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

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

or as a displacement from the vacuum

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

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

One may think of a non-linear coherent state {{cite journal | last1 = León-Montiel | first1 = R. de J. | authorlink2 = Héctor Manuel Moya Cessa | last2 = Moya-Cessa | first2 = H. | year = 2011 | title = Modeling non-linear coherent states in fiber arrays | doi = 10.1142/S0219749911007319 | journal = International Journal of Quantum Information | volume = 9 | issue = S1| pages = 349–355 }} by generalizing the

annihilation operator:

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

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.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{{cite journal|last1=Man'ko|first1=V I|last2=Marmo|first2=G|last3=Sudarshan|first3=E C G|last4=Zaccaria|first4=F|title=f-oscillators and nonlinear coherent states|journal=Physica Scripta|volume=55|issue=5|year=1997|pages=528–541|issn=0031-8949|doi=10.1088/0031-8949/55/5/004|arxiv=quant-ph/9612006|s2cid=250836172 }}