Boson operators systems

We next consider another example of quantum-mechanical relaxation. In this example an isolated harmonic mode, which is regarded as our system, is weakly coupled to an infinite bath of other harmonic modes. This example is most easily analyzed using the boson operator formalism (Section 2.9.2), with the Hamiltonian... 

The basic idea of the mapping approach is to change from the discrete representation employed in Eq. (53) to a continuous representation. There are several ways to do so, most of them are based on the representation of spin operators by boson operators. Well-known examples of such mappings are the Holstein-Primakoff transformation, which represents a spin system by a single nonlinear boson DoF, and Schwinger s theory of angular momentum,which represents a spin system by two independent boson DoF. 

Since the non-tilde operators describe physical variables, G(k (3)11 is the physical propagator to be used to treat the properties of the thermal bosonic system. It is interesting to observe that, except for the non-diagonal elements, this TFD-propagator is equal to the one introduced in the Schwinger-Keldysh approach, which is claimed to be (in this equivalence with TFD) a thermal theory describing linear-response processes only (H. Chu et.al., 1994). 

As discussed in Section VI.A for the case of spin systems, the formalism described above is not the only way to construct a mapping of a A -level system. First of all, it is clear that one may again eliminate one boson DoF by exploiting the operator (which corresponds to the identity operator in the physical... 

In order to rigorously describe the nonlinear interaction between the weak pulsed fields, we now turn to the fully quantum treatment of the system. The traveling-wave electric fields can be expressed through single mode operators as j(z, t) = dj(t) Cqz (j = 1, 2), where uj is the annihilation operator for the field mode with the wavevector kp + q. The singlemode operators a and aq possess the standard bosonic commutation relations... 

The generalization of the representation of paulion operators in terms of bosonic ones for the case of truncated oscillators of higher ranks is derived in the paper (15). The authors of this paper used this generalization to introduce a new constraint-free bosonic description of truncated oscillator systems. This result can be important in consideration of collective properties of Frenkel excitons in organic crystals with account of multilevel molecular structures and mixing of molecular configurations. 

Energy excitations in 1-d Fermi systems are effectively Bose excitations with zero mass. A suitable representation of Fermion field operators in terms of Bosons has been given by... 

The system-reservoir interaction operators are products of the type Vak = AR, where A represents a molecule operator (aj, a, B, B) and R a reservoir operator b, b ). Because of the BO approximation the electronic operators commute with the phonon operators. Moreover, the pseudolocalized phonon operators commute with those of the baind phonons because of the boson commutation rules. This amounts to the statement that all molecule variables commute with the reservoir variables, [A,R] = 0. 

Each term in Eq. (8.31) contains the creation and annihilation operators of at most two different particles. The energy due to the quasi-photons is additively composed of that of the individual quasi-photons, i.e. the quasi-photons constitute a system of independent Bosons. We may occupy the quasi-photon states according to the Bose distribution. 

---