# Dynamical Features of Isolated Two- and Three-Level Atoms Interacting with a Cavity Field

Authors: M. Y. Abd-Rabbou; E. M. Khalil; Fadhel Almalki
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Some dynamical features of two- and three-level atoms interacting locally with an ideal cavity field are investigated. These dynamical features are introduced by employing the statistical atomic inversion, entropy squeezing, and tomographic entropy. Our results show that the initial setting states play an essential role in the temporal evolution of the three quantities. The sensitivity of the two-level atomic state is less than that depicted by the three-level atom. The initial state has a small impact on the two types of entropies for the two-level atom. However, it has an appreciable effect in the case of the three-level atom for different regulations of the initial atomic state.


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## Body

## 1. Introduction

The interaction between different substances is still a hot topic in quantum optics and information issues. These issues concern the linear atom-field interaction [1] and nonlinear atom-field interaction [2]. The classical simulation of nonlinear atom-field interaction in photonic lattices has been realized [3]. In the ultra-strong coupling regime, the quantum simulation by applying a rotating wave approximation for the light-matter interaction has been introduced [4]. Some theoretical and practical phenomena of field-atom interaction, such as entanglement, revival, and collapse have been explained [5]. The entanglement of field-field interaction has been addressed [6]. Moreover, the effect of cross-Kerr nonlinear on the decoherence of a quantum system was studied [7]. The entanglement of time-dependent atom-atom interaction was discovered [8]. Some conical transformations are used to handle the atom-atom interaction with the presence of time dependence. For these interactions, some statistical and quantum information has been illustrated to analyze different phenomena [9]. Among these phenomena is the quantum entanglement [10], which is a cornerstone of quantum information theory [11]. Via employing the von Neumann entropy, the degree of entanglement has been studied for different substances, such as the entanglement of atom-field interaction in resonance and off-resonance cases, has been explored [12]. The degree of entanglement has also been discussed under the Unruh effect, for the two-qubit [13], qubit-qutrit [14], and two-qutrit [15]. Furthermore, the dynamics of entanglement of two isolated Jaynes-Cummings Hamiltonian have been studied, where the first atom interacting only with one cavity field and the second atom interacting with another cavity [16]. In addition, the entanglement of linear atom-field interaction has been illustrated under the influence of Kerr-like medium [17], degenerate parametric amplifier [18], vibrating graphene membrane [19, 20], external classical field [21], damping terms [22], Stark shift terms [23], and deformed cavity field [24]. A double Jaynes-Cummings models in the presence of non-Markovian environments [25], and Kerr medium [26] have been developed to simulate the entanglement dynamics. For high-dimensional atomic systems, the exact solution of double Tavis-Cummings model has been obtained to explain the dynamics of sudden death of entanglement [27]. On the other hand, the entanglement can be quantified by using some different indicators, for example, concurrence [28], the entanglement of formation [29], and negativity [11].The properties of Jaynes-Cummings model (JCM) were theoretically suggested and studied in the 1960s, and the results were confirmed in practice in the 1980s [30]. Therefore, the JCM is a familiar and important content for researchers in atomic and optical physics. Recently, JCM has been used to describe the evolution of entanglement between atoms and photons by researchers in quantum information science [31]. A thermal JCM and a discussion of the minimal entanglement between a two-level atom and thermal photons were investigated [32]. Moreover, the periods of sudden death and their association with the disentanglement of two separate atoms have been discovered [33]. Furthermore, we are working on our proposal to become one of the important candidates for a way to implement quantum logic in comparing the entanglement periods caused by the interaction of a two- or three-level atom with an ideal field.It is well known that the formula of the tomographic probability representation is used to describe the quantum states [34]. For the discrete spin observables, the tomographic probability and tomographic entropy have been reconstructed [35]. A comparative study of quantum Fisher information, tomographic entropy, and the von Neumann entropy for a single qubit and optical radiation field in the presence of excited and negative binomial distribution has been introduced [36, 37]. The interaction between a collection of an atomic quantum system and a quantized cavity field has been discussed [9].The paper is aimed at introducing a new interaction between an optical radiation field and two different dimensional atomic systems, where the two subsystems are two- and three-level atoms. The impact of the initial setting state of the two atoms on the temporal evolution of the physical phenomena is reported. The main task is to investigate the influence of initial atomic angles on some statistical information and quantum correlations of this system. The paper is organized as the following section briefly characterizes the physical Hamiltonian model and obtains the exact solution via the time-dependent Schrödinger equation. Accordingly, the final output atomic state of the two-level and three-level atoms is obtained. Section3 studies the dynamical behaviour of the atomic statistical inversion influenced by the initial state of the two different dimension atoms. Section 4 dedicates a comparative study of the two quadratures of entropy squeezing for the two atomic subsystems. We introduced the mathematical form of tomographic entropy and its temporal evolution in Section 5. Finally, Section 6 includes our conclusion.


## 2. The Physical Model

Let us briefly characterize a quantum system consisting of a single-mode cavity field coupled to isolated two- and three-level atoms, where the three-level atom is inΛ configuration (this system is proposed on Figure 1). The total Hamiltonian is expressed as (ℏ=1),
(1)H=ωfa∧†a^+∑j=e,g1,2,3ωjjj+λa^12+a^13+a^eg+h.c.,where ωjj=f,e,g,1,2,3 are the field, two-level atom, and three-level atom frequencies, respectively. Also, a^ (a∧†) is the common annihilation (creation) operator of a cavity field. λ denotes the coupling strength of atomic-field interaction.Figure 1
A sketch of two- and three-level atoms interact with a single mode of a quantized cavity field.At any timet>0, we assume that the temporal wave function ψt corresponding to the entire Hamiltonian is given by,
(2)ψt=∑n=0∞A1n,te−iη1te,1,n+A2n,te−iη2te,2,n+1+A3n,te−iη3te,3,n+1+A4n,te−iη4tg,1,n+1+A5n,te−iη5tg,2,n+2+A6n,te−iη6tg,3,n+2,where Ai represent the time-dependent probability amplitudes, whilst ηs are defined by,
(3)η1=ωe+ω1+nωf,η2=ωe+ω2+n+1ωf,η3=ωe+ω3+n+1ωf,η4=ωg+ω1+n+1ωf,η5=ωg+ω2+n+2ωf,η6=ωg+ω3+n+2ωf.Here, the initial state is considered as a separate state (product state), and then the entanglement periods resulting from the interaction of the field with a two-level atom or a three-level atom are detected via applying the time-dependent Schrödinger equationi∂ψt/∂t=Hψt [38]. Also, considering the initial state as a linear combination of product states between a superposition of atomic state Ξ and the coherent state α, which is represented by,
(4)ψ0=Ξ⊗α=cos2Θe,1+cosΘsinΘe,2+sinΘe,3⊗∑n=0∞Qnn,where Qn=e−α2/2α/n!−1/2, while Θ∈0,π.The case of resonance is considered for both the two-level atom and the three-level atom. For a two-level atom, the relation between atomic frequencies and field frequency is(5)ωe−ωg=ωf.For a three-level atom, the relationship between the atomic frequencies and the frequency of the field is(6)ω1−ω2=ωf,ω1−ω3=ωf.In the resonance case with the initial states (4), the probability amplitudes in Eq. (2) can be calculated as follows:
(7)A1n,t=a12R1X1+n,t+X1−n,t−ia2+a3ν12R1X2+n,t+X2−n,t,A2n,t=a2+a34R1X3+n,t+X3−n,t+2R1cosν2t−ia1ν12R1X2+n,t+X2−n,t,A3n,t=a2+a34R1X3+n,t+X3−n,t−2R1cosν2t−ia1ν12R1X2+n,t+X2−n,t,A4n,t=a2+a3R1Yn,t−ia1ν12R1X4+n,t+X4−n,t,A5n,t=2a1ν1ν2R1Yn,t−ia2+a322R1X4+n,t+X4−n,t−ia2−a32sinν2t,A6n,t=2a1ν1ν2R1Yn,t−ia2+a322R1X4+n,t+X4−n,t+ia2−a32sinν2t.Here,(8)X1±n,t=R1±3ν22−ν12costD∓2,X2±n,t=R1±ν22+3ν12D±sintD±2,X3±n,t=R1±ν22−ν12costD∓2,X4±n,t=R1±3ν22+5ν12D±sintD±2,Yn,t=ν12+ν22costD+2−costD−2,with
(9)a1=Qncos2Θ,a2=QncosΘsinΘ,a3=QnsinΘ,where
(10)R1=9ν14+ν24+14ν12ν22,D±=R1±3ν12+ν22,ν12=λn+12.By using Eqs. (2) and (7), the final output density operator of the total systems is given by
(11)ρA,B^,f=ψtψt.The main task of this manuscript is to discuss some statistical and dynamical aspects of the reduced density states for the two-level and three-level atoms. To get the reduced density operator, one can trace out the other two subsystems. The reduced density state of the two-level atomic system is defined byρ^A=TrB,fψtψt, which is obtained by
(12)ρ^A=∑n=0∞A1n,t2+A2n,t2+A3n,t2ee+A4n,t2+A5n,t2+A6n,t2gg+A1n+1,tA4∗n,t+A2n+1,tA5∗n,t+A3n+1,tA6∗n,teg+h.c..Likewise, the reduced density operator of the three-level atom is given by(13)ρ^B=∑n=0∞A1n,t2+A4n,t211+A2n,t2+A5n,t222+A3n,t2+A6n,t233+A1n+1,tA2∗n,t+A4n+1,tA5∗n,t12+A1n+1,tA3∗n,t+A4n+1,tA6∗n,t13+A2n,tA3∗n,t+A5n,tA6∗n,t23+h.c..


## 3. Atomic Inversion

Using the statistical population inversion, one can find the possibility of populating an atom in its excited or ground state. The population inversion of the two-level atom can be achieved by using the reduced density state (12) as follows:
(14)WAt=A1n,t2+A2n,t2+A3n,t2−A4n,t2−A5n,t2−A6n,t2.However, the statistical inversion of the three-level atom via employing the reduced density state (13) reads
(15)WBt=A1n,t2−A2n,t2−A3n,t2+A4n,t2−A5n,t2−A6n,t2.The influence of the initial atomic state on the statistical population inversion is displayed in Figure2, where the population of the two- and three-level atoms are shown by the blue and red curves, respectively. By setting the atomic system in the superposition state with Θ=π/4, Figure 2(a) shows that the collapse interval of the three-level atom is less than that is depicted for the two-level atom. The oscillation amplitudes of the two-level atom are greater than those displayed in the three-level atomic state. This means that the largest dimension is very sensitive to the field, so the interaction between the two-level atom and the cavity field is more potent. While standing Θ with the value π/6, Figure 2(b) discloses that the amplitude of the three-level atomic state decreases, and the two-level atom does not change. This illustrates that the three-level atomic state converges to the ground state and that its interaction with the field becomes fragile. By preparing the two atoms in the excited state, Figure 2(c) displays that the two atoms have the same amplitudes. The collapse intervals of the two-level atomic system are longer than that depicted for the three-level atom. If the higher dimensional atomic state is initially prepared in the ground state and the lower dimensional atomic system at the initial excited state, Figure 2(d) shows that the statistical inversion of the three-level atom always has a negative behaviour, while the collapse interval of the two-level atom increases. From this figure, we can deduce that the initial atomic state plays a central role in the statistical inversion, where the superposition state reduces the efficiency of the interaction between the two atoms and the field. The high atomic dimensionality is very sensitive to the cavity field. The sensitivity of the two-level state of the cavity field becomes fragile if we set the three-level atom in the ground state.Figure 2
The atomic inversion of two-level atomWAt (blue curve) and three-level atom WBt (red curve), where (a) Θ=π/4, (b) Θ=π/6, (c) Θ=0, and (d) Θ=π/2.
(a)(b)(c)(d)


## 4. Entropy Squeezing

In this section, we shall use the phenomena of entropy squeezing to investigate the information entropy of either the reduced two-level atomic system or the reduced three-level atomic system. It gave the condition of entropy squeezing inx and y directions by [39, 40]
(16)εlσi=expHlσi−γexpHlσz<0,l=A,B,andi=x,y,where the parameter γ equals 2 and 22 for two and three- level atom, respectively, and Hlσi are the observable Shannon information entropics with Hlσi=−∑n=jNPjlσilnPjlσi. For the two-level atom (N=2) the probabilities PjAσi are obtained by
(17)P1,2Aσx=121±Reρ12A,P1,2Aσy=121±Imρ12A,P1Aσz=ρ11A,P2Aσz=1−ρ11A,withρ12A=A1n+1,tA4∗n,t+A2n+1,tA5∗n,t+A3n+1,tA6∗n,t,ρ11A=A1n,t2+A2n,t2+A3n,t2.However, the probabilitiesPjBσi of the three-level atomic subsystem (N=3) are computed as follows:
(18)P1Bσx=12ρ11B+ρ33B−2Reρ13B,P2,3Bσx=141+ρ22B±2Re2ρ12B+ρ23B±ρ13BP1Bσy=12ρ11B+ρ33B+2Reρ13B,P2,3Bσy=141+ρ22B−2Reρ13B±22Imρ12B+ρ23B,P1Bσz=ρ11B,P2Bσz=ρ22B,P3Bσz=ρ33B,with
(19)ρ11B=A1n,t2+A4n,t2,ρ12B=A1n+1,tA2∗n,t+A4n+1,tA5∗n,t,ρ22B=A2n,t2+A5n,t2,ρ13B=A1n+1,tA3∗n,t+A4n+1,tA6∗n,t,ρ33B=A3n,t2+A6n,t2,ρ23B=A2n,tA3∗n,t+A5n,tA6∗n,t.The effect of initial atomic setting state of squeezing degree for the first quadratureεlσx is displayed in Figure 3, where we present the two-level atom εAσx in the blue solid curve and the three-level atom εBσx red dash curve. From Figure 3, we remarked that the two-level atom always has no squeezing along the scaled time, where the behaviour of εAσx>0 for different Θ. Squeezing of the three-level atom is depicted with the initial superposition states and the ground state. The entropy squeezing at Θ=π/3 is greater than that displayed for Θ=π/4, where εBσx<0 always with Θ=π/3. For the ground state, we disclose the entropy squeezing for a small period, where εBσx<0 at the scaled time λt∈0.05,0.06. Therefore, the superposition state generates large degrees of squeezing for the three-level atom.Figure 3
(a–d) The entropy squeezing inx direction of two-level atom εAσx (blue curve) and three-level atom εBσx (red curve), with the Θ parameter same as Figure 2.
(a)(b)(c)(d)On the other hand, Figure4 shows the general behaviour of the second quadrature εlσy. We noticed that the squeezing phenomenon of the two-level atom is generated after the first onset interaction, and it has a symmetric behaviour for different initial states. For the ground state, we disclose the entropy squeezing for a small period, where εBσx<0 at the scaled time λt∈0.05,0.06. Therefore, the superposition state generates large degrees of squeezing for the three-level atom. In contrast, the squeezing phenomenon of the three-level atom is dissipated according to the superposition state or the excited state. The reduced three-level atomic state generates a small-scale squeezing when the two-level atom is prepared in an excited state. From Figures 3 and 4, the two quadratures of entropy squeezing εlσx and εlσy are reflected behaviors. The robustness of entropy squeezing for the three-level atom is stronger than the two-level atom.Figure 4
(a–d) The entropy squeezing iny direction of two-level atom εAσy (blue curve) and three-level atom εBσy (red curve), with the Θ parameter same as Figure 2.
(a)(b)(c)(d)


## 5. Tomographic Entropy

For the atomic states, we constructed the tomographic entropy to indicate the entanglement of a quantum system. The spin tomogram entropy of an arbitrary spin-j is expressed by [35]
(20)Tα,β=2j+14π∫02π∫0π∑m=−jj−Qmα,βlnQmα,βsinαdαdβ,where Qiα,β represent the generic spin tomograms [35]. For the reduced two-level atom (j=1/2), the tomographic entropy is given by [37]
(21)TAα,β=12π∫02π∫0π∑m=12−QmAα,βlnQmAα,βsinαdαdβ.Here,
(22)Q1Aα,β=12+ρ11A−12cosα+sinαReρ12Aeiβ,Q2Aα,β=12+12−ρ11Acosα−sinαReρ12Aeiβ,where ρijA are defined in Eq. (17). Likewise, the tomographic entropy of the reduced three-level system (j=1) is computed as [35]
(23)TBα,β=34π∫02π∫0π∑m=03−QmBα,βlnQmBα,βsinαdαdβ,with the generic spin,
(24)Q1Bα,β=ρ11Bcos4α2+ρ22B2sin2α+ρ33Bsin4α2+sin2α2Reρ13Bexp2iβ+2sinαcos2α2Reρ12Bexpiβ+sin2α2Reρ23Bexpiβ,Q2Bα,β=ρ11B2sin2α2+ρ11Bcos2α+ρ33B2sin2α2−sin2αReρ13Bexp2iβ−12sin2αReρ12Bexpiβ−Reρ23Bexpiβ,Q3Bα,β=ρ11Bsin4α2+ρ22B2sin2α+ρ33Bcos4α2+sin2α2Reρ13Bexp2iβ−2sinαsin2α2Reρ12Bexpiβ+cos2α2Reρ23Bexpiβ.The tomographic entropy of the two- and three-level atomic systems is displayed in Figures5(a) and 5(b), respectively. For the two-level state, the initial atomic state has no clear influence on the tomographic entropy. The function TAα,β osculates between 2.55 and 3.5. The maximum bounds when the initial two atoms are prepared in the excited state are less than that shown in the superposition state. The upper bounds of the function TAα,β increase when the first atom is excited and the second in the ground state. For the three-level atomic state, Figure 5(b) shows that the parameter Θ plays a controlled role in tomographic entropy. The lower bounds of the function TBα,β at Θ=0,π/2 exit 4.7, while the upper bounds at Θ=0 are greater than that are displayed at Θ=π/2. Moreover, at superposition states with Θ=π/4,π/3, the lower and upper bounds of the tomographic entropy are convergent. The two-level atom is exchanged from a separable state into a partial (maximum) entangled state. However, the initial product state between the excited and ground state generates a maximally entangled state of the three-level atom.Figure 5
The tomographic entropy whereΘ=0 blue curve, Θ=π/2 red curve, Θ=π/3 green curve, and Θ=π/4 gray curve. (a) TAα,β two-level atom. (b) TBα,β three-level atom.
(a)(b)


## 6. Summary

In this paper, we have introduced a physical Hamiltonian comprising of an isolated two- and three-level atoms which are interacting locally with a single mode of a quantized cavity field. The exact solution of this system under different initial setting states by using the time-dependent Schrödinger picture is obtained. The temporal evolution of the statistical atomic inversion, entropy squeezing, and tomographic entropy of the two- and the three-level atoms are investigated. We remarked that the initial state has an active role in the dynamics of the statistical atomic inversion, where the superposition state suppresses the efficacy of the interaction between the atom and the field. The high-dimensional atomic system is sensitive to the cavity field. However, the sensitivity between the two-level state and the cavity field become more fragile if we set the three-level atom in the ground state. Also, the robustness of the entropy squeezing depends on the dimension of the atomic system, where the three-level state is stronger than the two-level atom. The general behaviours of the two quadratures of the entropy squeezing are reflected. By controlling the initial setting states, one can increase/suppress the temporal evolution of the tomographic entropy induced by the influence of the separability in the quantum system.



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*Source: 5258035-2022-07-21.xml*