Multiple Quantum Coherences(MQ)NMR and Entanglement Dynamics in the Mixed-Three-

Hamid Arian Zad

Young Researchers and Elite Club,Mashhad Branch,Islamic Azad University,Mashhad,Iran

1 Introduction

Low-dimensional Heisenberg spin models have been studied from both theoretical and experimental points of view in the last decades due to an interplay of strong quantum fluctuations and topology and many interesting quantum phenomena have been investigated for these models.[1−12]Quantum systems possess some additional correlations so that one can not find any classical counterpart for them.One of those correlations,called entanglement and it is probably one of the most amazing features of quantum mechanics.[13−16]Understanding the nature of entanglement has almost been a central issue in the discussion of the foundation of quantum information theory,where recently,it has been proved that entanglement can be appeared naturally at low temperature quantum spin chains.A great deal of attention has been attracted to the problem of disentanglement of the two-qubit quantum system in a finite time.Yu and Eberly[17−18]named entanglement sudden death(ESD)to the process of earlystage disentanglement and numerous research studies have been carried out in this issue.[19]

The concurrence as a measure of entanglement for an arbitrary bipartite quantum system AB was precisely introduced by Wootters.[20−21]Nonzero amount of the concurrence denotes that the parts A and B are entangled.Minimum amount of the concurrence C(AB)=0 corresponds to a separable state for the system and its maximum C(AB)=1 corresponds to a maximally entangled state.Entanglement also can be detected by means of a so-called entanglement witness(EW).[14−15,22−23]With characterizing separable state ρs(AB)and entangled state ρe(AB)for the bipartite quantum system,we can introduce a hermitian operator W as an entanglement witness(EW)such that Tr[Wρs(AB)]≥ 0 for all separable states,and Tr[Wρe(AB)]<0,which represents the existence of at least an entangled state in the circumstance,which the system is therein.

Mixed-spin systems with different exchange coupling constants have already been receiving an increasing amount of attention in the past few years.[24−29]Diamond chains as attractive structures among these spin models were exactly investigated from quantum correlations,phase diagrams and etc.view points.[30−34]In our previous works,we investigated the pairwise entanglement for the bipartite(sub)systems of the mixed-threespin(1/2,1,1/2)system in thermal equilibrium,[26−27]and then explained some magnetic properties,phase transitions,also,some entanglement witnesses were introduced.In the present work,we investigate pairwise entanglement and multiple quantum(MQ)NMR dynamics in the such mixed-three-spin(1/2,1,1/2)cell of a greater mixed-N-spin chain with XXX Heisenberg model in the thermal equilibrium state for which the nearest spins(1,1/2)have coupling constant J1and the spins(1/2,1/2)have coupling constant J2,moreover,a single-ion anisotropy property is considered for spin-1.

The method of MQ NMR dynamics in solids is so useful instrument for investigating of the solids struc-tures and dynamical processes(such as spin dynamics)therein.[35−38]Correlations between parts of the quantum systems lead to turning MQ coherences up,therefore,as it has already been proved[39]we can detect the MQ coherences in spin chains by this method.[35,40−41]Also in the corresponding mentioned references and some references therein,it was shown that MQ NMR allows us to find a relationship between the intensities of the MQ coherence of the zeroth and second orders and some measures of the quantum entanglement such as concurrence and quantum discord,[42]which are valuable resources of the quantum information processing.

Heretofore,many papers have been devoted to verify the MQ NMR dynamics for a one-dimensional two nuclear spin-1/2 system consisting dipole-dipole permanently interaction(DDI)in the thermal equilibrium and their main purpose isfinding a good relationship between this phenomenon and entanglement dynamics of such system.[43−46]For promoting this purpose,in the present paper we develop a phenomenological theory of a mixed-three-spin(1/2,1,1/2)XXX Heisenberg model in the MQ NMR experiment at low temperatures and calculate the zeroth and second orders MQ coherences in the preparation period by using the reduced density matrices of the pair spins(1,1/2)and spins(1/2,1/2).Then,we obtain the time evolution concurrences and fidelities for both spin pairs.Finally,we do some remarkable comparisons between these measures of entanglement and MQ coherences of the zeroth and second orders of any pairs and express some new outcomes.

2 Physical System and Theoretical Settings

2.1 Spin Hamiltonian of the Mixed-Three-Spin(1/2,1,1/2)XXX Heisenberg Model

We introduce the Hamiltonian of the selected mixedthree-spin anti-ferromagnetic XXX Heisenberg model in the vicinity of an external homogeneous magnetic field B,as the following form

where

M is the number of triangular cells,and we here consider M=1 also,=Bzthat denotes a homogeneous magnetic field applied in the z-direction.Note that,all of the introduced parameters B,ζ,J1and J2(also temperature T in the follows)are considered dimensionless.

Eigenvectors of the Hamiltonian are

where{|↑i,|↓i}are Szup and down states with eigenvalues 1/2 and−1/2 respectively.are Jzup,null,and down states with eigenvalues 1,0,–1 respectively.

And its eigenvalues are

In the standard qubit-qutrit basis states,we can characterize the total density matrix of the mixed-three-spin system in thermal equilibrium,by using Eq.(1)in the form

where β=1/T(we have setted kB=1)for which T is the temperature and Z=Tr[exp(−βH)]is the partition function of the system.Hence,the density matrix of the any pair of spins can be written as

where T1and T2are partial trace over basis states of the first and the second spins respectively. {α,δ,∆,η,δ,ξ,Σ,λ,γ}and{µ,κ,Ω,χ,ν}are functions of T,B,ζ,J2,and J1.

2.2 Concurrence

The concurrence that is a measure of entanglement,can be defined for bipartite spin systems[20−21]as

where λ =max{λ1,λ2,...,λk}(k=4 for the spins(1/2,1/2)and k=6 for the spins(1,1/2))and λiare square roots of the eigenvalues of the inner product R=ρ˜ρ,with spin- flip density matrix=(σy⊗σy)ρ†(σy⊗σy)for the spins(1/2,1/2),and=(jy⊗σy)ρ†(jy⊗σy)for the spins(1,1/2),where σyand jyare spin- flip operators and ρ†denotes the complex conjugation of the reduced density matrix ρ.

2.3 Fidelity

The fidelity is defined as the amount of statistical overlap between two distributions.Suppose that|ψi and|ϕi are two quantum states.The fidelity can be obtained from the inner product of such states|hϕ|ψi|2,which gives us amount of the overlapping between them.[24,41,47]

Let ρ =|ψihψ|and ρ′=|ϕihϕ|be density matrices of these states,hence the fidelity can be characterized as

Now,suppose that those states are pure,i.e.,ρ2= ρ and ρ′2= ρ′,therefore,andhence F(ρ,ρ′)=|hϕ|ψi|.From this one can realize that,0 ≤ F(ρ,ρ′) ≤ 1.F(ρ,ρ′)=1 denotes the both states are the same state and case F(ρ,ρ′)=0 denotes there is no overlap whatsoever.

In the case where operators ρ and ρ′commute with each other,they are diagonal in the same basis states socalled|qii,hence,we can write the fidelity in terms of the eigenvalues of these operators.If we rewrite ρ and ρ′in terms of the basis states|qii with probabilities uiand viin the k-dimensional Hilbert space as

then,we can easily gain the fidelity from the eigenvalues uiand vias

3 MQ NMR Dynamics of the Spins(1,1/2)and the Spins(1/2,1/2):Coherences Intensities at Low Temperature

At first,we consider MQ NMR experiment on the both aforementioned spins(1,1/2)and spins(1/2,1/2)with anti-ferromagnetic XXX Heisenberg model in an external homogeneous magnetic field B=Bzat low temperature T.At initial time τ=0 it is assumed that both(sub)systems be in the thermal equilibrium with general reduced density matrices introduced in Eq.(6).

There are schematically four distinct periods of time:preparation,evolution,mixing,and detection for the standard MQ NMR experiments.[39]MQ coherences are created by the multi-pulse sequence consisting of eight-pulse cycles on the preparation period.In the rotating reference frame,MQ dynamics in spin systems during the preparation period can be governed by the averaged non-secular Hamiltonian HMQas

We introduce this operator for the bipartite(sub)systems(1,1/2)and(1/2,1/2)in the next.

3.1 Bipartite(sub)system(1,1/2)

According to Eq.(11),we characterize 6×6 matrix HMQfor the pair spins(1,1/2)as

where J+=(1/2)(Jx+iJy)and σ+=(1/2)(σx+iσy)are raising operators of the spin-1 and spin-1/2 respectively,and J−=(1/2)(Jx− iJy)and σ−=(1/2)(σx− iσy)are lowering one withAt the end of the preparation period,the density matrix of the pair spins(1,1/2),is given by

where U(τ)=exp[−iHMQτ].

The two-spin Hamiltonian HMQcan be diagonalized in the representation of standard basisby means of its eigenvalues and the transformation operator(columns from left to right as corresponding eigenvectors)

Hence,at the end of the preparation period the density matrix ρ23(τ)may be written as

The intensities of MQ coherences of the zeroth order are lonely determined by the real part of the density matrix ρ23(τ),while,the intensities of MQ coherences of the second order are determined by the imaginary part of Eq.(15).Assume thatbe spectral intensities of order n as

here,the intensities of MQ coherences of the zeroth order,and the plus/minus second orders,can be expressed as

which by referring to Eq.(2),matrix representation ofis derived by taking the partial trace with respect to the first spin-1/2 of 12×12 matrix IWith a straightforward calculation one can find that the intensities of MQ coherences of the minus second order and the plus second order,are equal.However,in the experimental environment,there are some certain errors,which lead to measurement results for G−2(τ)and G+2(τ)not be the same.For more accuracy of these intensities and approaching experimental results to theoretical ones,we will use the sum of MQ coherences of the plus/minus second order G−2(τ)+G+2(τ).It is also worth to notice that in the usual MQ NMR experiments,the sum of the intensities of all MQ coherences does not depend on time,i.e.,

3.2 Bipartite(sub)System(1/2,1/2)

Here,we repeat the above statements for the pair spins(1/2,1/2).Again by using Eq.(11),we start characteriz-ing 4×4 matrix HMQfor the pair spins(1/2,1/2)as

At the end of the preparation period,reduced density matrix ρ13(τ),is given by

By diagonalizing corresponding Hamiltonian HMQin the representation of standard basis{|↑↑i,|↑↓i,|↓↑i,|↓↓i}using transformation operator

at the end of the preparation period,the density matrix ρ13(τ)will be evolved to

where,by using Eqs.(21)and(23),one can find that for the spins(1/2,1/2)

here,the intensities of MQ coherences of the zeroth orderand plus/minus second orderscan be expressed as

4 Numerical Analysis of the MQ NMR Intensities,Concurrences and Fidelities

After exact numerical calculations,we derived the intensities of MQ coherences of the zeroth and plus/minus second orders,concurrences and fidelities of the spins(1,1/2)and spins(1/2,1/2)at low temperature β=5,strong magnetic field B=4 and fixed values of J1=7π/5,J2=1 and ζ=5.Then,we compared them with each other and obtained novel results,which are expressed in this section.

In the considered region,we found that some components of the reduced density matrices Eq.(6)at initial time τ=0 are effectless in our calculations.For better understanding the subject,we depict matrix plots of the reduced density matrices at mentioned fixed values of the temperature,the magnetic field,the coupling constants and the single-ion anisotropy in Fig.1.So,we delete these components and use solely the must effective ones.Those are,{α,δ,∆,∆∗,η}for the spins(1,1/2)and{µ,κ,Ω,Ω∗,χ}for the spins(1/2,1/2).Indeed,those components are effective,which make first 1×1 and second 2×2 blocks on the density matrices diagonal.

Fig.1(Color online)Matrix plot of the density matrices introduced in Eq.(6),at low temperature β=5,strong magnetic field B=4 and fixed values of the coupling constants J1=7π/5,J2=1 and the single-ion anisotropy parameter ζ=5 for;(a)the pair spins(1/2,1/2);(b)the pair spins(1,1/2).

4.1 Bipartite(sub)System(1,1/2)

Using Eqs.(7),(10),(15),and(18),we depict the time dependence of the concurrence C(ρ23(τ)), fidelities F0=F(ρ23(τ),and F2=F(ρ23(τ),and the intensities of MQ coherences of the zeroth and second orders for the pair spins(1,1/2),at low temperature β=5,strong magnetic field B=4 and fixed values of J1=7π/5,J2=1 and ζ=5 in Fig.2.With regard to thisfigure,the concurrence as a measure of the pairwise entanglement is maximum at initial time τ=0 and by passing time,it decreases and reaches its first minimum at timethen periodically changes versus time with a spacial period.Fidelity F0is also maximum at τ=0 and periodically changes with the period of time almost the same for the concurrence.The behaviour of this quantity versus time is very similar to the concurrence.But on the other hand, fidelity F2is minimum at initial time τ=0 tain its position upto timeat which the concurrence and F0are minimum.and reaches its maximum at timeand will almost main-

Fig.2 (Color online)The time dependence of the concurrence C(ρ23(τ))×10(grey cross-diagonal curve), fidelities F0×10 and F2×10(respectively,black dotted and long-dashed lines),and intensities of the MQ coherences of the zeroth order(gold solid line)and second order(blue dot-dashed line)for the spins(1,1/2),at low temperature β=5,strong magnetic field B=4 and fixed values of J1=7π/5,J2=1,and ζ=5.

Moreover,in thisfigure it is clear that,the intensity of MQ coherence of the zeroth orderat initial time τ=0 is maximum and in the contrary,is miniand reaches its maximum value at timewhile simultamum.With pass of the time τ,initially increases neously,decreases and reaches its first minimum.

From these statements,one can precisely recognize that the period of the multi-pulse sequence at the preparation period of the concurrence and fidelities is double one for the intensities of the MQ NMR coherence.One can also verify the entanglement dynamics of the(sub)system via investigating the fidelitywhich denotes the amount of statistical overlap between two distributions ρ23(τ)andAnother interesting quantum phenomenon,which occurs versus time for this(sub)system,is the synchrony maxima of the fidelity F2and the intensity of the MQ coherence of the second ordernamely,when two distributions ρ23(τ)andhave maximum overlapping together atandthe intensity of the MQ coherence of the second orderis maximum.As it is explicitly seen,the behaviour of the fidelity F2is inverse the concurrence,in order words,when F2is maximum the concurrence becomes minimum,which denotes separable state for the(sub)system,and when F2reaches minimum the concurrence is maximum,which denotes state of the(sub)system is an entangled state.However since,the fidelity is a number that is ranged between 0 andone can use F2as a measure of pairwise entanglement for the spins(1,1/2)and detect separable states from entangled states.

Finally,as a new outcome from our investigation,it is suitable to introduceas a pairwise entanglement witness for the bipartite(sub)system(1,1/2)at certain time intervaland also another time intervals in which even maxima ofarise.Indeed,by referring the entanglement witness definition and inspecting Fig.2,we can see that the intensity of the MQ coherence of the zeroth orderis negative sometimes,which the concurrence is non zero.And it has maximum positive value atwhich the concurrence is vanished.Here,we can claim thatis positive for a convex set of separable states and is negative for at least an entangled state in the considered time intervals.

Fig.3 (Color online)Contour plots of the time dependence of the MQ coherence of the zeroth order×10−1for the spins(1,1/2),at low temperature β =5,strong magnetic field B=4 and fixed values of J2=1 and ζ=5,versus the coupling constant J1for regions;(a)0≤ J1≤ 7π/5 and 0≤ τ≤ 10;(b)0≤ J1≤ 7π/10 and 0≤τ≤5.

It is more interesting to investigate the intensity of the MQ coherence of the zeroth orderwith respect to time τ and the coupling constant J1.With change of J1(here,we consider special interval 0≤ J1≤ 7π/5),another ignored components of the density matrix of the spins(1,1/2)in Eq.(6)are involved in calculations,and they must be taken into account.We show contour plots of the time dependence ofwith respect to the coupling constant J1in Fig.3.As illustrated in Fig.3(a),the time dependence ofis very sensitive to the coupling constant J1,this is means that by increasing J1,the period ofis gradually changed.Indeed,the behaviour ofis similar to a 3-D sinusoidal wave,which its period depends on the two variable time and J1.For more simplicity,we divide the figure into four parts:(i)In region 0≤ J1≤ 7π/10 and 0≤ τ≤ 5,with the pass of time τ and increasing J1,the period ofis gradually restricted;(ii)In region 7π/10 ≤ J1≤ 7π/5 and 5 ≤ τ≤ 10,with the pass of time τ and increasing J1,its period be gradually expanded;(iii)In region 0≤ J1≤ 7π/10 and 5≤ τ≤ 10,with the pass of time τ and decreasing J1,the period ofis also gradually expanded and finally;(iv)In region 7π/10≤ J1≤ 7π/5 and 0≤ τ≤ 5,with the pass of time τ and decreasing J1,the period is gradually restricted.These restrictions and expansions are more clearly shown in Fig.3(b)for part(I)of the above division.

Fig.4 (Color online)The coupling constant J1dependence offor the spins(1,1/2)at low temperature β=5,strong magnetic field B=4 and fixed values of J2=1 and ζ=5 for various fixed values of the time τ.

We are now interested into introduce a fantastic quantum phenomenon that can be lonely occurred for the intensity of the MQ coherence of the zeroth orderfor J1>7π/5.This phenomenon is similar to the entanglement death,which was introduced in Refs.[7,26](also review Refs.[48–49]),and can be named “entanglement witness death”.Figure 4 depicts this phenomenon explicitly.With regard to thisfigure in detail,one can see that by the pass of time from τ=≈ 0.11 to τ=5,all amplitudes depending on the coupling constant J1restrict to the up and down solid-line curves related to times τ=≈0.11 and τ=≈0.25 respectively,and also,by increasing J1,the time-dependent amplitude damping ofoccurs,until at a unique point in the interval 5.5

4.2 Bipartite(sub)System(1/2,1/2)

Using Eqs.(7),(10),(24),and(27),we show the time dependence of the concurrence C(ρ13(τ)), fidelities F0=F(ρ13(τ),and F2=F(ρ13(τ),and the intensities of MQ coherences of the zeroth and second orders for the spins(1/2,1/2),at low temperature β=5,strong magnetic field B=4 and fixed values of J1=7π/5,J2=1,and ζ=5 in Fig.5.

Fig.5 (Color online)The time dependence of the concurrence C(ρ13(τ))× 10(grey cross-diagonal curve), fidelities F0×10,and F2×10(respectively,black dotted and long-dashed lines),and the intensities of the MQ coherences of the zeroth order (gold solid line)and second order (blue dot-dashed line)for the pair spins(1/2,1/2),at low temperature β=5,strong magnetic field B=4 and fixed values of J2=1,J1=7π/5 and ζ=5.

As before in here,the intensity of the MQ coherence of the zeroth orderis maximum at initial time τ=0,whileis minimum.With the pass of time,decreases and reaches its first minimum value after timewhereas,arises from zero and reaches its first maximum value at the same time.Also,the fidelities behaviour is similar to that one for the spins(1,1/2),namely,at initial time τ=0,F0=F(ρ13(τ),is maximum but F2=F(ρ13(τ),is minimum.Interestingly,the periodical behaviour of the fidelity F2and the intensity of the MQ coherence of the second orderversus time τ are the same.In conclusion,when two distributions ρ13(τ)andhave maximum overlapping together at(and all the next period times at which F2is maximum),the intensity of the MQ coherence of the second orderis maximum.Here,for the considered region of the temperature and the magnetic field,the concurrence C(ρ13(τ))has a permanent value C(ρ13(τ))× 10 ≈ 0.12.

5 Discussion and Conclusions

We have investigated in detail,both the MQ NMR and the pairwise entanglement dynamics of the Heisenberg spin chains studied in the previous works in to a mixedthree-spin(1/2,1,1/2)XXX Heisenberg model with singleion anisotropy property,at low temperature and strong magnetic field,from the theoretical and numerical perspectives.The fidelity was verified for both(sub)systems(1,1/2)and(1/2,1/2)as well.Also,some useful comparisons have been down between them.After investigations,we have obtained some interesting outcomes,which are listed in the follows:

(i) For the(sub)system(1,1/2)with the coupling constant J1,the time dependence of the fidelity F0=F(ρ23(τ),behaves similar to the time dependence of the concurrence C(ρ23(τ)),but the time dependence of the fidelity F2=F(ρ23(τ)behaves almost contrariwise the time dependence of the concurrence.In conclusion,one can investigate the pairwise entanglement dynamics of this(sub)system by verifying the time dependence of the fidelities F0and F2.

(ii) When the fidelity F2reaches its first and second maximum,becomes maximum simultaneously,meanwhile,becomes minimum one time when F2evolves from its first maximum to the second one.This means that,when two distributions ρ23(τ)andhave maximum overlapping together,the intensity of the MQ coherence of the second order is maximum,on the other hand,when F2vanishes,is zero.

(iii)By comparing the dynamics of the pairwise entanglement and the intensity of the MQ NMR coherence of the zeroth order by means of functions C(ρ23(τ))andone can realize that,in some special intervals of the time τbehaves as a pairwise entanglement witness,therefore,he can use it for detecting some entangled states for this(sub)system.

Moreover,we have investigated the time dependence of the MQ coherence of the zeroth order versus J1.We expressed that changes of J1is very effective on the timedependent periodical behaviour ofUltimately,we concluded that for all times between intervalfor interval 4≤J1≤6,increment of the coupling constant J1leads to the amplitude damping in the oscillations ofAll amplitudes specially restricted to the curves at times τ≈and τ≈Hence,one can interpret this interesting quantum phenomenon as a pairwise entanglement witness death.The considered values of the parameters for investigating the(sub)systems can be changed and another intervals obtained.

Finally,we have used the definition of the MQ NMR and pairwise entanglement dynamics for the pair spins(1/2,1/2)with the coupling constant J2.We found that,the behaviour of MQ coherences of the zeroth and second ordersandare opposite.The fidelitybehaviour is more analogous the concurrence thanOn the other hand,the fidelity F2behaviour is similar to the MQ coherence of the second orderAnd since,the fidelity is a kind of measure of how similar the two states are or how much overlap there is between them,we realized that when two distributions ρ13(τ)andhave maximum overlapping together,the MQ coherence of the second orderis maximum,and when the overlap between them is minimum(F2=0)vanishes simultaneously.

In this paper we restricted ourselves to a finite chain with mixed-three-spin(1/2,1,1/2)of a large mixed-N-spin chain,and because it is generally difficult to simulate a spin chain with large length(many body system),on the other hand,small-size systems can be good options to achieve some information about large-size systems.Fortunately,small-size cells of a large-size spin diamond chain can also depict some predictable properties of the large-size chain such as the thermal,the magnetic and the quantum correlation properties.Investigating the MQ NMR dynamics in the considered mixed-three-spin system with another Heisenberg models and also additional interactions such as Dzyaloshinskii–Moriya interaction and etc.,may be more stimulating and new windows might be opened on the quantum information processing science.However,if one investigates such model whether with longer chain of spins or added other mentioned options,his/her obtained results must be compatible with our results and interpretations in this paper.

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