^{1}

^{1}

^{1}

^{1}

^{*}

We have investigated numerically the dynamics of quantum Fisher information (QFI) and quantum entanglement (QE) for N-level atomic system interacting with a coherent field in the presence of Kerr (linear and non-linear medium) and Stark effects. It is observed that the Stark and Kerr effects play a prominent role during the time evolution of the quantum system. The evolving quantum Fisher information (QFI) is noted as time grows under the non-linear Kerr medium contrary to the QE for higher dimensional systems. The effect of non-linear Kerr medium is greater on the QE as we increase the value of Kerr parameter. However, QFI and QE maintain their periodic nature under atomic motion. On the other hand, linear Kerr medium has no prominent effects on the dynamics of N-level atomic system. Furthermore, it has been observed that QFI and QE decay soon under the influence of Stark effect. In short, the N-level atomic system is found prone to the change of the Kerr medium and Stark effect for higher dimensional systems.

Parameter estimation is a foundation of different branches of science and technology. The newly developed methods and techniques in measurement for parameter sensitivity have led to scientific revolutions and advancement in technology. A huge work and research has been done on phase estimation related to the practical problems of state generation, loss, and decoherence [

Quantum entanglement (QE) is a very famous and mysterious phenomenon of quantum mechanics that cannot be described completely. It has different applications in quantum mechanics and quantum information theory. Quantum correlations are observed in different physical systems. Quantum entanglement (QE) was first studied by Schrodinger [

The Jaynes-Cumming model (JCM) [

A Kerr-like medium is represented by a harmonic oscillator (HO) [

Recently, the JCM model was extended to consider the atomic motion along the axis of the cavity [

The interaction of light with matter is known as Stark effect and it is a very important phenomenon in the quantum optics [

von Neumann entropy (VNE) is considered the main entanglement measure for pure states. The correlation between quantum entanglement and Fisher information (FI) about a certain parameter in a quantum state has not been studied widely. Moreover, observations have been made to calculate the QE of pure state by using FI. In this regard, the entanglement evaluation and QFI of N-level atomic system under the influence of intrinsic decoherence is studied [

In this present paper our main focus is to investigate the QE and dynamics N-level moving atomic system which is interacting with coherent field in presence of Kerr medium and Stark shift. We have calculated VNE and QFI of the N-level atom system in the presence of atomic motion and without atomic motion.

The paper is arranged as follows. In Section 2, we present the background related to VNE and QFI. The system Hamiltonian and dynamics are presented in Section 3 for N-level atomic system affected by Kerr and Stark effect. Section 4 provides detailed results and numerical discussions. In Section 5, we present a brief conclusion.

The classical Fisher information (CFI) for any particular phenomenon with a one parameter θ which is unknown can be written as

I Φ = ∑ i p i ( Φ ) [ ∂ ∂ Φ ln p i ( Φ ) ] 2 , (1)

where p i ( Φ ) is representing the probability density having significant and prominent influence on the fixed parameter with the outcome { x i } of measurement for a distinct observable X. The CFI can be described with the help of the inverse variance of the asymptotic normality of a maximum-likelihood estimator. The QFI has a very important and effective part in quantum metrology and the large precise measurement of an unknown parameter can be gained which is having relation to inverse of the quantum fisher information and is described as [

F Φ = Tr [ ρ ( θ ) D 2 ] , (2)

where ρ ( Φ ) is describing the density matrix (DM) of the system, Φ is the parameter which will be estimated, and L is representing the quantum score (symmetric logarithmic derivative) which is written as

d ρ ( Φ ) d Φ = 1 2 [ ρ ( Φ ) D + D ρ ( Φ ) ] . (3)

here we are considering N-level atomic system with the density operator ρ ( Φ ) . The spectral decomposition of the DM is described as

ρ Φ = ∑ K λ K | k 〉 〈 k | , (4)

The QFI which is related to Φ for this DM is represented by [

F Φ = ∑ k ( ∂ Φ λ k ) 2 λ k + 2 ∑ k , k ′ ( λ k − λ k ′ ) 2 ( λ k + λ k ′ ) | 〈 k | ∂ Φ k ′ 〉 | 2 (5)

where λ k > 0 and λ k + λ k ′ > 0 . The first term in above equation is representing CFI and the second term describes QFI. The AQFI is calculated by taking the trace over the field. Therefore, we will be able to represent the AQFI of a bipartite density operator ρ A B which is related to Φ as [

I Q F ( t ) = I A B ( Φ , t ) = Tr [ ρ A B ( Φ , t ) { D ( Φ , t ) } 2 ] (6)

where D ( Φ , t ) is representing the quantum score [

∂ ρ A B ( Φ , t ) ∂ Φ = 1 2 [ D ( Φ , t ) ρ A B ( Φ , t ) + ρ A B ( Φ , t ) D ( Φ , t ) ] (7)

In the same way we will define the von Neumann entropy as

S A = − Tr ( ρ A ln ρ A ) = − ∑ i r i ln r i (8)

where r i represents the eigenvalues of the atomic DM ρ A = Tr B ( ρ A B ) .

We consider the model consisting of moving N-level atoms interacting with a single-mode cavity field. The total Hamiltonian of the system H ^ T under the RWA for atom-field system can be described as [

H ^ T = H ^ Atom-Field + H ^ I . (9)

where H ^ Atom-Field is representing the Hamiltonian for the non-interacting atom and field, and the interaction part is given by H ^ I . We will write H ^ Atom-Field as

H ^ Atom-Field = ∑ j ω j σ ^ j , j + Ω a ^ † a ^ , (10)

where σ ^ j , j = | j 〉 〈 j | are describing as population operators for the jth level. The interaction Hamiltonian of N-level atomic system for the case that is not resonant is written as [

H ^ I = ∑ s = 1 N Ω ( t ) [ a ^ e − i Δ s t σ ^ s , s + 1 + ( a ^ e − i Δ s t σ ^ s , s + 1 ) † ] . (11)

In the case of Kerr-like medium, the interaction Hamiltonian can be written as

H ^ I = ∑ s = 1 N Ω ( t ) [ a ^ e − i Δ s t σ ^ s , s + 1 + ( a ^ e − i Δ s t σ ^ s , s + 1 ) † ] + χ ( a ^ † a ^ ) q , (12)

and in case of linear Kerr-like medium q = 1 and in case of non-linear Kerr medium q = 2 , so the interaction Hamiltonian for linear and non-linear Kerr medium is given as

H ^ I = ∑ s = 1 N Ω ( t ) [ a ^ e − i Δ s t σ ^ s , s + 1 + ( a ^ e − i Δ s t σ ^ s , s + 1 ) † ] + χ ( a ^ † a ^ ) 1 , (13)

H ^ I = ∑ s = 1 N Ω ( t ) [ a ^ e − i Δ s t σ ^ s , s + 1 + ( a ^ e − i Δ s t σ ^ s , s + 1 ) † ] + χ ( a ^ † a ^ ) 2 , (14)

and when the Stark effect is included in the interaction Hamiltonian, it can be written as

H ^ I = ∑ s = 1 N Ω ( t ) [ a ^ e − i Δ s t σ ^ s , s + 1 + ( a ^ e − i Δ s t σ ^ s , s + 1 ) † ] + β a ^ † a ^ | g 〉 〈 g | . (15)

where g is representing the ground state of N-level atomic system and χ and β are the constants of Kerr-like medium and Stark effect.

We will describe the detuning parameter as

Δ s = Ω + ω s + 1 − ω s , (16)

and the coupling constant for atom and field is G, Ω ( t ) is representing the shape function of the cavity-field mode [

Ω ( t ) = G sin ( w π v t / L ) inthepresenceofatomicmotion , w ≠ 0 Ω ( t ) = G intheabsenceofatomicmotion w = 0 (17)

where the velocity of motion of atom is v and w denotes the wavelengths of half the number of the mode in the cavity and L describes the length of cavity along z direction. Now take the velocity of atom as v = λ L / π which gives us

Ω 1 ( t ) = ∫ 0 t Ω ( τ ) d τ = 1 w ( 1 − cos ( w π v t / L ) for w ≠ 0 (18)

= G t for w = 0. (19)

In order to find the phase shift parameter as precisely as possible, we consider the optimal input state as

| Ψ ( 0 ) 〉 Opt = 1 2 ( | 1 〉 + | 0 〉 ) ⊗ | α 〉 (20)

where | 1 〉 and | 0 〉 describe the states of atom and α is the coherent state of the input field given as

| α 〉 = ∑ n = 0 ∞ α n e − | α | 2 / n ! | n 〉 . (21)

In order to introduce the phase shift parameter ϕ , we consider a single-atom phase gate that introduces the phase shift as

U ^ ϕ = | 1 〉 〈 1 | + e i ϕ | 0 〉 〈 0 | , (22)

| Ψ ( 0 ) 〉 is obtained from the operation of the single-atom phase gate on | Ψ ( 0 ) 〉 Opt

U ^ ϕ | Ψ ( 0 ) 〉 Opt = | Ψ ( 0 ) 〉 (23)

= 1 2 ( | 1 〉 + e i ϕ | 0 〉 ) ⊗ | α 〉 (24)

After the operation of phase gate, the system will interact with a field. The precision of the estimation is strongly affected by the characteristics of the interaction between the field and moving N-level atomic system.

Now we can write the state | Ψ ( 0 ) 〉 as

| Ψ ( 0 ) 〉 = 1 2 ( | 1, n + 1 〉 + e i ϕ | 0, n 〉 ) (25)

where | 1, n + 1 〉 and | 0, n 〉 are allowable atom-field states. The states in which a number of photons are consistent with the atomic level are known as allowable atom field states. For our N-level atomic system, the allowable basis are given by

| 0, n 〉 , | 1, n + 1 〉 , | 2, n + 1 〉 , ⋯ , | N − 2, ( n + N − 2 ) 〉 , | N − 1, ( n + N − 1 ) 〉 , (26)

where n are number of photons initially present in the cavity and N − 1 are the number of levels in an atom. 0 and N − 1 represents the excited and ground state of the atom.

Let us consider the time-independent case, characterized by the transformation matrix U ^ ( t ) . The wave function

| Ψ ( t ) 〉 = U ^ ( t ) | Ψ ( 0 ) 〉 , (27)

In the next section, the influence of the parameters w, ϕ , χ and β on the evolution of the AQFI and VNE is presented.

In this section we will present the results of the time evolution of QFI and von Neumann entropy of a system of N-level atom interacting with a coherent field under the influence of Stark and Kerr effects. For the sake of simplicity, we scaled out the time t i.e. one unit of time is described by the inverse of the coupling constant G. Initially, we investigate the time evolution of QFI and Von Neumann entropy for N-level atomic system interacting with a non linear Kerr medium with and without atomic motion. In

In

In

In Figures 5-8 we plot QFI and von Neumann entropy as a function of time for N-level atomic system interacting with coherent field for | α | 2 = 6 , with linear Kerr-like medium χ = 1 , 3 , phase shift parameter ϕ = 0 , π / 4 with and without atomic motion i.e. w = 0 and 1. It is seen that the linear Kerr medium has no prominent effect on the dynamics of N-level system as compared to the effect of non-linear Kerr medium.

In Figures 9-12 we plot QFI and von Neumann entropy as a function of time for N-level atomic system interacting with coherent field for | α | 2 = 6 , χ = 1 , 3 , phase shift parameter ϕ = 0 , π / 4 , β = 1 , 3 and atomic motion parameter w = 0 and 1 respectively. It is seen that QFI and entanglement decay promptly

under the influence of Stark effect. Finally, we conclude that the N-level atomic system is fully prone to the change of the Kerr-like medium and Stark effect at higher dimensions.

We study the dynamical evolution of QFI and entanglement for N-level atomic system in the presence of Kerr (linear and non-linear medium) and Stark effects. The time evolution of the atomic system interacting with a coherent field under the influence of Stark and kerr effect is investigated numerically. It is shown that Stark and Kerr effect play a significant role during the time evolution of the quantum system. It is seen that QFI evolves as time grows contrary to the entanglement for higher dimensional systems under the influence of non-linear Kerr medium. However, the effect of non-linear Kerr medium is immense on the entanglement for larger values of Kerr medium. A sudden jump of entanglement is observed for N-level system (five-level atom) at χ = 3 because at larger values of non-linear Kerr parameter χ , five-level atom has more probability to interact with field as compared to three- and four-level atom. On the one hand, a periodic behaviour of QFI and entanglement is observed in the presence of atomic motion. On the other hand, linear Kerr medium has no prominent effects on the dynamics of N-level system. Furthermore, QFI and entanglement decay promptly under the influence of Stark effect. Finally, we conclude that the N-level atomic system is found fully prone to the change of the Kerr medium and Stark effect for higher-level atomic systems.

The authors declare no conflicts of interest regarding the publication of this paper.

Anwar, S.J., Ramzan, M., Usman, M. and Khan, M.K. (2019) Stark and Kerr Effects on the Dynamics of Moving N-Level Atomic System. Journal of Quantum Information Science, 9, 22-40. https://doi.org/10.4236/jqis.2019.91003