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.^[1]

One may think of a non-linear coherent state ^[2] 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.^[3]

References

1. R. J. Glauber "Coherent and Incoherent States of the Radiation Field", Physical Review 131, 2766 (1963). Coherent and Incoherent States of the Radiation Field. http://link.aps.org/doi/10.1103/PhysRev.131.2766
2. R. de J. León-Montiel and H. Moya-Cessa, International Journal of Quantum Information 9, (S1) 349 (2011). Modeling non-linear coherent states in fiber arrays. http://dx.doi.org/10.1142/S0219749911007319
3. V. I. Man'ko, G. Marmo, F. Zaccaria and E. C. G. Sudarshan, Proceedings of the IV Wigner Symposium, eds. N. Atakishiyev, T. Seligman and K. B. Wolf (World Scientific, Singapore, 1996), p. 421; Physica Scripta 55 (1997) 528.