Nonclassical Properties via Subtracting Photons from Displaced Thermal States and Decoherence Evolution for Amplitude Decay

Abstract

Using the integration method within an ordered product of operators, we present (anti-) normal products of the density operators of photon-subtracted displaced thermal states (PSDTSs) and their normalization factors, and investigate their nonclassical features via the photocount distribution and the negativity of Wigner distribution function. Also, we analytically and numerically investigate the time-evolution law and decoherence behaviors for amplitude decay. The results show that, for the PSDTSs, the smaller photon subtraction numbers m and thermal variance V and the appropriate displacement d can lead to the stronger nonclassicality, however the PSDTSs with large m show more robustness against decoherence because their initial negativity decreases more slowly.

Appendices

Appendix A: Derivation of Normalization Factor $\mathcal {N}_{m}$

In order to obtain the factor $\mathcal {N}_{m}$, we first use the formula

$$ \rho=\int \frac{d^{2}\alpha}{\pi}\left \langle -\alpha \right \vert \rho \left \vert \alpha \right \rangle {\vdots} \text{e}^{\left \vert \alpha \right \vert^{2} +\alpha^{\ast}a-\alpha a^{\dagger}+a^{\dagger}a}\vdots $$

(A1)

and the normal ordering product

$$ \rho=\frac{2\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V+1}}}{V+1} \colon \exp \left[ \frac{2}{V+1}\left( da^{\dag}+d^{\ast}a-a^{\dag}a\right) \right] \colon $$

(A2)

to calculate the antinormal ordering product of ρ as

$$ \rho=\frac{2\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V-1}}}{V-1} {\vdots} \exp \left[ \frac{2}{V-1}\left( da^{\dag}+d^{\ast}a-a^{\dag}a\right) \right] \vdots, $$

(A3)

thus the normalization factor $\mathcal {N}_{m}$ can be expressed as

$$ \mathcal{N}_{m}=\frac{2\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V-1}} }{V-1}\text{tr}\left( {\vdots} a^{m}\exp \left[ \frac{2}{V-1}\left( da^{\dag }+d^{\ast}a-a^{\dag}a\right) \right] a^{\dagger m}{\vdots} \right) . $$

(A4)

Inserting the completeness relation of the coherent states $\left \vert \alpha \right \rangle$ into Eq. A4 leads to

$$ \mathcal{N}_{m}=\frac{2\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V-1}} }{V-1}\int \frac{d^{2}\alpha}{\pi}\left \vert \alpha \right \vert^{2m} \text{e}^{\frac{2}{V-1}\left( d\alpha^{\ast}+d^{\ast}\alpha-\left \vert \alpha \right \vert^{2}\right) }, $$

(A5)

thus using the the mathematical integral formula

$$ \begin{array}{@{}rcl@{}} &&\int \frac{d^{2}\alpha}{\pi}\alpha^{n}\alpha^{\ast m}\exp \left( f\left \vert \alpha \right \vert^{2}+s\alpha+\eta \alpha^{\ast}\right) \\ &=&\left( -1\right)^{n+m+1}\sqrt{f^{-(n+m+2)}}\text{e}^{-s\eta/f} \text{H}_{m,n}(s/\sqrt{f},\eta/\sqrt{f}), \end{array} $$

(A6)

we can finally obtain the normalization factor$\mathcal {N}_{m}$ in Eq. 4

Appendix B: Derivation of Eq. 16

Inserting the coherent representation of single-mode Wigner operator, i.e.,

$$ {\Delta} \left( \alpha \right) =\int \frac{d^{2}z}{\pi}\left \vert \alpha +z\right \rangle \left \langle \alpha-z\right \vert \text{e}^{\alpha z^{\ast }-\alpha^{\ast}z} $$

(B1)

and the normal ordering product of ρ_m in Eq. 7 into Eq. 15, and using the generating functions for Hermite polynomials in Eq. 10, we obtain

$$ \begin{array}{@{}rcl@{}} w\left( \alpha \right) & =&\frac{\mathcal{D}}{\pi}\text{e}^{-\frac {2\left \vert d-\alpha \right \vert^{2}}{1+V}}\frac{\partial^{2m}}{\partial s^{m}\partial t^{m}}\text{e}^{-st+\left( gd^{\ast}+h\alpha^{\ast}\right) s+\left( gd+h\alpha \right) t}\\ && \times \int \frac{d^{2}z}{\pi}\text{e}^{-\frac{2V}{1+V}\left \vert z\right \vert^{2}}\text{e}^{\left[ \frac{2\left( d^{\ast}-\alpha^{\ast }\right) }{1+V}+ht\right] z}\text{e}^{\left[-\frac{2\left( d-\alpha \right) }{1+V}-hs\right] z}. \end{array} $$

(B2)

Further, using the integration formula in Eq. 11 leads to

$$ w\left( \alpha \right) =\frac{\mathcal{D}\left( 1+V\right) }{\pi2V} \text{e}^{-\frac{2\left \vert d-\alpha \right \vert^{2}}{V}}\frac{\partial^{2m} }{\partial s^{m}\partial t^{m}}\text{e}^{-\frac{1+V}{2V}st+\left( gd^{\ast }+h\alpha^{\ast}-\frac{h\left( d^{\ast}-\alpha^{\ast}\right) }{V}\right) s+\left( gd+h\alpha-\frac{h\left( d-\alpha \right) }{V}\right) t}. $$

(B3)

Finally, we use the generating functions in Eq. 10 to derive Eq. 16.

Appendix C: Derivation of Eq. 25

In order to obtain Eq. 25, substituting Eq. 16 into the formula Eq. 24 and using the generating functions for Hermite polynomials in Eq. 10, we have

$$ \begin{array}{@{}rcl@{}} \mathcal{W}(\alpha,t) &=&\frac{2\mathcal{D}\left( 1+V\right)^{m+1}} {\pi \mathcal{T}\left( 2V\right)^{m+1}}\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V}-\frac{2}{\mathcal{T}}\left \vert \alpha \right \vert^{2} }\frac{\partial^{2m}}{\partial s^{m}\partial t^{m}}\text{e}^{-st+\frac {2\left( Vg-h\right) d^{\ast}}{1+V}s+\frac{2\left( Vg-h\right) d}{1+V}t} \\ && \times \int \frac{d^{2}\alpha^{\prime}}{\pi}\text{e}^{-{\varkappa} \left \vert \alpha^{\prime}\right \vert^{2}+\left( 2ht+\frac{2\text{e}^{-\kappa t}}{\mathcal{T}}\alpha^{\ast}-\frac{2d^{\ast}}{V}\right) \alpha^{\prime }+\left( 2hs+\frac{2\text{e}^{-\kappa t}}{\mathcal{T}}\alpha-\frac{2d} {V}\right) \alpha^{\prime \ast}}. \end{array} $$

(C1)

Using the integration formula in Eq. 11, thus Eq. C1 becomes

$$ \begin{array}{@{}rcl@{}} \mathcal{W}(\alpha,t) & =&\frac{2\mathcal{D}\left( 1+V\right)^{m+1}} {\pi \mathcal{T}\left( 2V\right)^{m+1}\left( \frac{2\text{e}^{-2\kappa t}}{\mathcal{T}}+\frac{2}{V}\right) }\text{e}^{-\frac{2\left \vert d\right \vert^{2}}{V}-\frac{2}{\mathcal{T}}\left \vert \alpha \right \vert^{2}}\\ && \times \frac{\partial^{2m}}{\partial s^{m}\partial t^{m}}\text{e}^{-st+\frac{2\left( Vg-h\right) d^{\ast}}{1+V}s+\frac{2\left( Vg-h\right) d}{1+V}t}\\ && \times \exp \left[ \frac{\left( 2ht+\frac{2\text{e}^{-\kappa t} }{\mathcal{T}}\alpha^{\ast}-\frac{2d^{\ast}}{V}\right) \left( 2hs+\frac {2\text{e}^{-\kappa t}}{\mathcal{T}}\alpha-\frac{2d}{V}\right) } {\frac{2\text{e}^{-2\kappa t}}{\mathcal{T}}+\frac{2}{V}}\right] , \end{array} $$

(C2)

thus using the generating functions in Eq. 10, we finally obtain the result in Eq. 25.