Dynamics of a Single Two-Level System

This section considers a single asymmetric double-well potential. At low temperatures a quantum mechanical description is necessary, and only the lowest energy eigenstates will be relevant. If the energy asymmetry of the wells is not too great then it will be sufficient to describe the problem in terms of a two-state basis, where the two states are localized in each of the two wells of the potential. These two states compose the TLS. 

Whatever the mechanism, the simple first-order rate laws for the populations of the ground and excited TLS states, Pq i) and P t) respectively, are 

In order that the solutions to the above rate equations reach thermal equilibrium in the limit of long times the principle of detailed balance must be satisfied, which implies that 

Let me define Pa t) to be the conditional probability that the TLS will be in state P at time t given that it was in state a at time 0 (a, fS = 0, 1). These conditional probabilities are simply solutions to the rate equations subject to the appropriate initial conditions. They are given by 

For what follows it will be convenient to define a stochastic occupation variable for the TLS, such that when the system is in its ground state = 0 and when it is in its excited state = 1. The average value of any function of is given by 

---