Open quantum system simulation of time and frequency resolved spectroscopy

TL;DR

The paper addresses excitonic energy transfer in light-harvesting complexes under laser excitation and situates it within open quantum system theory using Frenkel excitons. It surveys weak-coupling Redfield, strong-coupling Förster, and nonperturbative HEOM approaches, detailing how each yields population and coherence dynamics and how optical spectra are computed. It emphasizes linear absorption and two-dimensional electronic spectroscopy (2DES), including isotropic averaging and static disorder, and demonstrates HEOM on the Fenna-Matthews-Olson (FMO) complex with DM-HEOM implementations. The work highlights the ability of HEOM to bridge regimes, accurately simulate 2DES signals, and interpret energy flow and coherence, albeit with significant computational requirements that motivate optimized, high-performance algorithms.

Abstract

The dynamics of excitonic energy transfer in molecular complexes triggered by interaction with laser pulses offers a unique window into the underlying physical processes. The absorbed energy moves through the network of interlinked pigments and in photosynthetic complexes reaches a reaction center. The efficiency and time-scale depend not only on the excitonic couplings, but are also affected by the dissipation of energy to vibrational modes of the molecules. An open quantum system description provides a suitable tool to describe the involved processes and connects the decoherence and relaxation dynamics to measurements of the time-dependent polarization.

Figures (5)

• Figure 1: Visualization of the integrand $c(\omega)=J(\omega)n(\omega,\beta){\rm e}^{{\rm i}\omega t}$ of $C(t)=\int_{-\infty}^\infty c(\omega) {\rm d}\omega$ in the complex $\omega$ plane in the domain $-10<\Re\omega<10$ and $0<\Im\omega<10$. To obtain $C(t)$, the integration path along the real $\omega$-axis is deformed and closed by a semi-circular contour in the positive imaginary $\omega$-plane. This new path encloses an infinite number of poles, which yield the result of the integration by a sum of the residues at the poles, multiplied by $2\pi{\rm i}$.
• Figure 2: Left panel: Spectral density $J_{\rm DL,0}$ (unshifted Drude-Lorentz form, $\lambda=35$ cm$^{-1}$ and $\nu^{-1}=50$ fs) and $J_{\rm SDL,420}$ (shifted Drude-Lorentz peak, $\Omega=420$ cm$^{-1}$, $\lambda=35$ cm$^{-1}$ and $\nu^{-1}=50$ fs). The arrow indicates the difference of eigenenergies of a two-site system $H_{\rm ex}=\left( -7510010075 \right)\;{\rm cm}^{-1}$, where by construction both spectral densities have the same value. Right panel: Relaxation of the diagonal element $\langle E_2| \rho(t) |E_2 \rangle$ to the thermal state (upper non-oscillatory graphs) and damped oscillations of the off-diagonal coherence ${\rm Re} (\langle E_1| \rho(t) |E_2 \rangle)$ at $T=277$ K. While both spectral densities give very similar relaxation rates, the off-diagonal coherence is much prolonged for $J_{\rm SDL,420}$ due to its small slope toward $\omega\rightarrow 0$. Reprinted from Kramer2014, Fig. 8, with the permission of AIP Publishing.
• Figure 3: Quantum dynamics starting from the highest eigenstate of the FMO Hamiltonian Adolphs2006 to the thermal state for various reorganization energies $\lambda$ at $T=277$ K. Solid line: HEOM method (exact) population in exciton basis of the highest and lowest eigenstate populations, dashed line: secular Redfield theory, dotted line: full Redfield. HEOM shows there exists an optimal value of $\lambda\approx 110$ cm${}^{-1}$ for fastest thermalization (as seen by the crossing of the populations of both states, vertical line), while in the secular Redfield approximation (not applicable at strong couplings) a higher coupling always increases the thermalization rate.
• Figure 4: Monomeric unit of the FMO complex with 7 bacteriochlorophylls with arrows indicating the directions of the transition dipoles. The protein scaffold keeping the bacteriochlorophylls in place is not shown.
• Figure 5: 2DES (sum of rephasing and non-rephasing pathways) for the FMO complex computed with DM-HEOM from left panels to right panels for increasing delay time $T_2=\{40,400,800\}$ fs at temperature $100$ K. Upper row: all parallel polarization $\langle 0^\circ ,0^\circ, 0^\circ ,0^\circ \rangle$. Lower row: double-crossed polarization $\langle 45^\circ ,-45^\circ, 90^\circ ,0^\circ \rangle$. Rotational averaging is performed, static disorder is not considered.