Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 60 / 5 / c / Solution

Put $K=\sqrt{\gamma }(a\otimes d{B}^{†}-a^{†}\otimes dB)$, with the time argument suppressed. Since the incoming field commutes with the adapted cavity operators,Applying the Gaussian quantum noise Ito table givesThe power series expansion $e^K=I+K+K^2/2+\cdots$ consequently becomes, through Ito order $dt$,This has precisely the required quadratic drift correction. The short-time ordering is essential: the linear noise term is order $\sqrt{\gamma dt}$ and the drift is order $\gamma dt$; higher stochastic orders are discarded in the differential limit. A remainder written merely as a power of $\gamma$ is a formal coupling expansion, not a dimensionally complete statement of the short-time error.

As a sign check, $K^{†}=-K$ and $Q^{†}=Q$. Thus $U_I^{†}{U}_I=I-\gamma Qdt-K^2=I$ through this order. The cross term is needed for stochastic unitarity.