Time-evolution operator quantum harmonic oscillator

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

InEq. (113), I)] (tfv is the IP time-evolution operator of the driven quantum harmonic oscillator interacting with the thermal bath,... 

Another possibility is to extract the reduced time evolution operator from the analytical solution obtained by Louisell and Walker for the reduced time-dependent density operator of a driven damped quantum harmonic oscillator. 

Appendix D shows that the IP time evolution operator of a driven quantum harmonic oscillator is given by Eq. (D.23), that is,... 

From this viewpoint, which is the most fundamental, the line shape as a whole is the sum of the diagonal matrix elements of the time evolution operator of the driven damped quantum harmonic oscillator in the IP representation with respect to the diagonal part of the Hamiltonian of this oscillator. According to Eq. (120), each diagonal element is a sum of time-dependent terms... 

As a consequence, because of Eqs. (D.7) and (D.23), the full-time evolution operator (D.2) of the driven quantum harmonic oscillator takes the form ... 

Here, pj is the Boltzmann density operator of the H-bond bridge viewed as a quantum harmonic oscillator, pe is the Boltzmann density operator of the thermal bath, and (t) are effective time-evolution operators governing the dynamics of the H-bond bridge depending on the excited-state degree k of the fast mode. They are given by Eq. (110), that is,... 

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