Collective bath mode

The parabolic barrier case demonstrates that the effect of the medium is to replace the original reaction coordinate q by a collective mode p along which the dynamics is trivial. It is useful to define a collective bath mode o orthogonal to the unstable mode p as... 

The second equality on the right-hand side follows directly from Eq. (46) for e = 0. This allows the collective bath mode frequency to be expressed in terms of the Laplace transform of the time-dependent friction (cf. Eqs. (39) and (43)), so it is well defined in the continuum limit. 

Note that in contrast to the ohmic friction, the normal mode friction function has memory. The corresponding spectral density of normal modes /(X) (cf. Eq. (49)) decays as X4, in a sense it is much better behaved than the ohmic spectral density J(o>), which increases without bound with u>. Finally, the collective bath mode frequency, fl (cf. Eq. (59)),... 

In the following the coordinate p will denote the generalized reaction coordinate (the analog of the coordinate / in the previous section) and the coordinate ct the collective bath mode. 

Recently, we have shown that whenever qubits collectively couple to common bath modes (e.g., in ion traps [119,120] or cavities [121-123]) local modulations of individual qubits can eliminate cross-decoherence. Our major result is that the modulation must be faster than the correlation time of the bath, but can be much slower than the (unmodulated) multipartite coherence time and still be effective. 

From the interaction Hamiltonian equation (15.3), we note that the Vb bath modes produce cumulative effects by their coupling to the discretized subsystem. This suggests that the interaction Hamiltonian can be formally re-written in terms of a set of collective coordinates such that... 

The basic features of ET energetics are summarized here for the case of an ET system (solute) linearly coupled to a bath (nuclear modes of the solute and medium) [11,30]. We further assume that the individual modes of the bath (whether localized or extended collective modes) are separable, harmonic, and classical (i.e., hv < kBT for each mode, where v is the harmonic frequency and kB is the Boltzmann constant). Consistent with the overall linear model, the frequencies are taken as the same for initial and final ET states. According to the FC control discussed above, the nuclear modes are frozen on the timescale of the actual ET event, while the medium electrons respond instantaneously (further aspects of this response are dealt with in Section 3.5.4, Reaction Field Hamiltonian). The energetics introduced below correspond to free energies. Solvation free energies may have entropic contributions, as discussed elsewhere [19], Before turning to the DC representation of solvent energetics, we first display the somewhat more transparent expressions for a discrete set of modes. 

To demonstrate the potential of two-dimensional nonresonant Raman spectroscopy to elucidate microscopic details that are lost in the ensemble averaging inherent in one-dimensional spectroscopy, we will use the Brownian oscillator model and simulate the one- and two-dimensional responses. The Brownian oscillator model provides a qualitative description for vibrational modes coupled to a harmonic bath. With the oscillators ranging continuously from overdamped to underdamped, the model has the flexibility to describe both collective intermolecular motions and well-defined intramolecular vibrations (1). The response function of a single Brownian oscillator is given as,... 

The feature of the aerodynamic regime is that collective effects in the bath are insignificant. In the next section the coupling to hydrodynamic modes in liquids is discussed. 

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