Nonclassicality and Decoherence of Photon-Subtraction Squeezing-Enhanced Thermal State

Abstract

We introduce a kind of non-Gaussian state—photon-subtracted squeezing-enhanced thermal state (PSSETS) characteristics by two-squeezing parameters (λ,r). Its normalization factor is a Legendre polynomial of two-squeezing parameters and average photon number $\bar{n}$ of the thermal state. The nonclassicality is investigated by using the negativity of Wigner function (WF). It is shown that the single PSSETS always has negative values when $\bar{n}<\cosh^{2}r\sinh^{2}\lambda$. The decoherence effect on PSSETS is then included by analytically deriving the time evolution of WF. For the single PSSETS, the characteristic time is longer than that of photon-subtracted squeezing thermal state.

Appendix: Derivation of Eq. (21)

Using the completeness relation and $\rho_{th}^{S}$’s normal ordering form in Eq. (14), as well as the overlap of coherent state,

$$ \langle \alpha |\beta \rangle =\exp \biggl[ -\frac{1}{2}\vert \alpha \vert ^{2}-\frac{1}{2}\vert \beta \vert ^{2}+ \alpha^{\ast}\beta \biggr] $$

(51)

we have

(52)

where, we used the integration formula of Eq. (16).

Noting that

$$ \frac{\partial^{2m}}{\partial t^{m}\partial \tau^{m}}\exp \bigl( -t^{2}-\tau^{2}+2xt\tau \bigr) |_{t,\tau =0}=2^{m}m!\sum_{n=0}^{ [ m/2] } \frac{m!}{2^{2n} ( n! )^{2} ( m-2n ) !}x^{m-2n} $$

(53)

one may rewrite Eq. (52) as

(54)

Recalling the newly found expression of the Legendre polynomial [its equivalence to the well-known Legendre polynomial’s (P _m (x))] expression is [20]

$$ P_{m} ( x ) =x^{m}\sum_{l=0}^{ [ m/2 ] } \frac{m!}{2^{2l} ( l! )^{2} ( m-2l ) !} \biggl( 1-\frac{1}{x^{2}}\biggr)^{l}, $$

(55)

we derive the compact form for C _m ,

(56)

where

$$ A_{1}=\mu -2\mu \omega +1,A_{2}=\mu \omega -1. $$

(57)

Equation (56) indicates that the normalization factor C _m is just related to Legendre polynomial.