Cat state

Time evolution of the probability distribution with quantum phase (color) of a cat state with α=3. The two coherent portions interfere in the center.

Wigner function of a Schrödinger cat state

In quantum optics, a cat state is defined as the coherent superposition of two coherent states with opposite phase:

${\displaystyle |\mathrm {cat} _{e}\rangle \propto |\alpha \rangle +|{-}\alpha \rangle }$,

where

${\displaystyle |\alpha \rangle =e^{-{|\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha ^{n} \over {\sqrt {n!}}}|n\rangle }$,

and

${\displaystyle |{-}\alpha \rangle =e^{-{|{-}\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{({-}\alpha )^{n} \over {\sqrt {n!}}}|n\rangle }$,

are coherent states defined in the number (Fock) basis. Notice that if we add the two states together, the resulting cat state only contains even Fock state terms:

${\displaystyle |\mathrm {cat} _{e}\rangle \propto 2e^{-{|\alpha |^{2} \over 2}}\left({\alpha ^{0} \over {\sqrt {0!}}}|0\rangle +{\alpha ^{2} \over {\sqrt {2!}}}|2\rangle +{\alpha ^{4} \over {\sqrt {4!}}}|4\rangle +\dots \right)}$.

As a result of this property, the above cat state is often referred to as an even cat state. Alternatively, we can define an odd cat state as

${\displaystyle |\mathrm {cat} _{o}\rangle \propto |\alpha \rangle -|{-}\alpha \rangle }$,

which only contains odd Fock states

${\displaystyle |\mathrm {cat} _{o}\rangle \propto 2e^{-{|\alpha |^{2} \over 2}}\left({\alpha ^{1} \over {\sqrt {1!}}}|1\rangle +{\alpha ^{3} \over {\sqrt {3!}}}|3\rangle +{\alpha ^{5} \over {\sqrt {5!}}}|5\rangle +\dots \right)}$.

Even and odd coherent states were first introduced by Dodonov, Malkin, and Man'ko in 1974.[10]