Boson operators time-evolution operator

Then, owing to Eq. (D.l), within the Bosons representation, the time-evolution operator (D.3) becomes... 

As a matter of fact, the Boson normal-ordering procedure allows us to get the formal solution of the IP time-evolution operator involving the Dyson timeordering operator P. Also, observe that within the Bosons representation, the IP coordinate is... 

The chapter is organized as follows in the Section 7.2, we first present some details of spectral collocation method to develop space-time evolution of polarization plots for overall view of classical breathers in Section 7.2.1, then we present the mathematical model for TPBS parameters after second quantization in Section 7.2.2.1 and finally second quantization on K-G lattice is done with Bosonic field operators in Section 12.2.2. In Section 7.3, the results and discussion are also presented in three parts for the above three cases, hr Section 7.4, the conclusions are given. 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

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