Exactly solvable quantum state reduction models with time-dependent coupling

Abstract

A closed-form solution to the energy-based stochastic Schrodinger equation with a time-dependent coupling is obtained. The solution is algebraic in character, and is expressed directly in terms of independent random data. The data consist of (i) a random variable H which has the distribution , where πi is the transition probability |ψ0|i|2 from the initial state |ψ0 to the Luders state |i with energy Ei, and (ii) an independent -Brownian motion, where is the physical probability measure associated with the dynamics of the reduction process. When the coupling is time independent, it is known that state reduction occurs asymptotically—that is to say, over an infinite time horizon. In the case of a time-dependent coupling, we show that if the magnitude of the coupling decreases sufficiently rapidly, then the energy variance will be reduced under the dynamics, but the state need not reach an energy eigenstate. This situation corresponds to the case of a 'partial' or 'incomplete' measurement of the energy. We also construct an example of a model where the opposite situation prevails, in which complete state reduction is achieved after the passage of a finite period of time.