System Embedded in Heat Bath

In Section 2, the density matrix method is applied to an isolated system. In this section, the system embedded in a heat bath shall be treated that is, the isolated system shall be divided into two parts, one is referred to as the system which is under observation and the other part the heat bath. 

Before presenting a rigorous method to treat this problem, a very useful and simple method, called the effective Hamiltonian method [17], shall be discussed as follows. 

The two-state system shall be considered as an example for illustration (see Eqs. (2.1)-(2.7)). According to the effective Hamiltonian method, the energies are complex, i.e., 

The density matrix elements for this system are given by 

From Eqs. (3.6) (3.8) one finds that the Liouville equation can be expressed as 

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In the above, the principles of how to study the dynamics of an isolated system by using the density matrix method have been shown. However, most experiments are performed for the system (or subsystem) embedded in a heat bath in this case the isolated system consists of the system plus heat bath. In the following the MEs shall be derived for the system embedded in heat bath. In this case, instead of pm, pnsnbfls b will be employed. Here s and b describe the system and heat bath, respectively. For the case in which the bath is much larger than the system, it may be assumed that the bath maintains thermal equilibrium and... 

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