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 $\mathrm{WS_2}$ as an example, we investigate the decoherence effect in the carrier transitions induced by the electron-phonon coupling with first-principle method.
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 as an example, we investigate the decoherence effect in the carrier transitions induced by the electron-phonon coupling with first-principle method.
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.
In quantum thermodynamics, the decomposition of energy exchanges into heat and work remains an open problem beyond weak-coupling and slow-driving regimes. Recent formulations have shown that quantum coherence introduces additional energy contributions whose thermodynamic interpretation is still under debate, raising fundamental questions about the structure of the quantum first law. In this work, we investigate this problem through a time-dependent perturbative framework applied to the first law of quantum thermodynamics. By expanding the thermodynamic quantities up to second order, we derive explicit perturbative corrections for work, heat, and coherence contributions. Our results show that the coherence term can be consistently decomposed into coherent heat and coherent work, demonstrating that quantum coherence does not require the introduction of an independent energetic contribution beyond heat and work. The formalism resolves inconsistencies associated with previous formulations of the quantum first law, including the interpretation of coherence contributions and their connection with entropy fluxes. At second order, the perturbative corrections become directly connected to transition rates governed by Fermi's golden rule, establishing a bridge between microscopic quantum transitions and macroscopic thermodynamic quantities. These results provide a physically transparent framework to investigate coherence-driven thermodynamic processes and offer new perspectives for the analysis of driven quantum systems and nonequilibrium quantum technologies.
We shall revisit the conventional adiabatic or Markov approximation, which --contrary to the semiclassical case-- does not preserve the positive-definite character of the corresponding density matrix, thus leading to highly non-physical results. To overcome this serious limitation, originally pointed out and partially solved by Davies and co-workers almost three decades ago, we shall propose an alternative more general adiabatic procedure, which (i) is physically justified under the same validity restrictions of the conventional Markov approach, (ii) in the semiclassical limit reduces to the standard Fermi's golden rule, and (iii) describes a genuine Lindblad evolution, thus providing a reliable/robust treatment of energy-dissipation and dephasing processes in electronic quantum devices. Unlike standard master-equation formulations, the dependence of our approximation on the specific choice of the subsystem (that include the common partial trace reduction) does not threaten positivity, and quantum scattering rates are well defined even in case the subsystem is infinitely extended/has continuous spectrum.
Fermi's golden rule is one of the most important tools of quantum mechanics. It gives the general formula for transition rates, the rates at which particles are ‘scattered’ from one state to another by a perturbation. ‘Scattered’ is in quotation marks because it is a much more general concept than one might guess. An obvious example is provided by impurities in a crystal, which scatter an electron from one Bloch state to another. They change its momentum but not its energy. Similarly, phonons (vibrations of the lattice) also scatter electrons, but in this case they change the energy of the electrons as well as their momentum. A less obvious example is the absorption of light, which can be viewed as a scattering process in which an electron collides with a photon. The converse process also occurs, where an electron loses energy to a photon, and gives rise to spontaneous and stimulated emission. Thus scattering is a remarkably general concept. The examples suggest that there are two broad classes of scattering processes that we should treat: (i) potentials that are constant in time, such as impurities in a crystal, which do not change the energy of the particle being scattered; (ii) potentials that vary harmonically in time as cos ω q t , such as phonons and photons, which change the energy of the particle by ± ħ ω q .
In interacting quantum systems, the single-particle Green's function is expected to decay in time due to the interaction induced decoherence of quasiparticles. In the limit of weak interaction strengths (Δ), a naive application of Fermi's Golden Rule (FGR) predicts an O(Δ2) quasiparticle decay rate. However, for 1d fermions on the lattice at T>0, this calculation gives a divergent result and the scaling of the quasiparticle lifetime with interaction strength remains an open question. In this work we propose a solution to this question: combining numerical simulations using the recently introduced dissipation-assisted operator evolution (DAOE) method, with non-perturbative diagrammatic re-summations, we predict a logarithmic enhancement of the quasiparticle decay rate τ−1∼Δ2logΔ−2. We argue that this effect is present in a wide variety of well-known weakly interacting quantum fermionic and bosonic systems, and even in some classical systems, provided the non-interacting limit has quasiparticles with a generic dispersion.
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.