Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 2, 2023



Article Number  46  
Number of page(s)  7  
DOI  https://doi.org/10.1051/jeos/2023044  
Published online  22 December 2023 
Research Article
Quantum coherence and entanglement of the system of a fivelevel atom in the presence of nonlinear fields
^{1}
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
^{2}
Imam Mohammad Ibn Saud Islamic University (IMSIU), College of Science, Department of Physics, Riyadh 11432, Saudi Arabia
^{3}
The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 MiramareTrieste, Italy
^{4}
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
^{5}
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow 119991, Russia
^{*} Corresponding author: sabotalb@tu.edu.sa
Received:
11
August
2023
Accepted:
27
November
2023
We investigate the atomfield system in the framework of harmonic oscillators based on deformed Heisenberg algebras. We explore the dynamic characteristics of the considered system under the effect of a nonlinear field. In particular, we consider the atomic population, atomic coherence, and atomfield entanglement for a system that comprises a single fivelevel atom interacting with a singlemode nonlinear field when the deformation effect is taken into account. We examine the time evolution of the quantum quantifiers in the presence of deformation when the initial state of the quantized field is defined to be a nonlinear coherent state (CS) or a superposition state.
Key words: Nonlinear fields / Quantum algebra / Deformed Heisenberg algebra / Superposition field state / Quantum correlation / Quantum coherence
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In the realm of quantum optics, the model that sees the most widespread use is known as the JaynesCummings model [1]. This model depicts how an atom reacts when it comes into contact with an electromagnetic field. A broad framework of a twolevel atom interacting with a onemode field with various kinds of nonlinearities has been developed [2]. In addition, many studied have been conducted on the interaction that occurs between a qubit and a twomode cavity field [3–5]. The interaction between a threelevel atom with various configurations (Ξ, Λ, and V) and a one or twomode field was studied [6–8]. Recent research has focused on investigating fourlevel systems in a variety of configurations to see how these systems interact with cavity modes [9]. In this context, a wide variety of fourlevel atomic system designs have been investigated and shown [10–17]. The subject of the interaction of fivelevel systems with Mtype and Ktype configurations has garnered a lot of attention in the domains of quantum optics and quantum information. The spontaneous emission in a fivelevel atomic system driven by four fields [18], the phasedependent optical properties of fivelevel Ktype atom systems [19], the effect of decoherence in a fivelevel atom system with Ξconfiguration [20], and the controllable entanglement of light in a fivelevel system [21] are some of the optical phenomena that have been investigated. On the other hand, a preliminary analysis of populations in a fivelevel atom with various photon distributions was considered [22].
Many quantum phenomena have been produced as resources for carrying out numerous tasks in the field of quantum optics and information, both theoretically and practically. This development took place over the course of a few years. In most cases, the coherent superposition of quantum states is regarded as an essential need for the existence of quantum correlations [23–29]. The idea of the EPR paradox [30] that was presented by Einstein, Podolsky, and Rosen was the impetus behind the establishment of this precondition. They described how the theory of quantum physics enables spooky activity at a distance Schrödinger viewed the problem as one in which it was possible for local observations to influence a distant system even if he did not have access to it [31]. After that, Bell came up with the socalled Bell inequality as a way to demonstrate that this “spooky” phenomenon was not a coincidence. Subsequently, Bell presented the socalled Bell inequality to demonstrate that this “spooky activity” causes a quantum correlation that defies any classical explanation [32]. This was done in order to prove that the existence of this quantum correlation cannot be explained using conventional methods. Quantum coherence is the fundamental concept that underpins a variety of quantum phenomena that occur in nanomaterials [33, 34], quantum measurements and quantum metrology [35–39], and applications of quantum mechanics to biological things [40–42]. In line with the fundamental significance of quantum coherence, it was not until recently that an accurate theory of coherence was emphasized with the constraints necessary to ensure that quantum coherence is a physical resource [43]. This was done in light of the fact that quantum coherence has been given increasing attention in recent years. As a result, many quantum measures that check these limitations have been developed, most notably those that are based on the l_{1} norm and relative entropy [43]. In addition, the quantum coherence may be measured by either the convexroof construction or nonlocal correlation [44, 45], and an operational theory of coherence has been developed [46].
Quantum entanglement is the aspect of a composite quantum system that most captivates my attention as a fascinating occurrence. If the combined state of two particles cannot be represented as a product of the states of their constituent subsystems, then the particles are said to be entangled [30]. A deeper comprehension of fundamental quantum processes may be attained through the production and manipulation of these entangled states. Examples of complicated entangled states that are utilized in tests of quantum nonlocality include triplets of particles named after Greenberger, Horne, and Zeilinger [47, 48]. In addition to these fundamental qualities, entanglement has developed into a crucial resource in quantum information processing, which is a field that has made significant strides in the past few years [49, 50]. In the present manuscript, we investigate the atomfield system in the framework of harmonic oscillators based on deformed Heisenberg algebras. We explore the dynamic characteristics of the considered system under the effect of a nonlinear field. We consider a system that comprises a single fivelevel atom interacting with a singlemode nonlinear field when the deformation effect is considered. We analyze the time evolution of the atomic population, atomic coherence, and atomfield entanglement in the presence of deformation when the initial state of the quantized field is defined to be a nonlinear CS or a superposition state.
The remaining parts of the article are structured as follows. In Section 2, a model of a fivelevel cascadetype atom and onemode system is shown to the reader. In addition to that, the wave function of the atomic system under consideration is shown below. The quantum quantifiers related to the atomfield system and the results of numerical simulations are discussed in Section 3. In Section 4, a summary and some conclusions are presented.
2 Physical model and dynamics
The states of the oscillator are characterized using Schrödinger cat states in deformed spaces. We choose an deformed algebra to define the Hamiltonian of an fdeformed oscillator as follows, assuming that the oscillator has a unitary frequency,$$H=\frac{\mathrm{\hslash}}{2}\left({\widehat{C}}^{+}\widehat{C}+\widehat{C}{\widehat{C}}^{+}\right),$$(1)where the operators Ĉ and Ĉ ^{+} are considered to obey the following commutator algebras$$\left[\widehat{C},\widehat{n}\right]=\widehat{C},\left[{\widehat{C}}^{\u2020},\widehat{n}\right]={\widehat{C}}^{+}.$$(2)
As for the standard operators of bosons that are nondeformed, it is not necessary for Ĉ and Ĉ ^{+} to be connected to photon number operator $\widehat{n}$ in the conventional manner; that is, in general, ${\widehat{C}}^{+}\widehat{C}=\left[\widehat{n}\right]\ne \widehat{n}$, where is called a box function that determines the field nonlinearity that CC ^{+} = [n + 1]. The definition of the commutation relation of operators Ĉ and Ĉ ^{+} is given as follows:$$\widehat{C}{\widehat{C}}^{+}{\widehat{C}}^{+}\widehat{C}=\left[\widehat{n}+1\right]\left[\widehat{n}\right].$$(3)
It is possible to see the deformed oscillators as particular examples of the socalled goscillators, in which the goscillator operators are specified as$$\widehat{C}=\widehat{c}g\left(\widehat{n}\right)=g\left(\widehat{n}+1\right)\widehat{c},$$(4) $${\widehat{C}}^{+}=g\left(\widehat{n}\right){\widehat{c}}^{\u2020}={\widehat{c}}^{\u2020}g\left(\widehat{n}+1\right),$$(5)and that$$\left[\widehat{C},{\widehat{C}}^{+}\right]=\left(\widehat{n}+1\right){g}^{2}\left(\widehat{n}+1\right)\widehat{n}\mathrm{g}\left(\widehat{n}\right),$$(6)where $g\left(\widehat{n}\right)$ represents an operator function of the Hermitian number operator $\widehat{n}={\widehat{c}}^{+}\widehat{c}$. The nonlinearity results from the dependence variable g. In the g → 1 limit, the deformed algebra will be reduced to the Heisenberg algebra.$$\left[\widehat{c},{\widehat{c}}^{+}\right]=1,\left[n,\widehat{c}\right]=\widehat{c}\mathrm{and}\left[n,{\widehat{c}}^{+}\right]={\widehat{c}}^{+}.$$(7)
The operators Ĉ and Ĉ ^{+} act on the Fock states as:$${\widehat{C}}^{\u2020}n\rangle =\sqrt{[n+1{]}_{g}}n+1\rangle ,\widehat{C}n\rangle =\sqrt{[n{]}_{g}}n1\rangle .$$(8)where n〉 form an orthonormal basis in the deformed Hilbert space.
In comparison to the Glauber states, the gCSs are defined in terms of eigenvectors of the deformed operator Ĉ $$\widehat{C}\xi ,g\rangle =\xi \xi ,g\rangle ,$$(9)where ξ is a complex eigenvalue of Ĉ. The deformed CSs are formulated as$$\xi ,g\rangle ={\mathcal{N}}_{g}\sum _{n=0}^{\infty}\mathrm{}\frac{{\xi}^{n}}{\sqrt{[n{]}_{g}!}}n\rangle ,\hspace{1em}{\mathcal{N}}_{g}={\left[\mathrm{ex}{\mathrm{p}}_{g}\left[\xi {}^{2}\right]\right]}^{\frac{1}{2}},$$(10)where we have introduced$$\mathrm{ex}{\mathrm{p}}_{g}\left[x\right]=\sum _{n=0}^{\infty}\mathrm{}\frac{{x}^{n}}{{\left[n\right]}_{g}!},{\left[n\right]}_{g}!=\left[n{g}^{2}\left(n\right)\right]\times \left[\left(n1\right){g}^{2}\left(n1\right)\right]\times \cdot \cdot \cdot \times \left[{g}^{2}\left(1\right)\right].$$(11)
The function exp_{ g } is a deformed version of the usual exponential function with exp_{ g }[x]exp_{ g }[y] ≠ exp_{ g }[x + y] and they coincide when the function g tends to the value one. Here, we use the effect of qdeformation of the box function [51]$$g\left(n\right)={\left[\frac{1}{n}\frac{1{q}^{n}}{q1}\right]}^{\frac{1}{2}},q\in \mathbb{R}.$$(12)
It is obvious that g(n) = 1 when q = 1, thus in this circumstance, nonlinear CSs become the standard CSs.
Now, we consider a fivelevel atom in the form of laddertype, as illustrated in Figure 1, with energy ω_{ j } (j = 1, 2, 3, 4, 5 and ω_{1} > ω_{1}>…>ω_{5}) as the energies of transition through each of the five level of the atom. The frequency and the creation (annihilation) operator â ^{+}(â) are the two variables that are used to characterize the field mode. The total Hamiltonian Ĥ for the system that is being investigated is expressed in the rotating wave approximation as$$\widehat{H}={\widehat{H}}_{AF}+{\widehat{H}}_{\mathrm{IN}},$$(13)where the Hamiltonian for the free field and the noninteracting atom is denoted by the Ĥ_{ A–F }, and the interaction component of the total Hamiltonian is given by$${\widehat{H}}_{AF}=\sum _{j}\mathrm{}{\omega}_{j}{\widehat{\sigma}}_{j,j}+\mathrm{\Omega}{\widehat{C}}^{\u2020}\widehat{C}.$$(14)where ${\widehat{\sigma}}_{j,j}=j\rangle \langle j$.
Fig. 1 Schematic diagram of a fivelevel laddertype atom interacting with a singlemode nonlinear field. 
In the meanwhile, we take into consideration the possibility that the connection between the atomic system and the field is mediated by fourphoton processes. These are transitions that need the use of four photons to complete. The interaction Hamiltonian of the system is formulated as$${\widehat{H}}_{\mathrm{IN}}=\sum _{k=1}^{4}\mathrm{}{\beta}_{k}\left[\widehat{C}{\widehat{\sigma}}_{k,k+1}+h.c.\right],$$(15)where β_{ k } is the atomfield coupling constant, and we consider the case of identical coupling (i.e., β_{1} = β_{2} = β_{3} = β_{4} = β).
We assume that the fivelevel atom starts in its upper state 1〉 and the field from the superposition of the gCS. The initial state of the whole system is$$\leftU\right(0)\rangle =\frac{1}{\sqrt{1+{r}^{2}+2r{\mathcal{N}}_{g}^{2}\mathrm{ex}{\mathrm{p}}_{g}\left[\xi {}^{2}\right]}}\left(\xi ,g,1\rangle +r\xi ,g,1\rangle \right)=\sum _{n}\mathrm{}{\varpi}_{n}n,1\rangle .$$(16)
The atomfield wave function at an arbitrarily scaled time T = βt is written in the form$$\leftU\right(T)\rangle =\sum _{n}\mathrm{}{\varpi}_{n}\left[\sum _{k=1}^{5}\mathrm{}{u}_{j}\left(T\right)n+k1\rangle \rangle \underset{}{\otimes}k\rangle \right],$$(17)where ${\varpi}_{n}=\mathrm{exp}\left(\frac{\stackrel{\u0305}{n}}{2}\right)\frac{{\alpha}^{n}}{\sqrt{n!}}$ in the limit q → 1 and r = 0 with $\stackrel{\u0305}{n}=\alpha {}^{2}$is the initial mean photon number for the field mode.
The timedependent coefficients u_{ i } are determined through solving the system of differential equations obtained from the Schrödinger equation$$i\frac{\mathrm{d}}{\mathrm{d}T}\left(\begin{array}{l}{u}_{1}\\ {u}_{2}\\ {u}_{3}\\ {u}_{4}\\ {u}_{5}\end{array}\right)=\left(\begin{array}{lllll}0& {B}_{1}& 0& 0& 0\\ {B}_{1}& 0& {B}_{2}& 0& 0\\ 0& {B}_{2}& 0& {B}_{3}& 0\\ 0& 0& {B}_{3}& 0& {B}_{4}\\ 0& 0& 0& {B}_{4}& 0\end{array}\right)\left(\begin{array}{l}{u}_{1}\\ {u}_{2}\\ {u}_{3}\\ {u}_{4}\\ {u}_{5}\end{array}\right),$$(18)where$${B}_{k}=\beta \sqrt{n+k}\hspace{1em}k=\mathrm{1,2},\mathrm{3,4}.$$
3 Quantum quantifiers and numerical results
This section reviews the basic definitions and concepts of quantumness measures considering the quantum coherence and entanglement. Moreover, we present and discuss the obtained results.
The fundamental properties of coherence depend on the diagonal elements of the system density operator. The l_{1} norm of quantum coherence is a measure of coherence through consideration of the absolute value of the nondiagonal elements. The measure of coherence based on entropy is defined as the distance between the state of interest and the closest incoherent state. The l_{1} norm of coherence is introduced as$${C}_{l}=\underset{\delta \in \mathcal{I}}{\mathrm{min}}{\left\right\rho \delta \left\right}_{{l}_{1}}=\sum _{i\ne j}\mathrm{}\left{\rho}_{\mathrm{ij}}\right,$$(19)where $\mathcal{I}$ is the set of incoherent states and i and j are, respectively, the row and column indices. In terms of the von Neumann entropy function$$S\left(\rho \right)=\mathrm{tr}\left(\rho \mathrm{ln}\rho \right),$$(20)the relative entropy coherence can be provided as$${C}_{R}=S\left(\rho \left\right{\rho}_{\mathrm{diag}}\right)=S\left({\rho}_{\mathrm{diag}}\right)S\left(\rho \right),$$(21)where the the operator ρ_{diag} describes the quantum incoherent state. ${C}_{{l}_{1}}$ and C_{ r } both achieve monotony for all quantum states. It has been shown that ${C}_{{l}_{1}}$ characterizes the upper bound of C_{ r } for a pure state.
To quantify the entanglement, we use the von Neumann function. This function value gets zero value for all pure states, which are states that fulfill the condition ${\widehat{\rho}}^{2}=\widehat{\rho}$ with $\widehat{\rho}$ is the density operator that describes a particular quantum system. In our situation, the entanglement of the atom (5LA)field (F) state can be determined through the von Neumann entropy of one of subsystems$${S}_{5\mathrm{LA}\left[\mathrm{F}\right]}={\mathrm{Tr}}_{\mathrm{A}\left[\mathrm{F}\right]}\left\{{\widehat{\rho}}_{5\mathrm{LA}\left[\mathrm{F}\right]}\mathrm{ln}\left[{\widehat{\rho}}_{5\mathrm{LA}\left[\mathrm{F}\right]}\right]\right\}=\sum _{j}\mathrm{}{\zeta}_{j}\mathrm{ln}{\zeta}_{j},$$(22)where the reduced density operators ${\widehat{\rho}}_{\mathrm{A}\left[\mathrm{F}\right]}$ are given by$${\widehat{\rho}}_{5\mathrm{LA}\left[\mathrm{F}\right]}=\mathrm{T}{\mathrm{r}}_{\mathrm{A}\left[\mathrm{F}\right]}\left\{U\left(T\right)\rangle \langle U\left(T\right)\right\}.$$(23)and ζ_{ j } denotes the eigenvalues of the atomic reduced density matrix (22).
We now examine the effects of the initial state of the field and deformation parameter on the population, von Neumann entropy, and quantum coherence in the dynamics of the considered system. The numerical results are presented in Figures 1–4. We assume that the initial field is in a CS or a superposition state with α = 5.
Figure 2 presents the variation in the population against the dimensionless time βt when the fivelevel atom starts from the upper state and the input field is defined in a CS or a superposition state for different values of the parameter q. Figures 2a and 2b, respectively, presents the results for r = 0 and r = 1 with q = 1. Figures 2c and 2d, respectively, presents the results for r = 0 and r = 1 with q = 1.5. Generally, the dynamics of the function ρ_{11} exhibits fast oscillations with amplitudes depending on q and r. These parameters may have a significant impact on the control of the inhibition decay of the excited state. Figure 2 shows that the presence of deformation increases the amplitude of the function ρ_{11} and reduces its oscillation. Furthermore, the singlemode field with r = 1 increases the amplitude of the population.
Fig. 2 Time evolution of the upper state populations ρ_{11} of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 
When considering a fivelevel atom that was first initialized in its upper state, the dynamics of the quantum coherence in the atomic system are analyzed when the singlemode field is specified by a CS or a superposition of CS that depend on q and r. Figures 3 and 4 show the dynamics of the functions C_{ R } and C_{ L } versus the dimensionless time T for different values of q with r = 0 and r = 1. In general, the quantum coherence of the fivelevel atom largely depends on the initial form of the field and its deformation. Figures 3 and 4 show that the quantum coherence of the atom can be generated through the fieldatom interaction in the time dynamics. It seems that the functions C_{ R } and C_{ L } have oscillatory behavior with the amplitude depending on the parameters r and q. Such input fields can thus help control the quantum coherence of the atomic system in the time dynamics, where the functions C_{ R } and C_{ L } have high amplitudes depending on the parameter q for r = 0 and r = 1. Moreover, Figures 3 and 4 show that the presence of deformation reduces the oscillations of the functions C_{ R } and C_{ L } and stabilizes their behavior with the time. Meanwhile, the superposition of the CSs for r = 1 enhances the oscillation of the functions C_{ R } and C_{ L } with more random behavior in the dynamics. Finally, Figure 5 presents the dynamics of the von Neumann entropy for the same conditions on the physical parameters of the quantum system. It is clear that the enhancement of entanglement of the atomfield state in the dynamics can be improved by a suitable choice of the deformed parameter for both cases r = 0 and r = 1. Furthermore, the entropy function behaves in a similar way as the quantum coherence with respect to the parameters of the quantum physics. Coherence is thus a good candidate for describing the nonlocal correlation, and its implementation in different areas of quantum information and optics is promising. The physical origin of the different behaviors of the quantum coherence might be the flow of information between the atom and nonlinear field as the correlations established during the interaction. In short, the results demonstrate that the quantum coherence and nonlocal correlation can be controlled and maintained through the deformation of the coherent field during the atomfield interaction and its initial state setting for describing factual experimental scenarios under optimal conditions.
Fig. 3 Time evolution of the quantum coherence C_{ R } of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 
Fig. 4 Time evolution of the quantum coherence C_{ L } of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 
Fig. 5 Time evolution of the von Neumann entropy S_{5LA} of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 
4 Conclusion
We have investigated the atomfield system within the framework of harmonic oscillators based on deformed Heisenberg algebras. We have explored the dynamic characteristics of the considered system under the effect of a nonlinear field. In particular, we have considered the atomic population, atomic coherence, and atomfield entanglement for a system that comprises a single fivelevel atom interacting with a singlemode nonlinear field when the deformation effect is taken into account. We have examined the time evolution of the quantum quantifiers in the case of deformation when the initial state of the quantized field is defined to be a nonlinear CS or superposition of CS. The results demonstrated that the quantum coherence and nonlocal correlation can be controlled and maintained through the deformation of the coherent field during the atomfield interaction and its initial state setting for describing factual experimental scenarios under optimal conditions. Here, we have considered the case of a closed systems, then it is very useful to introduce in the future the environment effect on the coherence and correlations considering for the case of open quantum systems with large levels.
Conflict of interest
The authors declare that there is no conflict of interest.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R225), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
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All Figures
Fig. 1 Schematic diagram of a fivelevel laddertype atom interacting with a singlemode nonlinear field. 

In the text 
Fig. 2 Time evolution of the upper state populations ρ_{11} of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 

In the text 
Fig. 3 Time evolution of the quantum coherence C_{ R } of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 

In the text 
Fig. 4 Time evolution of the quantum coherence C_{ L } of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 

In the text 
Fig. 5 Time evolution of the von Neumann entropy S_{5LA} of a fivelevel system for the field initially in (a) a CS with α = 5 and q = 1, (b) a superposition of CS with α = 5 and q = 1, (c) a CS with α = 5 and q = 1.5 and (d) a superposition of CS with α = 5 and q = 1.5. 

In the text 
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