A simple example—the quantum mechanical basis for macroscopic rate equations

2 A simple example—the quantum mechanical basis for macroscopic rate equations 

Consider two coupled multilevel systems L and 7 , characterizedby their spectrum of eigenvectors and eigenvalues. The Hamiltonian without the intersystem coupling is 

We assume that V, the operator that couples systems L and R to each other, mixes only / and r states, that is, = Vr,r — 0- We are interested in the transition between these two subsystems, induced by V. We assume that (1) the coupling V is weak coupling in a sense explained below, and (2) the relaxation process that brings each subsystem by itself (in the absence of the other) into thermal equilibrium is much faster that the transition induced by V between them. Note that assumption (2), which implies a separation of timescales between the L 7 transition and the thermal relaxation within the L and R subsystems, is consistent with assumption (1). 

In the absence of V the subsystems reach their own thermal equilibrium so that their density matrices are diagonal, with elements given by 

When V 0 transitions between L and R can take place, and their populations evolve in time. Defining the total L and R populations by our goal is to characterize the kinetics of the L R process. This is a reduced description because we are not interested in the dynamics of individual level /) and r), only in the overall dynamics associated with transitions between the L and R species. Note that reduction can be done on different levels, and the present focus is on Pl and Pr and the transitions between them. This reduction is not done by limiting attention to a small physical subsystem, but by focusing on a subset of density-matrix elements or, rather, their combinations. 

We assume that V, the operator that couples systems L and R to each other, mixes only / and r states, that is, F/// = = 0. We are interested in the transition 

---