Non-linear coherent states

Coherent states are quasi-classical states that may be defined in different ways, for instance as eigenstates of the annihilation operator

${\displaystyle a|\alpha \rangle =\alpha |\alpha \rangle }$,

or as a displacement from the vacuum

${\displaystyle |\alpha \rangle =D(\alpha )|0\rangle }$,

where ${\displaystyle D(\alpha )=\exp(\alpha a^{\dagger }-\alpha ^{*}a)}$ is the Sudarshan-Glauber displacement operator.^[1]

One may think of a non-linear coherent state ^[2] by generalizing the annihilation operator:

${\displaystyle A=af(a^{\dagger }a)}$,

and then using any of the above definitions by exchanging ${\displaystyle a}$ by ${\displaystyle A}$ . The above definition is also known as an ${\displaystyle f}$-deformed annihilation operator.^[3][4]