Loss of coherence in an optical cat state ($s_{\mathrm{env}}=e^{-2(1-\eta )|\alpha {|}^2}$)

A pure-loss channel of transmissivity $\eta$ maps the two branches to system amplitudes $\pm \sqrt{\eta }\alpha$ and environment amplitudes $\pm \sqrt{1-\eta }\alpha$. Tracing over the environment multiplies their cross terms by its branch overlap, the factor displayed. Large cat states can therefore lose interference rapidly even while their mean intensities change only moderately.