1 Introduction

The phenomenon of squeezing which is one of the signature of nonclassicality [1] was extensively studied in the past decades. Due to the attractive feature that the quantum fluctuations in one quadrature component of the field can be reduced below the standard quantum limit, the squeezed states of light provide potential applications including high-precision quantum measurements [2, 3], quantum communication [4], enhanced sensitivity in gravitational wave detectors [5], etc. In the quantum optical domain, squeezed light has been more commonly generated using nonlinear optical processes, including degenerate parametric amplification and degenerate four-wave mixing [6,7,8,9,10]. The authors of [11] showed that the squeezing of a single-mode quantized electromagnetic field could be achieved in the Jaynes–Cummings model [12] of a resonant two-level atom interacting with the field prepared initially in coherent state. Subsequently, it has been found that the significant amount of squeezing in the Jaynes–Cummings model can happen only when the mean photon number in the field is large enough [13].

It is worthwhile to mention that the amount of cavity field squeezing in the Jaynes–Cummings model can be enhanced even for low photon number by selective atomic measurements [14]. In addition, the time evolution of the squeezing in the Rabi model [15, 16] was realized numerically for a number of initial states [17, 18]. Besides, by adopting the initial state as a bipartite entangled state consisting of the coherent state in the oscillator subsystem, the squeezing was observed during its evolution [19, 20] in the Rabi model. The Rabi model reduces to the familiar Jaynes–Cummings model via the rotating-wave approximation which is solely applicable to the near resonance and weak coupling regime.

However, over recent decades the progress has been made towards the strong coupling regime of the radiation-matter interactions [21,22,23,24,25,26,27]. For example, by using circuit quantum electrodynamics the strong coupling of a single photon to a superconducting qubit has been studied experimentally [21], the realization of transmission spectra in a superconducting circuit QED system in ultra-strong-coupling regime [23], etc. Moreover, experimental observation of the Bloch–Siegert shift [24, 28] also assures the necessity of the counter rotating terms (CRT’s) in the description of the Jaynes–Cumming model. This reveals the importance of the CRT’s to comprehend the behaviour of full quantum Rabi model for all regimes of the coupling strengths [29,30,31,32,33,34,35,36,37,38,39,40,41]. In the recent works [42, 43], cavity mode squeezing in the strong coupling regime of the quantum Rabi model has been investigated. Thus, it naturally grows an interest to study the squeezing of the evolving qubit-oscillator state in the Rabi model with a parametric nonlinear term in the strong coupling domain.

To study the qubit-oscillator system under strong interaction where the Hamiltonian includes CRT’s, the authors of [44, 45] introduced an adiabatic approximation scheme that holds in the parameter domain where the oscillator frequency is much larger than the characteristics frequency of the qubit. Based on the separation of different time scales involved in the system, one can reduce the entire dynamics either to qubit or oscillator sector and evaluate the eigenstates of the system approximately [45]. To extend the parameter realm so that it includes both resonance as well as off-resonance, a generalization of the rotating wave approximation has been proposed [46]. This generalization exploits the basis states obtained in the adiabatic limit and the argument of excitation number conservation according to the rotating wave approximation is also applicable to the Hamiltonian in the new basis. The energy eigenvalues of the resultant block diagonalized Hamiltonian are now approximately valid for strong coupling strengths as well as a wide range of detunings [46].

Our objective in the present work is as follows. Within the framework of generalized rotating wave approximation, we study the squeezing phenomena in the Rabi model in presence of a parametric nonlinear term in the strong coupling regime. After approximate diagonalization of the system Hamiltonian, the time evolution of the initial state of the composite system is observed. By tracing over the qubit degree of freedom, we have the reduced density matrix corresponding to the oscillator subsystem. This reduced density matrix in turn yields the phase space quasi-probability distribution [47] such as Husimi Q-function. By exploiting the Q-function, we compute the quadrature variance by which the squeezing effect arising in this model is analysed. The work is organized as follows: In Sect. 2, the approximate diagonalization of the Hamiltonian is performed. In Sect. 3, the time evolution of the reduced density matrix corresponding to the oscillator degree of freedom and the Q-function is shown. In Sect. 4, the squeezing is studied by computing the quadrature variance. In Sect. 5, we illustrate the reconstruction of squeezed coherent states by analysing the near-null value of von Neumann entropy for the qubit degree of freedom via Hilbert–Schmidt distance measurement. Section 6 contains the summary of the work.

2 The approximate diagonalization of the Hamiltonian

The Rabi Hamiltonian [48,49,50,51] in the presence of a parametric nonlinear term [52,53,54,55] can be written as (\(\hbar =1\) herein)

$$\begin{aligned} H = \omega a^{\dag } a + \frac{\Delta }{2} \sigma _{z} + \lambda \sigma _{x} (a^{\dag } + a) + g ({a^{\dag }}^{2} + a^{2}). \end{aligned}$$
(1)

Here, the \((\sigma _{x}, \sigma _{z})\) are Pauli matrices for the qubit having a transition frequency \(\Delta \) and the single bosonic mode of frequency \(\omega \) is described by the annihilation and creation operators (\(a, a^{\dagger }| {\hat{n}} \equiv a^{\dagger } a\)). The coupling between the two subsystems is furnished through a term proportional to \(\lambda \) and the constant g corresponds to the strength of the parametric nonlinearity. The Fock states \(\{{\hat{n}} |n\rangle = n |n\rangle ,\,n = 0, 1,\ldots ;\;a \,|n\rangle = \sqrt{n}\,|n - 1\rangle , a^{\dagger }\, |n\rangle = \sqrt{n + 1}\,|n + 1\rangle \}\) provide the basis for the oscillator, whereas the eigenstates \(\sigma _x |\pm x\rangle = \pm \,|\pm x\rangle \) span the space of the qubit. The Hamiltonian (1) can be physically realizable in the atom–photon interacting system as reported in Refs. [56, 57]. To obtain the energy spectrum and eigenstates of the Rabi Hamiltonian, numerous approximation schemes have been advanced which are applicable to various ranges of parameters. For instance, to study the dynamical behaviour of the qubit-oscillator system we usually employ the well-known rotating wave approximation (RWA) [12] since it accurately describes the system in the regime where the oscillator and the qubit frequencies are nearly equal, and also for a weak qubit-oscillator coupling.

To explore the regimes outside the RWA, an adiabatic approximation scheme [44, 45] is introduced in the large detuning limit (\(\Delta \ll \omega \)). To overcome the limitations imposed by the adiabatic approximation which operates only in the large detuning regime, a new method has been proposed [46] known as the generalized rotating wave approximation that maintains a wide range of validity \((\lambda \sim O(\omega ), \Delta \lesssim \omega )\). We adopt the generalized rotating wave approximation to explicitly derive the approximate eigenenergies and eigenstates for the Hamiltonian (1) which will be conveniently employed in the study of nonclassical properties emerging from the dynamical evolution of the qubit-oscillator state in a bipartite system. To carry out the generalization of the standard rotating wave approximation, we begin with establishing a new set of basis in the adiabatic approximation which is exploited for representing the Hamiltonian (1) in the form of a direct sum of \(2 \times 2\) blocks along with an entry for uncoupled ground state.

In the adiabatic approximation, diagonalization of the Hamiltonian (1) is done by considering the qubit’s energy splitting \(\Delta \) smaller compared to the oscillator’s frequency \(\omega \), i.e. by allowing the initial energy eigenstate of the oscillator to adiabatically adjusts itself to any changes in the qubit’s state \(|\pm x\rangle \). Therefore, we neglect the qubit’s self-energy term in the Hamiltonian. This is achieved posing \(\Delta = 0 \) in (1) and allows to obtain the oscillator basis. Then, the Hamiltonian (1) is rewritten and truncated into a \(2 \times 2\) block-diagonal form in the aforementioned oscillator basis tensored with the qubit basis. This \(2 \times 2\) matrix block structure allows us to compute eigenenergies and eigenstates of the Hamiltonian which are known as adiabatic- energies and basis of our bipartite system. Hence, to start with the adiabatic approximation, the oscillator effective Hamiltonian deduced from the Hamiltonian (1) reads

$$\begin{aligned} H_{{\mathcal {O}}}=\omega a^{\dag }a \pm \lambda (a^{\dag } + a) + g ({a^{\dag }}^{2} + a^{2}). \end{aligned}$$
(2)

If \(g = 0\), the Hamiltonian \(H_{{\mathcal {O}}}\) is diagonalizable in the basis \(|{n_{\pm }}\rangle \) containing the degenerate eigenenergies \(E_{n} = \omega \big (n - \frac{\lambda ^{2}}{\omega ^{2}}\big )\), where the displaced number states read: \(|{n_{\pm }}\rangle = \mathrm {D}^{\dagger }\left( \pm \frac{\lambda }{\omega }\right) |{n}\rangle , \, \mathrm {D}\left( \alpha \right) = \exp \left( \alpha a^{\dagger }- \alpha ^{*}a \right) ,\, \alpha \in {\mathbb {C}}\). Within the adiabatic approximation, the composite state of the system consisting of displaced oscillator basis \(|{n_{\pm }}\rangle \) tensored with the qubit basis \(|{\pm x}\rangle \) are utilized to block-diagonalize the Rabi Hamiltonian which produces nondegenerate eigenspectrum. The overlap between the displaced number states [44] is given by

$$\begin{aligned} \langle {m_{-}|n_{+}}\rangle = {\left\{ \begin{array}{ll} (-1)^{m-n} \; \left( \frac{2\lambda }{\omega }\right) ^{m-n} \; \exp \big (-\frac{2\lambda ^{2}}{\omega ^{2}}\big ) \; \sqrt{n!/m!} \; L_n^{(m-n)}(\frac{4\lambda ^{2}}{\omega ^{2}}), &{} m \ge n \\ \left( \frac{2\lambda }{\omega }\right) ^{n-m} \; \exp \big (-\frac{2\lambda ^{2}}{\omega ^{2}}\big ) \; \sqrt{m!/n!} \; L_m^{(n-m)}(\frac{4\lambda ^{2}}{\omega ^{2}}) &{} m < n, \end{array}\right. } \end{aligned}$$
(3)

where the associated Laguerre polynomial reads \(L_{n}^{(j)}(x) = \sum _{k = 0}^{n} \,(-1)^{k}\, \left( {\begin{array}{c}n + j\\ n - k\end{array}}\right) \,\frac{x^{k}}{k!}\). The matrix element (3) leads to the identity: \(\langle {m_{-}|n_{+}}\rangle = (-1)^{n+m} \langle {n_{-}|m_{+}}\rangle \). In a similar way, for the case when nonlinear parametric term (\(g \ne 0\)) is present, we diagonalize the Hamiltonian \(H_{{\mathcal {O}}}\) with the aid of Bogoliubov transformation [58]. This corresponds to rewriting the Hamiltonian \(H_{{\mathcal {O}}}\) in terms of the new bosonic operators \(({\widetilde{a}},{\widetilde{a}}^{\dagger })\)

$$\begin{aligned} H_{{\mathcal {O}}}= \Omega \, {\widetilde{a}}^{\dagger }{\widetilde{a}} - \frac{1}{2}(\omega -\Omega ) - \frac{\lambda ^{2}}{\omega +2g}, \end{aligned}$$
(4)

where the operators \(({\widetilde{a}},{\widetilde{a}}^{\dagger })\) obeying the standard bosonic commutation relations are represented as

$$\begin{aligned} {\widetilde{a}} = \mathrm {S}^{\dag }(r) \mathrm {D}^{\dag }(\pm \eta ) a \mathrm {D}(\pm \eta ) \mathrm {S}(r), \; {\widetilde{a}}^{\dag } = \mathrm {S}^{\dag }(r) \mathrm {D}^{\dag }(\pm \eta ) a^{\dag } \mathrm {D}(\pm \eta ) \mathrm {S}(r). \end{aligned}$$
(5)

The squeezing operator given in (5) reads, \(\mathrm {S}(\xi )=\exp ((\xi {a^{\dag }}^{2}- \xi ^{*} a^{2})/2)\), \(\xi = r \exp (i \vartheta )\), \(\xi \in {\mathbb {C}}\), and it maintains the following unitary transformations:

$$\begin{aligned} \mathrm {S}^{\dag }(\xi ) a \mathrm {S}(\xi ) = \mu a + \nu a^{\dag }, \quad \mathrm {S}^{\dag }(\xi ) a^{\dag } \mathrm {S}(\xi ) = \mu a^{\dag } + \nu ^{*} a, \quad \mu = \cosh (r), \; \nu = \exp (i \vartheta ) \sinh (r), \end{aligned}$$
(6)

where we denote the abbreviations: \(\Omega = \sqrt{\omega ^{2}-4g^{2}}\), \(r=arc \cosh \left( \sqrt{\frac{\omega + \Omega }{2 \Omega }} \right) \) and \(\eta = \sqrt{\frac{\omega + \Omega }{2 \Omega }} \left( 1+ \frac{\omega - \Omega }{2g} \right) \frac{\lambda }{\omega + 2g} \). Now, the effective Hamiltonian \(H_{{\mathcal {O}}}\) (4) can be diagonalized in the following oscillator basis \(|{r,n_{\pm }}\rangle \)

$$\begin{aligned} H_{{\mathcal {O}}} |{r,n_{\pm }}\rangle {=} E_{n} |{r,n_{\pm }}\rangle , \quad E_{n}{=}(n{+}\frac{1}{2})\Omega {-} \frac{\omega }{2} {-} \frac{\lambda ^{2}}{\omega {+} 2g}, \quad |{r,n_{\pm }}\rangle {=} \mathrm {S}^{\dag }(r)\mathrm {D}^{\dag }(\pm \eta ) |{n}\rangle . \end{aligned}$$
(7)

Thereafter, by utilizing the oscillator basis (7) tensored with the qubit basis: \(|{\pm x ; r ,n_{\pm }}\rangle \equiv |{\pm x}\rangle \otimes |{ r ,n_{\pm }}\rangle \), the Hamiltonian (1) is truncated into \(2 \times 2\) blocks

$$\begin{aligned} \begin{pmatrix} E_{n} &{} \Delta _{n} \\ \Delta _{n} &{} E_{n} \end{pmatrix}, \; \Delta _{n} = \frac{\Delta }{2} \exp (-2 \eta ^{2}) L_{n}(4 \eta ^{2}), \; n\ge 0. \end{aligned}$$
(8)

From the above matrix representation (8), the adiabatic- energies and the basis are obtained:

$$\begin{aligned} E_{\pm ,n }=E_{n} \pm \Delta _{n}, \quad |{E_{\pm ,n}}\rangle = \frac{1}{\sqrt{2}}( |{x;r,n_{+}}\rangle \pm |{-x;r,n_{-}}\rangle ). \end{aligned}$$
(9)
Fig. 1
figure 1

Generalized rotating wave approximation (GRWA) energy levels (11) (dotted-dashed) are compared with numerically determined (solid) energies as a function of coupling strength \(\lambda /\omega \) a for the parameter values \(g = 0.35 \;\omega \) in the off-resonance \((\Delta = 0.5\, \omega )\) and b in the resonance \((\Delta =1.0 \, \omega )\) with \(g=0.2 \,\omega \). Our GRWA approach works well at far away from the resonance say \(\Delta \lesssim 0.5 \,\) in the parameters regime \( \lambda \lesssim 1.0 \; \omega \) and \(g \lesssim 0.35 \; \omega \), and also for higher value of \(g \lesssim 0.5 \; \omega \), the coupling strength \( \lambda \lesssim 0.5 \; \omega \). Similarly in the resonance, the admissible regime of the parameter \(g \lesssim 0.2 \; \omega \) which is evident from b. The spectral collapse is observed in the presence of parametric nonlinearity at far away from resonance c

Full size image

Furthermore, the adiabatic basis (9) is exploited towards approximate diagonalization of the Hamiltonian (1) via the generalized rotating wave approximation [46] and the resulting matrix elements truncated into \(2 \times 2\) blocks apart from the uncoupled ground state which can be written as

$$\begin{aligned} \begin{pmatrix} E_{+,n-1} &{} {\tilde{\Delta }}_{n} \\ {\tilde{\Delta }}_{n} &{} E_{-,n} \end{pmatrix}, \; {\tilde{\Delta }}_{n} = \frac{ \eta \Delta }{ \sqrt{n}} \exp \left( - 2\eta ^{2}\right) L^{(1)}_{n-1} \left( 4 \eta ^{2} \right) , \; n \ge 1. \end{aligned}$$
(10)

The uncoupled ground state energy and solutions for the simple block-diagonal form (10) of the doublets explicitly read

(11)

In addition, we observe that the energy spectrum in (7) and (11) can produce a spectral collapse [59, 60] in presence of two-photon term in the Hamiltonian (1) that does not happen in the standard quantum Rabi model. This spectral collapse can happen far away from the resonance when the parametric nonlinear term (g) approaches the value \(0.5\; \omega \) (Fig. 1c). The collapse of energy spectrum could be understood in terms of phase space variables. Defining the position and momentum operators of the oscillator mode as

$$\begin{aligned} x = \frac{1}{\sqrt{2 \omega }} (a + a^{\dagger }), \quad p = \frac{1}{i} \sqrt{\frac{\omega }{2}} (a - a^{\dagger }), \end{aligned}$$
(12)

respectively, we can re-write the Hamiltonian (2) as

$$\begin{aligned} H_{{\mathcal {O}}} = \frac{p^{2}}{2{\widetilde{m}}} + \frac{1}{2} {\widetilde{m}} \Omega ^{2} \Big ( x \pm \sqrt{\frac{2}{\omega }} \frac{\lambda }{(2g + \omega )} \Big )^{2} - \frac{\lambda ^{2}}{(2g+\omega )} - \frac{\omega }{2}, \end{aligned}$$
(13)

where \({\widetilde{m}} = \big (1 - \frac{2g}{\omega }\big )^{-1}\) and \(\Omega = \sqrt{\omega ^{2} - 4g^{2}}\). Apparently, for \(\frac{2g}{\omega } < 1\), the Hamiltonian (13) corresponds to a harmonic oscillator which is shifted in both position and energy, possesses a discrete spectrum. In the limit \(\frac{2g}{\omega } \rightarrow 1\), the oscillator approaches a free particle. Consequently, the spectrum starts to collapse and becomes continuous with Dirac-\(\delta \) normalizable eigenfunctions. For \(\frac{2g}{\omega } > 1 \), we have an inverted oscillator where the spectrum remains continuous and the eigenfunctions are Dirac-\(\delta \) normalizable [61].

The corresponding eigenstates from (10) are given by

$$\begin{aligned} |{{\mathcal {E}}}_0 \rangle\equiv & {} |{{E}_{-,0}}\rangle = \dfrac{1}{\sqrt{2}} \Big ( |{x;r,0_{+}}\rangle - |{-x;r,0_{-}}\rangle \Big ),\nonumber \\ |{{\mathcal {E}}}_{\pm ,n (\ge 1)} \rangle= & {} \zeta _{\pm ,n} | E_{+,{n-1}} \rangle \pm \dfrac{{\tilde{\Delta }}_{n}}{|{\tilde{\Delta }}_{n}|} \zeta _{\mp ,n} |E_{-,n} \rangle , \quad \zeta _{\pm ,n} = \sqrt{\frac{\chi _n \pm \varepsilon _{n}}{2\chi _n}}, \end{aligned}$$
(14)

here we abbreviate: \(\chi _n=\sqrt{{{\tilde{\Delta }}_{n}}^2+\varepsilon _{n}^2},\,\varepsilon _{n} =\frac{E_{+,n-1}-E_{-,n}}{2}\). The completeness relation of the orthonormal bipartite basis (14) now reads:

$$\begin{aligned} |{{\mathcal {E}}}_{0} \rangle \langle {{\mathcal {E}}}_{0}| + \sum _{n = 1}^{\infty } \left( |{{\mathcal {E}}}_{+,n} \rangle \langle {{\mathcal {E}}}_{+,n}| + |{{\mathcal {E}}}_{-,n} \rangle \langle {{\mathcal {E}}}_{-,n}|\right) = \sum _{n = 0}^{\infty } \left( | E_{+,n} \rangle \langle E_{+,n}| + |E_{-,n} \rangle \langle E_{-,n}|\right) = {\mathbb {I}}. \end{aligned}$$
(15)

3 Time evolution of the oscillator’s reduced density matrix and the Husimi Q-function

Upon completion of the above construction of the energy eigenstates (14) via the generalized rotating wave approximation, we further proceed to explore the role of parameter g on the nonclassical features of the oscillator degree of freedom, in particular, the Husimi Q-function and the squeezing through the dynamics of the bipartite system. The initial state of the qubit-oscillator system reads: \(|{\psi (0)}\rangle = |{-x}\rangle \otimes |{0}\rangle \), where \(|{0}\rangle \) is the vacuum state of the oscillator. The time evolution of the initial state is

$$\begin{aligned}&|{\psi (t)}\rangle = {\mathcal {C}}_{0} (t) |{{\mathcal {E}}_{0}}\rangle + \sum _{n=1}^{\infty } {\mathcal {C}}_{\pm ,n} (t) |{{\mathcal {E}}_{\pm ,n}}\rangle , \quad {\mathcal {C}}_{0}(t) = {\mathcal {C}}_{0} \exp (-i {\mathcal {E}}_{0} t),\nonumber \\&\quad {\mathcal {C}}_{\jmath , n}(t) = {\mathcal {C}}_{\jmath , n} \exp (-i {\mathcal {E}}_{\jmath } t), \quad \jmath \in \{\pm \}, \end{aligned}$$
(16)

where the coefficients read:

$$\begin{aligned} {\mathcal {C}}_{0}= & {} -\frac{1}{\sqrt{2 \mu }} \exp \left( -\frac{\eta ^{2}}{2} + \frac{\nu \eta ^{2}}{2 \mu } \right) , \; \frac{\nu }{\mu } = \frac{\omega - \Omega }{2g}, \nonumber \\ {\mathcal {C}}_{\pm ,n}= & {} -{\mathcal {C}}_{0} \left( \frac{-\nu }{2 \mu }\right) ^{ \frac{n}{2}} \left( \frac{\zeta _{\pm ,n}}{\sqrt{(n-1)!}} \left( \frac{-\nu }{2 \mu }\right) ^{ -\frac{1}{2}} \mathrm {H}_{n-1}\left( \frac{i(\mu -\nu )\eta }{\sqrt{2 \mu \nu }} \right) \mp \dfrac{{\tilde{\Delta }}_{n}}{|{\tilde{\Delta }}_{n}|} \frac{\zeta _{\mp ,n}}{\sqrt{n!}} \; \mathrm {H}_{n}\left( \frac{i(\mu -\nu )\eta }{\sqrt{2 \mu \nu }} \right) \right) ,\nonumber \\ \end{aligned}$$
(17)

here the Hermite polynomials are given by the exponential generating function [62]: \(\exp (2 \,{\mathsf {x}} {\mathsf {t}}-{\mathsf {t}}^{2}) = \sum _{n=0}^{\infty } \frac{\mathrm {H}({\mathsf {x}}) {\mathsf {t}}^{n}}{n!}\). To facilitate the construction of the time evolution of the initial state (16), we provide the following expansion of squeezed coherent state in the number state basis [63] together with the property below:

$$\begin{aligned}&\mathrm {S}(\xi ) \mathrm {D}(\alpha ) |{0}\rangle = \exp \left( -\frac{|\alpha |^{2}}{2} - \frac{\alpha ^{2} \nu ^{*}}{2 \mu }\right) \sum _{n=0}^{\infty } \frac{i^{n}}{\sqrt{n! \mu }} \left( \frac{\nu }{2 \mu }\right) ^{\frac{n}{2}} \mathrm {H}_{n}\left( \frac{-i\alpha }{\sqrt{2 \mu \nu }} \right) |{n}\rangle , \,\nonumber \\&\mathrm {D}(\alpha ) \mathrm {S}(\xi ) = \mathrm {S}(\xi ) \mathrm {D}(\alpha \mu - \alpha ^{*} \nu ). \end{aligned}$$
(18)

The above expressions are utilized to compute the coefficients of \(|{\psi (t)}\rangle \) given in (17). The normalization of the state \(|{\psi (t)}\rangle \): \(\langle {\psi (t)|\psi (t)}\rangle \equiv |{\mathcal {C}}_{0}(t)|^{2}+\sum _{n=1}^{\infty } |{\mathcal {C}}_{\pm ,n}(t)|^{2}=1\) can be shown by exploiting the following identity [64]

$$\begin{aligned} \sum _{n=0}^{\infty }\dfrac{{\mathsf {t}}^{n}}{2^{n}n!} \mathrm {H}_{n}({\mathsf {x}}) \mathrm {H}_{n}({\mathsf {y}}) = \dfrac{1}{\sqrt{1-{\mathsf {t}}^{2}}}\, \exp \left( -\dfrac{(\mathsf {tx})^{2}-2\mathsf {txy}+ (\mathsf {ty})^{2}}{ 1-{\mathsf {t}}^{2}} \right) . \end{aligned}$$
(19)

Therefore, the time evolution of the density matrix of the bipartite pure state can be represented as

$$\begin{aligned} \rho (t) \equiv |{\psi (t)}\rangle \langle {\psi (t)}|. \end{aligned}$$
(20)

The reduced density matrix for the oscillator is obtained by partial tracing over the qubit-Hilbert space, i.e.\(\rho _{{\mathcal {O}}} \equiv \mathrm {Tr}_{{\mathcal {Q}}} \rho \). Its explicit construction reads:

$$\begin{aligned} \rho _{{\mathcal {O}}}(t)= & {} |{\mathcal {C}}_{0}(t)|^{2} P_{0,0}^{(+)} + \sum _{n=1}^{\infty } \Big ( {\mathcal {C}}_{0}(t) {{\mathcal {A}}_{n}(t)}^{*} P_{0,n-1}^{(-)} + {{\mathcal {C}}_{0}(t)}^{*} {\mathcal {A}}_{n}(t) P_{n-1,0}^{(-)} + {\mathcal {C}}_{0}(t) {{\mathcal {B}}_{n}(t)}^{*} P_{0,n}^{(+)} \; \; \; \nonumber \\&+\, {{\mathcal {C}}_{0}(t)}^{*} {\mathcal {B}}_{n}(t) P_{n,0}^{(+)} \Big ) \,\, + \sum _{n,m=1}^{\infty } \Big ( {\mathcal {A}}_{n}(t) {{\mathcal {A}}_{m}(t)}^{*} P_{n-1,m-1}^{(+)} + {\mathcal {B}}_{n}(t) {{\mathcal {B}}_{m}(t)}^{*} P_{n,m}^{(+)} \nonumber \\&+\, {\mathcal {B}}_{n}(t) {{\mathcal {A}}_{m}(t)}^{*} P_{n,m-1}^{(-)} + {\mathcal {A}}_{n}(t) {{\mathcal {B}}_{m}(t)}^{*} P_{n-1,m}^{(-)} \Big ), \nonumber \\ {\mathcal {A}}_{n}(t)= & {} \zeta _{+,n} \, {\mathcal {C}}_{+,n}(t) + \zeta _{-,n} \, {\mathcal {C}}_{-,n}(t),\quad {\mathcal {B}}_{n}(t) = \frac{{\tilde{\Delta }}_{n}}{|{\tilde{\Delta }}_{n}|} \left( \zeta _{-,n} {\mathcal {C}}_{+,n}(t) - \zeta _{+,n} {\mathcal {C}}_{-,n}(t) \right) , \end{aligned}$$
(21)

where the projection operators read \(P_{n,m}^{(\pm )}= \frac{1}{2} \left( |{r,n_{+}}\rangle \langle {r,m_{+}}| \pm |{r,n_{-}}\rangle \langle {r,m_{-}}| \right) ,\,(n, m = 0, 1, \ldots )\). The density matrix (21) obeys the normalization condition: \(\mathrm {Tr}\,\rho _{{\mathcal {O}}}(t) = 1\).

The Husimi Q-function [47] 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-probability distributions. Being easily computable it has been extensively used [65, 66] in the study of the occupation on the phase space. For our reduced density matrix of the oscillator (21), the corresponding Q-function reads

$$\begin{aligned} Q(\beta ,\beta ^{*}) = \frac{1}{\pi } \langle {\beta }|\rho _\mathcal{O}|{\beta }\rangle , \; |{\beta }\rangle = \mathrm {D}(\beta ) |{0}\rangle , \; \beta \in {\mathbb {C}}. \end{aligned}$$
(22)

Our construction of the oscillator density matrix (21) now yields the time-evolution of the Q-function:

(23)

Here, the weight functions on the phase space read

(24)

where \(\beta _{\pm }=\beta \pm \eta (\mu - \nu )\). To arrive at the expression (23), we make use of the following inner products

$$\begin{aligned}&\langle {\beta |\xi ,n_{\pm }}\rangle = \frac{1}{\sqrt{\mu n!}} \left( -i \sqrt{\frac{\nu ^{*}}{2\mu }} \right) ^{n} \exp \left( - \frac{|\alpha |^{2}}{2} + \frac{\alpha ^{2}\nu ^{*}}{2\mu } - \frac{|\beta |^{2}}{2} - \frac{\beta ^{*2}\nu }{2\mu } \mp \frac{\alpha \beta ^{*}}{\mu }\right) \nonumber \\&\qquad \times \,\, \mathrm {H}_{n}\left( \frac{i (\pm \mu \alpha ^{*}\mp \alpha \nu ^{*}+ \beta ^{*})}{\sqrt{2 \mu \nu ^{*}}}\right) , \end{aligned}$$
(25)

with \(\langle {\beta |\xi ,n_{\pm }}\rangle \equiv \langle {0|D^{\dagger }(\beta ) S^{\dagger }(\xi ) D^{\dagger }(\pm \alpha )|n}\rangle \), and these inner products can be calculated by utilizing the expressions (18). The expression (23) can be shown to satisfy the normalization criteria, i.e. \(\int Q(\beta , \beta ^{*}) \mathrm {d}^{2}\beta = 1\) by employing the following integrals

$$\begin{aligned}&\int \mathrm {d}^{2}\beta \exp \left( -|\beta |^{2}-\frac{\nu }{2\mu }(\beta ^{2}+\beta ^{*2}) \mp \frac{\eta }{\mu } (\beta + \beta ^{*}) \right) \mathrm {H}_{n}\left( i \frac{\beta _{\pm }^{*}}{\sqrt{2 \mu \nu }}\right) \mathrm {H}_{m}\left( -i \frac{\beta _{\pm }}{\sqrt{2 \mu \nu }}\right) \nonumber \\&\quad = \pi \, \mu \, n! \left( \frac{2 \mu }{\nu }\right) ^{n} \exp \left( \frac{\eta ^{2}}{\mu } (\mu - \nu )\right) \, \delta _{n,m} \, , \end{aligned}$$
(26)

and it also maintains the bounds: \(0 \le Q(\beta ,\beta ^{*}) \le \frac{1}{\pi }\).

Another dynamical quantity that is useful in the study of squeezing is the polar phase density of the Husimi Q-function [67] obtained via its radial integration on the phase space:

$$\begin{aligned} \mathcal {Q(\theta )}= & {} \int \limits _{0}^{\infty } Q(\beta , \beta ^{*})\, |\beta |\, d|\beta |, \quad \beta \,= \, \vert \beta |\, \exp (i\theta ), \end{aligned}$$
(27)

which is a convenient tool for describing the splitting of the Q-function.

(28)
(29)

where \(\tau _{_\theta }= \mu + \nu \cos 2\theta \). To arrive at (28), we make use of the integral shown in “Appendix A”.

4 The quadrature squeezing

The quadrature operator is defined as \(X_{\phi }= \frac{1}{\sqrt{2}} (a \exp (-i\phi ) + a^{\dagger } \exp (i\phi ))\) where \(\phi \) is a real phase [68]. The squeezing effect is characterized by the variance

$$\begin{aligned} V_{\phi } = \langle {X^{2}_{\phi }}\rangle -\langle {X_{\phi }}\rangle ^{2}= \mathrm {Re} \left( (\langle {a^{2}}\rangle -\langle {a}\rangle ^{2}) \exp (-2i\phi ) \right) + \langle {a^{\dagger } a}\rangle - \; |\langle {a}\rangle |^{2} + \frac{1}{2}. \end{aligned}$$
(30)

Likewise, the so-called principal-quadrature squeezing [69,70,71,72] is characterized by

$$\begin{aligned} V_{\text {min}}= \min \limits _{\phi \in (0,2\pi )} (\langle {X^{2}_{\phi }}\rangle -\langle {X_{\phi }}\rangle ^{2}) = \frac{1}{2} + \; \langle {a^{\dagger } a}\rangle - \; |\langle {a}\rangle |^{2} - \; |\langle {a^{2}}\rangle -\langle {a}\rangle ^{2}|, \end{aligned}$$
(31)

which is the minimum value of \(V_{\phi }\) with respect to \(\phi \). For the vacuum state as well as coherent states the variances in (30) and (31) are equal to 0.5 which is called as the classical limit of the variance. The state of the field is said to be squeezed [1] if the corresponding variance is lesser than 0.5. The expectation values of the operators in the above variance can be conveniently computed via the Q-function through the following representation

$$\begin{aligned} \langle {a^{k}}\rangle= & {} \int \mathrm {d}^{2}\beta \; \beta ^{k} \; Q(\beta ,\beta ^{*}), \; \langle {a^{\dagger }a}\rangle = \langle {aa^{\dagger }}\rangle -1 = \int \mathrm {d}^{2}\beta \; |\beta |^{2} \; Q(\beta ,\beta ^{*})-1, \end{aligned}$$
(32)
$$\begin{aligned} \langle {a^{k}}\rangle \!\!= & {} \!\! \dfrac{1}{2} |{\mathcal {C}}_{0}(t)|^{2} G^{(k,+)}_{0,0} + \frac{1}{2} \left( {\mathcal {C}}_{0}(t)^{*} \sum _{n=1}^{\infty } \Big ( {\mathcal {A}}_{n}(t) G^{(k,-)}_{0,n-1} + {\mathcal {B}}_{n}(t) G^{(k,+)}_{0,n} \Big )\right. \nonumber \\&\left. + \,\,{\mathcal {C}}_{0}(t) \sum _{n=1}^{\infty } \Big ( {\mathcal {A}}_{n}(t)^{*} G^{(k,-)}_{n-1,0} + {\mathcal {B}}_{n}(t)^{*} G^{(k,+)}_{n,0} \Big )\right. \nonumber \\&\left. + \sum _{n,m=1}^{\infty } \! \Big ( {\mathcal {A}}_{n}(t)^{*} {\mathcal {B}}_{m}(t) G^{(k,-)}_{n-1,m} + {\mathcal {A}}_{n}(t) {\mathcal {B}}_{m}(t)^{*} G^{(k,-)}_{m,n-1}\right. \nonumber \\&\left. + \,\, {\mathcal {A}}_{n}(t)^{*} {\mathcal {A}}_{m}(t) G^{(k,+)}_{n-1,m-1} +{\mathcal {B}}_{n}(t)^{*} {\mathcal {B}}_{m}(t) G^{(k,+)}_{n,m} \Big ) \right) , \end{aligned}$$
(33)

where the weight functions on the phase space can be expressed by using the integral shown in “Appendix B” as

$$\begin{aligned} G^{(k,\pm )}_{n,m}= & {} \sqrt{n!m!} \left( \frac{\nu }{2 \mu }\right) ^{\!\! \frac{n+m}{2}} \sum _{\ell =0}^{k} ((-1)^{k-\ell }\mp 1) \left( {\begin{array}{c}k\\ \ell \end{array}}\right) (\eta (\mu - \nu ))^{k-\ell } \left( \frac{\mu \nu }{2} \right) ^{\ell } \sum _{p=0}^{n} \sum _{q=0}^{m} \frac{1}{p!q!} \left( {\begin{array}{c}p\\ n-p\end{array}}\right) \left( {\begin{array}{c}q\\ m-q\end{array}}\right) \qquad \nonumber \\&\times \!\!\!\! \sum _{s=0}^{\min (2q+\ell -m,2p-n)} \left( \frac{2 \mu }{\nu } \right) ^{s} s! \left( {\begin{array}{c}2q+\ell -m\\ s\end{array}}\right) \left( {\begin{array}{c}2p-n\\ s\end{array}}\right) \mathrm {H}_{2q+\ell -m-s}(0) \mathrm {H}_{2p-n-s}(0). \end{aligned}$$
(34)

Furthermore,

(35)

where

(36)

The explicit form of \(I_{n,m}^{(k,\ell )} \) is shown in “Appendix B”.

The enhancement of squeezing in the field mode is realized during the time evolution of the initial state of qubit-oscillator system in the presence of a parametric nonlinear term. In the strong coupling regime, the squeezing is noticed both at far away from resonance \((\Delta =0.3\,\omega )\) as well as at resonance \((\Delta =1.0\,\omega )\) (Fig. 2). The signature of the squeezing is observed when the variance \(V_{\phi }\) of the quadrature variable, say at \(\phi =0\) is rendered less than its classical value 1/2. It is noticed that in the absence of parametric nonlinear term \((g=0)\), the least value of the variance \(V_{\phi }\) reaches 0.4741 at the scaled time \(\omega t=220 \) for the coupling constant \(\lambda =0.1\,\omega \) (Fig. 2a ). The \({\mathcal {Q}}(\theta )\) and the polar plot of the variance \(V_{\phi }\) (Fig. 2b, c, respectively) represent this quadrature squeezing more prominently. In the presence of the parametric term, by comparing Fig. 2a, e it is apparent that the enhancement of squeezing is happening for the parameter \(g=0.35\,\omega \) with the identical coupling strength. It is important to note that the minimum value of the \(V_{\phi }\) in this case is 0.0954 at \(\omega t=264\). This enhancement also reflects within the \({\mathcal {Q}}(\theta )\) and polar plot of the \(V_{\phi }\) (Fig. 2f, g, respectively) at the same scaled time.

Fig. 2
figure 2

a The time evolution of the quadrature variance \(V_{\phi =0}\) (30) for the coupling constant \(\lambda =\, 0.1 \,\omega \) at far away from the resonance \((\Delta = 0.3 \;\omega )\) in the absence of parametric nonlinear term \((g = 0)\). The horizontal red line represents the classical limit of the variance \(V_{\phi } = 0.5\). The red circles in b, f, j indicate the polar phase density of the Q-function (28) for the vacuum state \(\rho _{{\mathcal {O}}} = |{0}\rangle \langle {0}|\), i.e. \({\mathcal {Q}}(\theta ) = \frac{1}{2\pi }\), whereas the red circles in c, g, k describe the classical limit of the \(V_{\phi } = 0.5\) at the scaled time \(\omega t = 0\). The \({\mathcal {Q}}(\theta )\) in \(({\mathsf {b}})\) (blue) and the polar plot of \(V_{\phi }\) in c (blue) are denoted at \(\omega t =220\). The least value of the variance is observed at \(\omega t=220\) equals 0.4741 a. The plot d indicates that the squeezing is reduced with the increase of the coupling strength when \(g =0\). The plot e illustrates the same as a in presence of the parametric nonlinear term \((g = 0.35 \, \omega )\). In this case, the minimum value of \(V_{\phi }\) equals 0.0954 which occurs at \(\omega t = 264\). The f and g describe the same as b and c at \(\omega t=264\) (blue), respectively, for nonzero value of the parametric nonlinearity. h The squeezing diminishes with the raising of the coupling strength even in presence of the parametric term. The plot i shows the time evolution of the \(V_{\phi }\) at the resonance \((\Delta = 1.0 \, \omega )\) for the coupling constant \(\lambda = 0.15\, \omega \) and the parameter \(g= 0.2\,\omega \). The \({\mathcal {Q}}(\theta )\) j and polar plot of the \(V_{\phi }\) k (blue) reveal the squeezing effect evidently also in the case of resonance at \(\omega t=254\)

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Fig. 3
figure 3

The time average of minimum variance \(V_{\text {min}}\) (31) is plotted with respect to coupling constant \(\lambda \) for various values of parametric nonlinear term (g) at a far away from resonance as well as b at resonance. The horizontal red line indicates the classical limit of \(V_{\text {min}}\)

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The physical reason for the occurrence of the revival–collapse phenomenon is due to the periodic exchange of energy between the qubit and oscillator mode. The dynamics of revival and collapse in the variance can be understood from the behaviour of mean photon number \((\langle a^{\dagger } a \rangle )\). It is observed that for \(\eta ^{2}\ll 1\), the mean photon number exhibits periodic revival and collapse. When \(\eta ^{2} \) increases, lots of incommensurate frequencies start to participate, and the dynamics become irregular. This makes the energy exchange between the qubit and oscillator no longer periodic.

We estimate the revival time under the condition \(\eta ^{2}\ll 1\). Firstly, the energy levels in (11) are approximated by retaining the terms up to the \(O(\eta ^{2})\) in the Laguerre polynomials. After some algebra, these are explicitly written as

$$\begin{aligned}&{\mathcal {E}}_{\pm ,n (\ge 1)} \approx \Big (n \Omega -\frac{\omega }{2} -\frac{\lambda ^{2}}{\omega +2g}\Big ) + \Delta \eta ^{2} \exp {(-2\eta ^{2})} \pm \Bigg ( \frac{\Omega }{2} -\frac{\Delta \exp {(-2\eta ^{2})}}{2} (1+2(1-2 n)\eta ^{2})\Bigg ).\nonumber \\ \end{aligned}$$
(37)

It is convenient to utilize the expression of qubit inversion \((\langle {\sigma _{z}}\rangle )\) in the study of Rabi oscillations for estimating the time period of revival. The time-dependent phase factors present in the qubit inversion are now approximated by Eq. (37). The revival time can be determined from the dominant frequency arising from the phase factors that are proportional to \(\eta ^{2}\) and linear in n. Following this, the order of time period of revival observed in the variance can be estimated as \(O\left( 2\pi /( \Delta (2\eta )^{2}e^{-2\eta ^{2}})\right) \). The successive revival times given up to a proportionality constant are \(t_{R} \sim 2\pi k/( \Delta (2\eta )^{2}e^{-2\eta ^{2}}), \; k = 0, \; 1, \; 2, ...\) For instance, in Fig. 2e, the proportionality constant for the estimated revival times at \(( t_{1}, t_{2}, t_{3})\) reads (0.392, 0.397, 0.396), respectively. The discrepancy in the revival times is observed to be \( \sim 2\%\).

The squeezing in this qubit-oscillator model can be understood by suitably transforming the Rabi Hamiltonian under a unitary operation which allows the construction of the effective Hamiltonian [73] in the dispersive limit, i.e. \(\lambda \ll \vert \Delta -\omega |\). The resulting effective Hamiltonian contains the two-photon terms \((a^{2}, {a^\dagger }^{2})\) that are responsible for the squeezing [42]. However, the squeezing generated in the system is inadequate which is evident from Fig. 3 illustrated by the black line. Hence, the enhancement of squeezing in the qubit-oscillator system can be achieved in the presence of a parametric nonlinear term which is evident from Fig. 2e and Fig. 3 (green and blue line). It is also noticed that the increase in the coupling strength in the strong coupling regime reduces the squeezing in the absence of nonlinear term as shown in Fig. 2d. This decrement in the squeezing happens due to the participation of other multiple photon terms in the effective Hamiltonian when we further increase the coupling strength. This higher-order multi-photon terms become more significant if compared to the two-photon terms and induce randomness in the phase relationship which considerably decreases the squeezing of the system.

The explicit presence of the parametric term in the system facilitates to overcome the aforesaid limitation, and it can be understood as follows. The strength of the two-photon term in the Hamiltonian [73] can be increased through the parameter g, and in this process, the two-photon terms are allowed to dominate the other multi-photon processes. This leads to more squeezing generated in the system which is evident from Fig. 2e. This argument is also applicable in the case of resonance (Fig. 2i).

5 The von Neumann entropy and reconstruction of the state

The reduced density matrix for the qubit is obtained by partial tracing over the oscillator-Hilbert space of the bipartite system as

$$\begin{aligned} \rho _\mathcal{Q} \equiv \hbox {Tr}_\mathcal{{O}} \, \rho = \begin{pmatrix} \varrho _{_+} &{}\quad \varrho _{_0} \\ \varrho _{_0}^{*} &{}\quad \varrho _{_-} \end{pmatrix}, \end{aligned}$$
(38)

where the matrix elements read

(39)

The reduced density matrix (38) satisfies the trace condition: \(\hbox {Tr} (\rho _\mathcal{{Q}}) = 1\). The qubit inversion is defined as \(\langle {\sigma _{z}}\rangle = \mathrm {Tr } (\sigma _{z} \rho _{{\mathcal {Q}}}) = \varrho _{_{+}} - \varrho _{_{-}}\). The pair of eigenvalues of the qubit density matrix (38) read: \(\frac{1}{2}\pm \Lambda \) where \( \Lambda = \frac{1}{2}\sqrt{1- 4(\varrho _{_+} \varrho _{_-} - \vert \varrho _{_0} |^{2})}\). The eigenvalues allow us to compute its von Neumann entropy \(S(\rho _\mathcal{{Q}}) \equiv - \hbox {Tr} (\rho _\mathcal{{Q}} \ln \rho _\mathcal{{Q}})\) as

$$\begin{aligned} S(\rho _\mathcal{{Q}}) = -\bigg (\frac{1}{2} + \Lambda \bigg ) \ln \bigg (\frac{1}{2}+ \Lambda \bigg ) -\bigg (\frac{1}{2} - \Lambda \bigg ) \ln \bigg (\frac{1}{2}- \Lambda \bigg ). \end{aligned}$$
(40)
Fig. 4
figure 4

a The von Neumann entropy \(S (\rho _\mathcal{{Q}})\)(40) with respect to time is shown. The minimum value of \(S (\rho _\mathcal{{Q}})\) is found to be 0.0008 which occurs at the scaled time \(\omega t = 1320\) indicated by the red-dashed vertical line in our chosen parameters regime. In the inset, it is described that with higher values of g, \(S (\rho _\mathcal{{Q}})\) becomes higher. In b, the Husimi Q-function is plotted on phase space for the scaled time \(\omega t = 1320\). c The time average of \(S (\rho _\mathcal{{Q}})\) is presented with respect to the parametric nonlinear term (g) for different values of coupling strength \(\lambda \). The horizontal line (blue) corresponds to von Neumann entropy for the maximally mixed state, i.e. \(\ln 2\)

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As our bipartite system is in a pure state, the entropies of individual subsystem are equal [74], i.e. \(S(\rho _\mathcal{{O}}) = S(\rho _\mathcal{{Q}})\). The entropy associated with the oscillator degree of freedom is therefore identical to that of the qubit sector. A decrease of \( S(\rho _\mathcal{{Q}})\) means that an individual subsystem of the bipartite system evolves towards a pure state, whereas a rise in \( S(\rho _\mathcal{{Q}})\) indicates that the two subsystems tend to be correlated or entangled. We investigate the role of parametric nonlinear term (g) to generate squeezed coherent states which we will illustrate with an example. Pure state in this system can be achieved by choosing the lower value of parametric nonlinearity because with higher values of g the entropy also becomes higher which signifies mixed state (Fig. 4a). By numerically minimizing the Hilbert–Schmidt distance, we reconstruct the state where the entropy approaches a near-null value (Fig. 4b. The Hilbert–Schmidt distance between the states characterized by the density matrices \(\rho _1\) and \(\rho _2\) is defined as [75]

$$\begin{aligned} d_{\text {HS}}(\rho _{1}, \rho _{2}) \equiv \sqrt{\hbox {Tr} (\rho _{1} -\rho _{2})^{2}}. \end{aligned}$$
(41)

By observing the values of entropy \(S(\rho _\mathcal{{Q}})\) and minimum variance \(V_{\text {min}}\) which are 0.0008 and 0.36, respectively, at the scaled time \(\omega t = 1320 \), we have made an educated guess that the fiducial marker state to be

$$\begin{aligned} \rho ' = |{\xi ',\alpha '}\rangle \langle {\xi ',\alpha '}|, \quad |{\xi ',\alpha '}\rangle = S(\xi ')D(\alpha ') |{0}\rangle , \quad \xi '= r' \exp (i \vartheta '),\; \xi ', \alpha ' \in {\mathbb {C}}. \end{aligned}$$
(42)

In our case, we assume \(\rho _{1}= \rho '\) in (42) and \(\rho _{2} = \rho _{{\mathcal {O}}} (t)\) in (21). As the marker state is a pure state, i.e. \(\hbox {Tr} (\rho '^{2}) = 1\), (41) can be written as

$$\begin{aligned} d_{\text {HS}}(\rho ', \rho _{{\mathcal {O}}}(t))= & {} \sqrt{ 1 + \hbox {Tr} (\rho _{{\mathcal {O}}}^2 (t)) - 2 \hbox {Tr} (\rho '\rho _{{\mathcal {O}}} (t))}. \end{aligned}$$
(43)

Now,

$$\begin{aligned}&\!\!\!\mathrm {Tr}(\rho _{{\mathcal {O}}}(t)^{2}) = \sum _{k,\ell =0}^{\infty } (\rho _{{\mathcal {O}}}(t))_{k,\ell }(\rho _{{\mathcal {O}}}(t))_{\ell ,k}\;, \; \mathrm {Tr}(\rho '\rho _{{\mathcal {O}}}(t) )= \sum _{k,\ell =0}^{\infty } \langle {k|\xi ',\alpha '}\rangle \langle {\xi ',\alpha '|\ell }\rangle (\rho _{{\mathcal {O}}}(t))_{\ell ,k} ,\nonumber \\ \end{aligned}$$
(44)

where \( \rho _{m,n} \equiv \langle {m|\rho |n}\rangle \). To evaluate the trace operations given in (44), we make use of the following inner product [76]

$$\begin{aligned}&\langle {m}| S(\xi )D(\alpha )|{n}\rangle \nonumber \\&\quad = \sqrt{\frac{m!}{n! \mu }} i^{m} \bigg (\frac{\nu }{2\mu }\bigg )^{\frac{m}{2}} \exp {\bigg (-\frac{|\alpha |^2}{2} -\frac{\alpha ^2 \nu ^{*}}{2 \mu }}\bigg ) \sum \limits _{k=0}^{\text {min} (m,n)} \left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{(-i)^k}{(m-k)!} \nonumber \\&\qquad \times \,\, \bigg ( \frac{2}{\mu \nu }\bigg )^{\frac{k}{2}} \bigg (\frac{\nu ^{*}}{2 \mu }\bigg )^{\frac{n-k}{2}} \; \mathrm {H}_{\text {m-k}} \bigg (\frac{-i \alpha }{\sqrt{2 \mu \nu }}\bigg ) \mathrm {H}_{\text {n-k}} \bigg (\frac{-\big ( \mu \alpha ^{*} + \nu ^{*} \alpha \big )}{\sqrt{2 \mu \nu ^{*}}}\bigg ). \end{aligned}$$
(45)

In the procedure of numerical minimization, we replace the converging infinite summations in (44) by an appropriate number of finite (\(k,\ell \)) sums. This assures the normalization of the density matrix. For our chosen parameters, we have taken the finite (\(k,\ell \)) sums up to 20 which allows the series to converge with 7-digit accuracy. The accuracy of our results is verified by including the succeeding terms in the series. This minimization procedure involves varying the parameters \(r', \vartheta '\) and \(\alpha '\) defining the marker state, and their appropriate values are found to be 0.17, 2.61 and 0.07i, respectively, and the corresponding Hilbert–Schmidt distance reads: \(d_{\text {HS}}|_{\text {min}} \sim 0.06\). The nonvanishing nature of the Hilbert–Schmidt distance for our marker state is due to the near-null entropy in our chosen parametric regime.

It is also noted that the time-averaged entropy \( \langle S(\rho _{{\mathcal {Q}}})\rangle \) reaches the maximally mixed state when the parametric nonlinearity approaches the value \( 0.5\; \omega \) which is evident from Fig. 4c. It can be argued that with larger g, the population of the higher modes are enhanced which gives rise to more mixedness in the system.

6 Conclusion

We have studied an interacting qubit-oscillator bipartite system in the presence of a parametric nonlinear term by employing the generalized rotating wave approximation in the strong coupling domain. A comparison is outlined between the analytically derived approximate energy spectrum with numerically computed spectrum of the full Hamiltonian to validate our approximation. For the initial state of the bipartite system, the time evolution of the reduced density matrix of the oscillator is obtained via the partial tracing over the qubit degree of freedom. On the oscillator phase space, its density matrix furnishes the Husimi Q-function with which we have derived the quadrature variance to realize the squeezing effect. It is observed that the squeezing gets reduced by increasing the coupling strength between the qubit and oscillator in the strong coupling limit. However, we have shown that the squeezing is enhanced significantly in the presence of a parametric term which corresponds to the two-photon process. Besides, by minimizing the Hilbert–Schmidt distance we have shown that the parametric term can facilitate the generation of squeezed coherent states.