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Entanglement dynamics of a pure bipartite system in dissipative environments
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  Entanglement dynamics of a pure bipartite system in dissipative environments This article has been downloaded from IOPscience. Please scroll down to see the full text article.2008 J. Phys. B: At. Mol. Opt. Phys. 41 205501(http://iopscience.iop.org/0953-4075/41/20/205501)Download details:IP Address: 111.68.99.56The article was downloaded on 24/09/2012 at 07:16Please note that terms and conditions apply.View the table of contents for this issue, or go to the  journal homepage for more HomeSearchCollectionsJournalsAboutContact usMy IOPscience  IOP P UBLISHING  J OURNAL OF  P HYSICS  B: A TOMIC,  M OLECULAR AND  O PTICAL  P HYSICS J. Phys. B: At. Mol. Opt. Phys.  41  (2008) 205501 (7pp)  doi:10.1088/0953-4075/41/20/205501 Entanglement dynamics of a purebipartite system in dissipativeenvironments Rabia Tahira 1 , Manzoor Ikram 1 , Tasnim Azim 1,4 and M Suhail Zubairy 1,2,3 1 Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan 2 Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station,TX 77843, USA 3 Texas A&M University at Qatar, Education City, PO Box 23874, Doha, Qatar  Received 1 April 2008Published 1 October 2008Online at stacks.iop.org/JPhysB/41/205501 Abstract We investigate the phenomenon of sudden death of entanglement in a bipartite systemsubjected to dissipative environments with arbitrary initial pure entangled state between twoatoms. We find that in a vacuum reservoir the presence of the state where both atoms are inexcited states is a necessary condition for the sudden death of entanglement. Otherwiseentanglement remains for an infinite time and decays asymptotically with the decay of individual qubits. For pure 2-qubit entangled states in a thermal environment, we observe thatthe sudden death of entanglement always happens. The sudden death time of the entangledstates is related to the temperature of the reservoir and the initial preparation of the entangledstates. 1. Introduction Entanglement can be viewed as a consequence of non-localcoherenceandisaquantum-mechanicalphenomenoninwhicha quantum state of two or more objects has to be describedwithreference to each, even though the individual objects maybe spatially separated. This leads to correlations betweenobservable physical properties of the system. Quantumcorrelations have attracted a lot of attention during thehistory of quantum mechanics. Bell [1] and Clauser   et al [2] have shown that these correlations violate inequalitiesthat must be satisfied by any classical local hidden variablemodel. Bell’s inequalities can be violated between entangledqubits because of correlation whose srcin lies in quantuminterference terms. Once these interference terms haveundergone decoherence, the remaining correlations betweenqubits obey Bell’s inequalities. The complex phenomenon of quantum entanglement has been studied extensively in recentyears because it represents an essential resource for quantuminformation processing [3 – 7]. The presence of decoherence in communication channel and computing devices presents a 4 Permanent address: Department of Physics, Quaid-i-Azam University,Islamabad, Pakistan. considerable obstacle, as it degrades the entanglement whenparticles propagate or the computation evolves.Decoherence and entanglement are closely connectedphenomena, sounderstandingthemechanismsofdecoherenceand disentanglement has both fundamental and practicalimplications. Real quantum systems are inevitably influencedby their surrounding environments, these unavoidable mutualinteractionsoftenresultinthedissipativeevolutionofquantumcoherence and loss of useful entanglement. Decoherencemay lead to both local dynamics, associated with singleparticle dissipation, diffusion and decay, and to globaldynamics, which may provoke the eventual disappearance of entanglement. The connection between disentanglement anddecoherence of bipartite systems has recently been analysedby Yu and Eberly [8]. In their system two non-interactingatoms are coupled to two separate cavities (environments). Asa result, the dynamic evolution of the atoms are independentand, depending on the initial state, the effect of spontaneousemission can drive the system to disentangle in finite time,calling it the ‘sudden death of entanglement’. Since thenmany studies on entangled state dynamics have been madefor different initial states ranging from pure Bell states tomixed states, and for different environments such as vacuum,thermal and squeezed vacuum etc [9 – 22]. In all these studies 0953-4075/08/205501+07$30.00  1  © 2008 IOP Publishing Ltd Printed in the UK  J. Phys. B: At. Mol. Opt. Phys.  41  (2008) 205501 R Tahira  et al Figure 1.  Two 2-level atoms, initially prepared in an entangledstate, have no directional interaction with each other butindependently interact with their local reservoirs. theauthorsshowthatentanglementsuddendeathissensitivetothe initial conditions of the 2-qubit system and the interactingenvironments. The sudden death of entanglement of a 2-qubitsystem under the influence of independent environments hasbeen experimentally demonstrated in all optical setups [23].Whenthetwoatomsareinthesamereservoirandcloseenoughtohavedipole–dipoleinteraction,TanasandFicek[24]showedthat the entanglement exhibits oscillatory behaviour.In this paper we consider another class of bipartiteentangled state of the form | ψ(A,B)  = 1  p,q = 0 C p,q | p,q  ,  (1)where  C p,q  are probability amplitudes with  1 p,q = 0 | C p,q | 2 = 1. It is the most general form of any bipartite entangledstate and contains the well-known Bell states with the proper choice of   C p,q . We consider the entanglement dynamicsusing the Wootters concurrence [25], when these kinds of entangledstatesintwoatomsinteractwithvacuumandthermalreservoirs. For the vacuum reservoir we get an analyticalexpressionwhichshowsthatthesuddendeathofentanglementhappens only when we have significant contribution of thestate where both the atoms are in excited states. In theabsence of this term the entanglement decays asymptotically.In the case of a thermal reservoir we always have finite-timedisentanglement. 2. The model We consider two 2-level atoms 1 and 2 which represent thebipartite system and interact independently with their localenvironments as shown in figure 1. The correlation between the atoms results only from an initial quantum entanglementbetween them. We also consider that atoms are far apartin the separate environments so there is no direct interactionbetween the atoms. Therefore, we neglect the dipole–dipoleinteractiontermsintheHamiltonian. Intheinteractionpicture,theHamiltonianoftheatom–fieldcoupledsystemhastheform[26]  ( ¯ h  =  1 )H   =  k  g ( 1 )k  e i (w − υ k )t  | a 1  b 1 | a k  +  H.c  +  k  g ( 2 )k  e i (w − υ k )t  | a 2  b 2 | b k  +  H.c  ,  (2)where  | a i  ,  and  | b i   are the excited and ground states of the i th atom,  ω  is the frequency separation between the atomicstates,  a k (b k )  is the annihilation operator for the photons of the reservoir surrounding atom 1 ( 2 )  in the mode  k ,  υ k  is  k  thmodefrequencyofthemode k and g (i)k  isthecouplingconstantof the interaction between the atom  i  and the local reservoir.In writing the Hamiltonian  ( 2 ) , it is assumed that the rotating-wave approximation is valid. We first consider the case thatthe local reservoir is in a vacuum state and extends the resultsto the case of a thermal reservoir. We find that in the case of a thermal field this configuration always leads to the suddendeath of entanglement.To describe the dynamic evolution of quantumentanglement we need a concrete measure of the degree of entanglement contained in a quantum state. For any bipartitesystem, Wootters’ concurrence [25] is particularly convenient.Any reliable measure of entanglement will yield the sameconclusion. The concurrence varies from  C  =  0 for aseparablestateto C  =  1foramaximallyentangledstate. Forabipartite system, the concurrence may be calculated explicitlyfrom the density matrix  ρ  for qubits 1 and 2 as C  =  Max { 0 ,   λ 1  −   λ 2  −   λ 3  −   λ 4 } ,  (3)where the quantities  λ i  are the eigenvalues in decreasing order of the matrix M   =  ρ  σ  1 y  ⊗  σ  2 y  ρ ∗  σ  1 y  ⊗  σ  2 y  ,  (4)where  ρ ∗ denotes the complex conjugation of the densitymatrix  ρ  in the standard basis  {| 00  , | 01  , | 10  , | 11 }  and  σ  y  isthe Pauli matrix expressed in the same basis as σ  y  =  0  − ii 0  .  (5) 3. Entanglement dynamics in a vacuum reservoir We first consider the dynamics of entanglement of the atomsin a vacuum reservoir. According to the general quantumreservoir theory, with the Hamiltonian (2), we can derive thefollowingequationofmotionforthereduceddensitymatrixof the atoms interacting with their local vacuum reservoirs [26] • ρ = − 12 γ  1  σ  1+ σ  1 − ρ  −  2 σ  1 − ρσ  1+  +  ρσ  1+ σ  1 −  −  12 γ  2  σ  2+ σ  2 − ρ  −  2 σ  2 − ρσ  2+  +  ρσ  2+ σ  2 −  ,  (6)where  γ  i  is the spontaneous emission rate of atom  i , and  σ  i ± are the raising and lowering operators of the atom  i , definedas  σ  i +  = | a i  b i |  and  σ  i −  = | b i  a i | . In deriving equation (6),we assume that the interaction between the atoms and thereservoirs is weak and there is no back-reaction effect of theatoms on the reservoirs.In this section we study in detail the time evolution of the concurrence for some class of initial states. Dependingon the initial state, concurrence can reach a value equal tozero asymptotically or at some finite time. It is interestingthat locally equivalent initial states with the same concurrencecan disentangle at different times. Most of the earlier workon entanglement dynamics is well discussed with the mixedstates as the initial states of the atoms. Here we consider theinitial state of the two atoms as a pure state in the form | ψ( 0 )  =  α 1 | 1   +  α 2  e i φ 1 | 2   +  α 3  e i φ 2 | 3   +  α 4  e i φ 3 | 4  ,  (7)where  | 1  = | a 1 ,a 2  , | 2  = | a 1 ,b 2  , | 3  = | b 1 ,a 2  , | 4  =| b 1 ,b 2  ,α i (i  =  1 , 2 , 3 , 4 )  are the probability amplitudes with  i α 2 i  =  1 and  φ i  are the relative phases. For this initial state,the solution of equation (6) can be written in full 4 × 4 matrix 2  J. Phys. B: At. Mol. Opt. Phys.  41  (2008) 205501 R Tahira  et al form as ρ  =  ρ 11  ρ 12  ρ 13  ρ 14 ρ 21  ρ 22  ρ 23  ρ 24 ρ 31  ρ 32  ρ 33  ρ 34 ρ 41  ρ 42  ρ 43  ρ 44  ,  (8)where the density matrix elements  ρ ij   are determined from ρ  = | ψ(t)  ψ(t) |  and are given in appendix A. For theentanglement dynamics, we determine the eigenvalues of thematrix  M   as λ 1  =  e − 4 tγ   α 41  −  2 α 41  e γt  + e 2 γt   α 21  −  α 21 α 22  −  α 21 α 23  +  α 22 α 23 − 2e 2 γt  α 1 α 2 α 3 α 4  cos ()  1 / 2 +e tγ   α 21 α 24  +  α 22 α 23  −  2 α 1 α 2 α 3 α 4  cos ()  1 / 2  2 ,λ 2  =  e − 4 tγ   α 41  −  2 α 41  e γt  + e 2 γt   α 21  −  α 21 α 22  −  α 21 α 23  +  α 22 α 23 − 2e 2 γt  α 1 α 2 α 3 α 4  cos ()  1 / 2 − e tγ   α 21 α 24  +  α 22 α 23  −  2 α 1 α 2 α 3 α 4  cos ()  1 / 2  2 ,λ 3  =  λ 4  =  e − 4 γt  α 41 ( − 1 + e tγ  ) 2 ,  (9)where, for the sake of simplicity, we consider that the atomicdecayratesareequal, i.e.,  γ  1  =  γ  2  =  γ  , and   =  φ 1 + φ 2 − φ 3 .The concurrence from these eigenvalues is determined to be C(t)  =  Max  0 , e − γt  C( 0 )  −  2 α 21  e − 2 γt  ( e γt  −  1 )  ,  (10)where  C( 0 )  is the initial concurrence given by C( 0 )  =  2   α 21 α 24  +  α 22 α 23  −  2 α 1 α 2 α 3 α 4  cos ().  (11)The sudden death time of the entanglement is t  d   = 1 γ  log   2 α 21 2 α 21  −  C( 0 )  .  (12)This expression shows that the presence of both atomsin excited states  | a 1 ,a 2   is necessary for the finite-timedisentanglement. For all other combinations excluding thisterm gives asymptotic decay of entanglement.However, if the term  | a 1 ,a 2   is included in our initialstate then the question arises as to how large  α 1  has to befor the entanglement sudden death. For     =  0, we see thatentanglement survives for  α 1  <  12  α 4  +   α 24  −  4 α 2 α 3   (13)and for all other values of   α 1  >  12  α 4  + √  α 24  −  4 α 2 α 3  ,  wehave a finite-time disentanglement. It is clear from the aboveexpression that if either of   α 2 ,α 3  or both are zero, then for the entanglement sudden death we should have  α 1  > α 4 .However, for  α 4  =  0, i.e., nocontributionfromthestatewhereboth atoms are in ground state, we should have  α 1  > α 2 α 3  for the entanglement sudden death.The EPR system plays a central role in quantumcommunications and is certainly of importance to study itsentanglement decay induced by the environment. So weconsider some special cases. ( 1 )  Consider the initial state  | ψ 2 ( 0 )  =  α 2 | a 1 ,b 2   + α 3  e i φ 2 | b 1 ,a 2  .  The eigenvalues of the matrix  M   are λ 1  =  4 α 22 α 23  e − 2 γt  , λ 2  =  λ 3  =  λ 4  =  0 (14)and the concurrence is C(t)  =  Max { 0 , 2 α 2 α 3  e − γt  } .  (15) (a)(b) Figure 2.  Entanglement dynamics of EPR states (a)  | ψ B ( 0 )  = 1 √  2 [ | a 1 ,b 2   +  | b 1 ,a 2  ] and (b)  | ψ A ( 0 )  =  1 √  2 [ | a 1 ,a 2   +  | b 1 ,b 2  ] atdifferent temperatures of thermal reservoirs. Solid for   n  =  0, dottedfor   n  =  0 . 1 and dashed for   n  =  0 . 2. It shows that initial concurrence is maximum at the equalvalues of   α 2  and  α 3 , where the entanglement decaysasymptotically and we never have entanglement sudden deathfor this kind of entangled state. ( 2 )  Consider the initial state  | ψ 1 ( 0 )  =  α 1 | a 1 ,a 2   + α 4  e i φ 3 | b 1 ,b 2   with the initial concurrence  C( 0 )  =  2 α 1 α 4 .  Asit decays, it gets entangled with the environment, graduallylosing its coherence and purity over time. The entanglementdecay dynamics depends on the relation between  α 1  and  α 4 . The eigenvalues of the matrix  M   are λ 1  =  α 21  e − 4 γt    α 21 ( 1  −  2e γt  )  + e 2 γt  +  α 4  e γt   2 ,λ 2  =  α 21  e − 4 γt    α 21 ( 1  −  2e γt  )  + e 2 γt  −  α 4  e γt   2 ,  (16) λ 3  =  λ 4  =  e − 4 γt  α 41 ( − 1 + e γt  ) 2 . With these eigenvalues, we get the concurrence as C(t)  =  Max { 0 , 2 (α 1 α 4  e − γt  −  α 21  e − 2 γt  ( e γt  −  1 )) } .  (17)It is obvious that, for   α 1    α 4 ,  entanglement disappearsonly when the individual qubits have completely decayed, thatis,decaywillalwaysbeasymptotic. However,for  α 1  > α 4 , wehaveafinite-timedisentanglement. Thus,weshowthatlocallyequivalent pure states with the same entanglement behavevery differently during the time evolution and simple localunitary operation performed on the initial state can changedisentanglement time from finite to infinite. 3  J. Phys. B: At. Mol. Opt. Phys.  41  (2008) 205501 R Tahira  et al (a) (b)(c) (d) Figure 3.  Entanglement dynamics of the states (a)  | ψ 1 ( 0 )  =  α 1 | a 1 ,a 2   +   1  −  α 21 | b 1 ,b 2   for vacuum reservoir and (b) | ψ 2 ( 0 )  =  α 2 | a 1 ,b 2   +   1  −  α 22 | b 1 ,a 2   for vacuum reservoir; (c) state is the the same as (a) but for   n  =  0 . 1 and (d) state is the same as (b)but for   n  =  0 . 1. 4. Entanglement dynamics in a thermal reservoir It is of interest to examine how the entanglement obtainedfor zero temperature (the case for a vacuum reservoir) inthe previous section changes when the temperature is finite.We consider the case of a thermal reservoir and derive thefollowing equation of motion for the reduced density matrixof the atoms interacting with their local thermal reservoirs of mean thermal photon numbers  m  and  n  as [26] • ρ = − 12 γ  1 (n  + 1 )  σ  1+ σ  1 − ρ  −  2 σ  1 − ρσ  1+  +  ρσ  1+ σ  1 −  −  12 γ  1 n  σ  1 − σ  1+ ρ  −  2 σ  1+ ρσ  1 −  +  ρσ  1 − σ  1+  −  12 γ  2 (m  + 1 )  σ  2+ σ  2 − ρ  −  2 σ  2 − ρσ  2+  +  ρσ  2+ σ  2 −  −  12 γ  2 m  σ  2 − σ  2+ ρ  −  2 σ  2+ ρσ  2 −  +  ρσ  2 − σ  2+  ,  (18)where γ  i  isthespontaneousemissionrateoftheatom i , and σ  i ± are the raising and lowering operators of the atom  i , definedas  σ  i +  = | a i  b i |  and  σ  i −  = | b i  a i | . In deriving equation (18),we assume that the interaction between the atoms and thereservoirs is weak and there is no back-reaction effect of theatoms on the reservoirs. It means that the reservoirs are at alltimes in the initial uncorrelated thermal equilibrium mixtureof photon number states. Here, we also assume that thecorrelation time between the atoms and the reservoirs is muchshorterthanthecharacteristictimeofthedynamicevolutionof the atoms such as spontaneous emission life and entanglementsudden death time so that the Markov approximation is valid.We determine the density matrix elements of  (18) for  the case of thermal reservoir as shown in appendix A. For the sake of simplicity, we assume that each qubit interactswith the same thermal bath, that is,  m  =  n , and bothatoms have the same decay rate. The exact solution of theconcurrence for the initial bipartite state (1) is unknown. So instead of considering initial state (1) we begin our analysiswith the cases of two qubits. To provide the completedescription of time dependence of the entanglement whenit interacts with the thermal environment, we consider theinitial 2-qubit states  | ψ 1 ( 0 )  =  α 1 | a 1 ,a 2   +  α 4  e i φ 3 | b 1 ,b 2  and  | ψ 2 ( 0 )  =  α 2 | a 1 ,b 2   +  α 3  e i φ 2 | b 1 ,a 2   and find their entanglement dynamics separately.Considertheinitialentangledstate | ψ 1 ( 0 )  =  α 1 | a 1 ,a 2  + α 4  e i φ 3 | b 1 ,b 2  , we get the eigenvalues of the density matrix  M  as follows λ 1  =  λ 2  = e − 4 ( 2 n +1 )γt  ( − 1 + e ( 2 n +1 )γt  ) 2 ( 2 n  + 1 ) 4 ×  n(n  + 1 ) e ( 2 n +1 )γt  +  n 2 +  ( 2 n  + 1 )α 21  2 ,λ 3  = e − 4 ( 2 n +1 )γt  ( 2 n  + 1 ) 4  (n  + 1 ) 2 ( 2 n e 3 ( 2 n +1 )γt  +  n 2 e 4 ( 2 n +1 )γt  − 2 (n  + 1 ) e ( 2 n +1 )γt  +  (n  + 1 ) 2 + e 2 ( 2 n +1 )γt  × ( 1  −  2 n(n  + 1 )))α 41  + e 2 ( 2 n +1 )γt  ( 4 n(n  + 1 ) cosh (γt) +4 n 2 (n  + 1 ) 2 cosh ( 2 γt)  + 1 + 4 n(n  + 1 ) × ( 1 + 3 n(n  + 1 )))α 21 α 24  +  n 2 ( 2 n e ( 2 n +1 )γt  − 2 ( 1 +  n) e 3 ( 2 n +1 )γt  +  n 2 +  ( 1 +  n) 2 e 4 ( 2 n +1 )γt  + ( 1  −  2 n( 1 +  n)) e 2 ( 2 n +1 )γt  )α 44  1   2 − α 1 α 4 ( 2 n  + 1 ) 2 e ( 2 n +1 )γt   2 λ 4  = e − 4 ( 2 n +1 )γt  ( 2 n  + 1 ) 4  (n  + 1 ) 2 ( 2 n e 3 ( 2 n +1 )γt  +  n 2 e 4 ( 2 n +1 )γt  − 2 (n  + 1 ) e ( 2 n +1 )γt  +  (n  + 1 ) 2 + e 2 ( 2 n +1 )γt  × ( 1  −  2 n(n  + 1 )))α 41  + e 2 ( 2 n +1 )γt  ( 4 n(n  + 1 ) cosh (γt) +4 n 2 (n  + 1 ) 2 cosh ( 2 γt)  + 1 + 4 n(n  + 1 )( 1 + 3 n(n  + 1 ))) 4
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