Electron transfer in confined electromagnetic fields: a unified Fermi's golden rule rate theory and extension to lossy cavities

TL;DR

This work develops a unified Fermi's golden rule rate theory for electron transfer in confined electromagnetic fields by starting from a polaron-transformed Hamiltonian. It derives analytic force–force correlation functions that remain valid across temperature regimes and cavity time scales, recovering Marcus and Marcus–Jortner limits and revealing the energy gap law at low temperatures. The authors extend the theory to lossy cavities via a Brownian-oscillator spectral density, enabling closed-form ET-rate expressions that capture cavity lifetimes. Numerical analyses illustrate resonance-enhanced ET and ET-induced photon emission, highlighting how nanophotonic environments can actively control charge-transfer dynamics with potential experimental observables.

Abstract

With the rapid development of nanophotonics and cavity quantum electrodynamics, there has been growing interest in how confined electromagnetic fields modify fundamental molecular processes such as electron transfer. In this paper, we revisit the problem of nonadiabatic electron transfer (ET) in confined electromagnetic fields studied in [J. Chem. Phys. 150, 174122 (2019)] and present a unified rate theory based on Fermi's golden rule (FGR). By employing a polaron-transformed Hamiltonian, we derive analytic expressions for the ET rate correlation functions that are valid across all temperature regimes and all cavity mode time scales. In the high-temperature limit, our formalism recovers the Marcus and Marcus-Jortner results, while in the low-temperature limit it reveals the emergence of the energy gap law. We further extend the theory to include cavity loss by using an effective Brownian oscillator spectral density, which enables closed-form expressions for the ET rate in lossy cavities. As applications, we demonstrate two key cavity-induced phenomena: (i) resonance effects, where the ET rate is strongly enhanced at certain cavity mode frequencies, and (ii) electron-transfer-induced photon emission, arising from the population of cavity photon Fock states during the ET process. These results establish a general framework for understanding how confined electromagnetic fields reshape charge transfer dynamics, and suggest novel opportunities for controlling and probing ET reactions in nanophotonic environments.

Figures (7)

• Figure 1: ET rates $k_{\text{D}\to \text{A}}$ obtained from FGR and various approximated rate expressions. (a) Model A under the high-temperature limit with $T = 3\times 10^5$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} (black open circles for outside cavity, and cyan dots for inside the cavity) are compared to Marcus results both outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and inside the cavity (Eq. \ref{['eq:Marcus-A']}, red dashed line) cases. (b) Model B under the high-temperature limit with $T = 3\times 10^4$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} are compared to Marcus results both outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and inside the cavity (Eq. \ref{['eq:Marcus-B']} in green dashed line, and Eq. \ref{['eq:Marcus-C']} in red dashed line) cases. (c) Model A under moderate temperature with $T = 300$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} are compared to Marcus results outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and MJ results inside the cavity (Eq. \ref{['eq:Marcus-A']}, red dashed line). (d) Model B under moderate temperature with $T = 300$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} are compared to Marcus results outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and MJ results inside the cavity (Eq. \ref{['eq:Marcus-B']}, red dashed line). (e) Model A under low temperature with $T = 0$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} are compared to Marcus results outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and MJ results inside the cavity (Eq. \ref{['eq:Marcus-A']}, red dashed line). (f) Model B under low temperature with $T = 0$ K. FGR results using Eq. \ref{['eq:FGR-LRT']} are compared to Marcus results outside cavity (Eq. \ref{['eq:Marcus']}, gray solid lines) and MJ results inside the cavity (Eq. \ref{['eq:Marcus-B']}, red dashed line). Note that in panels (e) and (f), $T = 0.01$ K has been applied to the Marcus / MJ rates in order to avoid singularities, and the EGL linear fitting results are shown in thin dashed lines (black for outside cavity, and cyan for inside cavity).
• Figure 2: Resonance effect of the cavity modified ET rates. We fix $T =$ 300 K. Panels (a)-(b) use the parameters of Model A while changing the cavity frequency $\omega$ and light-matter coupling strength $t'_{DA}$. Panel (a) shows the outside cavity ET rates ($k_\text{out}$) obtained using Marcus theory in Eq. \ref{['eq:Marcus']}, from which we pick up two specific $-\Delta G_0$ values in the Marcus inverted regime, 1.4 eV (red) and 2.0 eV (blue), to explore the resonance effect, where the cavity modified ET rate $k_\text{in}$ is obtained using the MJ rate expression in Eq. \ref{['eq:MJ_rate_A']}. Panel (b) shows the cavity rate modification $k_\text{in} / k_\text{out}$ by varying $\hbar \omega$. Similarly, panels (c)-(d) use the parameters of Model B while changing $\omega$ and $g'_{DA}$. We pick up $-\Delta G_0 =$ 0.4 eV (red) and 0.6 eV (blue) and their corresponding $k_\text{out}$ in panel (c) to explore the resonance effect, where the cavity modified ET rate $k_\text{in}$ is obtained using the MJ rate expression in Eq. \ref{['eq:MJ_rate_B']}. The corresponding cavity rate modification $k_\text{in} / k_\text{out}$ is shown in panel (d). The predicted resonance frequency is indicated by the red and blue dashed line(s).
• Figure 3: Quantum dynamics of the donor, acceptor states, as well as the photon number. Simulations are performed using the HEOM method. (a) Population dynamics of the donor state outside the cavity (blue) and inside the cavity (red). (b) Population dynamics of the photon-dressed acceptor states, $|\text{A}, 0 \rangle$ (gray), $|\text{A}, 1 \rangle$ (blue), $|\text{A}, 2 \rangle$ (green), and $|\text{A}, 3 \rangle$ (red). (c) Average photon number $N(t) = \langle \hat{a}^\dagger(t) \hat{a}(t) \rangle$ as a function of time.
• Figure 4: ET rate obtained from FGR using Eq. \ref{['eq:FFCF-3']} inside lossy cavities with various $\mathcal{Q}$ factors. Here, we fix $T =$ 300 K. The parameters are taken from (a) Model A, and (b) Model B, respectively.
• Figure 5: Visualization of typical correlation functions outside / inside the cavity. Upper panels: Model A (left for real part, right for imaginary part), where the short time behavior is also shown in the inner panels. Lower panels: Model B (left for real part, right for imaginary part).