Spin-boson models

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The following figures and data are our benchmarking results (for detailed discussion please see ref1).

 

Figure9

Figure 9. Results of the population difference D(t)=ρ11(t)ρ22(t) of spin-boson models with Ohmic spectral density at β=5. The first to fourth rows illustrate the models with parameters {α = 0.1, ωc = 1}, {α = 0.4, ωc = 1}, {α = 0.1, ωc = 2.5} and {α = 0.4, ωc = 2.5}, respectively. In panels (a1), (b1), (c1) and (d1), the green, blue and cyan solid lines represent the results produced by NaF-cc, NaF-cx and NaF-TW, respectively. Panels (a2), (b2), (c2) and (d2), the cyan, pink, orange and magenta solid lines denote the results produced by NaF-TW, NaF-TW2, NaF-HW and NaF-HW2, respectively. In panels (a3), (b3), (c3) and (d3), the orange dashed lines, purple dashed lines, brown dashed lines and cyan solid lines denote the results produced by SH-1, SH-2, SH-3 and NaF-TW, respectively. The numerically exact results produced by eHEOM2 are demonstrated by black points in each panel. data

 

Figure10

Figure 10. Similar to Figure 9, but illustrates the results of the modulus of the off-diagonal term |ρ12(t)|. data

 

 

 


1 B. Wu, B. Li, X. He, X. Cheng, J. Ren and J. Liu, "Nonadiabatic field: A conceptually novel approach for nonadiabatic quantum molecular dynamics ", J. Chem. Theory Comput. 21, 3775-3813 (2025).
2 Y. Tanimura and R. Kubo, "Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath", J. Phys. Soc. Jpn. 58, 101-114 (1989). Y.-A. Yan, F. Yang, Y. Liu and J. Shao, "Hierarchical approach based on stochastic decoupling to dissipative systems", Chem. Phys. Lett. 395, 216-221 (2004). R.-X. Xu, P. Cui, X.-Q. Li, Y. Mo and Y. Yan, "Exact quantum master equation via the calculus on path integrals", J. Chem. Phys. 122, 041103 (2005). J. Shao, "Stochastic description of quantum open systems: formal solution and strong dissipation limit", Chem. Phys. 322, 187-192 (2006). J. M. Moix and J. Cao, "A hybrid stochastic hierarchy equations of motion approach to treat the low temperature dynamics of non-Markovian open quantum systems", J. Chem. Phys. 139, 134106 (2013). Z. Tang, X. Ouyang, Z. Gong, H. Wang and J. Wu, "Extended hierarchy equation of motion for the spin-boson model", J. Chem. Phys. 143, 224112 (2015). C. Duan, Q. Wang, Z. Tang and J. Wu, "The study of an extended hierarchy equation of motion in the spin-boson model: the cutoff function of the sub-ohmic spectral density", J. Chem. Phys. 147, 164112 (2017).