General framework for quantifying dissipation pathways in open quantum systems. III. Off-diagonal system-bath couplings

Abstract

This paper extends the previously reported theory of dissipation pathways [J. Chem. Phys. 160, 214111 (2024)] to incorporate off-diagonal subsystem-bath coupling, which is often required to model molecular systems where the environment directly influences transitions and couplings between subsystem states. We systematically derive master equations for both population transfer and dissipation into individual bath components, for which we also rigorously prove energy conservation and detailed balance. The approach is based on second-order perturbation theory with respect to the subsystem-bath couplings, whose form is not limited to any specific model. The accuracy of the developed method is tested by applying it to diverse model Hamiltonians involving linearly coupled harmonic oscillator baths and comparing the outcomes against the hierarchical equations of motion (HEOM) method. Overall, our method accurately quantifies the contributions of specific bath components to the overall dissipation while significantly reducing the computational cost compared to numerically exact methods such as HEOM, thus offering a path to examine how vibronic interactions steer non-adiabatic processes in realistic chemical systems.

Figures (13)

• Figure 1: Schematic representation of the model systems. (a) Molecular dimer model where each system state ($E_1$, $E_2$) is connected with its own set of harmonic oscillators (represented by green circles). $V_{12}$ is the coupling connecting the system states. (b) Spin-boson model where the system states are connected to the same bath of harmonic oscillators (represented by blue circles). (c) Same as in (a) but for a molecular trimer model. (d) Shows the functional forms of the Drude-Lorentz spectral density (Eq. \ref{['eq:DL-SPD-5']}, green line) for the molecular dimer/trimer simulations and the Brownian Oscillator spectral density (Eq. \ref{['eq:BO-SPD']}, blue line) for the spin-boson simulations.
• Figure 2: Time-dependent population inversion, $\langle \hat{\sigma}_{z} (t)\rangle=P_{\alpha}(t)-P_{\beta}(t)$, for the molecular dimer in Simulation Set A. Results from MRT (cyan line) are compared with HEOM benchmarks (pink dashed line). Panels illustrate the dynamics for different reorganization energies ($\Lambda = \{0.05,0.2,0.5,1.0\}$) and energy gaps ($\Delta E= \{0.5,1.0,2.0\}$), with fixed parameters $T=1.0$, $V=0.25$, and a Drude-Lorentz cutoff frequency $\omega_{c}=0.5$.
• Figure 3: Time-dependent population inversion, $\langle \hat{\sigma}_{z} (t)\rangle=P_{\alpha}(t)-P_{\beta}(t)$, for the molecular dimer in Simulation Set B. Results from MRT (cyan line) are compared with HEOM benchmarks (pink dashed line). Panels illustrate the dynamics for different temperatures ($T = \{0.25,0.5,1.0\}$) and energy gaps ($\Delta E = \{0.5,1.0,2.0\}$), with fixed parameters $\Lambda=0.2$, $V=0.25$, and a Drude-Lorentz cutoff frequency $\omega_{c}=0.5$.
• Figure 4: Total steady-state dissipation density, $\mathcal{E}(\omega,\infty)$ for the molecular dimer in Simulation Set A. Results from MRT-D (cyan line) are compared with HEOM-D benchmarks (pink dashed line). Panels illustrate the dynamics for different reorganization energies ($\Lambda = \{0.05,0.2,0.5,1.0\}$) and energy gaps ($\Delta E= \{0.5,1.0,2.0\}$), with fixed parameters $T=1.0$, $V=0.25$, and a Drude-Lorentz cutoff frequency $\omega_{c}=0.5$.
• Figure 5: Total steady-state dissipation density, $\mathcal{E}(\omega,\infty)$ for the molecular dimer in Simulation Set B. Results from MRT (cyan line) are compared with HEOM benchmarks (pink dashed line). Panels illustrate the dynamics for different temperatures ($T = \{0.25,0.5,1.0\}$) and energy gaps ($\Delta E = \{0.5,1.0,2.0\}$), with fixed parameters $\Lambda=0.2$, $V=0.25$, and a Drude-Lorentz cutoff frequency $\omega_{c}=0.5$.