Cumulant Expansion for a System with Pre- and Post-Selection and a Weak Value of a Gaussian System

The generalized cumulant expansion is a suitable method for investigating a quantum system with pre- and post-selection. When pre- and post-selected states of a system to be measured and an initial state of a measurement device are Gaussian with respect to relevant observables, weak values can be obtained from measurement outcomes without any other restrictions such as weakness of a system-device coupling and large uncertainty of a pointer observable in the initial state since all the cumulants higher than the second order vanish. A Gaussian weak value is investigated for a single mode optical system prepared in a coherent-squeezed state, where the post-selection is done by a homodyne detection.