Squeezing-Induced Entanglement and Sub-Poissonian Statistics in an Extended Jaynes–Cummings Model with Pair Coherent Fields

Abstract

We present a two-mode squeezed Jaynes–Cummings model, built upon the formalism of pair coherent states (PCSs), to investigate the dynamics of a two-level atom interacting with a two-mode quantized field. By solving the time-dependent Schrödinger equation under the rotating-wave approximation, we elucidate the system’s quantum evolution, with particular emphasis on how the squeezing degree and photon number difference modulate atomic population inversion and entanglement. We further quantify the nonclassical traits of the two-mode squeezed PCSs via Mandel’s parameter and the violation of the Cauchy–Schwarz inequality, highlighting their sensitivity to model parameters. These findings illuminate the subtle interplay of squeezing, photon statistics, and entanglement in advanced quantum optical systems.

1. Introduction

In quantum optics, the Jaynes–Cummings model (J–CM) is one of the most widely used and fundamental models, despite its simplicity. The model characterizes the interaction between a two-level atom and a single quantized mode of the electromagnetic field [1]. Mathematically, the J–CM is straightforward and possesses an exact analytical solution [2,3]. The ability to create high-Q cavities through technological advancements has made it possible to experimentally implement this model for a Rydberg atom in a superconducting cavity [4]. Consequently, investigations of the J–CM have received a lot of attention over the last decades [5,6,7,8,9,10]. In the J–CM, a number of intriguing quantum phenomena have been predicted, including Rabi oscillations, the phenomena of collapse and revival, sub-Poissonian statistics, photon antibunching, etc. Under the rotating-wave approximation, the J–CM first described the interaction among undamped two-level atoms and nondecaying field modes. Nonetheless, there are several ways to expand or generalize the concept of J–CM including multi-level atoms and multi-photon transition during the process of interaction [11,12,13,14,15,16]. Another important generalization is the Tavis–CM [17], which extends the J–CM to describe the interaction between a quantized field and a collection of two-level atoms or qubits.

Quantum mechanics, as a precise and fundamental theory, reveals numerous phenomena that classical mechanics cannot explain. Among these, quantum entanglement is perhaps the most extraordinary, defying conventional intuition. In general, a quantum system that consists of two sub-quantum systems is known to be defined as in an entangled state if its overall state cannot be written as a product of the individual states of its subsystems. From a physical point of view, this means that information about one subsystem state may be obtained by measuring the other system state. The significance of the phenomenon goes well beyond basic issues in quantum theory, as the last decades have demonstrated. Indeed, entanglement is the fundamental component of several quantum information processing activities, including dense coding [18], quantum cryptography [19], and quantum teleportation [20]. To detect and generate entangled states, numerous schemes have been suggested in the literature [21,22,23,24,25]. All of the schemes used for generating the entangled state depend on indirect or direct interactions between quantum subsystems.

In quantum optics, nonclassical states of the radiation field are of fundamental importance. These states can be generated through nonlinear interactions within systems that operate under both resonant and non-resonant conditions. Two-photon coherent states, often referred to as squeezed states, stand out among the quantum states in the quantized field due to their unique nonclassical characteristics. The creation and characteristics of correlated two-mode states of the quantized field have drawn a lot of attention in the literature [26,27,28,29,30,31]. These states are interesting because they violate Bell’s inequality and the Cauchy–Schwarz inequality, and they exhibit nonclassical features like squeezing, sub-Poissonian features, and antibunching. In extending the J–CM, we introduce the two-mode squeezed J–CM, which describes a coupled system comprising a two-level atom and a radiation field within the context of two-mode PCSs. We investigate the dynamical characteristics of this quantum system as it interacts with the quantized field. Specifically, we analyze how the squeezing effect and the photon number difference in the two-mode field influence atomic population inversion and quantum entanglement during the interaction. Additionally, we explore the nonclassical properties of two-mode squeezed PCSs by examining Mandel’s parameter and the violation of the CSI in relation to the quantum model’s parameters. While prior extensions of the J–CM have incorporated squeezed states [26,27,28] or two-mode correlated fields [29,30,31], our work introduces a novel two-mode squeezed J–CM built specifically on PCSs. The primary innovation is the integration of squeezing into PCSs, enabling a systematic exploration of how the squeezing parameter $r$ and photon number difference $q$ collaboratively influence the system’s quantum dynamics. This includes enhanced insights into atomic population inversion, entanglement quantified via von Neumann entropy, and nonclassicality through Mandel’s parameter and CSI violations—effects that highlight squeezing’s role in promoting coherent interactions and reducing photon number fluctuations in ways not comprehensively studied in previous models.

This manuscript is organized as follows. Section 2 presents the quantum model and the fundamental theory for investigating the two-mode squeezed J–CM. Section 3 examines and discusses the impacts of squeezing and photon number differences in two-mode squeezed PCSs on the temporal variation in atomic inversion, quantum entanglement, and the nonclassical properties of the squeezed field. Finally, Section 4 provides a summary of the results obtained, with Appendix A offering detailed derivations of key quantities.

2. Model and Dynamics

In quantum optics, the interaction between matter and light is fundamental to understanding various nonclassical phenomena. Here, we propose an extended J–CM involving a two-level atom coupled to a two-mode quantized field under squeezing effects. The total Hamiltonian governing the system is expressed as

$$H=H_a+H_f+H_{af},$$

(1)

where $H_a$ describes the atomic subsystem, $H_f$ the free field, and $H_{af}$ their interaction. Explicitly, these terms are

$$H_f=\mathrm{\hslash }\left({\omega }_A{\widehat{n}}_A+{\omega }_B{\widehat{n}}_B\right),$$

(2)

$$H_a={\displaystyle \frac{\mathrm{\hslash }}{2}}\mathit{\Omega }{\widehat{\sigma }}_z,$$

(3)

$$H_{af}=\mathrm{\hslash }\beta \left(\widehat{a}\widehat{b}{\widehat{\sigma }}_{+}+{\widehat{a}}^{†}{\widehat{b}}^{†}{\widehat{\sigma }}_{-}\right).$$

(4)

The frequencies ${\omega }_A$ and ${\omega }_B$ correspond to the two field modes, while $\mathit{\Omega }$ is the atomic transition frequency. The number operators are ${\widehat{n}}_A={\widehat{a}}^{†}\widehat{a}$ and ${\widehat{n}}_B={\widehat{b}}^{†}\widehat{b}$, with ${\widehat{a}}^{†}\left({\widehat{b}}^{†}\right)$ and $\widehat{a}\left(\widehat{b}\right)$ being the creation and annihilation operators for mode A (B), satisfying $\left[\widehat{j},{\widehat{j}}^{†}\right]=\widehat{I}$ for $\widehat{j}=\widehat{a},\widehat{b}$. The atomic operators ${\widehat{\sigma }}_{+}$, ${\widehat{\sigma }}_{-}$, and ${\widehat{\sigma }}_z$ obey the Pauli algebra: $\left[{\widehat{\sigma }}_z,{\widehat{\sigma }}_{\pm }\right]=\pm 2{\widehat{\sigma }}_{\pm }$ and $\left[{\widehat{\sigma }}_{+},{\widehat{\sigma }}_{-}\right]={\widehat{\sigma }}_z$. Here, the atomic states are denoted |+⟩ (excited) and |−⟩ (ground), which are the eigenstates of the Pauli operator ${\widehat{\sigma }}_z$, with eigenvalues +1 and −1, respectively, satisfying ${\widehat{\sigma }}_z |+⟩ = |+⟩$ and ${\widehat{\sigma }}_z |-⟩ = -|-⟩$. The coupling constant $\beta$ quantifies the atom–field interaction strength.

We assume the system starts with the atom in its excited state $|+⟩$ and the field in a two-mode squeezed PCS $|r,\zeta ,q⟩$, defined as [26]

$$\left|r,\zeta ,q\right.⟩=\sum _{\mathrm{n}=0}^{\mathrm{\infty }}\sum _{\mathrm{m}=0}^{\mathrm{\infty }}{\mathrm{C}}_{\mathrm{n}}{\mathsf{{\rm P}}}_{\mathrm{m}}^{\left(\mathrm{n}\right)}\left|\mathrm{m}+\mathrm{q},\mathrm{m}\right.⟩=\sum _{m=0}^{\mathrm{\infty }}{D}_m\left|m+q,m\right.⟩,$$

(5)

where the coefficients $D_m$ are derived from the PCS formalism, incorporating the squeezing parameter $r$ and complex amplitude $\zeta$. Specifically,

$$P_m^{\left(n\right)}={\displaystyle \frac{{\left(\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}r\right)}^{n+m}}{{\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}r\right)}^{q+1}}}\sqrt{{\displaystyle \frac{n!\left(n+q\right)!}{m!\left(m+q\right)!}}}\sum _{k=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\left(n,m\right)}{\displaystyle \frac{{\left(-1\right)}^{m-k}{\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}r\right)}^{-2k}}{k!\left(m-k\right)!\left(n-k\right)!\left(q+k\right)!}},$$

(6)

with

$${\mathrm{C}}_{\mathrm{n}}={\mathrm{N}}_{\mathrm{q}}{\displaystyle \frac{{\mathsf{\zeta }}^{\mathrm{n}}}{\sqrt{\mathrm{n}!\left(\mathrm{n}+\mathrm{q}\right)!}}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{N}}_{\mathrm{q}}={\left[\sum _{\mathrm{n}=0}^{\mathrm{\infty }}{\displaystyle \frac{|{\mathsf{\zeta }|}^{2\mathrm{n}}}{\mathrm{n}!\left(\mathrm{n}+\mathrm{q}\right)!}}\right]}^{-1/2}.$$

This state satisfies $\widehat{a}\widehat{b}\left|r,\zeta ,q⟩=\zeta \right|r,\zeta ,q⟩$ and $\left({\widehat{n}}_A-{\widehat{n}}_B\right)\left|r,\zeta ,q⟩=q\right|r,\zeta ,q⟩$, highlighting its correlated nature with the fixed photon number difference $q$.

The initial state of the system is thus

$$\left|\psi \left(0\right)⟩=\right|+⟩\otimes \left|r,\zeta ,q\right.⟩=\sum _{m=0}^{\mathrm{\infty }}{D}_m|+,m+q,m⟩.$$

(7)

To find the time-evolved state $|\psi \left(t\right)⟩$, we solve the time-dependent Schrödinger equation

$$i\mathrm{\hslash }{\displaystyle \frac{\partial }{\partial t}}\left|\psi \left(t\right)⟩=H\right|\psi \left(t\right)⟩.$$

(8)

For simplicity, we set $\mathrm{\hslash }=1$ and assume resonance, $\mathit{\Omega }={\omega }_A+{\omega }_B$, to focus on the dominant interaction effects under the rotating-wave approximation. The resonance condition $\mathit{\Omega }={\omega }_A+{\omega }_B$ enables two-photon resonant transitions, where the atomic energy level splitting aligns with the combined energy of one photon from each field mode, promoting efficient atom–field energy exchange and coherent dynamics. The RWA is justified in the weak-coupling regime, where the coupling strength $\beta$ satisfies $\beta \ll \mathit{\Omega } \approx {\omega }_A + {\omega }_B$, allowing us to neglect rapidly oscillating counter-rotating terms that average to zero over relevant timescales. These approximations are valid in physical systems such as cavity QED with Rydberg atoms in superconducting microwave cavities or circuit QED platforms.

The interaction term $H_{af}$ preserves the photon number difference ${\widehat{n}}_A-{\widehat{n}}_B=q$, as each process annihilates or creates one photon in each mode simultaneously. This conservation allows us to decouple the dynamics into independent subspaces labeled by $m$.

We expand the wave function in the dressed basis,

$$|\psi \left(t\right)⟩=\sum _{m=0}^{\mathrm{\infty }}\left[A_m\left(t\right)\left|\hspace{0.17em}+,m+q,m⟩+B_m\left(t\right)\right|-,\hspace{0.17em}m+q+1,m+1⟩\right],$$

(9)

with initial conditions $A_m\left(0\right)=D_m$ and $B_m\left(0\right)=0$.

We substitute this expansion into the Schrödinger equation and project onto the basis states to obtain the coupled differential equations. For the coefficient of $|+,\hspace{0.17em}m+q,m⟩$,

$$i{\dot{A}}_m=\left[{\omega }_A\left(m+q\right)+{\omega }_{B}m+{\displaystyle \frac{\mathit{\Omega }}{2}}\right]A_m+\beta ⟨+,\hspace{0.17em}m+q,m\left|{\widehat{a}}^{†}{\widehat{b}}^{†}{\widehat{\sigma }}_{-}+\widehat{a}\widehat{b}{\widehat{\sigma }}_{+}\right|\psi ⟩.$$

(10)

The ${\widehat{\sigma }}_{+}$ term contributes zero to this projection because it maps to orthogonal atomic states. The ${\widehat{\sigma }}_{-}$ term acts on $B_k$ terms

$${\widehat{a}}^{†}{\widehat{b}}^{†}\left|-,\hspace{0.17em}k+q+1,k+1⟩=\sqrt{\left(k+q+2\right)\left(k+2\right)}\right|-,\hspace{0.17em}k+q+2,k+2⟩,$$

(11)

However, correctly matching, the projection yields $\beta \sqrt{\left(m+1\right)\left(m+q+1\right)}{B}_m$.

Similarly, for $B_m$,

$$i{\dot{B}}_m=\left[{\omega }_A\left(m+q+1\right)+{\omega }_B\left(m+1\right)-{\displaystyle \frac{\mathit{\Omega }}{2}}\right]B_m+\beta \sqrt{\left(m+1\right)\left(m+q+1\right)}{A}_m.$$

(12)

Under resonance $\mathit{\Omega }={\omega }_A+{\omega }_B$, the free evolution phases align. To eliminate the rapid oscillations, we transform to the interaction picture by defining

$$A_m\left(t\right)={\tilde{A}}_m\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\left[{\omega }_A\left(m+q\right)+{\omega }_{B}m+{\displaystyle \frac{\mathit{\Omega }}{2}}\right]t\right),$$

(13)

$$B_m\left(t\right)={\tilde{B}}_m\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\left[{\omega }_A\left(m+q+1\right)+{\omega }_B\left(m+1\right)-{\displaystyle \frac{\mathit{\Omega }}{2}}\right]t\right).$$

(14)

Since the exponents match on resonance, the transformed equations simplify to

$$i{\dot{\tilde{A}}}_m=\beta g_m{\tilde{B}}_m,$$

(15)

$$i{\dot{\tilde{B}}}_m=\beta g_m{\tilde{A}}_m,$$

(16)

where $g_m=\sqrt{\left(m+1\right)\left(m+q+1\right)}$ is the effective coupling, reflecting the intensity-dependent nature due to the two-mode process. Differentiating the first equation,

$${\ddot{\tilde{A}}}_m=i\beta g_m{\dot{\tilde{B}}}_m=i\beta g_m\left(-i\beta g_m{\tilde{A}}_m\right)=-{\left(\beta g_m\right)}^2{\tilde{A}}_m,$$

(17)

the general solution is ${\tilde{A}}_m\left(t\right)=C_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathit{\Omega }}_{m}t\right)+C_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathit{\Omega }}_{m}t\right)$, where ${\mathit{\Omega }}_m=\beta g_m$. Applying initial conditions ${\tilde{A}}_m\left(0\right)=D_m$ and ${\dot{\tilde{A}}}_m\left(0\right)=0$ (since ${\tilde{B}}_m\left(0\right)=0$) yields $C_1=D_m$, $C_2=0$.

Thus,

$${\tilde{A}}_m\left(t\right)=D_m\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta g_{m}t\right),$$

(18)

and from $i{\dot{\tilde{B}}}_m=\beta g_m{\tilde{A}}_m$,

$${\tilde{B}}_m\left(t\right)=-i{D}_m\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta g_{m}t\right).$$

(19)

Reverting to the Schrödinger picture, the full coefficients incorporate the phase factors, but for observables like populations and correlations, the moduli suffice due to phase cancelation.

3. Quantumness Measures and Results

The main findings of the quantum measures, grounded in the theoretical framework established in the previous section, are depicted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

3.1. Atomic Dynamics

Let us start by examining the dynamics of atomic inversion, which is determined through the expectation value of the operator ${\mathsf{\sigma }}_{\mathrm{z}}$ as

$$⟨{\widehat{\sigma }}_z\left(t\right)⟩=\sum _{m=0}^{\mathrm{\infty }}\left({\left|A_m\left(t\right)\right|}^2-{\left|B_m\left(t\right)\right|}^2\right)=\sum _{m=0}^{\mathrm{\infty }}{\left|D_m\right|}^2\mathrm{c}\mathrm{o}\mathrm{s}\left(2\beta g_{m}t\right).$$

(20)

This exhibits Rabi oscillations modulated by the distribution ${\left|D_m\right|}^2$, with frequencies depending on $m$ via $g_m$, leading to collapse–revival phenomena for broad distributions.

Figure 1 illustrates the temporal evolution of the atomic population inversion, represented by the expectation value $〈{\widehat{\sigma }}_z〉$, as a function of scaled time in the two-mode squeezed Jaynes–Cummings model. The field starts in a pair coherent state with fixed photon number difference $q = 0$ and amplitude $\left|\zeta \right| = 3$, across varying squeezing parameters $r$. The dynamics show a strong dependence on r, with all cases displaying periodic oscillations but differing in structure. At small r values, like $r = 0.01$, clear collapse–revival patterns emerge; the inversion oscillates rapidly before the envelope decays to zero during collapses, caused by destructive interference of probability amplitudes at various Rabi frequencies, then it re-emerges in revivals due to constructive interference. This occurs when the squeezed field approximates a standard coherent state (as $r$ approaches zero), with a wide spread in photon numbers leading to dephasing. In contrast, as $r$ grows larger, the collapses fade away, and $〈{\widehat{\sigma }}_z〉$ maintains sustained periodic oscillations without dipping to zero, reflecting how increased squeezing reduces photon number fluctuations and promotes more coherent atom–field interactions. Overall, these results demonstrate that the squeezing parameter $r$ critically shapes the atomic dynamics by altering the field’s nonclassical features, influencing the energy exchange and quantum correlations in the system.

Figure 1. Temporal evolution of the atomic population inversion $〈{\widehat{\sigma }}_z〉$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 1. Temporal evolution of the atomic population inversion $〈{\widehat{\sigma }}_z〉$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 2 illustrates the temporal evolution of the atomic population inversion, denoted as $〈{\widehat{\sigma }}_z〉$, as a function of scaled time T in the two-mode squeezed J–CM. The plots are generated for a fixed squeezing parameter r = 0.5 and PCS amplitude |ζ| = 3, while varying the photon number difference q across panels, as follows: (a) $q = 0$, (b) $q = 4$, (c) $q = 8$, and (d) $q = 12$. For small $q$ values, such as $q = 0$, the inversion exhibits smooth, periodic oscillations reminiscent of Rabi-like behavior, where the atom cyclically flips between its excited and ground states. At $q=4$, the dynamics display a particularly regular pattern with consistent amplitude and period, standing in clear contrast to the cases at $q=0,8,$ or $12$. As q increases further, the oscillations grow more intense and rapid, incorporating higher-frequency components that disrupt the simplicity seen at lower q. At a larger $q$, like $q=12$, the structure becomes increasingly intricate, with irregular amplitudes and varying periods that reflect interference from multiple photon pathways in the correlated field modes. These observations underscore how atomic dynamics are profoundly influenced by both the degree of squeezing and the photon number imbalance q, which modulates the field’s nonclassical statistics and alters the interaction strength in this quantum optical system.

Figure 2. Temporal evolution of the atomic population inversion $〈{\widehat{\sigma }}_z〉$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

Figure 2. Temporal evolution of the atomic population inversion $〈{\widehat{\sigma }}_z〉$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

3.2. Entropy of Subsystems and Entanglement

Recently, there have been many studies of entanglement dynamics, such as, for example, for the von Neumann entropy [32,33]. It provides a quantitative measure for the quantum state purity and the disorder of a system, which is defined as

$${\mathrm{S}}_{\mathrm{A}\left(\mathrm{F}\right)}=-\mathrm{T}\mathrm{r}\left({\mathsf{\rho }}_{\mathrm{A}\left(\mathrm{F}\right)}\ln {\mathsf{\rho }}_{\mathrm{A}\left(\mathrm{F}\right)}\right),$$

(21)

where ${\mathsf{\rho }}_{\mathrm{A}\left(\mathrm{F}\right)}$ is the atomic (field) density matrix. The function ${\mathrm{S}}_{\mathrm{A}\left(\mathrm{F}\right)}$ is a sensitive and convenient quantifier of the quantum entanglement of two interacting subsystems. Here, the Hamiltonian (1) that describes the system dynamics will give rise to entanglement between the two-mode squeezed field and the atom, and then we utilize the function ${\mathrm{S}}_{\mathrm{A}\left(\mathrm{F}\right)}$ for detecting the entanglement. The entropy of subsystems is related by the Araki and Lieb theorem which states ${\left|{\mathrm{S}}_{\mathrm{F}}-{\mathrm{S}}_{\mathrm{A}}\right|\le \mathrm{S}}_{\mathrm{A}\mathrm{F}}\le {\mathrm{S}}_{\mathrm{F}}+{\mathrm{S}}_{\mathrm{A}}$. The subsystems have identical entropies during their subsequent evolution if the entire system is defined in a pure state, which is one direct consequence of this inequality, which means that ${\mathrm{S}}_{\mathrm{A}}\left(\mathrm{t}\right)={\mathrm{S}}_{\mathrm{F}}\left(\mathrm{t}\right)$. In the following analysis, we only need to calculate the atomic entropy to investigate the time-dependent behavior of the quantum entanglement between the two-mode field and the atom.

In Figure 3, we present the study of atomic entropy with the squeezed effect when the two modes in the field have the same photon number $\mathrm{q}=0$. The dashed line is for $\mathrm{r}=0.01$, the solid line is for $\mathrm{r}=0.5$, and the dotted line is for $\mathrm{r}=0.8$. It reveals from our numerical results that the function $S_A\left(t\right)$ has period oscillations at $T=\pi$ for different values of r, and the period is equal to the period of the function $〈{\widehat{\sigma }}_z〉.$ This behavior is caused by the exchange of energy between the atom and two-mode squeezed field due to the relation between energy transfer and entanglement transfer [34,35,36]. We can observe that when the parameter $r$ increases, the maximum values of ${\mathrm{S}}_{\mathrm{A}}\left(\mathrm{t}\right)$ will be approximately closed to ln 2, which represents the maximal limit of the two-level system entropy. However, the amount of entanglement is identical in some time intervals with the same minimum value for various $r.$ These results illustrate that the squeezing effect in the TMF can enhance the value of entanglement in the bipartite state within the context of the two-mode squeezed J–CM. To analyze the effect of the photon number difference on the entanglement dynamics in the two-mode field–atom state, Figure 4 illustrates the temporal evolution of ${\mathrm{S}}_{\mathrm{A}}\left(\mathrm{t}\right)$ as a function of time $\mathrm{T}$ for various $q$ values, with $r=0.5$ and $\left|\mathsf{\zeta }\right|=3$. We can observe that the entanglement measure achieves period oscillation when the parameter $q$ achieves small values. Also, the dynamical behavior of the function ${\mathrm{S}}_{\mathrm{A}}\left(\mathrm{t}\right)$ experiences the same oscillation as the atomic inversion at $q=4$ and $r=0.5$. At large values of $q$, the oscillations’ structure of the function ${\mathrm{S}}_{\mathrm{A}}\left(\mathrm{t}\right)$ becomes random to amplitudes and periods that depend on $q.$

Figure 3. Temporal evolution of the von Neumann entropy ${\mathrm{S}}_{\mathrm{A}}$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 3. Temporal evolution of the von Neumann entropy ${\mathrm{S}}_{\mathrm{A}}$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 4. Temporal evolution of the von Neumann entropy ${\mathrm{S}}_{\mathrm{A}}$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

Figure 4. Temporal evolution of the von Neumann entropy ${\mathrm{S}}_{\mathrm{A}}$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

3.3. Statistical Properties of Two-Mode Squeezed Field

To test for sub-Poissonian statistics, we analyze the Mandel parameter for modes 1 and 2 in the TMSF, defined as follows:

$${\mathrm{Q}}_{\mathrm{A}}={\displaystyle \frac{〈{\mathrm{a}}^{+}\mathrm{a}{\mathrm{a}}^{+}\mathrm{a}〉-〈{\mathrm{a}}^{+}\mathrm{a}〉-{〈{\mathrm{a}}^{+}\mathrm{a}〉}^2}{〈{\mathrm{a}}^{+}\mathrm{a}〉}},$$

(22)

$${\mathrm{Q}}_{\mathrm{B}}={\displaystyle \frac{〈{\mathrm{b}}^{+}\mathrm{b}{\mathrm{b}}^{+}\mathrm{b}〉-〈{\mathrm{b}}^{+}\mathrm{b}〉-{〈{\mathrm{b}}^{+}\mathrm{b}〉}^2}{〈{\mathrm{b}}^{+}\mathrm{b}〉}}.$$

(23)

Sub-Poissonian statistics are indicated when $-1\le \mathrm{Q}<0$, while Poissonian statistics occur when $\mathrm{Q}=0$, and super-Poissonian statistics are observed when $\mathrm{Q}>0$.

Figure 5 and Figure 6 depict the time evolution of Mandel’s parameters, ${\mathrm{Q}}_{\mathrm{A}}$ and ${\mathrm{Q}}_{\mathrm{B}}$, which quantify the photon statistics in modes a and b of the two-mode squeezed field, plotted against scaled time T in the extended J–CM. Mandel’s parameter indicates sub-Poissonian statistics (nonclassical) when $Q < 0$, Poissonian statistics (random) when $Q = 0$, and super-Poissonian statistics (classical) when $Q > 0$. Figure 5 focuses on the symmetric case, with photon number difference $q = 0$ and pair coherent amplitude $\left|\zeta \right| = 3$, with the varying of the squeezing parameter $r$ across curves: red dash for $r = 0.01$, blue solid for $r = 0.5$, and black dotted dash for r = 0.8. Since $q = 0$ implies equal average photons in both modes, $Q_A = Q_B$ throughout. At small $r$ (near coherent state limit), the parameter dips to be deeply negative, with large-amplitude oscillations, signaling strong sub-Poissonian behavior due to enhanced quantum fluctuations and interference in the weakly squeezed field. As $r$ increases, the oscillations dampen and stay closer to zero, reducing the sub-Poissonian character as squeezing suppresses noise in one quadrature, leading to more Poissonian-like statistics over time.

In contrast, Figure 6 examines fixed $r = 0.5$ and $\left|\zeta \right| = 3$, but varies q across subpanels: (a₁, b₁) for q = 0 (where $Q_A = Q_B$), (a₂, b₂) for $q = 4$, (a₃, b₃) for $q = 8$, and (a₄, b₄) for $q = 12$, with the left columns showing $Q_A$ and right columns showing $Q_B$. For q = 0, the dynamics mirror Figure 5’s blue curve, with periodic negative excursions indicating sub-Poissonian statistics. As $q$ grows, the modes become asymmetric; $Q_A$ and $Q_B$ diverge, with one mode often displaying deeper sub-Poissonian dips while the other may cross into positive territory, reflecting super-Poissonian bursts. Higher $q$ introduces faster, more irregular oscillations, amplifying the interplay of photon imbalance and squeezing, which can switch modes between sub- and super-Poissonian regimes during evolution. These patterns highlight how squeezing ($r$) and photon difference ($q$) tune the field’s nonclassical features, with stronger squeezing favoring stability near Poissonian limits and larger imbalances promoting dynamic shifts in statistics, underscoring the model’s sensitivity to initial state parameters in quantum optical interactions.

Figure 5. Temporal evolution of the Mandel’s parameter ${\mathrm{Q}}_{\mathrm{A}}={\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 5. Temporal evolution of the Mandel’s parameter ${\mathrm{Q}}_{\mathrm{A}}={\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Finally, we introduce the CSI

$${\mathrm{I}}_0={\displaystyle \frac{{\left(〈{{\mathrm{a}}^{+}}^2{\mathrm{a}}^2〉〈{{\mathrm{b}}^{+}}^2{\mathrm{b}}^2〉\right)}^{1/2}}{\left|〈{\mathrm{a}}^{+}\mathrm{a}{\mathrm{b}}^{+}\mathrm{b}〉\right|}}-1,$$

(24)

which indicates a strong nonclassical correlation between the modes $\mathrm{A}$ and $\mathrm{B}$ when the inequality is violated for ${\mathrm{I}}_0<0.$

Figure 7 and Figure 8 depict the temporal evolution of the CSI violation measure, denoted as $I_0$, as a function of scaled time T in the two-mode squeezed J–CM. Figure 7 examines the symmetric case with photon number difference $q = 0$ and pair coherent amplitude $\left|\zeta \right| = 3$, varying the squeezing parameter $r$ across curves: red dash for $r = 0.01$, blue solid for $r = 0.5$, and black dotted dash for $r = 0.8$. At small $r$ (near the unsqueezed limit), $I_0$ remains nearly flat around zero, implying minimal violation and classical-like behavior due to weak quantum correlations in the field. As $r$ increases, the oscillations amplify with deeper negative excursions, demonstrating how stronger squeezing enhances inter-mode correlations and boosts nonclassical features through increased photon pairing. Figure 8, in contrast, fixes $r = 0.5$ and $\left|\zeta \right| = 3$, while varying $q$ across subpanels: (a) $q = 0$, (b) $q = 4$, (c) $q = 8$, and (d) $q = 12$. Starting with $q = 0$, $I_0$ oscillates symmetrically around −0.6 with moderate negative dips (to about −0.7), indicating baseline nonclassicality. As $q$ rises, the dynamics evolve: at $q = 4$, the amplitude grows, with deeper violations (likely to reach near −1 or more pronounced dips) reflecting amplified asymmetry in photon distribution that strengthens quantum interference and correlations. For higher $q$ ($8$ and $12$), the oscillations become more irregular, with varying depths that may peak in nonclassicality at intermediate $q$ before stabilizing, underscoring how photon imbalance tunes the field’s quantum statistics. Overall, these results highlight that optimal control of $r$ and $q$ maximizes CSI violations, revealing the pivotal role of squeezing and photon differences in fostering robust nonclassical inter-mode links essential for quantum information applications.

Figure 6. Temporal evolution of the Mandel’s parameters ${\mathrm{Q}}_{\mathrm{A}}$ and ${\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (${\mathbf{a}}_1,{\mathbf{b}}_1$) corresponds to $q=0$, panel (${\mathbf{a}}_2,{\mathbf{b}}_2$) to $q=4$, panel (${\mathbf{a}}_3,{\mathbf{b}}_3$) to q $=8$, and panel (${\mathbf{a}}_4,{\mathbf{b}}_4$) to $q=12$.

Figure 6. Temporal evolution of the Mandel’s parameters ${\mathrm{Q}}_{\mathrm{A}}$ and ${\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (${\mathbf{a}}_1,{\mathbf{b}}_1$) corresponds to $q=0$, panel (${\mathbf{a}}_2,{\mathbf{b}}_2$) to $q=4$, panel (${\mathbf{a}}_3,{\mathbf{b}}_3$) to q $=8$, and panel (${\mathbf{a}}_4,{\mathbf{b}}_4$) to $q=12$.

Figure 7. Temporal evolution of the violation factor ${\mathrm{I}}_0$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 7. Temporal evolution of the violation factor ${\mathrm{I}}_0$ as a function of the scaled interaction time $T=\beta t$ when the field initially in a two-mode squeezed PCS with photon number difference $q=0$ and amplitude $\mid \zeta \mid =3$, for different values of squeezing parameter $r$. The curves are as follows: red dash for $r=0.01,$ blue solid for $r=0.5$, and black dotted dash for $r=0.8$.

Figure 8. Temporal evolution of the Mandel’s parameters ${\mathrm{Q}}_{\mathrm{A}}$ and ${\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

Figure 8. Temporal evolution of the Mandel’s parameters ${\mathrm{Q}}_{\mathrm{A}}$ and ${\mathrm{Q}}_{\mathrm{B}}$ as a function of the scaled interaction time $T=\beta t$ when the field is initially in a two-mode squeezed PCS with squeezing parameter $r=0.5$ and amplitude $\mid \zeta \mid =3$, for different values of photon number difference $q$. Panel (a) corresponds to $q=0$, panel (b) to $q=4$, panel (c) to q$=8$, and panel (d) to $q=12$.

4. Conclusions

Based on the formalism of PCSs, we have introduced the two-mode squeezed J–CM, which describes a coupled system consisting of an atom and a quantum field in the context of two-mode squeezed PCSs. Based on the solution of the Schrödinger equation, we have thoroughly explored the dynamical characteristics of this quantum system under the influence of the quantized field. Our study focuses on several key aspects by considering the impact of squeezing and photon number differences. We have examined how variations in these parameters can affect the atomic population inversion and quantum entanglement during the interaction. This exploration provides insights into how these factors influence the behavior of the quantum system. Furthermore, we have investigated the nonclassical properties of the two-mode squeezed PCSs by analyzing the Mandel’s parameter and the violation of the CSI. These measures help in understanding the extent to which the squeezed states deviate from classical behavior and exhibit quantum characteristics. Overall, our results shed light on how the dynamics of the quantum system, including entanglement and statistical properties, are influenced by the squeezing effects and photon number differences in the two-mode squeezed J–CM. This comprehensive analysis contributes to the broader understanding of quantum systems and their nonclassical features.