Evolution of a hybrid micro–macro entangled state of the qubit–oscillator system via the generalized rotating wave approximation

It is well known that the coherent state representation is overcomplete [26]. Employing the overcompleteness property, it has been established [35, 36] that an arbitrary oscillator density matrix ${\rho }_{{ \mathcal O }}$ may be realized via diagonal coherent state projectors:

$$\begin{eqnarray}&&{\rho }_{{ \mathcal O }}=\int P(\beta ,{\beta }^{*})| \beta \rangle \langle \beta | {{\rm{d}}}^{2}\beta .\end{eqnarray} \tag{ 3.1 }$$

The normalization of the density matrix ${\rho }_{{ \mathcal O }}$ ensures the property

$$\begin{eqnarray}&&\int P(\beta ,{\beta }^{*}){{\rm{d}}}^{2}\beta =1,\end{eqnarray} \tag{ 3.2 }$$

while its Hermiticity implies $P(\beta ,{\beta }^{*})$ to be real. The P-representation, however, is a general distribution of indeterminate sign as it possibly includes highly singular derivatives of the δ-function [35]. For the coherent state, which corresponds as closely as possible to a classical harmonic oscillator state with a complex amplitude, the P-representation is a positive definite δ-function [26], which is a legitimate measure of the classical probability density. For this reason the coherent state is considered to be classical. On the other hand, if the diagonal kernel $P(\beta ,{\beta }^{*})$ cannot be considered a valid probability measure due to its negativity, the oscillator state given by the density matrix ${\rho }_{{ \mathcal O }}$ exhibits nonclassical features. The diagonal P-representation is known to be unique, even though it may be realized in different equivalent forms [37]. Towards inverting the diagonal representation (3.1) the Fourier transform is employed [26]:

$$\begin{eqnarray}P(\beta ,{\beta }^{*}) & = & \displaystyle \frac{\mathrm{exp}({| \beta | }^{2})}{{\pi }^{2}}\displaystyle \int \langle -\gamma | {\rho }_{{ \mathcal O }}| \gamma \rangle \;\mathrm{exp}({| \gamma | }^{2})\\ & & \times \mathrm{exp}(\beta {\gamma }^{*}-{\beta }^{*}\gamma ){{\rm{d}}}^{2}\gamma .\end{eqnarray} \tag{ 3.3 }$$

The invertibility of the relations (3.1, 3.3) suggests that both the density matrix ${\rho }_{{ \mathcal O }}$ and the diagonal representation $P(\beta ,{\beta }^{*})$ encapsulate equivalent informations. We now explicitly evaluate the pseudo probability distribution corresponding to the oscillator density matrix (2.19):

$$\begin{eqnarray}P(\beta ,{\beta }^{*}) & = & \displaystyle \frac{1}{2}{| {{ \mathcal C }}_{0}(t)| }^{2}{D}_{0,0}^{(+)}\\ & & +\mathrm{Re}\left({{ \mathcal C }}_{0}{(t)}^{*}\displaystyle \sum _{n=1}^{\infty }({{ \mathcal A }}_{n}(t){D}_{0,n-1}^{(-)}+{{ \mathcal B }}_{n}(t){D}_{0,n}^{(+)})\right.\\ & & +\left.\displaystyle \sum _{n,m=1}^{\infty }{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){D}_{n,m-1}^{(-)}\right)\\ & & +\displaystyle \frac{1}{2}\displaystyle \sum _{n,m=1}^{\infty }({{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){D}_{n-1,m-1}^{(+)}\\ & & +{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){D}_{n,m}^{(+)}),\end{eqnarray} \tag{ 3.4 }$$

where the arbitrarily singular derivatives of the δ-functions read

$$\begin{eqnarray}{D}_{n,m}^{(\pm )} & = & \displaystyle \frac{1}{\sqrt{n!m!}}\left(\mathrm{exp}({| {\beta }_{+}| }^{2}){\left(-\displaystyle \frac{\partial }{\partial {\beta }_{+}^{*}}\right)}^{n}{\left(-\displaystyle \frac{\partial }{\partial {\beta }_{+}}\right)}^{m}{\delta }^{(2)}({\beta }_{+})\right.\\ & & \left.\pm \mathrm{exp}({| {\beta }_{-}| }^{2}){\left(-\displaystyle \frac{\partial }{\partial {\beta }_{-}^{*}}\right)}^{n}{\left(-\displaystyle \frac{\partial }{\partial {\beta }_{-}}\right)}^{m}{\delta }^{(2)}({\beta }_{-})\right),\\ {\beta }_{\pm } & = & \beta \pm \displaystyle \frac{\sqrt{x}}{2}.\end{eqnarray} \tag{ 3.5 }$$

The diagonal P-representation (3.4) satisfies the normalization condition (3.2), and is obviously real. The initial $(t=0)$ hybrid entangled state (2.9) gives rise to a positive definite P-representation: $P(\beta ,{\beta }^{*}){| }_{t=0}$ $={2}^{-1}\;\mathrm{exp}(-({| \alpha | }^{2}-{| \beta | }^{2}))$ $({\delta }^{(2)}(\alpha -\beta )+$ ${\delta }^{(2)}(\alpha +\beta ))$. At an arbitrary time, however, a large number of interaction-dependent modes develop, and produce quantum interferences among themselves. The quantum properties in (3.4) are evident due to the appearance of rapidly oscillating singular derivatives of δ-function in the P-representation, which at time $t\gt 0$ is clearly not a nonnegative definite quantity. The existence of negativity of the diagonal P-representation (3.4) points towards the manifestations of specific quantum effects such as anti-bunching [38] and quadrature squeezing [39]. Other phase space pseudo probability functions such as Wigner W-distribution and the Husimi Q-function may be obtained from the diagonal P-representation after its suitable smoothing by the Gaussian kernels.

The Wigner phase space distribution for a quantum state given by the oscillator density matrix (2.19) is described [26] as

$$\begin{eqnarray}&&W(\beta ,{\beta }^{*})=\displaystyle \frac{1}{{\pi }^{2}}\int \mathrm{Tr}({\rm{D}}(\gamma ){\rho }_{{ \mathcal O }})\mathrm{exp}(\beta {\gamma }^{*}-{\beta }^{*}\gamma ){{\rm{d}}}^{2}\gamma ,\end{eqnarray} \tag{ 3.6 }$$

which is a real-valued function maintaining the normalization property:

$$\begin{eqnarray}&&\int W(\beta ,{\beta }^{*})\ {{\rm{d}}}^{2}\beta =1.\end{eqnarray} \tag{ 3.7 }$$

A characteristic of the Wigner W-distribution that distinguishes it from a classical probability density, is that its integral over a given subregion of the phase space may be negative or greater than one. As an efficient computational algorithm the W-distribution has been expressed [40] as a sum of an infinite series of the expectation values of the oscillator density matrix ${\rho }_{{ \mathcal O }}^{\phantom{A}}$ in the displaced number basis states:

$$\begin{eqnarray}W(\beta ,{\beta }^{*}) & = & \displaystyle \frac{2}{\pi }\displaystyle \sum _{k=0}^{\infty }{(-1)}^{k}\langle \beta ,k| {\rho }_{{ \mathcal O }}^{\phantom{A}}| \beta ,k\rangle ,\\ | \alpha ,n\rangle & = & {\rm{D}}(\alpha )| n\rangle .\end{eqnarray} \tag{ 3.8 }$$

On the other hand, the convolution of a smoothing Gaussian function of variance 1/2 on the phase space with the diagonal P-representation leads to the W-distribution that is free from the divergences brought forward by the derivatives of the δ-function contained in the expression (3.4) of the P-representation:

$$\begin{eqnarray}&&W(\beta ,{\beta }^{*})=\displaystyle \frac{2}{\pi }\int P(\gamma ,{\gamma }^{*})\mathrm{exp}(-2{| \beta -\gamma | }^{2})\ {{\rm{d}}}^{2}\gamma .\end{eqnarray} \tag{ 3.9 }$$

Employing the explicit evaluation (3.4) of the P-representation and the convolution integral (3.9) we now provide a direct derivation of the W-distribution. Towards this we employ the identity

$$\begin{eqnarray}&&\displaystyle \int \mathrm{exp}(-2{| \beta -\gamma | }^{2}+{| \gamma | }^{2}){\left(-\displaystyle \frac{\partial }{\partial {\gamma }^{*}}\right)}^{n}\\ &&\quad \times {\left(-\displaystyle \frac{\partial }{\partial \gamma }\right)}^{m}{\delta }^{(2)}(\gamma ){{\rm{d}}}^{2}\gamma \\ &&={2}^{n+m}{\beta }^{n}{\beta }^{*m}\mathrm{exp}(-2{| \beta | }^{2})\\ &&\quad \times {}_{2}{F}_{0}\left(-n,-m;{}_{-};-\displaystyle \frac{1}{4{| \beta | }^{2}}\right),\end{eqnarray} \tag{ 3.10 }$$

where the hypergeometric sum reads ${}_{2}{F}_{0}({\mathsf{x}},{\mathsf{y}};{}_{-};\tau )\ ={\sum }_{k=0}^{\infty }{({\mathsf{x}})}_{k}$ ${({\mathsf{y}})}_{k}\;\frac{{\tau }^{k}}{k!},{({\mathsf{x}})}_{k}={\prod }_{{\ell }=0}^{k-1}({\mathsf{x}}+{\ell })$. The above hypergeometric function with the negative integral numerator may be expressed [41] via the Charlier polynomial: ${{\rm{c}}}_{k}({\ell };\tau )$ $=\;{}_{2}{F}_{0}\left(-k,-{\ell };{}_{-};-\frac{1}{\tau }\right)$ $\forall \tau \gt 0$. To emphasize the equal footing of the integers $(k,{\ell })$ in (3.10) we, however, do not use this notation here. The integral representation (3.9) aided by the identity (3.10) now generate the W-distribution:

$$\begin{eqnarray}W(\beta ,{\beta }^{*}) & = & {| {{ \mathcal C }}_{0}(t)| }^{2}\ {{ \mathcal H }}_{0,0}^{(+)}\\ & & +2\ \mathrm{Re}\left({{ \mathcal C }}_{0}{(t)}^{*}\displaystyle \sum _{n=1}^{\infty }({{ \mathcal A }}_{n}(t){{ \mathcal H }}_{0,n-1}^{(-)}(\beta ,{\beta }^{*})\right.\\ & & +{{ \mathcal B }}_{n}(t){{ \mathcal H }}_{0,n}^{(+)}(\beta ,{\beta }^{*}))\\ & & +\left.\displaystyle \sum _{n,m=1}^{\infty }{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){{ \mathcal H }}_{n,m-1}^{(-)}(\beta ,{\beta }^{*})\right)\\ & & +\displaystyle \sum _{n,m=1}^{\infty }({{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){{ \mathcal H }}_{n-1,m-1}^{(+)}(\beta ,{\beta }^{*})\\ & & +{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){{ \mathcal H }}_{n,m}^{(+)}(\beta ,{\beta }^{*})),\end{eqnarray} \tag{ 3.11 }$$

where the weights involving the Gaussian functions are given by

$$\begin{eqnarray}{{ \mathcal H }}_{n,m}^{(\pm )}(\beta ,{\beta }^{*}) & = & \displaystyle \frac{{2}^{n+m}}{\pi \sqrt{n!m!}}({\beta }_{+}^{n}{\beta }_{+}^{*m}\mathrm{exp}(-2{| {\beta }_{+}| }^{2})\\ & & \times {}_{2}{F}_{0}\left(-n,-m;{}_{-};-\displaystyle \frac{1}{4{| {\beta }_{+}| }^{2}}\right)\\ & & \pm {\beta }_{-}^{n}{\beta }_{-}^{*m}\mathrm{exp}(-2{| {\beta }_{-}| }^{2})\\ & & \times \left.{}_{2}{F}_{0}\left(-n,-m;{}_{-};-\displaystyle \frac{1}{4{| {\beta }_{-}| }^{2}}\right)\right).\end{eqnarray} \tag{ 3.12 }$$

An alternate derivation of the Wigner W-distribution follows from sum rule (3.8). Towards this we list the matrix elements

$$\begin{eqnarray}\langle {n}_{\pm }| \alpha ,k\rangle & = & {(-1)}^{k}\displaystyle \frac{{\alpha }_{\pm }^{n}{\alpha }_{\pm }^{*k}}{\sqrt{n!k!}}\;\mathrm{exp}\left(-\displaystyle \frac{{| {\alpha }_{\pm }| }^{2}}{2}\mp {\rm{i}}{\phi }_{\alpha }\right)\\ & & \times {}_{2}{F}_{0}\left(-n,-k;{\phantom{}}_{-};-\displaystyle \frac{1}{{| {\alpha }_{\pm }| }^{2}}\right)\end{eqnarray} \tag{ 3.13 }$$

that facilitate the evaluation of the series (3.8). Another necessary tool for the current derivation is the identity

$$\begin{eqnarray}&&\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{(-1)}^{k}{\tau }^{k}}{k!}\\ &&\quad \times {}_{2}{F}_{0}\left(-n,-k;{}_{-};-\displaystyle \frac{1}{\tau }\right){}_{2}{F}_{0}\left(-k,-m;{}_{-};-\displaystyle \frac{1}{\tau }\right)\\ &&\quad ={2}^{n+m}\mathrm{exp}(-\tau )\ {}_{2}{F}_{0}\left(-n,-m;{}_{-};-\displaystyle \frac{1}{4\tau }\right)\end{eqnarray} \tag{ 3.14 }$$

that readily follows [41] from the bilinear generating function of the Charlier polynomials. The series sum (3.8), in conjunction with the results (3.13, 3.14), now reproduce the W-distribution obtained in (3.11). This serves as a consistency check on our derivations (3.4, 3.11) of the phase space distributions in the present model.

The negativity of the Wigner function is an indication towards the existence of nonclassical properties of the states. At t = 0 the W-distribution for our initial hybrid entangled state (2.9) is positive definite: $W(\beta ,{\beta }^{*}){| }_{t=0}$ $=\quad {\pi }^{-1}\;(\mathrm{exp}(-2{| \alpha -\beta | }^{2})$ $+\mathrm{exp}(-2{| \alpha +\beta | }^{2}))$. At a later time, however, interactions produce a large number of modes that interfere with each other. Interference fringes gives rise to appearance of phase space domains representing negative values of the W-distribution. A fruitful measure of negativity relate to the volume of the negative part of the Wigner function on the phase space [42]:

$$\begin{eqnarray}&&{\delta }_{W}=\int | W(\beta )| {{\rm{d}}}^{2}\beta -1.\end{eqnarray} \tag{ 3.15 }$$

We will return to the topic in section 4.

The Husimi Q-function [26] is a quasi-probability distribution defined as expectation value of the oscillator density matrix in an arbitrary coherent state. It assumes nonnegative values on the phase space in contrast to the other phase space quasi probabilities. Being easily computable it has been extensively employed [43, 44] in the study of the occupation on the phase space. For our reduced density matrix of the oscillator (2.19) the Q-function

$$\begin{eqnarray}&&Q(\beta ,{\beta }^{*})=\displaystyle \frac{1}{\pi }\langle \beta | {\rho }_{{ \mathcal O }}| \beta \rangle \end{eqnarray} \tag{ 3.16 }$$

maintains the normalization restriction: $\int Q(\beta ,{\beta }^{*}){{\rm{d}}}^{2}\beta =1$ and also the bounds: $0\leqslant Q(\beta ,{\beta }^{*})\leqslant {\pi }^{-1}$. Our construction of the oscillator density matrix (2.19) now yields the time evolution of the Q-function:

$$\begin{eqnarray}Q(\beta ,{\beta }^{*}) & = & \displaystyle \frac{1}{2}{| {{ \mathcal C }}_{0}(t)| }^{2}{H}_{0,0}^{(+)}(\beta ,{\beta }^{*})\\ & & +\mathrm{Re}\left({{ \mathcal C }}_{0}{(t)}^{*}\displaystyle \sum _{n=1}^{\infty }({{ \mathcal A }}_{n}(t){H}_{0,n-1}^{(-)}(\beta ,{\beta }^{*})\right.\\ & & +{{ \mathcal B }}_{n}(t){H}_{0,n}^{(+)}(\beta ,{\beta }^{*}))\\ & & \left.+\displaystyle \sum _{n,m=1}^{\infty }{{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){H}_{n-1,m}^{(-)}(\beta ,{\beta }^{*})\right)\\ & & +\displaystyle \frac{1}{2}\displaystyle \sum _{n,m=1}^{\infty }({{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){H}_{n-1,m-1}^{(+)}(\beta ,{\beta }^{*})\\ & & +{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){H}_{n,m}^{(+)}(\beta ,{\beta }^{*})),\end{eqnarray} \tag{ 3.17 }$$

where the weight functions on the phase space read

$$\begin{eqnarray}{H}_{n,m}^{(\pm )}(\beta ,{\beta }^{{}^{*}}) & = & \displaystyle \frac{1}{\pi \sqrt{n!m!}}({\beta }_{+}^{n}{\beta }_{+}^{*m}\mathrm{exp}(-{| {\beta }_{+}| }^{2})\\ & & \pm {\beta }_{-}^{n}{\beta }_{-}^{*m}\mathrm{exp}(-{| {\beta }_{-}| }^{2})).\end{eqnarray} \tag{ 3.18 }$$

A convolution integral of the diagonal P-representation with a Gaussian weight of unit variance produce [26] the well-behaved nonnegative Husimi Q-function as follows:

$$\begin{eqnarray}&&Q(\beta ,{\beta }^{*})=\displaystyle \frac{1}{\pi }\int P(\gamma ,{\gamma }^{*})\mathrm{exp}(-{| \beta -\gamma | }^{2})\ {{\rm{d}}}^{2}\gamma .\end{eqnarray} \tag{ 3.19 }$$

The explicit constructions of the P-representation (3.4) and that of the Q-function (3.17) obey the consistency check (3.19). Another convolution property that express the Q-function via the W-distribution reads [26]:

$$\begin{eqnarray}&&Q(\beta ,{\beta }^{*})=\displaystyle \frac{2}{\pi }\int W(\gamma ,{\gamma }^{*})\mathrm{exp}(-2{| \beta -\gamma | }^{2})\ {{\rm{d}}}^{2}\gamma .\end{eqnarray} \tag{ 3.20 }$$

Employing the integral representation (3.10) it follows that the phase space quasi probability densities (3.11, 3.17) maintain the integral sum rule (3.20). The convolution relations (3.9, 3.20) suggest an interesting feature. The diagonal P-representation containing singular derivatives of δ-functions is smoothed at two stages by a Gaussian weight of variance 1/2. After the first smoothing operation the W-distribution, which is nonsingular but not nonnegative, appears. Subsequent smoothing operation produces a nonsingular and nonnegative Husimi Q-function. If a single smoothing operation (3.19) is performed on P-representation via the Gaussian weight of variance 1, the Q-function is directly reproduced. We symbolically summarize this as ${[{[P]}_{1/2}]}_{1/2}={[P]}_{1}$, where the notation reads ${[{ \mathcal X }]}_{s}=\frac{1}{\pi \;s}\int { \mathcal X }(\gamma ,{\gamma }^{*})\mathrm{exp}\left(-\frac{{| \beta -\gamma | }^{2}}{s}\right){{\rm{d}}}^{2}\gamma$.

One of the utilities of the Q-function is that it provides a convenient evaluation of the expectation values of the operators expressed in their antinormal ordered form [26]. The first and second moments of the quadrature variable defined as

$$\begin{eqnarray}&&{X}_{\theta }=\displaystyle \frac{1}{2}(a\mathrm{exp}(-{\rm{i}}\theta )+{a}^{\dagger }\mathrm{exp}({\rm{i}}\theta ))\end{eqnarray} \tag{ 3.21 }$$

may be expressed [26] via the following phase space integrals over the complex plane:

$$\begin{eqnarray}&&\langle {X}_{\theta }\rangle \equiv \mathrm{Tr}({X}_{\theta }{\rho }_{{ \mathcal O }}(t))\\ &&\quad =\displaystyle \frac{1}{2}\displaystyle \int (\beta {{\rm{e}}}^{-{\rm{i}}\theta }+{\beta }^{*}{{\rm{e}}}^{{\rm{i}}\theta })Q(\beta ,{\beta }^{*}){{\rm{d}}}^{2}\beta ,\\ &&\langle {X}_{\theta }^{2}\rangle \equiv \mathrm{Tr}({X}_{\theta }^{2}{\rho }_{{ \mathcal O }}(t))\\ &&\quad =\displaystyle \frac{1}{4}\displaystyle \int ({(\beta {{\rm{e}}}^{-{\rm{i}}\theta }+{\beta }^{*}{{\rm{e}}}^{{\rm{i}}\theta })}^{2}-1)Q(\beta ,{\beta }^{*}){{\rm{d}}}^{2}\beta .\end{eqnarray} \tag{ 3.22 }$$

The Q-function (3.17) now allows explicit determination of the above quadrature moments:

$$\begin{eqnarray}&&\langle {X}_{\theta }\rangle =\mathrm{Re}({{\bf{G}}}_{1}(t)\mathrm{exp}(-{\rm{i}}\theta ))-\sqrt{x}\varrho \mathrm{cos}\theta ,\end{eqnarray} \tag{ 3.23 }$$

$$\begin{eqnarray}\langle {X}_{\theta }^{2}\rangle & = & \displaystyle \frac{1}{2}\;\mathrm{Re}({{\bf{G}}}_{2}(t)\mathrm{exp}(-{\rm{i}}2\theta )\\ & & -\sqrt{x}\ {{\bf{F}}}_{1}(t)\ (1+\mathrm{exp}(-{\rm{i}}2\theta )))\\ & & +\displaystyle \frac{1}{2}\left({{\bf{N}}}_{1}(t)-\displaystyle \frac{1}{2}\right)+\displaystyle \frac{x}{4}{\mathrm{cos}}^{2}\theta \end{eqnarray} \tag{ 3.24 }$$

where the time-dependent coefficients $\{{{\bf{G}}}_{k}(t),{{\bf{N}}}_{k}(t),{{\bf{F}}}_{k}(t)| k\in (1,2)\}$ expressed as sums over Fourier modes read:

$$\begin{eqnarray}&&{{\bf{G}}}_{k}(t)=\sqrt{k}\ {{ \mathcal C }}_{0}{(t)}^{*}{{ \mathcal B }}_{k}(t)+\displaystyle \sum _{n=1}^{\infty }(\sqrt{{(n)}_{k}}\ {{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{n+k}(t)\\ &&\qquad \quad +\sqrt{{(n+1)}_{k}}\ {{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{n+k}(t)),\\ &&{{\bf{N}}}_{k}(t)={| {{ \mathcal C }}_{0}(t)| }^{2}\\ &&\qquad \quad +\displaystyle \sum _{n=1}^{\infty }({n}^{k}\ {{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{n}(t)+{(n+1)}^{k}\ {{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{n}(t)),\\ &&{{\bf{F}}}_{k}(t)=k{{ \mathcal C }}_{0}{(t)}^{*}{{ \mathcal A }}_{2}(t)+\displaystyle \sum _{n=1}^{\infty }(\sqrt{n}\ {(n+1)}_{k-1}{{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal B }}_{n}(t)\\ &&\qquad \quad +\sqrt{n+1}{(n+2)}_{k-1}\ {{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal A }}_{n+2}(t)).\end{eqnarray} \tag{ 3.25 }$$

The variance of the quadrature variable is given by ${V}_{\theta }=\langle {X}_{\theta }^{2}\rangle -{\langle {X}_{\theta }\rangle }^{2},$ which we will later employ for studying emergence of the squeezed states during the time evolution. For later utilization we also quote here the mean photon number $\langle \hat{n}\rangle =\mathrm{Tr}({a}^{\dagger }a\;{\rho }_{{ \mathcal O }}(t))$ and its variance $\langle {({\rm{\Delta }}\hat{n})}^{2}\rangle =\mathrm{Tr}({\hat{n}}^{2}\;{\rho }_{{ \mathcal O }}(t))-{\langle \hat{n}\rangle }^{2}$:

$$\begin{eqnarray}&&\langle \hat{n}\rangle ={{\bf{N}}}_{1}(t)+\displaystyle \frac{x}{4}-\sqrt{x}\ \mathrm{Re}({{\bf{F}}}_{1}(t))-1\end{eqnarray} \tag{ 3.26 }$$

$$\begin{eqnarray}\langle {({\rm{\Delta }}\hat{n})}^{2}\rangle & = & {{\bf{N}}}_{2}(t)+\displaystyle \frac{x}{2}\left({{\bf{N}}}_{1}(t)+\mathrm{Re}({{\bf{G}}}_{2}(t))-\displaystyle \frac{1}{2}\right)\\ & & +\sqrt{x}(\mathrm{Re}({{\bf{F}}}_{1}(t))-2\mathrm{Re}({{\bf{F}}}_{2}(t)))\\ & & -{({{\bf{N}}}_{1}(t)-\sqrt{x}\mathrm{Re}({{\bf{F}}}_{1}(t)))}^{2}.\end{eqnarray} \tag{ 3.27 }$$

Another dynamical quantity that is useful in the study of 'kitten'-like states is the polar phase density of the Husimi Q-function [30] obtained via its radial integration on the phase space:

$$\begin{eqnarray}&&{ \mathcal Q }(\theta )={\displaystyle \int }_{0}^{\infty }Q(\beta ,{\beta }^{*})\;| \beta | \;{\rm{d}}| \beta | ,\quad \beta =| \beta | \mathrm{exp}({\rm{i}}\theta ),\end{eqnarray} \tag{ 3.28 }$$

which is a convenient tool for describing the splitting of the Q-function. Towards obtaining a series expansion of the phase density ${ \mathcal Q }(\theta )$ we define the weight factors depending on the polar angle:

$$\begin{eqnarray}&&{{ \mathcal H }}_{n,m}^{(\pm )}(\theta )={\displaystyle \int }_{0}^{\infty }{H}_{n,m}^{(\pm )}(\beta ,{\beta }^{{}^{*}})\;| \beta | \;{\rm{d}}| \beta | \end{eqnarray} \tag{ 3.29 }$$

that admit, via (3.18), explicit evaluation as follows:

$$\begin{eqnarray}{{ \mathcal H }}_{n,m}^{(\pm )}(\theta ) & = & \displaystyle \frac{1}{\pi \sqrt{n!m!}}\displaystyle \sum _{ \jmath =0}^{n}\displaystyle \sum _{{\ell }=0}^{m}\\ & & \mathrm{exp}({\rm{i}}( \jmath -{\ell })\theta )\mathrm{exp}\left(-\displaystyle \frac{x{\mathrm{sin}}^{2}\theta }{4}\right)\\ & & \displaystyle \times \left(\displaystyle \genfrac{}{}{0em}{}{n}{ \jmath }\right)\displaystyle \left(\displaystyle \genfrac{}{}{0em}{}{m}{{\ell }}\right){\left(\displaystyle \frac{x}{4}\right)}^{\frac{n+m- \jmath -{\ell }}{2}}\\ & & \times \left({P}_{n+m- \jmath -{\ell }}^{(\pm )}\ {\rm{\Gamma }}\left(\displaystyle \frac{ \jmath +{\ell }}{2}+1\right)\right.\\ & & \times {}_{1}{F}_{1}\left(-\displaystyle \frac{ \jmath +{\ell }+1}{2};\displaystyle \frac{1}{2};-\displaystyle \frac{x{\mathrm{cos}}^{2}\theta }{4}\right)\\ & & -{P}_{n+m- \jmath -{\ell }}^{(\mp )}\ \sqrt{x}\;\mathrm{cos}\theta \ {\rm{\Gamma }}\left(\displaystyle \frac{ \jmath +{\ell }+3}{2}\right)\\ & & \times \left.{}_{1}{F}_{1}\left(-\displaystyle \frac{ \jmath +{\ell }}{2};\displaystyle \frac{3}{2};-\displaystyle \frac{x{\mathrm{cos}}^{2}\theta }{4}\right)\right),\end{eqnarray} \tag{ 3.30 }$$

where we have used the integration given by

$$\begin{eqnarray}&&{\displaystyle \int }_{0}^{\infty }{r}^{p}\mathrm{exp}(-{(r\pm { \mathcal X })}^{2}){\rm{d}}r\\ &&\quad =\displaystyle \frac{1}{2}\;{\rm{\Gamma }}\left(\displaystyle \frac{p+1}{2}\right){}_{1}{F}_{1}\left(-\displaystyle \frac{p}{2};\displaystyle \frac{1}{2};-{{ \mathcal X }}^{2}\right)\\ &&\quad \mp \ { \mathcal X }\ {\rm{\Gamma }}\left(\displaystyle \frac{p}{2}+1\right){}_{1}{F}_{1}\left(\displaystyle \frac{1-p}{2};\displaystyle \frac{3}{2};-{{ \mathcal X }}^{2}\right).\end{eqnarray} \tag{ 3.31 }$$

The above compendium of expressions (3.28, 3.17, 3.30) now provides the evolution of the polar phase density ${ \mathcal Q }(\theta )$ for our initial state (2.9)

$$\begin{eqnarray}{ \mathcal Q }(\theta ) & = & \displaystyle \frac{1}{2}{| {{ \mathcal C }}_{0}(t)| }^{2}{{ \mathcal H }}_{0,0}^{(+)}(\theta )\\ & & +\mathrm{Re}\left({{ \mathcal C }}_{0}{(t)}^{*}\displaystyle \sum _{n=1}^{\infty }({{ \mathcal A }}_{n}(t){{ \mathcal H }}_{0,n-1}^{(-)}(\theta )\right.\\ & & +\left.{{ \mathcal B }}_{n}(t){{ \mathcal H }}_{0,n}^{(+)}(\theta ))+\displaystyle \sum _{n,m=1}^{\infty }{{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){{ \mathcal H }}_{n-1,m}^{(-)}(\theta )\right)\\ & & +\displaystyle \frac{1}{2}\displaystyle \sum _{n,m=1}^{\infty }({{ \mathcal A }}_{n}{(t)}^{*}{{ \mathcal A }}_{m}(t){{ \mathcal H }}_{n-1,m-1}^{(+)}(\theta )\\ & & +{{ \mathcal B }}_{n}{(t)}^{*}{{ \mathcal B }}_{m}(t){{ \mathcal H }}_{n,m}^{(+)}(\theta )).\end{eqnarray} \tag{ 3.32 }$$

The Wehrl entropy defined as [27]

$$\begin{eqnarray}&&{S}_{Q}=-\int Q(\beta ,{\beta }^{*})\;\mathrm{log}Q(\beta ,{\beta }^{*})\;{{\rm{d}}}^{2}\beta \end{eqnarray} \tag{ 4.1 }$$

is an information-theoretic measure estimating the delocalization of the system in the oscillator phase space. It is considered [45] as a count of an equivalent number of widely separated coherent states necessary for covering the existing phase space occupation of the coupled oscillator. Being subject to the restriction originating from the Heisenberg uncertainty principle, the Wehrl entropy (4.1) is a positive definite quantity [27]. In the present case we employ the definition (4.1) and the evolution (3.17) of the Q-function to numerically study the long-range time dependence of S_Q for various values of the coupling strength. We note that here and hereafter the time is measured in the natural unit: ${\omega }^{-1}$. We observe the following properties: (i) in the long time limit $t\gtrsim {({x}^{2}\tilde{{\rm{\Delta }}})}^{-1}$ the quasi periodicity of the Wehrl entropy is manifest in the coupling strength regime $\lambda \lesssim 0.05\omega$, where the Laguerre polynomials ${L}_{n}^{(j)}(x)$ are well-approximated by the terms up to their quadratic components $O({x}^{2})$. Frequency modes $O({x}^{2}\tilde{{\rm{\Delta }}})$ and their harmonics produced via the interaction now give rise to the quasi periodicity of S_Q, where the long-range time period maintains the property: ${T}_{\mathrm{long}}\propto {\lambda }^{-4}\;\mathrm{exp}(x/2)$ (figures 1 (a)–(c)). For a smaller value of the qubit frequency Δ the quasi periodic behavior persists for a comparatively higher coupling strength λ as an increase in the interaction-induced frequencies $(O({x}^{n}\tilde{{\rm{\Delta }}})| n=0,1,\ldots )$ quickly leads to randomization of phases, and consequent loss of the long-range periodicity. The time periods observed in figures 1 (a)–(c) are noted below: ${T}_{\mathrm{long}}=6.9914\times {10}^{6}$ (for $\lambda =0.008\;\omega$), ${T}_{\mathrm{long}}=2.9568\times {10}^{6}$ (for $\lambda =0.010\;\omega$), ${T}_{\mathrm{long}}=1.4743\times {10}^{6}$ (for $\lambda =0.012\;\omega$), respectively. The closeness of the values obtained for the product ${T}_{\mathrm{long}}\;{\lambda }^{4}\;\mathrm{exp}(-x/2)$ in the respective cases ($0.02863,0.02956,0.03056)$ validates our argument that the quantum fluctuations $O({x}^{2})$ produce the observed long-range time period. In the instance of nonlinear Kerr-like medium similar behavior in the time evolution of S_Q was previously noticed [29], where its local minima corresponded with the formations of finite superposition of coherent states. In the present bipartite interacting model, however, the qubit-oscillator interaction superimposes short time span fluctuations of the frequency $O(x\tilde{{\rm{\Delta }}})$ on the long-range oscillations that now act as an envelope of the total time evolution of the Wehrl entropy. This introduces important distinctions to the present model. We will discuss this in the sub 4.2. (ii) In the ultrastrong coupling regime $\lambda \gtrsim 0.1\omega$ all frequency modes $(O({x}^{n}\tilde{{\rm{\Delta }}})| n=0,1,\ldots )$ and their harmonics arise. Random phase differences between a large number of incommensurate modes cause the resultant interference to average out, while ensuring an effective stabilization of the occupation of the phase space after an initial build up (figure 1 (d)). The rapid high frequency $O(\omega )$ fluctuations are of small amplitude: $| {\rm{\Delta }}{S}_{Q}| /{S}_{Q}\ll 1$, and may be removed by using a suitable coarse graining process [46]. In this regime it is observed that for a fixed initial state parameter α the time-averaged value of the Wehrl entropy gradually increases with increasing coupling strength. Following (3.26) it is evident that higher coupling strength leads to an enhancement of the photon expectation value $\langle \hat{n}\rangle$ that causes a wider spread of the Q-function resulting in an increment in S_Q. (iii) Sufficiently localized states are transiently observed at large macroscopic values of the coherent state amplitude $\alpha \;$ even in the ultrastrong coupling domain. These states are realized (figure 2) at the local minimum of the Wehrl entropy S_Q. Physically, the energy exchange between the qubit and the oscillator degrees of freedom induces the revival and collapse of the qubit density matrix elements. A revival of the qubit matrix elements indicates that the oscillator has, reciprocally, less energy available to it. This acts as a constraint on its delocalization on the phase space. Moreover, a large value of the amplitude $\alpha \;$ signifies comparatively higher magnitude of energy residing in the interference pattern (figure 2 (b)) of the quantum oscillations. Qualitatively, when the energy associated with the stochastic randomization of the modes is less than the energy of the coherent interference pattern, a localization on the phase space takes place, and, consequently, a relative decrease in S_Q develops. The quantum interference pattern (figure 2 (b)) has the shape of a localized peak with 'twisted arms', where oscillations develop perpendicular to these arms causing a transport of energy necessary for the localization. The phase density ${ \mathcal Q }(\theta )$ diagram (figure 2 (d)) shows that the initial state (2.9) consisting of two maximally mixed Gaussian peaks coalesce at the local minimum of S_Q to produce a partially pure transient state of the oscillator with its von Neumann entropy given by S = 0.5883. (iv) The fast initial rise of S_Q in the ultrastrong coupling domain $\lambda \sim \omega$ may be understood as follows. Increased qubit-oscillator coupling leads to the generation of all high frequency quantum fluctuation modes. The phase randomization of the modes of incommensurate frequencies results in the rapid initial spreading on the phase space, and consequent fast production of S_Q. Once the modes are statistically populated, the Wehrl entropy S_Q, except for high frequency $O(\omega )$ quantum fluctuations of relatively small amplitude, maintains an almost stationary value. The initial production time of the Wehrl entropy ${T}_{\mathrm{ent}.\mathrm{prod}.}$ follows from the asymptotic behavior of associated Laguerre polynomials ${L}_{n}^{(j)}(x)$ at large $n\gg 1$, fixed j, and $x\gt 0$ [47]:

$$\begin{eqnarray}{L}_{n}^{(j)}(x) & = & \displaystyle \frac{{n}^{\frac{j}{2}-\frac{1}{4}}}{\sqrt{\pi }}\displaystyle \frac{\mathrm{exp}(x/2)}{{x}^{\frac{j}{2}+\frac{1}{4}}}\mathrm{cos}\left(2\sqrt{{nx}}-\displaystyle \frac{\pi }{2}\left(j+\displaystyle \frac{1}{2}\right)\right)\\ & & +O\left({n}^{\frac{j}{2}-\frac{3}{4}}\right).\end{eqnarray} \tag{ 4.2 }$$

The asymptotic limit of the energy eigenvalues (2.7) is now readily obtained:

$$\begin{eqnarray}{{ \mathcal E }}_{n(\gg 1)}^{(\pm )} & = & \omega \left(n\pm 1-\displaystyle \frac{1}{2}-\displaystyle \frac{x}{4}\right)\mp \displaystyle \frac{{\rm{\Delta }}}{\sqrt{\pi }}\;{({nx})}^{-\frac{1}{4}}\\ & & \times \mathrm{cos}\left(2\sqrt{{nx}}-\displaystyle \frac{\pi }{4}\right)+O\left({n}^{-\frac{1}{2}}\right),\end{eqnarray} \tag{ 4.3 }$$

where the leading interaction-generated part $O\left({n}^{-\frac{1}{4}}\right)$ on the rhs provides the effective statistical stabilization of the occupation in the phase space. Recognizing this, the coupling strength dependence of the typical time scale for the generation of S_Q is now given by ${T}_{\mathrm{ent}.\mathrm{prod}.}\;\frac{{\rm{\Delta }}}{\sqrt{\pi }}{({nx})}^{-\frac{1}{4}}\sim 1\Rightarrow {T}_{\mathrm{ent}.\mathrm{prod}.}\propto \sqrt{\lambda }$. The rapid initial increase of the Wehrl entropy is described in figure 3. For the data presented in figure 3 the proportionality constant ${T}_{\mathrm{ent}.\mathrm{prod}.}/\sqrt{\lambda }$ read $30.99,29.66,33.11$ for the coupling strengths $\lambda =0.9\;\omega ,1.1\;\omega ,1.3\;\omega$, respectively. The discrepancy ($\sim 6\%$) in the observed data occurs since the local fluctuations in the time evolution of S_Q make it difficult to precisely locate the onset $({T}_{\mathrm{ent}.\mathrm{prod}.})$ of stochastic stabilization. A suitable coarse-graining process [46] to smooth the high frequency fluctuations may be adopted for fuller agreement.

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For the Kerr-type nonlinear self-interacting photonic models, the local minima in the time evolution of the Wehrl entropy S_Q of an initial coherent state are associated [29] with transient formations of the superposition of a finite number of coherent states [28]–[31] maintaining a uniform angular separation on the complex plane. These superposition of coherent states are realized at rational submultiples of the time period of S_Q. Recently, such nonclassical superposition of multiple coherent states in a Kerr medium has been experimentally achieved [24]. Formation of cat-like states in a finite dimensional bosonic system that admits applying a displacement operator on its ground state has also been studied [48] in a Kerr medium.

In our model we study the emergence of these transitory 'kitten' states using the W-distribution and its smoothed analog the Q-function. In the strong coupling limit $\lambda \lesssim 0.05\;\omega$, and at specific times given by the rational submultiples of ${T}_{\mathrm{long}}\ :\{{T}_{p,q}=(p/q){T}_{\mathrm{long}}| (p,q)=1,p\leqslant q\}$ density matrices comprising of a finite number of macroscopic coherent states with uniform angular separation on the phase space are observed (figure 4). Starting with the initial hybrid Bell state (2.9) of the composite system, the evolution of S_Q in the long-range quasi periodic regime shows (figure 1 (b)) the existence of the local minima at rational submultiples of ${T}_{\mathrm{long}}$. The presence of numerous time scales due to the qubit-oscillator interaction in the present model, however, introduces another novel interference related feature. In particular, the interaction-dependent linear mode with frequency $O(x\tilde{{\rm{\Delta }}})$ causes an energy transfer, in a short time scale, between the qubit and the oscillator degrees of freedom. In the vicinity of the said instants ${T}_{p,q}$ the oscillations of the period ${T}_{\mathrm{long}}$ produce a locally minimum occupation on the phase space, whereas the short time period fluctuations engineer the spread of the occupation by splitting of the Gaussian peaks. This manifests as an ordered bifurcation (à la, say, figures 4(a₂, a₄) and (a₃, a₅)) of the q quasi-probability peaks to $2q$ peaks representing mixed state oscillator density matrices, while evolving from the local minima to the maxima of the short time period oscillations of the Wehrl entropy. These local variations of S_Q in the neighborhood of $(p/q){T}_{\mathrm{long}}$ are given in figures 4 (a₁)-(c₁), where we fix $p=1;q=1,2,3$, respectively. Moreover, the short-range time period (${T}_{\mathrm{short}}$) associated with the splitting and subsequent rejoining of the peaks at a particular rational fraction $(p/q){T}_{\mathrm{long}}$ scales inversely with q. For instance, from the figures 4 (a₁)-(c₁), respectively, we observe that ${T}_{\mathrm{short}}({T}_{\mathrm{long}})\approx 6750,{T}_{\mathrm{short}}({T}_{\mathrm{long}}/2)$ $\approx 3375,{T}_{\mathrm{short}}({T}_{\mathrm{long}}/3)\approx 2250$. This suggests the scaling relation ${T}_{\mathrm{short}}({T}_{\mathrm{long}}/q)\approx (1/q)\;{T}_{\mathrm{short}}({T}_{\mathrm{long}})$. In other words, due to the complex nature of the qubit–oscillator interaction in the strong coupling regime $\lambda \sim 0.05\;\omega$ the exchange of energy manifestly occurs between various interaction-dependent modes, and effectively the high frequency quantum oscillation $O(x\tilde{{\rm{\Delta }}})$ is frequency modulated by the low frequency component $O({x}^{2}\tilde{{\rm{\Delta }}})$. Moreover, the oscillator at the local minima of the Wehrl entropy (figures 4 (a₂, a₄), (b₂, b₄), (c₂, c₄)) is close to pure states. The respective von Neumann entropy S read $0.26616,0.17983,0.53918$, whereas the corresponding maxima (figures 4 (a₃, a₅), (b₃, b₅), (c₃, c₅)) describe almost maximally mixed states. The observed scaling property of ${T}_{\mathrm{short}}({T}_{\mathrm{long}}/q)$ follows from the operative quasi ${{\mathbb{Z}}}_{q}$ symmetry on the phase space engineered by the transient superposition of uniformly $(2\pi /q)$ phase-rotated q coherent states realized at ${T}_{\mathrm{long}}/q$. Using the adiabatic approximation scheme the bifurcation of the 'kitten' states was previously noticed [49] in the large decoupling regime ${\rm{\Delta }}\ll \omega$, where, for the Hamiltonian (2.1), only an even number of Gaussian peaks were found to be present. Our work based on the generalized rotating wave approximation broadens the validity of the splitting of the 'kitten' states.

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We also note the geometry of the domain on the phase space that supports the W-distribution (figures 4 (a₂)–(c₂), (a₃)–(c₃)). The interference pattern realized between two Gaussian peaks occurs in the intermediate phase space giving rise to oscillations in a direction perpendicular to the line joining the peaks. Alternate lines with the positive and the negative values of the W-distribution appear with relative phase differences of π. As the number of peaks increase, the interference pattern becomes more complex while being restricted within a regular polygon with peaks lying at its corners. The Gaussian peaks of the Q-functions (figures 4 (a₄)–(c₄), (a₅)–(c₅)) appear as smoothed versions of the W-distributions.

Recently such multiple 'kitten' states of the superconducting cavity resonators involving a large number of photons have been experimentally realized [22]-[25] as superposition of quasi orthogonal coherent states with controlled uniform phase shifts. In particular, employing a qubit state conditional cavity phase shift, in conjunction with a photon number conditional qubit rotation, it has been possible [25] to deterministically transfer arbitrary qubit states to quantum light states. This allows generation of complex nonclassical cavity states with any chosen amplitude and phase. Larger photon superpositions and consequent observation of more Gaussian peaks in such experiments may occur by increasing the displacement amplitudes characterizing the coherent states. The observed [25] W-distribution bears a strong resemblance with our multicomponent 'kitten' states obtained via the generalized rotating wave approximation. Moreover, finer resolution in the experiments may lead to observations of the doubling of the coherent peaks and the scaling behavior of ${T}_{\mathrm{short}}$ described above.

It is also of interest to study the quantum entropy based on the modulus of the Wigner distribution $| W(\beta ,{\beta }^{*})|$ [32] which is a nonnegative quantity:

$$\begin{eqnarray}&&{S}_{W}=-\int | W(\beta ,{\beta }^{*})| \;\mathrm{log}| W(\beta ,{\beta }^{*})| \ {{\rm{d}}}^{2}\beta .\end{eqnarray} \tag{ 4.4 }$$

As the W-distribution contains more information on the phase space structure of a quantum state than its smoothed analog the Q-function, a comparative study of the Wigner entropy S_W (4.4) and the Wehrl entropy S_Q (4.1) is expected to throw a light on the nonclassicality of the state. It is evident from the figures 5 (a), (b) that the time evolution of the Wigner entropy (4.4) and the negativity parameter (3.15) have close kinship with each other. This was observed for certain oscillator density functions in [32]. In this sense the Wigner entropy S_W reveals the extent of nonclassicality of a quantum density matrix. We distinguish between two possible scenarios depending upon the qubit-oscillator coupling strength. (i) In the strong coupling regime $(\lambda /\omega \lesssim 0.1)$ we observe (figure 5 (a)) a periodic structure that may be identified with the revival and collapse of the qubit density matrix elements reflecting the energy exchanges between the qubit and the oscillator degrees of freedom. The collapse, say, of the qubit density matrix elements coincides with a wider spread of the phase space distributions of the oscillator. The order of the time period of the revival and collapse of the qubit matrix elements, obtained via retaining up to the linear terms in the Laguerre polynomials, is $O(2\pi /x\tilde{{\rm{\Delta }}})$. Therefore the variables such as the Wigner entropy S_W, the negativity ${\delta }_{W}$, and the Wehrl entropy S_Q display similar periodic patterns in the said time scale. For a dominant value of ${\delta }_{W}$ the quantum interference effects are overwhelming, and we, expectedly, find ${S}_{W}\gt {S}_{Q}$, as an increased negativity necessitates an increment in the magnitude $| W(\beta ,{\beta }^{*})|$ for maintaining the normalization property (3.7). This, in turn, leads to increased value of the entropy S_W. On the other hand, for a low negativity domain ${\delta }_{W}\ll 1$ the inequality is reversed: ${S}_{W}\lt {S}_{Q}$. The underlying reason is that the Q-function is obtained from the W-distribution (3.20) after suitable smearing with a positive definite Gaussian kernel on the phase space, and therefore it incorporates less information on the quantum state than the latter [50]. (ii) In the ultrastrong coupling regime $(\lambda /\omega \gg 0.1)$ all modes for the qubit-oscillator interaction with incommensurate frequencies are excited and a fully randomized interference pattern evolves very quickly. Smearing the clear periodic structures observed earlier these large number of interaction-generated modes lead to quasi stationary values of the phase space observables (figure 5 (b)). However, despite the statistical stabilization of the occupation on the phase space the nonclassicality of the state remains prominent due to the interferences occurring between multiple modes. These interferences necessarily develop (figure 5 (c)) significant domains on the phase space with negative values of the W-distribution leading to a dominant negativity parameter ${\delta }_{W}$. The average value of ${\delta }_{W}$ increases with that of the coupling strength as more interfering modes come into existence. As mentioned before, this results in an increment of the entropy S_W. In the quasi stationary state the negativity ${\delta }_{W}$ is stochastically preserved. The stochastic stabilization of ${\delta }_{W}$ occurs after a suitable decoherence time. Therefore in figure 5 (b) we observe that except for a brief initial period the Wigner entropy S_W is consistently more than the Wehrl entropy S_Q, even though the nonnegative Q-function may be viewed as a smeared form of the W-distribution. Our results (figures 5 (a), (b)) suggest that when the negativity ${\delta }_{W}$ assumes more than a threshold value ${\delta }_{W}\gtrsim 0.4$ the quantum fluctuations ensure the entropy relation: ${S}_{W}\gt {S}_{Q}$.

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Another feature revealed in figures 5 (a) and (b) is that the relative fluctuations of the Wigner entropy S_W overwhelms that of the Wehrl entropy ${S}_{Q}\quad :| \delta {S}_{Q}/{S}_{Q}| \ll | \delta {S}_{W}/{S}_{W}|$. It signifies that the quantum interferences in the evolution of the W-distribution induce rapid reversal of its sign, whereas the fluctuations of different modes are significantly attenuated by the smoothing introduced towards obtaining the Q-function.

A feature of nonclassicality such as squeezing is observed during the time evolution of the state due to the quantum interferences between various modes. At relatively small value of the phase space separation $(\alpha \lesssim 1)$ and for the strong coupling regime $(\lambda \sim 0.1\;\omega )$, the squeezing is noticed both for the large detuning and the resonant frequencies (figure 6). The plot of the W-distribution (figure 6 (c)) makes the quadrature squeezing evident. The signature of the squeezing is observed (figure 6 (d)) when the variance ${V}_{\theta }$ of the quadrature variable, say at $\theta =0$, is rendered less than its classical value 1/4. It follows from the polar plot (figure 6 (d)) that at the scaled time $\omega t=28.80$ the quadrature variance ${V}_{\theta }$ reaches a minimum value 0.15967 at an angle $\theta =179.55^\circ$. The polar angle at which the minimum quadrature variance is realized varies with time and depends on the dynamical state of interference of the quantum modes. A qualitative understanding of the squeezing of the state may be described as follows. Passing to the interaction picture for the Hamiltonian (2.1), an effective Hamiltonian may be obtained à la [46] in an order by order perturbation theory. This effective Hamiltonian contains two photon terms $({a}^{2},{a}^{\dagger 2})$ at the order O(x) of the coupling strength. These two photon terms give rise to the squeezing of the state. An increase in the coupling λ first enhances the squeezing as it augments the strength of the two photon terms in the effective Hamiltonian. Multiple photon terms, however, soon appear in the effective Hamiltonian [46] with increased value of the coupling strength. The resulting randomness of the phase relationships of the higher order terms eliminates the squeezing property in the ultrastrong coupling limit. It is obvious in figure 6(a) that the quadrature squeezing, even though it may be present during part of the dynamical evolution of the oscillator state, is not realized throughout the oscillatory cycle. The quadrature variance remains above the threshold value: ${V}_{\theta =0}\gt 1/4$ during the major part of the evolution of the state. The instances of squeezing decreases with increasing λ. To illustrate the feature we plot (figure 6 (e)) the time-averaged quadrature variance ${\langle {V}_{\theta =0}\rangle }_{\mathrm{time}\mathrm{av}.}$ w.r.t. the coupling strength λ. We observe that, for the parametric regime studied here $\omega \gtrsim {\rm{\Delta }}$, the time-averaged quadrature variance has no squeezing property: ${\langle {V}_{\theta =0}\rangle }_{\mathrm{time}\mathrm{av}.}\gt 1/4$, and it smoothly increases with the rising λ. The effective Hamiltonian approach [46], however, suggests that the two photon terms $({a}^{2},{a}^{\dagger 2})$ underlying the squeezing of the state survive the smearing of the fast oscillatory terms only in the limit of the large qubit frequency: $\omega \ll {\rm{\Delta }}$. We will return to this topic elsewhere.

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It is interesting to note that the squeezed state described in figure 6 represents almost pure state of the oscillator at the given time as its von Neumann entropy $S{| }_{\omega t=28.80}=0.15690$ is much less than its maximal value. The almost pure state of the oscillator is reciprocated by the corresponding nearly pure state of the qubit. For the sake of completeness we include the qubit density matix (2.14) at the given time:

$$\begin{eqnarray}\left(\begin{array}{cc}0.90760 & -0.10668+i\ 0.19300\\ -0.10668-i\ 0.19300 & 0.09240\end{array}\right).\end{eqnarray} \tag{ 4.5 }$$

Its eigenvalues and corresponding eigenvectors read: ${\lambda }_{1}=0.96342$, ${\lambda }_{2}=0.03658;$ $| {\lambda }_{1}\rangle =(0.46897-i\ 0.84843)$ $| 1\rangle -0.24543| -1\rangle ,\ | {\lambda }_{2}\rangle =(-0.11873+i\ 0.21480)$ $| 1\rangle -0.96941| -1\rangle$. The magnitude of the largest eigenvalue of the qubit density matix (2.14) may be regarded as measure of the purity of the state. For smaller values of the coherent state amplitude $\alpha \lesssim 0.1$ the purity of the state may exceed 99%. Appearance of a squeezed state requires emergence of appropriate phase relations between various quantum modes that is facilitated by its proximity to a pure state. A statistical mixture of a large number of pure states is likely to destroy the phase relationships and, consequently, the squeezing property is eliminated.

These transient squeezed states obtained via the time evolution of our initial hybrid Bell states (2.9), however, incorporate a macroscopic number of photons for suitable values of the displacement variable α. An experimental realization of such states using a deterministic method may be useful in the area of optical quantum measurements towards improving the sensitivity of interferometric phase estimations [51].

In experiments allowing a direct detection of photons the Mandel parameter [33]

$$\begin{eqnarray}&&{Q}_{M}=\langle {({\rm{\Delta }}\hat{n})}^{2}\rangle /\langle \hat{n}\rangle -1\end{eqnarray} \tag{ 4.6 }$$

is a convenient tool for studying the classical-quantum boundary. For the coherent state the number operator follows the Poissonian statistics with its signature property: ${Q}_{M}=0$ that is regarded as the threshold for the classical characteristics. When the quantum correlations in the system suppress the fluctuations in the photon number, it assumes negative values: ${Q}_{M}\lt 0$ and captures the sub-Poissonian behavior of the photon statistics that acts as a measure of nonclassicality. The mean value (3.26) and the variance (3.27) of the photon number operator obtained in our model now provide a direct construction of the Mandel parameter Q_M. For the coupling strength $\lambda \lesssim 0.1\;\omega$ transient sub-Poissonian photon statistics is manifest (figure 7) during parts of the evolution of the system. The time evolution of the Mandel parameter Q_M in figure 7 (a), however, passes alternately between the classical and the nonclassical regimes. The said sub-Poissonian behavior within short sample intervals is offset by the super-Poissonian photon statistics: ${Q}_{M}\gt 0$ during most of the time evolution of the oscillator so that the averaged value over a time scale ${t}_{\mathrm{avg}}\gtrsim 2\pi /(x\tilde{{\rm{\Delta }}})$ characterising the qubit-oscillator energy fluctuations always remains nonnegative. The classicality of the photon statistics, therefore, emerge via the time-averaging procedure even though at a shorter time scale the underlying quantum nature of the photon emission process is evident. At higher coupling strengths figures 7 (b), (c) the randomization of the phases generated by a large number of incommensurate modes with the frequencies $\{O({x}^{n}\tilde{{\rm{\Delta }}})| n=1,2,\ldots \}$ partially erase the quantum correlations, while making the negativity of Q_M less common. The effective Hamiltonian approach [46], as noted in sub 4.4, produce multiple photon terms that give rise to the negativity of Q_M. Such terms, however, contribute after the smoothing of the rapidly fluctuating components only in the limit ${\rm{\Delta }}\gg \omega$. The time-averaged properties the Mandel parameter Q_M is summarized in figure 7(d), where, in the coupling regime $\lambda \lesssim 0.1\omega$ the time-averaged value ${\langle {Q}_{M}\rangle }_{\mathrm{time}\mathrm{av}.}\sim 0$ is maintained so that the overall photon statistics effectively remains to be Poissonian. After a brief transition zone the increasing coupling strength ($\lambda \gg 0.1\;\omega$) triggers a full randomization of the phase relations among the interfering modes, which now constitute an effective bath leading to progressive emergence of classical properties. The resultant stochasticity introduces a scaling behavior for the averaged value of the Mandel parameter: ${\langle {Q}_{M}\rangle }_{\mathrm{time}\mathrm{av}.}\propto (\lambda -{\lambda }_{0})$ with a suitable ${\lambda }_{0}$ that can be determined from the figure 7(d).

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