Non-classicality dynamics of Schrödinger cat states in a non-Markovian environment

Let us consider a quantum harmonic oscillator initially in Schrödinger cat state, which is represented by a coherent superposition of two coherent states $| \alpha \rangle$ and $| -\alpha \rangle$, as follows [67]

$$\begin{eqnarray}&&| {\rm{\Psi }}{\rangle }_{{scs}}=\displaystyle \frac{1}{N}\left[\cos (\phi )| \alpha \rangle +\sin (\phi )| -\alpha \rangle \right],\end{eqnarray} \tag{ 13 }$$

where $| \alpha \rangle$ denotes a coherent state with amplitude α, ϕ is the relative phase, and $N=\sqrt{1+{e}^{-2| \alpha {| }^{2}}\sin (2\phi )}$ is a normalization factor. In this paper, we take $\alpha \in {\mathbb{R}}$ for simplicity. Obviously, if ϕ = π/4, equation (13) is equal to the even coherent state; when ϕ = − π/4, then equation (13) describes the odd coherent state; and for $\phi =\tfrac{n}{2}\pi ,n\in {\mathbb{Z}}$, the state (13) is equal to the Yurke-Stoler coherent state. In the Fock-state representation, the even and odd coherent states can be written as [18, 19]

$$\begin{eqnarray}&&| {\rm{\Psi }}{\rangle }_{e}=\displaystyle \frac{1}{{\cosh }^{\tfrac{1}{2}}{\alpha }^{2}}\sum _{n=0}^{\infty }\displaystyle \frac{{\alpha }^{2n}}{\sqrt{(2n)!}}| 2n\rangle ,\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&| {\rm{\Psi }}{\rangle }_{o}=\displaystyle \frac{1}{{\sinh }^{\tfrac{1}{2}}{\alpha }^{2}}\sum _{n=0}^{\infty }\displaystyle \frac{{\alpha }^{2n+1}}{\sqrt{(2n+1)!}}| 2n+1\rangle ,\end{eqnarray} \tag{ 14b }$$

where the subscripts e and o denote the even and odd Schrödinger cat states, respectively.

Under the influence of non-Markovian environment (2), Schrödinger cat state evolves as

$$\begin{eqnarray}&&\begin{array}{l}{\rho }_{{scs}}(t)=\displaystyle \frac{1}{{N}^{2}}\left\{{\cos }^{2}(\phi )| \alpha (t)\rangle \langle \alpha (t)| \right.\\ \qquad +\,{\sin }^{2}(\phi )| -\alpha (t)\rangle \langle -\alpha (t)| \\ \qquad +\left.\,\displaystyle \frac{1}{2}\sin (2\phi )f({\boldsymbol{\alpha }},t)\left[| -\alpha (t)\rangle \langle \alpha (t)| +| \alpha (t)\rangle \langle -\alpha (t)| \right]\right\},\end{array}\end{eqnarray} \tag{ 15 }$$

where we have assumed $\alpha (t)=\alpha {e}^{-\tfrac{1}{2}{\rm{\Gamma }}(t)}$ and the decay factor $f({\boldsymbol{\alpha }},t)=\exp \left\{-2[1-{e}^{-{\rm{\Gamma }}(t)}]{\alpha }^{2}\right\}$.

It is well-known that the fidelity is meant to measure the degree of similarity between two quantum states. As such, it is a natural tool of quantifying the loss of coherence of a quantum state under a noisy quantum channel. The fidelity can be defined as

$$\begin{eqnarray}&&F(t)=\mathrm{Tr}[\rho (0)\rho (t)],\end{eqnarray} \tag{ 16 }$$

where ρ(0) is the density matrix at initial time and ρ(t) is the density matrix at later time. In ideal case, one would like to have F(t) = 1, corresponding to perfect, noiseless memory.

On substituting equations (13) and (15) into equation (16), we obtain for the fidelity of even and odd coherent states the following expression

$$\begin{eqnarray}\begin{array}{rcl}F(t) & = & \displaystyle \frac{2}{{N}_{\pm }^{4}}{e}^{-2{\alpha }^{2}[1+{e}^{-{\rm{\Gamma }}(t)}]}\cosh \left\{{\alpha }^{2}(1+{e}^{-{\rm{\Gamma }}(t)})\right\}\\ & & \times \left\{\cosh \left[2{\alpha }^{2}{e}^{-\tfrac{1}{2}{\rm{\Gamma }}(t)}\right]\pm 1\right\},\end{array}\end{eqnarray} \tag{ 17 }$$

where the plus (minus) sign denotes the even (odd) coherent state and ${N}_{\pm }=\sqrt{1\pm {e}^{-2| \alpha {| }^{2}}}$. Obviously, one sees from equation (17) that $F({\boldsymbol{\alpha }},t)\to 1$ at initial time t = 0. One also sees that the dynamical behavior of decoherence given by (17) for Schrödinger cat states will be determined by the fastest time scale of the environment ${t}_{c}\sim {\omega }_{c}^{-1}$ and the spectral function J(ω). Next, we will analyze this behavior in the sub-Ohmic, Ohmic, and super-Ohmic environments by means of the numerical simulation, respectively.

In figure 1, the dynamics of the fidelity F(α, t) of even and odd Schrödinger cat states in a zero-temperature non-Markovian environment is shown as a function of scaled time ω_ct for different values of α and s with γ₀ = 0.5. It was observed that the fidelity has two different dynamical regimes. For times t < ω⁻¹, it quickly damps with increasing of scaled time, irrespective of s, but it reaches asymptotically a steady-state value at times t > ω⁻¹. This steady-state value is determined by both α and s. It is interesting to see that if the amplitude α of the field mode is fixed, the larger the Ohmicity parameter s, the higher the steady-state value of fidelity. It shows that the super-Ohmic environment can help to protect quantum coherence of Schrödinger cat states compared with another two types of environment. On the other hand, the decoherence dynamics of Schrödinger cat states is significantly influenced by its amplitude. It is found that if the Ohmicity parameter s is fixed, the Schrödinger cat states with smaller amplitudes α can obtain a very high fidelity even in the long-time limit. For example, for a case of even coherent state with α = 0.2, we can obtain the steady-state fidelities as 0.9603 for s = 0.5, 0.9706 for s = 1, and 0.9888 for s = 3, respectively. This is completely different from the Markovian decoherence model. We can explain as follows. In the Markovian environment, we have $\alpha {e}^{-\gamma t}\to 0$ in the long-time limit [49], so the Schrödinger cat state is completely degraded, while for non-Markovian case, we have the decoherence function ${\rm{\Gamma }}(t){| }_{t\to \infty }\to \tfrac{2\eta \tilde{{\rm{\Gamma }}}(s)}{s-1}$ from equation (12) and then $\alpha {e}^{-{\rm{\Gamma }}(\infty )}\ne 0$, implying that the Schrödinger cat state is partly degraded. For Schrödinger cat state with larger amplitude, the fidelity quickly reduces and tends to a relatively small value(even almost to zero) in the long-time limit. Therefore. in the following cases, we will restrict ourselves to the case of the reservoirs with sub- and super-Ohmic spectral functions and mainly choose Schrödinger cat states with larger amplitude (i.e., α = 2) as initial state. That is because for some realistic quantum information processing, the fidelity with cat states at least as large as α > 1.22 needs to be remained at near-unit fidelity [68, 69].

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The Wigner function is one of the quasi-probability functions, which contains full information about the quantum system and can reconstruct via optical homodyne tomography [70]. The Wigner function of a quantum state ρ can be defined as

$$\begin{eqnarray}&&W({\boldsymbol{\eta }})=\displaystyle \frac{1}{{\pi }^{2}}\displaystyle \int {d}^{2}{\boldsymbol{\xi }}\exp \left(\eta {\xi }^{* }-{\eta }^{* }\xi \right)\chi ({\boldsymbol{\xi }}),\end{eqnarray} \tag{ 18 }$$

where $\chi ({\boldsymbol{\xi }})=\mathrm{Tr}[\rho \exp ({\hat{a}}^{+}\xi -\hat{a}{\xi }^{* })]$ is the symmetrically ordered characteristic function.

The Wigner function associated with such an noisy state (15) is

$$\begin{eqnarray}&&{W}_{{scs}}({\boldsymbol{\eta }};t)={W}^{(+\alpha )}({\boldsymbol{\eta }};t)+{W}^{(-\alpha )}({\boldsymbol{\eta }};t)+{W}_{I}({\boldsymbol{\eta }};t),\end{eqnarray} \tag{ 19 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{W}^{(+\alpha )}({\boldsymbol{\eta }};t) & = & \displaystyle \frac{2{\cos }^{2}(\phi )}{{N}^{2}\pi }\\ & & \times \exp \left[-2{\eta }_{i}^{2}\right]\exp \left\{-2{\left[{\eta }_{r}-\alpha {e}^{-\tfrac{{\rm{\Gamma }}(t)}{2}}\right]}^{2}\right\},\end{array}\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}\begin{array}{rcl}{W}^{(-\alpha )}({\boldsymbol{\eta }};t) & = & \displaystyle \frac{2{\sin }^{2}(\phi )}{{N}^{2}\pi }\\ & & \times \exp \left[-2{\eta }_{i}^{2}\right]\exp \left\{-2{\left[{\eta }_{r}+\alpha {e}^{-\tfrac{{\rm{\Gamma }}(t)}{2}}\right]}^{2}\right\},\end{array}\end{eqnarray} \tag{ 20b }$$

$$\begin{eqnarray}&&{W}_{I}({\boldsymbol{\eta }};t)=\displaystyle \frac{4\sin (2\phi )}{{N}^{2}\pi }\exp \left[-2| \eta {| }^{2}\right]\cos \left[4\alpha {\eta }_{i}{e}^{-\tfrac{{\rm{\Gamma }}(t)}{2}}\right],\end{eqnarray} \tag{ 20c }$$

where ${\eta }_{r}=\mathrm{Re}({\boldsymbol{\eta }})$ and ${\eta }_{i}=\mathrm{Im}({\boldsymbol{\eta }})$. As is known to all, the Wigner function for single-mode Schrödinger cat state consists of two Gaussian Bells corresponding to statistical mixture of individual composite states and interference fringes in between originating from the superposition between different components of the states, which vanishes for classical statistical mixture. So it can be viewed as a useful mark of the quantum to classical transition, which can be commonly quantified by the fringe visibility function [71]

$$\begin{eqnarray}&&V(t)=\displaystyle \frac{1}{2}\displaystyle \frac{{W}_{I}({\boldsymbol{\eta }};t){| }_{\mathrm{peak}}}{{[{W}^{(+\alpha )}({\boldsymbol{\eta }};t){| }_{\mathrm{peak}}{W}^{(-\alpha )}({\boldsymbol{\eta }};t){| }_{\mathrm{peak}}]}^{1/2}},\end{eqnarray} \tag{ 21 }$$

where ${W}_{I}({\boldsymbol{\eta }};t){| }_{\mathrm{peak}}$ is the value of the Wigner function at η = (0,0) and ${W}^{(\pm \alpha )}({\boldsymbol{\eta }};t){| }_{{peak}}$ are the values of the Wigner function at η = ( ± α,0), respectively. This quantity is responsible for the effect of environment-induced decoherence of quantum state.

Considering equation (19), we can obtain the fringe visibility of ESCS and OSCS as

$$\begin{eqnarray}&&V(t)=f({\boldsymbol{\alpha }},t).\end{eqnarray} \tag{ 22 }$$

One sees from equation (22) that these interference terms are decaying by the factor $f({\boldsymbol{\alpha }},t)=\exp \left\{-2[1-{e}^{-{\rm{\Gamma }}(t)}]{\alpha }^{2}\right\}$, which is closely related to the spectral structure of the environment. According to equation (12), one can see that for very short times $t\lt {\omega }_{c}^{-1}$, the super-Ohmic environment induces the fastest decoherence, but in the long-time limit, we can obtain the asymptotic expression of the fringe visibility as $V(t)=\exp \left\{-2[1-{e}^{-{\rm{\Gamma }}(\infty )}]{\alpha }^{2}\right\}$, It is easy to see that the asymptotic value has a very close connection with the type of the environment and is significantly larger for the super-Ohmic type than for other two types. As is well known, under the influence of Markovian environment, the two Gaussian peaks of the statistical mixture part of Schrödinger cat states move towards the origin and eventually merge into each other. A natural question arises: how does the wake-packet evolution of Wigner function for Schrödinger cat states in a non-Markovian environment?

In figures 2 and 3, we plot the time evolution of Wigner function for even and odd Schrödinger cat states in a non-Markovian environment at zero temperature, respectively. One sees that when Schrödinger cat states are expected in presence of non-Markovian environment, the two Gaussian peaks go to the origin and the oscillation behavior becomes weaker and weaker. It is interesting to see that in super-Ohmic environment, this oscillation doesn't disappear completely in the long-time limit, irrespective of even or odd Schrödinger cat state. This is completely different from the result obtained in [49]. In addition, within the effects of the environmental noise, the negativity of the Wigner functions, a signature of nonclassical effect of quantum state [72], is progressively attenuated, even up to a complete deletion at certain conditions. For example, sub-Ohmic environmental noise causes the disappearance of the negativity of Wigner function for even Schrödinger cat state, whereas the Wigner function of odd Schrödinger cat state can always take the negative values during the evolution, regardless of s. Compared figure 2(f) with figure 3(f), we see that Wigner functions of Schrödinger cat states have eventually different shapes in the sub-Ohmic reservoirs. For an even Schrödinger cat state, it becomes more and more similar to Gaussian form of vacuum, while odd Schrödinger cat state is close to the Wigner function of single-photon state. These can be clearly checked from equations (14b) and (15) with a special case of $\alpha \to 0$.

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On the other hand, we further investigate the non-classicality dynamics for single-mode Schrödinger cat states in non-Markovian environment by means of Wootters' concurrence [73]. When the evolving quantum state (15) is mixed with vacuum state at a 50:50 beam splitter, the resulting two-mode state becomes

$$\begin{eqnarray}\begin{array}{rcl}R(\rho ) & = & {\hat{U}}_{{BS}}[{\rho }_{{scs}}(t)\otimes | 0\rangle \langle 0| ]{\hat{U}}_{{BS}}^{+}\\ & = & \displaystyle \frac{1}{{N}^{2}}\left\{{\cos }^{2}\phi {\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{{scs}}\right.\\ & & \times \left\langle \displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\bigotimes {\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{00}\left\langle \displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\\ & & +{\sin }^{2}\phi {\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{{scs}}\left\langle -\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\bigotimes {\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{00}\left\langle -\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\\ & & +\displaystyle \frac{1}{2}\sin (2\phi )f({\boldsymbol{\alpha }},t){\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{{scs}}\\ & & \times \left\langle -\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\bigotimes {\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{00}\left\langle -\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\\ & & +\displaystyle \frac{1}{2}\sin (2\phi )f({\boldsymbol{\alpha }},t){\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{{scs}}\\ & & \left.\times \left\langle \displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\bigotimes {\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle }_{00}\left\langle \displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right|\right\},\end{array}\end{eqnarray} \tag{ 23 }$$

where we have used the following relation

$$\begin{eqnarray}&&\begin{array}{l}{\hat{U}}_{{BS}}[| \alpha {\rangle }_{a}\langle \beta | \otimes | 0{\rangle }_{b}\langle 0| ]{\hat{U}}_{{BS}}^{+}\\ \quad =\,{\left|\displaystyle \frac{1}{\sqrt{2}}\alpha \right\rangle }_{a}\left\langle \displaystyle \frac{1}{\sqrt{2}}\beta \right|\bigotimes {\left|\displaystyle \frac{1}{\sqrt{2}}\alpha \right\rangle }_{b}\left\langle \displaystyle \frac{1}{\sqrt{2}}\beta \right|,\end{array}\end{eqnarray} \tag{ 24 }$$

where ${\hat{U}}_{{BS}}=\exp \left[\tfrac{\pi }{2}({\hat{a}}^{+}\hat{b}-\hat{a}{\hat{b}}^{+})\right]$ corresponds to a 50:50 beam splitter.

In order to calculate the degree of entanglement of quantum state (23), we will need transform a two-mode quantum state into a two-qubit state and thus define the zero and one qubit states, respectively, as

$$\begin{eqnarray}&&| 0\rangle \equiv | \psi (t){\rangle }_{e}={N}_{+}(t)\left[\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle +\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle \right],\end{eqnarray} \tag{ 25a }$$

$$\begin{eqnarray}&&| 1\rangle \equiv | \psi (t){\rangle }_{o}={N}_{-}(t)\left[\left|\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle -\left|-\displaystyle \frac{1}{\sqrt{2}}\alpha (t)\right\rangle \right],\end{eqnarray} \tag{ 25b }$$

where the normalization factor ${N}_{\pm }{(t)}^{-1}=\sqrt{1\pm \exp [-2{\alpha }^{2}{e}^{-{\rm{\Gamma }}(t)}]}$. The two logical qubit states are orthogonal to each other even in the limit of $\alpha {e}^{-{\rm{\Gamma }}(t)}\to 0$. Then the density operator equation (23) for $\phi =\pm \tfrac{\pi }{4}$ can be written, in the standard basis $\{| 00\rangle ,| 01\rangle ,| 10\rangle ,| 11\rangle \}$, as

$$\begin{eqnarray}R(\rho )=\displaystyle \frac{1}{{N}_{\pm }^{2}}\left(\begin{array}{cccc}{\rho }_{11}(t) & 0 & 0 & {\rho }_{14}(t)\\ 0 & {\rho }_{22}(t) & {\rho }_{23}(t) & 0\\ 0 & {\rho }_{32}(t) & {\rho }_{33}(t) & 0\\ {\rho }_{41}(t) & 0 & 0 & {\rho }_{44}(t)\end{array}\right),\end{eqnarray} \tag{ 26 }$$

with

$$\begin{eqnarray}&&{\rho }_{11}(t)=[1\pm f({\boldsymbol{\alpha }},t)]{a}^{4},\end{eqnarray} \tag{ 27a }$$

$$\begin{eqnarray}&&{\rho }_{14}(t)={\rho }_{41}(t)=[1\pm f({\boldsymbol{\alpha }},t)]{a}^{2}{b}^{2},\end{eqnarray} \tag{ 27b }$$

$$\begin{eqnarray}&&{\rho }_{22}(t)={\rho }_{23}(t)={\rho }_{32}(t)={\rho }_{33}(t)=[1\mp f({\boldsymbol{\alpha }},t)]{a}^{2}{b}^{2},\end{eqnarray} \tag{ 27c }$$

$$\begin{eqnarray}&&{\rho }_{44}(t)=[1\pm f({\boldsymbol{\alpha }},t)]{b}^{4},\end{eqnarray} \tag{ 27d }$$

where $a=\tfrac{1}{2{N}_{+}(t)}$ and $b=\tfrac{1}{2{N}_{-}(t)}$.

The entanglement of the two qubits could be quantified by concurrence, which is defined as [73]

$$\begin{eqnarray}&&C({\rho }_{12})=\max \left\{0,{\lambda }_{1}-{\lambda }_{2}-{\lambda }_{3}-{\lambda }_{4}\right\},\end{eqnarray} \tag{ 28 }$$

where the λ_j are the square roots of the eigenvalues of the matrix ${M}_{{ab}}(\rho )=\rho ({\sigma }_{y}\otimes {\sigma }_{y}){\rho }^{* }({\sigma }_{y}\otimes {\sigma }_{y})$, in decreasing order, and σ_y the Pauli matrix. As usual for entanglement measures, concurrence ranges from zero (no entanglement) to one (maximally entanglement).

For a X-shape state (26), its concurrence can be simplified to

$$\begin{eqnarray}\begin{array}{rcl}C({\rho }_{12};t) & = & 2\max \left\{0,| {\rho }_{14}(t)| -\sqrt{{\rho }_{22}(t){\rho }_{33}(t)},\,| {\rho }_{23}(t)| \right.\\ & & \left.-\sqrt{{\rho }_{11}(t){\rho }_{44}(t)}\right\}.\end{array}\end{eqnarray} \tag{ 29 }$$

Figure 4 shows the entanglement potential dynamics for single-mode even and odd Schrödinger cat states in a non-Markovian environment. One sees that for these states, their entanglement shows the same behaviors as that of their decoherence during the evolution. That is, at shorter times $t\lt {\omega }_{c}^{-1}$, the entanglement deteriorates rapidly and ultimately tends to zero or does stay a steady-state value conditioned on the Ohmicity parameter s. In particular, it is shown that the entanglement is hardly decayed in super-Ohmic environment at longer times, clearly showing that there is some residual entanglement left in the steady state of light field modes, revealing that the non-classicality of these input Schrödinger cat states is very robust against super-Ohmic environmental noise. We want to point out that the phenomena of sudden death, revivals and oscillations of two-mode entanglement aren't observed at all in our non-Markovian noisy model, which can be explained with the fact that at the zero-temperature non-Markovian environment, the decay rate Γ(t) is always positive and the monotonic increasing functions of ω_ct. So this positivity ensures the monotonic decrease of C(t), but not oscillations.

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Finally, we study the dynamic of another two nonclassical properties, higher order antibunching (HOA) and sub-Poissonian photon statistics, of odd Schrödinger cat state in non-Markovian environment since no antibunching and sub-Poissonian photon statistics can exist for even Schrödinger cat state.

The phenomena of antibunching is an important aspect of quantum mechanical states, closely related to the photon statistics. Its concept was introduced by Lee in 1990 using the theory of majorization [74, 75]. Subsequently, Ba An [76] and Pathak and Garcia [77] modified Lee's criterion for HOA and presented a simplified counterpart for HOA. According to Pathak-Garcia criterion [77], a quantum state is said to be the lth order antibunching if it obeys the following inequality

$$\begin{eqnarray}&&D(l)=\langle {\left({\hat{a}}^{+}\right)}^{l+1}{\hat{a}}^{l+1}\rangle -\langle {\hat{a}}^{+}\hat{a}{\rangle }^{l+1}\lt 0;\end{eqnarray} \tag{ 30 }$$

otherwise, it does not exist higher-order antibunching. For l = 1, it recovers a usual antibunching criterion and for l ≥ 2, it corresponds to higher-order antibunching criterion.

Using the following identity

$$\begin{eqnarray}&&\langle {\hat{a}}^{+l}{\hat{a}}^{l}{\rangle }_{{\rho }_{{scs}}(t)}=\displaystyle \frac{1}{{N}_{-}^{2}}\left\{1+{\left(-1\right)}^{l+1}{e}^{-2{\alpha }^{2}}\right\}{\alpha }^{2l}{e}^{-l{\rm{\Gamma }}(t)},\end{eqnarray} \tag{ 31 }$$

we can obtain the higher-order antibunching of odd Schrödinger cat state in a non-Markovian environment as

$$\begin{eqnarray}&&D(l;t)=D(l;0){e}^{-(l+1){\rm{\Gamma }}(t)},\end{eqnarray} \tag{ 32 }$$

where $D(l;0)={\alpha }^{2(l+1)}$ $\left\{\tfrac{1}{{N}^{2}}{\left[1-{\left(-1\right)}^{l}{e}^{-2{\alpha }^{2}}]-\tfrac{1}{{N}^{2(l+1)}}[1+{e}^{-2{\alpha }^{2}}\right]}^{l+1}\right\}$ is the higher-order antibunching of odd coherent state at the initial time. It shows clearly that the odd coherent state satisfies always the inequality equation (29) since ${e}^{-(l+1){\rm{\Gamma }}(t)}\,\gt 0$ and $D(l;0)\lt 0$, and hence odd coherent state may exhibit the higher-order antibunching in non-Markovian environment. But the depth of higher-order antibunching has the same dynamical behavior as the decoherence obtained in section 3.1.

The indication of the non-classicality and statistics properties of the radiation field can also be revealed by the photon sub-Poissonian statistics, which does not have any classical counterparts. In quantum optics, a quantity that characterizes the departure from Poissonian photon statistics is given by the well-known Mandel's Q parameter defined as [78]

$$\begin{eqnarray}&&Q=\displaystyle \frac{\langle {\left({\rm{\Delta }}\hat{n}\right)}^{2}\rangle -\langle \hat{n}\rangle }{\langle \hat{n}\rangle }=\displaystyle \frac{\langle {\left({\hat{a}}^{+}\hat{a}\right)}^{2}\rangle -\langle {\hat{a}}^{+}\hat{a}{\rangle }^{2}}{\langle {\hat{a}}^{+}\hat{a}\rangle }-1,\end{eqnarray} \tag{ 33 }$$

where $\langle {\left({\rm{\Delta }}\hat{n}\right)}^{2}\rangle =\langle {\hat{n}}^{2}\rangle -\langle \hat{n}{\rangle }^{2}$. A state is defined to be sub-Poissonian when its photon number variance $\langle ({\rm{\Delta }}{\hat{n}}^{2})\rangle$ is smaller than its average photon number $\langle \hat{n}\rangle$. For Poissonian statistics, Q = 0. If Q < 0, the field mode is said to be sub-Poissonian; otherwise, it is super-Poissonian. However, we do not discuss it separately, as it can be obtained as a special case of higher order nonclassical criteria. In 2002, Kim et al has generalized the notion of sub-Poissonian statistics to higher-order sub-Poissonian photon one in terms of factorial moment [79]. They defined the higher-order Q^m-parameter as

$$\begin{eqnarray}&&{Q}^{(m)}=\displaystyle \frac{\langle {\left({\rm{\Delta }}\hat{n}\right)}^{2m}\rangle -{{\ell }}^{(2m)}\langle \hat{n}\rangle }{{{\ell }}^{(2m)}\langle \hat{n}\rangle },\end{eqnarray} \tag{ 34 }$$

where ${{\ell }}^{(2m)}\langle \hat{n}\rangle$ denotes the higher-order moment. Thus, the photon statistics is called mth-order sub-Poissonian if $-1\leqslant {Q}^{(m)}\lt 0$. For later calculations, we write the first four expressions for ${{\ell }}^{(2m)}\langle \hat{n}\rangle$ as

$$\begin{eqnarray}&&{{\ell }}^{(2)}\langle \hat{n}\rangle =\langle \hat{n}\rangle ,\end{eqnarray} \tag{ 35a }$$

$$\begin{eqnarray}&&{{\ell }}^{(4)}\langle \hat{n}\rangle =3\langle \hat{n}{\rangle }^{2}+\langle \hat{n}\rangle ,\end{eqnarray} \tag{ 35b }$$

$$\begin{eqnarray}&&{{\ell }}^{(6)}\langle \hat{n}\rangle =15\langle \hat{n}{\rangle }^{3}+25\langle \hat{n}{\rangle }^{2}+\langle \hat{n}\rangle ,\end{eqnarray} \tag{ 35c }$$

$$\begin{eqnarray}&&{{\ell }}^{(8)}\langle \hat{n}\rangle =105\langle \hat{n}{\rangle }^{4}+490\langle \hat{n}{\rangle }^{3}+119\langle \hat{n}{\rangle }^{2}+\langle \hat{n}\rangle .\end{eqnarray} \tag{ 35d }$$

Using the following identity

$$\begin{eqnarray}&&\left\langle {\left({\rm{\Delta }}\hat{n}\right)}^{N}\right\rangle =\sum _{k=0}^{N}{\left(-1\right)}^{k}\left(\displaystyle \genfrac{}{}{0em}{}{N}{k}\right)\left\langle {\hat{n}}^{N-k}\right\rangle {\left\langle \hat{n}\right\rangle }^{k},\end{eqnarray} \tag{ 36 }$$

and the relationship between $\langle {\hat{n}}^{l}\rangle$ and $\langle {\hat{a}}^{+m}{\hat{a}}^{m}\rangle$

$$\begin{eqnarray}&&\begin{array}{l}\left(\begin{array}{c}\langle \hat{n}\rangle \\ \langle {\hat{n}}^{2}\rangle \\ \langle {\hat{n}}^{3}\rangle \\ \langle {\hat{n}}^{4}\rangle \\ \langle {\hat{n}}^{5}\rangle \\ \langle {\hat{n}}^{6}\rangle \\ \langle {\hat{n}}^{7}\rangle \\ \langle {\hat{n}}^{8}\rangle \end{array}\right)=\,\left(\begin{array}{cccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 6 & 7 & 1 & 0 & 0 & 0 & 0\\ 1 & 10 & 25 & 15 & 1 & 0 & 0 & 0\\ 1 & 15 & 65 & 90 & 31 & 1 & 0 & 0\\ 1 & 21 & 140 & 350 & 301 & 63 & 1 & 0\\ 1 & 28 & 266 & 1050 & 1701 & 966 & 127 & 1\end{array}\right)\\ \left(\begin{array}{c}\langle {\hat{a}}^{+}\hat{a}\rangle \\ \langle {\hat{a}}^{+2}{\hat{a}}^{2}\rangle \\ \langle {\hat{a}}^{+3}{\hat{a}}^{3}\rangle \\ \langle {\hat{a}}^{+4}{\hat{a}}^{4}\rangle \\ \langle {\hat{a}}^{+5}{\hat{a}}^{5}\rangle \\ \langle {\hat{a}}^{+6}{\hat{a}}^{6}\rangle \\ \langle {\hat{a}}^{+7}{\hat{a}}^{7}\rangle \\ \langle {\hat{a}}^{+8}{\hat{a}}^{8}\rangle \end{array}\right),\end{array}\end{eqnarray} \tag{ 37 }$$

we can easily investigate the dynamics of higher-order sub-Poissonian photon statistics of odd Schrödinger cat state in a non-Markovian environment using equations (33)–(37).

Figure 5 shows the time evolution of higher-order Q^m-parameter for odd coherent state in a non-Markovian environment. It is clear that at the initial time, the value of ${Q}^{m}{\rm{s}}$ increases with the increase of the order m, indicating that the higher-order sub-Poissonian character is more pronounced than the usual sub-Poissonian behavior. However, one sees that for given amplitude $\alpha ,{Q}^{m}{\rm{s}}$ tend rapidly to zero under the influence of sub-Ohmic non-Markovian environment. That is, the higher-order sub-Poissonian of the initial Schrödinger cat state is very fragile against sub-Ohmic non-Markovian environmental noise. For super-Ohmic case, it is more favorable for non-classicality Q^m preservation. In particular, we can see from figure 5(b) that the higher-order sub-Poissonian photon statistics shows more depth of non-classicality than corresponding lower-order counterparts. Therefore, we conclude that the higher order sub-Poissonian photon statistics may be useful in detecting weak non-classicality.

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