Modified Fermi's golden rule rate expressions

TL;DR

The paper addresses fundamental ambiguities in applying $k_{\mathrm{FGR}}$ when final states are sparse or the Hamiltonian fluctuates in time, leading to divergences and ill-defined rates. It derives a general time-dependent rate $k(t)$ from second-order perturbation theory and clarifies its relation to the conventional FGR in the static-bath limit. For linearly coupled harmonic baths under the Condon approximation, it shows that the $t\to\infty$ limit can yield either a regular finite rate or a delta-function pathology, and it proposes two practical modified FGR schemes: a regularized $k_{\mathrm{m-FGR}}$ and a disorder-averaged variant that accounts for final-state uncertainty. Extending to general time-dependent Hamiltonians, the work develops a population-mechanics framework with time-dependent rates, and introduces further modified forms that incorporate Gaussian energy fluctuations and multiplicative coupling fluctuations, providing delta-free, ensemble-averaged rates suitable for modeling complex rate processes in condensed or disordered environments.

Abstract

Fermi's golden rule (FGR) serves as the basis for many expressions of spectroscopic observables and quantum transition rates. The utility of FGR has been demonstrated through decades of experimental confirmation. However, there still remain important cases where the evaluation of a FGR rate is ambiguous or ill-defined. Examples are cases where the rate has divergent terms due to the sparsity in the density of final states or time dependent fluctuations of system Hamiltonians. Strictly speaking, assumptions of FGR are no longer valid for such cases. However, it is still possible to define modified FGR rate expressions that are useful as effective rates. The resulting modified FGR rate expressions resolve a long standing ambiguity often encountered in using FGR and offer more reliable ways to model general rate processes. Simple model calculations illustrate the utility and implications of new rate expressions.

Figures (4)

• Figure 1: Real and imaginary parts, ${\mathcal{C}}_R^{(n)}(t)$ and ${\mathcal{C}}_I^{(n)}(t)$, for different bath spectral densities within the model of Eq. (\ref{['eq:bath-model']}), with $\lambda_n=\hbar\omega_c$ and $\theta=1$.
• Figure 2: Logarithms of dimensionless FGR rates, $\kappa= \hbar\sqrt{k_BT\lambda}/(\sqrt{\pi} J^2) k_{FGR}$, where $k_{FGR}$ is the $t\rightarrow \infty$ limit of Eq. (\ref{['eq:kt-2']}), versus $\Delta \tilde{E}=\tilde{E}_1 -\tilde{E}_2$, for different bath spectral densities within the model of Eq. (\ref{['eq:bath-model']}), with $\lambda_n=\hbar\omega_c$ and $\theta=1$. For the case of $n=3$, an additional damping factor $e^{-\gamma_d\omega_c t}$ was used. Results for two different values of $\gamma_d=0.01$ and $0.1$ are shown.
• Figure 3: Logarithms of dimensionless first modified FGR (m-FGR-1) rate, $\kappa= \hbar\sqrt{k_BT\lambda}/(\sqrt{\pi} J^2) k_{m-FGR}$, where $k_{m-FGR}$ is the modified FGR rate, Eq. (\ref{['eq:kfgm-1']}), versus $\Delta \tilde{E}=\tilde{E}_1 -\tilde{E}_2$. The super-Ohmic bath spectral density with $n=3$ is used for $\lambda_3=\hbar\omega_c$ and $\theta=1$. The original FGR rates with two different damping factors in the integrand, which were shown in Fig. 2, are also provided for comparison.
• Figure 4: Logarithms of dimensionless second modified FGR (m-FGR-2) rates, $\kappa= \hbar\sqrt{k_BT\lambda}/(\sqrt{\pi} J^2) k_{m-FGR}$, where $k_{m-FGR}$ is the modified FGR rate, Eq. (\ref{['eq:kfgr-fl-linear']}), versus $\Delta \tilde{E} =\tilde{E}_1-\tilde{E}_2$. The super-Ohmic bath spectral density with $n=3$ is used with $\lambda_3=\hbar\omega_c$, and $\theta=1$. Different values of $\gamma_e=0.1$, $1$, and $2$ were used while $\gamma_f=0$ and $\langle \delta E^2\rangle=0.1 (\hbar\omega_c)^2$. The m-FGR-1 rate without time dependent fluctuations, Eq. (\ref{['eq:kfgm-1']}), which was shown in Fig. 3, is also provided for comparison.