Introduction

Quantum entanglement, a phenomenon without any analog in classical physics, is one of the most fundamental aspects of quantum mechanics. It signifies an inseparable connection between parts of a quantum system, where the state of one component is irrevocably tied to the state of another, no matter how far apart they are1. This unique feature makes entanglement essential for many exciting technologies in quantum information science, like quantum computers2, secure communication3, teleportation4, and precise measurements5. Within the domain of quantum information processing, a challenge is that how to create and control entangled states. They are necessary to unlock the full power of quantum mechanics and perform tasks that classical systems cannot.

Among the various methods for creating such states, atom-field interactions in cavity quantum electrodynamics (QED) are particularly notable6. In these systems, the interaction between light and matter within a cavity naturally results in the formation of entangled states, specifically between the atom and the field7. This process not only illustrates the strong connection between quantum theory and physical systems but also highlights the practical importance of entanglement in advancing technology. The ability to exploit entanglement in such systems is crucial for realizing the full potential of quantum technologies, making it a significant subject of interest in the context of quantum optics.

In this sense, the Jaynes-Cummings model (JCM)8 is a foundational framework in quantum optics, which describes the interaction between two distinct quantum systems; a two-level system and a bosonic mode in an optical cavity. The JCM is a precise full quantum mechanical model that represents a basic form of atom-field interaction. It involves a dipole coupling between an undamped two-level atom and a one-dimensional harmonic oscillator under the rotating wave approximation. As a result of simplicity of the JCM, the model has found significant applications in quantum optics and quantum information processing9. Among them, generating W, EPR, and GHZ states10, trapped ion systems11, and dissipative processes12 can be mentioned. Furthermore, because of its accuracy in diverse physical systems from quantum optics to condensed matter, the JCM can be regarded as a key achievement in both theoretical and experimental quantum physics.

Numerous efforts have been made to generalize the JCM by incorporating various effects such as Kerr nonlinearity as well as f-deformed Kerr nonlinearity13,14,15, multi-level system16,17, multi-photon transition18,19, multi-atom interaction20,21, multi-mode field22,23, and intensity-dependent coupling24,25. Notably, the latter case has been experimentally confirmed through the observation of dressed state spectra in the interaction of a single two-level atom with a high-finesse cavity. This setup revealed nonlinear normal-mode splitting proportional to \(\sqrt{n}\)26. It is worth mentioning that, historically, this form of nonlinearity function was theoretically proposed by Buck and Sukumar, who linked the atom-field coupling to the intensity of light27.

To explain further, the concept of intensity-dependent coupling in atom-field interactions comes from the theory of nonlinear coherent states, which are quantum states of radiation fields with frequencies that depend on light intensity28. In this context, the intensity-dependent JCM acts as a bridge between the standard JCM and nonlinear coherent states29. Specifically, the nonlinear version of the standard JCM is a special case of the \(f\)-deformed model, where the bosonic operators of the radiation field are replaced by their nonlinear counterparts. It is also important to note that nonlinear coherent states are promising candidates for generating nonclassical light30,31. Therefore, integrating the nonlinear coherent states approach into the JCM has attracted significant interest32,33,34,35,36.

Many efforts have been made to compare intensity-dependent coupling with the constant coupling regime. An analytical study has examined a quantum system, including a double five-level atom interacting with a single-mode field in the presence of intensity-dependent coupling and a Kerr-like medium. This study explored the dynamical evolution of linear entropy and \(l_{1}\)-norm of coherence37. The dynamics of atomic mixedness and phase space quasi-probability was also investigated for two moving atoms interacting with a cavity mode under intensity-dependent coupling38. Additionally, a general formalism for two two-level atoms interacting with a single-mode cavity field was developed to analyze the temporal evolution of von Neumann entropy and average photon number in the presence of intensity-dependent coupling and Kerr-like medium39.

As another point, multi-photon transition has garnered significant attention due to its applications in atomic systems. These transitions highlight the nonclassical properties of emitted light in atomic interactions40, and contribute to dispersion interactions in atom-field diamagnetic interactions through two-photon processes41. Additionally, the interaction of single or two atoms with a single-mode field has been explored within a cavity embedded in a Kerr medium42. The dynamics and preservation of entanglement in the two-photon Jaynes-Cummings model have also been investigated43. Moreover, theoretical studies have examined multi-photon resonance in the quantum Rabi model under varying detuning conditions44.

Notably, enhanced entanglement due to two-photon transitions holds significant promise for advancing quantum technologies across various domains. In quantum communication, highly entangled photon pairs generated via two-photon transitions can enable robust quantum key distribution (QKD) and quantum teleportation, ensuring secure communication over long distances, such as in satellite-based quantum networks45,46. For quantum computing, these entangled states can improve the fidelity of quantum gates and error correction protocols, enhancing the performance of quantum processors based on trapped ions or superconducting qubits1,47. In quantum sensing and metrology, entangled photon pairs can push measurement precision beyond classical limits, benefiting applications like gravitational wave detection or biological imaging48,49. Additionally, two-photon transitions can facilitate quantum simulations of complex systems, such as molecular dynamics or condensed matter phenomena, providing insights into chemical reactions or material properties50,51. Quantum memories and repeaters can also leverage this enhanced entanglement to store and distribute entangled states over long distances, forming the backbone of a future quantum internet52,53. Furthermore, integrated quantum photonics can utilize two-photon transitions to create compact, on-chip devices for scalable quantum information processing54,55.

Studying quantum optical systems in an optical cavity within a Kerr medium is particularly relevant to the present work. For example, the influence of a Kerr medium on the dynamics of physical quantities has been analyzed in a system involving a four-level atom interacting with a nonlinear quantum field56. The presence of Kerr-like nonlinearity has been shown to enhance atomic Wehrl-Husimi entropy entanglement in a quantum system where a qubit interacts with a cavity field57. Additionally, the role of Kerr nonlinearity in the nonclassical behavior of optical systems has been investigated in a five-level atom coupled with a two-mode quantized field inside a high-Q cavity58.

The importance of the model we present can be understood by considering its potential applications in the field of quantum information. Quantum theory acknowledges that atomic systems are essential components for building future quantum technologies, such as those used for communication and cryptography45,59,60,61. Our research focuses on the interaction between atoms and light, which is crucial for developing these quantum technologies. By controlling this interaction, we can generate and manipulate the unique properties of quantum systems, such as entanglement. This is particularly important when considering systems with multi-atoms, as these can be used to create the basic building blocks for quantum computers. Importantly, the creation of entangled states between two atoms has already been achieved experimentally using techniques like trapping ions and cavity quantum electrodynamics62,63, demonstrating the feasibility of our approach.

Inspired by the above, in this paper we study the atomic population inversion and entanglement dynamics of a quantum system consisting of three identical two-level atoms, each interacting with a single-mode cavity field. The cavity is surrounded by a Kerr medium exhibiting f-deformed Kerr nonlinearity. We also consider a degenerate multi-photon transition in the nonlinear (intensity-dependent) regime. The effects of intensity-dependent coupling, detuning, Kerr and f-deformed Kerr nonlinearity, and multi-photon transition on the nonclassical properties of the system are analyzed.

The structure of this paper is as follows: Section “Theory and model” describes the quantum system in detail and derives the state vector using the time-dependent Schrödinger equation. Section “Atomic population inversion” analyzes the atomic population inversion over time. Section “Quantum entanglement” explores the entanglement dynamics between subsystems. Finally, Section “Summary and Conclusion” provides a summary and concluding remarks.

Theory and model

To analytically obtain the wave function of the system, it is essential to construct the appropriate Hamiltonian and fully understand all interactions affecting the system. We generalize the standard JCM by replacing the usual creation and annihilation operators in the atom-field interaction with f-deformed operators. Additionally, the Kerr effect is included as an intensity-dependent form. We also consider a single-mode field oscillating with frequency \(\omega _{0}\) interacting with three two-level atoms in a Kerr-like optical medium. The Hamiltonian, incorporating all relevant interactions and formulated within the electric-dipole and rotating wave approximations, is expressed as

$$\begin{aligned} \hat{H} = \hat{H}_0 + \hat{H}_1, \end{aligned}$$
(1)

with

$$\begin{aligned} \hat{H}_0 = \hbar \omega _0 \hat{n} + \sum _{j = 1}^{3}\frac{1}{2} \hbar \omega _j \hat{\sigma }_z^{(j)}, \end{aligned}$$
(2a)

and

$$\begin{aligned} \hat{H}_1 = \sum _{j = 1}^{3} \hbar \lambda _j \left( \hat{A}^k \hat{\sigma }_+^{(j)} + \hat{A}^{\dag \, k} \hat{\sigma }_-^{(j)} \right) + \hbar \chi \hat{R}^{\dag ^2} \hat{R}^2, \end{aligned}$$
(2b)

where \(\hat{n} = \hat{a}^\dag \hat{a}\) represents the number operator, and \(\hat{\sigma }_{\pm }^{(j)}\) denote the atomic raising/lowering operators defined as \(\hat{\sigma }_{+}^{(j)} = | e^{(j)} \rangle \langle g^{(j)} | = \hat{\sigma }_{-}^{(j) \dag }\) and \(\hat{\sigma }_{z}^{(j)} = \hat{\sigma }_{+}^{(j)} \hat{\sigma }_{-}^{(j)} - \hat{\sigma }_{-}^{(j)} \hat{\sigma }_{+}^{(j)}\). Here, \(| e^{(j)} \rangle\) and \(| g^{(j)} \rangle\) represent the excited and ground states of the j-th two-level atom, with energies \(\omega _{e}^{(j)}\) and \(\omega _{g}^{(j)}\), respectively. The atomic transition frequency is \(\omega _j = \omega _{e}^{(j)} - \omega _{g}^{(j)}\), and \(\lambda _j\) denotes the atom-field coupling constant. Additionally, k refers to the multi-photon transition process, \(\chi\) is proportional to the third-order nonlinear susceptibility of the Kerr medium, and the operators \(\hat{R} = \hat{a} \hat{g}(\hat{n})\) and \(\hat{A} = \hat{a} \hat{f}(\hat{n})\), with their Hermitian conjugates \(\hat{R}^{\dag } = \hat{g}(\hat{n}) \hat{a}^{\dag }\) and \(\hat{A}^{\dag } = \hat{f}(\hat{n}) \hat{a}^{\dag }\), respectively, describe the f-deformed bosonic operators in which \(g(\hat{n})\) and \(f(\hat{n})\) are distinct operator-valued nonlinearity functions that define the deformation of the Kerr medium and the intensity-dependent atom-field coupling. In the present paper, we employ the nonlinearity functions \(\sqrt{n}\) and \(1/\sqrt{n}\) throughout. Specifically, the nonlinearity function \(\sqrt{n}\) arises from the Hamiltonian describing the interaction between a two-level atom and a radiation field with intensity-dependent coupling64. Experimental validation of this nonlinearity function has been reported in26. Moreover, the nonlinearity function \(1/\sqrt{n}\) holds notable physical significance. This function, derived by Man’ko et al.28 through the nonlinear coherent states approach, corresponds to the class of nonlinear coherent states referred to as harmonious states by Sudarshan65. It is a widely recognized and extensively utilized nonlinearity function in the study of deformed bosonic operators within the framework of quantum optics66,67,68.

Note that for identical atoms where we have considered here, \(\lambda _j = \lambda\) and \(\omega _j = \omega\). For the next purpose, it is useful to express the Hamiltonian in the interaction picture, which is given by

$$\begin{aligned} \hat{\mathscr {V}} (t) = \hbar \lambda \sum _{j = 1}^{3} \left( \hat{\sigma }_{+}^{(j)} \hat{A}^k \text{e}^{- \text{i} \Delta t} + \hat{A}^{\dag k} \hat{\sigma }_-^{(j)} \text{e}^{ \text{i} \Delta t} \right) + \hbar \chi \hat{R}^{\dag ^2} \hat{R}^2, \end{aligned}$$
(3)

with \(\Delta = k \omega _{0} - \omega\). The wave function in the interaction picture is determined by solving the Schrödinger equation in the interaction picture, expressed as

$$\begin{aligned} \text{i} \hbar \frac{\partial }{\partial t} \left| \psi _I (t) \right\rangle = \hat{\mathscr {V}}(t) \left| \psi _I (t) \right\rangle . \end{aligned}$$
(4)

Before proceeding further, it is useful to introduce the matrix representation of the formalism.

Matrix presentation of atomic states and operators

For a single two-level atom, the Hilbert space is two-dimensional, where the atomic excited and ground states are represented by \(2 \times 1\) column vectors, and the operators are expressed as \(2 \times 2\) matrices

$$\begin{aligned}&\left| e \right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \left| g \right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \nonumber \\&\hat{\sigma }_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \hat{\sigma }_+ = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}. \end{aligned}$$
(5)

For a system of three two-level atoms, the Hilbert space becomes eight-dimensional. In this space, operators are expressed as \(8 \times 8\) matrices, while basis states are described by \(8 \times 1\) column vectors. These vectors are obtained via the tensor product of three \(2 \times 1\) vectors. In the tensor product, the first element corresponds to Atom 1, the second to Atom 2, and the third to Atom 3. The tensor product operates from left to right, as shown below:

$$\begin{aligned} \left| \mathfrak {u}_1 \right\rangle = \left| e_1 e_2 e_3 \right\rangle \equiv \left| e_1 \right\rangle \otimes \left| e_2 \right\rangle \otimes \left| e_3 \right\rangle = \begin{pmatrix} 1 \times \left| e_2 \right\rangle \\ 0 \times \left| e_2 \right\rangle \end{pmatrix} \otimes \left| e_3 \right\rangle = \begin{pmatrix} 1 \times \left| e_3 \right\rangle \\ 0 \times \left| e_3 \right\rangle \\ 0 \times \left| e_3 \right\rangle \\ 0 \times \left| e_3 \right\rangle \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \equiv \left( \delta _{m,1} \right) _{8 \times 1}. \end{aligned}$$
(6)

where \(\delta _{m,j}\) represents the Kronecker delta, and \(\left( \delta _{m,j} \right) _{8 \times 1}\) denotes an \(8 \times 1\) column vector where the m-th component \((m=1, 2, \dots , 8)\) equals \(\delta _{m,j}\). For the other basis states:

$$\begin{aligned} \hspace{-0.8cm}&\left| \mathfrak {u}_2 \right\rangle = \left| e_1 e_2 g_3 \right\rangle = \left( \delta _{m,2} \right) _{8 \times 1}, \quad \left| \mathfrak {u}_3 \right\rangle = \left| e_1 g_2 e_3 \right\rangle = \left( \delta _{m,3} \right) _{8 \times 1}, \quad \left| \mathfrak {u}_4 \right\rangle = \left| e_1 g_2 g_3 \right\rangle = \left( \delta _{m,4} \right) _{8 \times 1}, \quad \left| \mathfrak {u}_5 \right\rangle = \left| g_1 e_2 e_3 \right\rangle = \left( \delta _{m,5} \right) _{8 \times 1}, \nonumber \\ \hspace{-0.8cm}&\left| \mathfrak {u}_6 \right\rangle = \left| g_1 e_2 g_3 \right\rangle = \left( \delta _{m,6} \right) _{8 \times 1}, \quad \left| \mathfrak {u}_7 \right\rangle = \left| g_1 g_2 e_3 \right\rangle = \left( \delta _{m,7} \right) _{8 \times 1}, \quad \left| \mathfrak {u}_8 \right\rangle = \left| g_1 g_2 g_3 \right\rangle = \left( \delta _{m,8} \right) _{8 \times 1}. \end{aligned}$$
(7)

Utilizing the definition \((\sigma _z^{(m)})_{ij} = \langle \mathfrak {u}_i | \hat{\sigma }_z^{(m)} | \mathfrak {u}_j \rangle\) for the matrix elements of the population inversion operator of the m-th atom, the explicit forms of these operators can be determined as follows

$$\begin{aligned}&\hat{\sigma }_z^{(1)} = \begin{pmatrix} +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & +1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & +1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{pmatrix}, \qquad \hat{\sigma }_z^{(2)} = \begin{pmatrix} +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & +1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & +1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{pmatrix}, \nonumber \\&\hat{\sigma }_z^{(3)} = \begin{pmatrix} +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & +1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & +1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{pmatrix}. \end{aligned}$$
(8)

Consequently, the total operator of the atomic population inversion, \(\hat{\sigma }_z\), is written as

$$\begin{aligned} \hat{\sigma }_z = \dfrac{1}{3} \sum _{l=1}^{3} \hat{\sigma }_z^{(l)} = \begin{pmatrix} +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & +1/3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & +1/3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1/3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & +1/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1/3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1/3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{pmatrix}. \end{aligned}$$
(9)

Similarly, the raising and lowering operators for each atom are given by

$$\begin{aligned}&\hat{\sigma }_+^{(1)} = \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}= \left( \hat{\sigma }_{-}^{(1)} \right) ^\dag , \qquad \hat{\sigma }_+^{(2)} = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} = \left( \hat{\sigma }_{-}^{(2)} \right) ^\dag , \nonumber \\&\hat{\sigma }_+^{(3)} = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} = \left( \hat{\sigma }_{-}^{(3)} \right) ^\dag . \end{aligned}$$
(10)

Notably, all the above matrices have been given in the \(\left| \mathfrak {u}_{j} \right\rangle\) basis.

Wave function of the system

To find the exact form of the state vector for the whole system, we start by considering the Hamiltonian in Eq. (3). The wave function of the system at any time \(t > 0\) evolves as follows:

$$\begin{aligned} \left| \psi _I (t) \right\rangle&= \sum _{n=0}^{+\infty } \Bigg \{ A_n (t) \left| \mathfrak {u}_{1},n \right\rangle + B_n (t) \left| \mathfrak {u}_{2},n+k \right\rangle + C_n (t) \left| \mathfrak {u}_{3},n+k \right\rangle \nonumber \\&\qquad + D_n (t) \left| \mathfrak {u}_{4},n+2k \right\rangle + E_n (t) \left| \mathfrak {u}_{5},n+k \right\rangle + F_n (t) \left| \mathfrak {u}_{6},n+2k \right\rangle \nonumber \\&\qquad + G_n (t) \left| \mathfrak {u}_{7},n+2k \right\rangle + L_n (t) \left| \mathfrak {u}_{8},n+3k \right\rangle \Bigg \} \nonumber \\&= \sum _{n=0}^{+\infty } \begin{pmatrix} A_n (t) \left| n \right\rangle \\ B_n (t) \left| n+k \right\rangle \\ C_n (t) \left| n+k \right\rangle \\ D_n (t) \left| n+2k \right\rangle \\ E_n (t) \left| n+k \right\rangle \\ F_n (t) \left| n+2k \right\rangle \\ G_n (t) \left| n+2k \right\rangle \\ L_n (t) \left| n+3k \right\rangle \\ \end{pmatrix}. \end{aligned}$$
(11)

At this point, it is essential to express the time-dependent Schrödinger equation as

$$\begin{aligned} \text{i} \dfrac{\partial }{\partial \tau } \left| \psi _I (\tau ) \right\rangle = \dfrac{1}{\hbar \lambda } \hat{\mathscr {V}} (\tau ) \left| \psi _I (\tau ) \right\rangle \end{aligned}$$
(12)

in which

$$\begin{aligned} \hat{\mathscr {V}} (\tau ) = \hbar \lambda \begin{pmatrix} \chi ' \hat{\mathscr {R}} & \hat{\mathscr {A}} (\tau ) & \hat{\mathscr {A}} (\tau ) & 0 & \hat{\mathscr {A}} (\tau ) & 0 & 0 & 0 \quad \\ \hat{\mathscr {A}}^{\dag } (\tau ) & \chi ' \hat{\mathscr {R}} & 0 & \hat{\mathscr {A}} (\tau ) & 0 & \hat{\mathscr {A}} (\tau ) & 0 & 0 \quad \\ \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & \chi ' \hat{\mathscr {R}} & \hat{\mathscr {A}} (\tau ) & 0 & 0 & \hat{\mathscr {A}} (\tau ) & 0 \quad \\ 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & \hat{\mathscr {A}}^{\dag } (\tau ) & \chi ' \hat{\mathscr {R}} & 0 & 0 & 0 & \hat{\mathscr {A}} (\tau ) \quad \\ \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & 0 & 0 & \chi ' \hat{\mathscr {R}} & \hat{\mathscr {A}} (\tau ) & \hat{\mathscr {A}} (\tau ) & 0 \quad \\ 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & \chi ' \hat{\mathscr {R}} & 0 & \hat{\mathscr {A}} (\tau ) \quad \\ 0 & 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & \chi ' \hat{\mathscr {R}} & \hat{\mathscr {A}} (\tau ) \quad \\ 0 & 0 & 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & 0 & \hat{\mathscr {A}}^{\dag } (\tau ) & \hat{\mathscr {A}}^{\dag } (\tau ) & \chi ' \hat{\mathscr {R}} \end{pmatrix}, \end{aligned}$$
(13)

with \(\hat{\mathscr {A}} (\tau ) = \text{e}^{- \text{i} \Delta ' \tau } \hat{A}^k\) and \(\hat{\mathscr {R}} = \hat{R}^{\dag ^2} \hat{R}^2\). In addition, we have defined \(\tau = \lambda t\), \(\Delta = \lambda \Delta '\), and \(\chi = \lambda \chi '\). By substituting Eqs. (11) and (13) into Eq. (12) and carrying out detailed algebraic manipulations, we obtain the following coupled differential equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text{i} \dot{A}_n - \bf{Q}_0 A_n (\tau ) = \dfrac{P_1}{\sqrt{3}} \text{e}^{- \text{i} \Delta ' \tau } \Big ( B_n (\tau ) + C_n (\tau ) + E_n (\tau ) \Big ), \quad \\ \text{i} \dot{B}_n - \bf{Q}_1 B_n (\tau ) = \dfrac{P_1^{*}}{\sqrt{3}} \, \text{e}^{+ \text{i} \Delta ' \tau } A_n (\tau ) + \dfrac{P_2}{\sqrt{2}} \, \text{e}^{- \text{i} \Delta ' \tau } \Big ( D_n (\tau ) + F_n (\tau ) \Big ), \quad \\ \text{i} \dot{C}_n - \bf{Q}_1 C_n (\tau ) = \dfrac{P_1^{*}}{\sqrt{3}} \text{e}^{+ \text{i} \Delta ' \tau } A_n (\tau ) + \dfrac{P_2}{\sqrt{2}} \, \text{e}^{- \text{i} \Delta ' \tau } \Big ( D_n (\tau ) + G_n (\tau ) \Big ), \quad \\ \text{i} \dot{D}_n - \bf{Q}_2 D_n (\tau ) = \dfrac{P_2^{*}}{\sqrt{2}} \text{e}^{+ \text{i} \Delta ' \tau } \Big ( B_n (\tau ) + C_n (\tau ) \Big ) + \dfrac{P_3}{\sqrt{3}} \text{e}^{- \text{i} \Delta ' \tau } L_n (\tau ), \quad \\ \text{i} \dot{E}_n - \bf{Q}_1 E_n (\tau ) = \dfrac{P_1^{*}}{\sqrt{3}} \text{e}^{+ \text{i} \Delta ' \tau } A_n (\tau ) + \dfrac{P_2}{\sqrt{2}} \, \text{e}^{- \text{i} \Delta ' \tau } \Big ( F_n (\tau ) + G_n (\tau ) \Big ), \quad \\ \text{i} \dot{F}_n - \bf{Q}_2 F_n (\tau ) = \dfrac{P_2^{*}}{\sqrt{2}} \text{e}^{+ \text{i} \Delta ' \tau } \Big ( B_n (\tau ) + E_n (\tau ) \Big ) + \dfrac{P_3}{\sqrt{3}} \text{e}^{- \text{i} \Delta ' \tau } L_n (\tau ), \quad \\ \text{i} \dot{G}_n - \bf{Q}_2 G_n (\tau ) = \dfrac{P_2^{*}}{\sqrt{2}} \text{e}^{+ \text{i} \Delta ' \tau } \Big ( C_n (\tau ) + E_n (\tau ) \Big ) + \dfrac{P_3}{\sqrt{3}} \text{e}^{- \text{i} \Delta ' \tau } L_n (\tau ), \quad \\ \text{i} \dot{L}_n - \bf{Q}_0 L_n (\tau ) = \dfrac{P_3}{\sqrt{3}} \text{e}^{+ \text{i} \Delta ' \tau } \Big ( D_n (\tau ) + F_n (\tau ) + G_n (\tau ) \Big ), \end{array}\right. } \end{aligned}$$
(14)

where the \(\bf{Q}_m\)’s and the \(P_m\)’s are defined by

$$\begin{aligned}&\bf{Q}_{0} = \chi ' \mathscr {R}(n), \qquad \bf{Q}_{1} = \chi ' \mathscr {R} (n+k), \qquad \bf{Q}_{2} = \chi ' \mathscr {R} (n+2k), \qquad \bf{Q}_{3} = \chi ' \mathscr {R} (n+3k), \nonumber \\&P_1 = \sqrt{3} \, J(n + k), \quad \qquad P_2 = \sqrt{2} \, J(n + 2k), \quad \qquad P_3 = \sqrt{3} \, J(n + 3k), \nonumber \\&J(n) = \prod _{i=1}^k f(n-i+1)\sqrt{n-i+1} \quad ( n \ge k ), \qquad \mathscr {R}(n) = n (n - 1) g^{2}(n) g^{2}(n - 1). \end{aligned}$$
(15)

From Eq. (14), it follows that this leads to the following fourth-order differential equation for the coefficient \(A_n\), given by

$$\begin{aligned} \dfrac{\text{d}^4 A_n}{\text{d} \tau ^4} + \text{i} \beta _1 \dfrac{\text{d}^3 A_n}{\text{d} \tau ^3} + \beta _2 \dfrac{\text{d}^2 A_n}{\text{d} \tau ^2} + \text{i} \beta _3 \dfrac{\text{d} A_n}{\text{d} \tau } - \beta _4 A_n (\tau ) = 0, \end{aligned}$$
(16)

with

$$\begin{aligned}&\beta _1 = \bf{Q}_0 + \bf{Q}_1 + \bf{Q}_2 + \bf{Q}_3 + 6 \Delta ', \end{aligned}$$
(17a)
$$\begin{aligned}&\beta _2 = \left| P_1 \right| ^2 + \left| P_2 \right| ^2 + \left| P_3 \right| ^2 - \bf{Q}_0 \left( \bf{Q}_1 + \bf{Q}_2 + \bf{Q}_3 \right) -\bf{Q}_1 \left( \bf{Q}_2 + \bf{Q}_3 \right) \nonumber \\&\qquad \qquad - \bf{Q}_2 \bf{Q}_3 - \delta \left( 6 \bf{Q}_0 + 5 \bf{Q}_1 + 4 \bf{Q}_2 + 3 \bf{Q}_3 \right) - 11 \Delta '^{\;2}, \end{aligned}$$
(17b)
$$\begin{aligned}&\beta _3 = \left( \left| P_1 \right| ^2 - \bf{Q}_0 \bf{Q}_1 \right) \left( \bf{Q}_2 + \bf{Q}_3 \right) + \left| P_2 \right| ^2 \left( \bf{Q}_0 + \bf{Q}_3 \right) \nonumber \\&\qquad \qquad + \left( \left| P_3 \right| ^2 - \bf{Q}_2 \bf{Q}_3 \right) \left( \bf{Q}_0 + \bf{Q}_1 \right) + \Delta ' \left( 5 \left| P_1 \right| ^2 + 3 \left| P_2 \right| ^2 + \left| P_3 \right| ^2 \right) \nonumber \\&\qquad \qquad - \Delta ' \left( \bf{Q}_0 \left( 5 \bf{Q}_1 + 4 \bf{Q}_2 + 3 \bf{Q}_3 ) \, \right) + \left( \bf{Q}_1 ( 3 \bf{Q}_2 + 2 \bf{Q}_3 ) - \bf{Q}_2 \bf{Q}_3 \right) \right) \nonumber \\&\qquad \qquad - \Delta '^{\;2} \left( 11 \bf{Q}_0 + 6 \bf{Q}_1 + 3 \bf{Q}_2 + 2 \bf{Q}_3 \right) - 6 \Delta '^{\;3}, \end{aligned}$$
(17c)
$$\begin{aligned}&\beta _4 = \bf{Q}_3 \left( \bf{Q}_0 \left| P_2 \right| ^2 + \bf{Q}_2 \left| P_1 \right| ^2 \right) + \bf{Q}_0 \bf{Q}_1 \left( \left| P_3 \right| ^2 - \bf{Q}_2 \bf{Q}_3 \right) - \left| P_1 \right| ^2 \left| P_3 \right| ^2 \nonumber \\&\qquad \qquad + \Delta ' \left( \left| P_1 \right| ^2 ( 3 \bf{Q}_2 + 2 \bf{Q}_3 ) + \bf{Q}_0 \left( 3 \left| P_2 \right| ^2 + \left| P_3 \right| ^2 \right) - 3 \bf{Q}_0 \bf{Q}_1 \bf{Q}_2 \right) \nonumber \\&\qquad \qquad - \Delta ' \bf{Q}_0 \bf{Q}_3 ( 2 \bf{Q}_1 + \bf{Q}_2 ) + \Delta '^{\;2} \left( 6 \left| P_1 \right| ^2 - \bf{Q}_0 ( 6 \bf{Q}_1 + 3 \bf{Q}_2 ) \right) \nonumber \\&\qquad \qquad - 2 \Delta '^{\;2} \bf{Q}_0 \bf{Q}_3 - 6 \bf{Q}_0 \Delta '^{\;3}. \end{aligned}$$
(17d)

To solve the differential equation in Eq. (16), we substitute the proposed solution \(\text{e}^{- i X \tau }\) into the equation. This substitution leads to a fourth-order algebraic equation. Consequently, the solution to Eq. (16) can be expressed as a linear combination of the roots of this equation. At this stage, it is also essential to specify the initial state of the system. We assume that initially, the atoms are in the excited state, and the field is in the standard coherent state. Therefore, the initial state of the system is represented by the following equation

$$\begin{aligned} \left| \psi _I (0) \right\rangle = \left| e_{1}e_{2}e_{3} \right\rangle \otimes \left| \alpha \right\rangle = \sum _{n=0}^{\infty } q_n \left| \mathfrak {u}_{1},n \right\rangle , \end{aligned}$$
(18)

where

$$\begin{aligned} q_n = \exp \left( - \frac{| \alpha |^{2}}{2}\right) \frac{\alpha ^{n}}{\sqrt{n !}}. \end{aligned}$$
(19)

As a result, the coefficient \(A_n (\tau )\) is given by

$$\begin{aligned} A_n (\tau ) = \sum _{\ell = 1}^{4} a_{\ell } \, \text{e}^{- i X_\ell \tau } = q_n \sum _{\ell = 1}^{4} r_{\ell } \, \text{e}^{- i X_\ell \tau }, \qquad X_\ell = \Omega _\ell + \frac{\beta _1}{4}, \end{aligned}$$
(20)

where \(a_\ell = q_n r_\ell\) (with \(r_\ell\) being real) and \(q_n\) is given by Eq. (19). Additionally, the \(\Omega _\ell\) values (\(\ell = 1, 2, 3, 4\)) are the roots of the following fourth-order algebraic equation

$$\begin{aligned} \Omega ^4 - \left( \frac{3}{8} \beta _1^2 +\beta _2 \right) \Omega ^2 - \left( \frac{1}{8} \beta _1^3 +\frac{1}{2} \beta _1 \beta _2 \right) \Omega - \left( \frac{3}{256} \beta _1^4 +\frac{1}{16} \beta _1^2 \beta _2 - \frac{1}{4} \beta _1 \beta _3 + \beta _4 \right) = 0, \end{aligned}$$
(21)

which serves as the “characteristic equation” for the reduced differential equation associated with Eq. (16). Likewise, the other coefficients of the wave function in Eq. (11) are given by

$$\begin{aligned}&B_n (\tau ) = C_n (\tau ) = E_n (\tau ) = \dfrac{q_n}{\sqrt{3} \, P_1} \, \sum _{\ell = 1}^{4} b_\ell \, r_{\ell } \, \text{e}^{- i \left( X_\ell - \delta \right) \tau }, \qquad b_\ell = X_\ell - \bf{Q}_{0}, \end{aligned}$$
(22a)
$$\begin{aligned}&D_n (\tau ) = F_n (\tau ) = G_n (\tau ) = \dfrac{q_n}{\sqrt{3} \, P_1 P_2} \, \sum _{\ell = 1}^{4} d_\ell \, r_{\ell } \, \text{e}^{- i \left( X_\ell - 2 \delta \right) \tau }, \qquad d_\ell = b_\ell \left( X_\ell - \bf{Q}_1 - \delta \right) - \left| P_1 \right| ^2, \end{aligned}$$
(22b)
$$\begin{aligned}&L_n (\tau ) = \dfrac{q_n}{ P_1 P_2 P_3 } \, \sum _{\ell = 1}^{4} c'_\ell \, r_{\ell } \, \text{e}^{- i \left( X_\ell - 3 \delta \right) \tau }, \qquad c'_\ell = d_\ell \left( X_\ell - \bf{Q}_2 -2 \delta \right) - b_\ell \left| P_2 \right| ^2. \end{aligned}$$
(22c)

By applying the initial conditions of the system as \(A_n (0) = q_n\) and \(B_n (0) = C_n (0) = \cdots = L_n (0) = 0,\) the coefficients \(r_\ell\) are determined by solving the following matrix equation

$$\begin{aligned} \begin{pmatrix} 1 & 1 & 1 & 1 \\ b_1 & b_2 & b_3 & b_4 \\ d_1 & d_2 & d_3 & d_4 \\ c'_1 & c'_2 & c'_3 & c'_4 \end{pmatrix} \begin{pmatrix} r_1 \\ r_2 \\ r_3 \\ r_4 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \end{aligned}$$
(23)

Having obtained the explicit form of the state vector of the system, we can now proceed to examine its quantum properties.

Atomic population inversion

We are now prepared to investigate the dynamics of atomic population inversion representing the energy exchange between the atoms and the cavity field69. This quantity is typically defined as \(W(\tau ) = \text{Tr}(\hat{\rho }_{A} \hat{\sigma }_{z})\), where \(\hat{\rho }_{A}\) denotes the atomic density operator, \(\hat{\rho }_A \equiv \hat{\rho }_{A_{1} A_{2} A_{3}} = \text{Tr}_{\text {field}} \left( \hat{\rho }_{\text {total}} \right)\), where \(\hat{\rho }_{\text {total}} = \left| \psi _I(\tau ) \right\rangle \left\langle \psi _I(\tau ) \right|\) represents the total density operator of the system, and the partial trace is taken over the field states, and \(\hat{\sigma }_{z} = \left. (\hat{\sigma }_{z}^{(1)} + \hat{\sigma }_{z}^{(2)} + \hat{\sigma }_{z}^{(3)} \right. )/3\) indicates the normalized operator of atomic population inversion for the three atoms, therefore, we expect \(-1 \le W(\tau ) \le 1\). The matrix representation of \(\hat{\rho }_{A}\) is provided in Eq. (25). For the system under consideration, the atomic population inversion is given by

$$\begin{aligned} W (\tau ) = \rho _{11} + \rho _{22} - \rho _{44} - \rho _{88} = \sum _{n=0}^{+\infty } \left( \left| A_n (\tau ) \right| ^2 + \left| B_n (\tau ) \right| ^2 - \left| D_n (\tau ) \right| ^2 - \left| L_n (\tau ) \right| ^2 \right) . \end{aligned}$$
(24)

where the derivation of this equation is provided in the Appendix A1.

It is important to note that the parameter values were chosen based on well-established physical principles and relevant literature. The coupling parameter (\(\lambda\)) is typically on the order of \(10^8 \, \text {rad/s}\)70. The detuning parameter (\(\Delta\)) must satisfy \(\Delta \ll \omega , k\omega _0\)71 and is generally selected to be a few times \(\lambda\)72,73, comparable to the natural linewidth74. For the Kerr parameter (\(\chi\)), the ratio \(\chi /\lambda\) follows analytical formulations75,76 and is conventionally chosen within the range 0.001 to 577,78.

Figure 1 illustrates the dynamics of atomic inversion under the effects of intensity-dependent coupling (f(n)), Kerr nonlinearity (\(\chi '\)) and f-deformed Kerr nonlinearity (g(n)), and detuning parameter (\(\Delta '\)) in the presence of one-photon transition. Each plot contains two graphs, corresponding to two different values of detuning. The rows illustrate the impact of the Kerr strength and the intensity dependence of the Kerr nonlinearity, while the columns depict the role of intensity-dependent coupling, with \(f(n) = 1\) in the left column, \(f(n) = \sqrt{n}\) in the middle column, and \(f(n) = 1 / \sqrt{n}\) in the right column. In the first row, where \(\chi ' = 0\), the Kerr effect is absent, and the inversion dynamics are determined solely by the coupling \(f(n)\) and detuning \(\Delta '\), leading to relatively simple patterns that explicitly include the quantum phenomena of collapse and revival. The second row, \(\chi \ne 0\) and \(g(n) = 1\), shows the Kerr effect is introduced but is not intensity-dependent, resulting in collapse and revival for the case of \(f(n) = 1/ \sqrt{n}\). The third row introduces intensity dependence in the Kerr nonlinearity with \(g(n) = \sqrt{n}\), indicating chaotic oscillatory behavior, specially in two right columns. A comparison of the first and fourth rows reveals no significant difference in their qualitative behavior. The columns reflect the role of the intensity-dependent atom-field coupling f(n). It is found that the second column including \(f(n) = \sqrt{n}\) shows typical behavior with collapse and revival. Concerning detuning parameter, when \(\Delta ' = 0\), the system is near resonance, resulting in symmetric and regular oscillations. As detuning increases to moderate values (\(\Delta ' = 5, 20\)), asymmetry and modulations appear in the inversion patterns.

Fig. 1
figure 1

The time evolution of the atomic population inversion is analyzed as a function of the scaled time \(\tau\), assuming that the atoms and the field are initially in the excited state and coherent state, respectively, with \(|\alpha |^{2} = 25\), and one-photon transition. The study also explores the effects of Kerr nonlinearity (\(\chi '\)), \(f\)-deformed Kerr nonlinearity (\(g(n)\)), intensity-dependent coupling (\(f(n)\)), and detuning (\(\Delta '\)) on the dynamics of the population inversion.

Full size image

In Fig. 2, the population inversion is plotted for \(k=2\) with parameters similar to those in Fig. 1. Generally, the qualitative behavior of these two figures is similar. However, in some instances, the temporal behavior of the inversion changes more rapidly than in Fig. 1. Another noteworthy point is that in most graphs, the phenomena of collapse and revival are clearly observed.

Fig. 2
figure 2

The time evolution of the atomic population inversion is shown as a function of the scaled time \(\tau\), using parameters similar to those in Fig. 1, except for the inclusion of a two-photon transition.

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Quantum entanglement

The entanglement in a multipartite system emerges from the joint information among its subsystems. Since atom-field interactions are often considered a simple and effective way to generate entangled states, it is important to investigate the degree of entanglement (DEM) between different subsystems; specifically here, the three atoms and the field. In this section, we numerically evaluate the DEM between subsystems, including atom-field entanglement using von Neumann reduced entropy and atom-atom entanglement using concurrence. To calculate these entanglement measures, the reduced density matrix of the three atoms is required. This matrix can be expressed as \(\hat{\rho }_A = \text{Tr}_{\text {field}} \left( \hat{\rho }_{\text {total}} \right)\). Since \(B_n (\tau ) = C_n (\tau ) = E_n (\tau )\) and \(D_n (\tau ) = F_n (\tau ) = G_n (\tau )\) in the \(\left| \mathfrak {u}_j \right\rangle\)’s basis, the reduced density operator of the atoms can be written as

$$\begin{aligned} \hat{\rho }_{A} (\tau ) = \begin{pmatrix} \rho _{11} (\tau ) & \rho _{12} (\tau ) & \rho _{12} (\tau ) & \rho _{14} (\tau ) & \rho _{12} (\tau ) & \rho _{14} (\tau ) & \rho _{14} (\tau ) & \rho _{18} (\tau ) \quad\\ \rho _{12}^* (\tau ) & \rho _{22} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{24} (\tau ) & \rho _{28} (\tau ) \quad\\ \rho _{12}^* (\tau ) & \rho _{22} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{24} (\tau ) & \rho _{28} (\tau ) \quad\\ \rho _{14}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{44} (\tau ) & \rho _{48} (\tau ) \quad\\ \rho _{12}^* (\tau ) & \rho _{22} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{22} (\tau ) & \rho _{24} (\tau ) & \rho _{24} (\tau ) & \rho _{28} (\tau ) \quad\\ \rho _{14}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{44} (\tau ) & \rho _{48} (\tau ) \quad\\ \rho _{14}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{24}^* (\tau ) & \rho _{44} (\tau ) & \rho _{44} (\tau ) & \rho _{48} (\tau ) \quad\\ \rho _{18}^* (\tau ) & \rho _{28}^* (\tau ) & \rho _{28}^* (\tau ) & \rho _{48}^* (\tau ) & \rho _{28}^* (\tau ) & \rho _{48}^* (\tau ) & \rho _{48}^* (\tau ) & \rho _{88} (\tau ) \end{pmatrix}. \end{aligned}$$
(25)

The ten independent elements of this matrix are given by

$$\begin{aligned}&\rho _{11} (\tau ) = \sum _{n=0}^{+\infty } \left| A_n (\tau ) \right| ^2, & \rho _{12} (\tau ) = \sum _{n=0}^{+\infty } A_{n+k} (\tau ) B_n^*(\tau ), \nonumber \\&\rho _{14} (\tau ) = \sum _{n=0}^{+\infty } A_{n+2k} (\tau ) D_n^*(\tau ), & \rho _{18} (\tau ) = \sum _{n=0}^{+\infty } A_{n+3k} (\tau ) L_n^*(\tau ), \nonumber \\&\rho _{22} (\tau ) = \sum _{n=0}^{+\infty } \left| B_n (\tau ) \right| ^2, & \rho _{24} (\tau ) = \sum _{n=0}^{+\infty } B_{n+k} (\tau ) D_n^*(\tau ), \nonumber \\&\rho _{28} (\tau ) = \sum _{n=0}^{+\infty } B_{n+2k} (\tau ) L_n^*(\tau ), & \rho _{44} (\tau ) = \sum _{n=0}^{+\infty } \left| D_n (\tau ) \right| ^2, \nonumber \\&\rho _{48} (\tau ) = \sum _{n=0}^{+\infty } D_{n+k} (\tau ) L_n^*(\tau ), & \rho _{88} (\tau ) = \sum _{n=0}^{+\infty } \left| L_n (\tau ) \right| ^2. \end{aligned}$$
(26)

Entanglement between three atoms and optical field

In a multipartite quantum system described by a pure state, quantum entropy serves as an appropriate measure for quantifying the DEM between its subsystems79,80. The system under consideration consists of two components: three atoms and a single-mode field. Araki and Lieb81,82 showed that for a bipartite quantum system, the entropies of the total system and its subsystems, referred to as the atoms and the field, are bounded by the triangle inequality \(\left| S_A (\tau ) - S_F (\tau ) \right| \le S_T \le \left| S_A (\tau ) + S_F (\tau ) \right|\) where \(S_T\) represents the total entropy of the system, and the subscripts “\(A\)” and “\(F\)” stand for the atomic and field subsystems, respectively. For a system initialized in a pure state, such as the one under consideration, the total entropy \(S_T\) is zero and remains unchanged. This implies that at any time \(t> 0 \; (\tau > 0)\), the entropies of the atomic and field subsystems are equal, i.e., \(S_A(\tau ) = S_F(\tau )\), as discussed in83,84.

Therefore, to assess the DEM, it is sufficient to compute the reduced entropy of either the atomic or the field subsystem. For this purpose, the von Neumann entropy provides a way to calculate the reduced entropy of the atomic subsystem, which is given by

$$\begin{aligned} S_A (\tau ) = - \text {Tr}_A\left( \hat{\rho }_A (\tau ) \ln \hat{\rho }_A (\tau ) \right) , \end{aligned}$$
(27)

where \(\hat{\rho }_A (\tau )\) is the reduced density matrix of the atomic subsystem, as defined in Eq. (25). Expanding this expression, the entropy becomes:

$$\begin{aligned} S_A (\tau ) = - \sum _{j=1}^{8} \xi _j \ln \xi _j, \end{aligned}$$
(28)

where \(\xi _j\) are the eigenvalues of the reduced density matrix for the three-atom system.

Figure 3 investigates the entanglement dynamics between a single cavity mode and three atoms, under various coupling and detuning conditions. The y-axis represents the von Neumann entropy while the x-axis corresponds to scaled time \(\tau\). The panels are organized into rows labeled (a) (Fig. 3a1, a2, a3), (b) (Fig. 3b1, b2, b3), (c) (Fig. 3c1, c2, c3), and (d) (Fig. 3d1, d2, d3), corresponding to different conditions of Kerr nonlinearity and f-deformed Kerr nonlinearity via nonlinear function g(n). The columns (1), (2), and (3) illustrate entanglement dynamics for three specific atom-field coupling regimes: constant coupling (\(f(n) = 1\)) and two types of nonlinearity function, (\(f(n) = \sqrt{n}\) and \(f(n) = 1/\sqrt{n}\)). Red and blue plots in each panel show different detuning values (\(\Delta '\)), with red indicating zero detuning (\(\Delta ' = 0\)) and blue corresponding to non-zero detuning values (\(\Delta ' > 0\)). Looking at the rows, we observe that the Kerr effect and the intensity-dependent Kerr effect with nonlinearity of \(g(n) = \sqrt{n}\) significantly enhance the atoms-field entanglement. This enhancement becomes even more pronounced when operating in the intensity-dependent atom-field coupling regime with \(f(n) = 1 / \sqrt{n}\). Furthermore, it is evident from the figure that a nonzero detuning parameter (\(\Delta ' > 0\)) generally reduces the amount of entanglement, except in the plots of column (2) (Figs. 3a2, b2, c2, d2). A particularly notable case is shown in Fig. 3c2, where the system operates under intensity-dependent coupling and the f-deformed Kerr effect with \(f(n) = \sqrt{n} = g(n)\), leading to complex but rich entanglement dynamics. Overall, the figure emphasizes the controllability of entanglement dynamics through the tuning of parameters such as \(f(n)\), \(g(n)\), \(\chi '\), and \(\Delta '\). This controllability enables the tailoring of quantum correlations for potential applications in quantum computing, quantum information processing, and quantum communication systems.

Fig. 3
figure 3

The time evolution of the atoms-field entanglement is shown as a function of the scaled time \(\tau\), using parameters similar to those in Fig. 1.

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Figure 4 illustrates the entanglement dynamics between the atomic subsystem and the cavity mode for parameters similar to those in Fig. 3, except that the system now considers the degenerate two-photon transition (\(k=2\)). Comparing the two left columns of Fig. 3 and Fig. 4 reveals that, in the case of a two-photon transition, the detuning parameter (\(\Delta '\)) has no significant effect on the dynamics of entanglement. Examining Fig. 3a3, d3 and comparing them with the corresponding plots in Fig. 4 demonstrates that the two-photon transition process increases the amount of entanglement. Furthermore, focusing on the middle columns of Figs. 3 and 4, it is evident that in the case of \(k=2\), the atoms-field entanglement is significantly enhanced, and the entanglement dynamics fluctuate around a higher maximum value. These observations highlight the potential of the two-photon transition to improve entanglement and stabilize its dynamics in such quantum systems.

Fig. 4
figure 4

The time evolution of the atoms-field entanglement is shown as a function of the scaled time \(\tau\), using parameters similar to those in Fig. 2.

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Entanglement between atoms: Atom 1 (\(\mathrm {\mathbf {A_{1}}}\)) and Atom 2 (\(\mathrm {\mathbf {A_{2}}}\))

The concurrence85 is a commonly used measure to quantify entanglement in a two-level system (qubit). It is given by the expression \(C = \max \left[ 0, 2 \max \left[ \vartheta _{l}\right] - \sum _{l=1}^{4} \vartheta _{l}\right] ,\) where the \(\vartheta _{l}\)’s are the square roots of the eigenvalues of the non-Hermitian matrix \(\hat{\rho }(\hat{\sigma }_y \otimes \hat{\sigma }_y) \hat{\rho }^*(\hat{\sigma }_y \otimes \hat{\sigma }_y),\) with \(\hat{\rho }^*\) representing the Hermitian conjugate of \(\hat{\rho }\), and \(\hat{\sigma }_y\) being the Pauli matrix along the y-axis. In order to obtain the DEM between atoms \(\mathrm {A_1}\) and \(\mathrm {A_2}\), we need their reduced density operator. By taking the trace over the states of Atom 3 from the atomic density operator \(\hat{\rho }_A\) defined in Eq. (25), we arrive at the reduced density matrix for atoms \(\mathrm {A_1}\) and \(\mathrm {A_2}\), which is expressed as

$$\begin{aligned} \hat{\rho }_{A_1 A_2} = \begin{pmatrix} \rho _{11} + \rho _{22} & \rho _{12} + \rho _{24} & \rho _{12} + \rho _{24} & \rho _{14} + \rho _{28} \\ \rho _{12}^* + \rho _{24}^* & \rho _{22} + \rho _{44} & \rho _{22} + \rho _{44} & \rho _{24} + \rho _{48} \\ \rho _{12}^* + \rho _{24}^* & \rho _{22} + \rho _{44} & \rho _{22} + \rho _{44} & \rho _{24} + \rho _{48} \\ \rho _{14}^* + \rho _{28}^* & \rho _{24}^* + \rho _{48}^* & \rho _{24}^* + \rho _{48}^* & \rho _{44} + \rho _{88} \\ \end{pmatrix}, \end{aligned}$$
(29)

where the elements \(\rho _{ij}\) are defined in Eq. (26). It is worth mentioning that the measure of concurrence implies that a separable state corresponds to \(C = 0\), a maximally entangled state corresponds to \(C = 1\), and a partially entangled state lies in the range \(0< C < 1\).

Figure 5 depicts the dynamics of entanglement between Atom 1 (A\(_1\)) and Atom 2 (A\(_2\)), quantified by concurrence, for three atoms interacting with a single-mode cavity field under varying system parameters. The concurrence is plotted as a function of the scaled time \(\tau\), with different rows corresponding to the absence and presence of Kerr and f-deformed Kerr effects, and different columns representing variations in the intensity-dependent atom-field coupling. In Figs. 5a1,b1 ,c1 , d1 where the atom-field coupling is constant, \(f(n) = 1\), the concurrence varies between zero and its maximum value, gradually increasing over time. Introducing intensity-dependent coupling increases the maximum value of atom-atom entanglement (Figs. 5a2, b2, c2, d2). However, in the case of \(f(n) = 1 / \sqrt{n}\) (Figs. 5a3, b3, c3, d3), entanglement is completely suppressed. In row (b), where the Kerr nonlinearity (\(\chi ' = 0.03\)) is considered, it is observed that the Kerr effect decreases the entanglement in the cases of \(f(n) = 1\) and \(f(n) = \sqrt{n}\). However, in the intensity-dependent coupling regime with \(f(n) = 1 / \sqrt{n}\), the Kerr effect enhances the entanglement slightly. Furthermore, in row (c), the inclusion of f-deformed Kerr nonlinearity reduces the entanglement, regardless of the coupling regime or the form of \(f(n)\). Examining the overall behavior in Fig. 5, it is evident that the intensity-dependent atom-field coupling has a limited influence on the entanglement dynamics, except in cases where it completely suppresses the entanglement (e.g., \(f(n) = 1 / \sqrt{n}\)). Additionally, the detuning parameter \(\Delta '\) generally reduces the atom-atom entanglement, further highlighting the sensitivity of the system to tuning parameters. This figure emphasizes the interplay between Kerr effects, intensity-dependent coupling, and detuning in shaping atom-atom entanglement dynamics.

Fig. 6 illustrates the entanglement dynamics between Atom 1 (A\(_1\)) and Atom 2 (A\(_2\)) for parameters similar to those in Fig. 5, except that the system involves a two-photon transition (\(k=2\)). The figure explores the role of intensity-depndent coupling by applying \(f(n) = \sqrt{n}\) (Figs. 6a2, b2, c2, d2) and \(f(n) = 1/\sqrt{n}\) (Figs. 6a3, b3, c3, d3). Eliminating Fig. 6b3 and comparing Fig. 5 with Fig. 6 indicates that, in the case of a two-photon transition, the detuning parameter (\(\Delta '\)) has no noteworthy effect on the dynamics of atom-atom entanglement. Additionally, it can be observed that, in the linear regime (Figs. 6a1, b1, c1, d1), the two-photon transition process enhances the amount of entanglement compared to the single-photon transition case. This demonstrates the potential advantage of two-photon transitions in improving quantum correlations between atoms.

Fig. 5
figure 5

The time evolution of entanglement between Atom 1 (A\(_1\)) and Atom 2 (A\(_2\)) is shown as a function of the scaled time \(\tau\), using parameters similar to those in Fig. 1.

Full size image
Fig. 6
figure 6

The time evolution of entanglement between Atom 1 (A\(_1\)) and Atom 2 (A\(_2\)) is shown as a function of the scaled time \(\tau\), using parameters similar to those in Fig. 2.

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Summary and conclusion

This study investigated the atomic population inversion and entanglement dynamics of a quantum system consisting of three two-level atoms interacting with a single-mode cavity field, with the cavity surrounded by a Kerr medium exhibiting f-deformed Kerr nonlinearity. The paper also considered a degenerate multi-photon transition in the intensity-dependent regime, analyzing the effects of intensity-dependent coupling, detuning, Kerr and f-deformed Kerr nonlinearity, and multi-photon transition on the nonclassical properties of the system. The results demonstrated the complex interplay between these parameters and their influence on both atomic inversion and entanglement. For atomic inversion, the system exhibited collapse and revival phenomena under different coupling and detuning conditions, with detuning affecting the symmetry and oscillatory patterns of inversion. The introduction of Kerr nonlinearity and f-deformed Kerr nonlinearity significantly altered these dynamics, particularly in the presence of intensity-dependent coupling. Regarding entanglement, both the Kerr and f-deformed Kerr nonlinearities enhanced the entanglement between the atoms and the cavity field, with the entanglement dynamics being sensitive to the coupling function and detuning. The two-photon transition was found to notably increase and stabilize entanglement, suggesting its potential for enhancing quantum correlations. Overall, the study provided important insights into the role of nonlinearities and multi-photon transition in the quantum dynamics of this system. It highlighted the significant impact of Kerr and f-deformed Kerr nonlinearities on atomic inversion and entanglement dynamics. While detuning generally reduced entanglement, intensity-dependent atom-field coupling and two-photon transition offered promising avenues to control and enhance entanglement. These findings are crucial for tailoring quantum correlations, which are essential for applications in quantum computing, communication, and information processing, and emphasize the importance of considering multi-photon transition and nonlinearity when designing quantum systems for practical quantum technologies.