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arXiv:2610.29404

Bounds on Attainable Rate Modulation in Fermi's Golden Rule

Kaisei Ohata, Jun-Yu Dong, Yasutaka Kitahama, Akihito Ishizaki, Keisuke Goda

quant-phphysics.chem-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
v1 · Oct 21, 2026v2 · Oct 22, 2026
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Abstract

Controlling transition rates is a central objective across quantum optics, condensed-matter physics, and chemical physics, encompassing processes such as spontaneous emission, energy and charge transfer, spin relaxation, and chemical reactions. Although Fermi's golden rule provides a powerful framework for calculating such rates of weakly coupled transitions, conventional calculations require the interaction to be specified in advance; the range of rate responses attainable over a broader interaction space remains unknown. We formulate spectral-overlap golden-rule rates as quadratic forms of the interaction and derive rigorous bounds on rate modulation over a specified interaction space. As an example, application to energy transfer from a donor to a molecular vibration coupled to an optical cavity reveals that enhancement and suppression depend jointly on the interaction and cavity detuning, including continuous reversal of the rate response for a fixed interaction. The same approach may elucidate rate control for other golden-rule processes, including excitation-energy transfer, nonadiabatic electron transfer, nonradiative electronic transitions, vibrational and spin relaxation, and spontaneous emission.

Comments
6 pages, 2 figures

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Controlling transition rates is a central objective across quantum optics, condensed-matter physics, and chemical physics, encompassing processes such as spontaneous emission, energy and charge transfer, spin relaxation, and chemical reactions. Although Fermi's golden rule provides a powerful framework for calculating such rates of weakly coupled transitions, conventional calculations require the interaction to be specified in advance; the range of rate responses attainable over a broader interaction space remains unknown. We formulate spectral-overlap golden-rule rates as quadratic forms of the interaction and derive rigorous bounds on rate modulation over a specified interaction space. As an example, application to energy transfer from a donor to a molecular vibration coupled to an optical cavity reveals that enhancement and suppression depend jointly on the interaction and cavity detuning, including continuous reversal of the rate response for a fixed interaction. The same approach may elucidate rate control for other golden-rule processes, including excitation-energy transfer, nonadiabatic electron transfer, nonradiative electronic transitions, vibrational and spin relaxation, and spontaneous emission.
The simulation of electronically nonadiabatic dynamics and spectroscopy remains as a central challenge in computational chemistry. Amongst the reduced dynamic methods, various forms of the perturbative quantum master equation (QME) including Fermi's golden rule (FGR), Redfield equation, and Marcus theory enjoy widespread use for their efficiency in the perturbative regime, and provides important physical intuition for the underlying dynamics. In the first chapter of this dissertation, the accuracy of various QMEs that treat off-diagonal electronic coupling perturbatively (OD-QME) will be examined on a spin-boson model. OD-QME works in the interaction picture, limiting its application to population dynamics. A novel way of retrieving Schrodinger picture coherences from interaction picture dynamics will be introduced, and the resulting dynamics will be compared against the numerically exact multi-configuration time dependent Hartree method. It will be shown that OD-QME can generate accurate population and coherence in appropriate parameter ranges, and that the time-nonlocal QME is more accurate than time-local QME, especially if the QME uses a population only projection. In the second chapter, the aforementioned method of calculating electronic coherence via OD-QME is applied to the simulation of absorption spectra. The result will be compared against the numerically exact hierarchical equations of motion, as well as the Redfield time-local QME which is widely applied in the context of spectroscopy. It will be shown that OD-QME exhibits comparable accuracy to Redfield QME over a wide parameter range. Both QMEs fail to predict the spectra accurately in the case of a high-frequency bath at a low temperature, and this error is likely to be linked to the quantum behavior of the bath. In the third chapter, the FGR rates of electronically nonadiabatic reactions enhanced by a microcavity is calculated. It will be shown that the cavity-enhanced reaction rates can be evaluated from the cavity-free FGR correlation function with minimum additional effort, and this method will be applied to a model CT reaction. While a high-frequency cavity enhances reactions in the Marcus inverted region selectively, a low-frequency cavity enhances reactions in all regions uniformly. In the fourth chapter, the FGR charge transfer rate constants for a molecular triad system solvated in explicit solvent is calculated via linear semiclassical (LSC) approach. A Marcus-like expression can be derived after applying a hierarchy of approximations. The validity of such approximations will be confirmed for the triad model. It will also be shown that the charge transfer rate depends deeply on the conformation of the molecule, the fluctuation between which is the rate-limiting step of the charge transfer.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.
Fermi's golden rule which describes the transition rates between two electronic levels under external stimulations is used ubiquitously in different fields of physics. The original Fermi's golden rule was derived from perturbative time-dependent Schrödinger's equation without the direct contribution by decoherence effect. However, as a result of recent developments of quantum computing and ultra fast carrier dynamics, the decoherence becomes a prominent topic in fundamental research.Here, by using the non-adiabatic molecular dynamics which goes beyond the time-dependent Schrödinger's equation by introducing decoherence, we study the effect of decoherence on Fermi's golden rule for the fixed basis and the adiabatic basis, respectively. We find that when the decoherence time becomes short, there is a significant deviation from the Fermi's golden rule for both bases. By using monolayer WS2\mathrm{WS_2}WS2​ as an example, we investigate the decoherence effect in the carrier transitions induced by the electron-phonon coupling with first-principle method.
This chapter gives an overview of the field as it stands at the time of writing in mid 1995. It shows how the picture radically changes when there is more than one atom in the cavity. The simplest light-atom interaction one can think of is the coupling between a single atom and a single mode of the electromagnetic field. This system provides considerable insight into the fundamental physics of the light-atom coupling but the reader is warned that the more general many mode-many atom interactions are much richer in their dynamics. An atom in free space couples not only to one, but to a continuum of modes. The mathematics of the Weisskopf-Wigner theory of spontaneous emission tends to obscure the relatively simple physics of the emission process. An alternative, shorter way of calculating the single atom spontaneous emission rate is to use Fermi’s Golden rule, which is a prescription for calculating general transition rates to first order in perturbation theory.