Photon-Subtracted Two-Mode Squeezed Thermal State and Its Photon-Number Distribution

Abstract

We construct the photon-subtracted two-mode squeezed thermal state (PSTMSTS) by subtracting any number of photons from two-mode squeezed thermal state (TMSTS). It is found that the normalization factor of the density operator of PSTMSTS is a Jacobi polynomial of squeezing parameter λ and average photon number $\bar{n}$ of the thermal state. We investigate the photon-number distribution (PND) of PSTMSTS and find a remarkable result that it is a quotient of two Jacobi polynomials, as well as derive a corresponding character of Jacobi polynomial.

Appendices

Appendix A: Derivation of (4)

Using the operator identity [13–15]

$$e^{\sigma(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})} =\ :\!\exp\left[\left(e^{\sigma}-1\right)\left( a_{1}^{\dagger}a_{1} +a_{2}^{\dagger}a_{2}\right)\right]\!: $$

(29)

and the two-mode coherent state

$$\vert z_{1}z_{2}\rangle =\exp\biggl[-\frac{|z_{1}|^{2}}{2}-\frac {|z_{2}|^{2}}{2}+z_{1}a_{1}^{\dagger}+z_{2}a_{2}^{\dagger}\biggr] \vert 00\rangle, $$

(30)

where the symbol :: denotes normal ordering, we have

(31)

so ρ _2c is qualified to be a density operator.

Appendix B: Derivation of (8)

We employ the formula in which can put any operator F into its Weyl ordering

(32)

where |β〉 is the coherent state, the symbol$\genfrac{}{}{0pt}{}{:}{:} \genfrac{}{}{0pt}{}{:}{:}$denotes Weyl ordering [18, 19]. Substituting $F=e^{\sigma (a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})}$ into (32), we obtain

(33)

where

$$\everymath{\displaystyle} \begin{array}{lll}P_{1} &=& \frac{a_{1}-a_{1}^{\dagger}}{\sqrt{2}i},\qquad P_{2}=\frac{a_{2}-a_{2}^{\dagger}}{\sqrt{2}i}, \\Q_{1} &=& \frac{a_{1}+a_{1}^{\dagger}}{\sqrt{2}},\qquad Q_{2}=\frac{a_{2}+a_{2}^{\dagger}}{\sqrt{2}}.\end{array}$$

(34)

Then using the property that Weyl ordering is invariant under similar transformations and the following transform relations

$$ \begin{array}{lll}S_{2}Q_{1}S_{2}^{\dagger} &=& Q_{1}\cosh \lambda-Q_{2}\sinh \lambda,\qquad S_{2}P_{1}S_{2}^{\dagger}=P_{1}\cosh \lambda+P_{2}\sinh \lambda, \\[6pt]S_{2}Q_{2}S_{2}^{\dagger} &=& Q_{2}\cosh \lambda-Q_{1}\sinh \lambda,\qquad S_{2}P_{2}S_{2}^{\dagger}=P_{2}\cosh \lambda+P_{1}\sinh \lambda,\end{array}$$

(35)

we derive the Weyl ordering of two-mode squeezed thermal states

(36)

where we have set

$$A=\tanh \frac{\sigma}{2}\cosh2\lambda,\qquad B=\tanh \frac{\sigma}{2}\sinh2\lambda. $$

(37)

Then the classical Weyl correspondence function of ρ _2s is

$$\rho_{2s}\rightarrow4\tanh^{2}\frac{\sigma}{2}\exp \left[ A\left( p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\right) +2B\left( p_{1}p_{2}-q_{1}q_{2}\right) \right]. $$

(38)

According to the definition of Weyl correspondence rule which connects a classical function b(q ₁,p ₁;q ₂,p ₂) and its operator correspondence B(Q ₁,P ₁;Q ₂,P ₂),

$$B\left(Q_{1},P_{1};Q_{2},P_{2}\right) =\int_{-\infty}^{\infty}dp_{1}dq_{1}dp_{2}dq_{2}\Delta \left( q_{1},p_{1};q_{2},p_{2}\right)b\left(q_{1},p_{1};q_{2},p_{2}\right), $$

(39)

and the normal ordering form of two-mode Wigner operator [20],

$$\Delta \left( q_{1},p_{1};q_{2},p_{2}\right) =\frac{1}{\pi^{2}}:\exp \left[-\left( q_{1}-Q_{1}\right) ^{2}-\left( p_{1}-P_{1}\right) ^{2}-\left(q_{2}-Q_{2}\right) ^{2}-\left( p_{2}-P_{2}\right) ^{2}\right]:, $$

(40)

we can derive the normal ordering form of density operator ρ _2s ,

(41)

where

(42)