The diabatic potentials as defined in Eqs. (2) and (3) and parameters defined in Table I (state 1 depicted in black; state 2 in red). Also depicted are the initial coherent state (blue) on state 1 and snapshots of the densities for each electronic state at (state 1: full green line; state 2: dashed green line).
Pure system dynamics : (a) Lower state (state 2) population dynamics evaluated using the initial coherent state defined in Sec. IV. Also shown in red is the Rabi oscillation between the vibrational ground states of the two diabatic potentials; (b) electronic coherence —real and imaginary parts and absolute value; (c) position expectation values for the two states.
(a) Population dynamics for the upper state (state 1): The black curve depicts the populations computed using the system Hamiltonian only, the red curve depicts the populations computed using system and effective-mode Hamiltonian , the green curve depicts the populations computed from the Caldeira–Leggett equation as described in Eq. (16), and the blue curve depicts the populations computed from Eq. (1)–(6) by a 31 mode wavepacket calculation, where are depicted in part (b) below. For the dissipative examples, a value of was used for the friction coefficient. (b) Coupling coefficients defined in Eq. (6) that correspond to the spectral density of Eq. (31). The frequency spacing in (b) is .
Trajectories in the 2D system vs effective-mode plane for (a) weak , (b) medium , and (c) strong coupling . The black curves correspond to the upper state (state 1) trajectories, the red curves correspond to the lower state trajectories, and the green curve indicates the crossing seam. The blue crosses indicate the initial condition , .
System-plus-effective-mode dynamics : Densities for the lower state (blue) and upper state (red) captured at (upper figure), (middle), and (lower figure). The extension of the grid is from in and from in .
System-plus-effective-mode dynamics : (a) Upper state (state 1) population dynamics (blue) and electronic coherence—real and imaginary parts and absolute value—and (b) position expectation values for the two states.
(a) Population and (b) coherence dynamics for the example of Sec. ???, computed using different values of .
Time-dependent position expectation values for the two states computed for the nondissipative and, for different values, the dissipative case. The solid curves correspond to trajectories for the upper electronic state and the dashed curves correspond to trajectories for the lower electronic states. Due to the small amount of population transfer to the lower state for , the trajectory is not depicted for this state.
(a) Population, (b) coherence dynamics, and (c) position expectation values for the two states (dashed state), evaluated at different temperatures for the example of Sec. ???. The results were computed using a fixed value of for the Caldeira–Leggett equation as described in Eq. (16).
Population (a) and coherence (b) dynamics for the example of Sec. IV C using spectral densities approximated at the first, second, and third order levels of the effective-mode hierarchy , . For the bath a fixed value of was used. Part (c) depicts the couplings that are related to through Eq. (31).
System and effective-mode(s) parameters quoted in eVs.
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