User:AB1927/sandbox

In quantum optics, a cat state is defined as the coherent superposition of two coherent states with opposite phase.
 * $$|\mathrm{cat}\rangle \propto|\alpha\rangle+|{-}\alpha\rangle

$$, where
 * $$|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle

$$, and
 * $$|{-}\alpha\rangle =e^{-{|{-}\alpha|^2\over2}}\sum_{n=0}^{\infty}{({-}\alpha)^n\over\sqrt{n!}}|n\rangle

$$, are coherent states defined in the photon number (Fock) basis. Notice that if we add the two states together, the resulting cat state only contains even photon-number terms
 * $$|\mathrm{cat}\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^0\over\sqrt{0!}}|0\rangle+{\alpha^2\over\sqrt{2!}}|2\rangle+{\alpha^4\over\sqrt{4!}}|4\rangle+\dots\right)

$$.

As a result of this property, the above cat state is often refereed to as an even cat state. Alternatively, we can define an odd cat state as
 * $$|\mathrm{cat}\rangle \propto|\alpha\rangle+|{-}\alpha\rangle

$$, which only contains odd photon-number states
 * $$|\mathrm{cat}\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^1\over\sqrt{1!}}|1\rangle+{\alpha^3\over\sqrt{3!}}|3\rangle+{\alpha^5\over\sqrt{5!}}|5\rangle+\dots\right)

$$.