# Quantum Decoherence Technique for Two Two-level Interacting Atomic Engineering in Dissipative Field

Authors: Pengli Shu
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

In order to improve the in-depth understanding and research on the dissipative field of atomic engineering, the research object of this paper is the dissipative field of atomic engineering. Through the engineering of two two-level interacting atoms as the background, an in-depth study of two two-level atoms is carried out, so the decoherence factor of two two-level atoms is obtained. In this paper, the numerical simulation calculation experiment of the dissipative field of atomic engineering is carried out, and the evolution of the quantum coherent oscillation of the dissipative field to the quantum decoherence is discussed. The results indicate that the dissipation coefficient and strength of the atom-light field interaction not only affect the oscillation of the quantum coherent evolution of atomic states but also the periodicity of the evolution. All preliminary results throw light on the nature of two-level interacting atoms in the dissipative field of atomic engineering, which also provides a reference value for related researchers.


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## Body

## 1. Introduction

With the continuous social and economic development, the studies on quantum mechanics have become more and more in-depth gradually [1–5]. Quantum entanglement is a unique property of quantum mechanics and its essential application in the field of quantum information science, which has attracted the increasing attention of humans [6–8]. On the one hand, atoms can be used to store and process quantum information and photons can be used to transmit quantum information. However, nonlinear interaction is the physical foundation for quantum entanglement. Hence, the interaction between the light field and atoms is considered to be stored in the ideal carrier for processing quantum information [9–11]. The interaction between two atoms induced by the nonresonant interaction of the two atoms and the cavity field can be used to implement the Grover search. However, under the large detuning conditions, when the single-mode optical cavity has energy address loss, with respect to the evolution features of entropy in the single-mode radiation field and the interaction system of two two-level atoms, the impact of the cavity field loss on the dynamic is significant. Hence, it is of great significance to study the impact of the cavity field loss on the dynamic behavior of the light field interacting with two two-level atoms [12, 13].In recent years, as the term quantum has gradually moved from behind the scenes to the front, the practical application of quantum mechanics has also appeared in all aspects of human life. Quantum decoherence refers to the process in which a coherent quantum state becomes an incoherent quantum state under the influence of the outside world (such as environmental noise). It includes properties such as open quantum systems, quantum entanglement of quantum and environment, and loss of coherence over time. In this paper, the quantum decoherence technology in the dissipation field of two two-level interacting atoms engineering is analyzed based on the quantum theory. Through digital analysis and calculation, the impact of the coupling intensity coefficientg and the dissipation constant between the cavity field and the atom on the evolution of the quantum coherence in the atomic state is discussed [14–17].


## 2. Model and Solution

The system is radiation field. After the light field is quantized, from the perspective of the basic principle of the interaction between light and atoms, the general Hamiltonian of field interaction of the atom and light in the case of nonresonant interaction can be obtained. When it is far away from resonance, that is, in a large detuning situation (Δ=ω−ω0≫gn, in which Δ stands for the amount of atom-field detuning, ω stands for the frequency of the light field, after the rotation wave is approximated and subject to adiabatic eliminated. The effective Hamiltonian can be obtained as (η = 1):(1)H=λ∑j=1,21j,1aa+−0j,0a+a+S1+S2−+S1−S2+.In the above equation,λ=g2/Δ, in which a+a stands for the generation (annihilation) operator of the light field, and Sjj=1,2 stands for the artificial spin operator of the j-th atom.


### 2.1. The Characteristics of the Loss Field

If the cavity field loss is considered, the density operator of the system shall meet the master equation as the following:(2)dρtdt=−iH,ρt+Dρt.In the above equation,Dρt describes the energy loss of the cavity field, which can be written as the following:(3)Dρt=k2aρta+−a+aρt−ρta+a.In the above equation,k is the decay coefficient.It is assumed that at the initial moment, the light field is in a coherent state as the following:(4)φF0=α=e−1/2α2∑n=0∞αnn!n.The atoms are in an entangled state as the following:(5)φA0=sinθ0102+cosθ12.In the above equation,0 and 1 stand of a single atom, respectively. The density operator of the system at the initial moment is as the following:(6)ρF,A0=sin2θ00,αα,00|+sinθcosθ00,αα,00+cos2θ|α,1111,α.Under the initial conditions given by equation (6), in an interaction scene, through equation (2), the density can be obtained as the following:(7)ρF,At=α100,αte−2iλtαte−2iλtα,00|+a2|00,αte−2iλtαte−2iλtα,11+a311,αte−2iλtαte−2iλtα,00|+a4|11,αte−2iλtαte−2iλtα,11.The reduced density operator of the atom at timet can be obtained by tracing the light field as the following:(8)ρAt=trFρt=a10000|+a2exp−αt21−e−4iλt|0011+a3exp−αt21−e4iλt1100|+a4|1111.In general, the coherence phenomenon indicates that the nondiagonal element is not zero. However, the modulus of the nondiagonal element is of practical significance. Hence, from equation (8), it is not difficult to obtain the decoherence factor as the following:(9)fλ,α,k,t=e−2iλtexpkk+2iλ1−e−2k+2iλtα2×exp−α21−e−2ikt×exp−αt21−e−4iλt.After simplification, the following can be obtained:(10)fλ,α,k,t=expkα2k2+4λ2⋅k+e−2kt2λsin4λt−kcos4λt×exp−α21−e−2kt×exp−α2e−2kt1−cos4λt.From equation (10), the reduced density operator of the atom can be obtained as the following(11)ρat=Trfρt=C111,11,1+C22exp−αt21−e−2iΩt1,11,0+C13exp−αt21−e−2iΩt1,10,1+C14exp−αt21−e−4iΩt1,10,0+C21exp−αt21−e2iΩt1,01,1+C221,01,0+C231,00,1C24exp−αt21−e−2iΩt1,00,0+C31exp−αt21−e2iΩt0,11,1+C320,11,0+C330,10,1+C34exp−αt21−e2iΩt0,10,0+C41exp−αt21−e4iΩt0,01,1+C42exp−αt21−e−2iΩt0,01,0+C43exp−αt21−e2iΩt0,00,1+C440,00,0.From equation (10), the reduced density operator of the light field can be obtained as the following(12)ρft=Tταρt=C11αte−2iΩtαte−2iΩt+C22αtαt+C33αtαt+C44αte2iΩtαte2iΩt.From equation (10), the system can be obtained as the following(13)St=1−TTρ2t=12sin2θ1+sin2θ2−38sin2θ1sin2θ2−12sin2θ1+sin2θ2−sin2θ1sin2θ2×exp2k2k2+Ω21−e−2ktcos2Ωt+2kΩk2+Ω2e−2ktsin2Ωtα2exp−2α21−e−2kt−18sin2θ1sin2θ2exp2k2k2+4Ω21−e−2ktcos4Ωt+4kΩk2+4Ω2e−2ktsin4Ωtα2×exp−2α21−e−2kt.From equation (12), the linear entropy of the light field can be obtained as the following(14)Sft=1−Trfρf2t=12sin2θ1+sin2θ2−38sin2θ1sin2θ2−12sin2θ1+sin2θ2−sin2θ1sin2θ2×exp−2α2e−2kt1−cos2Ωt−18sin2θ1sin2θ2exp−2α2e−2kt1−cos4Ωt.Taking into consideration a system consisting of two entangled ideal cavities (C1 and C2) and two independent two-level Rydberg atoms (A1 and A2), atom A1 passes through one cavity C1, and atom A2 passes through the other cavity C2, as shown in Figure1 below. Based on the theory of the J-C model, the field composite system under the rotating wave approximation is shown as the following:(15)HJ=ga1σ+1+a1+σ−1+a2σ+2+a2+σ−2.Figure 1
System where two atoms in the ground state pass through the C1 and C2 cavities in the maximum entangled state at the same time.In the above equation,g stands for the interaction coupling intensity coefficient of the atomic cavity field; a+ and a stand of each cavity field; σ+σ− stand for the spin operators of each atom. Without considering the atom dissipation, if only the cavity field dissipation is taken into account, the dynamic evolution equation of the composite system in the interaction painting can be obtained based on equation (16) as the following:(16)ρ˙=ρ˙atorn−field+ρ˙field−reservoir.In which,(17)ρ˙atom−feild=−iH1,ρ=−iga1σ+1ρ+a1+σ−1ρ−ρa1σ+1−ρa1+σ−1+a2σ+2ρ+a2+σ−2ρ−ρa2σ+2−ρa2+σ−2ρ˙|field−reservoir=−k1a1+a1ρ−2a1ρa1++ρa1+a1−k2a2+a2ρ−2a2ρa2++ρa2+a2.In the above equation, k1 and k2 stand for the dissipation coefficients of the cavities C1 and C2, respectively. It is assumed that the eigenstates of the two single-mode cavity fields have only0 and 1, and they are initially ψt=0c1c2=1/20112+1002. The two atoms are initially in the respective ground states ψt=0A1=g1 and ψt=0A2=g2. In addition, the atom and the cavity field fall into the coherent resonance interaction. The initial state wave function of the aforesaid composite system is shown as the following:(18)ψt=0C1C2A1A2=1201121102⊗g1g2,where the high-order terms of k/g are ignored, the density operator of the system can be solved based on equations (15)–(18) as the following:(19)ρtC1C2A1A2=α10112g1g20112g1g2+α20112g1g20112g1g2+α31102g1g21102g1g2+α40112g1g20112g1g2+α50112g1g21102g1g2+α51102g1g20112g1g2+α60112g1g20112g1g2+α60112g1g20112g1g2.Based on equation (19), the reduced density matrix of the two atoms can be obtained as the following:(20)ρtA1A2=TrC1C2ρt=α1+α3g1g2g1g2+α2g1e2g1e2+α4e1g2e1g2+α6g1e2g1e2+α6e1g2g1e2.With the engineering of two two-level interacting atoms as the background, the quantum decoherence in the resonance interaction process of two two-level atoms with two entangled single-photons in the dissipative cavity field is studied to obtain the decoherence factor of two two-level atoms. Through the numerical simulation calculation, the evolution of the quantum coherence oscillation of the dissipation field to quantum decoherence is explored.


## 2.1. The Characteristics of the Loss Field

If the cavity field loss is considered, the density operator of the system shall meet the master equation as the following:(2)dρtdt=−iH,ρt+Dρt.In the above equation,Dρt describes the energy loss of the cavity field, which can be written as the following:(3)Dρt=k2aρta+−a+aρt−ρta+a.In the above equation,k is the decay coefficient.It is assumed that at the initial moment, the light field is in a coherent state as the following:(4)φF0=α=e−1/2α2∑n=0∞αnn!n.The atoms are in an entangled state as the following:(5)φA0=sinθ0102+cosθ12.In the above equation,0 and 1 stand of a single atom, respectively. The density operator of the system at the initial moment is as the following:(6)ρF,A0=sin2θ00,αα,00|+sinθcosθ00,αα,00+cos2θ|α,1111,α.Under the initial conditions given by equation (6), in an interaction scene, through equation (2), the density can be obtained as the following:(7)ρF,At=α100,αte−2iλtαte−2iλtα,00|+a2|00,αte−2iλtαte−2iλtα,11+a311,αte−2iλtαte−2iλtα,00|+a4|11,αte−2iλtαte−2iλtα,11.The reduced density operator of the atom at timet can be obtained by tracing the light field as the following:(8)ρAt=trFρt=a10000|+a2exp−αt21−e−4iλt|0011+a3exp−αt21−e4iλt1100|+a4|1111.In general, the coherence phenomenon indicates that the nondiagonal element is not zero. However, the modulus of the nondiagonal element is of practical significance. Hence, from equation (8), it is not difficult to obtain the decoherence factor as the following:(9)fλ,α,k,t=e−2iλtexpkk+2iλ1−e−2k+2iλtα2×exp−α21−e−2ikt×exp−αt21−e−4iλt.After simplification, the following can be obtained:(10)fλ,α,k,t=expkα2k2+4λ2⋅k+e−2kt2λsin4λt−kcos4λt×exp−α21−e−2kt×exp−α2e−2kt1−cos4λt.From equation (10), the reduced density operator of the atom can be obtained as the following(11)ρat=Trfρt=C111,11,1+C22exp−αt21−e−2iΩt1,11,0+C13exp−αt21−e−2iΩt1,10,1+C14exp−αt21−e−4iΩt1,10,0+C21exp−αt21−e2iΩt1,01,1+C221,01,0+C231,00,1C24exp−αt21−e−2iΩt1,00,0+C31exp−αt21−e2iΩt0,11,1+C320,11,0+C330,10,1+C34exp−αt21−e2iΩt0,10,0+C41exp−αt21−e4iΩt0,01,1+C42exp−αt21−e−2iΩt0,01,0+C43exp−αt21−e2iΩt0,00,1+C440,00,0.From equation (10), the reduced density operator of the light field can be obtained as the following(12)ρft=Tταρt=C11αte−2iΩtαte−2iΩt+C22αtαt+C33αtαt+C44αte2iΩtαte2iΩt.From equation (10), the system can be obtained as the following(13)St=1−TTρ2t=12sin2θ1+sin2θ2−38sin2θ1sin2θ2−12sin2θ1+sin2θ2−sin2θ1sin2θ2×exp2k2k2+Ω21−e−2ktcos2Ωt+2kΩk2+Ω2e−2ktsin2Ωtα2exp−2α21−e−2kt−18sin2θ1sin2θ2exp2k2k2+4Ω21−e−2ktcos4Ωt+4kΩk2+4Ω2e−2ktsin4Ωtα2×exp−2α21−e−2kt.From equation (12), the linear entropy of the light field can be obtained as the following(14)Sft=1−Trfρf2t=12sin2θ1+sin2θ2−38sin2θ1sin2θ2−12sin2θ1+sin2θ2−sin2θ1sin2θ2×exp−2α2e−2kt1−cos2Ωt−18sin2θ1sin2θ2exp−2α2e−2kt1−cos4Ωt.Taking into consideration a system consisting of two entangled ideal cavities (C1 and C2) and two independent two-level Rydberg atoms (A1 and A2), atom A1 passes through one cavity C1, and atom A2 passes through the other cavity C2, as shown in Figure1 below. Based on the theory of the J-C model, the field composite system under the rotating wave approximation is shown as the following:(15)HJ=ga1σ+1+a1+σ−1+a2σ+2+a2+σ−2.Figure 1
System where two atoms in the ground state pass through the C1 and C2 cavities in the maximum entangled state at the same time.In the above equation,g stands for the interaction coupling intensity coefficient of the atomic cavity field; a+ and a stand of each cavity field; σ+σ− stand for the spin operators of each atom. Without considering the atom dissipation, if only the cavity field dissipation is taken into account, the dynamic evolution equation of the composite system in the interaction painting can be obtained based on equation (16) as the following:(16)ρ˙=ρ˙atorn−field+ρ˙field−reservoir.In which,(17)ρ˙atom−feild=−iH1,ρ=−iga1σ+1ρ+a1+σ−1ρ−ρa1σ+1−ρa1+σ−1+a2σ+2ρ+a2+σ−2ρ−ρa2σ+2−ρa2+σ−2ρ˙|field−reservoir=−k1a1+a1ρ−2a1ρa1++ρa1+a1−k2a2+a2ρ−2a2ρa2++ρa2+a2.In the above equation, k1 and k2 stand for the dissipation coefficients of the cavities C1 and C2, respectively. It is assumed that the eigenstates of the two single-mode cavity fields have only0 and 1, and they are initially ψt=0c1c2=1/20112+1002. The two atoms are initially in the respective ground states ψt=0A1=g1 and ψt=0A2=g2. In addition, the atom and the cavity field fall into the coherent resonance interaction. The initial state wave function of the aforesaid composite system is shown as the following:(18)ψt=0C1C2A1A2=1201121102⊗g1g2,where the high-order terms of k/g are ignored, the density operator of the system can be solved based on equations (15)–(18) as the following:(19)ρtC1C2A1A2=α10112g1g20112g1g2+α20112g1g20112g1g2+α31102g1g21102g1g2+α40112g1g20112g1g2+α50112g1g21102g1g2+α51102g1g20112g1g2+α60112g1g20112g1g2+α60112g1g20112g1g2.Based on equation (19), the reduced density matrix of the two atoms can be obtained as the following:(20)ρtA1A2=TrC1C2ρt=α1+α3g1g2g1g2+α2g1e2g1e2+α4e1g2e1g2+α6g1e2g1e2+α6e1g2g1e2.With the engineering of two two-level interacting atoms as the background, the quantum decoherence in the resonance interaction process of two two-level atoms with two entangled single-photons in the dissipative cavity field is studied to obtain the decoherence factor of two two-level atoms. Through the numerical simulation calculation, the evolution of the quantum coherence oscillation of the dissipation field to quantum decoherence is explored.


## 3. Simulation Calculation and Discussion

When the two atoms are in the ground state (θ1=θ2=π) or the excited state (θ1=θ2=0) at the initial moment, the linear entropy of the atom-light field system, the atom, and the light field is always zero. The reason is that in the case of large detuning.The position of fluid particles is evenly distributed, and the velocity is randomly distributed according to the system temperature. The state of solid wall particles is given according to the specific boundary conditions. The macro flow parameters are obtained by statistical method, and the sampling time length is related to the equilibrium process. In this paper, 104 time steps are taken, the flow area is 20 × 20, 800 fluid particles, and 120 solid particles are arranged, which is recommended according to the literature. The DPD particle number density is between 2 and 10, which is taken as 2 in this paper. The horizontal motion velocities V of the upper plate are 012, 118, and 5, respectively.From equations (13)–(15), it can be known that the linear entropy of the atom-light field system, the atom, and the light field are all related, and the evolution of each linear entropy corresponding to various initial states of the atoms is also different with the extension of time. When the two atoms are in the ground state (θ1=θ2=π) or the excited state (θ1=θ2=0) at the initial moment, the linear entropy of the atom-light field system, the atom, and the light field is always zero. The reason is that in the case of large detuning, it can be known from equations (10) to (12) that when θ1=θ2=π, ρft=αte2iΩtαte2iΩt, ραt=0,00,0, ρt=ραt⊗ρft. From the above equations, the light field is at αte±2iΩt, and its amplitude decays exponentially.At the initial moment, ifθ1=0 and θ2=π, the following can be obtained based on (10)(21)ρt=121+cos2Ωt1,0,αt1,0,αt+isin2Ωt1,0,αt0,1,αt−isin2Ωt0,1,αt1,0,αt+1−cos2Ωt0,1,αt0,1,αt=cosΩt1,0−isinΩt0,1cosΩt1,0+isinΩt0,1⊗αtαt.IfΩt=π/4 is selected, the following can be obtained:(22)ρt=121,0−i0,11,0+i0,1|⊗αtαt=ψEPRψEPR⊗αtαt|.From the above analysis, it can be seen that under the aforesaid conditions, the prepared EPR pair does not decay in the whole evolution process. The reason is that the dipole interaction between atoms leads to entanglement between atoms (which can be observed from equation (1)), and there is always no photon exchange between the light field and the atomic system. Hence, the loss of the light field only results in the exponential decay of the amplitude of the light field but does not lead to the decay of the atomic EPR pair.At the initial moment, the two atoms are in a state of equal probability superposition; that is, when the sum of the distribution anglesθ1 and θ2 of the two atoms are both π/2, the linear evolution pattern of the atom-light field system, the atoms, and light fields over time can be obtained from equations (13) to (15). The linear entropy of the system is greater than zero at any time, except the initial time when it is zero, and it increases with the extension of time. When it increases to a certain level, it tends to stabilized to a specific value. The time from zero to a stable value is reduced with the increase of the decay constant. The magnitude of the stable value is related to the average photon number α2 of the light field and increases with the increase of α2; the linear entropy of the atom presents an oscillation situation with decreasing amplitude, and the oscillation stops after a period of time, which makes the linear entropy of the atom tend to be linear with the system at the stable value of the same entropy, and its oscillation time is quickly reduced with the increase of the decay constant. Due to the large detuning interaction between the atoms and the light field, the dynamic Stark shift and the atomic dipole-dipole interaction of the atomic system that is dependent on the light field occur in the atomic system. The shift of different atomic energy levels is not the same. As a result, the reduced density operator of the light field in the field interaction system evolves periodically between the pure state and the mixed state in atom-light interaction. Hence, the linear entropy of the light field shows periodic oscillation with decreasing amplitude, and the oscillation period is π/Ω. At td=nπ/Ωn=1,2,⋯, Sftd=0, Satd=Std>0, where the oscillation stops after a period of time. The linear entropy of the light field tends to reduce to zero; its oscillation time is related to the decay constant, reduced with the increase of the decay constant, and extended with the increase of the average photon number of the light field). From the above analysis, it can be seen that the system and atoms are in a mixed state at all times except at the initial moment when it is in a pure state, and the degree of mixing remains unchanged after a period of time. At the moment of td, the light field is completely decoupled from the atom, and the light field is reduced to the original pure state. However, the atoms are still in a mixed state.The nondiagonal element of the density matrix can be used to describe the coherence of the quantum system, in which the nondiagonal element is not 0, suggesting that there is a coherence phenomenon. The decoherence factor can be expressed by the modulus of the nondiagonal element. Hence, the decoherence factor between two atoms can be obtained based on equation (22) as the following:(23)Fg,k,t=12e−k1t/2singt−k1e−k1t/22g+k12g×e−k2t/2singt−k2e−k2t/22g+k22g.From equation (23), it can be easily observed that the decoherence factor is related to the atom-light field coupling constant and the cavity field dissipation coefficient. In order to understand the evolution of the decoherence factor f (g, k, t) over time more clearly, a numerical analysis was performed (shown in Figures 2–6).Figure 2
Diagram of the changes in the decoherence factor between two atoms with gt whenk1 = k2 = 0.Figure 3
Diagram of the changes in the decoherence factor between two atoms with gt whenk1 = k2 = 0.05 g.Figure 4
Diagram of the changes in the decoherence factor between two atoms with gt whenk1 = k2 = 0.1 g.Figure 5
Diagram of the changes in the decoherence factor between two atoms with gt whenk1 = 0.1 g, k2 = 0.05 g.Figure 6
Diagram of the changes in the decoherence factor between two atoms with gt whenk1 = 0.1 g, k2 = 0.2 g.As shown in Figure2, under the circumstance where the system does not have any dissipation (k1 = k2 = 0), the amplitude of the decoherence factor F remains unchanged with the extension of time, with a maximum value of 1 and a minimum value of 0. When the decoherence factor F = 1, the coherence is the most intense, and when F = 0, the coherence disappears. From Figures 3–6, the situation where there is dissipation in the cavity field can be observed (k/g≠0). From Figures 3–6, it can be seen that the decoherence factor presents the amplitude-reducing oscillation.After a certain period of time, it becomes stabilized to the value 0, when the phenomenon of complete decoherence appears. The system evolves from the initial pure state to the final mixed state. In addition, it can be found that although the decoherence factor presents an overall amplitude decay situation, it does not keep decaying all the time, but the amplitude may increase occasionally. Moreover, it can also be observed that the larger the value of k/g is, the shorter the time for the amplitude it takes to decay to 0, which suggests that the influence of the dissipation coefficient in the light field on the coherence is apparent.


## 4. Conclusion

In this paper, the evolution features of quantum coherence between the atoms and single-photons in the resonant atomic states in two entangled ideal cavities and the time evolution features of the atomic-light field system linear entropy, the atomic linear entropy, and the light field linear entropy are studied. The impacts of the initial state of the two atoms, the decay constant, and the average photon number of the light field on each linearity are discussed. The results indicate that the dissipation factor and the intensity of the atom-light field interaction can affect not only the oscillation of the quantum coherence evolution in the atomic state but also the periodicity of the evolution. The decoherence factor presents an amplitude-reducing oscillation and tends to be stabilized at a value of 0 after a certain period of time, with the occurrence of a complete decoherence phenomenon.



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