(d) (e) (f)
Figure 6. Qualitative trends in population dynamics of an open 2-level system as computed via HEOM. The variation of the dynamics with respect to: a, d) bath reorganization energy; b, e) decoherence rates; and c, f) temperature. All systems converged w.r.t Matsubara frequency and hierarchy level. The upper row corresponds to\(\Delta E=100\ cm^{-1}\) and the lower row corresponds to\(\Delta E=200\ cm^{-1}\).
As another illustration, we explore the qualitative dependence of population dynamics in a molecular dimer system on the choice of system and bath parameters. The molecular dimer Hamiltonian is given by:
\(H=\frac{1}{2}\Delta E*\sigma_{z}+V*\sigma_{x}\), (17)
where \(\Delta E\) is the energy gap, \(V=200\ cm^{-1}\), is the electronic coupling between the two states, and \(\sigma_{x}\) and\(\sigma_{z}\) are the Pauli matrices. For this example, we start with the base model given by Strumpfer and Schulten6 and vary the reorganization energy, \(\eta\) (Figure 6a, 6d), decoherence rates, \(\gamma\) (Figure 6b, 6e), and temperature (Figure 6c, 6f). For these examples, the other parameters are as follows: time step = 0.1 fs, number of steps = 10000, γ = 10 ps-1, temperature = 300 K, and η = 100 cm-1, unless otherwise specified. We observe that increasing \(\eta\), \(\gamma\), and/or temperature increases the rate of thermalization. These trends can be rationalized as follows. Reorganization energy quantifies how easy it is to perturb a system away from its equilibrium. A bath with a larger reorganization energy changes would tend to counteract the displacement away from equilibrium to a larger degree, and consequently would force faster thermalization. Small decoherence rates lead to a longer preservation of quantum coherences, which is in turn promotes persistent population transfer between involved states, which counteracts the idea of reaching thermal equilibrium. Note, this observation is also consistent with the behavior of decoherence-corrected surface hopping approaches.41 Finally, a higher temperature promotes more frequent “collisions” of the quantum system with the bath, which increases energy transfer rates, and in turn allows the system to reach thermal equilibrium more quickly.
In addition, we vary the energy gap magnitude, \(\Delta E\) to be either\(100\ cm^{-1}\) or \(200\ cm^{-1}\), for each of the bath parameters set. As expected, the equilibrium populations depend on the chosen\(\Delta E\). However, we also observe two notable trends. First, the rate of thermalization decreases with the increase of \(\Delta E\), which is consistent with the slowing down of the population transfer for larger energy gaps, expected from the simple Rabi oscillation consideration. This observation is also consistent with the recently reported surface hopping calculations on quantum systems embedded in effective baths.41 Second, we observe that the rates of thermalization become much more sensitive to the bath reorganization energy and decoherence rates for larger energy gap system (Figure 6, panels d-f).