1. Introduction
Over the past few decades, nonclassical light fields have attracted a great deal of attention and have been widely investigated. This interest stems from the useful nonclassical properties of this type of quantum state, including the photon antibunching effect, sub-Poissonian distribution and the squeezing effect, which could help to significantly advance quantum information research.
To prepare more nonclassical states with practical value in quantum physics, physicists have proposed various quantum operations, such as photon addition and subtraction and their coherent superpositions, as well as quantum catalysis and truncation [1–7]. In particular, photon addition and subtraction, which are respectively represented by Bose creation and annihilation operators (a†, a), are effective tools for generating nonclassical states. Agarwal and Tara first introduced the concept photon-added coherent states by applying photon addition for arbitrary times to a coherent state [1]. Since then, the use of photon addition and subtraction to prepare nonclassical states has become a research highlight and a large class of nonclassical states, such as photon-added/subtracted (squeezed) vacuum/thermal/coherent states, has successfully been introduced. Their properties and applications are discussed in detail in [8–11]. As a simple generalization, the elementary photon-modulated operation μa + $\upsilon$a† is applied to classical coherent and thermal states and the nonclassicality of their resultant states is studied, where the parameters μ, $\upsilon$ represent, respectively, the ratios of photon subtraction and addition, and ${\left|\mu \right|}^{2}\,+{\left|\upsilon \right|}^{2}=1$ [12]. In short, photon addition and subtraction operations have made significant contributions to the engineering of nonclassical and non-Gaussian quantum states.
As a continuation of the important work in [12], in this paper, we apply the general photon-modulated operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ with non-negative integer m to a displaced thermal state (DTS), and investigate the nonclassicality and non-Gaussianity of the output photon-modulated DTS, which can show the improvement of the operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ in causing nonclassical effects. It is worth pointing out that the original DTS can be seen as an intermediate quantum state between the coherent state and the thermal state, and the operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ can be a generalized case because some existing operations can be viewed as the operations ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ for some special values of the parameters μ, $\upsilon$ and m. In addition, the final states coming from the operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ can show some nonclassical properties that are completely different from the original states, making theoretical analysis particularly challenging.
The remainder of this paper is as follows. In section 2 , using the normal ordering form of ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ , we obtain the normal ordering product of the photon-modulated DTS that is related to the new special polynomials ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ . In section 3 , we obtain the compact normalization factor of the photon-modulated DTS, which is of particular interest for studying their nonclassicality. In sections 4 and 5 we examine the nonclassicality of the states in detail via the negativity of their Wigner functions and quantitatively measure the non-Gaussianity caused by the photon-modulated operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ . Finally, several main results of this paper are summarized in section 6 .
2. Preparation of the photon-modulated DTS
Theoretically, a displaced thermal state is defined as a pure coherent state $\left|\alpha \right\rangle $ immersed in a Gaussian noise
$\begin{eqnarray}{\mathfrak{P}}\left(V,d;\alpha \right)=\displaystyle \frac{2}{\pi \left(V-1\right)}{{\rm{e}}}^{-\tfrac{2{\left|\alpha -d\right|}^{2}}{V-1}},\end{eqnarray}$
where V is mixedness and yields the relation ${{\rm{e}}}^{{\hslash }\omega /{kT}}=\left(V+1\right)/\left(V-1\right)$ , ℏ, ω are, respectively, Planck’s constant and the frequency of the thermal field; that is, mixedness V increases with the the thermal field temperature T and d is the displacement in the phase space. In terms of [13–15], the integral form of the DTS reads $\begin{eqnarray}{\rho }_{{}_{V,d}}=\int {{\rm{d}}}^{2}\alpha {\mathfrak{P}}\left(V,d;\alpha \right)\left|\alpha \right\rangle \left\langle \alpha \right|.\end{eqnarray}$
For the case of mixedness V = 1, ${\rho }_{{}_{1,d}}\propto \left|d\right\rangle \left\langle d\right|$ (a pure coherent state) and ${\rho }_{{}_{V,0}}\propto $ ${{\rm{e}}}^{{a}^{\dagger }a\mathrm{ln}\tfrac{V-1}{V+1}}$ (a mixed thermal state) for d = 0, so ${\rho }_{{}_{V,d}}$ is an intermediate state between states ${\rho }_{{}_{1,d}}$ and ${\rho }_{{}_{V,0}}$ . What is even more special is that ${\rho }_{{}_{\mathrm{1,0}}}$ is a vacuum.Repeatedly applying the generalized operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ to the input state ${\rho }_{{}_{V,d}}$ , leads to the photon-modulated DTS, i.e.,appendix for details)
$\begin{eqnarray}{\rho }_{{}_{V,d}}^{m}={{\mathfrak{D}}}^{-1}{\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}{\rho }_{{}_{V,d}}{\left(\mu {a}^{\dagger }+\upsilon a\right)}^{m},\end{eqnarray}$
where ${{\mathfrak{D}}}^{-1}$ refers to the normalization factor of the state ${\rho }_{{}_{V,d}}^{m}$ . Specifically, for μ = 0, ${\rho }_{{}_{V,d}}^{m}\propto {a}^{\dagger m}{\rho }_{{}_{V,d}}{a}^{m}$ , is a multi-photon-added DTS, while for $\upsilon$ = 0, ${\rho }_{{}_{V,d}}^{m}\propto {a}^{m}{\rho }_{{}_{V,d}}{a}^{\dagger m}$ , is a multi-photon-subtracted DTS. Using the normal ordering product of ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ , which is of the form [5, 16] $\begin{eqnarray}{\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}={\left(-\mu \varrho \right)}^{m}:{{\rm{H}}}_{m}\left(\varkappa a+\varrho {a}^{\dagger }\right):,\end{eqnarray}$
with ϰ = ${\rm{i}}\sqrt{\mu /(2\upsilon )}$ and ϱ = ${\rm{i}}\sqrt{\upsilon /(2\mu )}$ , where :: denotes normal ordering [17–20], and the single-variable Hermite polynomials ${{\rm{H}}}_{m}\left(x\right)$ , we can obtain the specific form of the photon-modulated DTS (see the $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{V+1}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V+1}}\\ & & \times \,:{{ \mathcal F }}_{m,m}\left(\varsigma ,{\varsigma }^{\dagger };\displaystyle \frac{2\left(V-1\right)u}{\left(V+1\right)\upsilon }\right)\\ & & \times \,{{\rm{e}}}^{\tfrac{2d}{V+1}{a}^{\dagger }}{{\rm{e}}}^{-\tfrac{2}{V+1}{a}^{\dagger }a}{{\rm{e}}}^{\tfrac{2{d}^{* }}{V+1}a}:,\end{array}\end{eqnarray}$
which is indeed the normal ordering product of the state ${\rho }_{{}_{V,d}}^{m}$ , and where $\begin{eqnarray}\varsigma =\displaystyle \frac{4d\varkappa }{V+1}+\displaystyle \frac{2\left(V-1\right)\varkappa }{V+1}a+2\varrho {a}^{\dagger },\end{eqnarray}$
and we have used a new two-index, three-variable special polynomial [21] $\begin{eqnarray}\begin{array}{rcl} & & {{ \mathcal F }}_{n,m}\left(x,y;\theta \right)\\ & = & {\partial }_{s}^{n}{\partial }_{t}^{m}{\left.\exp \left(-{s}^{2}-{t}^{2}+\theta {xy}+{sx}+{ty}\right)\right|}_{s=t=0}\\ & = & \displaystyle \sum _{l=0}^{\min (n,m)}\left(\begin{array}{c}n\\ l\end{array}\right)\left(\begin{array}{c}m\\ l\end{array}\right)l!{\theta }^{l}{{\rm{H}}}_{n-l}\left(\displaystyle \frac{x}{2}\right){{\rm{H}}}_{m-l}\left(\displaystyle \frac{y}{2}\right),\end{array}\end{eqnarray}$
with the generating function ${{\rm{e}}}^{-{s}^{2}-{t}^{2}+\theta {xy}+{sx}+{ty}}$ . As a special case, ${{ \mathcal F }}_{\mathrm{0,0}}\left(\varsigma ,{\varsigma }^{\dagger };\tfrac{2\left(V-1\right)u}{\left(V+1\right)\upsilon }\right)=1$ for m = 0. Thus ${\rho }_{{}_{V,d}}^{m}$ becomes the normal ordering product of ${\rho }_{{}_{V,d}}$ . For μ = 0 and $\upsilon$ = 0, the state ${\rho }_{{}_{V,d}}^{m}$ respectively becomes the multi-photon-added and -subtracted DTS.3. Normalization
The normalization factor is important for describing the success probability in quantum state preparation and for examining the nonclassicality and applications of the state. For the photon-modulated DTS ${\rho }_{{}_{V,d}}^{m}$ , its normalization factor can be given by tr${\rho }_{{}_{V,d}}^{m}=1$ . Thus, substituting the completeness relation of coherent states $\left|\beta \right\rangle $ into equation (A3 ), we obtain
$\begin{eqnarray}\begin{array}{rcl}\mathrm{tr}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{V-1}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V-1}}{\partial }_{s}^{m}{\partial }_{t}^{m}{{\rm{e}}}^{-{s}^{2}-{t}^{2}}\\ & & \times \,\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{2}\alpha }{\pi }\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{2}\beta }{\pi }\left\langle \beta \right|:{{\rm{e}}}^{-\tfrac{V+1}{V-1}{\left|\alpha \right|}^{2}}{{\rm{e}}}^{\left(\tfrac{2{d}^{* }}{V-1}+2s\varkappa +{a}^{\dagger }\right)\alpha }\\ & & \times \,{{\rm{e}}}^{\left(\tfrac{2d}{V-1}+2t{\varkappa }^{* }+a\right){\alpha }^{* }}{{\rm{e}}}^{2s\varrho {a}^{\dagger }}{{\rm{e}}}^{-{a}^{\dagger }a}{{\rm{e}}}^{2t{\varrho }^{* }a}:\left|\beta \right\rangle .\end{array}\end{eqnarray}$
Further, using the integral formula (A4) and the generating function for ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ , we finally obtain $\begin{eqnarray}\mathrm{tr}{\rho }_{{}_{V,d}}^{m}={{\mathfrak{D}}}^{-1}{\left(\displaystyle \frac{\mu \upsilon V}{2}\right)}^{m}{{ \mathcal F }}_{m,m}\left(\chi ,{\chi }^{\dagger };\sigma \right),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\chi & = & \displaystyle \frac{{\rm{i}}\sqrt{2}\left(d\mu +{d}^{* }\upsilon \right)}{\sqrt{\mu \upsilon V}},\\ \sigma & = & \displaystyle \frac{\left(V-1\right){\mu }^{2}+\left(V+1\right){\upsilon }^{2}}{\mu \upsilon V}.\end{array}\end{eqnarray}$
Therefore, we have $\begin{eqnarray}{\mathfrak{D}}={\left(\displaystyle \frac{\mu \upsilon V}{2}\right)}^{m}{{ \mathcal F }}_{m,m}\left(\chi ,{\chi }^{\dagger };\sigma \right),\end{eqnarray}$
which is just related to the new special polynomials ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ . In particular, for the case of μ = 0 (or $\upsilon$ = 0), the normalization factor ${\mathfrak{D}}$ of ${\rho }_{{}_{V,d}}^{m}$ reduces to that of the multi-photon-added (or -subtracted) DTS.4. Negativity of the Wigner function
Negativity of the Wigner function can be seen as a signal of whether quantum states exhibit nonclassicality from the perspective of phase space. As a general rule, for any quantum state ρ, its Wigner function is given by w(β) = tr $[\rho {\rm{\Delta }}\left(\beta \right)]$ , where ${\rm{\Delta }}\left(\beta \right)$ is the coherent state form of a single-mode Wigner operator.
Substituting equation (A3 ) and the Wigner operator
$\begin{eqnarray}{\rm{\Delta }}\left(\beta \right)={{\rm{e}}}^{2{\left|\beta \right|}^{2}}\int \displaystyle \frac{{{\rm{d}}}^{2}z}{{\pi }^{2}}\left|z\right\rangle \left\langle -z\right|{{\rm{e}}}^{-2\left(z{\beta }^{* }-{z}^{* }\beta \right)},\end{eqnarray}$
into the original definition of w(β), and using the inner product of two different coherent states, i.e., $\left\langle z\right|\left.{z}^{{\prime} }\right\rangle \,=$ ${{\rm{e}}}^{-{\left|z\right|}^{2}/2-{\left|{z}^{{\prime} }\right|}^{2}/2+{z}^{* }{z}^{{\prime} }}$ [22, 23] and the integral formula (A4), we obtain the Wigner function for the output state ${\rho }_{{}_{V,d}}^{m}$ as $\begin{eqnarray}w(\beta )={w}_{0}(\beta ){w}_{m}(\beta ),\end{eqnarray}$
where $\begin{eqnarray}{w}_{0}(\beta )=\displaystyle \frac{1}{\pi V}{{\rm{e}}}^{-\tfrac{2{\left|d-\beta \right|}^{2}}{V}}\end{eqnarray}$
is just the Wigner function for the DTS, and the term $\begin{eqnarray}{w}_{m}(\beta )={{\mathfrak{D}}}^{-1}{\left(\displaystyle \frac{\mu \upsilon }{2V}\right)}^{m}{{ \mathcal F }}_{m,m}\left(\varsigma ,{\varsigma }^{\dagger };\rho \right)\end{eqnarray}$
comes from the photon-modulated operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ , and $\begin{eqnarray}\begin{array}{rcl}\varsigma & = & \displaystyle \frac{{\rm{i}}\sqrt{2}\left\{\left[\left(V-1\right)\beta +d\right]\mu +\left[\left(V-1\right){\beta }^{* }-{d}^{* }\right]\upsilon \right\}}{\sqrt{\mu \upsilon V}},\\ \rho & = & \displaystyle \frac{\left(V-1\right){\mu }^{2}-\left(V+1\right){\upsilon }^{2}}{\mu \upsilon }.\end{array}\end{eqnarray}$
Specifically, for the case of μ = 0 (or $\upsilon$ = 0), the Wigner function w(β) becomes that of the multi-photon-added (or -subtracted) DTS.In figure 1, we plot the Wigner functions for the states ${\rho }_{{}_{V,d}}^{m}$ for different parameters m, V, d and μ. Obviously, for V = 1 or d = 0, the functions w(β) are respectively the Wigner functions for the photon-modulated coherent states or thermal states. As the parameters m, V, d and μ increase, the multipeak structure of Wigner function w(β) becomes a single peak and its main peak value slowly decreases. In addition, the upward or downward direction of the single peak depends on the sufficiently large even or odd m.
Figure 1. Wigner function w(β) for the state ${\rho }_{{}_{V,d}}^{m}$ for different values of the parameters m, V, d and μ, where the values of (m, V, d, μ) are, respectively, (a) (1, 2, 0.5, 0.5), (b) (2, 2, 0.5, 0.5), (c) (4, 2, 0.5, 0.5), (d) (5, 2, 0.5, 0.5), (e) (1, 3, 0.5, 0.5), (f) (1, 2, 1, 0.5), (g) (1, 2, 2, 0.5), (h) (1, 2, 0.5, 0.9). |
In order to more clearly observe the influences of the parameters m, V, d and μ on the nonclassicality of the states ${\rho }_{{}_{V,d}}^{m}$ , we use the formula [24]
$\begin{eqnarray}{\rm{\Delta }}=\displaystyle \frac{1}{2}\int {\rm{d}}q{\rm{d}}p\left[\left|w(p,q)\right|-w(p,q)\right]\end{eqnarray}$
to calculate the the negative volumes of the functions w(β) that can quantify the nonclassiclity of ${\rho }_{{}_{V,d}}^{m}$ , where w(p, q) is the Wigner function for a single-mode quantum state. In figure 2, we plot the changes of the negative volume of w(β) as parameters m, V, d and μ increase. Without loss of generality, we have taken some special parameter values here. Obviously, for m = 1, the function w(β) has the best negativity. With the increase of even or odd m, the negativity of w(β) always decreases without the other parameters. In addition, with the parameters d or μ, the negativity always decreases gradually, but first increases and then decreases with the parameter V when m is odd. More meaningfully, the negativity of w(β) for odd m has stronger negativity than that for even m+k (k being odd). In summary, states ${\rho }_{{}_{V,d}}^{m}$ with smaller m, V, d and μ have stronger nonclassicality, and the states ${\rho }_{{}_{V,d}}^{m}$ for odd m have stronger nonclassicality than those for even m+k.
Figure 2. The negative volume of the Wigner function w(β) for the state ${\rho }_{{}_{V,d}}^{m}$ with the parameter d for V = 2 and μ = 0.5 in (a) and (a0), the parameter μ for V = 2 and d = 0.5 in (b) and (b0), the parameter V for μ = 2 and d = 0.5 in (c) and (c0), where m = 1 (thick, black line), 2 (dashed, thick, black line), 3 (thick, red line), 4 (dashed, thick, red line), 5 (thick, green line), 6 (dashed, thick, green line). It should be noted that, in order to show the variations of negative volumes with parameters d, μ, V more clearly, we took a part of figures (a), (b) and (c), which correspond to figures (a0), (b0) and (c0), respectively. |
5. Non-Gaussianity
In order to quantify the non-Gaussianity of quantum states, we introduce the concept of the input-output fidelity that is generally defined as ${ \mathcal F }\,=$ tr (ρoρi)/ tr${\rho }_{i}^{2}$ , where ρi is the input state and ρo is its corresponding output state [3]. Considering that the photon-modulated operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ fully contributes to the appearance of the non-Gaussianity of the output state ${\rho }_{{}_{V,d}}^{m}$ for taking the Gaussian DTS as the input state, by using the definition of the fidelity ${ \mathcal F }$ , we can quantitatively measure the non-Gaussianity of the photon-modulated DTS.
Using the normal ordering product of ${\rho }_{{}_{V,d}}$ , i.e.,
$\begin{eqnarray}{\rho }_{{}_{V,d}}=\displaystyle \frac{2}{V+1}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V+1}}:{{\rm{e}}}^{\tfrac{2d}{V+1}{a}^{\dagger }}{{\rm{e}}}^{-\tfrac{2}{V+1}{a}^{\dagger }a}{{\rm{e}}}^{\tfrac{2{d}^{* }}{V+1}a}:,\end{eqnarray}$
and the completeness relation of coherent states, as well as the integral formula (A4), we find $\begin{eqnarray}\mathrm{tr}{\rho }_{{}_{V,d}}^{2}=\displaystyle \frac{1}{V}.\end{eqnarray}$
On the other hand, using the normal ordering product of ${\rho }_{{}_{V,d}}^{m}$ , that is $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{V+1}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V+1}}{\partial }_{s}^{m}{\partial }_{t}^{m}{{\rm{e}}}^{-{s}^{2}-{t}^{2}}\\ & & \times \,{{\rm{e}}}^{\tfrac{4\left(V-1\right){\left|\varkappa \right|}^{2}}{V+1}{st}}:{{\rm{e}}}^{\left(\tfrac{4d\varkappa }{V+1}+\tfrac{2\left(V-1\right)\varkappa }{V+1}a+2\varrho {a}^{\dagger }\right)s}\\ & & \times \,{{\rm{e}}}^{\left(\tfrac{4{d}^{* }{\varkappa }^{* }}{V+1}+\displaystyle \frac{2\left(V-1\right){\varkappa }^{* }}{V+1}{a}^{\dagger }+2{\varrho }^{* }a\right)t}\\ & & \times \,{{\rm{e}}}^{\tfrac{2d}{V+1}{a}^{\dagger }}{{\rm{e}}}^{-\tfrac{2}{V+1}{a}^{\dagger }a}{{\rm{e}}}^{\tfrac{2{d}^{* }}{V+1}a}:,\end{array}\end{eqnarray}$
and the completeness relationship of coherent states twice, as well as the generating function for ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ , we finally obtain $\begin{eqnarray}\mathrm{tr}({\rho }_{{}_{V,d}}^{m}{\rho }_{{}_{V,d}})=\displaystyle \frac{{{\mathfrak{D}}}^{-1}}{V}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{ \mathcal F }}_{m,m}\left(\lambda ,{\lambda }^{\dagger };\tau \right),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\lambda & = & \displaystyle \frac{{\rm{i}}2\sqrt{V}\left(d\mu +{d}^{* }\upsilon \right)}{\sqrt{\left({V}^{2}+1\right)\mu \upsilon }},\\ \tau & = & \displaystyle \frac{{V}^{2}-1}{\left({V}^{2}+1\right)\mu \upsilon }.\end{array}\end{eqnarray}$
In figure 3, we plot the changes of the input-output fidelity ${ \mathcal F }$ with the ratio μ for different parameters m, V and d. Figure 2 shows that, for all m, V and d, the fidelity ${ \mathcal F }$ slowly increases with increasing ratio μ. For any ratio μ, the fidelity ${ \mathcal F }$ gradually decreases with the increase of m, but increases as d increases. As the mixedness V is increased, the fidelity F increases first and then decreases for a smaller ratio μ, but always decreases for a larger ratio μ. In summary, for the states ${\rho }_{{}_{V,d}}^{m}$ , a smaller d and a larger m, V can lead to stronger non-Gaussianity.
Figure 3. The input-output fidelity ${ \mathcal F }$ as a function of the ratio μ for different values of parameters m, V and d, where the values of (m, V, d) are, respectively, (1, 1.1, 0.5) (black solid line), (2, 1.1, 0.5) (red solid line), (4, 1.1, 0.5) (green solid line), (1, 2, 0.5) (blue solid line), (1, 3, 0.5) (black dashed line), (1, 2, 1) (red dashed line), (1, 2, 2) (green dashed line). |
6. Conclusions
In summary, after applying the photon-modulated operation ${\left(\mu a+\upsilon {a}^{\dagger }\right)}^{m}$ to the Gaussian DTS, we have proposed a new non-Gaussian quantum state, i.e., photon-modulated DTS ${\rho }_{{}_{V,d}}^{m}$ . Making full use of the operator ordering method, we derive the normal ordering form of the density operator of the output state ${\rho }_{{}_{V,d}}^{m}$ , which is demonstrated to be related to the multi-variable polynomial ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ , which is a new and interesting result.
In terms of this normal ordering product, we obtain the analytical forms of its normalization, Wigner function and the input-output fidelity, and we find that their analytical expressions are connected to the polynomial ${{ \mathcal F }}_{n,m}\left(x,y;\theta \right)$ , which can be convenient for numerically investigating the nonclassicality and non-Gaussianity of the state ${\rho }_{{}_{V,d}}^{m}$ . The results show that the states ${\rho }_{{}_{V,d}}^{m}$ have stronger nonclassicality for smaller m, V, d and μ, the states ${\rho }_{{}_{V,d}}^{m}$ for odd m have stronger nonclassicality than those for even m+k, and the states ${\rho }_{{}_{V,d}}^{m}$ with smaller d and larger m, V have stronger non-Gaussianity.
Acknowledgments
This project is supported by the Collaborative Innovation Project of University, Anhui Province (Grant No. GXXT-2022-088).
Appendix. Derivation of the photon-modulated DTS
Substituting equations (2 ) and (3 ) and using the operator identity (4 ), we obtain7 ), the normalization factor ${\mathfrak{D}}$ can be obtained.
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{\pi \left(V-1\right)}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}\displaystyle \int {{\rm{d}}}^{2}\alpha {{\rm{e}}}^{-\tfrac{2{\left|\alpha -d\right|}^{2}}{V-1}}\\ & & \times \,{{\rm{H}}}_{m}\left(\varkappa \alpha +\varrho {a}^{\dagger }\right)\left|\alpha \right\rangle \left\langle \alpha \right|{{\rm{H}}}_{m}\left({\varkappa }^{* }{\alpha }^{* }+{\varrho }^{* }a\right).\end{array}\end{eqnarray}$
Using the single-variable Hermite polynomials $\begin{eqnarray}{{\rm{H}}}_{m}\left(x\right)={\left.{\partial }_{s}^{m}{{\rm{e}}}^{-{s}^{2}+2{sx}}\right|}_{s=0},\end{eqnarray}$
we have $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{\pi \left(V-1\right)}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{\partial }_{s}^{m}{\partial }_{t}^{m}\displaystyle \int {{\rm{d}}}^{2}\alpha \\ & & \times {{\rm{e}}}^{-{s}^{2}+2s\left(\varkappa \alpha +\varrho {a}^{\dagger }\right)}:{{\rm{e}}}^{-{\left|\alpha \right|}^{2}+\alpha {a}^{\dagger }+{\alpha }^{* }a-{a}^{\dagger }a}:\\ & & \times {{\rm{e}}}^{-\tfrac{2{\left|\alpha -d\right|}^{2}}{V-1}}{{\rm{e}}}^{-{t}^{2}+2t\left({\varkappa }^{* }{\alpha }^{* }+{\varrho }^{* }a\right)}\\ & & =\displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{V-1}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V-1}}{\partial }_{s}^{m}{\partial }_{t}^{m}{{\rm{e}}}^{-{s}^{2}-{t}^{2}}\\ & & \times \displaystyle \int \displaystyle \frac{{{\rm{d}}}^{2}\alpha }{\pi }{{\rm{e}}}^{-\tfrac{V+1}{V-1}{\left|\alpha \right|}^{2}}:{{\rm{e}}}^{\left(\tfrac{2{d}^{* }}{V-1}+2s\varkappa +{a}^{\dagger }\right)\alpha }\\ & & \times {{\rm{e}}}^{\left(\tfrac{2d}{V-1}+2t{\varkappa }^{* }+a\right){\alpha }^{* }}{{\rm{e}}}^{2s\varrho {a}^{\dagger }}{{\rm{e}}}^{-{a}^{\dagger }a}{{\rm{e}}}^{2t{\varrho }^{* }a}:.\end{array}\end{eqnarray}$
Further, using the following integral formula $\begin{eqnarray}\int \displaystyle \frac{{{\rm{d}}}^{2}z}{\pi }{{\rm{e}}}^{\varrho {\left|z\right|}^{2}+\varkappa z+\omega {z}^{* }}=-\displaystyle \frac{1}{\varrho }{{\rm{e}}}^{-\tfrac{\varkappa \omega }{\varrho }},\mathrm{Re}\varrho \lt 0,\end{eqnarray}$
we obtain $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{}_{V,d}}^{m} & = & \displaystyle \frac{2{{\mathfrak{D}}}^{-1}}{V+1}{\left(\displaystyle \frac{\mu \upsilon }{2}\right)}^{m}{{\rm{e}}}^{-\tfrac{2{\left|d\right|}^{2}}{V+1}}{\partial }_{s}^{m}{\partial }_{t}^{m}{{\rm{e}}}^{-{s}^{2}-{t}^{2}}\\ & & \times {{\rm{e}}}^{\tfrac{4\left(V-1\right){\left|\varkappa \right|}^{2}}{V+1}{st}}:{{\rm{e}}}^{\tfrac{2d}{V+1}{a}^{\dagger }}{{\rm{e}}}^{\left(\tfrac{4d\varkappa }{V+1}+\tfrac{2\left(V-1\right)\varkappa }{V+1}a+2\varrho {a}^{\dagger }\right)s}\\ & & \times {{\rm{e}}}^{\left(\tfrac{4{d}^{* }{\varkappa }^{* }}{V+1}+\displaystyle \frac{2\left(V-1\right){\varkappa }^{* }}{V+1}{a}^{\dagger }+2{\varrho }^{* }a\right)t}{{\rm{e}}}^{-\tfrac{2}{V+1}{a}^{\dagger }a}{{\rm{e}}}^{\tfrac{2{d}^{* }}{V+1}a}:.\end{array}\end{eqnarray}$
Finally, using the new multi-variable special polynomials in equation (