(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).