^{1}, Jian Xu

^{2}, Jie Hu

^{2}, Rui-Xue Xu

^{1,a)}and YiJing Yan

^{1,2,b)}

### Abstract

Hierarchical equations of motiontheory for Drude dissipation is optimized, with a convenient convergence criterion proposed in advance of numerical propagations. The theoretical construction is on the basis of a Padé spectrum decomposition that has been qualified to be the best sum-over-poles scheme for quantum distribution function. The resulting hierarchical dynamics under the *a priori* convergence criterion are exemplified with a benchmark spin-boson system, and also the transient absorption and related coherent two-dimensional spectroscopy of a model exciton dimer system. We combine the present theory with several advanced techniques such as the block hierarchical dynamics in mixed Heisenberg-Schrödinger picture and the on-the-fly filtering algorithm for the efficient evaluation of third-order optical response functions.

Support from the NNSF of China (21033008 & 21073169), the National Basic Research Program of China (2010CB923300 & 2011CB921400), and the Hong Kong RGC (604709) and UGC (AoE/P-04/08-2) is gratefully acknowledged.

I. INTRODUCTION

II. FORMALISM

A. Optimal hierarchy for Drude dissipation

B. Accuracy control criterion

III. NUMERICAL DEMONSTRATIONS

A. Spin-boson dynamics

B. Nonlinear spectroscopy of a model exciton dimer system

IV. SUMMARY

### Key Topics

- Excitons
- 8.0
- Correlation functions
- 7.0
- Nonlinear spectroscopy
- 7.0
- Random noise
- 7.0
- Absorption spectroscopy
- 6.0

## Figures

The deviation spectrum function δ*C* _{ N }(ω) of Drude bath, plotted in terms of δ*C* _{ N }(ω)/Δ_{ N } versus ω/Γ_{ N }(γ) for some selected values of {*N*, βγ}, where Δ_{ N } is given via Eq. (8) and Γ_{ N }(γ) is by the approximate of Eq. (12).

The deviation spectrum function δ*C* _{ N }(ω) of Drude bath, plotted in terms of δ*C* _{ N }(ω)/Δ_{ N } versus ω/Γ_{ N }(γ) for some selected values of {*N*, βγ}, where Δ_{ N } is given via Eq. (8) and Γ_{ N }(γ) is by the approximate of Eq. (12).

βΓ_{ N }(γ) and versus βγ.

βΓ_{ N }(γ) and versus βγ.

Evolution of the reduced spin system density matrix elements, with the same parameters as Fig. 8 of Ref. 45; i.e., ε/*V* = 1, λ/*V* = 0.25, γ/*V* = 5, and β*V* = 50. The accuracy control parameters are {Γ_{ N }/Ω_{ s }, κ_{ N }} = {2.6, 3.6} for *N* = 4 and {4.7, 8.5} for *N* = 8, as indicated. The converged dynamics are identical to that of *N* = 10 whose {Γ_{ N }/Ω_{ s }, κ_{ N }} = {6.3, 12.0}.

Evolution of the reduced spin system density matrix elements, with the same parameters as Fig. 8 of Ref. 45; i.e., ε/*V* = 1, λ/*V* = 0.25, γ/*V* = 5, and β*V* = 50. The accuracy control parameters are {Γ_{ N }/Ω_{ s }, κ_{ N }} = {2.6, 3.6} for *N* = 4 and {4.7, 8.5} for *N* = 8, as indicated. The converged dynamics are identical to that of *N* = 10 whose {Γ_{ N }/Ω_{ s }, κ_{ N }} = {6.3, 12.0}.

Dispersed transient absorption coefficient signals of the model dimer system (see text): (a) Linear absorption (A) and emission (E) signals; (b) Nonlinear absorptive component; (c) Nonlinear emissive component. The pump field is a transform-limited 50 fs-pulse of finite intensity (see text), centered at ω = ε; i.e., Δω ≡ ω − ε = 0, as indicated by the arrow in panel (a). At *T* = 298 K, *N* = 1 is in the numerically accurate range, with {Γ_{ N }/Ω_{ s }, κ_{ N }} = {8.3, 8.7} as indicated.

Dispersed transient absorption coefficient signals of the model dimer system (see text): (a) Linear absorption (A) and emission (E) signals; (b) Nonlinear absorptive component; (c) Nonlinear emissive component. The pump field is a transform-limited 50 fs-pulse of finite intensity (see text), centered at ω = ε; i.e., Δω ≡ ω − ε = 0, as indicated by the arrow in panel (a). At *T* = 298 K, *N* = 1 is in the numerically accurate range, with {Γ_{ N }/Ω_{ s }, κ_{ N }} = {8.3, 8.7} as indicated.

Same as Fig. 4, but at *T* = 77 K, where *N* = 1 is in the semi-quantitative range, with {Γ_{ N }/Ω_{ s }, κ_{ N }} = {2.4, 2.4}, as indicated.

Same as Fig. 4, but at *T* = 77 K, where *N* = 1 is in the semi-quantitative range, with {Γ_{ N }/Ω_{ s }, κ_{ N }} = {2.4, 2.4}, as indicated.

Two-dimensional spectroscopy of the same dimer system of Fig. 4 at *T* = 298 K, reported in Δω_{1} = ω_{1} − ε and Δω_{3} = ω_{3} − ε. All pulsed fields are operated in weak and impulsive limit.

Two-dimensional spectroscopy of the same dimer system of Fig. 4 at *T* = 298 K, reported in Δω_{1} = ω_{1} − ε and Δω_{3} = ω_{3} − ε. All pulsed fields are operated in weak and impulsive limit.

The Δω_{1} = 0 slices of the absorptive and emissive panels of Fig. 6. This figure resembles the impulsive pump counterpart of Fig. 4.

The Δω_{1} = 0 slices of the absorptive and emissive panels of Fig. 6. This figure resembles the impulsive pump counterpart of Fig. 4.

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