Dynamics of N two-level moving atoms under the influence of the non-linear Kerr medium

Dynamical evolution of global quantum discord (GQD) and von Neumann entropy (VNE) is studied for N two-level atomic system (TLS), namely two, three and four TLS. The system interacts with the single mode Fock field and the cavity is filled with the non-linear Kerr medium (NLKM). It is seen that for large values of the Kerr parameter, the GQD and VNE show periodic behavior which leads to the sudden death and birth of the quantum entanglement (QE). It is found that the oscillations of the GQD is suppressed for the initial mixed state. The VNE has non-zero values of quantum interference throughout the dynamics of the system for initial mixed states. It is also found that the multipartite systems show sustained response to the QE during the time evolution. Moreover, the GQD and VNE are notably affected in the presence of the NLKM and affected for the entire range of Kerr values as we increase the number of photons in the cavity. Furthermore, for both initial pure and mixed states, the periodic behavior of the GQD and VNE is almost the same for the case of moving TLS. The sudden death and birth of QE is more prominent for moving TLS case. The atomic motion is favorable to sustain the QE in the atomic systems in a NLKM.


Introduction
Successful quantum computation and quantum information rely on the controlled evolution and precise measurements of many TLS Bennett and DiVincenzo (2000).Many phenomena in the field such as entanglement and multipartite quantum correlations using TLS and spin models are studied and explored Amico et al. (2008); Osborne and Nielsen (2002); Vidal et al. (2003); Osterloh et al. (2002); Yang et al. (2019).However, quantifying entanglement and multipartite quantum correlations for the multipartite systems remains a theoretical challenging task.Quantum correlations are more meaningful to study in many-body quantum systems as compared to bipartite entanglement Rulli and Sarandy (2011).Moreover, for QIP, the QE serve as a resource of quantum teleportation Bennett et al. (1993) and superdense coding Agrawal and Pati (2006).Moreover, Extended author information available on the last page of the article

The model
We consider a similar approach as introduced in the simplest Tavis-Cummings (TC) model Schlicher (1989) where N TLS is surrounded by a NLKM in a single-mode cavity.The cavity mode is coupled to the Kerr Medium and N TLS.The total Hamiltonian of the system, ĤT under the RWA can be written as where â and â † are the field annihilation and creation operators, σz i is defined as atomic inversion operator of ith atomic system.The operators σ+ i and σ− i are the atomic raising and lowering operators of an ith TLS.The energies of the atomic system and the field is 0 and , respectively (with ℏ = 1 ), and g is called the coupling parameter between the atom and the field.We assume that N atoms resonantly interact with the single-mode field.The interaction part of the Hamiltonian is given by where represents the Kerr nonlinearity.The field mode shape function of the cavity is Ω(t) = g sin( vt/L).With the atomic motion, we have (i.e.≠ 0) and Ω(t) = g without the atomic motion (i.e.= 0) .While, v is the velocity of the atoms moving along the z-axis and is the parameter of the atomic motion and L is the cavity length.We consider the atomic velocity as v = gL∕ , we have This equation describes the distribution probability of the TLS inside the cavity.The moving atomic system moves inside the cavity with variable Ω and without the atomic motion it is gt.We take the initial state of the system interacting with the Fock state of the input field as where �n⟩ represents the Fock state and n is the number photons inside the cavity.The state vector � ⟩ can be written as where p is the statistical mixture parameter that makes the initial atomic state pure or mixed with limits as given below The allowable basis set states (2) ) =gt for = 0. (5) � ⟩ = cos ( )�g 1 g 2 .....g N ⟩ + sin ( )�e 1 e 2 .....e N ⟩, (7) 0 ≤ p ≤ 1 and 0 ≤ ≤ .The final state of the system at time t is According to the Markovian approximation, the system's time evolution is given by Milburn (1991), where is the density matrix and the intrinsic decoherence coefficient is .For → 0 , Eq. ( 10) we get von Neumann equation describing the Schrodinger equation.The solution of Eq. ( 10) is given by, with where ρ(0) is the initial state of the system.For = 0 , the final state in terms of eigenval- ues can be written as, where E i , E j and � i ⟩, � j ⟩ are the eigenvalues and eigenvectors of the density matrix.In order to compute the correlations, we use the GQD expression as given below where where �k⟩ are the eigenstates of ⊗ N j=1 σz j and rotational local qubit operator is given as R which acts on the jth qubit as given below with The VNE is given by ( 8) where the eigenvalues of are r i and is the VNE of the subsystem jth and the total atomic system.

Numerical discussion
We investigate the dynamical behavior of the GQD and VNE a moving N TLS (N=2,3, and 4).The system interacts with a single-mode Fock field and it is placed inside the cavity filled with the NLKM.In Figs. 1 and 2, the dynamics of the quantifiers for a two TLS (i.e.N = 2 ), in the presence of a NLKM with = 0 , is presented.The other parameter in Fig. 1 is p = 0 , the initial pure atomic state is prepared with = 0 and = 3 ∕4 as the ground state and a superposition state, respectively.The time t is taken in scaled time units.The behavior of both the quantifier is analyzed for different Kerr factor values.For an initial ground atomic state, the dynamical evolution of the GQD under the NLKM exhibits a periodic behaviour.The amplitude of the GQD decreases if is increased.It is seen that the period of oscillations is reduced and its amplitude is suppressed for = 3 .Moreover, not- ing the g ratio, the smaller ratio corresponds to more coupling between the atomic system and the field leading to increase in the entanglement, while the large ratio values point out that the field and the atom are almost decoupled leading to a decreased value of entanglement.Very strong nonlinear interaction of the Kerr-like medium with the cavity fields, entanglement magnitude vanishes between the atoms and the field.The dynamics of the VNE for the initial ground state ( = 0) also shows periodic oscillations.On the other hand, for an initial superposition state, the GQD and VNE dynamics show the revivals in both quantifiers at = 1 .These revivals are more prominent at = 1 which means the sys- tem shares the quantum correlations and entanglement between two TLS.For the initial superposition state, slow a periodic variations are seen in the GQD at = 3 and for = 0.3 , the system shows varying amplitude between the maximum and minimum quan- tum correlations among the TLS.Therefore, the amplitude of periodic oscillations of the GQD is decreased for = 0.3 to = 3 in the two TLS.This dynamical behavior is also observed for the case of the VNE.However, the frequency of these oscillations show different behavior in the system dynamics i.e. the frequency of oscillations increases by increasing the value of .In Fig. 2, the quantifiers are plotted for two different initial mixed states.A periodic behavior, which is oscillatory in nature, is seen.The oscillations are comparatively suppressed for the GQD in case of mixed state.
Figures 3 and 4 show the behavior of the quantifiers, for different values and with = 0 , for the three TLS (i.e.N = 3 ) for p = 0 and p = 0.5 , respectively.The dynamics of the quantifiers have constant and periodic varying amplitude for = 0.3 and = 0 .Both the GQD and VNE show constant oscillations at = 1 .If we further increase of NLKM value i.e. = 3 , rapid oscillations are observed in the dynamics of both quantifiers.For = 3 ∕4 , the three TLS has revivals in the GQD and VNE for all the whole range of .The oscillations in these quantifiers reflect the sharing of quantum correlations among the atomic systems throughout the dynamics.Non-zero value of the VNE for this initial superposition pure atomic states suggests that the atomic systems have non-vanishing quantum interference in the system due to availability of photons and the accessible microstates of atomic states all the time.The dynamics is also computed with two different initial mixed states and both the quantifiers have shown the periodic oscillations.The dynamics of VNE shows non-vanishing quantum interference in the atomic system for both mixed states.The dynamics of the quantifiers with = 0 for the four TLS (i.e.N = 4 ) is shown in Figs. 5 and  6.From Fig. 5, the rapid oscillations can be observed in the quantifiers as is increased.However, higher values suppresses the dynamics of of the quantifiers.Furthermore, the dynamical behavior of both the quantifier is found highly prone to the NLKM.In Fig. 7, the effects of the NLKM parameter with zero photons case with = 0 , is studied for an initial mixed state and the results are plotted by taking = 3 ∕4.When no photons are available in the system, the two, three and four TLS have periodic behavior in both the quantifiers.Unlike the non-zero photon case, the GQD shows no rapid oscillations for all the range of Kerr parameter .Furthermore, without the presence of the photons in the system, the magnitude of the GQD and VNE in the dynamics is not suppressed by the same fashion as with non-zero photons case (see Figs. (1,2, 3, 4, 5 and 6).The dynamics of the GQD suggest that for N = 2 , the NLKM does not have a prominent effect on the magnitude and behavior of the GQD.For N = 3 and N = 4 , the Kerr parameter effects  the dynamics of the GQD and a prominent difference in the magnitude and behavior is observed for different values.For zero photon case, the higher value decreases the VNE in the system, and hence decreases QE.A more periodic behavior in the VNE is observed for = 3.
In Figs. 8 and 9 we have plotted the dynamics of the quantifiers for two, three and four moving TLS in the presence of the NLKM for an initial state corresponding to p = 0.5 .In Fig. 8, we take = 0.3 , = 0 (left panel), 3 ∕4 (right panel), = 1 and n = 10, the dynamics of both the quantifiers exhibit periodic oscillations.In Fig. ( 9), Kerr parameter is  taken as = 3 .For = 3 , a rapid oscillatory behavior of the quantifiers is observed.In the case of moving TLS, and for = 0 and = 3 ∕4 , the behavior of the VNE shows the non- zero value for the entire range of Kerr parameter values due to accessible states in the system.It is investigated for = 1 , increasing reduces the magnitude of the GQD and VNE and causes rapid oscillations in these quantifiers.For = 3 , the GQD (see Fig. 9), three and four TLS has rapid oscillations than two TLS and this increase in oscillations is not observed for the case of = 0.3 (Fig. 8).We can conclude that the systems with more than two TLS are more prone to the Kerr parameter value .Therefore, the motion of atoms is favorable to sustain the QE in the atomic systems.The effect of intrinsic decoherence on the dynamics of the GQD and VNE for two TLS is shown in Fig. 10.We have taken initial mixed state with p = 0.5 and = 3 ∕4 .The maximum value of the GQD and VNE decreases with the increase in the NLKM parameter.In the presence of the intrinsic decoherence, the GQD and VNE oscillations are decayed.For this case, the amplitude of the GQD and VNE reduces with increasing values.For = 1 and = 3 , the correlations of the quantifiers grow a station- ary value in the given scaled time.For all the values, the quantifiers eventually get a quasi-steady value of non-classical correlations and entanglement.The sudden death and birth of the GQD and VNE disappear due to the intrinsic decoherence effects.For

Conclusions
Multipartite quantum correlations and entanglement plays a crucial role in quantum computing and QIP.QE is thought to be a resource of quantum information.The study of how multipartite quantum correlations and entanglement change over time in open quantum systems interacting with environment and under more realistic cavity conditions has recently received a lot of attention from researchers.The potential use of such TLS in the construction of quantum computers provides motivation for the study of multipartite quantum correlations and the entanglement of many TLS.
We have studied the dynamical behavior of the GQD and VNE for N TLS (two, three and four TLS) is studied.The system interacted with the singled mode Fock field.The cavity was filled the NLKM.We observed that for higher Kerr parameter values, the GQD and VNE show periodic behaviour which leads to the collapses and revivals in the QE.The mixed states have comparatively suppressed oscillations of the quantifiers and has non-zero values of quantum interference throughout the dynamics for the mixed states which leads to the sustained response to the QE during the time evolution.The dynamical character of both the quantifier is found highly prone to the NLKM.Moreover, the presence of photons leads to a notable effect on the GQD and VNE for the entire range of Kerr values and photons are responsible for the rapid oscillations in the quantifiers.Furthermore, for the moving TLS case, the periodic behavior was seen in the of the quantifiers.This behavior was almost the same for initial pure and mixed atomic states.Hence the atomic motion is favorable to sustain the QE in the systems in a NLKM.In the presence of intrinsic decoherence, the sudden death and birth of the quantifiers disappear.

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Page 4 of 13 Fig. 1 (color online) The GQD and VNE as a function of time for two (N=2) TLS is shown.We have taken p = 0 , n = 10 and = 0.

Fig. 2
Fig. 2 (color online) The GQD and VNE as a function of time for two (N=2) TLS is shown.We have taken p = 0.5 , n = 10 and = 0.

Fig. 3
Fig. 3 (color online) The GQD and VNE as a function of time for three (N=3) TLS is shown.We have taken p = 0 , n = 10 and = 0.

Fig. 4
Fig. 4 (color online) The GQD and VNE as a function of time for three (N=3) TLS is shown.We have taken p = 0.5 , n = 10 and = 0.

Fig. 5
Fig. 5 (color online) The GQD and VNE as a function of time for four (N=4) TLS is shown.We have taken p = 0 , n = 10 and = 0.

Fig. 6
Fig. 6 (color online) The GQD and VNE as a function of time for four (N=4) TLS is shown.We have taken p = 0.5 , n = 10 and = 0.

Fig. 7
Fig. 7 (color online) The GQD and VNE as a function of time for two, three and four TLS interacting with single-mode Fock field in the NLKM for p = 0.5 , = 3 ∕4 , n = 0 and = 0.

Fig. 8
Fig. 8 (color online) The dynamics of the quantifiers for two, three and four moving TLS is shown.We have taken = 0.3 , p = 0.5 , n = 10 and = 1 at = 0 (left panel) and = 3 ∕4 (right panel)