Schematic illustration of the model used for mixed quantum-classical dynamics. Each site n comprises a quantum two-state unit with transition energy ω n . Such unit couples to a classical oscillator x n , which in turn interacts with a stochastic environment corresponding to a temperature T. Arrows indicate couplings, purely quantum mechanical (J n, m ), quantum-classical (λ n ), and classical-stochastic (γ).
Double sided Feynman diagrams illustrating the six Liouville pathways that contribute to the 2D optical signal. Shown are the diagrams for ground state bleach (GB), stimulated emission (SE), and excited state absorption (EA), where |g⟩, |e⟩, and |f⟩ denote the quantum ground state and excitations in the singly and doubly excited manifold, respectively. Dashed lines represent interactions with a light pulse. The arrows on the left-side indicate the time-direction, and serve to specify the interaction times τ1, τ2, τ3, and τ4, as well as the intervals t 1, t 2, and t 3. The upper row shows the rephasing diagrams, whereas the nonrephasing variants are demonstrated in the bottom row.
Decomposition of the NR-SE diagram into contributions from populations |ϕ k ⟩⟨ϕ k | and interstate coherences |ϕ k ⟩⟨ϕ l | (k ≠ l).
Real part of the calculated 2D spectra for a dimer system at waiting times t 2 = 0 ps (top row), 1.5 ps (middle row), and 15 ps (bottom row). Left column displays results obtained using the conventional NISE method, neglecting quantum feedback. Results for the surface hopping approaches are shown in the second and third columns, where the response is obtained through the primary and auxiliary wavefunctions, respectively. The outcome of HEOM is demonstrated in the right column. Contours indicate levels for every 10% of the maximum absolute value. This value is used to normalize each spectrum. The labels I and II in the top-left plot indicate the two cross-peaks (see text).
(a) Calculated transfer of population from the higher to the lower-energy adiabatic state (curves), as a function of waiting time t 2. Also shown is the intensity of cross-peak I (circles), taken from simulated 2D spectra. Results are demonstrated for the conventional NISE method (green) and for surface hopping using the primary (blue) and auxiliary (red) wavefunction. The black curve indicates the Boltzmann factor, as derived from the instantaneous adiabatic energies in the course of the surface hopping calculations. Note that the peak intensities are rescaled so as to overlap the curves at t 2 = 1 ps and 15 ps. The corresponding scaling factors are reported in Table I . (b) Results for cross-peak II and population transfer from the lower to the higher-energy adiabat.
Time scales, obtained by fitting the data from Fig. 5(a) to the exponential C 1 + C 2 exp (−t 2/t c ). Also tabulated are factors used to rescale the cross-peak intensities in this figure.
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