Non-local dynamics of Bell states in separate cavities
aa r X i v : . [ qu a n t - ph ] N ov Non-local dynamics of Bell states in separate cavities
Jun Jing ∗ , Zhi-guo L¨u , Guo-hong Yang Department of Physics, Shanghai University, Shanghai 200444, China Department of Physics, Shanghai Jiaotong University, Shanghai 200240, China (Dated: November 24, 2018)
Abstract
We present non-local dynamics of Bell states in separate cavities. It is demonstrated that (i)the entanglement damping speed will saturate when the cavity leakage rate γ ≥ .
4; (ii) thesynchronism relationship between the fidelity and the concurrence depends on the initial state;(iii) if the initial state is 1 / √ | i + | i ), the dynamics of entropy is opposite to that of fidelity. PACS numbers: 75.10.Jm, 03.65.Bz, 03.67.-a ∗ Email address: [email protected] . INTRODUCTION In contrast with the extensively investigated static entanglement [1, 2, 3, 4, 5, 6],dynamic entanglement under the influence of variant environments is one of the mostimportant and largely unexplored problems in the field of quantum teleportation, quantumcomputation and quantum communication [7, 8, 9, 10]. It is not only involved with thefoundation of quantum mechanics, but also a fundamental issue in creating, quantifying,controlling, distributing and manipulating the entangled quantum bits, which are composedof spin-1 / / / s and s ) and two single-modecavities (labelled 1 and 2 correspondingly). The atom s j ( j = 1 or 2) is embedded in andcoupled only with the cavity mode j , which could be regarded as its bath or environment.The two cavities are so far departed that there is no direct interaction between them as wellas the two atoms. Initially, the two qubits are prepared as a most-entangled states (Bellstates). The focus of interest is their degrading quantum evolution, which are measured bythe concurrence [35, 36], the fidelity [37] and the entropy exchange [38, 39]. The calculationsand physical arguments will be carried out in two conditions: (i) there is leakage of photonsfor the cavities, which are in the vacuum states from the beginning; (ii) the cavities are soperfect that the loss of photons from them could be neglected and the two single modes areinitialled in a thermal equilibrium state with the same temperature. The rest of this paperis organized as following. In Sec. II we begin with the model Hamiltonian and its analysisderivation; and then we introduce the numerical calculation procedure about the evolutionof the reduced matrix for the subsystem. Detailed results and discussions can be found inSec. III. We will make a conclusion in Sec. IV. II. MODEL AND METHOD
The master equation for a two-level atom in a single-mode cavity [40], as one of the twopartitions in our model, can be taken as i dρ j dt = [ H j , ρ j ] + iγ j (cid:18) a j ρ j a † j − a † j a j ρ j − ρ j a † j a j (cid:19) . (1)For density matrix ρ j , j refers to s or s ; for the mode operator a j or a † j , j (1 or 2) representsthe photon mode coupling with the corresponding atom. γ j is the leakage rate of photonsfrom the cavity j . H j describes the Hamiltonian for a subsystem of one atom and one cavity3 j = 1 , H j = ω j σ zj + (1 + ǫ j ) ω j a † j a j + g j ω j ( a † j + a j ) σ xj . (2)where ω j is the energy level difference of atom s j in cavity j . ǫ j is the detuning parame-ter measuring the deviation of the photon j energy from ω j . g j is introduced as anotherdimensionless parameter which suggests the coupling strength between qubit s j and mode j . The x and z components of σ are the well-known Pauli operator. The two qubits areembedded in remote cavities without direct interaction. Therefore the whole Hamiltonianfor this two-atom-two-cavity problem is H = H + H . (3)The whole state of the total system is assumed to be separable before t = 0, i.e. ρ (0) = ρ S (0) ⊗ ρ b (0) , (4) ρ S (0) = | ψ (0) ih ψ (0) | , (5) ρ b (0) = ρ b (0) ⊗ ρ b (0) . (6)The initial state | ψ (0) i for the two qubits is one of the Bell states. And the two cavitiesare in their (i) vacuum states ρ bj (0) = | j ih j | (in this case, we will consider γ j = 0) or(ii) thermal equilibrium states ρ bj (0) = e − H B /k B T /Z (in this one, we set γ to be zero todistinguish the effect of temperature from that of γ ), where H B is the pure bath part ofthe whole Hamiltonian and Z = Tr (cid:0) e − H B /k B T (cid:1) is the partition function and the Boltzmannconstant k B will be set to 1 for the sake of simplicity.For the former case, Eq. 1 will be exploited to calculate ρ ( t ). For the latter one, Eq. 1is reduced to ρ ( t ) = exp( − iHt ) ρ (0) exp( iHt ) . (7)To determine the dynamics of the density matrix for the whole system, two factors needto be considered. The first one is the expression of the thermal bath state. In numericalcalculations [41], we have to expand ρ bj (0) ( j = 1 ,
2) to a summation of its eigenvectors withcorresponding weights determined by its eigenvalues: ρ bj (0) = X m | φ mj i ω mj h φ mj | , ω mj = e − E mj /T Z j (8)4hen for the two single-modes, we have ρ b (0) ⊗ ρ b (0) = X mn | φ m i| φ n i ω mn h φ n |h φ m | , ω mn = e − ( E m + E n ) /T Z Z (9)where the subscripts m and n refer to mode 1 and 2 respectively. The second importantfactor is the evaluation of the evolution operator U ( t ) = exp( iHt ). A polynomial expansionscheme proposed by us in Ref. [42, 43, 44] is applied into the computation, U ( t ) = (cid:18)
11 + it (cid:19) α +1 ∞ X k =0 (cid:18) it it (cid:19) k L αk ( H ) , (10) L αk ( H ) is one type of Laguerre polynomials as a function of H , where α ( − < α < ∞ )distinguishes different types of the Laguerre polynomials and k is the order of it. Thescheme is of an efficient numerical algorithm motivated by Ref. [45, 46], which is prettywell suited to many quantum problems, open or closed. Additionally, it could give resultsin a much shorter time compared with the traditional methods under the same numericalaccuracy requirement, such as the well-known 4-order Runge-Kutta algorithm. After thedensity matrix ρ ( t ) for the whole system is obtained, the reduced density matrix ρ S ( t ) forthe two atoms can be derived by tracing out the degrees of freedom of the two single-modecavities. III. SIMULATION RESULTS AND DISCUSSIONS
We discuss three important physical quantities which indicate the time evolution of thesubsystem. (i) The concurrence. It is a very good measurement for the intra-entanglementbetween two qubits and monotone to the quantum entropy of the subsystem when thesubsystem is in a pure state. It is defined as: C = max { λ − λ − λ − λ , } , (11)where λ i are the square roots of the eigenvalues of the product matrix ρ S ( σ y ⊗ σ y ) ρ ∗ S ( σ y ⊗ σ y )in decreasing order. (ii) The fidelity. It is defined as F ( t ) = Tr S [ ρ ideal ( t ) ρ S ( t )] . (12)where ρ ideal ( t ) represents the pure state evolution of the subsystem only under H S , withoutinteraction with the environment. In this study, H S = ω σ z + ω σ z . The fidelity is a5easurement for decoherence and depends on ρ ideal . It achieves its maximum value 1 only if ρ S ( t ) equals to ρ ideal ( t ). (iii) The entropy exchange En is defined as En = − Tr( ρ S log ρ S ).It is the von Neumann entropy of the joint state of the subsystem as composed of the twoqubits in our model. It measures the amount of the quantum information exchange betweenthe subsystem and the environment. For the subsystem consisted by two two-level atoms(its Hilbert space is 4 × (4) = 2 .
0. When it reaches itsmaximum value, it means all the quantum information is cast out of the subsystem or thequantum subsystem degenerates to a classical state.
A. Dynamics at different γ ω t C ( t ) (a) C ( t ) ω t F ( t ) (b) F d ( t ) FIG. 1: Time evolution for (a) Concurrence, (b) Fidelity with the subsystem starting from1 / √ | i + | i ) at different values of anisotropic parameter: γ = 0 (solid curve), γ = 0 . γ = 0 . γ = 0 . | i | i . In order to discuss the effect of γ (We suppose the two cavities have the same loss degree: γ = γ = γ .) and T , all the other parameters are fixed for the sake of simplicity andwithout loss of generality: ω = ω = ω = 0 . ,ǫ = ǫ = ǫ = − . ,g = g = g = 0 . . ω t C ( t ) (a) C ( t ) ω t F ( t ) (b) F d ( t ) FIG. 2: Time evolution for (a) Concurrence, (b) Fidelity with the subsystem starting from1 / √ | i + | i ) at different values of anisotropic parameter: γ = 0 (solid curve), γ = 0 . γ = 0 . γ = 0 . | i | i . ω t E n ( t ) (a)1 / √ | i + | i ) ω t E n ( t ) (b)1 / √ | i + | i ) FIG. 3: Time evolution for entropy of the subsystem from (a) 1 / √ | i + | i ), (b) 1 / √ | i + | i ) at different values of anisotropic parameter: γ = 0 (solid curve), γ = 0 . γ = 0 . γ = 0 . | i | i . And we choose 1 / √ | i + | i ) and 1 / √ | i + | i ) as two different initial states forthe subsystem. 7e first discuss the effect of the photon loss rate γ . It is evident that with a larger γ ,the entanglement degree of the subsystem will decrease in a faster speed, which could beverified by Fig. 1(a) and Fig. 2(a). The tendency of the two cases is similar. If γ = 0,the concurrence will oscillate periodically with time and will not be dissipated; but thepeak value of it will never reach 1 .
0. On the whole, the curves of the concurrence are notperfectly harmonic, which is a little different with the results gotten in previous works. It isdue to the stochastic and irrelevant microscopical processes (the spins drop from the excitedstate by emitting a photon or jump to the excited state by absorbing a phonon) inside thetwo different cavities. And the dynamics of the concurrence stems from such numerousprocesses, so the evolution is approximately harmonic but not perfect. When γ >
0, theconcurrence drops abruptly to zero in a short time. It coincides with the description aboutentanglement sudden death (ESD) in Ref. [15, 47] that “after the concurrence goes abruptlyto zero, it arises more or less from nowhere”. This is an example of ESD. The photonsleaking out of the cavities greatly reduced the nonlocal connection between the two qubits.When γ is bigger than 0 .
4, the speed of ESD is saturated and we almost cannot distinguishthe curve of γ = 0 . γ = 0 .
8. The state 1 / √ | i + | i ) seems more robustthan 1 / √ | i + | i ). It is verified that, for example, in the condition of γ = 0 .
2, theconcurrence of the former state (to see the dot dashed line in Fig. 2(a)) decreases to zeroat ωt = 17 .
472 for the first time, while the latter one does at ωt = 15 . ωt = 37 . F ( t ) = 0 . ωt ∈ [5 . , .
0] along the solid linein Fig. 1(a) are just fake phenomena: although the entanglement degree between thetwo subsystem atoms is high, but the state of the subsystem is different from the initialone. The tendency of γ = 0 . γ = 0 . F = 0 . ∼ .
4. While the curve of γ = 0 . F = 0 . γ = 0, the fidelity is synchronous with the concurrence.Thus the fidelity of the state | i + | i is more robust than that of | i + | i under8he same environment. These contrasts between the two initial state can be noticedin other works. It implies the physics essence of them is different, although both ofthem are of the most entangled states. Yet the other three cases with leakage γ > F = 0 . ωt = 40 . γ = 0, the evolutionsof the two Bell states are a little different but the oscillation periods of them are almost thesame. Although there is no leakage, but the bath, the two single-mode cavities, will absorbsome of the quantum information inside the subsystem, which means the entropy never goesback to zero as initialled. To compare the solid line in Fig. 3(b) (Fig. 3(a)) with that inFig. 1(a) (Fig. 2(a)), we notice that the tendency of the entropy is opposite to that of theconcurrence. The dynamics difference from the two initial states can almost be removed byintroducing non-zero γ as it is shown in the comparison of Fig. 3(a) and Fig. 3(b). It isevident that with a larger γ , more quantum information of the subsystem is transferred intothe bath. B. Dynamics under different T In this subsection, we turn to the effect of temperature of the cavity modes. It is foundthat when the temperature is comparatively low, T = 0 . ω , the entanglement degreeoscillates but will not corrupt to a sudden death (to see the dot dashed line in Fig. 4(a) andFig. 5(a)). When it goes up to a moderate temperature T = 0 . ω , the ESD happens. Thefirst moment at which the concurrence decreases to zero and does not revive immediatelyis dependent on the initial state. For 1 / √ | i + | i ), it takes place at ωt = 4 . / √ | i + | i ), ωt = 11 . | i + | i is slower than that of state | i + | i ). It is hinted that the bath influenceon the former state is comparatively inapparent. Yet both of them can still revive to acertain extent 0 . ∼ . T ≥ . ω ,9 ω t C ( t ) (a) C ( t ) ω t F ( t ) (b) F d ( t ) FIG. 4: Time evolution for (a) Concurrence, (b) Fidelity with the subsystem starting from1 / √ | i + | i ). The two cavities are initialed in thermal states at different temperature: T = 0(solid curve), T = 0 . ω (dot dashed curve), T = 0 . ω (dashed curve), T = 1 . ω (dotted curve). ω t C ( t ) (a) C ( t ) ω t F ( t ) (b) F d ( t ) FIG. 5: Time evolution for (a) Concurrence, (b) Fidelity with the subsystem starting from1 / √ | i + | i ). The two cavities are initialed in thermal states at different temperature: T = 0(solid curve), T = 0 . ω (dot dashed curve), T = 0 . ω (dashed curve), T = 1 . ω (dotted curve). it can not revive in the future after the first sudden death happens for both Bell states.Till the temperature is as high as T = 1 . ω , the concurrence of both cases falls with a veryquick speed to zero. Obviously, the temperature will destroy the initial most-entangledstates even if it is much lower than the energy bias ω of the two-level atoms. After ESDhappened, the quantum oscillation from the local thermal bath may help to entangle the10 ω t E n ( t ) (a)1 / √ | i + | i ) ω t E n ( t ) (b)1 / √ | i + | i ) FIG. 6: Time evolution for entropy of the subsystem from (a) 1 / √ | i + | i ), (b) 1 / √ | i + | i ) on the condition that the two cavities are initialed in thermal states at different values oftemperature: T = 0 (solid curve), T = 0 . ω (dot dashed curve), T = 0 . ω (dashed curve), T = 1 . ω (dotted curve). two qubits, but this positive effect is neglectable when the temperature is high enough.Then the entanglement between the Bell states is damped forever.Under variant temperature, the different dynamics of fidelity, which rely on the initialstate, are shown in Fig. 4(b) and Fig. 5(b). In a short interval after ωt = 0, highertemperature means faster damp speed for both initial states. Yet in a long time scale, theiractions are totally different. For T ≤ . ω , the four curves evolves pseudo-periodically,but the period of 1 / √ | i + | i ) is much larger than that of 1 / √ | i + | i ). In Fig.4(b), the dotted curve of T = 1 . ω fluctuates with time, whose amplitude damps from thebeginning time and gets some revival when ωt > .
0. In Fig. 5(b), the oscillation evolutionat temperature T = 1 . ω is in the same manner as those at T = 0 . ω and T = 0 . ω .And their amplitudes and peak values of the fidelity decrease with increasing temperature.Obviously, an environment at higher temperature destroys the fidelity of the subsystemeven stronger.In Fig. 6, we give the dynamics of the entropy exchange under different temperatures.Although there are some differences between the two sub-figures even when T is as high as11 . ω , but the tendency of the two cases are almost the same. With higher T , the entropyincreases with faster speed and behaves an oscillation evolution with smaller amplitude. Itis important to find that the entropy exchange in Fig. 6(a) exhibits an opposite behaviorin comparison with that of the fidelity in Fig. 5(b). The periods of the two evolutionsare the same and when the fidelity experiences a peak value, (For instance, for the dotdashed curves in the two figures, when T = 0 . ω , the four peaks appear at ωt = 8 . ωt = 18 . ωt = 27 .
776 and ωt = 37 .
696 in the given interval) the entropy is at thecorresponding valley point and vice versa. In Ref. [48], the authors found the entropyexchange exhibit the behavior opposite to that of the concurrence. That coincides withour results. In our model, when the temperature is not too high and the initial Bell stateis 1 / √ | i + | i ), the concurrence and fidelity evolve synchronously, then the dynamicsof entropy is also opposite to that of the concurrence. However, when the temperature ishigh enough, the ESD will make this synchronization relationship invisible. So the oppositerelationship between the entropy and concurrence is lost. Therefore we have to concludethat the more quantum information of the subsystem transfers to its bath, the more fidelityof that decreases during the time interval if | ψ (0) i = 1 / √ | i + | i ). IV. CONCLUSION
In conclusion, we investigate the dynamics of two distinct uncoupled qubits embeddedrespectively in two single-mode cavities with leakage, which constitute the environment inour model. The subsystem consisting of the two qubits is initially prepared as one of theBell states and the cavities as vacuum states or thermal equilibrium states. Under these twosituations, the concurrence, the fidelity and the entropy exchange are used to portrait thesubsystem dynamics from different views. Polynomial expansion method is applied into thenumerical calculation. It is found that (i) for the leaky cavities, when the loss rate γ ≥ . cknowledgments We would like to acknowledge the support from the National Natural Science Foundationof China under grant No. 10575068, the Natural Science Foundation of Shanghai MunicipalScience Technology Commission under grant Nos. 04ZR14059 and 04dz05905 and the CASKnowledge Innovation Project Nos. KJcx.syw.N2. [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cam-bridge University Press, Cambridge, 2000).[2] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A , 2046 (1996).[3] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, arXiv:quant-ph/0702225 (2007).[4] R. F. Werner, Phys. Rev. A , 4277 (1989).[5] D. M. Greenberger, M. Horne, and A. Zeilinger, Bells Theorem, Quantum Theory, and Con-ceptions of the Universe (Kluwer Academic Publishers, Dordrecht, 1989).[6] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A , 062314 (2000).[7] T. Yu and J. H. Eberly, Phys. Rev. B , 165322 (2003).[8] L. Diosi, in Irreversible Quantum Dynamics , edited by F. Benatti and R. Floreanini (Springer,New York, 2003), pp. 157-163.[9] T. Yu and J. H. Eberly, Phys. Rev. Lett. , 140404 (2004).[10] A. R. R. Carvalho, F. Mintert, and A. Buchleltner, Rev. Lett. , 230501 (2004).[11] D. Loss, and D. P. DiVincenzo, Phys. Rev. A , 120 (1998).[12] B. E. Kane, Nature (London) , 133 (1998).[13] R. Tana´s and Z. Ficek, J. Opt. B , 90 (2004).[14] A. Shimony, Ann. N.Y. Acad.Sci, 755 (1995).[15] T. Yu and J. H. Eberly, Opt. Commun. 264, 393 (2006).[16] Z. Ficek and R. Tana´ s , Phys. Rev. A , 024304 (2006).[17] L. M. Liang, J. Yuan, and C. Z. Li, J. Phys. B , 4539 (2006).[18] C. A. Sackett, et. al, Nature, , 515 (2000).[19] S. B. Zheng and G. C. Guo, J. Mod. Opts. , 963 (1997).[20] A. Rauschenbeutel, et. al, Science, , 2024 (2000).
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