Multi-mode entanglement of N harmonic oscillators coupled to a non-Markovian reservoir
aa r X i v : . [ qu a n t - ph ] J un Multi-mode entanglement of N harmonic oscillatorscoupled to a non-Markovian reservoir Gao-xiang Li , Li-hui Sun and Zbigniew Ficek E-mail: [email protected] Department of Physics, Huazhong Normal University, Wuhan 430079, P. R. China The National Centre for Mathematics and Physics, KACST, P.O. Box 6086, Riyadh11442, Saudi Arabia
Abstract.
Multi-mode entanglement is investigated in the system composed of N coupled identical harmonic oscillators interacting with a common environment. Wetreat the problem very general by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoirto the oscillators. We invoke an N -mode unitary transformation of the position andmomentum operators and find that in the transformed basis the system is representedby a set of independent harmonic oscillators with only one of them coupled to theenvironment. Working in the Wigner representation of the density operator, we findthat the covariance matrix has a block diagonal form that it can be expressed in termsof multiples of 3 × × Submitted to:
J. Phys. B: At. Mol. Phys. ulti-mode entanglement of N harmonic oscillators
1. Introduction
Controlled dynamics and preservation of an initial entanglement encoded into acontinuous variable system of harmonic oscillators coupled to a noisy environment arechallenging problems in quantum information technologies [1, 2]. The coupling inducesdecoherence phenomena, such as decay and dissipation that reduce and even can destroythe initial entanglement over a finite evolution time [3, 4, 5]. Dynamics of an openquantum system are usually studied in terms of the master equation of the reduceddensity operator whose structure depends on the nature of the environment to whichthe system is coupled. It has been noted that the dynamics crucially depend on whetherthe oscillators interact with a common or independent local environments. In the latercase the interaction usually leads to a degradation of the entanglement whereas in theformer, the environment can not only create decoherence, as it usually does, but mayact as a source of coherence that not only preserves the initial entanglement but alsocreates an additional entanglement. A series of papers accounts these properties for thecase of two coupled harmonic oscillators being in contact with a Markovian thermalreservoir and the work of Liu and Goan [6], Maniscalco et al. [7] and H¨orhammerand B¨uttner [8] accounts for a non-Markovian thermal bosonic reservoirs. Detaileddiscussions and extensive reference lists devoted to the decoherence of two harmonicoscillators can be found in Refs. [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Non-Markovian quantum dynamics of open systems has been discussed by others, notably byBreuer and Vacchini [22], who provide the memory kernel treatment and illustrate it forvarious examples and applications. In all these studies a general conclusion made is thatentanglement dynamics depends on the form of the reservoir and the non-Markoviannature of the reservoir preserves entanglement over a longer time.More important in the quantum technologies is the characterization and the studyof dynamics of a large number of harmonic oscillators that are crucial for the study ofquantum coherence, entanglement, fluctuations and dissipation of mesoscopic systems.The correct understanding of the mechanism responsible for entanglement evolution inthe system is essential for designing N -atom systems for quantum information processingand quantum computation. The key problem is to find the master equation for N harmonic oscillators coupled to an environment that can be solved in a simple andeffective way. How to treat such a composed system in the most effective way and howto understand its complicated dynamics are challenging questions that still have notbeen resolved.In this paper, we pursue a research that especially addresses these questions. In theapproach, we treat the problem very general, fully accounting the non-RWA dynamicsand considering the environment as a non-Markovian reservoir to the oscillators. Weintroduce an N -mode unitary transformation of the position and momentum operatorsand find that in the transformed basis the system is represented by a set of independentharmonic oscillators with only one of them coupled to the environment. This fact makesthe problem remarkably simple that the relaxation properties of N harmonic oscillators ulti-mode entanglement of N harmonic oscillators
2. The model
We consider a system composed of N mutually coupled identical harmonic oscillators ofmass M and frequency Ω that are simultaneously interacting with a common thermalbath environment (reservoir). The system is determined by the Hamiltonian, which interms of the position q i and momentum p i operators can be written as H = H s + H ε + V s + V, (1)where H s = N X i =1 (cid:18) p i M + 12 M Ω q i (cid:19) (2)is the free Hamiltonian of the harmonic oscillators, H ε = X n (cid:18) p n m n + 12 m n ω n q n (cid:19) (3)is the Hamiltonian of the common reservoir to which the oscillators are coupled, V s = λ N X i =1 X j>i q i q j (4)is the interaction between the oscillators, and V = X n N X i =1 λ n q n q i (5)is the interaction between the oscillators and the reservoir.In equations (1)-(5), the parameter λ stands for the coupling constant between theoscillators, and λ n is the coupling strength of the oscillators to the reservoir. We modelthe environment as an ensemble of harmonic oscillators of mass m n and frequency ω n that interact bilinearly through their position operators q n with the oscillators.The system of harmonic oscillators coupled to an environment is usually describedin terms of a reduced density operator ˆ ρ , which is obtained by tracing the densityoperator of the total system over the reservoir operators. Instead of working in the bare ulti-mode entanglement of N harmonic oscillators q i , p i ), we introduce an N -mode unitary transformation of the systems’ positionoperators ˜ q k = r N − kN − k + 1 " q k − N − k N X j = k +1 q j , k = 1 , , , . . . , N − , ˜ q N = 1 √ N N X k =1 q k , (6)and the same for the momentum operators. We note that the transformations involveanti-symmetrical (˜ q i , ˜ p i ) and symmetrical (˜ q N , ˜ p N ) combinations of the position and themomentum operators, a close analog of the symmetric and antisymmetric multi-atomDicke states [25, 26, 27].In order to derive the master equation for the density operator ˆ ρ of the system,we use the standard method involving the Born approximation that corresponds tothe second-order perturbative approach to the interaction between the oscillatorsand the environment, but we do not make the rotating-wave (RWA) and Markovianapproximations. We find that in terms of the transformed operators the reduced densityoperator ˆ ρ satisfies the master equation˙ˆ ρ ( t ) = − i ~ (cid:20) ˜ H s + 12 M ˜Ω N ( t )˜ q N , ˆ ρ (cid:21) − i ~ γ ( t ) [˜ q N , { ˜ p N , ˆ ρ } ] − D ( t )[˜ q N , [˜ q N , ˆ ρ ]] − ~ f ( t )[˜ q N , [˜ p N , ˆ ρ ]] (7)in which the Hamiltonian ˜ H s of the coupled oscillators is of the form˜ H s = N X i =1 (cid:18) ˜ p i M + 12 M Ω i ˜ q i (cid:19) , (8)where Ω i ≡ Ω F = r Ω − λM , i = 1 , , . . . , N − , Ω N = r Ω + ( N − λM , (9)are the effective frequencies of the oscillators. Note that the frequency Ω N of theoscillator coupled to the environment differs from that of the remaining independentoscillators. It means that the reservoir affects the evolution of only one of the oscillatorsleaving the remaining N − N − th oscillator that is affected by the reservoir are determined by the following time-dependent coefficients˜Ω N ( t ) = − M Z t dt cos(Ω N t )Π( t ) , (10)represents a shift of the frequency of the oscillator due to the interaction with theenvironment. It includes the frequency renormalization that leads to a finite Lambshift [28]. ulti-mode entanglement of N harmonic oscillators γ N ( t ) = 1 M Ω N Z t dt sin(Ω N t )Π( t ) (11)is the dissipation coefficient, and D N ( t ) = 1 ~ Z t dt cos(Ω N t ) ν ( t ) , (12) f N ( t ) = − M Ω N Z t dt sin(Ω N t ) ν ( t ) , (13)are diffusion coefficients.The time dependent functions Π( t ) and ν ( t ) appear as the dissipation and noisekernels, respectively, and are given byΠ( t ) = 12 ~ X n λ n h [ q n ( t ) , q n (0)] i = Z ∞ dωJ ( ω ) sin( ω t ) , (14) ν ( t ) = 12 ~ X n λ n h{ q n ( t ) , q n (0) }i = Z ∞ dωJ ( ω ) cos( ω t )[1 + 2 ¯ N ( ω )] , (15)where J ( ω ) is the spectral density of the modes of the environment. For a Gaussian-typespectral density J ( ω ) = 2 π γ ωM (cid:16) ω Λ (cid:17) n − e − ω / Λ , (16)where Λ is cut-off frequency that represents the highest frequency in the environment, γ is proportional to the coupling strength between the N − th oscillator and theenvironment, and n determines the type of the reservoir. For n = 1, the environmentis called an Ohmic reservoir, for n > n < λ n .In the transformed basis, the Hamiltonian of the system exhibits interestingproperties. First of all, we observe that the system is represented by a set of N independent oscillators with only one of them being coupled to the environment. Theoscillator effectively coupled to the environment is that one corresponding to thesymmetric combination of the position and momentum operators. In addition, theeffective frequency Ω N of the oscillator coupled to the environment differs from that ofthe remaining independent oscillators. The oscillators effectively decoupled from theenvironment may be regarded as composing a relaxation-free subspace. It should bestressed that the subspace is not a decoherence-free subspace. We shall demonstratethat the subsystem of the ”relaxation free” oscillators still can evolve in time that maylead to decoherence. We will recognize that only a part of the subspace can be regardedas a decoherence-free subspace. ulti-mode entanglement of N harmonic oscillators
3. Covariance matrix
We study dynamics of the system in terms of the Wigner characteristic function, whichfor an N -mode Gaussian state can be written in terms of a covariance matrix as [29] χ ( X ) = exp (cid:18) − ~XV ~X T (cid:19) , (17)where ~X = col(˜ q , ˜ p , ˜ q , ˜ p , . . . , ˜ q N , ˜ p N ) is an 2 N dimensional column vector of thetransformed operators, and V is the covariance matrix whose elements are defined as V i,j = Tr ( { ∆ X i , ∆ X j } ˆ ρ ) , (18)with ∆ X i = X i − h X i i , { ∆ X i , ∆ X j } = 12 (∆ X i ∆ X j + ∆ X j ∆ X i ) , (19)and X i is the i th component of the vector ~X .The covariance matrix is composed of 4 N elements. However, due to thesymmetrical property that V ij = V ji , it is enough to find the diagonal elements andthose off-diagonal elements with i < j to completely determine the matrix. Thus, thenumber of elements that have to be found is equal to N (2 N +1). Technically, it is done byusing the definition (18) and the master equation (7) form which one finds the equationsof motion for the covariance matrix elements that then are solved for arbitrary initialconditions. However, the equations form a set of coupled linear differential equationswhose number is large even for a small number of oscillators. Therefore, the dynamics ofcoupled harmonic oscillators have usually been studied by employing numerical methods.We propose a different approach that illustrates the advantage of working in thebasis of the transformed position and momentum operators. As we shall see, theapproach allows to determine the covariance matrix elements in an effectively easy wayrequiring to solve separate sets of equations composed of only a small number of coupleddifferential equations.From equation (18) and the master equation (7), we find a set of inhomogeneousdifferential equations for the covariance matrix elements, which can be written in amatrix form as˙ ~V N ( t ) = C N ( t ) ~V N ( t ) + ~ ~F N ( t ) , (20)where ~V N ( t ) = col( V , V , V , . . . , V N − , N − , V N − , N , V N, N ) (21)is a column vector of the covariance matrix elements, ~F N ( t ) = col(0 , , , . . . , − f N ( t ) , ~ D N ( t )) (22)is a column vector composed of the inhomogeneous time-dependent terms, and C N ( t )is an N (2 N + 1) × N (2 N + 1) block diagonal matrix of the time-dependent coefficients. ulti-mode entanglement of N harmonic oscillators C N ( t ) is a direct sum of small size matrices C N ( t ) = " N M n =2 (cid:16) A (0) ⊕ A ( N − n )4 (0) ⊕ A ( t ) (cid:17) ⊕ A ( t ) , (23)where A (0) = M − − M Ω F M − − M Ω F , (24) A ( t ) = M − − M ¯Ω N ( t ) − γ ( t ) M − − M ¯Ω N ( t ) − γ ( t ) , (25) A ( t ) = M − M − − M ¯Ω N ( t ) − γ ( t ) 0 M − − M Ω F M − − M Ω F − M ¯Ω N ( t ) − γ ( t ) , (26)and A (0) = M − M − − M Ω F M − − M Ω F M − − M Ω F − M Ω F . (27)with ¯Ω N ( t ) = Ω N + ˜Ω N ( t ) and γ ( t ) = 2 γ N ( t ). The superscript ( N − n ) in A ( N − n )4 (0) isunderstood as the number of the A (0) matrices appearing in the direct sum. Thus, for N = 2, no matrix A (0) is involved in C N ( t ), one matrix A (0) is involved for N = 3,and so on.There are several interesting and important conclusions arising from equation (23).Firstly, the equations of motion group into decoupled subsets of smaller sizes involvingonly three and four equations. In other words, the block diagonal matrix C N ( t ) iscomposed of 3 × × N > N = 2 oscillators [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20]. Thirdly, thematrices A (0) and A (0) are independent of time. This means that the time evolutionof the covariance matrix elements whose dynamics are determined by A (0) and A (0)can be found in an exact analytical form. Fourthly, the matrices A (0) and A (0) areindependent of the relaxation coefficient γ . Thus, they reflect features of the N − relaxation-free subspace . Finally, the matrices A ( t ) and A ( t ) are explicitly dependent on time through the relaxation terms γ ( t ). Therefore,they represent dynamics of the oscillator effectively coupled to the environment. Theexplicit time dependence of the matrices (25) and (26) results from the non-Markovian ulti-mode entanglement of N harmonic oscillators A (0). It is easy to note that the determinantof the matrix A (0) is equal to zero. Mathematically, it means that among the threematrix elements involved, V , V and V , there is a linear combination whose equationof motion is decoupled from the remaining equations. It is easy to show that the linearcombination V +11 = M Ω F V + 1 M V ,V − = M Ω F V − M V (28)obeys ˙ V +11 = 0 and the remaining elements form a set of two coupled equations˙ V − = 4Ω F V , ˙ V = − V − , (29)where V − = M Ω F V − (1 /M ) V .The property of ˙ V +11 = 0 indicates that the linear combination V +11 is a constantof motion, i.e. V +11 ( t ) = V +11 (0). In other words, V +11 ( t ) does not change in time andretains its initial value for all times. Physically, if initially the system was preparedin a state such that V +11 (0) = 0 and with the other elements of the covariance matrixequal to zero, it would remain in that state for all times. For example, if the initialstate is an entangled state, the initial entanglement of the system will remain constantin time. Therefore, the subspace composed of the V +11 ( t ) element can be regarded as a decoherence-free subspace .The remaining matrix elements V − and V can undergo a temporal evolution.Since there is no damping involved in the equations of motion (29), the solution wouldlead to the matrix elements continuously oscillating in time. It is easy to find that thesolution of equation (29) has a simple form V − ( t ) = V − (0) cos(2Ω F t ) + 2Ω F V (0) sin(2Ω F t ) ,V ( t ) = V (0) cos(2Ω F t ) − V − (0)2Ω F sin(2Ω F t ) , (30)from which we see the matrix elements continuously oscillate in time with frequency 2Ω F .This indicates that the system will never reach a stationary time-independent stateunless V − (0) = V (0) = 0. We stress that the continuous in time oscillations are notrelated to the non-Markovian nature of the reservoir as the matrix A (0) determinesdynamics of the oscillators that are not coupled to the reservoir.It is also found that the determinant of the matrix A (0) is equal to zero. Thus,following the above analysis, we can find that the set of the equations of motion forthe covariance matrix elements determined by the matrix A (0) can be reduced to ulti-mode entanglement of N harmonic oscillators N − constants of motion and a smaller size set of coupled equations determined by a matrix C ′ N ( t ) = A ( N ( N − ) (0) ⊕ A ( N − ( t ) ⊕ A ( t ) , (31)where A (0) is a 2 ×
4. Multi-mode entanglement and squeezing
We have already shown that due to the presence of the constants of motion in theevolution of the covariance matrix elements, the dynamics of the system, even aftera long time, may strongly depend on the initial state. Since we are interested in theevolution of an initial entangled state and it is well known that multi-mode squeezedstates are examples of entangled states, We have already shown that due to the presenceof the constants of motion in the evolution of the covariance matrix elements, thedynamics of the system, even after a long time, may strongly depend on the initialstate. Since we are interested in the evolution of an initial entangled state and it is wellknown that multi-mode squeezed states are examples of entangled states, we considertwo experimentally realizable initial squeezed vacuum states with markedly differentsqueezing behaviors. We also demonstrate that with the two specific initial states, theproblem of treating the dynamics of N harmonic oscillators simplifies to analysis of theproperties of only those constants of motion and the matrices which involve only thediagonal elements of the covariance matrix.In the first example, we consider the most familiar multi-mode continuous variableGreenberger-Horne-Zeilinger (GHZ) entangled state [30, 31] | ψ i = U N Y i =1 | b i i , (32)with U = exp ( − r " N X i = j =1 (cid:16) b † i b † j − ( b † i ) (cid:17) − H . c . , (33)where r is the squeezing parameter and the ket | b i i represents the state with zerophotons in each of the N modes. This GHZ state for N = 3 has been realizedexperimentally by two groups [33, 34].In the second example, we assume that the system is initially prepared in a purenon-symmetric multipartite squeezed state of the form | ψ i = U N Y i =1 | b i i , (34) ulti-mode entanglement of N harmonic oscillators U = exp ( r N − X i =1 b † i b † N + 12 r s N − X i = j =1 b † i b † j − H . c . ) . (35)Here, each of the N − r with thedamped mode, whereas the relaxation-free modes are correlated between themselvesto a degree r s . Practical schemes for generation of such a state have recently beendiscussed [35]. For example, it could be created by use of concurrent interactions ina second-order nonlinear medium placed inside an optical resonator, which might berealized experimentally in periodically poled KTiOPO [36]. We will use this exampleto demonstrate the dependence of stationary entanglement on the amount of correlationsinitially encoded into the relaxation-free modes.One manifestation of the squeezed properties of the states is entanglement betweendifferent modes. We examine this entanglement property shortly, but first we examinethe manifestation of the squeezed correlations in the form of the covariance matrix.The choice of the initial state (32) is a consequence of the diagonal form of the initialcovariance matrix. In particular, the initial values of the covariance matrix elements ofthe two-mode case are V (0) = V (0) = 12 e − r , V (0) = V (0) = 12 e r , (36)whereas for the three-mode case the initial elements are V (0) = V (0) = V (0) = 12 e − r ,V (0) = V (0) = V (0) = 12 e r , (37)With the asymmetric squeezed state (34), the initial covariance matrix is notdiagonal and has the following symmetric form V (0) = V (0) 0 V (0) 0 V (0) 00 V (0) 0 V (0) 0 V (0) V (0) 0 V (0) 0 V (0) 00 V (0) 0 V (0) 0 V (0) V (0) 0 V (0) 0 V (0) 00 V (0) 0 V (0) 0 V (0) , (38)where the explicit expressions for the non-zero matrix elements are given in theAppendix A.Before proceeding further with the analysis of the entangled and squeezingproperties of the system, we return for a moment to the solutions for the covariancematrix elements. We have seen that with the states (32), the initial covariance matrixis diagonal. An immediate consequence of the diagonal form of the initial covariancematrix is that for t >
0, the diagonal elements will be different from zero and only thoseoff-diagonal elements whose the equations of motion were coupled to the equations ofmotion for the diagonal elements. It is easy to see from equation (20) that the non-zero ulti-mode entanglement of N harmonic oscillators A (0) and A ( t ). Thus, dynamics of a systemcomposed of N harmonic oscillators can be readily determined from properties of thetwo simple 3 × q i meets the inequality ∆(˜ q i ) < /
2, then we say the state exhibits ordinary multi-modesqueezing. The minimum variance corresponds to the optimal multi-mode squeezing.However, the ordinary squeezing is produced by both one and two-mode correlations,whereas entanglement is solely related to the two-mode correlations [32]. Thus, theordinary squeezing does not necessarily mean entanglement. We may distinguishbetween the contributions of the one and two-mode correlations to the variances and thendetermine the multi-mode squeezing by performing suitable unitary transformations ofthe mode operators.We illustrate this procedure for the case of three modes since the GHZ state (32)for N = 3 is an example of multipartite entangled state whose entanglement isshared by more than two parties. Moreover, the three-mode GHZ state has beenrealized experimentally [33, 34] and also has been successfully applied to demonstratequantum teleportation [31, 37] and quantum dense coding [33]. We will demonstratethe equivalence between the three-mode squeezing and the negativity criterion forentanglement. The two-mode case, N = 2, has been extensively studied in theliterature [38]. First, we make a local squeezing transformation on each of the modes,which results in transformed annihilation operators of the form [39]˜ a = a e iθ ( u u − e i ( ϕ − θ ) v v ) + a † e − iθ ( u v − e − i ( ϕ − θ ) v u ) , ˜ a = ˜ a = a e iθ ( u u + e i ( ϕ − θ ) v v ) − a † e − iθ ( u v + e − i ( ϕ − θ ) v u ) , (39)where u i = cosh( r i ) and v i = sinh( r i ) ( i = 1 ,
2) and the transformation has been madewith the squeezing parameter r and the phase angle ϕ on the mode 1, and with r andthe phase angle θ on the modes 2 and 3.We use the Wigner characteristic function, which in terms of the above specificallychosen transformation can be written in a Gaussian form as χ ( ~µ, t ) = exp (cid:26) − ~µ G ~µ T (cid:27) , (40)where ~µ = (˜ y , ˜ x , ˜ y , ˜ x , ˜ y , ˜ x ) is a vector composed of the real ˜ y j and imaginary˜ x j ( j = 1 , ,
3) parts of the phase-space variables corresponding to operator ˜ a j , and G is the correlation matrix of the form G = a c c b d dc a c d b dc c a d d b . (41)Note, the matrix G involves only four parameters that are a = − f e r , b = − f e − r , ulti-mode entanglement of N harmonic oscillators c = 2 h e r , d = 2 h e − r , (42)where h = ( | f | − f ) and h = − ( | f | + f ), with f = m u + m ∗ e iϕ v + 2 m e iϕ u v ,f = (cid:0) m e − iϕ + m ∗ e iϕ (cid:1) u v + m (1 + 2 v ) ,f = 4 | m | u v + m (1 + 2 v ) , (43)and m i are linear combinations of the covariance matrix elements V ′ ij given in the barebasis m = 12 ( V ′ − V ′ − iV ′ ) ,m = V ′ − V ′ − i ( V ′ + V ′ ) ,m = V ′ + V ′ , m = V ′ + V ′ . (44)The entangled nature of the three-mode squeezed states is clearly exhibited by thepresence of the off-diagonal terms in the correlation matrix G .The squeezing parameters r , r and the phase angles ϕ, θ appearing in thetransformation of the field operators can be carefully chosen to match the form of thecorrelation matrix G with the form of the covariance matrix V ′ in the bare basis. In thisway we can achieve the equivalence between three-mode squeezing and entanglement.This can be done by choosing the squeezing parameters ase r = (cid:18) m − | m | m + 2 | m | (cid:19) , e r = (cid:18) | h | + f | h | + f (cid:19) , (45)with m = | m | exp(2 iϕ ) and f = | f | exp(2 iθ ).Having available the time dependent solutions for the covariance matrix elements,we then can easily find the characteristic function that allows us to compute variancesof the position operators and momentum operators˜ X k = s − k − k ) " ˜ a k − − k X j = k +1 ˜ a j ! + H . c . , ˜ Y k = − i s − k − k ) " ˜ a k − − k X j = k +1 ˜ a j ! − H . c . , (46)for k = 1 ,
2, and˜ X = r X j =1 (cid:16) ˜ a j + ˜ a † j (cid:17) , ˜ Y = − i r X j =1 (cid:16) ˜ a j − ˜ a † j (cid:17) . (47)The variances are involved in the criterion for multi-mode squeezing that fluctuations ofthe correlations between three modes are squeezed if and only if the sum of the variances h (∆ ˜ X i ) i and h (∆ ˜ Y j ) i with i = j satisfies the following inequality [23] h (∆ ˜ X i ) i + h (∆ ˜ Y j ) i < , i, j = 1 , , . (48)Among the permutations of the variances involved on the left-hand side of equation (48),there might be more than one satisfying inequality condition for multi-mode squeezing.In this case, we choose the combination that reflects the largest squeezing. ulti-mode entanglement of N harmonic oscillators j V ′ ( t )Γ j + 12 iσ, j = 1 , , , . . . (49)where Γ j is the partial transpose matrix with the transposition made on the j thmode block and σ is a block diagonal symplectic matrix. It has been shown thatmulti-mode Gaussian states are not completely separated when for all j there arenegative eigenvalues of the matrix (49). The eigenvalues can be degenerated or non-degenerated. However, for a system of identical oscillators the covariance matrix V ′ ( t )is permutational symmetric, so all the negative eigenvalues are degenerated. We denotethem by a parameter η − and call it as the negativity criterion for entanglement.
5. Temporal evolution of squeezing and entanglement
We now perform numerical analysis of time evolution of multi-mode squeezing andentanglement in a system of two and three mutually interacting harmonic oscillatorssimultaneously coupled to an environment. We will illustrate the advantage of workingin the transformed basis to obtain a simple interpretation of the results. In particular,to understand short time non-Markovian dynamics of entanglement and to provideconditions for optimal and stable long time entanglement. In addition, we comparethe time evolutions of the variances and the negativity to find if the condition forthree-mode squeezing could be used as the necessary and sufficient condition for threemode entanglement. In all cases considered here, we assume that the oscillators interactwith an Ohmic reservoir ( n = 1) of temperature k B T = 10 ~ Ω with the Boltzmanndistribution of photons characterized by the mean occupation number ¯ N (Ω) = 9 . λ = 0, butinteracting with the environment. Figure 1 shows the negativity and variances as afunction of time for the initial symmetric squeezed state | ψ i with different degreeof squeezing r . First of all, we note that at times where squeezing occurs there isentanglement, and vice versa, at times where entanglement occurs, there is squeezing. Inaddition, we see a threshold value for the degree of squeezing r at which a continuous intime entanglement occurs. The threshold that corresponds to entanglement undergoingthe phenomenon of sudden death, occurs at r = 1 . r has been predicted for the two-mode case [17].The presence of the threshold value for r at which continuous in time entanglement ulti-mode entanglement of N harmonic oscillators t/ Ω < ( ∆ ˜ X ) + ( ∆ ˜ Y ) > η − Figure 1.
Time evolution of the negativity η − and the combined variance h (∆ ˜ X ) i + h (∆ ˜ Y ) i for γ = 0 . , Λ = 100 , n = 1 , λ = 0 and different r : r = 1 . r = 1 .
498 (dashed line), r = 2 . | ψ i . occurs has a simple interpretation in terms of the covariant matrix elements. Considerthe threshold in the long time limit in which we may consider the evolution under theMarkov approximation, but retaining the non-RWA terms. Under this approximation,we can put γ ( t ) → γ which then allows us to obtain a simple analytical solution forthe threshold condition for entanglement.It is easy to show that the threshold for two mode entanglement occurs at V ( t ) V ( t ) = 1 / , (50)so that the two modes are entangled when V ( t ) V ( t ) < /
4, otherwise are separable.Note that the covariance matrix element V ( t ) is associated with the relaxation freemodes whereas the element V ( t ) is associated with the mode that is coupled to thereservoir and thus undergoes the damping process. Under the Markov approximation,we find from equations (20)-(26) that in the long time limit of t ≫ γ − , the element V ( t ) reaches the stationary value equal to the level of the thermal fluctuations V ( t ) → N + 1 , (51)whereas V ( t ) retains its time dependent behavior which depends on the initial values V ( t ) = V (0) cos Ω F t + V (0) M (cid:18) sin Ω F t Ω F (cid:19) . (52) ulti-mode entanglement of N harmonic oscillators V ( t ) is due to the presence of the constant of motion V +11 .Averaging equation (52) over a long period of oscillations, the thresholdcondition (50) simplifies to2 V (0) (cid:0) N + 1 (cid:1) = 1 . (53)We see that the threshold behavior of entanglement depends on the initial value of thecovariance matrix element V (0). In other words, the entanglement behavior can becontrolled by the suitable choosing of the initial state. For example, with the initialstate (32), we find from equations (53) and (36) that continuous entanglement occursfor the degree of squeezing r = 12 ln (cid:0) N + 1 (cid:1) . (54)With the parameter value k B T = 10 ~ Ω, we find that the threshold value for r equalsto 1 .
498 that is the same found numerically in figure 1. We should point out herethat the same condition for the threshold value of r has been found under the RWAapproximation [17]. Thus, we may conclude that the threshold value for continuousentanglement is not sensitive to the RWA approximation.We now proceed to discuss the dependence of the long time entanglement on therelaxation rate γ . An example of this feature is shown in figure 2. It is interesting tonote that under the relaxation the entanglement oscillates in time and the amplitudeof the oscillations increases with increasing γ leading to a better entanglement whenthe oscillators are strongly damped. It is a surprising result as one could expectthat entanglement should decrease with increasing γ . Again, a straightforwardinterpretation of this effect can be gained from a qualitative inspection of the propertiesof the transformed covariance matrix.It is easy to see from equations (24) and (25) that in the limit of vanishing damping, γ ( t ) → λ = 0, the matrix A ( t ) reduces to A (0). One could argue that inthis limit the covariance matrix elements determined by the matrix A ( t ) coincidencewith the elements determined by the matrix A (0). Of course, their time evolution isdetermined by the same equations, but there is a subtle difference in their initial values.For example, the initial values of the elements whose evolution is determined by thematrix A (0) are V ± (0) = 12 (cid:18) M Ω F e − r ± M e r (cid:19) , (55)whereas that one determined by the matrix A are V ± (0) = 12 (cid:18) M Ω F e r ± M e − r (cid:19) . (56)The initial elements are significantly different that what appears as a squeezedcomponent in V ± (0), the counterpart in V ± (0) appears as an anti-squeezed component.This is a crucial difference that has a significant effect on the evolution of anentanglement. These two contributions cancel each other that results in no oscillations ulti-mode entanglement of N harmonic oscillators t/ Ω < ( ∆ ˜ X ) + ( ∆ ˜ Y ) > η − Figure 2.
Time evolution of the negativity η − and combined variance h (∆ ˜ X ) i + h (∆ ˜ Y ) i for Λ = 100 , n = 1 , λ = 0 , r = 1 . γ : γ = 0 .
05 (solid line), γ = 1 . γ = 5 . | ψ i . in the entanglement evolution when γ ≪
1. On the other hand, for a large γ thecovariance matrix elements determined by A ( t ) are rapidly damped to their stationaryvalues leaving the elements determined by A (0) continuously oscillating in time. Theseoscillations lead to the continuous oscillation of the entanglement seen in figure 2.One can interpret these results in terms of collective symmetric and antisymmetricstates of an N -atom Dicke model [25, 26, 27]. The symmetric and antisymmetricstates correspond to the atomic dipole moments oscillating in-phase and out-of-phase,respectively. The most interesting is that in the case of the atoms coupled to a commonreservoir, the antisymmetric states do not decay, whereas the symmetric states decaywith an enhanced rate N γ , where γ is the single atom decay rate. Hence, in theabsence of the damping, oscillations induced by the symmetric and antisymmetric statescancel each other as they occur with opposite phases. When damping is included, theoscillations induced by the symmetric states are damped in time whereas the oscillationsinduced by the antisymmetric states remain unaffected. The oscillations induced by thesymmetric states die out on the time scale of t ∼ / ( N γ ) leaving the oscillations inducedby the antisymmetric states unaffected.Figure 3 shows the evolution of entanglement and squeezing when the oscillatorsare coupled to each other. In this case there is no continuous stationary entanglement. ulti-mode entanglement of N harmonic oscillators t/ Ω < ( ∆ ˜ X ) + ( ∆ ˜ Y ) > η − Figure 3.
Time evolution of the negativity η − and the combined variance h (∆ ˜ X ) i + h (∆ ˜ Y ) i for γ = 0 . , Λ = 100 , n = 1 , λ = 0 . r : r = 1 . r = 1 .
498 (dashed line), r = 2 . | ψ i . Thus, the interaction between the oscillators has a destructive effect on the stationaryentanglement. However, for a large squeezing, entanglement re-appears in some discreteperiods of time, exhibiting periodic sudden death and revival of entanglement. Inother words, the threshold behavior of entanglement is a periodic function of time.As before, this feature has a simple interpretation in terms of the covariance matrixelements. According to equation (50), for a given temperature the threshold value forentanglement depends on the covariance matrix element V ( t ) which, on the other hand,depends on λ through the frequency parameter Ω F . We see from equation (9) that Ω F decreases with increasing λ . Thus, according to equation (52) for interacting oscillatorsthe matrix element V ( t ) oscillates slowly in time. The averaging over the oscillations isnot justified and thus the threshold condition for entanglement is the oscillating functionof time even in a long time regime.Finally, in figure 4 we illustrate the evolution of entanglement for two different casesof the initial asymmetric state | ψ i . As we have shown in section 3, more constants ofmotion are then involved than in the symmetric case which, on the other hand, maylead to a better stationary entanglement. In the first case, we plot the negativity η which describes entanglement between the mode 2 and the pair 1 ↔
3. We see fromfigure 4(a) that the stationary entanglement appears only when r < r s . Otherwise, the ulti-mode entanglement of N harmonic oscillators t/ Ω η η (a)(b)(a)(b)(a)(b) Figure 4.
Time evolution of the negativity (a) η and (b) η for the initial asymmetricstate | ψ i with γ = 0 . , Λ = 100, λ = 0 , r s = 1 .
489 and different r : r = 1 . r = 1 .
489 (dashed line), r = 2 . initial entanglement rapidly decays to zero and disappears after a finite time. Again, thisfeature can be easily explained in terms of the transformed oscillators. When r < r s ,the pair of modes 1 and 2 that is decoupled from the environment is more stronglycorrelated than the pairs 1 ↔ ↔
3, which involve the mode coupled to theenvironment. This preserves the entanglement in the system. In the opposite case of r > r s , a large entanglement is initially encoded into the pairs that are damped dueto the coupling to the environment. This results in the loss of the correlations andentanglement. Quite different properties exhibits entanglement between the mode 3,which is coupled to the environment, and the remaining pair 1 ↔
2. In this case,illustrated in figure 4(b) there is no stationary entanglement. This can be interpretedas the result of the coupling of the mode 3 to the reservoir that leads to the continuousdissipation of the initial correlations r .
6. Conclusion
We have analyzed dynamics of a set of N harmonic oscillators coupled to a non-Markovian reservoir in terms of the covariance matrix. By performing a suitabletransformation of the position and momentum operators of the system oscillators, wehave shown that the set of coupled differential equations for the covariance matrix ulti-mode entanglement of N harmonic oscillators N oscillatorscan be completely determined by properties of 4 × × Acknowledgments
We acknowledge financial support from the National Natural Science Foundation ofChina (Grant No. 60878004), the Ministry of Education under project SRFDP (GrantNo. 200805110002), the National Basic Research Project of China (Grant No. 2005CB724508).
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Phys. Rev. A J. Phys. B: At. Mol. Opt. Phys. Phys.Rev. A Phys. Rev. Lett. Appendix A. Initial values of the covariance matrix
In this appendix, we list the non-zero elements of the initial covariance matrix for thecase of the asymmetric initial state (34). The diagonal elements are of the form V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e r s r + q sinh ¯ r ) ,V (0) = e − r s r − q sinh ¯ r ) , (A.1) ulti-mode entanglement of N harmonic oscillators V (0) = − e − r s √ (cid:2) − e r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = − e r s √ (cid:2) − e − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = √ V (0) = − e r s ( r − r s ) sinh ¯ r √ r ,V (0) = √ V (0) = e − r s ( r − r s ) sinh ¯ r √ r , (A.2)where ¯ r = p r + r s and q = (8 rr
In this appendix, we list the non-zero elements of the initial covariance matrix for thecase of the asymmetric initial state (34). The diagonal elements are of the form V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e r s r + q sinh ¯ r ) ,V (0) = e − r s r − q sinh ¯ r ) , (A.1) ulti-mode entanglement of N harmonic oscillators V (0) = − e − r s √ (cid:2) − e r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = − e r s √ (cid:2) − e − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = √ V (0) = − e r s ( r − r s ) sinh ¯ r √ r ,V (0) = √ V (0) = e − r s ( r − r s ) sinh ¯ r √ r , (A.2)where ¯ r = p r + r s and q = (8 rr + rr
In this appendix, we list the non-zero elements of the initial covariance matrix for thecase of the asymmetric initial state (34). The diagonal elements are of the form V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e − r s (cid:2) r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = e r s (cid:2) − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = e r s r + q sinh ¯ r ) ,V (0) = e − r s r − q sinh ¯ r ) , (A.1) ulti-mode entanglement of N harmonic oscillators V (0) = − e − r s √ (cid:2) − e r s (3 cosh ¯ r − q sinh ¯ r ) (cid:3) ,V (0) = − e r s √ (cid:2) − e − r s (3 cosh ¯ r + q sinh ¯ r ) (cid:3) ,V (0) = √ V (0) = − e r s ( r − r s ) sinh ¯ r √ r ,V (0) = √ V (0) = e − r s ( r − r s ) sinh ¯ r √ r , (A.2)where ¯ r = p r + r s and q = (8 rr + rr s ) / ¯ rr