Full-counting statistics and quantum information of dispersive readout with a squeezed environment

Motivated by the importance of dispersive readout in quantum technology, we study a prototypical dispersive readout setup that is probed by a squeezed vacuum in a time-reversal-symmetric fashion. To this end, we develop a full-counting-statistics framework for dispersive readout and analyze its measurement information, accompanied by a generalized mean-field approach suitable to deal with non-unitary dynamics. Distinct from conventional input-output theory, our full-counting-statistics approach enables the direct calculation of arbitrary-order cumulants for the measured cumulative (i.e., time-integrated) photonic distribution while maintaining applicability to nonlinear systems. The corresponding Fisher information exhibits an exponential dependence on the squeezing parameter and a robustness against residual nonlinearity, which can even approach the quantum Fisher information, setting an upper limit. This work introduces a conceptually streamlined and computationally efficient framework for continuous quantum measurements, making it well suited for widespread adoption in quantum technologies.