Weak measurement on a quantum system in contact with a thermal reservoir: projection operator method

Weak measurement is studied for a quantum system which is in contact with a thermal reservoir, where pre- and postselection on the quantum system are also performed under the influence of the thermal reservoir. A single parameter which completely characterizes time dependence of statistical properties of weak measurement is derived by means of the projection operator method. The parameter consists of three parts. One is a part that determines the statistical properties when correlation between reservoir’s states before and after the weak measurement and initial correlation between the quantum system and the thermal reservoir are ignored. Another is a correction due to the correlation between the reservoir’s states. The other represents the effect of the initial correlation on the weak measurement. When the reduced time evolution of the quantum system is Markovian, the latter two contributions become negligible. The result shows how to calculate the parameter once a model of a quantum system and a thermal reservoir is given. Furthermore an explicit expression is given for the weak value of an observable influenced by a thermal reservoir.