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.