Schrödinger cat state

A coherent superposition of distinct optical coherent amplitudes, for example $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. It differs from the corresponding incoherent mixture through cross terms in observables and its phase-space interference. For large $|\alpha |$, branch distinguishability is large while ordinary quadrature squeezing can become exponentially small.

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.

For real $\alpha$, put $s=e^{-2{\alpha }^2}$ in $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. Its mean second quadrature is $-\alpha s\sin \varphi /(1+s\cos \varphi )$, which is generally nonzero. Subtracting its square from the second moment gives the displayed variance and squeezing precisely when $\cos \varphi >-s$, for nonzero finite amplitude. A useful counterexample to confusing moment and variance is $\varphi =\pi /2$.

The difference $N_{-}(|\alpha ⟩-|-\alpha ⟩)$ contains only odd photon numbers and tends to a one-photon state as $\alpha \to 0$. Exactly at zero amplitude the unnormalized difference vanishes; the normalized limiting state is obtained only by taking a limit.

The normalized sum $N_{+}(|\alpha ⟩+|-\alpha ⟩)$ contains only even photon numbers, since the two coherent-state expansions cancel every odd term. It tends to vacuum at small amplitude. With real $\alpha$, its second quadrature can be squeezed below the vacuum variance.