_{1}

^{*}

In this study, our goal is to obtain the entanglement dynamics of trapped three-level ion interaction two laser beams in beyond Lamb-Dicke parameters. Three values of LDP,
*η*=0.09, *η*=0.2 and
*η*=0.3 are given. We used the concurrence and the negativity to measure the amount of quantum entanglement created in the system. The interacting trapped ion led to the formation of phonons as a result of the coupling. In two quantum systems (ion-phonons), analytical formulas describing both these measurements are constructed. These formulas and probability coefficients include first order terms of final state vector. We report that long survival time of entanglement can be provided with two quantum measures. Negativity and concurrence maximum values are obtained N = 0.553 and for LDP = 0.3. As a similar, the other two values of LDP are determined and taken into account throughout this paper. For a more detailed understanding of entanglement measurement results, “contour plot” was preferred in Mathematica 8.

Quantum states as usual are evident in itself with laws in quantum information theory [

Quantum entanglement has dramatically increased during the last two decades due to the emerging field of quantum information theory [

C d = C d I O N ⊗ C d p (1)

where d I O N = 3 and d p = 4 represent the dimensions at three-level ion and photons, respectively. We characterize quantum correlations using concurrence (C) [

The deep Lamb-Dicke regime (LDR) described with LDP of small, η ≪ 1 . LD limit is not accordingly established with common experiments [

We report analytical results of quantum entanglement for system via N and C for the LDR and 12-Dimensional (D) of Hilbert space. We focus the quantum correllations in N and C [

The rest of the study is coordinated as follows. Section 2 discusses growth for two unentangled qubits and analitical solutions in the quantum system. Section 3 describes how to obtain highly N and C of two quantum systems by the LDR. The results and comments are given in Section 4.

For section 2, flow chart is:

· In this section, the Hamiltonian and its dynamics are given between Equations (1)-(5).

· In Λ configuration, U transformation matrix processes evolved in Equations (6)-(11).

· The initial state of the system has written by Equations (12)-(22).

· Equation (3) is the final state of the ion-phonons system.

· In Equations (24)-(32), the probability applitudes are given.

We propose a trapped atomic ion interacting with two laser beams. In this system, the Hilbert space dimension is 12. The quantum dynamics of trapped ion-phonons system is emerged by previous investigation [

H I o n = ω g | g 〉 〈 g | + ω r | r 〉 〈 r | + ω e | e 〉 〈 e | + p x 2 2 m + 1 2 m υ 2 x I o n 2 . (2)

The e-level energy is ω e = 0 , r-level is ω r , and g-level is ω g . The reason for ω e to be zero is the following: As can be seen in Equation (12), the excited level | e 〉 is removed in the first quantum state. Here H 1 and H 2 are Hamiltonians of these interactions for excited-ground and excited-raman:

H e − g = H 1 = Ω 2 e i ( k 1 x I o n − ω t ) | e 〉 〈 g | + h . c . (3)

H e − r = H 2 = Ω 2 e i ( − k 2 x I o n − ω t ) | e 〉 〈 r | + h . c . (4)

where ℏ = 1 , p x and x I o n are momentum and the x-component of position of ion center of mass movement. The movement of ion in the system is along the x-axis (one-D). Atomic levels are shown: | e 〉 → trapped ion excited level, | r 〉 → raman level, and | g 〉 → ground level. Trapped ionmass center is given

with standard harmonic-oscillator of H i o n in p x = i 1 2 m υ ( a + − a ) and x I o n = 1 2 m υ ( a + a + ) . Here, a is annihilation operator and a + creation operator for two laser beams. Laser frequencies are ω 1 and ω 2 , and Rabi frequency is Ω . Trapped ion-phonons total Hamiltonian is written ( ℏ = 1 ):

H = ( Ω 2 e i η ( a + + a ) | e 〉 〈 g | + υ a + a − δ | e 〉 〈 e | + Ω 2 e − i η ( a + a + ) | e 〉 〈 r | ) + h . c . , (5)

here, LDP is η = k / 2 m υ , υ is trap frequency of harmonic, and delta function is δ = υ η 2 . We have taken the base vectors as follow:

| e 〉 = ( 1 0 0 ) , | r 〉 = ( 0 1 0 ) , | g 〉 = ( 0 0 1 ) (6)

In this study, important transformed Hamiltonian is H ˜ = U + H U . Hamiltonian in Equation (5) is found after transmission action. Λ model is given by a cascade Ξ scheme in two phonons. Ion-two phonons system was covered by unitary transformation. Matrix of transformation, namely U is performed [

U = 1 2 ( 0 2 2 − 2 B [ η ] B [ η ] − B [ η ] 2 B [ − η ] B [ − η ] − B [ − η ] ) . (7)

Here displacement operators of Glauber, B ( η ) = e ( i η ( a + a + ) ) , B ( − η ) = e ( − i η ( a + a + ) ) are achieved. H ˜ is performed H ˜ = H ˜ 0 + V ˜ , here

H ˜ 0 = υ ( | r 〉 〈 r | − | g 〉 〈 g | ) + υ η 2 + υ a + a (8)

V ˜ = − i 2 δ η 2 ( a + | e 〉 〈 r | − a + | e 〉 〈 g | + h . c . ) . (9)

In our system, the LDR is performed between the values 0.09 and 0.3 of LDP. By using unitary transformation method [

| ψ ( t ) 〉 = U 0 + U e − i t H ˜ 0 K ( t ) U + | ψ ( 0 ) 〉 , (10)

where K ( t ) is typical vector for time-independent Hamiltonian; e ( − i t H ˜ 0 ) is the exponencial function, and U 0 = exp ( − i ω t | e 〉 〈 e | ) is the transformation matrix [

K ( t ) = 1 2 ( C o s ( Λ t ) − ε S a + − ε S a ε a S 1 + ε 2 a G a + ε 2 a G a ε a + S ε 2 a + G a + 1 + ε 2 a + G a ) , (11)

here ε = υ η / 2 , Λ = ε 2 a + a + 1 , G = cos ( Λ t ) Λ 2 and S = sin ( Λ t ) Λ . We take υ = 10 6 Hz and ω e g = 5 × 10 14 Hz for frequencies. In the system, we take a = 1 and b = 0.005 . Normalization condition of ion is certainly [ 1 2 ] 2 + [ − 1 2 ] 2 = 1 , and normalization condition of two phonons is ‖ a ‖ 2 + ‖ b ‖ 2 = | 1 | 2 + | 0.005 | 5 ≅ 1 , approximately. So, the earliest of trapped ion-phonon states system is given as

| ψ ( 0 ) 〉 = 1 2 [ | g 〉 − | r 〉 ] ⊗ ( a | 0 〉 + b | 1 〉 ) , (12)

here, the phonon levels are 〈 0 | = ( 1 , 0 ) , and 〈 1 | = ( 0 , 1 ) . a and b are the probability amplitudes of the first and the second phonon. New equation for ion-two phonons is performed as

| ψ ( 0 ) 〉 = 1 2 [ | g 〉 − | r 〉 ] ⊗ ( ∑ n = 0 ∞ F n ( b ) | n 〉 ) . (13)

It is used by η 0 and η 1 are zero and first-order indication of LDP, respectively. Beside, both of them, η 2 and η 3 are ignored. Ion-phonons system is evolved to an initial unentangled state,

| ψ K ( t ) 〉 = | ψ ˜ ( 0 ) 〉 = U + | ψ ( 0 ) 〉 = ∑ σ , m N σ , m ( t ) | σ , m 〉 . (14)

In Equation (12), our system is produced in respect of ∑ σ , m N σ , m ( t ) | σ , m 〉 . As a result of advanced mathematical transformations between Equation (10)-(14), 12 of significiant coefficients are

s c N e 0 ( t ) = [ cos ( 1 2 t ) + η i 2 sin ( 1 2 t ) ] exp [ − t i / η ] (15)

s c N e 1 ( t ) = b cos ( 3 2 t ) exp [ − t i / η ] (16)

s c N e 2 ( t ) = − η i 5 sin ( 5 2 t ) exp [ − 2 t i / η ] (17)

s c N r 0 ( t ) = b 3 sin ( 3 2 t ) exp [ − t i / η ] (18)

s c N r 1 ( t ) = η i 2 [ 3 2 + 2 5 cos ( 5 2 t ) ] exp [ − 2 t i / η ] (19)

s c N g 1 ( t ) = [ sin ( 1 2 t ) − η i 2 cos ( 1 2 t ) ] exp [ − t i / η ] (20)

s c N g 2 ( t ) = b 2 3 sin ( 3 2 t ) exp [ − t i / η ] (21)

s c N g 3 ( t ) = − 3 5 η i [ 1 − cos ( 5 2 t ) ] exp [ − 2 t i / η ] (22)

and four of significiant coefficients are zero:

s c N e 3 ( t ) = s c N r 2 ( t ) = s c N r 3 ( t ) = s c N g 0 ( t ) = 0 . For Equations from (15) to (22), index σ is positioned in the states of atomic ( g , r , e ) , index m is positioned by vibrational numbers ( 0 , 1 , 2 , 3 ) . Vibrational phonon states are located by a Hilbert 4D-space H_{phonons} and subsystem of trapped ion-phonons is located in a Hilbert 3D-space H_{Ion}. Thus, two quantum systems are in Hilbert 12D-space. Here, t is dimensionless and scaled with υ η . What does υ η dimensionless mean? Accordingly in

calculation is as follows; for η = 0.2 , υ η = 0.2 × 10 6 , 1 υ η = 5 × 10 − 6 = 5 ms . The state vector is

| ψ f i n a l ( t ) 〉 = ∑ m = 0 3 ( A m ( t ) | e , m 〉 + B m ( t ) | r , m 〉 + C m ( t ) | g , m 〉 ) . (23)

The coefficients A m ( t ) , B m ( t ) and C m ( t ) are shown by state vector amplitudes of Λ and Ξ models. 12 of the probability amplitudes of the vector are

A m ( t ) = 1 2 e − i ω t / υ η [ N r m ( t ) + N g m ( t ) ] , ( m = 0 , 1 , 2 , 3 ) , (24)

B 0 ( t ) = − 1 2 N e 0 ( t ) + 1 2 N r 0 ( t ) − i η 2 N g 1 ( t ) (25)

B 1 ( t ) = − i η 2 N e 0 ( t ) − 1 2 N r 1 ( t ) + 1 2 N r 1 ( t ) − 1 2 N g 1 ( t ) (26)

B 2 ( t ) = − 1 2 N e 2 ( t ) − i η 2 N g 1 ( t ) − 1 2 N g 2 ( t ) (27)

B 3 ( t ) = − 1 2 N g 3 ( t ) (28)

C 0 ( t ) = 1 2 N e 0 ( t ) + 1 2 N r 0 ( t ) + i η 2 N g 1 ( t ) (29)

C 1 ( t ) = − i η 2 N e 0 ( t ) + 1 2 N r 1 ( t ) + 1 2 N r 1 ( t ) − 1 2 N g 1 ( t ) (30)

C 2 ( t ) = 1 2 N e 2 ( t ) + i η 2 N g 1 ( t ) − 1 2 N g 2 ( t ) (31)

C 3 ( t ) = − 1 2 N g 3 ( t ) (32)

here ω e g is frequency e-g levels and ω = ω e g − η 2 υ for Equation (24). i is complex number, and i is ion index.

We plotted N and C of two quantum systems as l ⊗ l ′ ( l ≤ l ′ ) in Figures 2-7

and

Hilbert spaces are l = 4 for two-phonons, l ′ = 3 for ion. It is used a simplified density matrix ρ i o n = T r p h o n o n ( ρ i o n − p ) by Equation (33). Fully density matrix ρ i o n − p is performed with 12 × 12 matrix with respect to the bases | i , p 〉 . With tracing, 3 × 3 -simplified density matrix, ρ i o n is performed

ρ i o n = T r p ( ρ i − p ) = ( T r | e 〉 〈 e | T r | e 〉 〈 r | T r | e 〉 〈 g | T r | r 〉 〈 e | T r | r 〉 〈 r | T r | r 〉 〈 g | T r | g 〉 〈 e | T r | g 〉 〈 r | T r | g 〉 〈 g | ) (33)

where diagonal terms, | e 〉 〈 e | , | r 〉 〈 r | and | g 〉 〈 g | are a 4 × 4 -matrix. For help to Equation (32), fully density matrix of two quantum system is written as:

ρ i o n − p h o n o n = ( | Z 〉 〈 Z | ) (34)

where | Z 〉 〈 Z | is a 12 × 12 -square matrix and Hilbert 12-space in qauntum mechanic. The initial state in second section derive in Hilbert 12-space H = H i ⊗ H p . In state vector | ψ ( t ) 〉 , fully density matrix of system is given by ρ i o n − p h o n o n = | ψ ( t ) 〉 〈 ψ ( t ) | = | Z 〉 〈 Z | in Equation (33). Negativity is first reported in literature as a quantum entanglement measurement in [

In this part, we examine if the state is entangled how much quantum entanglement it involves. They are analyzed quantum correlations with concurrence and negativity [

| ψ 〉 = ∑ j μ j | x j 〉 | y j 〉 (35)

where μ j , ( j = 1 , ⋯ , k ) are Schmidt coefficients abbreviated as SCs, x j and y j are orthogonal basis in H X and H Y [

Therefore, three SCs are the three eigenvalues of the matrix in Equation (33), μ j [

Negativity of any quantum system is written as [

N ( | ψ 〉 ) = 2 k − 1 ( ∑ i < j μ i μ j ) (36)

N ( | ψ 〉 ) = 2 3 − 1 ( μ 1 μ 2 + μ 1 μ 3 + μ 2 μ 3 ) (37)

Concurrence is developed as a quantum entanglement measurement for bipartite system of two qubits [

C ( | ψ 〉 ) = 2 ( ∑ i < j μ i μ j ) (38)

C ( | ψ 〉 ) = 2 ( μ 1 μ 2 + μ 1 μ 3 + μ 2 μ 3 ) (39)

As shown in Figures 2-5, LDPs are taken between 0.09 and 0.30. It is understood that taking these adjustable values of LDP is an appropriate choice, because the N and C values have seen with the maximums. This leads to higher dimensional entanglement with η . In

We reported entanglement via negativity in the LDR discretely from other papers [

η = 0.09 | η = 0.2 | η = 0.3 | |
---|---|---|---|

Negativity, t = 10.16 ms, | 0.493 | 0.512 | 0.553 |

Concurrence, t = 10.16 ms, | 0.978 | 0.992 | 1.000 |

We show the quantum correlations with N and Cfor coupling parameters. We found seperate dynamic features in N in reaction to increasing η . In

We concentrated on quantum entanglement of two quantum systems in the Hilbert 12-space. We investigate the negativity through the definition of variance LDR. Some physical correlations have been that we measure N and C. Our analysis has discovered maximally entangled state. The family is equal to a group of quantum measurements. To more detailed understanding of entanglement measurement results N and C, “contour plot” was preferred in Mathematica 8 in

These plots are obtained by N and C with quantum corrections. Entanglement is compared and is analyzed by two quantum measures which are N and C. Quantum correlations and interactions between ion and two phonons are investigated. Because, the discussion on physical properties of trapped ion-two phonos interaction is an important subject for quantum information.

The main contribution and novelty of my work has been explained with concluding remarks shown below:

· In our system, quantum entanglement is shown to have the capacity and degree of N and C are N = 0.553, C = 1.000;

· N bases on three different LDPs;

· This extracts that such entanglement is connected with η . We achieved long-lived entanglement in LDR;

· Maximally entangled states as presented by means of ion-two phonons system can be important for researchers with trapped ions;

· Extending the life time can be succeeded by using Rabi frequencies and η . This study and similar studies based on quantum measurement will lead to a better understanding of quantum physics and quantum entanglement.

This work is supported by Afyon Kocatepe University project number: 18-Kar- iyer.64.

The author declares no conflicts of interest regarding the publication of this paper.

Dermez, R. (2020) Investigation of Quantum Entanglement through a Trapped Three Level Ion Accompanied with Beyond Lamb-Dicke Regime. Journal of Quantum Information Science, 10, 23-35. https://doi.org/10.4236/jqis.2020.102003