and the many-body theory of such forces based on dyadic Green functions are analyzed from the perspective of Langevin noise, vacuum field fluctuations, and zero-point energy. There are numerous historical sidelights throughout the book, and approximately seventy exercises.
This article addresses the problem of whether it is possible to describe the effect of the atoms of material media on the field only in terms of a dielectric constant, in the regime where the field has to be treated quantum mechanically. Using a simple model of a linear lossless material medium, we start from first principles and determine the validity of the approximations required to obtain such a quantum analogue of classical macroscopic electrodynamics. This theory is derived here from the fundamental microscopic QED description of a medium, in terms of its constituent atoms in the vacuum, by taking macroscopic averages of the dynamical variables. The condition of the validity of the macroscopic approximation is obtained as the proviso for neglecting the contribution of the atoms of the medium to the quantum noise of the field. We show that macroscopic averaging is compatible with a quantum theory and does not imply any smoothening of the intrinsic quantum fluctuations of the field. Although this theory is based on a simple one-dimensional model of a single-mode cavity, it is able to describe the frequency dependence of the dielectric constant.
The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.