=Paper=
{{Paper
|id=Vol-1904/paper40
|storemode=property
|title=Detuning and dipole-dipole interaction effects on the entanglement of two qubits interacting with quantum fields of resonators
|pdfUrl=https://ceur-ws.org/Vol-1904/paper40.pdf
|volume=Vol-1904
|authors=Eugene K. Bashkirov
}}
==Detuning and dipole-dipole interaction effects on the entanglement of two qubits interacting with quantum fields of resonators ==
https://ceur-ws.org/Vol-1904/paper40.pdf
Detuning and dipole-dipole interaction effects on the entanglement of two qubits
interacting with quantum fields of resonators
Eugene K Bashkirov∗1
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
Abstract
We investigate the entanglement dynamics between two dipole-coupled qubits interacting with vacuum or thermal
fields of lossless resonators. Double Jaynes-Cummings model and two-atom Jaynes-Cummings model are considered
taking into account detuning and direct dipole-dipole interaction. Using the dressed-states technique we derive the
exact solutions for models under consideration. The computer modeling of the time dependence of qubit-qubit nega-
tivity is carried out for different strength of the dipole-dipole interaction and detuning. Results show that dipole-dipole
interaction and detuning may be used for entanglement operating and controlling.
Keywords: Entanglement, Superconducting qubits, Detuning, Dipole-dipole interaction, Vacuum field, Thermal field
1. Introduction
Quantum computers are devices that store information on quantum variables such as spins, photons, and atoms,
and that process that information by making those variables interact in a way that preserves quantum coherence . To
perform a quantum computation, one must be able to prepare qubits in a desired initial state, coherently manipulate
superpositions of a qubits two states, couple qubits together, measure their state, and keep them relatively free from
interactions that induce noise and decoherence [1]. Qubits have been physically implemented in a variety of systems,
including cavity quantum electrodynamics, superconducting qubits, atoms and ions in traps, quantum dots, spins and
hybrid systems [2]. The connection between qubits can be arranged through their interaction with quantum fields of
resonators. Basic protocols of quantum physics calculations are based on the use of entangled states [1]. Therefore,
great efforts have been made to investigate entanglement characterization, entanglement control, and entanglement
production in different systems. It is well known that the Jaynes-Cummings model (JCM) [3] is the simplest possible
physical model that describes the interaction of a natural or artificial two-level atom (qubit) with a single-mode cavity
[2], and has been used to understand a wide variety of phenomena in quantum optics and condensed matter systems,
such as superconducting circuits, spins, quantum dots, atoms or ions in a cavity [2]. In order to explore a wider range of
phenomena caused by the interaction of the qubits with the quantum fields in resonators the numerous generalizations
of the JCM have been investigated in recent years (see references in [4]-[8]). Yönac et al. [9] have proposed the
so-called double JCM (DJCM), consisting of two two-level atoms and two resonator modes, provided that each atom
interacts only with one field of the resonator, and investigated the pairwise entanglement dynamics of this model.
Recently, the DJCM have been extensively investigated [10]-[17].
The direct dipole-dipole interaction between the qubits is the natural mechanism of entanglement producing and
controlling. It’s very important that the effective dipole-dipole interaction for superconducting Josephson qubits may
be much greater than the coupling between the qubit and cavity field [18, 19]. The numerous references to the
theoretical papers devoted to investigation of entanglement in two-qubit systems taking into account the dipole-dipole
interaction are cited in our works [20]-[24]. In this paper, we considered two two-atom Jaynes-Cummins models
taking into account the direct dipole-dipole interaction between qubits. We concerned our attention on two-atom
double JCM and two-atom JCM with common cavity field. We investigated the entanglement between qubits, and
∗ Corresponding author. Tel.: +7-846-334-5434; fax: +7-846-335-1836.
Email address: bash@ssau.ru (Eugene K Bashkirov )
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239
Mathematical Modeling / E.K. Bashkirov
discussed the dependence of the entanglement on the parameters of the considered systems, such as the intensity of
dipole-dipole interaction and the detuning between the atomic transition frequency and the cavity field frequencies.
2. Double Jaynes-Cummings model
In this section we consider two identical superconducting qubits labeled A and B, and two cavity modes of coplanar
resonators labeled a and b. Qubit A not-resonantly interacts with a single-mode cavity field a, and qubit B not-
resonantly interacts with a single-mode cavity field b. Due to the randomness of the qubits positions in the cavity, it is
very difficult to control the couplings between different atom-cavity systems to be the same. Therefore the coupling
constants between the atoms and cavities are assumed to be unequal. For superconducting qubits interacting with
microwave coplanar resonators or LC superconducting circuits the intensity of effective dipole-dipole interaction can
be compared with the atom-cavity coupling constant. In this case the dipole-dipole interaction should be included in
the model Hamiltonian. Therefore the Hamiltonian for the system under rotating wave approximation can be written
as
H = (~ω0 /2) σzA + (~ω0 /2) σzB + ~ωa a+ a + ~ωb b+ b + ~γa (σ+A a + a+ σ−A ) + γb (σ+B b + b+ σ−B ) + ~J(σ+A σ−B + σ−A σ+B ), (1)
where (1/2)σzi is the inversion operator for the ith qubit (i = A, B), σ+i = |+⟩ii ⟨−|, and σ−i = |−⟩ii ⟨+| are the transition
operators between the excited |+⟩i and the ground |−⟩i states in the ith qubit, a+ and a are the creation and the
annihilation operators of photons of the cavity mode a, b+ and b are the creation and the annihilation operators of
photons of the cavity mode b, γa is the coupling constant between qubit A and the cavity field a and γb is the coupling
constant between qubit A and the cavity field a, δa = ωa − ω0 and δb = ωb − ω0 are the detunigs for mode a and b and
J is the coupling constant of the dipole interaction between the qubits A and B. Here ω0 is the qubit frequency and ωa
and ωb are the frequencies of the cavities modes.
Firstly we take two qubits initially in the Bell-like pure state of the following form
|Ψ(0)⟩A = cos θ|+, −⟩ + sin θ|−, +⟩, (2)
where 0 ≤ θ ≤ π and the cavity fields initially are in vacuum states |0, 0⟩ = |0⟩ ⊗ |0⟩. We take into account that optimal
temperature at which the superconducting qubits are used for quantum computing is mK. For such temperature the
influence of thermal photons of the microwave cavity field on the dynamics of qubits can be neglected.
Then the full initial state is
|Ψ(0)⟩ = (cos θ|+, −⟩ + sin θ|−, +⟩) ⊗ |0, 0⟩. (3)
The evolution of the system is confined in the subspace |−, −, 0, 1⟩, |−, −, 1, 0⟩, |−, +, 0, 0⟩, |+, −, 0, 0⟩. To obtain
the time-dependent wave function of considered model one can use the so-called dressed states or eigenvectors of the
Hamiltonian (1). We have obtained these for general case when parameters of the Hamiltonian (1) take the arbitrary
values. But the general expressions for eigenvectors are too cumbersome to display here. Therefore, we present below
the eigenvectors and eigenvalues of the Hamiltonian (1) for special case when δa = −δb = δ and γa = γb = γ.
In this case the eigenvectors of the Hamiltonian (1) in a frame rotating with the qubit frequency ω0 can be written
as
|Φi ⟩ = ξi (Xi1 |−, −, 0, 1⟩ + Xi2 |−, −, 1, 0⟩ + Xi3 |−, +, 0, 0⟩ + Xi4 |+, −, 0, 0⟩) (i = 1, 2, 3, 4),
where √
ξi = 1/ |Xi1 |2 + |Xi2 |2 + |Xi3 |2 + |Xi4 |2
and
√ √ √
2α 2 −α2 − ∆2 + B + 2∆ A − B
X11 = √ √ , X12 = √ √ , X13 = ( √ √ ) , X14 = 1;
2 2
α + ∆ − B + 2∆ A − B ∆ 2− A−B α −2∆ + 2 A − B
√ √ √
2α 2 α2 + ∆2 − B + 2∆ A − B
X21 = √ √ , X22 = √ √ , X23 = ( √ √ ) , X24 = 1,
2 2
α + ∆ − B − 2∆ A − B ∆ 2+ A−B α 2∆ + 2 A − B
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√ √ √
2α α2 + ∆2 + B − 2∆ A + B 2
X31 = √ √ , X32 = √ √ √ , X34 = 1,
√ , X33 =
2 2
α + ∆ + B + 2∆ A + B ∆ 2− A+B 2α∆ − 2α A + B
√ √ √
2α 2 α2 + ∆2 + B + 2∆ A + B
X41 = √ √ , X42 = √ √ , X43 = ( √ √ ) , X44 = 1,
α2 + ∆2 + B − 2∆ A + B ∆ 2+ A+B α 2∆ + 2 A + B
√
where ∆ = δ/γ, α = J/γ and A = 2 + α2 + ∆2 , B = α4 + 4∆2 + ∆4 − 2α2 −2 + ∆2 .
( )
The corresponding eigenvalues are
√ √ √ √ √ √ √ √
E1 = −~γ A − B/ 2, E2 = γ~ A − B/ 2, E3 = −~γ A + B/ 2, E4 = ~γ A + B/ 2.
For entanglement modeling we can obtain the time dependent wave function
|Ψ(t)⟩ = e−ıHt/~ |Ψ(0)⟩. (4)
Using the eigenvalues and eigenvectors of Hamiltonian (1) and the initial state (3) we can derive from (4)
|Ψ(t)⟩ = C1(1) (t)|−, −, 0, 1⟩ + C2(1) (t)|−, −, 1, 0⟩ + C3(1) (t)|−, +, 0, 0⟩ + C4(1) (t)|+, −, 0, 0⟩, (5)
where
C1(1) = cos θ Z11 + sin θ Z12 , C2(1) = cos θ Z21 + sin θZ22 ,
C3(1) = cos θ Z31 + sin θ Z32 , C4(1) = cos θ Z41 + sin θ Z42
and
Z11 = e−ıE1 t/~ ξ1 Y41 X11 + e−ıE2 t/~ ξ2 Y42 X21 + e−ıE3 t/~ ξ3 Y4n X31 + e−ıE4 t/~ ξ4 Y44 X41 ,
Z12 = e−ıE1 t/~ ξ1 Y31 X11 + e−ıE2 t/~ ξ2 Y3n X21 + e−ıE3 t/~ ξ3 Y33 X31 + e−ıE4 t/~ ξ4 Y34 X41 ,
Z21 = e−ıE1 t/~ ξ1 Y41 X12 + e−ıE2 t/~] ξ2 Y42 X22 + e−ıE3 t/~ ξ3 Y43 X32 + e−ıE4 t/~ ξ4 Y44 X42 ,
Z22 = e−ıE1 t/~ ξ1 Y31 X12 + e−ıE2 t/~] ξ2 Y32 X22 + e−ıE3 t/~ ξ3 Y33 X32 + e−ıE4 t/~ ξ4 Y34 X42 ,
Z31 = e−ıE1 t/~ ξ1 Y41 X13 + e−ıE2 t/~ ξ2 Y42 X23 + e−ıE3 t/~ ξ3 Y43 X33 + e−ıE4 t/~ ξ4 Y44 X43 ,
Z32 = e−ıE1 t/~ ξ1 Y31 X13 + e−ıE2 t/~ ξ2 Y32 X23 + e−ıE3 t/~ ξ3 Y33 X33 + e−ıE4 t/~ ξ4 Y34 X43 ,
Z41 = e−ıE1 t/~ ξ1 Y41 X14 + e−ıE2 t/~ ξ2 Y42 X24 + e−ıE3 t/~ ξ3 Y43 X34 + e−ıE4 t/~ ξ4 Y44 X44 ,
Z42 = e−ıE1 t/~ ξ1 Y31 X14 + e−ıE2 t/~ ξ2 Y32 X24 + e−ıE3 t/~ ξ3 Y33 X34 + e−ıE4 t/~ ξ4 Y34 X44 ,
where Yi j = ξ j X ∗ji .
We also can consider an another type of Bell-like pure initial state of two qubits
|Ψ(0)⟩A = cos θ|+, +⟩ + sin θ|−, −⟩.
For this initial atomic state and vacuum cavities fields the full initial state of the system is
|Ψ(0)⟩ = (cos θ|+, +⟩ + sin θ|−, −⟩) ⊗ |0, 0⟩. (6)
For initial state (6) the time-dependent wave function can be written in the form
|Ψ(t)⟩ = C1(2) (t)|+, +, 0, 0⟩ + C2(2) (t)|+, −, 0, 1⟩ + C3(2) (t)|−, +, 1, 0⟩ + C4(2) (t)|+, −, 1, 0⟩+
+C5(2) (t)|−, +, 0, 1⟩ + C6(2) (t)|−, −, 2, 0⟩ + C7(2) (t)|−, −, 0, 2⟩ + C8(2) (t)|−, −, 1, 1⟩ + C9(2) (t)|−, −, 0, 0⟩. (7)
The coefficients Ci (t) may be obtained by using the way which is described in previous case. But these are too
cumbersome. Therefore, we will use below the numerical results for coefficients under consideration.
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For two-qubit system described by the reduced density operator ρA (t), a measure of entanglement or negativity
can be defined in terms of the negative eigenvalues µ−i of partial transpose of the reduced atomic density matrix ρTA1
[26, 27] ∑
ε = −2 µ−i . (8)
Using the wave functions (5) or (7) one can obtain the density operator for the whole system as
ρ(t) = |Ψ(t)⟩⟨Ψ(t)|. (9)
Taking a partial trace over the field variable one can obtain from (9) the reduced atomic density operator in the two-
qubit basis |+, +⟩, |+, −⟩, |−, +⟩, |−, −⟩ for initial state (3) in the form
0 0 0 0
0 V(t) H(t) 0
ρA (t) = . (10)
0 H(t)∗ W(t) 0
0 0 0 R(t)
The matrix elements of (10) are
V(t) = |C4(1) (t)|2 , W(t) = |C3(1) (t)|2 , R(t) = |C1(1) (t)|2 + |C2(1) (t)|2 , H(t) = C4(1) (t)C3 (t)∗ .
The partial transpose of the reduced atomic density matrix (10) is
0 0 0 H(t)∗
0 V(t) 0 0
T1
ρA (t) = . (11)
0 0 W(t) 0
H(t) 0 0 R(t)
From equation (11), we obtain four eigenvalues. Three of them are always positive. The eigenvalue µ−4 = 1/2(R −
√
4H 2 + R2 ) is always negative. As a result, the negativity can be written as
√
ε(t) = R(t)2 + 4|H(t)|2 − R(t). (12)
The partial transpose of the reduced atomic density matrix ρTA1 for initial state (6) has the form
U1 (t) 0 0 H̃1 (t)∗
0 V1 (t) H1 (t)∗ 0
ρTA1 (t) = . (13)
0 H1 (t) W1 (t) 0
H̃ (t)
1 0 0 R1 (t)
where one can obtain with using (7)
U1 (t) = |C1(2) (t)|2 , H1 (t) = C1(2) (t)C9(2) (t)∗ , H̃1 (t) = C2(2) (t)C5(2) (t)∗ + C4(2) (t)C3(2) (t)∗ ,
V1 (t) = |C2(2) (t)|2 + |C4(2) (t)|2 , W1 (t) = |C3(2) (t)|2 + |C5(2) (t)|2 , R1 (t) = |C6(2) (t)|2 + |C7(2) (t)|2 + |C8(2) (t)|2 + |C9(2) (t)|2 .
Two eigenvalues of matrix (13) may be negative. Then, the negativity can be written as a superposition of two terms
√ √
ε(t) = (U1 (t) − R1 (t))2 + 4|H̃1 (t)|2 − U1 (t) − R1 (t) + (V1 (t) − W1 (t))2 + 4|H1 (t)|2 − V1 (t) − W1 (t). (14)
The first term is taken into account if and only if |H̃1 |2 > U1 R1 and the second term is taken into account if and only
if |H1 |2 > V1 W1 .
3rd International conference “Information Technology and Nanotechnology 2017” 242
3. Two-atom Jaynes-Cummings model
In this section we consider two-atom JCM with common thermal cavity field. We have two identical qubits A and
B (spins, quantum dots etc.) non-resonantly interacting with common one-mode quantum electromagnetic field of
resonator. As in a previous case we assume that the direct dipole-dipole interaction between qubits takes place. But
in contrast with previous case we investigated the entanglement induced by a thermal field. In a frame rotating with
the field frequency, the Hamiltonian for the system under rotating wave approximation can be written as
B
∑
H = ~δσzA + ~δσzB + ~γ (σ+i a + a+ σ−i ) + ~J(σ+A σ−B + σ−A σ+B ). (15)
i=A
Here σzA and σzB are the inversion operators for qubit A and B respectively, δ = ω − ω0 is detuning, where ω is the
cavity field frequency and ω0 is the atom transition frequency. The other notations are similar to these used in Section
2. Let us note that concurrence dynamics for system with Hamiltonian (15) without dipole-dipole interaction has been
earlier investigated by Zhang [25].
We consider two type of initial atomic states: separable state |+, −⟩ (or |−, +⟩) and entangled state (2). The initial
∑
cavity mode state are assumed to be the thermal one-mode state ρF (0) = pn |n⟩⟨n|, where the weight functions are
n
pn = n̄n /(1+ n̄)n+1 . Here n̄ is the mean photon number in a cavity mode, n̄ = (exp[~ωi /kB T ]−1]−1 , kB is the Boltzmann
constant and T is the equilibrium cavity temperature.
Before considering the interaction between two qubits and thermal field, it is straightforward to first study two
qubits simultaneously interacting with Fock state. Suppose that the excitation number of the atom-field system is n
(n ≥ 0). The evolution of the system is confined in the subspace
|−, −, n + 2⟩, |+, −, n + 1⟩, |−, +, n + 1⟩, |+, +, n⟩.
On this basis, the eigenfunctions of the Hamiltonian (15) can be written as
|Φin ⟩ = ξin (Xi1n |−, −, n + 2⟩ + Xi2n |+, −, n + 1⟩ + Xi3n |−, +, n + 1⟩ + Xi4n |+, +, n⟩) (i = 1, 2, 3, 4),
where
X11n = 0, X12n = −1, X13n = 1, X14n = 0,
√ √ √
2 1+n 2+n 1 + n (2∆ + Ein )
Xi1n = − 2
, X i2n = − 2
,
4 + 2n + 2∆ + Ein − 2∆ Ein − Ein 4 + 2n + 2∆ + Ein − 2∆ Ein − Ein
√
1 + n (2∆ + Ein )
Xi3n = − 2
, Xi4n = 1 (i = 2, 3, 4).
4 + 2n + 2∆ + Ein − 2∆ Ein − Ein
The corresponding eigenvalues are
E1n = −~γ α, E2n = (1/3) ~γ (α + An /Bn + Bn ) ,
[( √ ) ( √ ) ]
E3n = (1/6) ~γ Re 2α − 1 + i 3 An /Bn + i i + 3 Bn ,
[ ( √ ) ( √ ) ]
E4n = (1/6) ~γ Re 2α + i i + 3 An /Bn − 1 + i 3 Bn .
Here
An = 18 + 12n + α2 + 12∆2 ,
( ( ) 1√ ( )1/3
3 2 2 2
)3 ( 3 ( 2
))2
Bn = α − 54∆ + 9α 3 + 2n − 4∆ + −4 18 + 12n + α + 12∆ + 4 α − 54∆ + 9α 3 + 2n − 4∆ .
2
To derive the full dynamics of our model one can consider also the basis states |−, −, 1⟩, |+, −, 0⟩, |−, +, 0⟩.
Assume that the system is initially prepared in the state |+, −, n⟩ (n ≥ 0), then at time t, the whole system will
evolve to
|Ψ(t)⟩ = Z12,n |−, −, n + 2⟩ + Z22,n |+, −, n + 1⟩ + Z32,n |−, +, n + 1⟩ + Z42,n |+, +, n⟩. (16)
Mathematical Modeling / E.K. Bashkirov
Here
Z12,n = e−ıE1n t/~ ξ1n Y21n X11n + e−ıE2n t/~ ξ2n Y22n X21n + +e−ıE3n t/~ ξ3n Y23n X31n + e−ıE4n t/~ ξ4n Y24n X41n ,
Z22,n = e−ıE1n t/~ ξ1n Y21n X12n + e−ıE2n t/~] ξ2n Y22n X22n + e−ıE3n t/~ ξ3n Y23n X32n + e−ıE4n t/~ ξ4n Y24n X42n ,
Z32,n = e−ıE1n t/~ ξ1n Y21n X13n + e−ıE2n t/~ ξ2n Y22n X23n + e−ıE3n t/~ ξ3n Y23n X33n + e−ıE4n t/~ ξ4n Y24n X43n ,
Z42,n = e−ıE1n t/~ ξ1n Y21n X14n + e−ıE2n t/~ ξ2n Y22n X24n + e−ıE3n t/~ ξ3n Y23n X34n + e−ıE4n t/~ ξ4n Y24n X44n ,
where Yi jn = ξ jn X ∗jin .
If the initial state of our system is |+, −, 0⟩, the time dependent wave function takes the form
|Ψ(t)⟩ = Z12 |−, −, 1⟩ + Z22 |+, −, 0⟩ + Z32 |−, +, 0⟩, (17)
where
( )
Z12 = −2ıe−ı(α−2∆)t/2 sin(Ωt/2)/Ω, Z22 = e−ı(α−2∆)t/2 eı(3α−2∆)t/2 + Ω cos(Ωt/2) − 2ı sin(Ωt/2) /(2Ω),
( )
Z32 = e−ı(α−2∆)t/2 −eı(3α−2∆)t/2 + Ω cos(Ωt/2) − 2ı sin(Ωt/2) /(2Ω)
√
and Ω = 8 + (α + 2∆)2 .
For initial state |−, +, n + 1⟩ (n ≥ 0) the time-dependent wave function is
|Ψ(t)⟩ = Z13,n |−, −, n + 2⟩ + Z23,n |+, −, n + 1⟩ + Z33,n |−, +, n + 1⟩ + Z43,n |+, +, n⟩. (18)
Here
Z13,n = e−ıE1n t/~ ξ1n Y31n X11n + e−ıE2n t/~ ξ2n Y32n X21n + e−ıE3n t/~ ξ3n Y33n X31n + e−ıE4n t/~ ξ4n Y34n X41n ,
Z23,n = e−ıE1n t/~ ξ1n Y31n X12n + e−ıE2n t/~] ξ2n Y32n X22n + e−ıE3n t/~ ξ3n Y33n X32n + e−ıE4n t/~ ξ4n Y34n X42n ,
Z32,n = e−ıE1n t/~ ξ1n Y31n X13n + e−ıE2n t/~ ξ2n Y32n X23n + e−ıE3n t/~ ξ3n Y33n X33n + e−ıE4n t/~ ξ4n Y34n X43n ,
Z42,n = e−ıE1n t/~ ξ1n Y31n X14n + e−ıE2n t/~ ξ2n Y32n X24n + e−ıE3n t/~ ξ3n Y33n X34n + e−ıE4n t/~ ξ4n Y34n X44n .
If the initial state is |−, +, 0⟩, the time dependent wave function takes the form
|Ψ(t)⟩ = Z13 |−, −, 1⟩ + Z23 |+, −, 0⟩ + Z33 |−, +, 0⟩, (19)
where Z13 = Z12 , Z23 = Z22 , Z33 = Z32 .
Now we go back to the theme of this Section. Using the equations (16)-(19) one can obtain the density operator
for the whole system. Taking a partial trace over the field variables one can obtain the reduced atomic density operator
and partial transpose of the reduced atomic density matrix ρTA1 . For initial atomic state |+, −⟩ the partial transpose of
the reduced atomic density operator has the form
U2 (t) 0 0 H2 (t)∗
0 V2 (t) 0 0
T1
ρA (t) = . (20)
0 0 W2 (t) 0
H2 (t) 0 0 R2 (t)
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where
∞
∑ ∞
∑
U2 (t) = pn |Z42,n (t)|2 , V2 (t) = pn |Z22,n−1 (t)|2 + p0 |Z22 (t)|2 ,
n=0 n=1
∞
∑ ∞
∑
W2 (t) = pn |Z32,n−1 (t)|2 + p0 |Z32 (t)|2 , R2 (t) = pn |Z12,n−1 (t)|2 + p0 |Z12 (t)|2 , (21)
n=1 n=1
∞
∑
H2 (t) = pn Z22,n−1 (t)Z32,n−1 (t)∗ + p0 Z22 (t)Z32 (t)∗ .
n=1
Only one of the eigenvalues of matrix (20) may be negative. Therefore the negativity can be written in the form
√
ϵ(t) = (|R2 (t)| − |U2 (t)|)2 + 4|H2 (t)|2 − |R2 (t)| − |U2 (t)|.
For initial atomic state |−, +⟩ the partial transpose of the reduced atomic density operator has the form (20). Its matrix
elements may be obtained from (21) by replacing the coefficients Zi2n with Zi3n , where i = 1, 2, 3, 4. For entangled
initial atomic state (2) the partial transpose of the reduced matrix also has the form (20). The elements of this matrix
may be obtained by combining the elements of two partial transpose matrix for initial states |+, −⟩ and |−, +⟩.
(a) (b)
(c) (d)
Figure 1: The negativity as a function of γt for double JCM and initial state (3) with θ = π/4. Parameters δa = δb = 0, γ b = γa
(a), δa = −δb = 5, γb = γa (b), δa = δb = 5, γb = γa (c) and δa = δb = 0, γa = 2γb (d). The strength of dipole interaction
α = 0 (dotted), α = 3 (dashed) and α = 5 (solid).
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(a) (b)
(c) (d)
Figure 2: The negativity as a function of γt for double JCM and initial state (6) with θ = π/4. Parameters δa = δb = 0, γ b = γa
(a), δa = −δb = 5, γb = γa (b), δa = δb = 5, γb = γa (c) and δa = δb = 0, γa = 2γb (d). The strength of dipole interaction
α = 0 (dotted), α = 3 (dashed) and α = 5 (solid).
(a) (b)
Figure 3: The negativity as a function of γt for two-atom JCM with common field and initial atomic state |+, −⟩. The strength of
dipole interaction δ = 0 (a) and δ = 0.5 (b). The detuning δ = 0 (solid) and δ = 1 (dashed). Mean photon number n̄ = 0.1.
4. Modeling of qubits entanglement dynamics
The results of calculations of entanglement parameter (12) for double JCM and initial state (3) are shown in
Figs. 1(a)-(d). Results of calculations of entanglement parameter (14) for initial state (6) are displayed in Figs. 2(a-
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Mathematical Modeling / E.K. Bashkirov
(a) (b)
Figure 4: The negativity as a function of γt for two-atom JCM with common field and entangled initial atomic state (2) with θ = π/4.
The strength of dipole interaction α = 0 (a) and α = 0.5 (b). The detuning δ = 0 (solid) and δ = 5 (dashed). Mean photon number
n̄ = 0.1.
d). Fig. 1(a) shows that for exact resonance the negativity evolves periodically between 0 and 1, but the period is
affected by the coupling constant. For resonance interaction the inclusion of the dipole-dipole interaction leads to a
stabilization of entanglement behavior. Figs. 1(b)-1(d) show the effect of dipole-dipole interaction on negativity for
non-resonant interaction and different couplings. When qubits A and B interact with a single-mode cavities fields
via not-zero detuning the presence of dipole-dipole interaction with intermediate strength leads to increasing of the
amplitudes of the negativity oscillations. But for large values of dipole-dipole interaction strength one can see the
stabilization of entanglement oscillations as in the case of exact resonance. Figs. 2(a)-(d) show the time dependence
of negativity for initial state (6) and different strength of dipole-dipole interaction. Fig. 2(a) gives the entanglement
behavior for exact resonance. This behavior is different from that obtained for initial state (3) in resonance regime. The
dipole-dipole interaction does not lead to stabilization of the entanglement, but has only an effect on the periods and
amplitudes of the oscillations of entanglement. However, for non-resonant interaction between dipole-coupled qubits
and fields the reverse behavior of atom-atom entanglement is true. For large values of the dipole-dipole interaction
strength we have to deal with the stabilization of entanglement. Figs. 3 and 4 show the influence of detuning and
dipole-dipole strength on atom-atom entanglement for two atoms interacting with common thermal field of resonator.
Fig. 3(a) shows the entanglement time behavior for different couplings and separable atomic state |+, −⟩ ignored the
dipole-dipole interaction. One can easily find that as the detuning increases, higher entanglement is obtainable. Zhang
[25] earlier discovered such behavior and noted that when the atom-field detuning is large enough, the atoms tend to
exchange energy with each other instead of with the field, and the field, which acts as a medium, is virtually excited
during the atom-atom coupling process. Fig. 3(b) shows the negativity behavior for dipole-coupled qubits. For this
case the reverse behavior of the entanglement is true. It seems like the negativity for qubits decreases as the detuning
increases. We can also consider the negativity behavior for entangled initial state (2). In this case the inclusion of the
detuning leads to a stabilization of entanglement behavior both to the model with dipole-dipole interaction and to the
model without such interaction.
5. Conclusion
In this paper, we investigated the entanglement between two qubits interacting with fields of resonators in the
framework of two type of JCM: double JCM with different coupling constants and detunings and two-qubit JCM
with common cavity field taking into account the direct dipole-dipole interaction. For double JCM we discussed the
influence of dipole-dipole interaction on qubit-qubit entanglement for resonance and not resonance interactions. The
results showed that these parameters have great impact on the amplitude and the period of the atom-atom entanglement
evolution. In addition, the presence of sufficiently large dipole-dipole interaction leads to stabilization of entanglement
for all Bell-types initial qubits states and different couplings and detuning. For two-qubit JCM with common field we
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247
Mathematical Modeling / E.K. Bashkirov
investigated the entanglement dynamics taking into account the dipole-dipole interaction for separable and entangled
initial qubits states and thermal cavity field. For dipole coupled qubits prepared in a separable state the entanglement
decreases as the detuning increases. For dipole uncoupled qubits the reverse behavior of the entanglement is true. For
entangled initial states the inclusion of the detuning leads to a stabilization of entanglement behavior.
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