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

$a|\alpha \rangle =\alpha |\alpha \rangle$,

or as a displacement from the vacuum

$|\alpha \rangle =D(\alpha )|0\rangle$,

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

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

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

and then using any of the above definitions by exchanging $a$ by $A$ . The above definition is also known as an $f$-deformed annihilation operator.