Violation of Svetlichny inequality in Triple Jaynes-Cummings Models

We study the genuine tripartite nonlocality of some qubit states in a triple JCM. In this model, each atom state (A, B or C) was initially prepared with an independent cavity (a, b or c). By using two kinds of GHZ-like states as the atomic initial states, we investigate the genuine tripartite nonlocality as the time evolutions for the non-interaction three-qubit subsystems. We also study the genuine tripartite nonlocality of the subsystems by using the Svetlichny inequality. For the subsystems of three atoms ABC and three cavity modes abc, we show that they are genuinely nonlocal at certain period intervals of time. The states of all the other inequivalent subsystems satisfy the Svetlichny inequality for two types of GHZ-like states.


Results
the model and genuine nonlocality of general three-qubit state. We consider the model of three two-level atoms A, B and C put in three single-mode near-resonant cavities a, b, c. There are no interactions among the subsystems Aa, Bb and Cc. Initially, three cavities are in unexcited state and three atoms entangled state. The local atom-cavity interaction is depicted by the JCM, the local Hamiltonian of the system is  where k ω is the energy difference between the two states of an atom and ν k is the frequency of the corresponding field, g k is the atom-field coupling constant between atom and cavity mode, a a ( ) . An important way to detect the genuine tripartite nonlocality is to analyze through the amount of maximal violation of the Svetlichny inequality. The amount of maximal violation of the Svetlichny inequality is a widely used measure for quantifying the genuine tripartite nonlocality. As a matter of fact, even in the the simplest multipartite systems, the genuine nonlocality of three-qubit states is not completely understood.
In this paper, our purpose is to analyze the genuine nonlocality in a triple JCM with the two types of GHZ-like states as initial states.
To be read conveniently, we briefly review the tripartite Bell scenario and Svetlichny inequality, see ref. 16 for more details.
Suppose there is a three-qubit quantum system shared by three parties, say, Alice, Bob and Clare. We assume the two measurement observables for Alice are X x σ = ⋅ and σ are unit vectors and σ σ σ σ = ( , , ) 1 If there exists a Svetlichny operator S such that the inequality is violated for some three-qubit state, then this state is genuinely nonlocal. For the GHZ state | 〉 = | 〉 + | 〉 GHZ eee ggg ( ) / 2, the Svetlichny inequality is maximally violated and the maximal violation value is 16 as follows: As for the calculating of S( ) ρ , we need the following results. For any three-qubit state ρ, it can be represented by Pauli matrix basis: The coefficients can be used to form three matrices which are called correlation matrices 16 , that is, = T t ( ) k i jk indexed by i and j for = k 1, 2, 3. For a three-dimensional vector z z z z ( , , ) is called the correlation cube 16 . In order to calculate ρ S( ), we need two more vectors: The following theorem and lemma are essential for our calculating. Theorem 1. Suppose 16 ρ is the density operator of a three-qubit system whose correlation matrices are T 1 , T 2 and T 3 . Then, the genuine tripartite nonlocality of the state ρ is and the maximum is taken over all possible measurement variables y, y′, z and z′. Lemma 1. Assume 19 ρ is a three-qubit state with correlation matrices T 1 , T 2 and T 3 . Then where the nine coefficients C i ( 1 9) i = … are not given here because they are the equal to Eq. (9) in ref. 13 except the phase factors do not affect the final results.
Next, we will investigate the genuine nonlocality of the subsystems of atoms and cavities. Since the atoms A, B and C in the GHZ-like state (2) are permutationally invariant, the whole six-qubit system with highly symmetry only has four inequivalent non-interaction subsystems abc, abc, ABc and Abc. We only focus on these four inequivalent non-interaction subsystems ABC, abc, ABc and Abc. The states of the corresponding subsystems are determined by the density at time t and these operators can be calculated by tracing over other qubits in the state ρ have the following X-form: As for the explicit entities can be obtained from ref. 13 and we leave them out here. Now we calculate the value S( ) ρ for the three-qubit states t ( ) to analyze their genuine nonlocality. At first, the correlation matrices T T T , , 1 2 3 and the correlation cube can be calculated directly. We write them down here.
where the maximum is taken over all possible measurement variables ′ ′ y y z z , , , . By the definition of λ 0 and 1 λ , we have Since we only need the maximum value of 0 y y z z 0 1 , , , 1  2  1  2  3  2  3   1  1  3  3  2  2  1  2  1  2  3  2  3 , , ,  18 where ρ stands for ρ t ( ) Similarly we have . In Fig. 1, we plot S(ρ ABC ), S(ρ abc ), S(ρ ABc ) and S(ρ Abc ) as a function of gt for the case of = = = g g g g A B C and /4, /6, /12 θ π π π = . As we can see from the picture, there exist some time intervals in which ρ abc and ρ abc violate the Svetlichny inequality for /4, /6 θ π π = . And in other time intervals, the corresponding subsystems all satisfies the Svetlichny inequality as the time evolutions. After a simple calculation by (27)  and t he ot her c as es c an b e der ive d in t he simi l ar way. Since . In ref. 14 the authors find that the three-tangle magnitudes of the subsystem ABc and Abc are smaller than those of the subsystem ABC and abc. When we investigate the genuine tripartite nonlocalities of these subsystems, maybe the correlations of the Abc ( ) and Abc ( ) systems are not able to be tested by the Svetlichny inequalities. From a physical point of view, we don't know very well this peculiar phenomenon that happened among these subsystems. This results may implicit that the correlation among three atoms or among three cavities is more intense than the subsystems of atoms and cavities mixed together.
In Fig. 2, we plot S( ) ABC ρ and ρ S( ) abc for |Φ 〉 ABC as a function of gt for both the cases of equal ( Fig. 2(a,b)) and different (Fig. 2(c,d) ) coupling constants. We next analyze the genuine nonlocality of the corresponding atoms and cavities and obtain that the nonlocality losses (gains) of atomic subsystems while the nonlocality gains (losses) of cavities. Set ρ = S( ) 4 ABC , after a simple calculate by expression (27), we see that the genuine tripartite nonlocality changes in the time interval T which is defined as ≤ ≤ π T g t : 0 max 2 , where = g g g g max{ , , } A B C max . Formally, one can treat the cavity fields during T as a dissipation factor for the atoms and map a dissipative evolution (refs. 13,20,21 ). This dissipative evolution can be looked upon as a decaying process obeyed by an exponential rule exp t ( ) γ − ′ ; onto the JC dymamics by identifying between the t and t′ as exp γ − ′ = t g t ( ) cos ( ) 2 max 22 . Then, t′ → ∞ is corresponding to g t max 2 → π . Comparing Fig. 2(a-d)), we find that the atoms (cavities) are genuine tripartite nonlocality for some time interval before = π gt 2 , no matter whether the coupling constants are equal or different. Comparing Fig. 2(a-d)), roughly speaking, the genuine tripartite nonlocality of atoms decreases while the nonlocality of cavities increases and vice versa. are not written here since they are the same as Eq. (18) in ref. 13 except the phase factors do not affect the final results.

Genuine nonlocality of the |Ψ
As the atoms A and B in the GHZ-like state (3) are permutationally invariant, the whole highly symmetry six-qubit system only has six inequivalent non-interaction subsystems ABC, abc, ABc, Abc, ACb and Cab. The www.nature.com/scientificreports www.nature.com/scientificreports/ nonzero entities of the density matrices of the six subsystems can be found in refs. 13,14 and we also leave these data out here.
As for the calculation of ρ S( ), it is similarly to the last section. We only write the results down here  and θ π π π = /4, /6, /12. In the case of GHZ-like state |Ψ 〉 ABC we have a similar picture to the former case. As we can see from the picture, there exist some time intervals in which ρ ABC and abc ρ violate the Svetlichny inequality for /4, /6 θ π π = . These time intervals are also periodic similar to the GHZ-like state ABC |Φ 〉. While the exact intervals such that the corresponding state is genuinely nonlocal is coincide with the situation of GHZ-like state |Φ 〉 ABC as initial state, since we choose the same coupling constants. And in other cases the states all satisfy the Svetlichny inequality as the time evolutions.
In Fig. 4, we plot S( ) ABC ρ and S( ) abc ρ for ABC |Ψ 〉 as a function of gt for both the cases of equal ( Fig. 4(a,b)) and different (Fig. 4(c,d) with g g g g /3 A B C = = = ) coupling constants. Easily, we find that Fig. 4 is similar with Fig. 2 for the chosen coupling constants. Therefore, we get the corresponding analysis for Fig. 4 as before in the dissipation language.

Discussion
We investigate the genuine nonlocality dynamics in a triple Jaynes-Cummings model with the two types of GHZ-like states as initial states based on the violation of the Svetlichny inequality. We calculate and get the exact analytical expressions for all inequivalent non-interaction subsystems. For the two types of GHZ-like states as initial states, we know there are certain time intervals and angles θ that the corresponding states violate the Svetlichny inequality for the subsystems ABC and abc. This means the corresponding subsystems are genuine tripartite nonlocality. Since the quantum state of three atoms and of three cavities all do not display nonlocality via the MABK inequality as shown in ref. 15 , it seems that MABK inequality is not always the optimum way to test the local realism according to our results. As for the other cases, we know the Svetlichny inequalities of the corresponding subsystems hold for all possible Svetlichny operators. Since ρ > S( ) 4 is a sufficient condition for the three-qubit state ρ being genuinely nonlocal, we don't know whether the subsystems are genuinely nonlocal or Scientific RepoRtS | (2020) 10:6621 | https://doi.org/10.1038/s41598-020-63236-9 www.nature.com/scientificreports www.nature.com/scientificreports/ not for the cases which the Svetlichny inequalities hold. Physically speaking, the subsystem of three atoms or three cavities may be more intense correlation than the other subsystems consist of atoms and cavities.

Methods
For any three-qubit state ρ, the genuine tripartite nonlocality S( ) ρ can be calculated by Theorem 1 in ref. 16 ρ = S FT T T ( ) 2 ( , , ) , and the maximum is taken over all possible measurement variables y, y′, z and z′. If we detect that ρ > S( ) 4, then we know the corresponding state ρ is genuinely nonlocal by the violation of Svetlichny inequality. .