M Times Photon Subtraction-Addition Coherent Superposition Operated Odd-SchrH{o}dinger-cat State: Nonclassicality and Decoherence

We introduce a new non-Gaussian state, generated by m times coherent superposition operation $a\cos \theta +a^{\dagger }e^{i\varphi }\sin \theta $ (MCSO) on odd-Schrodinger-cat state (OSCS). Its normalized constant is turned out to be related with the Hermite polynomial. We further investigate the nonclassical properties of the MCSO-OSCS through Mandel's Q-parameter, quadrature squeezing, the photocount distribution and Wigner function (WF). It is shown that the nonclassicality of the MCSO-OSCS is influenced by the number of times (m) of coherent superpositon operation, the angle $\theta $ and the amplitude of the coherent state (|$\alpha _{0}$|). Especially the volume of negative region of WF increases with the increment of parameters m, $\theta $ and $\alpha _{0}$. We also investigate the decoherence of the MCSO-OSCS in terms of the fadeaway of the negativity of WF in a thermal environment.