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]

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. León-Montiel, R. de J.; Moya-Cessa, H. (2011). "Modeling non-linear coherent states in fiber arrays". International Journal of Quantum Information. 9 (S1): 349–355. doi: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
4. Man'ko, V I; Marmo, G; Sudarshan, E C G; Zaccaria, F (1997). "f-oscillators and nonlinear coherent states". Physica Scripta. 55 (5): 528–541. arXiv:quant-ph/9612006. doi:10.1088/0031-8949/55/5/004. ISSN 0031-8949.