Noiselessly amplified thermal states and after multi-photon addition or subtraction

In this paper, we introduce a noiselessly amplified thermal state (ATS), by operating the noiseless amplification operator ($g^{\widehat{n}}$) on the thermal state (TS) with corresponding mean photon number (MPN) $\bar{n}$. Actually, the ATS is an new TS with MPN $\bar{N}=g^2\bar{n}\mathrm{/}[1-\bar{n}(g^2-1)]$. Furthermore, we introduce photon-added-ATS (PAATS) and photon-subtracted-ATS (PSATS) by operating $m$-photon addition (${\widehat{a}}^{\text{†}m}$) and $m$-photon subtraction (${\widehat{a}}^m$) on the ATS, respectively. We study photon number distributions (PNDs), purities, and Wigner functions (WFs) for all these states.