System-bath interactions

Another concern regards the initial conditions. Here we have assumed a factorized initial state of the system and bath. This prevents us from taking into account system-bath interactions that may have occurred prior to that time. In particular, if the system is in equilibrium with the bath, their states are entangled or correlated [24, 94]. 

The initial temperature of the molecule is very low and certainly is much lower than hcajk of totally symmetric modes. We may divide the modes of the molecule into those which are optically active (predominantly totally symmetric or relevant R) and those which are not excited directly by the laser (bath modes B). This division of the system-bath interactions accounts for dephasing and energy relaxation by T2 and Ti time constants as discussed elsewhere. ... 

The Hamiltonian in question is the sum of the system (S), reservoir bath (B) and system-bath interaction (7) terms,... 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

The system-bath interaction term in (8.48) is xf, where f = cjqjis, the force exerted by the thermal environment on the system. The random force 7 (t), Eq. (8.56) is seen to have a similar form. 

Comparing Eqs (J.liy-ij.19 ) we see that Z t) is essentially the Fourier transform of the spectral density associated with the system-bath interaction. The differences are only semantic, originating from the fact that in Eqs (7.77)-(7.79) we used mass renormalized coordinates while here we have associated a mass nij with each harmonic bath mode j. 

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations ... 

What did we achieve so far We have an equation, (10.133) or (10.134), for the time evolution of the system s density operator. All terms in this equation are strictly defined in the system sub-space the effect of the bath enters through correlation functions of bath operators that appear in the system-bath interaction. These correlation functions are properties of the unperturbed equilibrium bath. Another manifestation of the reduced nature of this equation is the appearance of... 

As discussed in Sections 10.4.8 and 10.4.9, these eigenstates may be defined in terms of a system Hamiltonian that contains the mean system-bath interaction. 

The rationale behind this choice of system-bath interaction is that it represents the first term in the expansion of a general interaction between the... 

Choosing a physically motivated representation is useful in developing physically guided approximation schemes. A commonly used approximation for the model (12,4)—(12.6) is to disregard tenns with j j in the system-bath interaction (12.5b). The overall Hamiltonian then takes the fonn... 

In this transformed Hamiltonian Hq again describes uncoupled system and bath the new element being a shift in the state energies resulting from the system-bath interactions. In addition, the interstate coupling operator is transformed to... 

Under the assumption of weak system-bath interactions, going into the Markovian limit, the probabilities P to occupy the n state of the molecular oscillator satisfy the master equation [31] ... 

We explain here the operation principles of simple molecular devices, a thermal rectifier [20] and a heat pump [21]. First we present the heat current in the anharmonic (TLS) model. Figure 12.2 demonstrates that the current ino-eases monotonicaUy with AT, then saturates at high tanperature differences. It can be indeed shown that dJ/dAT > 0, which indicates that negative differential thermal conductance (NDTC), a decrease of J with increasing AT, is impossible in the present (bilinear coupling) case. As shown in Ref [19], NDTC requires nonlinear system-bath interactions, resulting in an effective temperature-dependent molecule-bath coupling term. 

Considering the system in the contact with the thermal bath (thermal reservoir) the same assumption about neglecting system-bath interaction leads to the existence of canonical (Gibbs) distribution for the probabilities to find the system in the state with energy E ... 

A possibility to overcome this limitation of the above conical-intersection models, at least in a quahtative manner, is to consider anhar-monic couplings of the active degrees of freedom of the conical intersection with a large manifold of spectroscopically inactive vibrational modes. The effect of such a couphng with an environment has been investigated for the pyrazine model in the weak-coupling limit (Redfield theory) in Ref. 19. The simplest ansatz for the system-bath interaction, which is widely employed in quantum relaxation theory assumes a coupling term which is bilinear in the system and bath operators... 

Despite the fact that the exact j or TZ can be formally expressed in terms of an infinite series expansion, its evaluation, however, amounts to solve the total composite system of infinite degrees of freedom. In practice, one often has to exploit weak system-bath interaction approximations and the resulting COP [Eq. (1.2)] and POP [Eq. (1.3)] of QDT become nonequivalent due to the different approximation schemes to the partial consideration of higher order contributions. It is further noticed that in many conventional used QDT, such as the generalized quantum master equation, Bloch-Redfield theory and Fokker-Planck equations, there involve not only... 

---