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Optimizing hierarchical equations of motion for quantum dissipation and quantifying quantum bath effects on quantum transfer mechanisms
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Image of FIG. 1.
FIG. 1.

Friction spectrum J(ω)/ω, via Eq. (3) by a common reorganization energy λ, as function of ω (in unit of ωBO), for the critically damped (r BO = 1, black), critically weakly underdamped (r BO = 0.5, red), and strongly underdamped (r BO < 0.5, blue) BO cases, respectively. The inset plots the corresponding J(ω) curves.

Image of FIG. 2.
FIG. 2.

The parameters and in the [N + 1/N] Bose function approximant [Eq. (12)] as functions of N. Note that is negative and scales approximately as , for large N, while behaves similarly as the [N/N] counterpart, R N = 1/[4(N + 1)(2N + 3)] (black dash curve).

Image of FIG. 3.
FIG. 3.

The residue function δC N (ω)/λ vs. βω using the [N + 1/N] X (black), [N + 1/N] (red), (blue dash), and MSD (black dot) schemes, with N = 1 in all. The BO parameters are r BO = 0.04 and βωBO = 10. The scaled accuracy control parameters are {16.9,22}, {16.3,21.1}, and {15.5,19.9}, for [2/1] X , , and MSD N = 1, respectively.

Image of FIG. 4.
FIG. 4.

The residue spectrum function δC N (ω), for the [N/N] (solid black), [N/N] X (red dash), and MSD (blue dot) schemes, with N = 1 and the four specified values of βγD. See Table I for the scaled accuracy control parameters .

Image of FIG. 5.
FIG. 5.

The scaled residue function δC N (ω)/Δ N vs. ω/Γ N using the [N + 1/N] scheme, for the BO parameters r BO = 0.5, 1, 2 and βωBO = 1 or 10, at some selected values of N.

Image of FIG. 6.
FIG. 6.

The scaled accuracy control parameters using the [N + 1/N] scheme, as functions of βωBO for r BO = 0.5 (black), 1 (red), and 2 (blue).

Image of FIG. 7.
FIG. 7.

Dynamic results of ET systems at T = 298 K, subject to the strongly underdamped BO bath with r BO = 0.25 (black), the critically damped case with r BO = 1 (red), and overdamped case with r BO = 3 (blue), respectively. Panel (a) shows the relative entropy related quantity φ(t) [Eq. (26)]. Panels (b) and (c) illustrate the population evolution P α(t) = ραα(t), and the state resolved measure of interference ϕα(t) [Eq. (27)], respectively, for the acceptor (thick curves) and bridge (thin curves) sites.

Image of FIG. 8.
FIG. 8.

Same as in Fig. 7, but with the temperature T = 77 K.

Image of FIG. 9.
FIG. 9.

The delocalized populations (left panels) and the delocalized state-resolved interference measures (right panels), recasting Figs. 8(b) and 8(c) in the eigenstate representation of the reduced system.


Generic image for table
Table I.

The scaled accuracy control parameters , corresponding to each residue function in Fig. 4.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Optimizing hierarchical equations of motion for quantum dissipation and quantifying quantum bath effects on quantum transfer mechanisms