GENERATION AND CONTROL OF QUANTUM ENTANGLEMENT IN PHYSICAL SYSTEMS DAI LI (B.Sc., Soochow University) A thesis submitted for the Degree of Doctor of Philosophy Supervisor Professor Feng Yuan Ping Professor Kwek Leong Chuan DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgments I would like to thank all the people who have helped and inspired me during my PhD study. First I would like to express my deepest appreciation to my supervisors, Professor Feng Yuan Ping and Professor Kwek Leong Chuan, who have supported me throughout my doctoral study with their patience and knowledge. This dissertation would not have been possible if without their guidance and persistent help. Their untiring enthusiasm in research inspires me to develop a good understanding of quantum information science. I thank Professor Dimitris G. Angelakis for the collaboration on the research subject of coupled atom-cavity arrays which constitutes an important part of the dissertation. I thank Professor Oh Choo Hiap and Professor Valerio Scarani for introducing me to the field of quantum information science in my early days of PhD study. Their profound knowledge and thorough study on quantum information science are crucial for leading me to the research field. I also thank Professor Dagomir Kaszlikowski and Professor Vlatko Vedral for their wonderful lectures on quantum information and solid-state physics respectively. These lectures helped me considerably in the grasp of fundamental knowledge on physics of quantum information science, which is a requisite for the subsequent studies. I thank many people for having interesting discussions that helped me shape ideas in the research works and for being helpful in various ways. These include Alastair Kay, Stefano Mancini, Kavan Modi, Ying Li, Mingxia Huo, Man-Hong Yung, Junhong An, Huangjun Zhu, Lin Chen and Qi Zhang. I would like to acknowledge the financial support of the Research Scholar- ship and the President’s Graduate Fellowship offered by National University of Singapore, of the Research Assistantship offered by Centre for Quantum Technologies, Singapore, and of the National Research Foundation and Ministry of Education, Singapore. Lastly, but by no means least importantly, I would like to thank my parents for their unconditional support and for their love. ii Contents Summary vii List of Figures xi List of Symbols xiii Introduction 1.1 1.2 1.3 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Quantum Entanglement . . . . . . . . . . . . . . . . . 1.1.3 Separability criteria and measures of entanglement . . 1.1.4 Quantum discord . . . . . . . . . . . . . . . . . . . . . 1.1.5 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . 12 Review of basic models . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 The spin chain model . . . . . . . . . . . . . . . . . . . 16 1.2.2 Coupled atom-cavity arrays . . . . . . . . . . . . . . . 17 Objectives of the Study . . . . . . . . . . . . . . . . . . . . . . 19 Entanglement generation in a spin chain 2.1 Spin Chain 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 PST in a Spin Chain with Constant Couplings . . . . . 25 2.1.2 PST in a Spin Chain with Engineering Couplings . . . 26 2.1.3 Entanglement Generation in a Spin Chain . . . . . . . 27 2.2 Entanglement generation with pre-engineered couplings . . . . 28 2.3 A physical interpretation of the solution . . . . . . . . . . . . 32 2.4 Applications: quantum cloning . . . . . . . . . . . . . . . . . . 35 iii Contents 2.5 The subspace of multi-excitations . . . . . . . . . . . . . . . . 42 2.6 A more general result . . . . . . . . . . . . . . . . . . . . . . . 45 2.7 Realizing single-spin operations in a chain of quadrilaterals . . 45 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Multiparticle HBT interferometer in a spin network 55 3.1 Bosonic HBT interferometers . . . . . . . . . . . . . . . . . . 55 3.2 Fermionic HBT interferometers and our setup . . . . . . . . . 58 3.3 The HBT effect and the proof . . . . . . . . . . . . . . . . . . 63 3.4 The situation of multi-excitations . . . . . . . . . . . . . . . . 67 3.5 Physical mechanisms of the HBT effect . . . . . . . . . . . . . 68 3.6 The enhancement of the HBT effect . . . . . . . . . . . . . . . 70 3.7 Special cases and experimental realizations . . . . . . . . . . . 73 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Entanglement control in coupled atom-cavity arrays 77 4.1 Coupled cavity arrays and Entanglement . . . . . . . . . . . . 78 4.2 The setup and the Hamiltonian . . . . . . . . . . . . . . . . . 80 4.3 The dynamics of the system . . . . . . . . . . . . . . . . . . . 83 4.4 The steady state of the system . . . . . . . . . . . . . . . . . . 86 4.5 An alternative setup and comparison . . . . . . . . . . . . . . 90 4.5.1 A two-coupled-cavity setup with three driving fields . . 90 4.5.2 Comparison of different setups . . . . . . . . . . . . . . 92 4.6 Entanglement witness . . . . . . . . . . . . . . . . . . . . . . . 94 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Thermalization of the atom-cavity system 5.1 99 The system revisited . . . . . . . . . . . . . . . . . . . . . . . 100 iv Contents 5.2 The weak coupling regime . . . . . . . . . . . . . . . . . . . . 102 5.3 The moderate and strong coupling regimes . . . . . . . . . . . 104 5.4 The structure of the steady state . . . . . . . . . . . . . . . . 110 5.5 Graphical illustration . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Conclusion and future work 117 Bibliography 119 A Transformation for coupling constants 131 B The proof based on the special PST solution 133 C A more general proof for an odd spin chain 141 D The expressions for xi ’s 145 E Proof of the lemma 147 F Anti-correlated atom-cavity states for a strong driving field 149 v Summary This thesis examines four topics in the field of generation and control of quantum entanglement in physical systems. The first topic concerns the maximal entanglement generation in a spin chain of XX model with specially engineered couplings. For a spin chain with odd number of spins, the solution of the couplings is obtained by only modifying the two central coupling of the optimal perfect-state-transfer solution, and the modification is proved to be universal for any given couplings that permit perfect state transfer. The introduction of asymmetric couplings is analogous to an insertion of a beam splitter. A generalized spin chain which is composed of quadrilaterals is discussed, and it is found that single-spin unitary operations can be realized by varying the electric and magnetic fields that are applied to the quadrilateral. The second topic discusses the realization of the multiparticle Hanbury Brown-Twiss interferometer in a spin network comprising multiple spin chains. It is proved that for an N -particle system, the interference effect is manifested only in the N th-order correlation function of parity-preserving observables. This effect is enhanced through a post-selection process in which the multipartite Greenberger-Horne-Zeilinger entanglement is generated and tested with Svetlichny inequality. A possible experimental realization: Nitrogen-Vacancy color centers in diamond crystals is proposed and discussed. The third topic studies the coherent control of the steady-state entanglement in lossy and driven coupled atom-cavity systems. It is found that the steady-state entanglement can be coherently controlled through the tuning of the phase difference between the driving fields. Furthermore, for an array of three coupled atom-cavity systems, the maximal of entanglement for vii Summary any pair is achieved when their corresponding direct coupling is much smaller than their individual couplings to the third party. This effect is reminiscent of the coherent trapping of the Λ−type three-level atoms using two classical coherent fields. The fourth topic studies the thermalization of a single atom-cavity system, i.e. the relation between the steady state and a thermal equilibrium state of the system when the parameters such as the reservoir temperature and the driving strength are varied. It is found that the atom-cavity quantum correlation (quantum discord) appears to be a suitable quantity to characterize the degree of thermalization. Publication List [1] Li Dai, Yuan Ping Feng and Leong Chuan Kwek. Engineering quantum cloning through maximal entanglement between boundary qubits in an open spin chain, Journal of Physics A: Mathematical and Theoretical, 43 (2010) 035302. [2] Dimitris G. Angelakis, Li Dai and Leong Chuan Kwek. Coherent control of long-distance steady-state entanglement in lossy resonator arrays, arXiv:0906.2168v2. Europhysics Letters, 91 (2010) 10003; [3] Li Dai, Dimitris G. Angelakis, Leong Chuan Kwek and Stefano Mancini. Correlations and thermalization in driven cavity arrays, arXiv:1104.3422. Proceedings of International Symposium on 75 Years of Quantum Entanglement: Foundations and Information Theoretic Applications, American Institute of Physics, Conference Proceedings, 1384 (2011) 168. [4] Li Dai and Leong Chuan Kwek. Realizing the Multiparticle Hanbury Brown-Twiss Interferometer Using Nitrogen-Vacancy Centers in Diamond Crystals, Physical Review Letters, 108 (2012) 066803. viii matrix form1 (Bi = 0):  H =  Xm+1 Y where m = N Y T , (B.8) Zm and m is odd. This matrix must be equal to the matrix (B.7) with some R1 , R1 , D1 and D2 . Therefore, Y in (B.8) must be 0. From this, we get: N −3 ), √ N −1 = ( + 1)Ji+1 , (i = ). Ji = JN −i , (1 ≤ i ≤ Ji (B.9) (B.10) Using (B.9) and (B.10) to simplify (B.8), we obtain  H =  Xm+1 Zm , (B.11) where  Xm+1      =       J1 . J1 J2 . . J2 . . . . √ − 2Jm √ − 2Jm            (B.12) Here, we not present the explicit form of the matrix elements of H . But √ it√can be −Ji +JN −i N −3 √ observed that the matrix Y contains terms: , for ≤ i ≤ , and 2+ [(1 − 2 √ N −1 2)Ji + Ji+1 ], for i = . 135 Appendix B. The proof based on the special PST solution  Zm      =       J1 . J1 J2 . . J2 . . . . Jm−1 Jm−1            (B.13) For even m, exchange Xm+1 and Z m of (B.11). In order to get the solution for the couplings, one needs to find R1 , R1 , D1 and D2 in Eq. (B.7), which are not unique. We conjecture that D1 = diag[mπ, −mπ, (m − 2)π, −(m − 2)π, · · · , π, −π], D2 = diag[(m − 1)π, −(m − 1)π, (m − 3)π, −(m − 3)π, · · · , 0]. If m is even, replace m with m − in D1 , and replace m with m + in D2 . Actually, the diagonal terms of D1 and D1 are the eigenvalues of the Hamiltonian (2.1). These eigenvalues are same to those of the Hamiltonian for perfect state transfer (Hpst ) with √ i(N −i) Ji = π. This property, as can be seen later, helps to solve our prob2 lem. Rewrite Hpst in the representation of the eigenvectors of e−iHpst , we get, for N = 2m + 1, if m is odd,  Hpst =   Km+1 Lm , (B.14) 136 where,   J1pst Km+1 .   pst  J1 J2pst .   pst = .  J2  . √ .  . pst . . 2Jm  .  √ pst 0 2Jm  (B.15)  J1pst Lm            .   pst  J1 J2pst . . .   pst = .  J2  .  . pst . Jm−1 . .  .  pst 0 Jm−1            (B.16) If m is even, exchange Km+1 and Lm in (B.14). The general solution to U2 = e−iHpst is (Hpst is only a special solution to U2 ):  H2 =  R D1 R †  R D2 R † , (B.17) where the parameters are similar to those in Eq. (B.7). Since we have conjectured that the eigenvalues of (2.1) are same to those of Hpst (i.e. D1 = D1 and D2 = D2 ) and the structure of (B.14) is same to that of (B.11), we conclude that R1 = R1 and R2 = R2 . Therefore, the matrix (B.14) is equal 137 Appendix B. The proof based on the special PST solution to the matrix (B.11). Thus, Ji = Jipst , (1 ≤ i ≤ m − 1), √ √ pst − 2Jm = 2Jm . (B.18) (B.19) This ends the proof of first solution for odd N . The second solution for odd N can be obtained similarly. The only difference is that U0 slightly changes and Ji exchanges with Ji+1 for i = (N − 1)/2. For the first solution of even N, the U0 in (B.1) and (B.2) does not work, which suggests we need to add some relative phase between antipodal qubits (i.e. qubit and qubit N , qubit and qubit N − 1, etc.). N U0 |j = |j + i(−1) |N + − j √ . (B.20) Using the method in the proof for odd N , the counterpart of (B.7) is (N = 2k)  H2k =  R Pk R †  R2 Qk R2 † , (B.21) where π Pk = diag[qmin , · · · , − π, − , π, · · · , qmax ], 4 π Qk = diag[ − qmax , · · · , − π, , π, · · · , −qmin ], 4 k−1 π k π 2π, qmax = − + 2π. qmin = − − 4 138 The counterpart of (B.11) is:   H2k =  Rk Sk , (B.22) where       Rk =        J1 . J1 J2 . . J2 . . . . Jk−1 (B.23) Jk−1 (−1)k Jk        Sk =                  J1 . J1 J2 . . J2 . . . . Jk−1            (B.24) Jk−1 (−1)k−1 Jk The eigenvalues of Pk and Qk in (B.21) are shifted +π/4 and −π/4 respectively relative to the corresponding blocks of Hpst of even N , which makes the problem different from the odd N case. Here, we only give a calculation method. It can be verified that the eigenvalues of Rk and Sk only differ in a minus sign, which accords with those of Pk and Qk respectively. Thus we only need to deal with (B.23). Since the eigenvalues of Rk are same with those of Pk , 139 Appendix B. The proof based on the special PST solution we have N k Det[Rk − λIk ] = (−1) (λ − λi ), (B.25) i=1 k T r[Rk ] = (−1) Jk = T r[Pk ], (B.26) where, λi ’s are the eigenvalues of Pk in (B.21) (i.e. diagonal elements). The left hand side of (B.25) can be calculated using a recursion relation: Det[Rk − λIk ] = [−λ + (−1)k Jk ]Det[Rk−1 − λIk−1 ]|Jk−1 =0 − Jk−1 Det[Rk−2 − λIk−2 ]|Jk−2 =0 Using (B.25) and (B.26), we can calculate Ji ’s for any even N . Then, the general formula i.e. the second part of Eq. (2.5) could be obtained by mathematical induction. For the second solution of even N , the proof is very similar which is omitted here. 140 Appendix C A more general proof for an odd spin chain As mentioned earlier in the footnote on page 29, in an odd spin chain, there is a close relation between the solution of couplings for PST and that for maximal entanglement generation of boundary qubits. I.e., the modification factor for an odd spin chain in Eq. (2.5) is universal (not only for the specific PST solution π i(N − i)/2). Here, we give a proof1 . First, we rewrite the Hamiltonian (2.1) in the form of Dirac notation. H= N N N −1 (−2Bk + Jk (|k k + 1| + |k + k|) + k=1 k where Jk is assumed to be real and positive (cf. the eigenvalues of H as Ei with eigenfunctions |ψi Bj )|k k|, (C.1) j=1 Appendix A). Denote = N j=1 Ci,j |j (i = 1, 2, · · · , N ). It can be proved that Ji = JN −i and Bi = BN +1−i for all i if H realizes PST between boundary qubits [83]. In this situation, H commutes with the symmetry operator P = N i=1 |i N + − i| so that the eigenfunctions of H are either symmetric or anti-symmetric. We rewrite the symmetric eigen1 More recently, an elegant proof has been presented in Ref. [83]. 141 Appendix C. A more general proof for an odd spin chain functions as |ψis = N j=1 s s s s |j , (Ci,j = Ci,N Ci,j +1−j ) with eigenvalues Ei and N j=1 the anti-symmetric eigenfunctions as |ψia = a a a Ci,j |j , (Ci,j = −Ci,N +1−j ) with eigenvalues Eia . The eigenvalue equations are written as follows. J1 Ci,2 = (Ei − f1 )Ci,1 (C.2) Jk Ci,k+1 + Jk−1 Ci,k−1 = (Ei − fk )Ci,k , (k = 2, 3, · · · , N − 1) (C.3) JN −1 Ci,N −1 = (Ei − fN )Ci,N , where fk = −2Bk + N j=1 (C.4) s a Bj and Ci,j represents Ci,j or Ci,j . Let us multiply s the above equations by some factors using the following rules. If Ci,j = Ci,j , multiply Eqs. (C.2) and (C.3) by 1+ √1 n (an exception is that when k = (N + 1)/2, multiply J N +1 in the corresponding Equation by − √1n and J N −1 by + √1 ), n and multiply Eq. (C.4) by 1− √1 . n It can be seen that the new s s = Ci,N set of equations is equivalent to the original one (Note that Ci,j +1−j ). s through the The multiplication factors can be incorporated into Jk and Ci,j following redefinition. Ji = fN,i Ji , fN,i = s s Ci,j = Ci,j 1 + (δi, N −1 − δi, N +1 ) √ , 2 n (C.5) 1 + (θ N −1 ,j − θj, N +3 ) √ , 2 n (C.6) a where θi,j = if i ≥ j and otherwise. If Ci,j = Ci,j , the multiplication rules are same to the above except that (1) √1 n is replaced by − √1n and (2) when k = (N + 1)/2, multiply the corresponding equation by − n1 . In this way, the set of equations is still invariant and the corresponding redefinition is that a a = Ci,j Ji = fN,i Ji with fN,i defined before and Ci,j − (θ N −1 ,j − θj, N +3 ) √1n . By performing above operations, we actually obtain the eigenvalues and 142 N −1 k eigenfunctions of a new Hamiltonian H = N k=1 (−2Bk + N j=1 Jk (|k k + 1| + |k + k|) + Bj )|k k|. The eigenvalues are same to the original Hamil- tonian H for PST. But the eigenfunctions of H are changed to |ψis N j=1 s Ci,j |j with eigenvalues Eis and |ψia N j=1 = = a Ci,j |j with eigenvalues Eia . Let us further see what exp(−iH ) is. k1 −iH e = −iEis e k2 |ψis ψis i=1 a e−iEi |ψia |+ ψia |. (C.7) i=1 In particular, −iH 1|e |1 = (1 + √ ) n k1 −iEis e s |Ci,1 | i=1 + (1 − √ ) n k2 a a e−iEi |Ci,1 |, i=1 (C.8) k1 N |e−iH |1 s k2 s e−iEi |Ci,1 | − = ( i=1 a a e−iEi |Ci,1 | ) 1− i=1 , n (C.9) a a s s . By using = −Ci,1 and Ci,N = Ci,1 where we have used the property that Ci,N the fact that H realizes PST, i.e., 1|e−iH |1 = and N |e−iH |1 = exp(iφ1 ) (φ1 is an arbitrary phase factor). We can calculate the two matrix elements for e−iH . 1|e−iH |1 eiφ1 = √ , n N |e−iH |1 = eiφ1 Using the unitary property of e−iH that N i=1 (C.10) 1− . n (C.11) | i|e−iH |1 |2 = 1, we conclude that i|e−iH |1 = for i = 2, 3, · · · , N −1. Actually, H is the Hamiltonian we want. If we replace √1 n s s by − √1n (with a and Ci,j in the redefinition for Jk , Ci,j s little change for the multiplication rules that for the case of Ci,j = Ci,j , when k = (N + 1)/2, multiply J N +1 in the corresponding Equation by + √1 n and 143 Appendix C. A more general proof for an odd spin chain J N −1 by − √1 )), n we can obtain a new Hamiltonian H for which e−iH is very similar to e−iH . In particular, one only needs to multiply the right-hand side of Eq. (C.11) by a minus sign. Choose n = 2, we prove the claim in the footnote on page 29. 144 Appendix D The expressions for xi’s The expressions for xi ’s are as follows. x0 =γb e−iφ1 /2 [cos(kso l1 ) cos(kso l2 ) − sin(kso l1 ) sin(kso l2 ) cos(θ1 + θ2 )] + γc eiφ2 /2 [cos(kso l3 ) cos(kso l4 ) − sin(kso l3 ) sin(kso l4 ) cos(θ3 + θ4 )], x1 =γb e−iφ1 /2 [sin(kso l1 ) cos(kso l2 )(sin θ1 − sin θ2 )] − γc eiφ2 /2 [sin(kso l3 ) cos(kso l4 )(sin θ3 − sin θ4 )], x2 = − iγb e−iφ1 /2 [cos(kso l1 ) sin(kso l2 ) cos θ2 + sin(kso l1 ) cos(kso l2 ) cos θ1 ] − iγc eiφ2 /2 [cos(kso l3 ) sin(kso l4 ) cos θ4 + sin(kso l3 ) cos(kso l4 ) cos θ3 ], x3 =iγb e−iφ1 /2 [sin(kso l1 ) sin(kso l2 ) sin(θ1 + θ2 )] − iγc eiφ2 /2 [sin(kso l3 ) sin(kso l4 ) sin(θ3 + θ4 )], where γb = J1 J ε−εb and γc = J J4 . ε−εc 145 Appendix E Proof of the lemma For each term of the expansion (3.9), we define a set f = {f1 , f2 , · · · , fN } with fk = {ak , ak }, where ak = 0, denotes the state of the spin sk1 and ak = − ak is for skm . Then after the sequential swap operations the set becomes g = {g1 , g2 , · · · , gN } with gk = {ak+1 , ak }, (aN +1 ≡ a1 ). The corresponding position set mentioned in the Lemma is v. For f, f , we have g, g and v, v . If f, f differ in k0 th element, e.g. ak0 = ak0 , then gk = gk . For v = v , we must have ak+1 = ak+1 , and thus gk = gk = {ak+1 , ak }. Continuing the process to analyze gk0 −1 , gk0 −1 , etc, we find that gk = gk i.e. gk = gk for all k if v = v . In fact, the two sets f, f for which v = v must have the property that fk = fk for all k. Moreover, the 2N possibilities of f can be divided into 2N −1 pairs of such f, f for which v = v . 147 Appendix F Anti-correlated atom-cavity states for a strong driving field It is sufficient to show that the numerator of Eq. (5.11) is negative. This quantity can be calculated straightforwardly by using Eq. (5.14): (a† a − a† a )(σ † σ − σ † σ ) = (n − n)(s2 − s2 ) − s2 (1 − s2 ), (F.1) where v is the average value of the variable v over the probabilistic distribution {pn,i = λi |αn,i |2 } with αn,i = n|D(α0 )|i − , n = 1, 2, 3, · · · and s2 = sin2 + − θi,n −θi,n . Note that the fluctuation correlation between the vari- ables n and s2 i.e. (n − n)(s2 − s2 ) ≈ since n is an element of the index set {n, i} and it is a slowly varying function over the index {n, i} while s2 is a rapidly oscillating function (see also some examples of the expressions ± for θi,n in the context below Eq. (5.15)). Also ≤ s2 ≤ 1. Therefore, (a† a − a† a )(σ † σ − σ † σ ) ≈ −s2 (1 − s2 ) ≤ 0. This quantity is generally nonzero since s2 ∈ / {0, 1} for a general {pn,i }. 149 [...]... dynamics of a single excitation in an N = 61 spin chain 33 2.5 A beam splitter in a spin chain 34 2.6 The dynamics of a wave packet in a spin chain 35 2.7 A Mach-Zehnder interferometer in a spin chain 36 2.8 The fidelity of the phase covariant quantum cloning 38 2.9 Alternative methods for quantum cloning in a spin chain 39 2.10 Waveforms for the clone -and- swap... Dai and Leong Chuan Kwek Generation of boundary entanglement in a chain of coupled double quantum dots, accepted to be published in Laser Physics ix List of Figures 1.1 The set of zero-discord states 12 2.1 The factor fN,i for even-number spin chains 30 2.2 Patterns of Ji ’s for spin chains 31 2.3 The dynamics of a single excitation in an N = 60 spin chain... chain containing an odd number of spins for producing maximal entanglement As an interesting application, we also demonstrate the possibility of realizing a multiparticle interference effect: the Hanbury BrownTwiss interferometer in a spin star network comprising multiple spin chains Throughout the study, the spin chain of XX model is considered, while other models such as Heisenberg Model and Kitaev... quantum information processing, since only one spin chain is needed to fulfill the two tasks In contrast, performing the two tasks usually needs two permanently-coupling spin chains One of the aims of the present study was to find such solutions to realize both PST and maximal entanglement between boundary qubits Our study also reveals the universality of asymmetrical central couplings in a spin chain... Section 1.2.1, we show the possible realization of a spin chain of XX model in a linear array of coupled cavities 15 Chapter 1 Introduction 1.2.1 The spin chain model The spin chain model describes a one-dimensional chain of two-level spins which are short-range coupled Typically, only nearest-neighbor interactions are considered An important class of the spin chain model is the Heisenberg model The Hamiltonian... so-called Einstein-Podolsky-Rosen 1 Chapter 1 Introduction (EPR) paradox [7] which questioned the completeness of quantum mechanics and caused great debates among scientists including Einstein and Bohr Later, in 1964, Bell devised the Bell Inequality [8] which was used by Aspect to verify and confirm in experiment [9] the correctness of quantum mechanics However, the mystery of quantum entanglement, which... many scientists In 1994, Popescu and Rohrlich devised the Popescu-Rohrlich (PR) Box [10] which shows the similarities and differences between quantum correlations and classical ones Taking the controversial and difficult understanding of quantum entanglement aside, scientists have come to a consensus on the importance of finding ways to generate and control quantum entanglement, especially in practical environment... This topic is the main field of the thesis and will be discussed in later chapters As for this chapter, preliminary concepts and mathematical tools such as qubits, quantum entanglement, measures of entanglement will be introduced and discussed 1.1 1.1.1 General Concepts Qubit A qubit is an abbreviation for a quantum bit, the quantum analog of a bit in classical information It is a quantum- mechanical... σi (1.23) i We will mainly discuss the XX model in Chapter 2 This model can be experimentally realized through manipulations of control lasers and detuning in coupled atom-cavity arrays [46, 47] or through controlling external voltage in linear arrays of tunnel-coupled quantum dots [48] The former realization 16 1.2 Review of basic models will be discussed in detail in the following section 1.2.2 Coupled... three interacting atom-cavity systems 81 4.2 The polar plot of the maximum concurrence against the driving fields’ intensity ratio 88 4.3 The coherent trapping of correlations in a 3-cavity system 89 4.4 The coherent effect of the entanglement in a 3-cavity system 90 4.5 Schematic diagram of the two atom-cavity systems 91 4.6 The coherent effect of the entanglement in a 2-cavity . Spin Chain with Constant Couplings . . . . . 25 2.1.2 PST in a Spin Chain with Engineering Couplings . . . 26 2.1.3 Entanglement Generation in a Spin Chain . . . . . . . 27 2.2 Entanglement generation. entanglement in physical systems. The first topic concerns the maximal entanglement generation in a spin chain of XX model with specially engineered couplings. For a spin chain with odd number of spins,. GENERATION AND CONTROL OF QUANTUM ENTANGLEMENT IN PHYSICAL SYSTEMS DAI LI (B.Sc., Soochow University) A thesis submitted for the Degree of Doctor of Philosophy Supervisor Professor Feng