Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Cox, Timothy
Doctor of Philosophy - PhD
University of British Columbia
We present a theoretical discussion of various aspects of the dynamics of information in quantum systems of qubits. We begin by reviewing ideas about entanglement and information in systems of qubits, different models for realistic qubits coupled to an external environment, how this causes the loss of information stored in qubits, and some practical designs for qubits that have been created. Then we study how the state of many-body systems can be decomposed in terms of its different subsystems and derived a hierarchy of equations describing the motion of these different parts. We show that in a qubit system, the central objects in this hierarchy are correlators between different components of different qubits. Systems where the environment coupled to a central qubit is a "spin bath'' of environmental qubits are then considered in detail and we find that information lost by the central qubit is transferred via a cascade to higher and higher order correlations with the environment. We discuses this process in the realistic example of an "Feâ'' magnetic molecule qubit. Finally we discuss simple models of the dynamics of many entangled qubits.
Aspects of Decoherence in QubitSystemsbyTimothy CoxB.Sc. (Hons), Victoria University of Wellington, 2007M.Sc., McGill University, 2010A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019cÂ© Timothy Cox, 2019iiThe following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Aspects of Decoherence in Qubit Systemssubmitted by Timothy Cox in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Physics.Examining Committee:Professor Philip C. E. Stamp, PhysicsSupervisorProfessor Mona I. Berciu, PhysicsSupervisory Committee MemberProfessor Gordon W. Semenoff, PhysicsSupervisory Committee MemberProfessor Jim Bryan, MathematicsUniversity ExaminerProfessor Ian Keith Affleck, PhysicsUniversity ExaminerAdditional Supervisory Committee Members:Associate Professor Joshua Folk, PhysicsSupervisory Committee MemberProfessor Fei Zhou, PhysicsSupervisory Committee MemberiiiAbstractWe present a theoretical discussion of various aspects of the dynamics of informationin quantum systems of qubits. We begin by reviewing ideas about entanglement andinformation in systems of qubits, different models for realistic qubits coupled to anexternal environment, how this causes the loss of information stored in qubits, andsome practical designs for qubits that have been created. Then we study how thestate of many-body systems can be decomposed in terms of its different subsystemsand derived a hierarchy of equations describing the motion of these different parts.We show that in a qubit system, the central objects in this hierarchy are correlatorsbetween different components of different qubits. Systems where the environmentcoupled to a central qubit is a âspin bathâ of environmental qubits are then consi-dered in detail and we find that information lost by the central qubit is transferredvia a cascade to higher and higher order correlations with the environment. Wediscuses this process in the realistic example of an âFe8â magnetic molecule qubit.Finally we discuss simple models of the dynamics of many entangled qubits.ivLay SummaryA quantum computer will be able to solve problems much faster than a traditionalcomputer. Quantum computers rely on the quantum behaviour of matter, typicallyseen at the atomic scale. So far only small quantum computers, consisting of afew dozen quantum bits or âqubitsâ, have been built and tested. Information isstored and used in a quantum computer when qubits have correlations betweenthemselves and within the qubit. The main obstacle to development of a largequantum computer is that information leaks out of the system when the qubits buildcorrelations with the environment. We have studied a number of different theoreticalmodels of realistic systems consisting of various multiple qubits arrangements (andtheir environment). For these models we have worked out the exact nature of thespurious correlations that arise with the environment, and have thereby developeda picture of how these systems lose information to the environment.vPrefaceFigure 3.8 is reproduced from [40] with permission, figure 6.2 was provided by PhilipC. E. Stamp. The rest of the material in this thesis is the original work of the author,Timothy Cox. All work was supervised by Philip C. E. Stamp including guidance inanalyses and project direction. Chapters 2-4 and appendices A-C, of this thesis areadapted from a draft of an original manuscript published in a peer-reviewed journalwith Timothy Cox as the primary author [18]. All other chapters are original work,first published in this document.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Quantum State . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Quantifying and Detecting Quantum Entanglement . . . . . . . . . . 61.4.1 Bipartite Entanglement . . . . . . . . . . . . . . . . . . . . . 71.4.2 Multipartite Entanglement . . . . . . . . . . . . . . . . . . . 91.5 Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Models of the Quantum Environment . . . . . . . . . . . . . . . . . 101.7 The Environment as a Quantum System . . . . . . . . . . . . . . . . 101.7.1 Oscillator bath models . . . . . . . . . . . . . . . . . . . . . . 111.7.2 Spin Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Practical Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.1 The Josephson Junction Circuit Element . . . . . . . . . . . 201.9.2 Coupling to the Environment . . . . . . . . . . . . . . . . . . 221.9.3 Flux Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.9.4 Transmon Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 281.10 Spin-based qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.10.1 Semiconducting Spin Qubits . . . . . . . . . . . . . . . . . . 301.10.2 Insulating spin qubits . . . . . . . . . . . . . . . . . . . . . . 31vii1.11 Conclusions and Outline of the Rest of the Thesis . . . . . . . . . . 322 Partitioned Density Matrices and Their Correlations . . . . . . . 342.0.1 Definition of Correlated Density Matrices . . . . . . . . . . . 342.0.2 A 4-cell Example . . . . . . . . . . . . . . . . . . . . . . . . . 362.0.3 General Properties of Entanglement Correlated Density Ma-trices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Structure of the Many Qubit Density Matrix . . . . . . . . . . . . 433.1 General Results for N coupled Qubits . . . . . . . . . . . . . . . . . 433.1.1 Spin Representations . . . . . . . . . . . . . . . . . . . . . . . 433.1.2 General Results for N qubits . . . . . . . . . . . . . . . . . . 443.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 A Pair of Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.2 Three Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.3 N-qubit states . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Dynamics of Partitioned Density Matrices . . . . . . . . . . . . . . 524.1 Result for N -partite System . . . . . . . . . . . . . . . . . . . . . . . 524.2 Hierarchy of Equations for Reduced Density Matrices N Qubits . . . 544.2.1 General Form of Hierarchy . . . . . . . . . . . . . . . . . . . 554.2.2 One- and Two-qubit Correlators . . . . . . . . . . . . . . . . 554.2.3 Relationship to Schwinger-Dyson Hierarchy . . . . . . . . . . 574.3 Entanglement Correlators . . . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Example: Dynamics of a Single Qubit . . . . . . . . . . . . . 594.3.2 Example: Entanglement Correlator Dynamics: Two Qubits . 594.4 Remarks on a General Formulation . . . . . . . . . . . . . . . . . . . 644.4.1 Two Coupled Systems . . . . . . . . . . . . . . . . . . . . . . 654.4.2 Perturbation Expansions . . . . . . . . . . . . . . . . . . . . . 674.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Integrating a Spin Bath Out of the Hierarchy . . . . . . . . . . . . 695.1 Integrating Out the Bath Spins . . . . . . . . . . . . . . . . . . . . . 725.1.1 Product State Initial Conditions . . . . . . . . . . . . . . . . 745.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 Overview of Simple Single Qubit Models . . . . . . . . . . . . . . . 756.1 Example, Degeneracy Blocking Spin Bath . . . . . . . . . . . . . . . 756.1.1 Example: Gaussian Bias Distribution . . . . . . . . . . . . . 776.1.2 Return Probability . . . . . . . . . . . . . . . . . . . . . . . . 796.1.3 Decay of the Central Qubit Polarisation . . . . . . . . . . . . 79viii6.2 Example: Motion of Spin in Time Dependent Bias . . . . . . . . . . 956.3 Randomly Fluctuating Force . . . . . . . . . . . . . . . . . . . . . . 966.3.1 Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.2 Example: Low frequency noise, diffusing bias . . . . . . . . . 976.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997 The Spin Bath Influence Functional and Precessional Decoherence1017.1 The Influence Function for Qubit Systems Coupled to a Spin Bath . 1017.2 Evaluating the Influence Functional for Precessional Decoherence . . 1057.2.1 Canonical Transformation of the Bath Spin Variables . . . . 1057.2.2 The Orthogonality Blocking Approximation. . . . . . . . . . 1067.2.3 Calculation of the Reduced Density Matrix . . . . . . . . . . 1077.2.4 Dynamics of Decoherence in the Canonical Variables . . . . . 1147.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198 Application: The Fe8 Qubit . . . . . . . . . . . . . . . . . . . . . . . . 1208.1 The âFe8â Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.1.1 The Coupling of the Fe8 Qubit to the Spin Bath . . . . . . . 1228.2 The Fe8 Molecule in Real Systems . . . . . . . . . . . . . . . . . . . 1278.3 Dynamics of Dechoerence Due to 1H Nuclei . . . . . . . . . . . . . . 1278.4 Dynamics of Decoherence Due to 57Fe Nuclei . . . . . . . . . . . . . 1308.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 Evolution of a Large âCat Stateâ . . . . . . . . . . . . . . . . . . . . 1339.1 Quantities of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.2 Motion in Applied Longditudional Fields . . . . . . . . . . . . . . . . 1389.3 Motion in General fields . . . . . . . . . . . . . . . . . . . . . . . . . 1399.3.1 The Effect of Disorder . . . . . . . . . . . . . . . . . . . . . . 1449.4 Degeneracy Blocking for a Large Cat State . . . . . . . . . . . . . . 1499.5 General averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.5.1 Spherically Symmetric Degeneracy Blocking . . . . . . . . . . 1619.5.2 Precessional Decoherence . . . . . . . . . . . . . . . . . . . . 1639.5.3 Degeneracy Blocking with a Lorentzian Field Distribution . . 1649.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16510 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A Properties of Entanglement Density Matrices . . . . . . . . . . . . 179A.1 Proof of Eqtn. (2.23) . . . . . . . . . . . . . . . . . . . . . . . . . . . 179A.2 Proof that any partial trace of ÏÂ¯CAn is zero . . . . . . . . . . . . . . . 180ixB Derivation of Equations of Motion hierarchies . . . . . . . . . . . . 182B.1 Equation of Motion for N -Partite system . . . . . . . . . . . . . . . 182B.2 Equation of Motion for N -Qubit system . . . . . . . . . . . . . . . . 189C Matrix Propagator for 2-Spin System . . . . . . . . . . . . . . . . . 191D Numerical Calculation of Single Spin Evolution in a Time Depen-dent Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193D.1 Treatment of Stochastic Forces . . . . . . . . . . . . . . . . . . . . . 194E Details of precessional decoherence calculations: . . . . . . . . . . 195E.1 Calculating the full reduced density matrix for the central spin. . . . 196E.2 Next to leading order contributions to ãÏx(t)ã . . . . . . . . . . . . . 198E.3 Calculating correlators between bath and central spins. . . . . . . . 201E.3.1 Correlators Involving Transverse Components of the Bath Spins201E.3.2 Correlators Involving Longditudional Components of the BathSpins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209E.3.3 Calculations of Correlators When There is a Small Bias onthe Central Qubit . . . . . . . . . . . . . . . . . . . . . . . . 210F Some Steepest Descents Integrals . . . . . . . . . . . . . . . . . . . . 216F.1 Gaussian Average of a Green function . . . . . . . . . . . . . . . . . 216F.2 Loretzian Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218xList of Figures1.1 The potential energy for an idealised effective qubit as a function ofQ. There are two minima at QÂ±. If we assume the two potential wellsare deep, we can expand the potential around these minima and findan aproximate lowest energy states with energies Â± in each of theminima, these states act as our qubit states. . . . . . . . . . . . . . . 51.2 The field on the jâth bath spin in the precessional decoherence regime.Shown in blue we have the field acting on the bath spin when thecentral spin is up Î³+j or down Î³âj . The angle Î²j is shaded in red. . . 181.3 The density of states of the possible biases on the central qubit in theprecessional decoherence model. The shaded (blue) areas shows thedensity of biases in the case of a bath with a Gaussian distributionof biases around Ï0. The thin (black) lines show the position of deltafunction contributions when all bath spins have the same coupling tothe central spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 The Josephson junction circuit component. (a) The Josephson junctionis made up of an insulating layer (black) separating two supercon-ducting leads. (b) The schematic circuit element. (c) An equivalentcircuit containing a capacitor in parallel with the non linear voltagecurrent relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 A simple rf squid, (a) the squid consists of a superconducting ringwith an insulating junction, (b) the effective circuit diagram consistsof a Josephson junction and inductance as the loop can enclose flux. 241.6 The potential energy for an idealised flux qubit as a function of Ïthere are two minima at ÏÂ±. If we assume the two potential wells aredeep, we can expand the potential around these minima and find anaproximate lowest energy states with energies Â± in each of the minima. 241.7 The three Josephson junction qubit proposed by Mooij et al.. (a) Thedevice consists of a super conducting ring brocken into three seperateislands by the junctions 1,2, and 3, junction 3 has a larger area.(b)the equvilant circuit diagram for the qubit including its capacitivecoupling to external enviromental voltages VA and VB . . . . . . . . 261.8 The potential energy U(Ï+, Ïâ) from equation (1.62) with Ïe = pi andÎ± = 0.75 the two minima of the two wells are marked with bullets â¢.States in these two wells define the two qubit states. . . . . . . . . . 27xi1.9 The effective circuit diagram of a transmon circuit. . . . . . . . . . . 281.10 The three lowest energy levels of the Hamiltonian (1.65). Left thethree lowest energy levels as a function of the parameter EJ/EC andthe gate charge ng. Right the three lowest energy levels as a functionof ng with EJ = 10EC . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 The different partitions of a system containing n = 4 distinguishablecells, with each cell denoted by a black dot. The total number ofpartitions is B4 = 15, where Bn is the Bell number. . . . . . . . . . 362.2 Diagrammatic representation of some of the terms in the 4-cell densitymatrix. In (a) we show the term ÏÂ¯134ÏÂ¯2 appearing in equation (2.17);in (b) we show the term ÏÂ¯C134ÏÂ¯2, also appearing in equation (2.17);and in (c) we show the term ÏÂ¯CC134 ÏÂ¯2, appearing in equation (2.19). . 392.3 Diagrammatic representation of the expansion of Ï1234 into densitymatrices for the four sub-systems, as expressed in equation (2.13) . . 402.4 Diagrammatic representation of the expansion of the 4-cell densitymatrix into the B4 = 15 different cumulant density matrices for thesub-systems, given in equation (2.19) . . . . . . . . . . . . . . . . . . 402.5 A representation of the sets used in equation (2.23). The set An is asubset of the whole system S, and contains n members. The set Cm,which contains m members, is a subset of An. . . . . . . . . . . . . . 414.1 An illustration of the terms in the sum in equation (4.5). The set A,a subset of the total system S, is shown in blue in (a), along with fourother sets 1, 2, 3, 4 distinct from A. Then in (b) in green we show thefour different sets that can be made from the union of A and one ofthe other sets. Each dotted line represents a possible term in the sum(4.5) due to an interaction potential. We have omitted the subscriptson the set variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Graphical representation of a term in the expansion of G(t) as anexponential power series (cf. equation (4.39)). In this term all theentries in the matrix M appear (these entries are given in equation(4.28)). The correlators ãÏ 1ã, ãÏ 2ã, and ãÏ 1 â Ï 2ã are shown as redvertices, the interaction matrices U1,p, U2,p, Up,1, and Up,2 are shownas directed lines, and the rotation matrices L1, L2, and Lp are shownas undirected lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62xii5.1 The various terms appearing in equation (5.3) when there are threecentral qubits and three bath spins: Right, an illustration of a pos-sible choices of the set A in red, which contains some central qubits(represented by green dots) and some bath qubits (represented byblue dots). The left hand figure shows the terms on the right hadside of equation (5.3). Terms are in the same order in the equation asin the figure. In each term, a correlator between qubits in the shadedset, the local field acting on circled qubits, and the interaction u (V )acting between qubits linked by green (dark red) dashed lines enterinto the corresponding term in equation (5.3). Note the secound,fourth, and seventh terms on the right hand side of equation (5.3)are each represented by two diagrams in the figure, as each of theseterms has a sum over two correlators. . . . . . . . . . . . . . . . . . 716.1 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins,their couplings are all Ï0 =120â0 (so12âNÏ0 =14â0), and the initialreduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In allfigures the solid red curve is ãÏxânj Ïzj ã, the dashed green curve isãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã . The left handfigure shows the first eight correlators and the right hand figure showshigher order correlators. . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins,their couplings are all Ï0 =15â0 (so12âNÏ0 = â0), and the initialreduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In allfigures the solid red curve is ãÏxânj Ïzj ã, the dashed green curve isãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã . The left handfigure shows the first eight correlators and the right hand figure showshigher order correlators. . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins,their couplings are all Ï0 =45â0 (so12âNÏ0 = 4â0), and the initialreduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In allfigures the solid red curve is ãÏxânj Ïzj ã, the dashed green curve isãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã . The left handfigure shows the first eight correlators and the right hand figure showshigher order correlators. . . . . . . . . . . . . . . . . . . . . . . . . 85xiii6.4 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins,their couplings are all Ï0 â 0.1581â0 ( 12âNÏ0 =14â0), and theinitial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).In all figures the solid red curve is ãÏxânj Ïzj ã, the dashed green curveis ãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã. . . . . . . . 866.5 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins,their couplings are all Ï0 â 0.6325â0 ( 12âNÏ0 = â0), and theinitial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).In all figures the solid red curve is ãÏxânj Ïzj ã, the dashed green curveis ãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã . . . . . . . . 876.6 Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins,their couplings are all Ï0 â 2.529â0 ( 12âNÏ0 = 4â0), and theinitial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).In all figures the solid red curve is ãÏxânj Ïzj ã, the dashed green curveis ãÏyânj Ïzj ã, and the dotted blue curve is ãÏ zânj Ïzj ã . . . . . . . . 886.7 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =120â0 (so12âNÏ0 =14â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). 896.8 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =15â0 (so12âNÏ0 = â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). 906.9 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =45â0 (so12âNÏ0 = 4â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). 916.10 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N =10 bath spins, their couplings are all Ï0 â 0.1581â0 ( 12âNÏ0 =14â0), and the initial reduced density matrix for the system is ÏÂ¯S(0) =12 (1 + Ïx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92xiv6.11 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N =10 bath spins, their couplings are all Ï0 â 0.6325â0 ( 12âNÏ0 =â0), and the initial reduced density matrix for the system is ÏÂ¯S(0) =12 (1 + Ïx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.12 The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N =10 bath spins, their couplings are all Ï0 â 2.529â0 ( 12âNÏ0 =4â0), and the initial reduced density matrix for the system is ÏÂ¯S(0) =12 (1 + Ïx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.13 Plots of some solutions to the diffusing bias model discussed in section6.3.2 calculated using the methods described in appendix D. In bothcases the parameters are chosen so that â0 = 0.1Î and Î¾0 = 0.In both figures the solid red curve is ããÏ z(t)ããÎ¾, the dashed greencurve is ããÏy(t)ããÎ¾, and the dotted blue curve is ããÏx(t)ããÎ¾ computedusing the method described in appendix D.1. In the top figure theinitial configuration of the central spin is ãÏ (0)ã = yË and the backcurve is the â0 solution for ããÏy(t)ããÎ¾ calculated using equation (6.56).In the bottom figure the initial configuration of the central spin isãÏ (0)ã = zË and the back curve is the solution calculated using theintegro-differential equation (6.63). The inset figure on the bottemshows the considerably smaller ããÏx(t)ããÎ¾, and ããÏy(t)ããÎ¾ componentsof the polarisation. The thin black curve in the inset plot showsthe y compenent of the polarisation computed numerically from theequation ddtããÏ zããÎ¾ = â0ããÏ zããÎ¾ and the numerical values of the zcomponent rather than the direct method in the appendix. . . . . . 988.1 Structure of the Fe8 molecule. The crosshatched circles are the Fe3+ions, the hatched circles are the oxygen atoms, and the empty circlesrepresent, in order of decreasing size, nitrogen and carbon atoms.[Reprinted figure with permission, from D Gatteschi, A. Caneschi,L., R. Sessoli, Science, 265, 1054 (1994), [40]. Copyright 1994 by theAmerican Association for the Advancement of Science.]. . . . . . . . 121xv8.2 The energy of the electronic spin vector in the Fe8 molecule. Boththe main plot and the inset plot shaded colour shows the semicassicalenergy of the electronic spin in the Fe8 molecule as a function of itsdirection on the Bloch sphere, when the applied field is Âµ0Hâ¥ = 2.5TyË. The scale shown on the right gives the energy/kB in Kelvins.The black bullets (â¢) mark the positions of the energy minima wherethe qubit states are localised. The dotted black line marks the semi-classical tunneling route. The inset plot is the same as the main plotbut viewed from a different angle. . . . . . . . . . . . . . . . . . . . . 1238.3 The field dependence of the Fe8 qubitâs tunneling amplitude â0(Hâ¥),when the field Hâ¥ is pointing in the yË direction. Data for this plotis obtained from [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.4 The time scale associated with decoherence due to phonons in the Fe8qubit. The field dependence of Ïph as a function of the applied field,when the field Hâ¥ is pointing in the yË direction. Data for this plot isobtained from [73], using equation 8.5 and assuming a temperatureof T = 0.5K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.5 A histogram of the different hyperfine coupling strengths. The blueboxes show the hyperfine coupling strengths for the H nuclei and thered histrogram shows those for 57Fe nuclei (in the case where all Fe3+ions are 57Fe). All data is binned into 0.5 MHz intervals. Data forthis plot is taken from [99] . . . . . . . . . . . . . . . . . . . . . . . 1268.6 A plot of the variable Îº parametrising the bath of 1H nuclei in theFe8. The ploted data is obtained from [99] and it corrisponds to thecase where all of all of the hydrogen atoms in the molecule are 1H. . 1288.7 Plots of the correlators for the case of a single Fe8 molecule coupledto its 1H nuclear spin bath, as decribed in section 8.3. The top graphshows the time dependence of the components of the central spinpolarisation, ãÏ ã, the solid red curve is ãÏxã, the dashed green curveis ãÏyã, and the dotted blue curve is ãÏ zã. The bottom left graph showscomponents of correlators involving two bath spins, i and j, wherespin i is intialy up and j is intialy down. The bottom right graphshows components of the same correlators (when they are non-zero)in the case that spins i and j are intialy aligned. In the main plots onthe bottom, the dotted blue curve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã, the dashedgreen curve is ãÏyÏxj Ïxj ã, the solid red curve is ãÏyÏxi Ïxj ã = ãÏyÏyi Ïyj ã,and in the inset the thick solid red curve is ãÏxÏxi Ïyj ã, the dashed redcurve is ãÏxÏyi Ïxj ã, and the thin black curve is ãÏxi Ïxj ã. . . . . . . . . 129xvi8.8 A plot of correlators between the bath spins and the central spin, forthe case of a single Fe8 molecule coupled to its1H nuclear spin bath,when we have averaged over the different choices for sites i and j.The dotted blue curve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã and the dashed greencurve is ãÏyÏxj Ïxj ã. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.9 The initial stages of decoherence in an Fe8 qubit, when we have avera-ged over the different choices for bath spins i and j. The top figureshows components of the central qubit ãÏ ã, the solid red curve is ãÏxã,the dashed green curve is ãÏyã, and the dotted blue curve is ãÏ zã. Thebottom figure is plot of correlators between the pairs of bath spinsand the central spin, the dotted blue curve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ãand the dashed green curve is ãÏyÏxj Ïxj ã. . . . . . . . . . . . . . . . 1329.1 Some plots of pc(t) for cat states in systems where all qubits have thesame field. The solid (blue) line shows the exact expression (9.42) andthe dotted (red) line shows the approximation (9.51), in (a) N = 10,Î¸0 = 0, in (b) N = 10, Î¸0 =pi4 , in (c) N = 10, Î¸0 =38pi, in (d) N = 20,Î¸0 = 0, in (e) N = 20, Î¸0 =pi4 , and in (f) N = 20, Î¸0 =38pi. In allcases we have set Ï = pi8 . Note that the approximate formula worksbest close to the peaks but fails in between in particular the phaseof the oscillating function is not well described by the approximateformula in between the peaks. Not shown is the case where Î¸0 =pi2where we get a pure sinusoidal waveform. . . . . . . . . . . . . . . . 1439.2 Some plots of off diagonal correlators for cat states in system withN = 20 central qubits where all qubits feel a central field. Thesolid (blue) line shows the exact expression (9.42) and the dotted(red) line shows the approximation (9.71). Plot (a) showsâ©âNa Ïxaâªwithout disorder, (b) showsâ©âNa Ïxaâªwhere the central field strengthhas gaussian disorder characterised by the mean Î´Ï2 = 0.05Ï0, (c)showsâ©âNa Ïyaâªwithout disorder and (d) showsâ©âNa Ïyaâªwhere thecentral field strength has gaussian disorder characterised by the meanÎ´Ï2 = 0.05Ï0 . In all cases we have set Ï = 0 and all Î¸a = Î¸0 =pi4 . . . 147xvii9.3 Some plots of pc(t) for cat states in systems when there is a disorderedcentral field strength. The solid line shows the exact expression (9.62)and the dashed line shows the approximations (9.70), in (a)âÎ´Ï2=0.01Ï0,Î¸0 = 0, in (b)âÎ´Ï2= 0.01Ï0, Î¸0 =pi4 , in (c)âÎ´Ï2= 0.01Ï0,Î¸0 =38pi, in (d)âÎ´Ï2= 0.05Ï0, Î¸0 = 0, in (e)âÎ´Ï2= 0.05Ï0,Î¸0 =pi4 , and in (f)âÎ´Ï2= 0.05Ï0, Î¸0 =38pi. In all cases we have setÏ = pi8 and N = 20. The approximate formula works best from timessuch that Î´Ï2t2 1, increasing N makes the approximation betterat longer times as it kills off the smaller oscillations. Not shown isthe case where Î¸0 =pi2 where we get a pure sinusoidal waveform. . . 1489.4 The configuration of the bath discused in section 9.4, each centralsystem qubit {Ï a for i = 1 . . . N} is coupled to its own set of of bathspins Ba = {Ïai for i = 1 . . . |Ba|}. We have illustrated the case wherethere are two central spins.. . . . . . . . . . . . . . . . . . . . . . . . 1499.5 Some plots of pc(t) for cat states in systems with a simple degeneracyblocking spin bath. The solid line shows the exact expression (9.86),the dotted red line shows the long time approximation (9.94), and thedashed black line shows the long time small Î´Î¾ approximation (9.100).In (a) Î´Î¾ = 0.2â0 and N = 1, in (b)Î´Î¾ = 0.2â0,N = 5, and the insetis the same plot magnified at earlier times , in (c) Î´Î¾ = 0.1â0 andN = 1, in (d) Î´Î¾ = 0.1â0,N = 5, and the inset is the same plotmagnified at earler times. In all cases we have set Ï = pi4 . Note thedifference in scale on the time axes between (a-b) and (c-d). We seein general the approximation (9.100) does a much better job when Î´Î¾is small for smaller times. . . . . . . . . . . . . . . . . . . . . . . . . 1549.6 Some plots of pc(t) for cat states in systems with a simple degeneracyblocking spin bath. The solid line shows the exact expression (9.86)and the dotted red line shows the long time small Î´Î¾ approximation(9.100) and the dashed black line shows the approximation to therecurrence peak height (9.101). In (a) Î´Î¾ = 0.1â0 and N = 25,in (b)Î´Î¾ = 0.1â0,N = 50, in (c) Î´Î¾ = 0.1â0 and N = 100, in(d) Î´Î¾ = 0.05â0,N = 25, in (e) Î´Î¾ = 0.05â0,N = 50, and in (f)Î´Î¾ = 0.05â0,N = 100. In all cases we have set Ï =pi4 . We see ingeneral the approximation (9.100) does a much better job when Î´Î¾ issmall for smaller times. . . . . . . . . . . . . . . . . . . . . . . . . . 156xviii9.7 Some plots of off diagonal correlators for cat states in system. Thesolid (blue) line shows the exact expression and the dotted (red) lineshows the approximation (9.106). Plot (a) showsâ©âNa Ïxaâªwith N =25 central qubits at Î´Î¾ = 0.1â0, (b) showsâ©âNa Ïyaâªwith N = 25central qubits at Î´Î¾ = 0.1â0, (c) showsâ©âNa Ïxaâªwith N = 101central qubits at Î´Î¾ = 0.1â0, and (d) showsâ©âNa Ïyaâªwith N = 101central qubits at Î´Î¾ = 0.1â0. In all cases we have set Ï =pi4 . . . . . . 1589.8 Some plots of pS(t) for cat states in systems with a simple degene-racy blocking spin bath. In (a) Î´Î¾ = 0.2â0 and N = 5, in (b)Î´Î¾ =0.2â0,N = 25, and the inset is the same plot magnified at earliertimes , in (c) Î´Î¾ = 0.1â0 and N = 25, in (d) Î´Î¾ = 0.1â0,N = 100. . 1599.9 A plot of gzza1(Ï) precessional decoherence with various values of Îº, asindicated in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . 1639.10 A plot of pS(t) for cat states in systems with a simple degeneracyblocking spin bath with a Lorentzian field distribution. The solid(blue) line shows the exact value and the dotted (red) line shows anapproximation derived as in the appendix F.2. We have Î³ = 0.01â0and N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.1 The different classes of interaction involving A. In (i) we have inte-ractions entirely between cells inside A; in (ii) we have interactionsbetween clls inside A and cells outside; and in (ii) the interactionsare entirely between cells outside A. The interactions are denoted bythe wavy line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183F.1 The contour used in the steepest descents integral. The shaded colourshows the real part of the exponent in equation (F.3). The dashedline is deformed to the filled line along which the real part of theexponent decreases most rapidly. The contribution from the verticalline vanishes as it is moved to positive infinity along the real axis . . 217xixList of TablesE.1 Parameters used in the transition expansion calculation of the redu-ced density matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197xxAcknowledgementsFirst I acknowledge the support and encouragement from my wife, Amanda Par-ker throughout my program. The support and encouragement of my parents, andGrandmother has been vital as well. I would thank my supervisor, Professor PhilipStamp, for his direction, advice, encouragement on this project, as well as his painfulproof-reads of early versions of this document. I have benefited from many discus-sions with the other members of our research group past and present, particularly(in no particular order): Ryan McKenzie, Zhen Zhu, Jordan Wilson, Colby DeLisle,Lara Thompson, Julien Froustey, Leon Ruocco, Dan Carney, AÂ´lvaro GoÂ´mez-LeoÂ´n,and Jean-Sebastien Bernier. Finally Iâd like to thank the support and companyfrom the group of friends Iâve made in Vancouver without whom my stay wouldhave pretty bleak.1Chapter 1IntroductionThis thesis concerns the dynamics and stability of information contained in qubits,a quantum analog of the classical bit, which stores information in computers [78].There is a large effort underway in the design and construction of quantum com-puters containing a large number of qubits. Such a computer would be able tosolve complicated problems significantly faster than a classical computer [34]. Themain difficulty in constructing a quantum computer is that quantum information isfragile, it is lost as qubits become entangled with the outside world (the environ-ment)[24, 25].The key questions addressed in this thesis are: How can we divide the informationstored in a many qubit system? How does is that information lost into the envi-ronment? Where does the information go when it lost? And how does the rate ofinformation lost depend on the size of our system.Before we turn to these questions, we need to discuss some more background mate-rial, which is presented in the remainder of this chapter. First the density matrixdescription of the quantum state, qubits, quantum entanglement, and how one canuse quantum mechanics speed up computations is briefly discussed. In particularwe discuss the link between âinformationâ, entanglement and correlators of qubitcomponents. In section 1.6 previous models for the coupling of a âcentral systemâof qubits to an environment are discussed. In section 1.8 two different qubit designsare reviewed, paying particular attention to their coupling to the environment whichwe hope to model. Following that we outline the remaining content of the thesis.1.1 The Quantum StateIn quantum mechanics the notion of a âstateâ is richer than that in classical me-chanics. This is because of the superposition principle, which means that any linearcombination of basis states can define a state. Consider a prototypical example of ad-level quantum system, a single particle which may hop between d âsitesâ labeledj = 1, 2, . . . , d. Classically, to specify the state of the particle one would simply haveto give its position (one integer). While quantum mechanically, the wave function|Ïã which defines the state of the particle, can in general be in any superposition in2the d-dimensional Hilbert space[91]|Ïã =dâj=1cj |jã, (1.1)with the restriction thatâdj=1 |cj |2 = 1, and states related by an arbitrary phaserotation are identical. So in the quantum case one needs to specify d complexnumbers {cj} with one constraint. So one needs 2dâ2 real numbers to fully describethe state.The extra information contained in a quantum state and the fact that it can beall manipulated âin parallelâ is the basis for the operation of a quantum computer[21]. Currently there is a large amount of research being done to produce such acomputer, which would be able to solve problems which would be intractable on atraditional âclassicalâ computer.The wave function such as (1.1) describes a pure state of a completely isolatedsystem. If we are studying a system which can be entangled with an externalâenvironmentâ (this is strictly true for all experimental systems), then the stateof the system S is defined by its reduced density matrix ÏÂ¯S . The reduced densitymatrix ÏÂ¯S can be any operator on the Hilbert space that satisfies the axioms [91]:1. The density matrix has trace one, trÏÂ¯S = 1.2. The density matrix is Hermitian, ÏÂ¯â S = ÏÂ¯S .3. The density matrix is positive semi definite, all its eigenvalues Î»n â¥ 0.Specifying a d Ã d Hermitian matrix requires specifying d2 real numbers, so thatin general the reduced density matrix of a system can be specified by d2 â 1 realnumbers as its trace is fixed to one. Of course some possible choices for these dreal numbers will be invalid, they will result in a density matrix which has negativeeigenvalues. So there is a region of this d2 â 1 dimensional space which give validdensity operators. We see that the quantum state is much more complicated than aclassical state. We can construct a set of observables which this extra informationspecifies. If we define the sets of Hermitian operators,PËj â¡|jããj| (1.2)rËjk â¡|jããk|+ |kããj| (j 6= k) (1.3)cËjk â¡â i|jããk|+ i|kããj| (j 6= k). (1.4)Together these form a basis for the real Hilbert space of possible d Ã d Hermitianmatrices and then the density matrix takes can be written in terms of these opera-tors,ÏÂ¯S =dâj=1â©PËjâªPËj +12dâj=1jâ1âk=1ãrËjkãrËjk + 12dâj=2jâ1âk=1ãcËjkãcËjk. (1.5)3In the expression (1.5) the density matrix is written in terms of expectations ofa complete set of d2 observables {PËj , rËjk, sËjk}, where only d2 â 1 are independentdue to the trace conditionâdk=1ãPËkã = 1. If we allowed our classical system to bedescribed by a probability distribution with one probably for each site, we would onlyrequire d â 1 real values to specify this probability distribution. This is equivalentto specifying the values of the diagonal values of the density matrix {ãPË ã} onlyand gives us an interpretation to these values. The values of expectations of theoff diagonal operators cËjk and rËjk are intrinsically quantum and are present whenthe system is in a coherent superpositions of the different states. A picture ofthe classical limit is that there is some ânatural basisâ in which coupling to anenvironment means that these off diagonal expectations die down rapidly with time.This process is called decoherence.1.2 QubitsMost practical implementations of a quantum computer will store and operate onquantum information stored in the quantum states of qubits. Qubits are the quantumanalogs of classical bits. Classical bits are hypothetical systems that may be in oneof two states which we will refer to as -1 and 1 (this is slightly different than thetraditional convention quantum information convention of 0 and 1 but it will makethe notation we use in the study of the dynamics simpler). Qubits or quantum twolevel systems on the other hand are in general described by a density matrixÏ =( ã1|Ï|1ã ã1|Ï| â 1ããâ1|Ï|1ã ãâ1|Ï| â 1ã)(1.6)whose diagonal elements give the probability of finding the qubit in either of thestates and whose off diagonal elements describe coherent quantum superposition ofthese states.There are a variety of designs for creating qubits for use in quantum information. Allpractical qubits will be the result of arranging a complicated system in such a waythat only two states are accessible to the system. One way to achieve this is to takea system which naturally has two levels such as an electronic or nuclear spin andisolate it from the environment, for example spins of magnetically trapped ions [15]or electronic spins in defect in a semiconductor [54]. In such a system the two qubitstates are the state where the spin is up |1ã = | âã and down |â1ã = | âã with respectto the quantisation axis. Sometimes in this thesis we will use the spinâ12 languageto refer to qubits which are not necessarily half spins, referring to | â 1ã as âspindownâ and |1ã as âspin upâ. Another approach is to isolate two low lying energylevels of a system so that exciting the system at a specific frequency only causestransitions between these levels. Examples of qubit designs utilising this approachinclude the various different qubit designs based on superconducting circuits [64]and magnetic molecules [100], which we discuss in more detail in section 1.8.4For a single qubit we can build a basis of all possible operators which operate the twodimensional qubit Hilbert space of states using the Pauli matrices and the identity,Ïx = |1ããâ1|+ | â 1ãã1| = rË1â1 (1.7)Ïy = i| â 1ãã1| â i|1ããâ1| = cË1â1 (1.8)Ïz = |1ãã1| â | â 1ããâ1| = PË1 â PËâ1 (1.9)I = |1ãã1|+ | â 1ããâ1| = PË1 + PËâ1. (1.10)(1.11)The single qubit density matrix can be conveniently represented in terms of thisbasis[31, 32, 78, 82]Ï =12(1 + ãÏã Â· Ï). (1.12)This representation is discussed in more detail and generalised to systems withmultiple qubits in chapter 3. The various terms in the above representation forthe density matrix have the following interpretations when the z direction is ourquantisation axis: The ãÏzã information tells us about classical probabilities of beingin states |Â±1ã which are pÂ± = 12(1Â±ãÏzã) and the off-diagonal correlators ãÏxã, ãÏyãtell us about quantum coherent quantum superposition which are intrinsically non-classical.We can also write any possible effective Hamiltonian which govern the dynamics ofa single qubit system in terms of the Pauli operators. Suppose the energy of thestate | Â± 1ã is Â± then the diagonal part of the effective Hamiltonian isHd = +|1ãã1|+ â| â 1ããâ1| = + + â2+(+ â â2)Ïz. (1.13)If we allow tunneling between the two states with an amplitude of â0/2 accompaniedby a phase shift of Ï (this is the most general tunneling element), then the tunnelingHamiltonian is,Ht =12(â0eiÏ| â 1ãã1|+ â0eâiÏ|1ããâ1|)=â02(cosÏÏx + sinÏÏy) . (1.14)The total effective Hamiltonian is then H = Hd + Ht. We see that in the absenceof an energy splitting the ground state of the qubit system is a superposition of |1ãand | â 1ã.The qubit levels could be either two levels of a ânaturalâ two level system (e.g.the spin of spinâ12 particle) or two low lying energy levels a more complicatedsystem. One important example this second type of qubit, which we will call aneffective qubit, is a continuous system where the co-ordinate Q has a double welltype potential like that shown in figure 1.1. To see how this pans out in practice wehave to consider an actual system, we will do this in section 1.8.5Figure 1.1: The potential energy for an idealised effective qubit as a function of Q.There are two minima at QÂ±. If we assume the two potential wells aredeep, we can expand the potential around these minima and find anaproximate lowest energy states with energies Â± in each of the minima,these states act as our qubit states.1.3 EntanglementThere is another intrinsically quantum phenomena that can occur when we havemore than one quantum systems interacting, entanglement. Here we only give arudimentary introduction and give a short explanation of what entanglement is,how it is used in quantum computing, and how it can be quantified and measured.Entanglement in qubit systems is discussed in more details in chapter 3.Quantum entanglement occurs when systems made up of multiple parts (forexample particles) are in a superposition of states, for which each of the parts ofthe system is different. For example consider two qubits in the state|Ïeã = 1â2|+ 1 + 1ã+ 1â2| â 1â 1ã, (1.15)which is a superposition of two states for which both the qubits are different. Thismeans the properties of qubit one and two are intrinsically linked. It has been shownthat this kind of entanglement is vital for the fast operation of a quantum computer[53]. The density matrix representing the entangled state with wave function, |Ïeãin equation 1.15 is,Ï12 =12(|11ãã11|+ | â 1â 1ããâ1â 1|+ |11ããâ1â 1|+ | â 1â 1ãã11|) . (1.16)6From equation (1.16) which we see that the entangled qubits âshareâ off diagonalelements in the density matrix. Suppose we traced the second qubit out, then thereduced density matrix, which encodes the information stored in the first qubit aloneis,ÏÂ¯1 =12(|1ãã1|+ | â 1ããâ1|) = 12I. (1.17)The reduced state of the first qubit is completely described by a classical probabilitydistribution, with the | Â± 1ã states having equal probability. So the expectation ofany component of the first qubitâs polarisation ãÏ1ã is zero. The same applies tothe second qubit. However if both qubits are measured, one will find that they arealways aligned with one another along the z axis and a measurement of one of thequbits with respect to this axis, will completely predict the outcome of the measu-rement of the other. In this way the two spins are entangled. These kind of states,particularly in the case where the two qubits are spins that are spatially separated,were at the heart of early controversies about quantum mechanics [30].We will review some of the formal theory of quantifying entanglement in the nextsection in more detail. But for now we consider the qualitative features of an arbi-trary many qubit quantum state consisting of N , qubits. Define s = (s1, s2, . . . , sN ),where si = Â±1, an arbitrary N qubit states wave function can be written|Î¨N ã =âsas|sã. (1.18)Where there are 4N â 1 free real parameters (the â1 accounts for the fact the wavefunction must be normalised) in the coefficients {as}. It is clear that all statesfor which more than one of the {as} coefficients are non-zero have entanglement,and moreover states where more than two of these coefficients are non-zero haveentanglement involving at least three qubits. Continuing this line of argument,one can see that generic N qubit states are highly entangled. So that much ofinformation contained in the wave functions specifies these entanglements. In thenext section we discuss ways to quantify and measure this entanglement.1.4 Quantifying and Detecting QuantumEntanglementIt is desirable to quantify the âamountâ of entanglement in different states. In asystem consisting of multiple subsystems there are in general many different typesof entanglement possible. There is a large amount of literature on quantifying en-tanglement, see the reviews [3, 43, 47, 113] for an introduction. Here we discusssome of the simpler formal definitions of entanglement, measures of entanglement,and touch on some of the complexity of the problem in the multipartite case.71.4.1 Bipartite EntanglementWe start by considering a pure state, of a system which consists of two subsystemsA and B. Subsystems A and B are said to be entangled if the total density matrixÏAB cannot be written as a product state [47]. That is, no state vectors |ÏAã and|ÏBã in the Hilbert spaces for the subsystems exist such thatÏAB = |ÏAããÏA||ÏBããÏB|. (1.19)A natural measure of how entangled the subsystem A is with the subsystem B isthe von Neumann entropy S(ÏAB) defined by,S(ÏAB) = âtrÏÂ¯A log ÏÂ¯A = âtrÏÂ¯B log ÏÂ¯B. (1.20)Where ÏÂ¯A and ÏÂ¯B are the reduced density matrices on A and B respectively, definedby tracing out the complementary set,ÏÂ¯A =trBÏAB (1.21)ÏÂ¯B =trAÏAB. (1.22)It can be shown that von Neumann entropy can be interpreted in terms of the in-formation contained in the reduced density matrix. The von Neumann of a purequantum state is equal to the number qubits necessary to transmit the quantuminformation contained in the reduced density matrix ÏÂ¯A (or equivalently ÏÂ¯B) [93].When A and B are fully entangled ÏÂ¯A contains no information and S(ÏAB) = 0.This is the case for the qubit example in section 1.3.Note that, while the von Neumann entropy has a nice information theoretic inter-pretation, there are many other quantities[111] which behave monotonically underlocal transformations and can be used to characterise entanglement. Such quantitiesare called entanglement monotones. One such monotone, which is in general easierto calculate than the von Neumann entropy, is the linear entropy SL(ÏAB),SL(ÏAB) â¡ 1â trÏÂ¯2A. (1.23)When the linear entropy is zero, the reduced density matrices ÏÂ¯A and ÏÂ¯B representpure states and the sub systems are not entangled. We note for future reference thatthe quantity tr(ÏÂ¯2A), which is one when ÏÂ¯A represents a pure state and less than oneotherwise, is called the purity of ÏÂ¯A.Now consider the case where ÏÂ¯AB represents a mixed state of the bipartite systemsystem AB. We say that a density matrix represents a genuinely entangled, stateif it cannot be written as an ensemble average of Ns separable states |ÏiAã|ÏiBã(i = 1, . . . , Ns), with probabilities {pi}. That is if ÏÂ¯AB cannot be written,ÏÂ¯AB =Nsâi=1pi|ÏiAããÏiA||ÏiBããÏiB|, (1.24)8for any ensemble {pi, |ÏiAã|ÏiBã}, then the subsystems A and B are genuinely en-tangled. This definition of entanglement distinguishes between, states which areensembles of classical states in which A and B are correlated and states which havegenuine quantum entanglement. The appropriate generalisation of the von Neumannentropy to mixed states is the entropy of formation SF (ÏÂ¯A),SF (ÏÂ¯A) = inf{pi,|ÏiAã|ÏiBã}âipiS(|ÏiAããÏiA||ÏiBããÏiB|). (1.25)Here inf{pi,|ÏiAã|ÏiBã} denotes the infimum (or equivalently the greatest lower bound)over all possible decompositions of the density matrix which are of the form (1.24).This makes the entropy of formation difficult to calculate in general. There aresome cases where simpler formulae may be used to calculate the entanglement offormation, one such case is where A and B both consist of single qubits [119]. Theentanglement of formation has the same interpretation in terms of information asthe von Neumann entropy does for pure states.Detecting Bipartite EntanglementEntanglement can be detected by measuring quantities called entanglement wit-nesses. For example consider a system consisting of two qubits Ï1 and Ï2. TheoperatorWË (aË, aËâ², bË, bËâ²) defined by [104] ,WË (aË, aËâ², bË, bËâ²) â¡14[2I â (aË Â· Ï1)(bË Â· Ï2)â (aË Â· Ï1)(bËâ² Â· Ï2)+ (aËâ² Â· Ï1)(bËâ² Â· Ï2)â (aËâ² Â· Ï1)(bË Â· Ï2)], (1.26)for a given set of unit vectors aË, aËâ², bË, and bËâ². Measuring a value ofâ©WË (aË, aËâ², bË, bËâ²)âª<0 indicates a violation of the Bell inequalities [6], in the form derived by Clauser et.al.[16]. So WË (aË, aËâ², bË, bËâ²) is an entanglement witness asâ©WË (aË, aËâ², bË, bËâ²)âª< 0 indi-cates the two qubits are entangled. Note thatâ©WË (aË, aËâ², bË, bËâ²)âª< 0 for a given setof aË, aËâ², bË, and bËâ² is a sufficient condition for entanglement, but not a necessary con-dition. For example with the parameters [78] aË = zË, bË = â ( xË+zË2 ) , aËâ² = xË, bËâ² = zËâxË2 ,for the entangled state |Ïã = 1â2(|11ã+ | â 1â 1ã) one finds,â©WË(zË, xË,â(xË+ zË2),zË â xË2)âª= 12(1 +â2)> 0. (1.27)So the witness WË(zË, xË,â ( xË+zË2 ) , zËâxË2 ) fails to detect the entanglement in |Ïã. Ingeneral different witnesses may detect different âtypesâ of entanglement. As entang-lement witnesses are always represented by Hermitian operators in a multi-qubit sy-stem any witness can be constructed from correlators of different spin components.91.4.2 Multipartite EntanglementNow consider a multipartite system S, composed of N subsystems {i}, i = 1, . . . , N .Suppose the (in general mixed) state of S is represented by the density matrix ÏÂ¯S ,then the state is said to be fully separable and therefore not genuinely entangled ifit can be written [27],ÏÂ¯S =Nsâa=1paâiâSÏÂ¯ai . (1.28)That is, if ÏÂ¯S represents a genuinely entangled state, it cannot be written as anensemble average over a number, Ns of states, ÏÂ¯aS =âiâS ÏÂ¯ai (a = 1, . . . , Ns), whichare a product of local density matrices .A fully separable state contains no entanglement between any of its subsystems. Astate can fail to be fully separable in a large number of ways. For instance, a stateof a system containing a large number of subsystems, can fail to be fully separable ifjust two of its subsystems (say subsystems â1â and â2â) are entangled. Such a statewould be partially separable, in the sense that its density matrix could be writtenâNsa=1 paÏÂ¯a12âiâS{1,2} ÏÂ¯ai . In general we can ask about the separability of a state ofthe system S with respect to any partition of S into number of disjoint subsystems{A`}. A density matrix ÏÂ¯S is separable with respect to the partition {A`} if it canbe written as an ensemble average of product states over that partition [27],ÏÂ¯S =Nsâa=1paâ`ÏÂ¯aA` . (1.29)The number BN of partitions containing N âelementary subsystemsâ into whichS, can be divided is called the Bell number and is well studied in the field ofcombinatorics[36]. The Bell number grows super exponentially with the N [36]. Sowe see, that even just considering the separability of a many party density matrix,characterising the entanglement of a multipartite system can be very complicated.A large number of entanglement witnesses have been derived to characterise differenttypes of multipartite entanglement, see [43] for a review. In this thesis we will avoidmost of the complications of characterising the many different types of multipartiteentanglement, by focusing on the study of the dynamics of the correlation functions,which can be used to construct any entanglement witnesses.1.5 Quantum ComputingQuantum algorithms have been designed, which would be able to solve many largeproblems much faster than traditional classical algorithms [78]. These algorithmswould have to be run on a quantum computer. Quantum computing has the poten-tial to revolutionist the field of computation allowing the fast solutions to problems,10which would be unsolvable with a classical computer.We saw in section 1.3, that most of the information in many qubit systems,is stored in the entanglements between the various qubits. There is good reasonto believe that the information stored in the entanglements is vital for quantumcomputing. This because of quantum parallelism [21], the ability for a quantumcomputer to be able perform a set of operations on a state which is the superpo-sition of multiple input states and produce the desired output for all the inputs atonce [82] (i.e. repeat the calculation multiple times in parallel).In most designs for a quantum computer the information is stored and the cal-culations are preformed using a set of qubits. This type of computer would consistof a set of qubits, an apparatus that could manipulate these qubits, and preformprojective measurements on the qubits. DiVincenzo described a list of requirementsfor the operation of such a computer [25]. The relevant criteria for the work in thisthesis are that the information stored on the set of qubits survives long enough sothat the computation can take place.In later chapters of this thesis we will consider the dynamics systems consistingmultiple qubits, and try and understand how they loss their information to theenvironment. In order to prepare for this work, in the next section we will discussprevious models for qubits coupled to an environment.1.6 Models of the Quantum EnvironmentHere we review several models commonly used to describe the dynamics of a âcen-tralâ quantum system coupled to an âenvironmentâ. As with the rest of this thesiswe are interested in how the central system losses information to the environmentand the problem of decoherence. In particular we consider the case where the centralsystem consists of a set of qubits.We will discuss two models where the full quantum dynamics of the environmentare modeled as a âbathâ containing a large number of environmental modes. Weconsider the cases where the environment consists of a bath of oscillator modes insection 1.7.1 and a bath of qubit modes in section 1.7.2.1.7 The Environment as a Quantum SystemAs the environment is a system in its own right, the most complete model of a centralsystem coupled to an environment will treat both the system and the environmenton an equal footing, using an effective Hamiltonian which couples the system modesand environment.11In general the low energy effective Hamiltonian describing the coupled systemand environment, is obtained by âintegrating outâ dynamics which occur at higherenergies. This results in a description of the system in terms of a number of ef-fective variables, which are important at the energy scales in which the system isprobed. For a central system of NS qubits {Ï a}, a â {1, 2, . . . NS}, in many casesthis procedure leads to models where the low energy dynamics of the environmentis modeled by âbathsâ containing two different types of modes, (i) delocalised mo-des such phonons which may be represented as oscillators[13, 65] and (ii) localisedmodes such as spin and charge defects which may be represented as qubits or spins[83, 86, 109, 122]. In this section we review some of the work that has been done onthese models. First we discuss the oscillator bath and how it can be treated usingthe influence functional method developed by Feynman and Vernon [33]. Then wediscuss the spin bath and its effect on both âemergentâ two level systems and spinhalf systems.1.7.1 Oscillator bath modelsThe Hamiltonian for a set of qubits coupled to a âbathâ consisting of a large number,No of harmonic oscillators, with coordinates {Qi}, and momenta {Pi} (for i =1, . . . Np) is,H =12NSâa=1ha Â· Ï a + 12NSâa=1NSâb 6=auabÂµÎ½ÏÂµa ÏÎ½b +NSâa=1Noâi=1ÎaiÂµ ÏÂµaQi +Noâi=1(P 2i2mi+miÏ2iQ2i2).(1.30)The oscillators are coupled to the central qubits with the constants ÎÂµia, the iâthoscillator has natural frequency Ïi (in the absence of the influence from qubits),mass mi and we have allowed the central qubits to have a pairwise coupling uabÂµÎ½ .The problem of a single qubit coupled to an oscillator bath has been studiedextensively[13, 65, 114], using the influence function method originally developedby Feynman and Vernon [33], which is described below. More general environmentsfor example a spin bath may be mapped to the problem of a bath of oscillatorscoupled to a central spin[86], however such mappings are generally only valid whenthe coupling to the environmental modes are weak[86]. If the oscillators are modesof a delocalised field (eg the phonon field in a crystal hosting the qubit) this kind ofcoupling weak emerges naturally.Influence FunctionalThe influence functional method is based on the path integral expression for thedensity matrix Ï(q,Q; qâ²,Qâ²; t), as a function of the general system co-ordinatesq,qâ² and bath co-ordinates Q,Qâ². At time t the density matrix is related to the12initial density matrix by [33],Ï(q,Q; qâ²,Qâ²; t) =â«dQËâ«dQËâ²â«dqËâ«dqËâ²U(q,Q, t; qË, QË, 0)Ï(qË, QË; qËâ², QËâ²; 0)Uâ(qâ²,Qâ², t; qËâ², QËâ², 0).(1.31)U(q,Q, t; qË, QË, 0) is a propagator matrix element, which has the following pathintegral representation. Each path accumulates a phase according to the actionS[q,Q] of that path, then these paths are summed over in a path integral,U(q,Q, t; qË, QË, 0) =â« x(t)=qx(0)=qËDxâ« X(t)=QX(0)=QËDXeâiS[x,X]/~. (1.32)Assuming that the initial density matrix is separable, so thatÏ(q,Q; qâ²,Qâ²; 0) = ÏÂ¯S(q; qâ²; 0)ÏÂ¯B(Q; Qâ²; 0) (1.33)and that the action separates S[q,Q] = S0[q] +SB[q,Q], then one has the followingexpression for the reduced density matrix of the system,ÏÂ¯S(q,qâ²; t) =â«dqËâ«dqËâ²â« x(t)=qx(0)=qËDxâ« xâ²(t)=qâ²xâ²(0)=qËâ²Dxâ²eâ i~ (S0[x]âS0[xâ²])ÏÂ¯S(qË, qË)F [x,xâ²].(1.34)Equation (1.34) can be interpreted as follows. The reduced density matrices propa-gation consists of two sums over paths, one evolving forward in time and the otherevolving backward in time. In the absence of the environment each of these pathsaccumulates phase independently according to the action S0[q] which depends onthe path of the system alone. However in the presence of an environment these twopaths are correlated by the influence functional F [q,qâ²], which is defined by,F [x,xâ²] =â«dQâ«dQËâ«dQËâ²â« X(t)=QX(0)=QËDXâ« Xâ²(t)=QXâ²(0)=QËâ²DXâ² (1.35)eâi~ (SB[x,X]âSB[xâ²,Xâ²])ÏÂ¯B(QË, QËâ²; 0),So the influence functional encodes the influence of the environmental co-ordinateson these different paths. So far we have said nothing about the form of the environ-ment (or the system), next we describe the exact expressions derived by Feynmanand Vernon[33] for the case where the environment is one or many harmonic oscil-lators linearly coupled to a general central system.Influence Functional for a Single Harmonic OscillatorSuppose there is just one environmental coordinate, Q1, which has the action of aharmonic oscillator with mass m1 and natural frequency Ï1 and is linearly coupled13to the central system co-ordinates with a coupling C1. That is the action for thebath isSB[q, Q] =â« t0ds[m12QË21 âm1Ï22Q21 âC1 Â· qQ12]. (1.36)If the initial state for Q1 is thermal, then the influence functional F1[q,qâ²] is[33]F1[q,qâ²]= exp{â1~â« t0dtâ²â« tâ²0dtâ²â²[q(tâ²)â qâ²(tâ²)] Â· (Re[Î³1(tâ² â tâ²â²)]Â·[q(tâ²â²)â qâ²(tâ²â²)]â iIm[Î³1(tâ² â tâ²â²)]Â·[q(tâ²â²) + qâ²(tâ²â²)])}(1.37)withÎ³ab1 (tâ² â tâ²â²) = Ca1Cb12m1Ï1[eâiÏ1(tâ²âtâ²â²) +2 cosÏ1(tâ² â tâ²â²)eÎ²~Ï1 â 1]. (1.38)The influence functional is of form F1[q,qâ²] = eiÎ¦[q,qâ²]âÎ¨[q,qâ²]. The imaginary partof the exponent, Î¦[q,qâ²] has the form of a non-local (in time) addition to the actionof the central system. While the real part of the exponent, Î¨[q,qâ²] decreases thecontributions from paths where the density matrix is off diagonal for large periodsof time.Multiple Harmonic OscillatorsSuppose now the bath consists of No (non-interacting) harmonic oscillators with co-ordinates {Qi}, frequencies {Ïi}, masses {mi}, and linear couplings to the centralsystem Ci. So the bath action is,SB[q, {Qi}] =âiâ« t0ds[mi2QËi2 â miÏ2i2Q2i âCi Â· qQi2]. (1.39)Then the influence functional is a product of single harmonic oscillator influencefunctionals [33],F [q,qâ²] =âiFi[q,qâ²]. (1.40)When the initial state of the bath is thermal Fi[q, qâ²] is as in equation (1.37), withÎ³1 â Î³i. Where Î³i is defined by equation (1.38) but with the substitutions Ï â Ïi,m1 â mi and C1 â Ci. The full influence functional can then be convenientlydescribed in terms of the spectral function J(Ï), defined by,Jab(Ï) =pi2Nâi=1Cai CbimiÏiÎ´(Ï â Ïi). (1.41)14The full influence functional is thenF [q,qâ²] = exp{â 1pi~â« t0dtâ²â« tâ²0dtâ²â²[q(tâ²)â qâ²(tâ²)] Â· (L2(tâ² â tâ²â²)Â·[q(tâ²â²)â qâ²(tâ²â²)]â iL1(tâ² â tâ²â²)[q(tâ²â²) + qâ²(tâ²â²)])}(1.42)with,L1(t) =â« â0dÏ J(Ï) sinÏt (1.43)L2(t) =â« â0dÏ J(Ï) cosÏtcothÎ²~Ï2. (1.44)So we see that as far as the dynamics of the central system are concerned, thestructure of the oscillator environment can be described by a single function, thespectral density. The spectral density for a small number of oscillator modes is asum over a small number of delta function contributions. A continuous spectraldensity can be obtained when there are a large number No of oscillators, with aneffectively continuous frequency spectrum. In the limit No â â, for J to remainfinite, the coupling constants Ci must be of order O(Nâ12o ).We now briefly discus some examples of spectral densities for different physicalenvironments. The review article by Leggatt et. al. [64] and the book by Weiss [114]as well as the references therein, give a large number of examples . If the oscilla-tors are acoustic phonon modes in a three dimensional crystalline solid, then at lowfrequencies J(Ï) â¼ Ï3 or Ï5 depending on the crystal symmetry. When the bathconsists of conduction electron hole typed excitations, it may be mapped to a modelwhere the bath may be treated as an oscillator bath with a spectral density J(Ï) â¼ Ïfor small frequencies. Optical phonons can give a give a spectral density which iszero up to a specific gap frequency, then a continuous function at higher frequencies.Influence Functional for the Spin-Boson ProblemThe influence functional approach described above can be adapted very easily totreat a qubit coupled to an oscillator bath with a Hamiltonian of the formH =â02Ïx +Î¾02Ï z +Nâi=1Caiq0ÏzQi +Nâi=1(P 2i2mi+miÏ2iQ2i2). (1.45)This Hamiltonian is equivalent to the single qubit version of the spin-boson Hamil-tonian (1.30), with the restriction that the interaction with the oscillator bath isin the spins z component, ÎÂµi = zËÂµCiq0 (the field ha appearing in the Hamiltonian15(1.30), is of the form â0xË+ Î¾0zË in equation (1.45), any field can be put in this formso long as the interaction term only depends on Ï z). The co-ordinate of the centralsystem is then a discrete variable q(t) = q0sz(t), where sz(t) = Â±1. The paths whichare summed over in the evolution of the central systems are specified by the timeswhen the central spin flips. The influence functional is exactly as in equation (1.42)and can then be evaluated for each path (see [65, 114], and the references therein).When there is a traverse coupling, the problem requires some more work.Leggett et. al. describe the dynamics of the spin boson model with the Ha-miltonian (1.45) and a spectral density that is is a power law at low frequencies,cut off exponentially at frequencies much higher than a cut off frequency Ïc. Thatis J(Ï) = Î·Ïph(ÏÏph)seâÏ/Ïc . They find that ãÏ z(t)ã undergoes coherent under-damped oscillation for s > 2, there is a crossover temperature for 2 > s > 1 wherecoherent under-damped oscillation gives way to over-damped decay, and a morecomplicated set of possible behaviours for 0 < s â¤ 1.1.7.2 Spin BathA natural low energy model for localised modes interacting with the central qubitsis one where the environmental modes consist of two level systems, ( a âspin bathâ).Supposing the bath consists of a N qubits with operators {Ïi} for i = 1, . . . , N thenan appropriate Hamiltonian describing the central system and environment isH =NSâa=1ha Â·Ï a+NSâa=1NSâb6=auabÂµÎ½ÏÂµa ÏÎ½b +NSâa=1Nâi=1V aiÂµÎ±ÏÂµa ÏÎ±i +Nâi=1bi Â·Ïi+Nâ1âi=1Nâj=i+1vijÎ±Î²ÏÎ±i ÏÎ²j .(1.46)This Hamiltonian is the most general Hamiltonian including local fields on the cen-tral {ha}, bath spins {bi}, pairwise interaction potentials among the different cen-tral spins {uabÂµÎ½}, bath spins {vijÎ±Î²}, and between the bath spins and the central spins{V aiÂµÎ±}.Of particular interest is the case where the coupling to the bath spins is notweak and the system cannot be mapped to an appropriate oscillator bath model.We will study aspects of this model in chapters 4-9.In the following sections we will examine the relevant low energy Hamiltonianpreviously obtained by truncating the full Hamiltonian of a âdouble wellâ type qubit,where the two qubit states are based in two. We will see that in some cases, thisHamiltonian can include multiparty interactions not included in (1.46). We willthen review some of the results previously obtained for the problem of the centralspin coupled to a spin bath (the so called âcentral spinâ problem), we will buildthese results on in the rest of this thesis.16Effective Qubit SystemsConsider the case where the system qubits are effective qubit system of the typediscussed in section 1.2. The effective Hamiltonian of such systems can be derivedusing the instanton method (see [65]). When such a system is coupled to a spin bathone obtains an effective Hamiltonian for a single central qubit and its spin bath isof the form [84, 86â88, 109],Heff =â02Ï+ cosï£«ï£­Ï0 +âjÎ±j Â· Ïjï£¶ï£¸+ h.c. (1.47)+Î¾02Ï z +Ï z2âiÏâi Ïzi +12âiÏâ¥i Ïxi +Nâ1âi=1Nâj=i+1vijÎ±Î²ÏÎ±i ÏÎ²j (1.48)In this Hamiltonian â0 cosÏ0 is the âbareâ amplitude for a central qubit to tunnelbetween its two levels, this may be accompanied by flips of any number of the bathspins with respect to their some axis determined by the Î±j vector for each bathspin. The vectors {Î±i} may have complex components representing the fact thatbath spins may cause fluctuations in the tunneling probability and modify the phaseaccumulated on each tunneling event of the central qubit. The central qubit systemhas an energy difference (bias) of Î¾0 between its two levels. The bath spins alignmentwith what we have defined as the z axis can effect the bias with a coupling constantÏâj , so that the total bias felt by the central spin is Î¾Ë = Î¾0 +âj Ïâj . The bath spinscan flip (with respect to their z axis) without the influence of the central spin withamplitude Ïâ¥j . Finally there can be a small pairwise interaction vijÎ±Î² between thebath spins.Hamiltonians of the form (1.48) have been derived for the specific cases of mag-netic molecule qubits [83, 84, 109] and superconducting flux qubits [87].Prokofâev and Stamp [86] studied this model in detail, they focused on calcu-lating the return probability for the central spin which is initially up. This Returnprobability prâ(t) can be defined asprâ(t) â¡ tr[12(1 + Ïz)UE(t)12(1 + Ïz)U â E(t)], (1.49)where UE(t) is the time evolution operator. Prokofâev and Stamp found the effectof the bath on the return probability in most regimes of the model (1.48). In theregimes they studied they found tracing the bath out in such a problem could resultin the motion of the central spin being averaged in a combination of four differentways. So the dynamics of the return probability are of the formprâ(t) =â«d0PDB(0)â«dXP (X)â«dÏPTop(Ï)â«DÎ¾(t)P [Î¾(t)]prâ(t;X,Ï, 0, Î¾(t)].(1.50)17Here prâ(t;X,Ï, Î¾0, Î¾(t)] is the return probability as a function of the auxiliary va-riables X,Ï, the initial bias 0, and a functional of a time dependent bias Î¾(t). Thefour averages each have an associated probability density function P (X), PTop(Ï),PDB(Î¾0), or functional P [Î¾(t)]. Each of these averages is associated with a distincteffect and set of terms in the Hamiltonian:(i) The static bias average or degeneracy blocking. If the bath spins haveno mechanism by which they can flip their z components, then the Hamiltonianreduces to H = 12â0Ïz +âj ÏâjÏzj and the central qubit feels a static bias Ë0 =Î¾Ë = Î¾0 +âi Ïâi Ïzi . If the bath is initialised so either, the bath is initially in astate state that has an indeterminate bias Î¾Ë (that is [Î¾Ë, Ï(0)] 6= 0) or measurementsare performed on an ensemble of systems with different biases (e.g. the physicallyimportant case where the bath initially is in a thermal state). Then the resultingmotion of the central spin will be averaged over the distribution of possible biasesPDB(0). This is referred to as degeneracy blocking. We discuss the dynamics in thisregime in more detail in section 6.1.(ii) Topological decoherence. If the main cause of the flipping of the bathspins is the tunneling term of the Hamiltonian and the bias acting on the centralspin is negligible, then we will get topological decoherence. In which case the mainaffect of the bath spins is that they can exchange phase with the central spin everytime it flips. Each time the central spin flips even numbers of bath spin may flipand averaging over the number and possible orientations of these flips (clockwiseor anticlockwise) leads to decoherence. In this thesis we are mainly concerned withsystems, in which this effect is negligible so we refer the reader to the references[83, 86] for more details on this type of decoherence.(iii) Precessional decoherence. If the most important terms in the Hamiltonianwhich influence the dynamics of the bath spin are those with coefficients Ïâj and Ïâ¥j .Then every time the central spin flips, the effective field Î³j acting on the jâth bathspin, jumps between two distinct values, Î³Â±j (illustrated in figure 1.2). This causesprecessional decoherence, where the bath spin states spread out overtime as theyare processing under the influence of different fields depending on what the centralqubit does.We can define an angle Î²j for each of the bath spins cos 2Î²j â¡ âÎ³+j Â·Î³âj /(|Î³+j ||Î³âj |).If Ïâj Ïâ¥j then this field jumps between two almost perpendicular values and theproblem is tractable, even in the non-trivial case where the coupling is strong. Withthe strong coupling limit in mind we plot the density of states of the possible bi-ases on the central spinâj ÏâjÏz in figure 1.3, for a case where the distributionof coupling constants {Ïâj } is sharply distributed around mean value Ï0 (this kindof distribution is natural, for example if the bath spins share a common hyperfine18Figure 1.2: The field on the jâth bath spin in the precessional decoherence regime.Shown in blue we have the field acting on the bath spin when the centralspin is up Î³+j or down Î³âj . The angle Î²j is shaded in red.coupling with the central spin). We see that the states are split into different pola-risation groups, each with a different value for the bath z polarisation MË =âj Ïzj .When this coupling is strong transitions that conserve the total interaction energiesare resonant with respect to the interacting Hamiltonian and are therefor dominant.These transitions occur between polarisation groups of opposite sign (if Î¾0 is smallcompared to Ï0) and a sum over possible resonant transitions leads to averagingover an auxiliary variable X. We will explain this in more detail in section 7.2,where we re-derive and extend the results obtained by Prokofâev and Stamp, usinga slightly different method to what they used.(iv) Average over a dynamic bias. When the coupling vijÎ±Î² between the bathspins is weak compared to Ïâj . The interaction term has two effects: the diagonalpart vijzz causes further splitting of the polarisation groups (by an amount â¼âNv0,where v0 is the mean |vijÎ±Î²|), and the off diagonal parts such as V ijxx cause pairwise flipsamong the bath spins which causes the bath to magnetisation to diffuse incoherently.Therefor an average over the randomly diffusing bias must be computed, we treatan example of this in section 6.3.2.In a more general case some of these averaging effects may be combined, see [86]for more details.19-5 -4 -3 -2 -1 0 1 2 3 4 5Figure 1.3: The density of states of the possible biases on the central qubit in theprecessional decoherence model. The shaded (blue) areas shows thedensity of biases in the case of a bath with a Gaussian distribution ofbiases around Ï0. The thin (black) lines show the position of deltafunction contributions when all bath spins have the same coupling tothe central spin.201.8 Practical QubitsIn this chapter we discuss some practical examples of qubits, which are currentlybeing investigated. We consider superconducting qubits, magnetic molecule basedqubits, and semi-conducting spin qubits as examples.We use superconducting qubits as an example, to show some of the varioustechniques which can be used to make qubits less susceptible to the influence ofthe environment. As such in section 1.9, we will review the basic operation ofsuperconducting qubits, their coupling to the environment and some modern designs,which aim to minimise the effect of the environment.In section 1.10 discuss a particular example of a magnetic molecule qubit, the Fe8qubit. We will discuss this because its spin environment and its exact coupling to thecentral qubit are well understood. This will allow make some concrete predictionsfrom the theory presented later in the thesis.Finally we will give a short discussion of semiconductor spin qubits. Becausethese are another experimentally relevant type of qubit, have a well understoodenvironment, and a well developed theory to describe it.1.9 Superconducting qubitsOne of the most successful approaches to building qubits for use in quantum in-formation systems has been to utilise superconducting electronics[22, 23, 115, 116].First the basic superconducting circuit element, the Jospehson junction is introdu-ced, then the coupling of such a system to the environment is discussed, and finallythree different superconducting circuit qubits are discussed: A SQUID flux qubit, aTransmon qubit, and a three Josephson junction squid qubit.1.9.1 The Josephson Junction Circuit ElementThe basic building block of most superconducting qubits are Josephson junctions[52] which are illustrated in figure 1.4 and consist of a layer of insulator sandwichedbetween two superconducting leads. The low energy dynamics of the circuit can bedescribed in terms of the phase difference Ï of the superconducting order parameteracross the junction [52] and has the Lagrangian[2, 52, 61] (neglecting couplings ofthe pair excitations to normal electrons in the circuit, phonons, charge defects, andother modes which make up the environment)L(Ï, ÏË) =~24EC(ÏËâ QrC)2â EJ(1â cosÏ). (1.51)Which can be described by an equivalent circuit shown in figure 1.4 (b). This circuitof a capacitance in parallel with a nonlinear Josephson element.21Figure 1.4: The Josephson junction circuit component. (a) The Josephson junctionis made up of an insulating layer (black) separating two superconductingleads. (b) The schematic circuit element. (c) An equivalent circuitcontaining a capacitor in parallel with the non linear voltage currentrelationship.The kinetic term in the Lagrangian is the capacitive energy of the capacitor,which depends on the energy EC , which depends on C the capacitance of the junctionand the charge of a Cooper pair (2e),EC =(2e)22C. (1.52)Qr is the residual charge left on the junction when the circuit is constructed. Inthe absence of this charge, the voltage drop V0 across the junction with no currentflowing through it is related to the phase difference by ÏË = 2e~ V0 [52, 105], so thatthe kinetic terms can be written~24ECÏË2 = 12CV20 . (1.53)Which is the familiar expression for the energy stored on a capacitor.The potential term UJ = EJ(1â cosÏ) is the Josephson energy associated withthe current through the junction[52] (see also [105]). The energy scale EJ is relatedto the critical current Ic (which depends on the magnitude of the superconductingorder parameter and the resistance of the normal layer in the junction),EJ =~2eIc. (1.54)We can get a Hamiltonian from the circuit Lagrangian 1.51 using the usualprocedure; the canonical variable to Ï ispÏ â¡âLâÏË=(~2e)2C(ÏËâ QrC)= ~n. (1.55)H =pÏÏËâ L = EC(nâ nr)2 â EJ cosÏ. (1.56)22Here nr = âQr/(2e). n can be interpreted as the net number of Cooper pairs thathave crossed the junction. Now one can find the energy levels of this Hamiltonianby making the canonical substitution pÏ â âi~âÏ and solving for the eigenvaluesof the Hamiltonian operator.If the Josephson junction is wired into a circuit in such a way that that there isa loop which can enclose magnetic flux Î¦, an inductive term UL must be added tothe potential (and therefore the Lagrangian L = T â U),UL =EL2(Ïâ Ïe)2 . (1.57)EL sets the scale for the inductive energy of the circuit, which depends on theenclosed magnetic flux through the circuit due an applied magnetic field Î¦e throughÏe =2e~ Î¦e. EL is set by the inductance, L of the loopEL =Î¦204pi2L(1.58)Î¦0 =h2e. (1.59)Î¦0 defines a quantum of flux and the total flux Î¦ through the loop is the sum of thatdue to the applied field and the induced flux due to the current Î¦ = â2e~ (ÏeâÏ). Acircuit which contains a Josephson junction in a loop that encloses a magnetic fluxhas a HamiltonianH = pÏÏËâ L = EC(nâ nr)2 â EJ cosÏ+ EL2(Ïâ Ïe)2. (1.60)In general to construct a qubit one wants a system of two isolated energy levelsthat can be excited at a specific frequency, without exciting other levels. There areseveral ways to achieve this; the most experimentally relevant to modern applicationsare discussed later in this section, but first we consider the effect of coupling a circuitcontaining a junction to the environment.1.9.2 Coupling to the EnvironmentIn a real system the superconducting circuit described by the Hamiltonian (1.60)will be coupled to the environment. There are several ways which the Hamiltonian(1.60) can be coupled to the environment which are particularly important whendesigning qubits [22]:1. Charge fluctuations in the environment can lead to Qr depending on the stateof the environment resulting in time dependent charge noise.2. Spins in the environment can cause stray magnetic fields in any inductingloops so that the applied flux Ïe may vary with time (flux noise). So Ïe is ingeneral an operator on the environment.233. The critical current Ic (and therefore EJ) may vary due to impurities in thejunction or its coupling to phonons (critical current noise)The different noise sources 1-3 above have the effect of promoting the variablesQr, Ïe, and Ic to operators acting on the environment. Fluctuations in these ânoisevariablesâ are well characterised experimentally (see [44, 60, 71, 76, 81, 110, 125] andthe references therein). It all cases the noise is strongly peaked at low frequencies andscales as some power of inverse frequency and is believed to be caused by fluctuatingtwo level systems, see [81] for a review. In the cases of charge, and critical currentnoises these are thought to be because of localised electrons on defects which mayhop between different sites. This can cause fluctuations in dipole moments in thejunction, leading to a fluctuating junction resistivity and therefore a fluctuatingcritical current. These localised electrons can also fluctuate the charge on a leadand cause charge noise. The microscopic details of the couplings causing chargenoise and critical current noise are not well understood.Experiments have shown that charge noise [76] and flux noise [44] are the keylimiting factors for coherence in the simplest qubit designs so that the more com-plicated designs discussed below are designed around reducing these noise sources.1.9.3 Flux QubitsA simple flux qubit [64] for isolating two energy levels of the Hamiltonian (1.60) isto set up the potential U(Ï) â¡ âEJ cosÏ + EL2 (Ï â Ïe)2 as in into the double wellconfiguration described in section 1.2. An RF-SQUID qubit is an example of this.RF-SQUIDThe simplest way to achieve a potential like that in figure 1.6 is to apply a fluxthrough a superconducting ring with a Jospheson junction (illustrated in figure 1.5(a)). If the flux is a half integer multiple of the flux quantum so that Ïe = pi then thepotential U(Ï) shown in figure 1.6, has the desired form, with two distinct energywells at Ï = ÏÂ±, so that a linear combination of the lowest energy level of each of thewells form the states of the qubits. In this case ignoring the possibility of tunnelingbetween the two wells, each will have a ground states which we can call | Â± 1ã, withenergies Â±. Because the phase can tunnel between the wells with amplitude â0 theHamiltonian of the effective low energy qubit Hamiltonian will beH = 12(+ â â)Ïz + âÏx. (1.61)This type of qubit is known as a Superconducting QUantum Interference Device(SQUID) because it is essentially a miniaturised version the device of the samename which was used to make accurate magnetic field measurements[105]. The firstexperimental demonstration of an rf-SQUID was achieved in the 60âs [50], while thefirst observation of a rf-SQUID in a superposition of flux states was by Friedman et24Figure 1.5: A simple rf squid, (a) the squid consists of a superconducting ring withan insulating junction, (b) the effective circuit diagram consists of aJosephson junction and inductance as the loop can enclose flux.Figure 1.6: The potential energy for an idealised flux qubit as a function of Ï thereare two minima at ÏÂ±. If we assume the two potential wells are deep, wecan expand the potential around these minima and find an aproximatelowest energy states with energies Â± in each of the minima.25al.[37]. Notably rf-SQUID qubits are used in the D-wave quantum annealing device[45, 51]. The main cause of decoherence in rf-SQUID qubits is fluctuations in theexternal flux [44]. There are several types of flux Qubit which are designed to reducethe coupling to the external flux. In the following section we describe one of these,3-junction SQUID qubits.3-Junction SQUID qubitsThe 3-junction SQUID qubit first proposed by Mooij et al. [72, 79] is a flux qubitwith low inductance and therefore is less sensitive to flux noise. The geometry of thisqubit is shown in figure 1.7. Instead of having one Josephson junction in the SQUIDthere are three. Two of these (junctions one and two, say) are identical, while thethird junction has an area larger by a factor of Î±â1 and therefore has a smallercapacitance and smaller Junction energy. In this case the three phase differencesÏ1, Ï2, Ï3 form the canonical position variables in the Lagrangian for the circuit.The parameter Î± characterises the difference between the third junction and theother two so that the Josephson energies satisfy EJ3 = Î±EJ1 = Î±EJ2 â¡ Î±EJ (andlikewise for the capacitance). The inductance is small enough that we can assumethe effect of the inductive energy term is to fix the sum of the phase differences tothe total external flux Ï1 + Ï2 + Ï3 = Ïe. Then the remaining potential term UJ inthe Lagrangian comes from the Josephson energies of the junctions and in terms ofthe variables ÏÂ± = (Ï1 Â± Ï2) /2 it is,UJ(Ï+, Ïâ, Ïe) = âEJ (Î± cos(Ïe â 2Ï+) + 2 cosÏ+ cosÏâ) . (1.62)With the parameters Ïe = pi and Î± = 0.75 the potential (1.62) is shown in figure1.8 and has two potential wells near the origin. The states in these two wells definethe low lying qubit states.The âkineticâ term T in the Lagrangian comes from the electrostatic energy onthe effective capacitors. As well as the capacitances from the Josephson junction thethree different superconducting âislandsâ can be coupled to charges in the environ-ment; if we can treat the island between the junctions 1 and 2 as if it were grounded(we can always define voltage so that it has a zero voltage) then the effective circuitdiagram for the qubit is as in figure 1.7. There are two effective âgateâ capacitancesÎ³AC and Î³BC on the other islands which couple these islands to bias voltages VAand VB which depend on the state of the environment. The total kinetic term canthen be written as a sum of the electrostatic energies on each of the junctions aswell as the work required to charge the two gate capacitances which have chargeQgA and QgB. This givesT =12C{V 21 + V12 + Î±(V1 â V2)2 + Î³AV 2gA + Î³BV 2gB}âQgAVA âQgBVB. (1.63)Using the relation between the junction voltages and phase differences, i.e., Vj =Ï0ÏËj for j = 1, 2, and the relations QgA = Î³ACVgA = Î³AC(VA â V1), and QgB =26Figure 1.7: The three Josephson junction qubit proposed by Mooij et al.. (a) Thedevice consists of a super conducting ring brocken into three seperateislands by the junctions 1,2, and 3, junction 3 has a larger area.(b) theequvilant circuit diagram for the qubit including its capacitive couplingto external enviromental voltages VA and VB27Figure 1.8: The potential energy U(Ï+, Ïâ) from equation (1.62) with Ïe = pi andÎ± = 0.75 the two minima of the two wells are marked with bullets â¢.States in these two wells define the two qubit states.28Figure 1.9: The effective circuit diagram of a transmon circuit.Î³BCVgB = Î³BC(VB â V2) for the gate charges we haveT =C2Ï20(ÏË1 ÏË2)(1 + Î±+ Î³A âÎ±âÎ± 1 + Î±+ Î³B)(ÏË1ÏË2)â Î³AC2V 2A âÎ³BC2V 2B. (1.64)Thus the environmental bias voltages enters only as an unimportant co-ordinate in-dependent contribution to the Lagrangian L = TâU , so that the qubit is insensitiveto the bias voltages caused by charge noise.In the original paper Mooij et al.[72] report the production and experimentalresults of a 3 Josephson junction qubit; unfortunately they were not able to set theflux to the optimum value and their qubit was limited by flux noise. Since thena variant of this design containing four junctions has been tested [102] and it wasfound that the main source factor limiting coherence is charge noise. Even morerecent experiments [120] further develop the four qubit version reducing the chargenoise to the point where photon shot noise from the cavity they use for read out isthe main source of noise.1.9.4 Transmon QubitsThis section describes the operation of the transmon qubit first presented by Kochet al.[58]. The transmon qubit is a type of charge qubit as the two states are distin-guished by the charge on one side of a Josephson junction. Variants of the transmonqubit are used in Googleâs [77] and IBMâs [1] prototype quantum computers. Thetransmon qubit has the effective circuit diagram shown in figure 1.9. It consistsof SQUID ring containing two Josephson junctions shunted with a large bias capa-citance CB connected via a gate capacity with capacitance Cg to an applied gatevoltage source Cg. The inductance of the SQUID ring is low enough so that theflux is fixed to the applied flux and the phases across both Josephson junctions, aresimply related. The effective Hamiltonian for the entire circuit reduces to[58]H = 4EC(nâ ng)2 â EJ cosÏ (1.65)29Figure 1.10: The three lowest energy levels of the Hamiltonian (1.65). Left the threelowest energy levels as a function of the parameter EJ/EC and the gatecharge ng. Right the three lowest energy levels as a function of ng withEJ = 10EC .Here EC depends on the total effective capacitance. One can solve for the quantumenergy levels by putting nâ âiâÏ and then solving for the eigenvalues m(ng, EJ , EC)for m = 0, 1, 2, . . .. On dimensional grounds we can writem(ng, EJ , EC) = ECEm(ng, EJ/EC). (1.66)Koch et al. pointed out that when EJ EL the dependence of the energy levels onthe charge (ng) is exponentially small. Figure 1.10 shows the lowest three energylevels of the Hamiltonian and we see it is indeed the case that the dependence ofthe energies on ng, rapidly decreases as EJ/EC increases. Koch et al. also givea nice physical explanation for the small energy dispersion with ng, which is asfollows. The Hamiltonian (1.65) is identical to that of a pendulum of mass m, withcharge q, and length `, under the influence of gravity and in a constant magneticfield perpendicular to the plane of the pendulum when the following identificationsare made: the canonical momentum ~n is identified with the angular momentumLz = Lz, ~ng with 12qB`2, EC with~28m`2, and the Josephson energy EJ with thegravitational potential mg`. Thus when EJ/EC 1 the gravitational potential isstrong, so one expects the pendulum to spend most of its time near the bottom wherethe potential is approximately harmonic. The magnetic field enters the Hamiltonianas a gauge field a change of gauge can eliminate it locally near Ï = 0. It is onlywhen the pendulum rotates right around that the effect of the magnetic field is feltby the pendulum as the gauge field cannot be globally removed by a gauge change.This is when the Ï variable is tunneling to another potential well at some integermultiple of 2pi away so that the magnetic field enters only in the tunneling elementand its magnitude can be estimated using a WKB type argument leading to a factorâ¼ eâcâEJ/EC which is exponentially suppressed.30When operated with a large EJ/EC we see that the transmon qubitâs energylevels are unaffected by charge fluctuations. However if EJ/EC is too large thetransmon fails to act as a qubit, as the energy gaps âE10 = 1â0 and âE21 = 2â1between the ground state and the first excited state and between the first excitedstate and the second tend to the same value as EJ/EC â â, which provides anupper limit for EJ/EC . Fortunately EJ/EC can be chosen to be big enough tosuppress the effect of charge fluctuations while still having a significant differencebetween âE21 and âE10.An additional advantage of the transmon qubit design is that its coupling toan microwave cavity allows the qubit to be controlled with microwave pulses[10],rather than using leads or gates which might host large numbers of defects whichcan couple to the qubit(this is what gives the transmon qubit its name it acts likea transmission line).The first transmon qubits were constructed and tested soon after the initialproposal [48, 92]. These experiments showed that the effect of charge noise wasgreatly reduced compared to other charge qubits. With charge noise reduced themain cause of decoherence was the interaction of their qubits with the resonantmodes of the cavity. In redesigning the cavity later experiments were able to improvethe coherence time even further, and produce a qubit which is mainly limited byphoton shot noise from the cavity [80, 89, 96].State of the art transmon qubitsinclude the âXâ shaped âXmonâ qubit[4, 70] and âgatemonâ qubits [62, 68] whereit is believed the key cause of decoherence is two level systems in the dialectricmaterials [68, 70].1.10 Spin-based qubitsAnother promising approach to the construction of qubits, is to use spin states insolid state systems. Here we discuss two broad classes of such qubits, spin qubits insemiconducting systems and spin qubits in insulating systems. We introduce thesesystems and discuss how the qubits interact with their environment.1.10.1 Semiconducting Spin QubitsOne promising type of spin qubit, consists of either an electronic spin [75] ornuclear spin [54], of a dopant atom in a semiconductor (e.g. 31P in a Siliconsemiconductor[75]). These qubits are controlled by applying a voltage through agate on the surface of the semiconductor that can effect the spin via the Zeemaneffect [54]. Typically the dopant is spinâ1/2, so that qubit levels are the spin states(although there are more complicated designs where the qubit levels are mixturesof the spin half states and other modes, see for example [63, 74]).Typically the dominant source of environmental noise in such qubit is the nuclearspins inside the semiconductor. The effective Hamiltonian for a semiconducting31qubit spin Ï coupled to the surrounding bath of nuclear spins {Ïj} is[5, 17, 20, 90,121, 122]H = 12â¦zÏz + 12âjÏjÏzj â 14âjajÏ Â· Ïj +âjâi 6=jHij . (1.67)â¦z and Ïj are the Zeeman splittings for the electronic spin and nuclear spins re-spectively, aj is the hyperfine coupling between the nuclear and electronic spin, andHij is the interaction between nuclear spins which includes, dipolar interactions.The theory describing decoherence in these type of qubits is well developed anddifferent to that described in 1.7.2, so we describe this briefly here for contrast.The only term in the Hamiltonian (1.67) which can flip the central spin is theoff diagonal part of the hyperfine interaction 14âj aj(ÏxÏxj + ÏyÏyj). This fact iskey to most sophisticated treatments of this model[5, 17, 121, 122]. For examplethe structure of the Hamiltonian (1.68) allows the problem to be mapped (via acanonical transformation) to a relatively simple effective spin bath problem [122],with an effective Hamiltonian of the formHeff = 12âj{ÏjÏzj+âi 6=j12BijÏ+i Ïâj +âi 6=j12DijÏzi Ïzj}+12Ïzâj{EjÏzj+âi 6=j12AijÏ+i Ïâj}(1.68)For the parameters Aj , Bj , Dj , and Ej see cited references. The effective Hamil-tonian (1.68) does not contain any terms that flip the central spin. So while thedynamics of the bath spin in this model may be complicated, the model is traceableand can be treated with a variety of approximations[20, 67, 90, 123]. The maincause of decoherence comes from the different states of the central qubit causing theinteractions between bath spins to have different strengths [67].1.10.2 Insulating spin qubitsA wide variety of spin qubits in insulating systems have also been studied. Fromrare-earth insulators [59] to crystals of single molecule magnets [9, 14, 41, 100].These systems have an advantage when studying decoherence, in that the cou-pling of the qubit to the environment is relatively easy to understand. This is notthe case for superconducting qubits, where most important environmental couplingcomes from impurities in the superconductor and substrate. We should note thatthe technology required to operate a quantum information system based of singlemolecule magnet or rare earth spin qubits is not as developed as for superconductingqubits. Nevertheless there have been significant advances making magnetic moleculequbits which are protected from ânoiseâ in ways analogous to the super conducting32qubits discussed earlier, see for example [97].In general these qubits these come in regular crystalline arrays, which may havea complicated unit cell including many spins electronic spins, but at low energiesthey reduce to two level systems that may be described by relatively simple Hamil-tonians. The qubits then couple to each other via dipolar and exchange interactions.Decoherence of the qubit causes via their coupling to collective magnon modes cau-sed by these interactions, their coupling to phonons in the lattice, and their couplingto the bath of nuclear spins present in the sample. At low temperatures there aremany situations where the dominant cause of decoherence is the spin bath.1.11 Conclusions and Outline of the Rest of the ThesisWe discussed in section 1.5 the construction of a quantum computer, which wouldbe a major advance of technology. We explained that for such a computer to ope-rate the information stored in a many qubit system needs to be maintained. Insections ?? and 1.4, we saw most of the information stored in many qubits systemsis contained in the multi-partite entanglements between the qubits. But until nowthere has been no good way quantify this mulitpartite entanglement. This leadsto one of the key questions we seek to solve in this thesis, is there a useful way todivide up the information in a multipartite system, particularly when its parts arequbits? Chapters 2 and 3 are devoted to answering this question. In chapter 2 weshow how a density matrix representing a multipartite system may be decomposedin terms of reduced density matrices on its subsystems and correlated parts, whichcontain the entanglement. Then in chapter 3 we apply these results to a system ofqubits and show that the multipartitie entanglement is stored in correlators betweencomponents of these qubits.In sections 1.6-1.10 of this chapter we discussed how real systems of qubits arecoupled to an environment. Important questions one can ask about the effect ofthe environment are: how fast are the various multipartite entanglements lost tothe environment? and where does the information go when it is lost âto the envi-ronmentâ? The remainder of the thesis is focused on these questions. We saw insections 1.9 and 1.10 that, for many practical qubit systems, at low temperaturesthe environment is either a real or effective âspin bathâ, which can be described as alarge number of qubits. So it makes sense to study the dynamics of entanglement inqubit systems which are coupled to a spin bath. To this end, in Chapter 4 we con-sider the dynamics of reduced density matrix, both in a general many body systemand in a system consisting of many qubits and derive a Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY)[11, 12, 56, 57, 124] type hierarchy for the equations ofmotion of reduced density matrices and spin correlators. This heirarchy shows how33the dynamics of different mulitipartite entanglements are linked. Then in chap-ter 5 we study the effect of formally âintegrating outâ an environmental spin bathfrom the hierarchy of equations of motion for a system of qubits. The resultinghierarchy of equations of motion are then used to study simple models of a singlequbit interacting with an enviromental spin bath bath in chapter 6. We see thatfor a simplified model, the information carried by the central system cascades intohigher and higher order correlations between the central system and the bath. Acomplementary approach to studying qubit dynamics in a spin bath is the subjectof chapter 7, where we derive the influence function in the form of what we calla âtransition expansionâ. We then use this method to show that this cascade ofinformation still occurs in a more realistic model. Chapter 8 discusses the resultsfrom chapter 7 applied to a realistic model of an experimental system. Chapter 9then studies the dynamics of highly entangled âcat statesâ containing large numbersof non-interacting qubits and the decay of the different correlations which diagnoseentanglement is studied. Finally we present an overview the results in chapter 10.34Chapter 2Partitioned Density Matricesand Their CorrelationsIn this chapter we consider the structure of the many body density matrix. Weconsider some general quantum system S which has its degrees of freedom dividedinto N sub-systems Ïj , with j = 1, 2, . . . , N . We wish to characterise the behaviourof S in terms of correlations over the N sub-systems. We will show how one maydescribe the properties of any non-relativistic many-body quantum system S interms of a sum over all the BN different possible partitions of subsets of S, offunctions defined for each of these partitions. Here Bn, for a set of n distinguishablesub-systems, is the Bell number; and the functions involve various correlated andreduced density matrices defined for each different partition. We will also writethis sum in terms of a complete set of entanglement correlators for the system.Having done this we will then, in the following chapter, derive a hierarchy of coupledequations of motion for these correlators.In what follows we first define a set of correlated density matrices in terms ofthe full (unreduced) density matrix of the entire system S we are dealing with.To make intuitively clear what these correlated density matrices are, we discuss insome detail the example of a system partitioned into 4 sub-systems. Then we give ageneral expression for the correlated density matrices for some part An of the entiresystem containing n sub-systems; and we discuss one of the key defining propertiesof the entanglement correlated density matrices.2.0.1 Definition of Correlated Density MatricesConsider a system S made up of some number N of distinguishable disjoint subsys-tems (which we will often call âsitesâ, âelementary cellsâ, or just âcellsâ for short).We may then enumerate all possible different ways of partitioning S into groups ofsubsets - this list constitutes a set PS . As an example, in Fig. 2.1 we show thevarious partitions for the case N = 4. We can also enumerate all possible subsets ofS; this list forms another set PS .The two sets PS and PS are not the same. Thus, suppose we have N elementarycells. The set PS of all partitions of S then contains BN members, where BN isthe Bell number (see for example [36]); we will label the different members by pÂµ,with Âµ = 1, 2, ....BN , noting that one of the partitions pÂµ contains only S itself.The number BN grows super-exponentially with N (we have B1 = 1, B2 = 2, B3 =355, B4 = 15, B5 = 52, and already B15 â¼ 1.4 Ã 109). We will not, need to knowanything more about BN .The set PS , on the other hand, simply has as members the different subsets of S;it is usually called the âpower setâ of S. If S has N members, then the total numberof members of PS is just 2N ; these are easily enumerated. We will sometimes labelthe members of PS by AÎ±, where aÎ± = 1, 2, ...2N , for a set S containing N members.Notice that any given partition of S is made up of a specific group of subsets ofS (thus, eg., the partition (12|3|4) of a set S of 4 members - depicted as the 2nd ofthe 15 members of the partitions of this set in Fig. 2.1 - is made up of the subsets(12), (3), and (4) of S). We can write this statement as S = âªAâpÂµA.With these distinctions in mind, we would like in what follows to find an expres-sion for the total density matrix of the system in terms of all the different reduceddensity matrices for the different subsets of S, and of all the different entanglementcorrelated density matrices.We will give a precise definition of these entanglement correlated density matricesbelow. The reduced density matrices are defined in the usual way, i.e., we define thereduced density matrix ÏÂ¯AÎ± for some specific subset AÎ± of S as the partial trace ofthe full density matrix over those other subsystem cells i 6â AÎ±. We shall write thisdefinition asÏÂ¯AÎ± â¡ trSAÎ± ÏS (2.1)where SAÎ± denotes the set containing all cells except those in the subset AÎ±; hereand from now on a bar over a density matrix indicates it is a reduced density matrix.We can then write the full density matrix in the formÏS =âAâSï£«ï£­âj 6âAÏÂ¯jï£¶ï£¸ ÏÂ¯CA (2.2)that is, as the sum over all subsets A of S (including the sets â and S) of a âcor-related partâ ÏÂ¯CA multiplied by the reduced density matrices ÏÂ¯j on those remainingindividual cells not contained in A. The above expression should be read with thefollowing conventions:ÏÂ¯Câ = 1 (2.3)âjââ ÏÂ¯j = 1 (2.4)ÏÂ¯Ci = 0 (2.5)ie., we have that (i) the correlated part of the density matrix ÏÂ¯Câ over a set containingno cells is 1; (ii) the product of the reduced density matrices taken over no cells istaken to be 1; and (iii) the correlated part of the density matrix for a single cell iszero. Consider, for example, some system with a number N > 3 cells; and consider36Figure 2.1: The different partitions of a system containing n = 4 distinguishablecells, with each cell denoted by a black dot. The total number of parti-tions is B4 = 15, where Bn is the Bell number.the terms in the sum (2.2), in the cases where (i) A = â , (ii) A = {1, 2, 3} and (iii)A = S. These terms are then given byÏS =âiâSÏÂ¯i (A = â ) (2.6)ÏS =ï£«ï£­ âi 6â{1,2,3}ÏÂ¯iï£¶ï£¸ ÏÂ¯C123 (A = {1, 2, 3}) (2.7)ÏS = ÏÂ¯CS (A = S) (2.8)respectively.There are 2 properties of the entanglement correlated parts ÏÂ¯CA that make themuseful. First, equation (2.2) is a linear expansion of the full density matrix interms of the ÏÂ¯CA. Second, we will take it as one of the defining conditions for theentanglement correlated density matrices that if we trace any single cell out of ÏÂ¯CAwe get zero; ie., we have for any i â A thattriÏÂ¯CA = 0 (âi â A) (2.9)Now equations (2.2) and (2.9), taken together, define the correlated parts ÏÂ¯CA uni-quely. However one needs to unpack these equations to see what they really mean;and we would also like to have an explicit expression for ÏÂ¯CA. In what follows wefirst see how to understand (2.2) with simple examples; and we then find the desiredexpression for ÏÂ¯CA.2.0.2 A 4-cell ExampleThe kind of thing we are talking about can be simply understood by looking at asystem S composed of 4 sub-systems. In what follows we do this, introducing adiagrammatic representation of the results, and showing how the expansion over en-tanglement correlators can also be related to one over âcumulant density matricesâ.37Expansion Over Entanglement CorrelatorsLet us begin by looking at only 2 sub-systems (what we will call a â2-cellâ system).The total density matrix ÏS for S is thenÏS â¡ Ï12 = ÏÂ¯1ÏÂ¯2 + ÏÂ¯C12 (2.10)where ÏÂ¯1 and ÏÂ¯2 are the reduced density matrices for sub-systems 1 and 2 respecti-vely, and ÏC12 is that part of ÏS in which there are correlations between the twosub-systems. We write ÏS = Ï12 here to indicate the system now is just made up oftwo sub-systems 1 and 2.Notice that (2.10) actually defines what we mean by ÏÂ¯C12, ie., we have defined ÏC12asÏÂ¯C12 = Ï12 â ÏÂ¯1ÏÂ¯2. (2.11)in terms of ÏS , ÏÂ¯1, and ÏÂ¯2. The generalization of (2.10) to a 3-cell system is simple,Ï123 = ÏÂ¯C123 + ÏÂ¯C12ÏÂ¯3 + ÏÂ¯C23ÏÂ¯1 + ÏÂ¯C13ÏÂ¯2 + ÏÂ¯1ÏÂ¯2ÏÂ¯3. (2.12)A system consisting of 4 sub-systems, whose partitions were already shown inFig. 2.1, turns out to be more interesting. Then (2.2) readsÏ1234 = ÏÂ¯1ÏÂ¯2ÏÂ¯3ÏÂ¯4 + ÏÂ¯C12ÏÂ¯3ÏÂ¯4 + ÏÂ¯C13ÏÂ¯2ÏÂ¯4 + ÏÂ¯C14ÏÂ¯2ÏÂ¯3+ ÏÂ¯C23ÏÂ¯1ÏÂ¯4 + ÏÂ¯C24ÏÂ¯1ÏÂ¯3 + ÏÂ¯C34ÏÂ¯1ÏÂ¯2 + ÏÂ¯C123ÏÂ¯4+ ÏÂ¯C234ÏÂ¯1 + ÏÂ¯C134ÏÂ¯2 + ÏÂ¯C124ÏÂ¯3 + ÏC1234. (2.13)Let us first notice how we get the lower reduced density matrices from this. Wecan immediately trace out cell 4, to get ÏÂ¯123; then, because tr4ÏÂ¯C14 = tr4ÏÂ¯C24 = . . . =tr4ÏÂ¯C124 = . . . = tr4ÏÂ¯C1234 = 0, we haveÏÂ¯123 â¡ tr4Ï1234= ÏÂ¯1ÏÂ¯2ÏÂ¯3 + ÏÂ¯3ÏÂ¯C12 + ÏÂ¯2ÏÂ¯C13 + ÏÂ¯1ÏÂ¯C23 + ÏÂ¯C123 (2.14)which is just equation (2.12) (except in the present case {1, 2, 3} is a subset of thesystem so that ÏÂ¯123 is a reduced density matrix). We can then trace out cell 3, aswell, to getÏÂ¯12 â¡ tr{3,4}Ï1234 = ÏÂ¯1ÏÂ¯2 + ÏÂ¯C12. (2.15)which is just equation (2.10).Analogous expressions exist for ÏÂ¯C23 and ÏÂ¯C13; substituting these into expression(2.14) and rearranging we then findÏÂ¯C123 = ÏÂ¯123 â ÏÂ¯12ÏÂ¯3 â ÏÂ¯13ÏÂ¯2 â ÏÂ¯32ÏÂ¯1 + 2ÏÂ¯1ÏÂ¯2ÏÂ¯3. (2.16)38so that finally we get an expression for the fourth order correlated part of the densitymatrix asÏÂ¯C1234 =Ï1234 â ÏÂ¯C123ÏÂ¯4 â ÏÂ¯C234ÏÂ¯1 â ÏÂ¯C134ÏÂ¯2 â ÏÂ¯C1234ÏÂ¯3+ ÏÂ¯C12ÏÂ¯3ÏÂ¯4 + ÏÂ¯C13ÏÂ¯2ÏÂ¯4 + ÏÂ¯C14ÏÂ¯2ÏÂ¯3 + ÏÂ¯C23ÏÂ¯1ÏÂ¯4+ ÏÂ¯C24ÏÂ¯1ÏÂ¯3 + ÏÂ¯C34ÏÂ¯1ÏÂ¯2 â 3ÏÂ¯1ÏÂ¯2ÏÂ¯3ÏÂ¯4. (2.17)At this point it is very useful to introduce a diagrammatic representation forthe various functions involved. We represent the different cells or sub-systems withâbulletsâ (ie., by the symbol â¢), and the reduced density matrix for a group of cellsis shown by linking these cells with a thick line. Then, for example, the expressionÏÂ¯134ÏÂ¯2 is represented as shown in Fig. 2.2(a).We now represent the entanglement correlated density matrices, like ÏÂ¯C12, ÏÂ¯C123, . . .,by double lines linking the cells. Then, in the 4-cell example, we have for the rela-tion between the full density matrix Ï1234 and the entanglement correlated densitymatrices ÏC , given above in (2.13), the diagrammatic representation shown in Fig.2.3.Before continuing with the analysis, we remark on two things about these results:(i) we are not summing over different partitions to get these results, but overdifferent subsets of the 4-site system, ie., over the power set.(ii) the number of different terms shown in Fig. 2.3 is not 24 = 16, as one mightnaively expect for the power set of our 4-site system. Instead it is 24â 4 = 12. Thisis because the 4 subsets made from single individual sites gives no contribution - thecorrelated part of a single site density matrix is zero, as specified in equation (2.5).Thus in general we expect a total number of diagrams 2N âN to contribute to theexpansion (2.2).Expansion Over Cumulant MatricesAs just noted, the expansion (2.2) is not an expansion over the different partitionsof the total set S, but over the power set. However one can also do an expansiondefined directly in terms of these partitions, rather than by the zero trace conditionin equation (2.9).Suppose we take the set PS of all partitions of S, and then for each one ofthese partitions we factorise the result into reduced density matrices for single cellsuncorrelated with the rest, and a set of âcumulant reduced density matricesâ ÏCCfor the other cells. The expansion of the total density matrix in terms of thesecumulant matrices then has the same structure as a cumulant expansion of a jointprobability function or functional; ie., we can writeÏS =âpÂµâPSâAâpÂµÏÂ¯CCA . (2.18)39Figure 2.2: Diagrammatic representation of some of the terms in the 4-cell densitymatrix. In (a) we show the term ÏÂ¯134ÏÂ¯2 appearing in equation (2.17); in(b) we show the term ÏÂ¯C134ÏÂ¯2, also appearing in equation (2.17); and in(c) we show the term ÏÂ¯CC134 ÏÂ¯2, appearing in equation (2.19).Equation (2.18) can be used to inductively to define ÏÂ¯CCA , with the convention thatfor a single elementary subsystem, the cumulant matrix ÏÂ¯CCi is defined to be thereduced density matrix, ie., ÏÂ¯CCi â¡ ÏÂ¯i.The relation between this cumulant expansion and the power set expansion weare using here, which is given in terms of entanglement correlated density matrices,is easily illustrated for the 4-cell problem, for which we find the cumulant expansionÏ1234 =ÏÂ¯CC1234 + ÏÂ¯CC123 ÏÂ¯4 + ÏÂ¯CC124 ÏÂ¯3 + ÏÂ¯CC134 ÏÂ¯2 + ÏÂ¯CC234 ÏÂ¯1+ ÏÂ¯CC12 ÏÂ¯CC34 + ÏÂ¯CC14 ÏÂ¯CC23 + ÏÂ¯CC13 ÏÂ¯CC24 + ÏÂ¯CC23 ÏÂ¯1ÏÂ¯4+ ÏÂ¯CC13 ÏÂ¯2ÏÂ¯4 + ÏÂ¯CC14 ÏÂ¯2ÏÂ¯3 + ÏÂ¯CC12 ÏÂ¯3ÏÂ¯4 + ÏÂ¯CC24 ÏÂ¯1ÏÂ¯3+ ÏÂ¯CC34 ÏÂ¯1ÏÂ¯2 + ÏÂ¯1ÏÂ¯2ÏÂ¯3ÏÂ¯4 (2.19)for Ï1234 in terms of the ÏCC .One can of course invert the relation (2.18) as well. Thus, for example, the4th-order cumulant density matrix is given in terms of the entanglement correlatedmatrices ÏC and the reduced density matrices byÏCC1234 â¡ ÏÂ¯C1234 â ÏC12ÏC34 â ÏC14ÏC23 â ÏC13ÏC24 (2.20)which when expanded out givesÏCC1234 â¡ ÏÂ¯1234 â ÏÂ¯C123Ï4 â ÏÂ¯C234Ï1 â ÏÂ¯C134Ï2 â ÏÂ¯C1234Ï3â ÏÂ¯C12ÏÂ¯C34 â ÏÂ¯C14ÏÂ¯C24 â ÏÂ¯C13ÏÂ¯C24+ 2(ÏÂ¯C12ÏÂ¯3ÏÂ¯4 + ÏÂ¯C13ÏÂ¯2ÏÂ¯4 + ÏÂ¯C14ÏÂ¯2ÏÂ¯3 + ÏÂ¯C23ÏÂ¯1ÏÂ¯4+ÏÂ¯C24ÏÂ¯1ÏÂ¯3 + ÏÂ¯C34ÏÂ¯1ÏÂ¯2) â 6ÏÂ¯1ÏÂ¯2ÏÂ¯3ÏÂ¯4. (2.21)We can also illustrate the cumulant expansion diagramatically. If we representthe cumulant reduced density matrices ÏÂ¯CC12 , . . . by single lines between the relevantcells (compare Figs. 2.2(b) and 2.2(c)). Then, for the relation between the full den-sity matrix Ï1234 and the cumulant density matrices ÏCC , we have the diagrammaticrepresentation shown in Fig. 2.4.We see that the relationship between the full density matrix ÏS and the cumulantdensity matrices ÏCC is the same as that in a typical cumulant expansion, and socan be derived in the usual way for any value of n.40Figure 2.3: Diagrammatic representation of the expansion of Ï1234 into density ma-trices for the four sub-systems, as expressed in equation (2.13)Figure 2.4: Diagrammatic representation of the expansion of the 4-cell density ma-trix into the B4 = 15 different cumulant density matrices for the sub-systems, given in equation (2.19)2.0.3 General Properties of Entanglement Correlated DensityMatricesAs we have just seen, the relationship between ÏS and the cumulant density matricesÏCC is relatively straightforward. On the other hand, the relationship between ÏSand the entanglement correlated density matrices ÏC is not so obvious - we still donot have a general expression for the correlated part of the total density matrix. Toproperly understand things we now turn to the general case.What we wish to show is how, for a general subset A(n)Î± of n cells of a totalsystem S containing N cells, the correlated part of the reduced density matrix canbe written as a sum over terms involving the reduced density matrices for all subsetsC(m)Âµ â A(n)Î± . The notation used here labels the specific subsets C(m)Âµ and A(n)Î± by thesubscripts Âµ and Î±; the superscripts m and n tell us how many cells are contained inthese subsets. This is illustrated in Fig. 2.5. The key result we find can be written41Figure 2.5: A representation of the sets used in equation (2.23). The set An is asubset of the whole system S, and contains n members. The set Cm,which contains m members, is a subset of An.asÏÂ¯CA(n)Î±=nâm=2(â1)(nâm)âC(m)Âµ âA(n)Î±ï£«ï£¬ï£­ÏÂ¯Cm âjâA(n)Î± C(m)ÂµÏÂ¯jï£¶ï£·ï£¸â (â1)n(nâ 1)âjâA(n)Î±ÏÂ¯j . (2.22)which says that the entanglement correlated density matrix ÏÂ¯CA(n)Î±for the specific setA(n)Î± of cells can be written as a sum over entanglement correlated density matricesfor all the different subsets C(m)Âµ of A(n)Î± , multiplied by the product of the reducedmatrices for all the cells j that are not included in the subset C(m)Âµ (this being thefirst term in (2.22)), minus a term which is simply the product of all the individualcell reduced density matrices for all the cells in A(n)Î± .To reduce somewhat the profusion of indices in this expression, we will henceforthwrite expressions of this kind without the Greek indices labelling the specific subsets- thus (2.22) becomesÏÂ¯CAn =nâm=2(â1)(nâm)âCmâAnï£«ï£­ÏÂ¯Cm âjâAnCmÏÂ¯jï£¶ï£¸â (â1)n(nâ 1)âjâAnÏÂ¯j . (2.23)The simplest way to demonstrate the result in equations (2.22) and/or (2.23) isto construct an inductive proof - this is done in Appendix A. This result shows howone can define n-cell entanglement explicitly in terms of all possible combinationsof m-cell entanglements over the different subsets of the n cells, âm < n, alongwith products of single cell reduced density matrices. We shall see in the next twosections how we can employ eq.(2.23) to define a set of correlation functions whichexhaustively characterise all the different kinds of entanglement that exist at thenth level, i., for a set of n entangled cells.42As noted above, a key property of the entanglement correlation density matricesÏÂ¯C is that any partial trace over ÏÂ¯CAn in (2.23), ie., one in which we trace out anyi â An, gives zero - compare equation (2.9). In the discussion above, we treated thisequation as a defining property of the ÏÂ¯C . However, one can also derive the resultexplicitly from the expression (2.23). The derivation is given in Appendix A.2.2.1 ConclusionWe have investigated the decomposition of the density matrix in a system madeup of many distinguishable cells. We have defined two natural ways to decomposethe system: (i) as a sum over all possible subsets of the system of terms involvingentanglement correlated density matricies ÏÂ¯CA and (ii) as a sum over all possiblepartitions of the system of terms which are products of cumulant density matricesÏÂ¯CCA . We saw that these two decomposition are distinct when the number of cellsN â¥ 4.In the next chapter we will discuss how in systems where the elementary cells arequbits, the reduced density matrices and different decomposition can be describedin terms of correlations between different spin components. Then in chapter 4 useÏÂ¯CA to derive a hierarchy of equations of motion linking reduced density matricesof different sizes. We leave further exploration of the properties and uses of thecumulent density matrices ÏÂ¯CCA for future research.43Chapter 3Structure of the Many QubitDensity MatrixNow we consider the structure of the density matrix when the system S consists ofN qubits. In this case, our âelementary cellsâ become much simpler - each cell isa single spin-1/2 degree of freedom. Because these cells are irreducible, ie., can nolonger be split into a set of smaller âsub-cellsâ, we will refer in this case to the cellsas âsitesâ.Apart from discussing the general N -qubit case, we also look in detail at pairsand triplets of spins (N = 2, 3). The results are useful - in particular, they teachus that the easiest way to understand the hierarchy of entanglement at the level ofdifferent qubits is just to look at the different partitioned correlated density matrices.3.1 General Results for N coupled QubitsIn what follows we wish to write some of the results of the last section for a set ofN qubits - these results will hold regardless of what kinds of interaction may existbetween the qubits, or what external fields may be acting on them.3.1.1 Spin RepresentationsWe begin by establishing some notation. In dealing with a set of N spin-12s we writePauli matrices for each spin as {ÏÂµi } (where i â {1, 2, . . . , N} labels the site andin the âCartesianâ representation Âµ â {x, y, z} denotes the Cartesian components).Then for a single spin we have the density matrix in the Bloch representation [31]Ï =12(1 + ãÏã Â· Ï) . (3.1)so that the purity of the density matrix istrÏ2 =12(1 + ãÏã2), (3.2)and for a pure state the polarization ãÏã sits on the Bloch sphere, with |ãÏã| = 1;otherwise |ãÏã| < 1. Notice the tracetrÏÂµÏÎ½ = 2Î´ÂµÎ½ (3.3)44, so that the coefficient of operators in any operators Bloch expansion can be easilycalculated.In what follows we will denote the eigenstates of ÏËz by | âã, | âã, so that| âããâ | = 12(1 + Ïz) (3.4)| âããâ | = 12(1â Ïz) (3.5)and for a pure state at some angle Ï in the xy-plane,ÏÏÏâ² =12(| âã+ eiÏ| âã)(ãâ |+ eâiÏãâ |)= 12(1 + cosÏÏx + sinÏÏy). (3.6)with Ï, Ïâ² = Â±1 labelling the rows and columns of the density matrix.3.1.2 General Results for N qubitsWe assume a system of N qubits {Ïj}, with j = 1, 2, ....N . Let us write the densitymatrix for this system S in the formÏS =12NâCâSâ©âiâCÏÂµiiâªâiâCÏÂµii . (3.7)in which the density matrix contains contributions from all 2N distinct subsets Cof the set S. The contribution to the density matrix from a given cluster C isdetermined by the correlation tensor for those spins contracted into a product ofthe Pauli matrices then multiplied by a normalisation factor.Clearly ÏS , composed entirely of Pauli matrices, must be Hermitian. The traceof ÏS comes from the contribution in which C is the empty set (because all thePauli matrices are traceless) which is 2âN tr(I) = 1 as required. One can verify thattr (ÏÂµ1 ÏS) = ãÏÂµ1 ã etc. by using the relation ÏÂµ1ÏÎ±1 = Î´ÂµÎ±I1 + iÂµÎ±Î³ÏÎ³1 and using thetraceless property of the Pauli matrices (so that any term in the sum which containsa Pauli matrix after it has been multiplied by ÏÂµ1 gives zero). In general the densitymatrix must be positive semidefinite, although this is a hard condition to get ahandle on using the representation (3.7), as it depends on the spectrum of ÏS . If ÏSrepresents a pure state then Ï2S = ÏS , which can be used to derive those relationsamong the correlation functions which hold for pure states (see section 3.2.1 belowfor examples). More generally we havetrÏ2S =12NâCâSâ©âiâCÏÂµiiâªâ©âiâCÏÂµiiâªâ¤ 1. (3.8)As noted above, there are 2N possible C â S. When one takes the partial traceof (3.7) we see that the expression for a reduced density matrix on a set A â Scontaining n spins is of the same form as (3.7), viz.,ÏA =12nâCâAâ©âiâCÏÂµiiâªâiâCÏÂµii . (3.9)453.2 Some ExamplesThe following simple examples are useful in that they not only illustrate much ofthe general theory discussed so far, but they also indicate some of the ways in whichit can be further developed.3.2.1 A Pair of QubitsConsider a pair of qubits Ï1,Ï2, for which the density matrix is [32]Ï12 =14(1 +âj=1,2ãÏjÂµãÏjÂµ + ãÏ1ÂµÏ2Î½ãÏ1ÂµÏ2Î½)(3.10)We can split this up to a correlated and uncorrelated part, according toÏ12 = Ï1Ï2 + ÏC12=14âj(1 + ãÏjÂµãÏjÂµ) + 14ããÏ1ÂµÏ2Î½ããÏ1ÂµÏ2Î½ (3.11)where we have definedããÏ1ÂµÏ2Î½ãã = ãÏ1ÂµÏ2Î½ã â ãÏ1ÂµããÏ2Î½ã. (3.12)Now Ï12 is a 4 Ã 4 hermitian matrix with unit trace, and as such has 16-1=15free real parameters, viz., 3 components of ãÏ1ã and ãÏ2ã each, and 9 componentsof ãÏ1ÂµÏ2Î½ã. In the case of a single qubit in a pure state, the spin had to lie on theBloch sphere. In the two-qubit case things are more complicated; for a pure stateone requires Ï212 = Ï12, which leads to the following constraints on the correlators,3 = ãÏ1ã2 + ãÏ2ã2 + ãÏ1ÂµÏ2Î½ããÏ1ÂµÏ2Î½ã (3.13)ãÏÂµ1 ã = ãÏÂµ1ÏÎ²2 ããÏÎ²2 ã (3.14)ãÏÂµ2 ã = ãÏÎ²1ÏÂµ2 ããÏÎ²1 ã (3.15)ãÏÂµ1ÏÎ½2 ã = ãÏÂµ1 ããÏÎ½2 ã â 12ÎµÂµÎ±Î»ÎµÎ½Î²Î³ãÏÎ±1 ÏÎ²2 ããÏÎ»1ÏÎ³2 ã. (3.16)This gives 1 + 3 + 3 + 9 = 16 constraint equations on the correlators for a purestate - obviously only 10 of these are independent, since there is a 6-dimensional setof real numbers which describes the possible pure states (8 real numbers describea 2-qubit ket |Ïã, reduced by two by the requirements of normalization and theinvariance of Ï12 = |ÏããÏ| under phase rotations). For the pure state,|Ïã =âÏÏâ²aÏÏâ²eiÏÏÏâ² |ÏÏâ²ã (3.17)where Ï, Ïâ² = | âã, | âã; the normalization condition is then âÏÏâ² a2ÏÏâ² = 1.46For a general mixed state of two qubits, equations (3.13-3.16) are replaced bya set of three independent inequalities, which ensure the positivity of the densitymatrix[38]. This reflects the fact that a mixed state density matrix requires 15independent real parameters (the 16 required to define an arbitary 4Ã 4 hermitianmatrix, minus one because the matrix must be traceless) rather than the eightrequired to define a pure state.Of particular interest for qubit pairs are âcat statesâ, which are fully entangled.An example of such a state is |Î¨C2 ã with wave-function and density matrix given by|Î¨C2 ã â¡ 1â2(| ââã+ eiÏââ | ââã)(3.18)|Î¨C2 ããÎ¨C2 | =14(1 + cosÏââ [Ïx1Ïx2 â Ïy1Ïy2 ]+ sinÏââ [Ïy1Ïx2 + Ïx1Ïy2 ] + Ïz1Ïz1)(3.19)When we come to look at entanglement dynamics, it is then the correlated part ofthese functions which will interest us.Let us now consider the relationship between ÏC12 and the different types ofentanglement. There is some subtlety in this [47], especially in the case of mixedstates. Consider, for instance, a mixed state which is an incoherent mixture of thestate | ââã, with spins are polarised in the z direction, and the state | ââã, withboth spins polarised in the x direction, so thatÏ12 =12(| ââããââ |+ | ââããââ |)=14[1 + 12(xË+ zË) Â· (Ï1 + Ï2) + 12 (Ïx1Ïx2 + Ïz1Ïz2)]. (3.20)Now Ï12 has non-zero correlation functions; we haveããÏx1Ïx2 ãã = ããÏz1Ïz2ãã= âããÏx1Ïz2ãã = âããÏz1Ïx2 ãã =14. (3.21)On the other hand, since Ï12 is an incoherent mixture of two separable states, it haszero entanglement of formation[8]. This is not the only measure of entanglement;and for a general mixed state the formulae for different entanglement measures maybe quite complicated.This example shows nicely that it makes sense to consider directly the set ofcorrelators, instead of the different entanglement measures. Because the full setof 15 correlators completely specifies the density matrix, all information about en-tanglement between the pair of qubits is then contained in these correlators. Sinceany entanglement witness [43, 46, 104] used to detect entanglement is necessarily aHermitian operator, it follows that its expectation can also be written as a weightedsum over the correlators. Thus we can simply use the correlators themselves as theprimary quantities, whose behaviour is to be determined.473.2.2 Three QubitsFor a system with three qubits, the general density matrix is written as a sum overcorrelators asÏ123 =18(1 +âjãÏ1ÂµãÏjÂµ +âi<jãÏiÂµÏjÎ½ãÏiÂµÏjÎ½+ ãÏ1ÂµÏ2Î½Ï3Î»ã Ï1ÂµÏ2Î½Ï3Î»). (3.22)We now have a number of different types of entangled state. Consider as an examplethe three different states|Î¨a3ã = 1â2 (| âââã+ | âââã) (3.23)|Î¨b3ã = 1â2 (| âââã+ | âââã) (3.24)and|Î¨c3ã â¡ 1â3âÏ1,Ï2,Ï3|Ï1Ï2Ï3ã Î´[(âjÏj) + 1]= 1â3(| âââã+ | âââã+ | âââã) (3.25)For each of these states we can find the non-zero expectation values for the correla-tors in the density matrix representation (3.22). Consider first |Î¨a3ã, for which|Î¨a3ã : ãÏz3ã = ãÏy1Ïy2ã = ãÏx1Ïx2Ïz1ã = ãÏz1Ïz2Ïz1ã = â1ãÏx1Ïx2 ã = ãÏz1Ïz2ã = ãÏy1Ïy2Ïz2ã = 1 (3.26)We that |Î¨a3ã does not have 3-qubit entanglement, because we can write |Î¨a3ã =1â2(| ââã+ | ââã) â | âã, and this is reflected in the fact that the correlated part ofthe three point function, ããÏÂµ1ÏÎ½2ÏÎ»3 ãã defined byããÏÂµ1ÏÎ½2ÏÎ»3 ãã â¡ tr(ÏÂµ1ÏÎ½2ÏÎ»3 ÏÂ¯C123)(3.27)= ãÏÂµ1ÏÎ½2ÏÎ»3 ã â ããÏÂµ1ÏÎ½2 ãããÏÎ»3 ã â ãÏÂµ1 ãããÏÎ½2ÏÎ»3 ãã â ããÏÂµ1ÏÎ»3 ãããÏÎ½2 ã â ãÏÂµ1 ããÏÎ½2 ããÏÎ»3 ã(3.28)is zero. However it does have 2-qubit entanglement and single qubit polarization.Now consider the other two states, for which we have|Î¨b3ã : ãÏz1Ïz2ã = ãÏz1Ïz2ã = ãÏz1Ïz3ã = ãÏz2Ïz3ã= ãÏx1Ïx2Ïx3 ã = 1ãÏx1Ïy2Ïy3ã = ãÏy1Ïx2Ïy3ã = ãÏy1Ïy2Ïx3 ã = â1 (3.29)48for the second state, and|Î¨c3ã : ãÏzi ã = ãÏzi Ïzj ã = â13 , ãÏxi Ïxj ã = ãÏyi Ïyj ã = 23ãÏz1Ïz2Ïz3ã = 1, ãÏxi Ïxj Ïz` ã = ãÏyi Ïyj Ïz` ã = 23 ,(for i, j, ` distinct â {1, 2, 3}). (3.30)for the third state. Both |Î¨b3ã and |Î¨c3ã do have three qubit entanglement, as thecorrelated 3-qubit functions are non-zero (this especially obvious in the case of |Î¨b3ã,which is the superposition of two terms, each of which is obtained from a triple spinflip of the other). Both states also have 2-qubit entanglement, and |Î¨c3ã also hassingle qubit polarisation. It can be shown that the states |Î¨b3ã and |Î¨c3ã are membersof the only two different classes of fully entangled 3-qubit states [28], and all otherfully entangled states can be obtained from them by local operations assisted withclassical communication.We observe that neither of the states |Î¨b3ã, |Î¨c3ã has a full â3-qubit entanglementâin the way that |Î¨C2 ã has full â2-qubit entanglementâ. For |Î¨C2 ã all the single qubitcorrelators are zero, whereas for the 3-qubit system it is impossible for the followingthree conditions to hold at once:ãÏÂµi ã = 0 âi â {1, 2, 3} (3.31)ãÏÂµi ÏÎ½j ã = 0 âi 6= j â {1, 2, 3} (3.32)Ï123 represents a pure state. (3.33)To show this, we note that the first two conditions imply Ï123 =18(I + ãÏ1ÂµÏ2Î½Ï3Î»ãÏÂµ1ÏÎ½2ÏÎ»3).We can then calculate Ï2123 and we find that the âÏ1Ï2Ï3â component is132ãÏ1ÂµÏ2Î½Ï3Î»ãÏÂµ1ÏÎ½2ÏÎ»3 6=18ãÏ1ÂµÏ2Î½Ï3Î»ãÏÂµ1ÏÎ½2ÏÎ»3 (3.34)where the inequality 6= holds for any non-zero value of ãÏ1ÂµÏ2Î½Ï3Î»ã. Thus the statecanât be pure.3.2.3 N-qubit statesThere are still simple questions one can ask about N -qubit states; for example,whether an analogue of the statements (3.31-3.33) be true when we have N qubits.In other words, one can ask: does the N -qubit density matrixÏ12...N =12N(I + ãÏÂµ11 ÏÂµ22 . . . ÏÂµNN ãÏÂµ11 ÏÂµ22 . . . ÏÂµNN)(3.35)represent a valid pure state? The answer is that this is true only if N = 1 or N = 2.For N = 3 we have just seen that is not a pure state, and proofs for the non-existenceof pure states of the form (3.35) for N â¥ 4 are given in the literature ([49], and refstherein).49N-qubit Cat StatesConsider the class of N spin cat states,|ÏNc (Ï)ã =1â2(| ââ . . . âã+ eiÏ| ââ . . . âã)(3.36)|ÏNc (Ï)ããÏNc (Ï)| =12(| ââ . . . âããââ . . . â |+ | ââ . . . âããââ . . . â | (3.37)+ eâiÏ| ââ . . . âããââ . . . â |+ eiÏ| ââ . . . âããââ . . . â |).Here Ï is an angle which specifies the phase difference between the two superimposedstates. We will consider the dynamics of these states and their generalisations inchapter 9. These are highly entangled states that posses many body entanglement[43]. In the Bloch vector representation (sums over sets here include a zero setterm),| ââ . . . âããââ . . . â | = 12NâCâSâiâCÏzi (3.38)| ââ . . . âããââ . . . â | = 12NâCâSâiâC(âÏzi ) (3.39)| ââ . . . âããââ . . . â | =âiâC| âiããâi | =âiâCS+i =12NâS(Ïxi + iÏyi ). (3.40)So the only non zero correlators containing x and y spin components in the densitymatrix are those of the formâ©âC ÏxiâSC Ïyiâª(i.e. they contain correlation betweencomponents all qubits in the cat state) and one has|ÏNc (Ï)ããÏNc (Ï)| =12Nï£±ï£´ï£´ï£²ï£´ï£´ï£³âCâS|C| evenâiâCÏzi +âCâScos(Ï+pi|C|2)âSCÏxiâCÏyiï£¼ï£´ï£´ï£½ï£´ï£´ï£¾ . (3.41)The genuine N -partite entanglement of this state manifests itself in the order Noff diagonal correlators. One could also consider a generalised cat state which isconveniently described using the vector s = (s1, s2, . . . , sN ) where all si = Â±1 define|ÏNcs(Ï)ã â¡1â2(|sã+ eiÏ| â sã) (3.42)50we have| ââ . . . âããââ . . . â | = 12NâCâSâiâCsiÏzi (3.43)| ââ . . . âããââ . . . â | = 12NâCâSâiâC(âsiÏzi ) (3.44)| ââ . . . âããââ . . . â | =âiâC|siããâsi| = 12NâS(Ïxi + isiÏyi ) (3.45)|ÏNcs(Ï)ããÏNcs(Ï)| =12Nï£±ï£´ï£´ï£²ï£´ï£´ï£³âCâS|C| evenâiâCsiÏzi +âCâScos(Ï+pi|C|2)âSCÏxiâCsiÏyiï£¼ï£´ï£´ï£½ï£´ï£´ï£¾ .(3.46)Since N = 4 is the smallest number of qubits for which ÏCS is distinct from ÏCCS , itis instructive to compute these two quantities for the cat state. For the generalisedcat state (3.46) with N = 4 we haveÏC1234 =116{âCâScos(Ï+pi|C|2)âSCÏxiâCsiÏyi + s1s2s3s4Ïz1Ïz2Ïz3Ïz4 (3.47)â3âi=1âj>isisjÏzi Ïzj}ÏCC1234 =116{âCâScos(Ï+pi|C|2)âSCÏxiâCsiÏyi â 2s1s2s3s4Ïz1Ïz2Ïz3Ïz4 (3.48)â3âi=1âj>isisjÏzi Ïzj}.3.3 ConclusionWe have seen how the density matrix in a system of many qubits can be decomposedin terms of correlations between different spin components of all the possible clustersof qubits. We have then discussed examples of states in systems of qubits, focusingon the complicated relationship between the different types of entanglement andthese correlators.When we deal with the full complexity of N -qubit states, it is hard to get very farin the analysis of entanglement, beyond simple statements of the kind presented here.The number of possible partitions of the system becomes immense, growing super-exponentially as the Bell number, and to characterise the entanglement properties51is clearly going to be very complicated. There is a large body of literature on thedifferent types of multipartite entanglement, along with several reviews [43, 47, 113].However, again, even for N spins, any observable witness we build to diagnosethis entanglement can be expressed as a sum of different clusters of Pauli operators.Thus again it makes sense to go back to the study the dynamics of these correlators,in order to understand the dynamics of entanglement - this is perhaps the mainlesson of the examples just examined.The remainder of this thesis is therefor devoted to studying the dynamics ofthese correlators in many qubit systems.52Chapter 4Dynamics of PartitionedDensity MatricesOne of our main objectives is to derive the dynamics of the entanglement correlateddensity matrices. For a system S made up from N sub-systems or âcellsâ, this meansfinding the equations of motion for each of the reduced density operators ÏÂ¯A , aswell as the correlated density operators ÏÂ¯CA,which describe the different sub-sets Aof S. Now, unless the Hamiltonian for S is trivially non-interacting (ie., it consistsof a simple sum of terms over each cell, with no interactions between the cells), it isclear that these equations of motion will actually couple the different ÏÂ¯CA, since anysub-set A will have interactions with cells not contained in that sub-set (unless ofcourse A = S). Thus we will end up with set of coupled equations of motion, whichtakes the form of a hierarchy of coupled differential equations.In what follows we begin by deriving the hierarchy for a general closed systemin which all interactions between the different cell subsystems are pairwise. Then,in order to see how things look for a specific example, we derive the hierarchy forthe system of N qubits discussed in the previous section, with a set of local fieldson each qubit as well as pairwise interactions between them.4.1 Result for N-partite SystemIn the most common kind of Hamiltonian in physics, one has (i) a âfreeâ or trivialpart which only acts inside individual cells, along with (ii) an interacting part whichcontains pairwise terms between cells. The Hamiltonian then takes the formHS â¡ H0S +HIS =âjâSï£«ï£­H0i + 12 âi 6=jâSVijï£¶ï£¸ (4.1)We make no assumptions for the moment about the nature of the cells, or ofthe interactions between them, except those assumptions already noted in the Intro-duction, viz., that we refer to distinguishable sets of degrees of freedom for each cell(so that there are no âexchange termsâ between cells), and the system is assumednon-relativistic.The equation of motion for the system density matrix isi~âtÏS = [HS , ÏS ]. (4.2)53Starting from this equation, and taking its trace over all cells except those con-tained in An, one can derive an equation of motion for the reduced density matrixÏÂ¯An which takes the formi~âtÏÂ¯An =[HÂ¯An , ÏÂ¯An]+â`6âAntr`ï£«ï£­âjâAn[Vj`, ÏÂ¯Anâª{`}]ï£¶ï£¸ . (4.3)where we have defined an effective local Hamiltonian (ie., one entirely restricted toAn), byHÂ¯An =âjâAnï£«ï£­H0i + 12 âi 6=jâAnVijï£¶ï£¸ (4.4)Although equation (4.3) apparently has a fairly simple form, its derivation isactually quite lengthy, and we have found no way to shorten it. This derivationappears in appendix B.1.We can interpret (4.3) by noting first that the time evolution of ÏÂ¯An is determi-ned both by the local Hamiltonian HAn , acting solely on An, and by the effect ofinteractions on all possible sets containing An along with one other member.One can think of the local effective Hamiltonian as one in which all interactionterms act solely on pairs of cells within An, ie., it is an âinternalâ effective Hamilto-nian for An. The second âinteractionâ term in (4.3) is then one in which Vjl couplesÏÂ¯An to âlargerâ reduced density matrices ÏÂ¯Anâª{`} which involve not only all the cellsin An but also one other cell ` from S that is outside An; we then sum over all thedifferent cells {`} that are outside An. That there is only one other cell involvedfollows because we have only pairwise interactions in the original Hamiltonian.To see how this works let us consider a simple example. Suppose one has anN -cell system S, and we define a specific sub-set A(n)Î± of S by removing 4 designatedcells from S (so that n = N â 4). This example is illustrated in Fig. 4.1. Writingout the sum over ` in equation (4.3) explicitly we have (omitting the subscripts onthe set variables),i~âtÏÂ¯A =[HÂ¯A, ÏÂ¯A]+âjâA(tr1[Vj1, ÏÂ¯Aâª{1}]+ tr2[Vj2, ÏÂ¯Aâª{2}](4.5)+ tr3[Vj3, ÏÂ¯Aâª{3}]+ tr4[Vj4, ÏÂ¯Aâª{4}]).and we see explicitly how the equation of motion for the (N â 4)-cell system A(n)Î±involves a coupling between the (Nâ4)-cell density matrix ÏA(n)Î± and a set of (Nâ3)-cell density matrices ÏÂ¯A(n)Î± âª{`}, with ` = 1, 2...4.In the next sub-section we discuss the example of a system of qubits; this willallow us to work out expressions like this explicitly.54Figure 4.1: An illustration of the terms in the sum in equation (4.5). The set A,a subset of the total system S, is shown in blue in (a), along with fourother sets 1, 2, 3, 4 distinct from A. Then in (b) in green we show thefour different sets that can be made from the union of A and one of theother sets. Each dotted line represents a possible term in the sum (4.5)due to an interaction potential. We have omitted the subscripts on theset variables.As already noted above, there is a loose analogy here with the Schwinger-Dysonequations in quantum field theory and in non-relativistic many-body theory, in thatwe end up with a chain of coupled integro-differential equations for the ÏÂ¯A(n)Î± (here werestore the indices Î± and n, to emphasise that we are dealing in all these equationswith a specific subset of S in which n denotes the number of cells involved, and Î±the specific set of n cells that has been chosen).4.2 Hierarchy of Equations for Reduced DensityMatrices N QubitsFor our set of N qubits, the {Ïi}, the cells again become individual sites, eachwith its own qubit. We wish to find the dynamics of the various spin correlators,following the general theory given in section 4.1. For this we need a Hamiltonianfor the N -qubit system. The general pairwise interaction Hamiltonian for this caseis:H =âi12hi Â· Ïi +Nâi=1âj<i12VÂµÎ½ij ÏÂµi ÏÎ½j . (4.6)In this Hamiltonian each qubit feels a local field hj , and we have a pairwiseinteraction V ÂµÎ½ij between the qubits. Commonly used examples are (i) the quan-55tum Ising model, for which hi = hoxË and VÂµÎ½ij ÏÂµi ÏÎ½j = Vzzij Î´ÂµzÎ´Î½z, and (ii) thenearest-neighbour Heisenberg model, where hi = h is a uniform external field, andV ÂµÎ½ij ÏÂµi ÏÎ½j = JoÎ´ÂµÎ½ , with i, j restricted to be nearest neighbours.In what follows we first derive the general hierarchy of equations of motion, andthen look at some simple special cases.4.2.1 General Form of HierarchyWe derive the equations of motion for the various spin correlators from the reduceddensity matrix equation of motion we have found in (4.3). Again, we pick a specificsubset A of the total N -qubit system; we will therefore be interested in the timeevolution of expectation values of products of spin operators for spins in A.The result of the calculation can be read off from the general equation of motionin (4.3); the commutators are evaluated in Appendix B.2, and we findddtâ©âiâAÏÂµiiâª=âiâAÎµÂµiÎ±Î½hÎ±iâ©ÏÎ½iâjâA{i}ÏÂµjjâª+âiâAâjâA{i}ÎµÂµiÎ±Î½VÎ±Âµjijâ©ÏÎ½iâkâA{i,j}ÏÂµkkâª+âiâAâ`6âAÎµÂµiÎ±Î½V Î±Î»i`â©ÏÎ»` ÏÎ½iâjâA{i}ÏÂµjjâª(4.7)in which we see the characteristic form of a coupled hierarchy of differential equa-tions: the time derivative of the correlator is given in terms of correlators betweenspins in A and correlators among all possible subsets of A with one spins removed,as well as all possible sets made from adding one spin to A. The local field termmixes up the different correlators between qubits in the cluster of qubits A, whilethe interaction terms âtransfers correlationsâ to clusters which contain either oneless or one more qubit.The result (4.7) is still rather forbidding, mainly because it describes the dyna-mics of correlators for all of the spins contained in A. To make it more transparent,we now consider two special cases of this general result.4.2.2 One- and Two-qubit CorrelatorsTo simplify equation (4.7), we can make the subset A small. We consider the twosimplest cases, where A includes one or two sites.Single-site A: Suppose A is just a single qubit - without loss of generality wecall this âsite 1â. Then there is only one correlator, given by the expectation value56ãÏÂµ1 (t)ã; the equation of motion, read off from (4.7), is justddtãÏÂµ1 ã = ÎµÂµ1Î±Î²ï£«ï£­hÎ±1 ãÏÎ²1 ã+â`6=1V Î±Î»1` ãÏÎ»` ÏÎ²1 ãï£¶ï£¸ (4.8)where we recall that V Î±Î²ii = 0, ie., there is no on-site interaction apart from the localfield hi, and we note again that the product over an empty set just gives unity forthe 3rd term in (4.7). In vector notation eq. (4.8) readsddtãÏ1ã = (h1 + VË1)Ã ãÏ1ã (4.9)where the total field VË1 acting on Ï1 from all the other qubits, via the interaction,has componentsVË Î±1 =â`6=1V Î±Î»1` ãÏÎ»` ã (4.10)Thus (4.9) is simply telling us that spin 1 is precessing in a total field coming fromthe local external field plus the field on site 1 generated by all the other spins, viathe interaction.This result is of course well known, and can be derived trivially starting directlyfrom the Hamiltonian. The second term in (4.9) can be thought of as a âHartreeâmean field interaction term.Two-site A: Slightly less trivial is the result we get when A incorporates a pairof sites, which we call site 1 and site 2. We are then interested in the dynamics ofthe pair correlator ãÏÂµ11 ÏÂµ22 ã, and we findddtãÏÂµ11 ÏÂµ22 ã =âj 6=jâ²=1,2ÎµÂµjÎ±Î²[hÎ±j ãÏÎ²j ÏÂµjâ²jâ² ã+ VÎ±Âµjâ²12 ãÏÎ²j ã+â`6=1,2V Î±Î»j` ãÏÎ»` ÏÎ²j ÏÂµjâ²jâ² ã](4.11)whereâj 6=jâ²=1,2 means that we sum over both j and jâ², with the restriction thatj 6= jâ². This result contain both the fields we already saw for the single-site correlator(but now acting on both spins) plus a term - the 2nd term on the RHS in (4.11)above - which accounts for the interaction between the two spins.We can now see intuitively how the results will develop as one goes to correlatorsincluding larger numbers of spins in A. It is also interesting to see how thingssimplify if we look at a very small total system. Thus, eg., suppose system Scomprises only N = 2 spins. Then the sub-system A is just the whole system, andwe expect the result to be trivial. Writing out all terms explicitly, we haveddtãÏÂµ11 ÏÂµ22 ã = ÎµÂµ1Î±Î²(hÎ±1 ãÏÎ²1ÏÂµ22 ã+ V Î±Âµ2o ãÏÎ²1 ã)+ ÎµÂµ2Î±Î²(hÎ±2 ãÏÎ²2ÏÂµ11 ã+ V Î±Âµ1o ãÏÎ²2 ã) (4.12)57where we have written V12 = Vo for the interspin interaction; the role of the effectivefields acting on the one- and two-spin correlators is now transparent.4.2.3 Relationship to Schwinger-Dyson HierarchyThe Schwinger-Dyson hierarchy [29, 69, 94, 95] exists in both relativistic and non-relativistic forms - it is an infinite chain of coupled equations of motion for n-pointcorrelators, whose specific form depends on the interactions in the theory beingtreated. Its general form is similar to the classical Bogoliubov Born Green KirkwoodYvon (BBGKY) hierarchy.To see how this related to what we have done, consider the Schwinger-Dysonhierarchy for a simple scalar field Lagrangian of formL = 12ÏKËâ1o Ï â V (Ï) (4.13)where KËo is the free field propagator. Here x is a spacetime coordinate; and to bedefinite let us assume a simple local âpairwiseâ interaction, of formV (Ï) =g4!Ï4(x) (4.14)Then the Schwinger-Dyson hierarchy for the n-point correlation functionsGn({xj}),with j = 1, Â· Â· Â·n, is given byKâ1o (x, x)Gn(x, xâ²1, . . . , xâ²nâ1)âg6Gn+2(x, x, x, xâ²1, . . . , xâ²nâ1)= i~nâ1âj=1Î´(xâ xâ²j)GËnâ2({xâ²j})(4.15)where Kâ1o (x, xâ²) = ãx|KËâ1o |xâ²ã. If we multiply (4.15) through by KËo, we haveGn(x, xâ²1, Â· Â· Â· , xâ²nâ1)+g6â«d4z Ko (xâ z)Gn+2(zzz, xâ²1, Â· Â· Â· , xâ²nâ1)+ i~nâ1âj=1Ko(xâ xâ²j)GËnâ2({xâ²j})= 0 (4.16)In both of these equations we define the âreducedâ correlator GËnâ2({xâ²j})byGËnâ2 ({xj}) = Gnâ1(xâ²1, Â· Â· Â·xâ²jâ1, xâ²j+1, Â· Â· Â· , xâ²nâ1)(4.17)from which the external legs with coordinates xâ²j and xâ²n have been removed.The hierarchical form of equation (4.16), in which correlators Gn are coupledto both higher and lower correlators, is very clear. Physically, one describes thisequation by saying that if we have an excitation propagating from xâ²j to x in the58presence of a set of mutually interacting excitations propagating between the pointsxâ²1, Â· Â· Â·xâ²jâ1, xâ²j+1, Â· Â· Â· , xâ²nâ1, then it can do so with or without interacting with theother excitations.Mathematically, we see that the main differences between the Schwinger-Dysonhierarchy and the one we have derived here are:(i) Here we are not dealing with the propagation of correlators like Gn betweendifferent spacetime intervals, but instead with time-local correlators in which spacedoes not appear (in its place we have cell or site indices i, j, Â· Â· Â· ).(ii) In contrast with field theory where equations simplify because the variablesare indistinguishable, the variables considered here on different cells or sites aredistinguishable, and each such variable has to be identified explicitly in the equationsof motion. This makes the equations more complex.One can of course integrate equations like (4.3), (4.5), or (4.7), over time - inanalogy with the passage from (4.15) to (4.16). The resulting form can be seen bychoosing simple examples, such as the spin examples in (4.9) or (4.11). The sameinterpretation applies - the spins in A can evolve with or without interacting withother spins outside A.Such an approach is very useful when dealing with regular lattices of, eg., spins;then we can apply decoupling techniques to the resulting hierarchy very similar tothose used for the Schwinger-Dyson equations.However, in dealing with the general case, we would like to develop other appro-aches, to which we now turn.4.3 Entanglement CorrelatorsAlthough the hierarchy of equations governing the dynamics of the different densitymatrices has a clear physical interpretation, and allows us to formulate the idea ofdifferent levels of entanglement, the equations of motion in the form given are notall that convenient to solve.In what follows we set up a more useful description. The basic idea is fairlysimple - we define a âsupervectorâ whose components are an ordered list of all thedifferent time-dependent correlation functions. We then derive a linear first-orderdifferential equation for the time dependence of this vector.To more easily explain the development, we do things first for a simple 2-spinproblem, and then discuss some aspects of a general formulation of this kind - inparticular, we describe how one treats a pair of coupled systems, and how to treatthe equation of motion perturbatively, when there is a small parameter.594.3.1 Example: Dynamics of a Single QubitFor a single qubit Ï the most general Hamiltonian (upto an arbitrary constant term)is,H = 12h Â· Ï , (4.18)here h is an effective applied field acting on the qubit. The equations of motion forthe polarisation vector ãÏ (t)ã are,ddtãÏ (t)ã = hÃ ãÏ (t)ã =ï£«ï£­ 0 âhz hyhz 0 âhxâhy hx 0ï£¶ï£¸ ãÏ (t)ã. (4.19)The solution to the equation of motion can then be written conveniently in termsof the Green function for the equation of motion g(t),ãÏ (t)ã =g(t)ãÏ (0)ã. (4.20)g(t) can be evaluated by taking the Laplace transform of the equation of motion(4.19)g(z) =ï£®ï£°zIâï£«ï£­ 0 âhz hyhz 0 âhxâhy hx 0ï£¶ï£¸ï£¹ï£»â1 (4.21)where I is the 3Ã 3 identity matrix. Which leads to the explicit formula for g(z)gÂµÎ½(z) =hËÂµhËÎ½z+Î´ÂµÎ½ â hËÂµhËÎ½ + iÎµÂµÎ³Î½ hËÎ³2(z â ih) +Î´ÂµÎ½ â hËÂµhËÎ½ â iÎµÂµÎ³Î½ hËÎ³2(z + ih), (4.22)where we have written the vector h in terms of its magnitude h and direction hË.Inverting the Laplace transform then leads togÂµÎ½(t) =Î(t)hËÂµhËÎ½ +eiht2(Î´ÂµÎ½ â hËÂµhËÎ½ + iÎµÂµÎ³Î½ hËÎ³)+eâiht2(Î´ÂµÎ½ â hËÂµhËÎ½ â iÎµÂµÎ³Î½ hËÎ³)(4.23)=Î(t){hËÂµhËÎ½ + cos(ht)(Î´ÂµÎ½ â hËÂµhËÎ½)+ sin(ht)ÎµÂµÎ½Î»hËÎ»}. (4.24)Here Î(t) is the Heaviside step function. Equation (4.24) shows that g(t) is a matrixdescribing the rotation around the axis of the applied field, hË by an angle of ht.4.3.2 Example: Entanglement Correlator Dynamics: Two QubitsFor an arbitrary quantum system, the set of all possible observables is usually rathercomplicated. However in the case of spin systems, one can make an exhaustive list.For a single spin Ï , the spin dynamics is completely defined by giving, as a functionof time, the expectation values of all 3 components ãÏÂµ(t)ã. For a pair of spins Ï 160and Ï 2, 15 different correlators are required, viz., ãÏ 1(t)ã, ãÏ 2(t)ã, and ãÏ 1 â Ï 2ã,where this last contains components ãÏÂµ1 (t)ÏÎ½2 (t)ã. For a set of N qubits, we need22N â 1 correlators.To see how the general idea works, we go back to the example of two qubits,with HamiltonianH =2âa=112ha Â· Ï a + 12VÂµÎ½ÏÂµ1 ÏÎ½2 (4.25)in which the orientation of the 2 static fields h1, h2 is arbitrary. This is just theHamiltonian (4.6), for a pair of spins.Now, suppose we arrange the all the information contained in the 2-qubit densitymatrix (compare equation (3.10)) in the form of a 15-component âsupervectorâ Xin the âspace of possible correlatorsâ, according toX =ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­X1X2X3X5X6X7X8...ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸=ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ãÏx1 ããÏy1 ããÏ z1 ããÏx2 ããÏy2 ããÏ z2 ããÏx1 Ïx2 ããÏx1 Ïy2 ã...ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸â¡ï£«ï£­ ãÏ 1ããÏ 2ããÏ 1 â Ï 2ãï£¶ï£¸ . (4.26)We can then rewrite the hierarchy of equations of motion for the 2-qubit densitymatrix in the formddtX = MX (4.27)or, written out explicitly, in the block structureddtï£«ï£­ ãÏ 1ããÏ 2ããÏ 1 â Ï 2ãï£¶ï£¸ =ï£«ï£­ L1 0 U1,p0 L2 U2,pUp,1 Up,2 Lpï£¶ï£¸ï£«ï£­ ãÏ 1ããÏ 2ããÏ 1 â Ï 2ãï£¶ï£¸ . (4.28)Looking first at the diagonal matrix elements of M, we see that L1, L2 are 3Ã 3matrices which give an infinitesimal rotation of ãÏ 1ã, ãÏ 2ã around the applied fields.The 9 Ã 9 matrix Lp rotates the pair correlator ãÏ 1Ï 2ã around the applied fields,and can also be written as a rank 4 tensor (the lowered indices in the followingexpressions are understood to be contracted to the right in equation (4.28)). Thuswe have:LÂµ1 Î½ =hÎ»1ÎµÂµÎ»Î½ (4.29)LÎ±2 Î² =hÎ³2ÎµÎ±Î³Î² (4.30)LÂµÎ±p Î½Î² =LÂµ1 Î½Î´Î±Î² + Î´ÂµÎ½LÎ±2 Î². (4.31)61Turning now to the non-diagonal interaction matrices, we have terms U1,p,U2,pwhich are 3 Ã 9 matrices, and which create single qubit coherences from the paircorrelator; the corresponding terms Up,1,Up,2 are 9 Ã 3 matrices which create paircoherences from the single qubit coherences. All of these interaction matrices maybe represented as rank 3 tensors:U1,pÂµÎ½Î² =VÎ»Î²ÎµÂµÎ»Î½ (4.32)Up,1ÂµÎ±Î½ =VÎ»Î±ÎµÂµÎ»Î½ (4.33)U2,pÎ±Î½Î² =VÎ½Î³ÎµÎ±Î³Î² (4.34)Up,2ÂµÎ±Î² =V ÂµÎ³ÎµÎ±Î³Î². (4.35)We see that the matrix M is fully anti-symmetric and has eigenvalues which areeither zero or pure imaginary. One can divide these into two classes, as follows:(i) there are at least 3 zero eigenvectors of M, which are linear combinationsof the eigenstates |nããn| of the Hamiltonian (the dimensionality of the system ofequations is one less than the number of components of the density matrix, becausethe equations automatically preserve the trace of the density matrix).(ii) The other eigenvalues of M occur at every difference En â Em in the eigen-values of the Hamiltonian and their eigenvectors are off-diagonal elements of thedensity matrix |nããm|.In general we can define a set of Green functions {gij} with i, j â {1, 2, p} forthe solution to the equations of motion (4.28), so that the solution to the equationsof motion for the vector X can be writtenX(t) = G(t)X(0) (4.36)where the total propagator has the block formG(t) =ï£«ï£­g11(t) g12(t) g1p(t)g21(t) g22(t) g2p(t)gp1(t) gp2(t) gpp(t)ï£¶ï£¸ (4.37)A formal solution for this Green function is found by Laplace transforming; writingf(z) =â«â0 dt f(t)eâzt, we haveG(z) = [zIâM]â1 . (4.38)so that G(z) has poles at along the imaginary axis at all the differences betweenthe energy eigenvalues Â±iâE as well as a pole at zero with a degeneracy of at leastfour.In the time domain the Green function is justG(t) = exp (Mt) =âân=0Mntnn!. (4.39)62U1pUp2Up1U2pL1 L2LpÏ1< > Ï2< >< >Ï2Ï1 XFigure 4.2: Graphical representation of a term in the expansion of G(t) as an expo-nential power series (cf. equation (4.39)). In this term all the entries inthe matrix M appear (these entries are given in equation (4.28)). Thecorrelators ãÏ 1ã, ãÏ 2ã, and ãÏ 1 â Ï 2ã are shown as red vertices, the in-teraction matrices U1,p, U2,p, Up,1, and Up,2 are shown as directed lines,and the rotation matrices L1, L2, and Lp are shown as undirected lines.This series can be represented graphically (see Fig. 4.2). We define a graph whosevertices are the possible correlators, having (directed) links between them whichrepresent the block components of M. Then we an n-th order term in the sumis represented by a âwalkâ (ie., sequence of n hops) across n links between nodes;multiplying each term by tn/n! we get the Green function.It is important to get an idea of what these expressions look like in practice.Suppose we look first at a very simple case, where the Hamiltonian isH = 12 [â1Ïx + â2Ïx + ÏÏ zÏz] (4.40)having energy eigenvalues Â±1,Â±2 with1 =12âÏ2 + (â1 + â2)2 (4.41)2 =12âÏ2 + (â1 ââ2)2. (4.42)The 225 elements in the 15Ã15 matrix M can now be written out directly, usingeqtns. (4.29) - (4.32). The large majority of the elements are zero; the non zero63eigenvalues of M for this case areÏ10 = (1 â 2) (4.43)Ï20 = 1 + 2 (4.44)Ï30 = 21 (4.45)Ï21 = 22. (4.46)The different components of G(t), ie., the 9 different matrix Green functionsgiven in equation (4.37) are then multiperiodic functions containing these 4 fre-quencies. Their explicit expressions are of course quite lengthy to write out; in App.C the explicit results for G(z) are written in full.The general 2-qubit Hamiltonian (4.25) is not much more complicated than this.In particular, the matrix M has the key property that it is rather sparse, ie., mostelements are still zero. To see this, we write the interaction tensor in diagonal form,ie., VÂµÎ± = VxxxËÂµxËÎ± + VyyyËÂµyËÎ± + Vzz zËÂµzËÎ±; note that there is always a co-ordinatesystem where VÂµÎ± is of this form, which can be obtained using the singular valuedecomposition of VÂµÎ±. Then the sub-matrices which make up M can be written64explicitly asL1 =ï£«ï£­ 0 âhz1 hy1hz1 0 âhz1âhy1 hz1 0ï£¶ï£¸ (4.47)L2 =ï£«ï£­ 0 âhz2 hy2hz2 0 âhz2âhy2 hz2 0ï£¶ï£¸ (4.48)Lp =ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­0 âhz2 hy2 âhz1 0 0 hy1 0 0hz2 0 âhx2 0 âhz1 0 0 hy1 0âhy2 hx2 0 0 0 âhz1 0 0 hy1hz1 0 0 0 âhz2 hy2 âhx1 0 00 hz1 0 hz2 0 âhx2 0 âhx1 00 0 hz1 âhy2 hx2 0 0 0 âhx1âhy1 0 0 hx1 0 0 0 âhz2 hy20 âhy1 0 0 hx1 0 hz2 0 âhy20 0 âhy1 0 0 hx1 âhy2 hx2 0ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸(4.49)U1p =ï£«ï£­0 0 0 0 0 âVzz 0 Vyy 00 0 Vzz 0 0 0 âVxx 0 00 âVyy 0 Vxx 0 0 0 0 0ï£¶ï£¸ (4.50)U2p =ï£«ï£­0 0 0 0 0 Vyy 0 âVzz 00 0 âVxx 0 0 0 Vzz 0 00 Vxx 0 âVyy 0 0 0 0 0ï£¶ï£¸ (4.51)Up1 =â UT1p (4.52)Up2 =â UT2p. (4.53)By making ha = xËâa, and using a purely longitudinal coupling, we get back thesimpler Hamiltonian in (4.40). In any case, we see that most elements in thesematrices are zeroes.More generally, so long as there are only local fields and pairwise interactions,it is evident that the âsparsenessâ of the matrix M will increase rapidly with thenumber of qubits.4.4 Remarks on a General FormulationLet us now consider how this might go for more complicated systems. The gene-ralization of the 2-spin results to N spins is clear - now the supervector X hasdN = 22N â 1 entries, growing very rapidly with N .More generally one may have to deal with systems in the thermodynamic limit,having an infinite number of degrees of freedom. Moreover, most degrees of freedomin Nature are usually described by continuous variables, and this automatically leads65to an infinite set of possible correlators (such as the set ãxã, ãx, xâ²ã, ãx, xâ², xâ²â²ã, Â· Â· Â· ,etc, for a single coordinate degree of freedom); and as noted in the introduction, thesystem may be composed of indistinguishable particles.We will not deal with all these complications here - but it is still useful tounderstand some more general features of problems involving distinguishable spins.In what follows we look at two key questions, viz., (i) how do things work when wehave 2 coupled spin systems, and (ii) if there is a small parameter in the problem,how do we make perturbation expansions for the entanglement correlators?4.4.1 Two Coupled SystemsThe special case of two separate but coupled systems is of interest for several reasons.Most notably, it forms the basis for a discussion of a central system coupled to someenvironment; and it is also useful when one comes to analyze how entanglementdevelops between any pair of systems.Consider a pair of systems S1 and S2, which may or may not interact, and whichare in general entangled. We again define an abstract supervector Î¾ which containsall possible correlators for the pair of systems, in the formÎ¾ =ï£«ï£­X1YX2ï£¶ï£¸ (4.54)where X1 is the vector containing all the correlators of operators acting on S1 alone,X2 likewise for S2, and Y refers to all âjointâ operators, acting on both systemstogether.As an example one can consider a pair of qubit systems, one containing n1 spins{Ï i}, and the other n2 spin-1/2 degrees of freedom {Ïj}, with the total number ofspins being N = n1 + n2. We then haveX1 =ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ãÏÎ±1 ããÏÎ±2 ã...ãÏÎ±1 ÏÎ²2 ããÏÎ±1 ÏÎ²3 ã...ãÏÎ±1 ÏÎ²2 Ï Î´3 ã...ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸; X2 =ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ãÏÎ±1 ããÏÎ±2 ã...ãÏÎ±1 ÏÎ²2 ããÏÎ±1 ÏÎ²3 ã...ãÏÎ±1 ÏÎ²2ÏÎ´3ã...ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸(4.55)for the supervectors of S1 and S2 respectively; the supervector Y on the other hand66has the entriesY =ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ãÏÂµ1 ÏÎ±1 ããÏÂµ1 ÏÎ±2 ã...ãÏÂµ2 ÏÎ±1 ããÏÂµ2 ÏÎ±2 ã...ãÏÂµ1 ÏÎ½2 ÏÎ±1 ã...ãÏÂµ1 ÏÎ±1 ÏÎ²2 ã...ï£¶ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£·ï£¸(4.56)The number of components of these different vectors are then given bydX1 = 22n1 â 1 dX2 = 22n2 â 1dY = (22n1 â 1)(22n2 â 1)dX1+X2 â¡ dN = 22N â 1 (4.57)where N = (n1 + n2).Let us now take the Laplace transform of Î¾(t), defined as before byÎ¾(z) =â« â0dteâztÎ¾(t) (4.58)The equations of motion can then be written in the following formÎ¾(z) = G(z)Î¾(0) (4.59)where Î¾(0) is the initial value of Î¾ and G(z) is a matrix, whose inverse has thefollowing block structure:G(z)â1 =ï£«ï£­gâ11 (z) âV1M 0âVM1 gâ1M (z)â VMM âVM20 âV2M gâ12 (z)ï£¶ï£¸ (4.60)where the âmixedâ propagator gM in the middle matrix element MMM is given bygâ1M (z) = gâ11 (z)I2 + gâ12 (z)I1 â zI1I2 (4.61)In these equations Xj(z) = gj(z)X(0) is the solution to the equations of motionfor the individual system j (with j = 1, 2) in the absence of any coupling betweenthem; Ij is the identity acting on system j, and the interaction matrix V has theform, in the same dN -dimensional space,V =ï£«ï£­ 0 V1M 0VM1 VMM VM20 V2M 0ï£¶ï£¸ . (4.62)67The elements of the sub matrices of V can be obtained as needed by readingthem off from the equations of motion (for which of course we require a specificHamiltonian).In general the G(z) will have poles at z = i(Ïn â Ïm) for all n,m where Ïn isthe nth energy eigenvalue of the Hamiltonian. The pole at z = 0 will be of at leastof order 2n1ân2 â 1, with larger orders occurring when the system has degenerateenergy levels.When the systems are large it does not make sense to be enumerating all thepoles and their residues. Instead we simply define a spectral function which givesus the density of the poles along the imaginary axis; we writeA(Ï) =12pi[G(iÏ + )âG(iÏ â )] (4.63)where we choose  to be small but still larger than the typical separation betweenpoles. For sufficiently large systems the poles will become so close that we can treatthem as defining a branch cut along the imaginary axis, with magnitude A(Ï).4.4.2 Perturbation ExpansionsSuppose we have solved the full hierarchy in some specific case, and we add a smallterm to the Hamiltonian, - this could be, eg., to each of the bath spin local fields,or to the interaction between the central systems and the bath spins. The questionis how a perturbation theory will be structured.We do not give a full treatment here, since it is rather messy. The simplest caseis the one in which we treat the interaction term V as a perturbation. We can thenwrite an equation for the full Green function, G(z) as an expansion about the V = 0Green function, G0(z), where in this case one hasG0(z) â¡ï£«ï£­g1(z) 0 00 gM(z) 00 0 g2(z)ï£¶ï£¸ . (4.64)A Dyson series for G(z) may then be obtained in through the usual manipulati-ons,G(z) =[Gâ10 (z)â V]â1(4.65)=G0(z)âân=0(VG0(z))n (4.66)where the matrix being raised to the n-th power is justVG0(z) =ï£«ï£­ 0 V1MgM(z) 0VM1g1(z) VMMgM(z) VM2g2(z)0 V2MgM(z) 0ï£¶ï£¸ (4.67)Note that care needs to be taken when this expansion is performed near the highorder poles of G0(z), to ensure that the corrections are still small.684.5 ConclusionWe have used the entanglement correlated part of the reduced density matrix toderive a hierarchy of equations of motion linking the motion for reduced densitymatrices. This hierarchy links the reduced density matrix ÏÂ¯A on a subset of thesystem A to reduced density matrices on all sets obtained by adding one cell to A.Then we derived a linear hierarchy of equations for systems of qubits, linkingdifferent orders of correlators. We found that the time of a derivative correlatorcontaining pauli matrices acting on a cluster A of qubits, depends on correlatorson clusters obtained by adding or removing one qubit from A. Then we presentedresults for a single qubit and a pair of qubits, and saw how the equations could besolved for Green functions which propagate different correlators.The structure of the hierarchy of equations for spin correlators was then inves-tigated in more detail. In particular we saw how the evolution of correlators in asystem consisting of two distinct sets of qubits. We discuss this case in more detailin the next chapter, with the case in mind where one of the sets of qubits is theenvironment. We will formally âintegrate outâ this set of environmental qubits andexamine the structure of the resulting solution for the central qubitsâ motion.Then in following chapters, the hierarchy of equations for qubit correlators willbe used, to gain an understanding of the process by which central qubits lose infor-mation into correlations with the environment.69Chapter 5Integrating a Spin Bath Out ofthe HierarchyIn this chapter we consider the effect a set B of âenvironmental spinsâ on the remai-ning system spins. We use the structure of the hierarchy of equations of motion forqubit correlators described in chapter 4, when there is an environmental spin bathpresent. We eliminate the bath spins and derive formal expressions for the Greenfunctions describing the evolution of correlators between system qubits only. Theresult is a formal expression for the dynamics of the system correlators, which givesus insight into how the bath spins can affect the motion of the central spins. Thenwe consider the special case where the environment and the bath are initially in aproduct state.We consider NS âcentralâ 12âspins {Ï a} (a = 1, . . . NS) coupled to NB bathspins {Ïi} (i = 1, 2, . . . NB). In general the full density matrix Ï can be written,Ï =12NA+NB{1 +NSâa=1ãÏÂµa ãÏÂµa +NBâi=1ãÏÎ±i ãÏÎ±i + ãÏÂµ1 ÏÎ½2 ãÏÂµ1 ÏÎ½2+Nâi=1NSâa=1ãÏÂµa ÏÎ±i ãÏÂµa ÏÎ±i +NBâi=1âj<iãÏÎ²j ÏÎ±i ãÏÎ²j ÏÎ±i + . . .}(5.1)in the above and in what follows, i, j, k . . . â {1, 2, . . . , NB} index bath spins anda, b, c . . . â {1, 2} index central spins. In the above the terms â. . .â involve sums overall distinct clusters of spin operators. All information about the system at a giventime (including the bath) is contained in the density matrix. We can construct asuper vector Î¾(t) of these correlators just like in section 4.4.1, where in this casesystem one is the central system S and system two is the bath B.Assume the Hamiltonian is of the following form,H = 12NSâa=1ha Â· Ï a + 12NSâa=1âb<auabÂµÎ½ÏÂµa ÏÎ½b +122âa=1NBâi=1V aiÂµÎ±ÏÂµa ÏÎ±i +12Nâi=1bi Â· Ïi. (5.2)For completeness we give the hierarchy of equations of motion in this case. Let Cbe a subset of the central system containing m spins (the form written here willbe valid for a central system containing an arbitrary number of spins) and A be70a subset of the bath containing n spins the equation of motion for the equal timecorrelatorâ©âaâC ÏÂµaaâiâA ÏÎ±iiâªis,ddtâ©âaâCÏÂµaaâiâAÏÎ±iiâª=âaâCÎµÂµaÎ»Î½{haÎ»â©ÏÎ½aâbâC{a}ÏÂµbbâiâAÏÎ±iiâª+âbâC{a}uabÎ»Âµbâ©ÏÎ½aâcâC{a,b}ÏÂµccâiâAÏÎ±iiâª+âbâSCuabÎ»Î³â©ÏÎ½a ÏÎ³bâcâC{a}ÏÂµccâiâAÏÎ±iiâª+âjâAV ajÂµÎ±jâ©ÏÎ½aâbâC{a}ÏÂµbbâiâA{j}ÏÎ±iiâª+âjâBAV ajÂµÎ²â©ÏÎ½a ÏÎ²jâbâC{a}ÏÂµbbâiâAÏÎ±iiâª}+âjâAÎµÎ±jÎºÎ²{bjÎºâ©ÏÎ²jâaâCÏÂµaaâiâA{j}ÏÎ±iiâª+âbâCV bjÂµbÎºâ©ÏÎ²jâaâC{b}ÏÂµaaâiâA{j}ÏÎ±iiâª+âbâSAV bjÎ½Îºâ©ÏÎ½b ÏÎ²jâaâCÏÂµaaâiâA{j}ÏÎ±iiâª}. (5.3)The various terms in equation (5.3) are illustrated in figure 5.1.The equation of motion (5.3) is of the formdÎ¾dt= MÎ¾, (5.4)when written in terms of the supervector of correlators. After we take the Laplacetransform of Î¾(t), we find that the equation of motion may indeed be written in theform of equation (4.59), which we repeat here for clarity,Î¾(z) = G(z)Î¾(0). (5.5)Where Î¾(0) is the initial value of Î¾ and G(z) is a matrix, whose inverse has thefollowing block structure:G(z)â1 =ï£«ï£­gâ1S (z) âVSM 0âVMS gâ1M (z)â VMM âVMB0 âVBM gâ1B (z)ï£¶ï£¸ (5.6)and gâ1M is as in equation (4.61). Note that G(z)â1 is of the formG(z)â1 = zIâM. (5.7)Also in this case gS and gB are (2NS â 1) Ã (2NS â 1) and (22NB â 1) Ã (22NB â1) matrices respectively. All the terms VAC (A, C â {S,M,B}) are due to theinteraction V aiÂµÎ± between the system and the bath. The matrix elements relating tothe various V submatricies in can be read of the equations of motion. For example,71Figure 5.1: The various terms appearing in equation (5.3) when there are three cen-tral qubits and three bath spins: Right, an illustration of a possiblechoices of the set A in red, which contains some central qubits (repre-sented by green dots) and some bath qubits (represented by blue dots).The left hand figure shows the terms on the right had side of equation(5.3). Terms are in the same order in the equation as in the figure. Ineach term, a correlator between qubits in the shaded set, the local fieldacting on circled qubits, and the interaction u (V ) acting between qubitslinked by green (dark red) dashed lines enter into the corresponding termin equation (5.3). Note the secound, fourth, and seventh terms on theright hand side of equation (5.3) are each represented by two diagramsin the figure, as each of these terms has a sum over two correlators.72VMM acts on mixed correlatorsâ©âaâC ÏÂµaaâjâA ÏÎ±jjâªand produces correlatorswhich have one less or one more spin. So there are some matrix elements in VMMwhich give a contribution to the time derivative of the correlator with one of thebath spins (say i) removed ddtâ©âaâC ÏÂµaaâjâA{i} ÏÎ±jjâª, this contribution isâcâCÎµÂµcÎ»Î½V ciÎ»Î²â©ÏÎ½c ÏÎ²iâaâCÏÂµaaâjâA{i}ÏÎ±jjâª. (5.8)5.1 Integrating Out the Bath SpinsThe bath spins can be formally integrated out by inverting the matrix G(z)â1 =zIâM, there are multiple ways to do this, in what follows we describe two.In the first approach one inverts (zI âM), by performing elementary row re-duction to reduce (zIâM) to an upper triangular matrix, then complete the processof row reduction to invert the matrix. When the (zIâM) is inverted in this mannerthis the top âblock-rowâ (which is what is needed to obtain X(z) the correlatorsspecifying the state of the central system in terms of all of the initial correlators) of(zIâM)â1 can be written,GSS(z) â¡ GSS = gS[IS â VSMGËMMVTSMgS+ VSMGËMMVMBGBVTMBGËMMVTSMgS](5.9)GSM(z) = gSVSMGËMM[IM â VMBGBBVTMBGËMM](5.10)GSB(z) = gSVSMGËMMVMBGBB, (5.11)withGËMM =[(gM)â1 â VMM â VTSMgSVSM]â1(5.12)GBB =[(gB)â1 â VTMBGËMMVMB]â1. (5.13)Note that GBB is an exact sub-matrix of G(z). Also note that in the above we coulddefine the {GSS ,GSM, . . .} as the Laplace transforms of the following quantities,GAC(t) =âÎ¾A(t)âÎ¾C(0)â£â£â£â£Î¾(0)=0A, C â {S,M,B}. (5.14)Where we have defined the super vectors Î¾S â¡ XS , Î¾B â¡ XB, and Î¾M â¡ Y .We can interpret the results (5.9-5.13) as follows. GËMM in equation (5.12) isthe Green function which gives a contributions to mixed correlators from mixed cor-relators. The âself energyâ term VTSMgSVSM includes all the terms where mixed73correlators evolve into system only correlators then back into mixed correlators (soGËMM includes the back-reaction terms from XS) . GËMM can then be used to cal-culate the exact bath-bath Green function(5.13), because bath correlators are onlysourced by mixed correlators or by bath correlators. The last term in (5.13) canbe seen as the effect of all the mixed and system correlators on the dynamics onbath only correlators. Once one has the exact bath-bath Green function one can getthe exact the exact bath-system Green function (5.11) by including the propagationof correlations through the mixed correlators to the central system. One can alsothen calculate the mixed Green function (5.10) using GËMM to propagate mixedcorrelators from to system correlators including the possibility that the mixed cor-relators back-react through the bath correlators (the second term in equation (5.10)is the self energy for this effect). Then the full system-system Green function (5.9)is given by the âbareâ system system Green function corrected by the possibilitiesof backreaction; through the set of mixed correlators and through the set of bathcorrelators via the mixed correlators (the second to last and last term respectivelyin equation (5.9)).In the second approach inverts (zIâM) by performing elementary row reductionto reduce (zIâM) to an lower triangular matrix as a first step, after the inversionis complete one gets,GSM = GSSVSMG2 (5.15)GSB = GSMVMBgB (5.16)withG2 =[(gM)â1 â VMBgBVTMB]â1(5.17)GSS =[(gS)â1 â VSMG2VTSM]â1. (5.18)The representation of the inverse in equations (5.15-5.18) is in a sense the opposite tothat (5.9-5.13). The effective mixed-mixed Green function including back-reactiononly through the bath correlators is calculated first in equation (5.17) (instead ofincluding only back-reaction through system correlators in the previous approach).From which one can immediately calculate the exact system-system Green function(5.18) by including back-reaction through the mixed correlators. It is then no troubleto get the mixed-system (5.15) and bath-system (5.16) Green functions.The different sets of formulae (5.15-5.18) and (5.9-5.13) for the Green functionsare of course only two of many such forumulae.Now we discuss the structure of the different Green functions in the complexplane. As discussed in section 4.3.2 G(z) has poles along the imaginary axis, atpoints determined by the differences between eigenvalues of the Hamiltonian (5.2).In general this means all the sub matrices of G(z) have poles at these frequencies,though the weights of these poles may be very different. For instance the Greenfunctions GSB and GSM are zero when there is no interaction between the bath andcentral system.745.1.1 Product State Initial ConditionsOften we will be interested in the case where initially the system is in a productstate Ï(0) = ÏS(0)â ÏB(0) this implies that initially we will be able to write Y (0) =XS(0) â XB(0) â¡ X0S â X0B. In this case the solution for the vector XS whichcontains all the information about the central system can be written:XS(z) =[GSS +GSMX0Bâ]X0S +GSBX0S (5.19)=GSS[IS + VSMG2X0Sâ]X0S +GSSVSMG2VMBgBX0B (5.20)â¡GXS (z, {Ï0k})X0S + JS(z, {Ï0k}). (5.21)So we can see that the correlators in the central system are propagated by a pro-pagator which is renormalised by the effect of the bath and the correlations in thebath can also generate correlations in the central system. We can write a definitionfor GXS (z, {Ï0k}), JS(z, {Ï0k}) as followsGXS (z, {Ï0k}) â¡âXS(t)âXB(0)â£â£â£â£Î¾(0)=(XS(0), XS(0)âXB(0), 0)T(5.22)JS(z, {Ï0k}) â¡âXS(t)âXB(0)â£â£â£â£Î¾(0)=(0, 0, XB(0))T(5.23)5.2 ConclusionWe have seen that when studying the dynamics of a qubit system coupled to a spinbath the set of correlators can divided into system, mixed, and bath correlators. Wecan integrate out the bath spins and find effective dynamics for the central systemcorrelators, including terms that are sourced by the mixed and bath correlators.We have also seen how the Green functions for the correlators inherit theirsingularities from the inverses of the dynamical matrix and how that depends onfixed block structure of that matrix. Finally we have simplified the results forcase with product state initial conditions and we see the propagation off the systemcorrelators still in general depends on a âsourceâ term in which the initial conditionson the bath can source system correlators.In following chapters we will turn to more practical applications and exampleswhere we can use the hierarchy of equations to model the interaction of centralqubits coupled to an environment.75Chapter 6Overview of Simple SingleQubit ModelsNow we consider some simple single central qubit models and discuss their dynamics.Most of the work in this chapter is an original take on models and results previouslystudied in the literature. We also derive some new results for these models.The first model we consider is the of a central qubit coupled to a spin bath inthe degeneracy blocking regime introduced in section 1.7.2, we develop a picture forthe loss of information by the central system into correlators with the environment.Then we discuss a central qubit under the influence of a time varying field caused byan external source. We then treat a case when this time varying force is stochasticand can be averaged (we discuss performing the bias average introduced in section1.7.2).6.1 Example, Degeneracy Blocking Spin BathConsider a central spin Ï coupled to a spin bath consisting of N spins {Ïi} with theHamiltonian [19, 86, 88],H =â02Ïx +Nâj=112ÏjÏzÏzj . (6.1)So we have a central qubit which may flip with an amplitude â0 and the states havean energy difference set by their coupling to bath spin variables. The bath spinvariables cannot flip with respect to the z axis. Prokofâev and Stamp named themechanism of the central qubits decay this model âdegeneracy blockingâ, becausein states with ââNj=112ÏjÏzj â0â the central spin is blocked from flipping by thelack of degeneracy with the state they would flip into. As Ïzj commutes with theHamiltonian this is simple to solve.Define the bath states |Î·ã = âNj=1 |Î·jã specified by a length N vector Î· =(Î·1, Î·2, . . . , Î·N ), whose entries, Î·i = Â±1 specify the orientation of the bath spinswith respect to the z axis. In terms of this basis the Hamiltonian isH =âÎ·(â02Ïx +Nâj=112ÏjÎ·jÏz)|Î·ããÎ·|. (6.2)76Thus the Hamiltonian can be viewed as a sum over different system Hamiltoniansfor each bath state. We can define hÎ· to be the effective âmagnetic fieldâ acting ona qubit when the bath is in the state |Î·ã,hÎ· â¡ â0xË+âjÏjÎ·j zË. (6.3)The energy eigenvalues associated with a specific bath configuration |Î·ã are,Â± hÎ· = Â±12âââââ20 + ( Nâj=112ÏjÎ·j)2. (6.4)So that the evolution operator U(t) can be written in the same form as the Hamil-tonian (6.2),U(t) =âÎ·eâi2hÎ· Â·Ï t|Î·ããÎ·|. (6.5)Therefore if the total density matrix is initially Ï(0), then the density matrix aftertime t is thenÏ(t) =âÎ·âÎ·Ë|Î·ãeâ i2hÎ· Â·Ï tãÎ·|Ï(0)|Î·Ëãe i2hÎ·Ë Â·Ï tãÎ·Ë|. (6.6)Tracing out the bath, one then obtains the exact reduced density matrix ÏÂ¯S(t) forthe central system,ÏÂ¯S(t) =âÎ·eâi2hÎ· Â·Ï tãÎ·|Ï(0)|Î·ãe i2hÎ· Â·Ï t. (6.7)So the reduced density matrix for the central system is the sum over contributionsfrom each of the possible configurations of the bath spins. In each of these contri-butions effective magnetic field on the central is determined by the configurationof the bath spins. The operator ãÎ·|Ï(0)|Î·ã (which acts on the system Hilbert spaceonly) appearing in the expression (6.7) can be written in terms of correlators,ãÎ·|Ï(0)|Î·ã = 12N+1âSâB(â©âjâCÏzjâª+ Ï Â·â©ÏâjâCÏzjâª)âjâCÎ·j . (6.8)Note that the expression for the systemâs reduced density matrix (6.7) is true re-gardless of the initial state. In particular it still holds if the central spin is initiallyentangled with the bath. However if the system is initially separable from the bathÏ(0) = ÏÂ¯S(0)ÏÂ¯B(0) then the solution (6.7) simplifies to an average over an ensembleof static biases,ÏÂ¯S(t) =â« âââdÎ¾ PDB(Î¾) eâ i2h(Î¾)Â·Ï tÏÂ¯S(0)ei2h(Î¾)Â·Ï t, (6.9)77where h(Î¾) â¡ â0xË+ Î¾zË when the bias is Î¾ and PDB(Î¾) is the distribution of biases.The distribution of biases is set by the baths initial state,PDB(Î¾) â¡âÎ·Î´ï£«ï£­Î¾ ââjÏjÎ·jï£¶ï£¸ ãÎ·|ÏÂ¯B(0)|Î·ã. (6.10)The delta functions in equation (6.10) can be summed to give an almost continuousbias distribution if either: the bath state represents an ensemble of a large numberof states or if there are a large number of bath spins and the bath is initially in a(potentially pure) state with out a definite bias, Î¾Ë =âi ÏiÏzi . The decay of differentelements of the density matrix in this model is caused by interference of componentswith different biases. Note that distribution of biases (6.10) only depends on thediagonal parts of the bath density matrix. So that for instance the system evolutionwill be the same whether the bath is prepared in the pure stateÏÂ¯B(0) =âj(| âãj + | âãj)(ãâ |j + ãâ |j) (6.11)or if the bath is prepared in the impure stateÏÂ¯B(0) = 12N I. (6.12)The polarisation ãÏ (t)ã of the central spin is also averaged in the same way asthe central spin reduced density matrix,ãÏ (t)ã =â« âââdÎ¾ PDB(Î¾) g(t; Î¾) Â· ãÏ (0)ã. (6.13)Here g(t; Î¾) is the single qubit Green function from equation (4.24) with the fieldh = h(Î¾).6.1.1 Example: Gaussian Bias DistributionHere we consider the specific case of a Gaussian bias distribution with standarddeviation Î´Î¾ centred around Î¾ = 0, that isPDB(Î¾) =eâÎ¾2/(2Î´Î¾2)â2piÎ´Î¾. (6.14)This distribution naturally arises from the law of large numbers when many uncor-related random factors contribute to the bias. For example if the bath is initiallyprepared in either of the states (6.11) or (6.11), all the couplings Ïi = Ï0 are thesame, there are a large number N of bathspins, and the coupling is small enough.Then the distribution of biases will be approximately that given by equation (6.14)78with Î´Î¾ = 14âNÏ0. The Green function is the averaged of the Green functions foreach bias,gÂµÎ½(t) â¡â« âââdÎ¾ PDB(Î¾) gÂµÎ½(t; Î¾) (6.15)=Î(t)â« âââdÎ¾ PDB(Î¾) hËÂµ(Î¾)hËÎ½(Î¾) (6.16)+ Reâ« âââdÎ¾ PDB(Î¾) eih(Î¾)t(Î´ÂµÎ½ â hË(Î¾)ÂµhË(Î¾)Î½ + iÎµÂµÎ³Î½ hËÎ³(Î¾)).Using the symmetry of the bias distribution we havegÂµÎ½(t) =Î(t)â« ââ0dÏA(Ï)[zËÂµzËÎ½ +â20Ï2(xËÂµxËÎ½ â zËÂµzËÎ½)]+ Reâ« ââ0dÏA(Ï)eiÏt(Î´ÂµÎ½ â[zËÂµzËÎ½ +â20Ï2(xËÂµxËÎ½ â zËÂµzËÎ½)]+ iâ0ÏÎµÂµ1Î½)(6.17)with A(Ï) =â2piÎ´Î¾2eâ (Ï2ââ20)2Î´Î¾2â1â (â0Ï )2 . (6.18)So the pole that we had before in the Laplace transform of the single qubit Greenfunction is smeared out into a branch cut and long time behavior of (6.18) is de-termined by the singularity at the branch point Ï = â0. At long times gÂµÎ½(t) willoscillate around gÂµÎ½â ,gÂµÎ½â =â« ââ0dÏA(Ï)[zËÂµzËÎ½ +â20Ï2(xËÂµxËÎ½ â zËÂµzËÎ½)](6.19)= zËÂµzËÎ½ +âpi2|â0|2Î´Î¾eâ202Î´Î¾2 erfc( |â0|â2Î´Î¾0)[xËÂµxËÎ½ â zËÂµzËÎ½ ] . (6.20)Where erfc denotes a complementary error function [26]. The long time oscillationswill have a frequency â0 and the different components to the oscillation amplitudeswill die down like,gÂµÎ½(t) â¼ gÂµÎ½â + Re{eiâtâipi4âpiÎ´Î¾â2t(zËÂµzËÎ½ + yËÂµyËÎ½ + iÎµÂµ1Î½ +34â2t2xËÂµxËÎ½)}(6.21)as t â +â (in this expression I have kept the leading order term for each inde-pendent tensor component). Examining the long time solution we see we only getcoherent rotation of the central qubit polarisation vector around the x axis and thatthe components of the polarisation vector in the z â y plane die down slower thanthose in the x direction. This is because averaging over both positive and negativebiases means that all rotation of components around the z axis due to the bias fieldis averaged out.796.1.2 Return ProbabilityOf particular interest to us is the probability pr(t) that the central spin will returnto its initial state after a given time, this can be calculated in our formalism. LetC stand for the central system and B stand for the bath then if the central systemstarts in a state |Ïã unentangled with the bath state |Î¦Bã so that the total wavefunction |Î¨(0)ã = |Ïã â |Î¦Bã then the return probability ispr =trB(ãÏ|U(t)|Î¨ããÎ¨|U â (t)|Ïã) (6.22)=trC(|ÏããÏ|ÏÂ¯C(t)) (6.23)=14trC[(I + ãÏ (0)ã Â· Ï ) (I + ãÏ (t)ã Â· Ï )] (6.24)=12(1 + ãÏ (0)ã Â· ãÏ (t)ã) (6.25)=12(1 + ãÏÂµ(0)ããÏÎ½(0)ãgÂµÎ½(t)). (6.26)So the symmetric part of gÂµÎ½(t) determines the return probability in one qubitsystems and in the case of degeneracy blocking for long times we havepr(t) â¼12{1 + ãÏx(0)ã2(gxxâ +3 cos(âtâ pi4)4â2piÎ´Î¾â30t5/2)(6.27)+ ãÏy(0)ã2(gyyâ +cos(â0tâ pi4)â0piÎ´Î¾â2t)+ ãÏ z(0)ã2(gzâz +cos(â0tâ pi4)â0piÎ´Î¾â2t)}.6.1.3 Decay of the Central Qubit PolarisationIn this model the polarisation of the central qubit (and therefore the purity of itsdensity matrix) decays. Indeed for generic separable initial conditions for the totaldensity matrix Ï(0) = 12(1 + ãÏ (0)ã Â· Ï )ÏÂ¯B(0), in the case discussed in the previoussection where ÏÂ¯B(0) results in a Gaussian distribution for the bias, the long timecentral qubit polarisation,ãÏ (t)ã2 = (gâãÏ (0)ã)2 +O(tâ12 ). (6.28)So that the central qubitâs polarisation decays to an initial condition dependentconstant, with the slowest decaying components decaying with tâ1/2.The decay of the polarisation indicates that information is lost to the environmentalbath. It is reasonable to ask, how does this information decay into the environment?80We can start by using the hierarchy of equations of motion (5.3) to derive an ex-pression for the rate of change of the central qubit polarisationddtãÏ ã2 = 2âjÏj(zË Ã ãÏÏzj ã) Â· ãÏ ã. (6.29)Thus the decay of the central qubit polarisation depends on the correlators ãÏxÏzj ãand ãÏxÏzj ã. In fact in this this model the only correlators involving bath spinswhich affect the dynamics of the central spin are of the formâ©Ï (t)âjâC Ïzj (t)âª(forC â B). This is because there is a closed subset of the hierarchy of equations,ddtâ©Ï (t)âjâCÏzj (t)âª=â0xËÃâ©Ï (t)âjâCÏzj (t)âª(6.30)+â`6âCÏ`zË Ãâ©Ï (t)Ïz` (t)âjâCÏzj (t)âª+â`âCÏ`zË Ãâ©Ï (t)âjâC{`}Ïzj (t)âª,which involve only these correlators. In fact we can calculate these correlatorsexactly, â©Ï (t)âjâCÏzj (t)âª=â{Î·`}âjâCÎ·zjg(t; Î¾({Î·`})) Â· ãÏ(0)ããÎ·|ÏÂ¯B(0)|Î·ã. (6.31)In the case where there is an equal likelihood of any given initial bath spin configu-ration (that is ãÎ·|ÏÂ¯(0)|Î·ã = 12N) and all bath spins have the same couplings Ïj = Ï0,then we have,â©Ï (t)âjâCÏzj (t)âª=12NNâ|C|ân=0(N â |C|n)(6.32)Â·|C|âm=0(|C|m)(â1)mg(t;âNÏ0 + 2Ï0(m+ n)) Â· ãÏ(0)ã.Before we discuss what these solutions actually look like, we note that correlators ofthe form (6.32) are the only non-zero correlators that arise in the evolution of thedensity matrix from the chosen initial conditions. Before we prove this statement,we note that it implies equation (6.32) actually gives us exact expressions for theconnected correlators (whether we define these as the coefficients in the Pauli ex-pansion of either ÏÂ¯CC or ÏÂ¯CCC , discussed in chapter 2). We now prove that the only81non-zero correlators are of the formâ©Ï(t)âjâC Ïzj (t)âª, for some subset of the bathC. First note that any possible correlator can be written in the either of the forms,â©ÏÂµâiâAÏaiiâjâCÏzjâª(6.33)â©âiâAÏaiiâjâCÏzjâª(6.34)for appropriate choices of the disjoint bath subsets A, C (which may be the zeroset), the index Âµ, and the indices {ai} which may take the values ai = 1 (x), 2 (y).Therefore the entire hierarchy of equations of motion can be written,ddtâ©ÏÂµâiâAÏaiiâjâCÏzjâª=â0ÎµÂµ1Î½â©ÏÎ½âiâAÏaiiâjâCÏzjâª(6.35)+â`âAÏ`zËÂµÎµa`3Î²â©ÏÎ²`âiâA{`}ÏaiiâjâCÏzjâª+âk 6âAâªBÏkÎµÂµ3Î½â©ÏÎ½âiâAÏaiiâjâCâª{k}Ïzjâª+âqâAÏqÎµÂµ3Î½â©ÏÎ½âiâAÏaiiâjâC{q}Ïzjâªddtâ©âiâAÏaiiâjâCÏzjâª=â`âAÏ`Îµa`3Î²â©Ï zÏÎ²`âiâA{`}ÏaiiâjâCÏzjâª. (6.36)Notice that in equations (6.35) and (6.36), the only values of the index Î² give nonzero contributions are Î² = 1, 2. This means that all correlators appearing on theright hand sides of equations (6.35-6.36) contain off-diagonal bath Pauli matricesacting on qubits in the set A. So for a fixed set A, the set of equations (6.35-6.36)with all applicable choices for the indices and set C, constitute a closed subset of thefull hierarchy of equations of motion. Therefore for each A 6= â , there is a closedsubset of linear equations of motion for the correlators of the form (6.33-6.34) andbecause all these correlators are zero initially, the unique solution to this subset ofequations is zero. When A = â the equation (6.36) reduces toddtâ©âjâCÏzjâª= 0, (6.37)and as these correlators are initially zero, they remain zero. We have now shown allthe correlators except those of the formâ©ÏÂµâjâC Ïzjâªare zero.82We have plotted increasing orders of correlators of this form with a range differentvalues of the parameters â0, Ï0, and N in figures 6.1-6.6. We have plots with 100bath spins (figures 6.1-6.3) andN = 10 (figures 6.4-6.6) bath spins. We show systemswhere the standard deviation of the biases on the central spin are 12âNÏ0 =14â0(figures 6.1 and 6.4), 12âNÏ0 = â0 (figures 6.2 and 6.5), and12âNÏ0 = 4â0 (figures6.3 and 6.6), one can think of these as systems with weak, intermediate, and strongcoupling respectively. We focus on the initial decay behaviour of the solutions, attimes before 2pi/Ï0, when there are approximate recurrences. What we see is, inall cases, as the central qubitâs polarisation decays there is a cascade of coherence.The coherence lost by the central qubit correlators is then is transferred to thesecond order correlators, then some portion of this is transferred to the third ordercorrelators and so on. This process occurs rapidly, correlators containing sevenbath spins are populated quickly. In the systems with larger baths and smallercouplings, correlators of all orders appear to tend toward a âsteady stateâ value(until recurrences). Note that the order of magnitudes of different sized correlatorscan vary drastically. In particular it appears from the figures 6.1-6.6, that the morepossible choices there are for correlators of a given order (e.g. are(N2)third ordercorrelators of the form ãÏÂµÏzi Ïzj ã) the smaller in magnitude correlators of that orderwill be. So it is desirable to consider a measure of the total strength of correlationsof a given order, which is what we turn to now.Recall from chapter one the purity of the full density matrix is p(t) = trÏ2(t) andas discussed in chapter 1 it gives a measure of how much information is containedin the full density matrix. The purity of the full density matrix is conserved (thisis easy to show: pË = âitr([H, Ï]Ï + Ï[H, Ï]) = 0). We can therefor get an idea ofhow much information is stored in correlators of different orders by computing theircontribution to the purity. Using the definition of the purity, the Pauli expansionof the density matrix (5.1), and our knowledge of the symmetry of the non-zerocorrelators in this case we have,p =12N+1{1 +âCâBâ©ÏâiâCÏziâª2}(6.38)=12N+1{1 +Nân=0(Nn)â©Ïnâi=1Ïziâª2}. (6.39)So we can define the part of the purity contained in the nâpartite correlations cnby,cn â¡(Nnâ 1)â©Ïnâ1âi=1Ïziâª2, (6.40)(we will call this the n-partite correlation strength). In figures 6.7-6.12 we haveplotted the time dependence of cnâs of different orders for the same cases described in8300.51-0.100.1-10-2010-2-10-3010-3-10-4010-4-3 10-503 10-5-5 10-605 10-60 20 -2 10-602 10-6-3 10-503 10-5-3 10-803 10-8-3 10-1003 10-10-3 10-1203 10-12-3 10-1303 10-13-10-14010-14-3 10-1503 10-150 20-5 10-1605 10-16Figure 6.1: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins, theircouplings are all Ï0 =120â0 (so12âNÏ0 =14â0), and the initial redu-ced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã . The left hand figure showsthe first eight correlators and the right hand figure shows higher ordercorrelators.8400.51-0.0500.050-10-3010-30 5 10 15 20 25 3002 10-44 10-504 10-5-10-500 5 -2 10-602 10-6-4 10-504 10-5-10-70-5 10-1005 10-10010-11-4 10-1304 10-13-2 10-140 2 10-14-5 10-1505 10-150 5 -10-150 10-15Figure 6.2: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins,their couplings are all Ï0 =15â0 (so12âNÏ0 = â0), and the initial re-duced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã . The left hand figure showsthe first eight correlators and the right hand figure shows higher ordercorrelators.8500.5100.050-10-3002 10-405 10-5-10-500 5 /8-4 10-6005 10-5-10-70-4 10-100010-1105 10-13-5 10-140-10-1400 5 /80 2 10-15Figure 6.3: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 100 bath spins,their couplings are all Ï0 =45â0 (so12âNÏ0 = 4â0), and the initial re-duced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã . The left hand figure showsthe first eight correlators and the right hand figure shows higher ordercorrelators.8600.51-0.200.2-0.04-0.020-0.0200.02-5051010-3-0.0100.01-0.0100.01-0.0100.010.02-0.0200.02-0.100.10 2 4 6-0.200.2Figure 6.4: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins, theircouplings are all Ï0 â 0.1581â0 ( 12âNÏ0 =14â0), and the initial re-duced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã.8700.51-0.200.2-0.06-0.04-0.020-0.04-0.0200.0200.010.020.03-0.0100.010.02-0.02-0.010-0.0500.05-0.0200.020.040.06-0.2-0.100.10 /2 3 /2-0.6-0.4-0.200.2Figure 6.5: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins, theircouplings are all Ï0 â 0.6325â0 ( 12âNÏ0 = â0), and the initial redu-ced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã .8800.51-0.200.2-0.1-0.050-0.0500.0500.020.04-0.0200.02-0.03-0.02-0.010-0.0500.0500.050.1-0.200.20 3 /8-1-0.50Figure 6.6: Plots of the correlators of the form ãÏÂµânj Ïzj ã for the degeneracyblocking model described in section 6.1.2.Values are computed fromequation (6.32). In the case plotted there are N = 10 bath spins, theircouplings are all Ï0 â 2.529â0 ( 12âNÏ0 = 4â0), and the initial redu-ced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx). In all figuresthe solid red curve is ãÏxânj Ïzj ã, the dashed green curve is ãÏyânj Ïzj ã,and the dotted blue curve is ãÏ zânj Ïzj ã .890 10 2010-610-510-410-310-210-1100261014182226303438Correlator order:Figure 6.7: The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =120â0 (so12âNÏ0 =14â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).the previous paragraph. From these figures we see that a portion of the correlationstrength is cascaded between different orders. In the cases where we can see adefinite âsteady stateâ emerge (those with N = 100 and strong to intermediatecoupling strength), the steady state nâpartite correlation strength decreases withthe order (at least for orders n < N/2).Before we move on to other examples, an obvious question one can ask is, doesthe picture we have developed in this section of the decay central qubitâs polarisationas a cascade into higher order correlations hold in more complicated models thatdescribe decoherence? We attempt to answer this for one model of decoherence inchapter 7.900 2.5 5 10-610-510-410-310-210-1100261014182226303438Correlator order:Figure 6.8: The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =15â0 (so12âNÏ0 = â0), and theinitial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).910 5 /810-610-510-410-310-210-1100261014182226303438Correlator order:Figure 6.9: The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 100bath spins, their couplings are all Ï0 =45â0 (so12âNÏ0 = 4â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).920 2 4 610-610-510-410-310-210-1100246810Correlator order:Figure 6.10: The nâpartite correlation stregth for the degeneracy blocking mo-del described in section 6.1.2. Values are computed from equation(6.32) and the definition for cn (6.40). In the case plotted thereare N = 10 bath spins, their couplings are all Ï0 â 0.1581â0 (12âNÏ0 =14â0), and the initial reduced density matrix for the sy-stem is ÏÂ¯S(0) = 12 (1 + Ïx).930 /2 3 /210-610-510-410-310-210-1100246810Correlator order:Figure 6.11: The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 10bath spins, their couplings are all Ï0 â 0.6325â0 ( 12âNÏ0 = â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).940 3 /810-610-510-410-310-210-1100246810Correlator order:Figure 6.12: The nâpartite correlation stregth for the degeneracy blocking modeldescribed in section 6.1.2. Values are computed from equation (6.32)and the definition for cn (6.40). In the case plotted there are N = 10bath spins, their couplings are all Ï0 â 2.529â0 ( 12âNÏ0 = 4â0), andthe initial reduced density matrix for the system is ÏÂ¯S(0) = 12 (1 + Ïx).956.2 Example: Motion of Spin in Time Dependent BiasIn the previous section we studied a model where the effect of an external envi-ronment on the central spin is to average the energy bias on the central qubit. Inthis section we will consider the case where an external bias can result in a timedependent bias on the central system, which will prepare us to tackle the problem ofaveraging over a random dynamics bias in the next section. These kinds of modelsshould be used with caution as the neglect quantum correlations built up betweenthe central system and the environment. Consider the Hamiltonian describing a twolevel system with constant flipping probability and a time dependent biasH =12â0Ïx +12Î¾(t)Ï z. (6.41)The equation of motion for ãÏ (t)ã is,ddtãÏ (t)ã =ï£®ï£°â0ï£«ï£­0 0 00 0 â10 1 0ï£¶ï£¸+ Î¾(t)ï£«ï£­0 â1 01 0 00 0 0ï£¶ï£¸ï£¹ï£» ãÏ (t)ã. (6.42)While this equation of motion is not exactly solvable we can write down a formalsolution, which we now describe. Begin by solving the problem in the â0 = 0 case,this leads us to the Green function g0,g0(t, tâ²) = expï£«ï£­ââ« ttâ²dsÎ¾(s)ï£«ï£­0 â1 01 0 00 0 0ï£¶ï£¸ï£¶ï£¸ =ï£«ï£­ cos Î¸(t, tâ²) sin Î¸(t, tâ²) 0â sin Î¸(t, tâ²) cos Î¸(t, tâ²) 00 0 1ï£¶ï£¸(6.43)with Î¸(t, tâ²) =â« ttâ²dsÎ¾(s). (6.44)In the time between tâ² and t the zero â0 Green function rotates the qubits polari-sation around the z axis by an angle Î¸(t, tâ²). The final total Green function g(t, tâ²)is related to this by, the usual Dyson seriesâ =â0ï£«ï£­0 0 00 0 â10 1 0ï£¶ï£¸ (6.45)g(t, tâ²) =g0(t, tâ²) +â« ttâ²dsg0(t, s)âg(s, tâ²) (6.46)=g0(t, tâ²) +âân=1(â« ttâ²dsng0(t, sn)â)(â« sntâ²dsnâ1g0(sn, snâ1)â). . .(â« s2tâ²ds1g0(s2, s1)â)g0(s1, tâ²). (6.47)96This formal series solution is the best we can do without specifying more about thetime dependence of the bias field. We note that the series (6.47) solution to thisproblem is equivalent to writing the Green function in terms of the time orderedexponential T expg(t, tâ²) = T exp{â« ttâ²ds [â + Î¾(s)]}, (6.48)withÎ¾(t) â¡ Î¾(t)ï£«ï£­0 â1 01 0 00 0 0ï£¶ï£¸ . (6.49)Numerical evaluation of these kind of formulae is discussed in appendix D.6.3 Randomly Fluctuating ForceWe now consider models where the bias is fluctuating randomly over time. Thismay be because the system is coupled to some fluctuating bath variable. In thatcase the model is only applicable where the central systems dynamics do not effectthe bath dynamics in any appreciable way. We are talking about the case where theHamiltonian is (6.41) but where the functions Î¾(t) are drawn from some probabilitydistribution described by a probability distribution functional P [Î¾(t)]. We denoteaverages over this probability distribution by ã. . .ãÎ¾, the average of quantity Q[Î¾] isgiven by the functional integral (see for example [35]),ãQ[Î¾]ãÎ¾ â¡â«DÎ¾ P [Î¾] Q[Î¾]. (6.50)Obviously there are many different specifications for the probability distributionwhich are interesting, below we discuss one specific case.6.3.1 Gaussian NoiseConsider the case where Î¾(t) has a Gaussian probability distribution function definedby a correlator K(t, tâ²):P [Î¾] â¡ N exp{â12â«dÏ1â«dÏ2Î¾(Ï1)Kâ1(Ï1, Ï2)Î¾(Ï2)}(6.51)K(t, tâ²) = ãÎ¾(t)Î¾(tâ²)ãÎ¾ (6.52)Then one can use the well known relationship [35] ,ãeiâ« t0 dsÎ¾(s)ãÎ¾ = exp[iâ« t0ds1ãÎ¾(s1)ãÎ¾ ââ« t0ds1â« t0ds2K(s1, s2)](6.53)to calculate the exact Green function in the absence of flipping. Now we work witha specific form for noise correlator.976.3.2 Example: Low frequency noise, diffusing biasNow consider the case [86] where Î¾(t) is randomly fluctuating withã|Î¾(t)â Î¾(tâ²)|2ãÎ¾ = Î3|tâ tâ²|. (6.54)That is the bias field under goes Brownian motion. Then the Green function in theabsence of flipping averaged over the bias, ãg0(t)ãÎ¾ is thenãg0(t)ãÎ¾ =ï£«ï£­ cos(Î¾0t)eâÎ3|t|3/6 sin(Î¾0t)eâÎ3|t|3/6 0â sin(Î¾0t)eâÎ3|t|3/6 cos(Î¾0t)eâÎ3|t|3/6 00 0 1ï£¶ï£¸ . (6.55)Consider the regime where the diffusion is fast so that Î |â0|. The averaged solu-tion to the equation of motion is shown in figure 6.13. We see that the componentsof the polarisation in the x â y plane decay more rapidly than the z component,there is however some coherent oscillation between the y and z components.We can understand this behaviour analytically. Components of the central qubitpolarisation components ããÏxããÎ¾ and ããÏyããÎ¾, which are perpendicular to the diffusingfield die down rapidly. To leading order we have(ããÏxããÎ¾ããÏyããÎ¾)=(cos(Î¾0t)eâÎ3|t|3/6 sin(Î¾0t)eâÎ3|t|3/6â sin(Î¾0t)eâÎ3|t|3/6 cos(Î¾0t)eâÎ3|t|3/6)(ãÏx(0)ããÏy(0)ã)+O(â20). (6.56)For the decay of ãÏ z(t)ã to occur the flipping term is necessary. We will now at-tempt to calculate the rate of this slower decay. We define an âinteraction pictureâpolarisation vector ãÏ (t)ãI , which is the measured polarisation in a frame rotatedaccording to the bias field alone,ãÏ (t)ãI = g0(0, t)ãÏ (t)ã. (6.57)Then the equation of motion for ãÏ (t)ãI isddtãÏ (t)ãI = g0(0, t)âgT0 (0, t)ãÏ (t)ãI . (6.58)The formal solution to equation (6.58) isãÏ (t)ãI =â« t0dt1g0(0, t1)âgT0 (0, t1)ãÏ (t1)ãI + ãÏ (0)ã. (6.59)Substituting the right hand side of equation (6.59) back into itself gives,ãÏ (t)ã =â« t0dt2g0(t, t2)ââ« t20dt1g0(t2, t1)âãÏ (t1)ã (6.60)+â« t0dt1g0(t, t1)âg0(t1, 0)ãÏ (0)ã+ g(t1, 0)ãÏ (0)ã.980 1 2 3010 5 100.810 2 4 6 8 10-0.2-0.100.1Figure 6.13: Plots of some solutions to the diffusing bias model discussed in section6.3.2 calculated using the methods described in appendix D. In bothcases the parameters are chosen so that â0 = 0.1Î and Î¾0 = 0. Inboth figures the solid red curve is ããÏ z(t)ããÎ¾, the dashed green curve isããÏy(t)ããÎ¾, and the dotted blue curve is ããÏx(t)ããÎ¾ computed using themethod described in appendix D.1. In the top figure the initial confi-guration of the central spin is ãÏ (0)ã = yË and the back curve is the â0solution for ããÏy(t)ããÎ¾ calculated using equation (6.56). In the bottomfigure the initial configuration of the central spin is ãÏ (0)ã = zË andthe back curve is the solution calculated using the integro-differentialequation (6.63). The inset figure on the bottem shows the considerablysmaller ããÏx(t)ããÎ¾, and ããÏy(t)ããÎ¾ components of the polarisation. Thethin black curve in the inset plot shows the y compenent of the polari-sation computed numerically from the equation ddtããÏ zããÎ¾ = â0ããÏ zããÎ¾and the numerical values of the z component rather than the directmethod in the appendix.99Differentiating and taking the z component givesddtãÏ (t)ã =â« t0ds1âg0(t, s1)âãÏ (s1)ã (6.61)ddtãÏ z(t)ã = ââ2â« t0ds1 cos(â« ts1ds2Î¾(s1))ãÏ z(s1)ã. (6.62)Now take the average over the rapidly varying field Î¾(t). Assuming that (6.62) canbe written as an integral of a product of independent averages,ddtããÏ z(t)ããÎ¾ = ââ2â« t0ds1Re(eiÎ¾0(tâs1)â16Î3(tâs)3)ããÏ z(s1)ããÎ¾. (6.63)The integro-differential equation (6.63) can be solved numerically (we use the ID-SOLVER package [42] for MATLAB), this solution is compared against the fullnumerical solution to the motion in figure 6.13. We see that the integro-differentialdifferential equation (6.63) is only accurate for short times. The reason for this isas follows. The central spin will only flip between its two z states when the bias isnot much greater than the tunneling element â0 and because this element is smallcompared to the time scale for the diffusion of the bias field, this will only occurrarely when the bias diffuses to zero. If the bias is zero at some time t0, it is thensignificantly more likely to be zero again at some later time than if the bias hadbeen large at t0. So as ããÏ z(t)ããÎ¾ can change via tunneling, ããÏ z(s1)ããÎ¾ is correlatedwith cos(â« ts1ds2Î¾(s1))and that correlation grows as t increases, making the ap-proximation (6.63) invalid over long times.So far we have calculated features of the model with diffusing bias in the limitwhere the diffusion is fast compared to the tunneling time scale. We could obtainfurther results for other regimes in this model. Instead as we are mainly interestedin the dynamics of entanglement of the system with a bath, which is not capturedin this model, we note the salient features of this model in the regime we havediscussed and move on. We see that in this model in the regime studied ããÏxããÎ¾decays rapidly at a time scale set by the Îâ1. ããÏ zããÎ¾ decays over a larger time scalethat depends on both the tunneling amplitude â0 and Î. ããÏyããÎ¾ is approximatelythe sum of two components, a component that decays rapidly on the fast timescale Îâ1 and a component that decays on the slower timescale which comes fromcoherent oscillations around the x.6.4 ConclusionIn this chapter we have explore some simple models of the interaction of a centralqubit coupled to an environment.100We have derived new results for correlators between the central qubit and spinbath in the degeneracy blocking model in section 6.1 and we have seen that informa-tion lost by the central qubit is transferred to higher and higher order correlationswith bath spins. In chapter (7) will provide evidence for a similar cascade in a modelof precessional decoherance, where the bath spins are allowed to flip.In section 6.2 we derived methods for dealing with qubits under the influence ofa dynamic bias. Finally in section 6.3 we study the effect of time dependent randombias acting on a qubit and saw some of the subtitles involved in using the equationsof motion, when different components evolve on different different timescales.101Chapter 7The Spin Bath InfluenceFunctional and PrecessionalDecoherenceWe now derive an influence functional approach to the motion of a system of centralqubits coupled to a bath, analogous to that discussed in section 1.7.1 for the oscillatorbath. This is a complementary approach the hierarchy of equations of motion. Theform of the spin bath influence functional that we derive is well suited to studyingsystems with a well defined discreet classical Hamiltonian and few different possibletransitions or âjumpsâ which can occur.We then use the influence functional to re-derive and extend the results obtainedby Prokofâev and Stamp [86], for precessional decoherence. In order to understandthe dynamics of decoherence we calculate evolution of correlations between the cen-tral qubit and the bath spins in this model.7.1 The Influence Function for Qubit SystemsCoupled to a Spin BathConsider the case in section 1.7.2 were we have a set of system spins and bath spinsgoverned by the Hamiltonian,H =12NSâa=1ha Â· Ï a + 12âaâb 6=auabÂµÎ½ÏÂµa ÏÎ½b +12NBâj=1bj Â· Ïj + 12âaâjV ajÂµÎ±ÏÂµa ÏÎ±j . (7.1)Now we chose the âclassicalâ basis of the total Hilbert space in terms of a list of theclassical values of the qubits s = (s1, s2, . . . , sNS ), Î· = (Î·1, Î·2, . . . , Î·NB) are binaryvalues sa, Î·i = Â±1 so that our kets areâ£â£s Î·âª â¡âaâS|saãâiâB|Î·iã. (7.2)It is then natural to split the Hamiltonian into parts H = Hâ + Hâ¥ which arediagonal Hâ in and completely off diagonal Hâ¥ in this basis. For the Hamiltonian102(7.1) we haveHâ =12NSâa=1hzaÏza +12âaâb 6=auabzzÏza Ïzb +12NBâj=1bzjÏzj +12âaâjV ajzz ÏzaÏzj (7.3)Hâ¥ =H âHâ. (7.4)â£â£s Î·âª is by construction an eigenstate of Hâ with an eigenvalue hsÎ· which is a classicalenergy associated with the classical bit configuration,hsÎ· â¡h0s + hSBsÎ· (7.5)h0s â¡12NSâa=1hzasa +12âaâb 6=auabzzsasb (7.6)hSBsÎ· â¡12NBâj=1bzjÎ·j +12âaâjV ajzz Î·jsa. (7.7)Where as Hâ¥ defines a set of possible âjumpsâ or more precisely transition ampli-tudes between different states in the{|sÎ·ã} basis. We now proceed to derive theinfluence functional. Start by defining the complete set of projectors psÎ·,psÎ· â¡â£â£s Î·âª â©s Î·â£â£ . (7.8)For each configuration of sÎ· the projection operator psÎ· projects onto a distinctorthogonal direction in Hilbert space, so we have the following completeness andorthogonality conditionspsÎ·psâ²Î·â² =psÎ·Î´ssâ²Î´Î·Î·â² (7.9)âÎ·âspsÎ· =1 (7.10)and we can write the diagonal Hamiltonian and the associated evolution operatorU0(t, tâ²) = eâiHâ(tâtâ²) asHâ =âÎ·âshsÎ·psÎ· (7.11)Hâ¥ =âÎ·âsâÎ·â² 6=Î·âsâ² 6=spsâ²Î·â²Hâ¥psÎ· (7.12)U0(t, tâ²) =âÎ·âseâihsÎ·(tâtâ²)psÎ·. (7.13)103Then expanding in the usual way the Dyson series [91] for the evolution operator inpowers off diagonal HamiltonianU(t) =âân=0â{Î·`,s`}(â« t0dtn . . .â« t20dt1)exp{âinâ`=0(t`+1 â t`)hs`Î·`}(7.14)psnÎ·nHâ¥psnâ1Î·nâ1Hâ¥ . . . ps1Î·1Hâ¥ps0Î·0where tn+1 = t in each term in the sum (7.14). We can interpret the expression forthe evolution operator (7.14) as a âsum over all pathsâ expression, here the pathsconsists of transitions contained in Hâ¥ between different states of the form |sÎ·ã, attimes {t`}, with accumulation of phase according to Hâ in the time between thetransitions events. Now one can use the expression (7.14) to derive an expressionfor the elements of the system reduced density matrix ÏÂ¯s sËS â¡ ãs|ÏÂ¯S |sËã in terms of theinitial full density matrix Ï(0)ÏÂ¯s sËS (t) =âân=0âânË=0â{Î·`,s`}{Î·Ë`,sË`}(â« t0dtn . . .â« t20dt1â« t0dtËn . . .â« tË20dtË1)(7.15)expï£±ï£²ï£³âinâ`=0(t`+1 â t`)hs`Î·` + inËâË`=0(tËË`+1 â tËË`)hsËË`Î·Ë Ë`ï£¼ï£½ï£¾ãsÎ·n|psÎ·nHâ¥psnâ1Î·nâ1Hâ¥ . . . ps1Î·1Hâ¥ps0Î·0Ï(0)psË0Î·Ë1Hâ¥psË1Î·Ë1 . . . Hâ¥psËnËâ1Î·ËnËâ1Hâ¥psËÎ·n |sËÎ·nã.We will refer to (7.15) as the transition expansion for the system reduced densitymatrix, it involves the sum of two sets of possible paths one with the system evolvingforward in time with transitions occurring at the times {t`} and the other evolvingbackwards in time with transitions occurring at the times {tËË`}.The transition expansion expression for ÏÂ¯S can be made to look a bit more likethe influence functional expressions obtained when there is an oscillator bath. Wesplit the projection operators psÎ· = pSs pBÎ· , into parts acting on the bath pBÎ· = |Î·ããÎ·|and central system pSs = |sããs|. Then we assume that the off diagonal Hamiltoniancan be written as a sum of products between system âtransitionâ operators {tËa} andbath âtransitionâ operators {TËa}Hâ¥ =âatËaTËa. (7.16)Note âtransitionâ appeared in quotation marks in that last sentence because theway we have defined it Hâ¥ may have terms that only cause transitions either the104system or bath. Then if we assume that the initial density matrix is separable,Ï(0) =ÏÂ¯S(0)ÏÂ¯B(0), we can write the transition expansion (7.15),ÏÂ¯s sËS (t) =âân=0âânË=0â{Î·`,s`}{Î·Ë`,sË`}â{a`}{aËË`}(â« t0dtn . . .â« t20dt1â« t0dtËn . . .â« tË20dtË1)(7.17)expï£±ï£²ï£³âinâ`=0(t`+1 â t`)h0s` + inËâË`=0(tËË`+1 â tËË`)h0sËË`ï£¼ï£½ï£¾F [s(t), sË (tË) , {a`}, {aËË`}]ãs|pSs tËanpSsnâ1 tËanâ1 . . . pSs1tËa1pSs0ÏÂ¯S(0)pSsË0 tËâ aË1pSsË1 . . . tËâ aËnËâ1pSsËnËâ1 tËâ aËnËpSsË |sËã.Where F[s(t), sË(tË), {a`}, {aËË`}]is a functional of the two paths for the spin variabless(t), sË(t) by the transition times {t`}, {tËË`} and the values {s`}, {sËË`} in between thetransitions, sos(t) ={s` for t`â1 < t < t` (â` = 1, . . . , n)}(7.18)sË(Ët) ={sËË` for tËË`â1 < tË < tËË` (âË`= 1, . . . , nË)}. (7.19)The influence functional also depends on the sequences of different parts of the offdiagonal Hamiltonian that result in these paths, specified by {a`}, and {aËË`}. Theinfluence functional is,F[s(t), sË(tË), {a`}, {aËË`}]=â{Î·`}{Î·Ë Ë`}expï£±ï£²ï£³âinâ`=0(t`+1 â t`)hSBs`Î·` + inËâË`=0(tËË`+1 â tËË`)hSBsËË`Î·Ë Ë`ï£¼ï£½ï£¾(7.20)ãÎ·n|TËanpÎ·Î·nâ1TËanâ1 . . . pBÎ·1TËa1pBÎ·0ÏÂ¯B(0)pBÎ·Ë0TË â aË1pBÎ·Ë1. . . TË â aËnËâ1pBÎ·ËnËâ1TË â aËnË |Î·nã.In general transition expansion is a complicated expression for the time evolutionof the reduced density matrix or influence functional. The complexity of the differentsums depend on the off diagonal part of the Hamiltonian (indeed when Hâ¥ = 0 theexpression is trivial) which specifies all possible transitions. A strategy that canmake the transition expansion manageable is to identify which sets of transitionsare most important to the coherent evolution of the density matrix and then workwith this reduced set of possible transitions, while either ignoring those that areless important or including them perturbatively. We will take this approach inthe next section using the specific example of the precessional decoherence problemintroduced in section 1.7.2.1057.2 Evaluating the Influence Functional forPrecessional DecoherenceConsider the precessional decoherence model discussed in section 1.7.2 with a singlecentral spin. The central spin has a purely longitudinal Ïâi coupling to bath spins{Ïi}, the bath spins and the central spins both have transition amplitudes â0 andâiÏâ¥i (we have rotated our frame, from that in section 1.7.2). We assume that thetransition amplitudes for the bath spins are small.H =12â0Ïx + 12Î¾0Ïz â 12âiâBÏâ¥i Ïyi +12âiâBÏâi Ïzi Ïz (7.21)â¡12â0Ïx + 12Î¾0Ï z +âiâBBËi (7.22)The precessional decoherence Hamiltonian is more complicated than the simple de-generacy blocking Hamiltonian discussed in the chapter 6, in the processional casethe energy splitting of the central spin is no longer static as the bath spins canflip. So each transition in the transition expansion the central spin or any of thebath spins may flip making the expansion in its most straightforward form unwieldy.However there is a transformation which simplifies things considerably for this case.7.2.1 Canonical Transformation of the Bath Spin VariablesThe part of the Hamilonain BËi which acts on a single bath spin i may be writtenBËi =12Ïz(Ïâi Ïzi â Ïâ¥i Ï zÏyi)= 12Ïieâ i2Î²iÏzÏxi Ï zÏzi ei2Î²iÏzÏxi (7.23)with ({Î²i} are simply the angles defined in section 1.7.2)Ïi cosÎ²i =Ïâi (7.24)Ïi sinÎ²i =Ïâ¥i . (7.25)Thus BËi may be viewed as a longitudinal coupling in a frame where the bath spinsare rotated by an angle depending on the central spins state. Hamiltonian can thenbe writtenH = 12â0Ïx + 12 UËâ (Î¾0Ïz +âiÏiÏzÏzi)UË (7.26)whereUË â¡âiâBei2Î²iÏxi Ïz(7.27)(not be confused with U(t) the evolution operator). Now perform a canonical trans-formation on Hilbert space |Ïã(old) â UË |Ïã(new) then the Hamiltonian in this new106basis isH = 12â0UËÏxUË â + 12Î¾0Ïz + 12âiÏiÏzÏzi . (7.28)In this new basis the transformed Hamiltonian (7.28) action of the off diagonal partof the Hamiltonian Hâ¥ = 12â0UËÏxUË â always flips the central spin. So that the newbath spins variables only flip when the central spin also flips. This will help whenusing the transition expansion for the reduced density matrix of the central spin.The off diagonal part of the Hamiltonian can be written in a more intuitive formusing the âladderâ operators ÏÂ± = | Â± 1ããâ1|Hâ¥ =12â0Ï+eiâiâB Î²iÏxi + 12â0ÏâeâiâiâB Î²iÏxi (7.29)=12â0Ï+âCâBi|C|âj /âCcosÎ²jâiâCsinÎ²iÏxi + h.c. (7.30)Equations (7.29) and (7.30) can be interpreted as follows, the terms that flip ofthe central spin rotate the bath spins around their x axis in either the positiveor negative orientation depending on the initial state of the central spin. As suchthe transition Hamiltonian contains terms that co-flip any number of bath spinswith a central spins with amplitudes depending on the angles {Î²i}. Also, as thereare off diagonal interaction terms in (7.30) containing products of Pauli operatorsbelonging to all subsets of the spin bath, the hierarchy of equations of motion forthe qubits in this basis will link all orders of correlators, making the hierarchy lessuseful in this transformed basis.7.2.2 The Orthogonality Blocking Approximation.Now we describe an approximation first presented by Prokofâev and Stamp [86, 88],which will reduce the number of components of Hâ¥ that we need to consider. Westart with a mathematical explanation of this approximation before returning to aphysical motivation. Begin by writing the transition expansion for the evolutionoperator (7.14) for this case in the slightly different formU(t) =âân=0â{Î·`,s`}(â« t0dtn . . .â« t20dt1)exp{âi12nâ`=1t`(Î¾Î·`â1s`â1 â Î¾Î·`s`)}(7.31)psnÎ·nHâ¥psnâ1Î·nâ1Hâ¥ . . . ps1Î·1Hâ¥ps0Î·0 ,whereÎ¾Î· â¡ Î¾0 +âiÏiÎ·i, (7.32)is the bath spin dependent bias on the central spins. Equation (7.31) can be viewedas a perturbation expansion in Hâ¥ or equivalently in â0. In such an expansion theterms can be classed as either secular or non-secular [7]. The secular terms diverge107as some power of t as t â â, while the non-secular terms remain bounded for allt. The secular terms occur when one of the dt` integrals has an integrand with azero frequency component. One can obtain an approximation which is valid for longtimes by summing all the most secular terms for every order of â0. That is keepingthe terms in the series (7.31) in which the frequency is zero in every integrand, viz,Î¾`â1s`â1 = Î¾Î·`s`, (7.33)for all `, this is the orthogonality blocking approximation. Physically this amountsto keeping only transitions between resonant levels of Hâ, as these are the dominanttransitions. Before we explore fully the validity of this approximation and use it tocalculate the systems reduced density matrix we will specify some details involvedin problem.7.2.3 Calculation of the Reduced Density MatrixBefore completing the calculation we need to specify the initial density matrix andparameters.For simplicity we will assume that the bias field Î¾0 = 0, so that all the bias comesfrom the bath spins. We will also assume the couplings can be written as,Ïj = Ï0 + Î´Ïj , (7.34)and are some small deviation Î´Ïj from a mean value of Ï0. We begin by assumingÎ´Ïj = 0 for simplicity.We assume the system and the bath are in a product state to begin with so thatÏ(0) = ÏS(0)ÏÂ¯B(0), where the initial state for the central system is some arbitrarypure state ÏÂ¯S(0) which has been prepared by some process. Furthermore assumethat as the bath spins do not interact directly, the bath initial state is a productstate ÏÂ¯B(0) =âiâB ÏÂ¯i(0) where ÏÂ¯i(0) = | Â± 1ããÂ±1|. We will make the further as-sumption that the number of bath spins which are initially up and the numberwhich are initially down are equal. This along with the Î¾0 assumption correspondsto operating the qubit at its so called âsweet spotâ, where the rate of change of thecentral qubit energy with Î¾ is zero. Operating qubits at such a point is a commonlyused method to prevent decoherence, see for example [55, 97, 106, 112].The fact that we are working in a basis which is transformed from the physicalbath states complicates things a little here but we assume that the central spin stateis prepared in such a way that the separation in the transformed basis is nearly ex-act. We will consider the effect of relaxing this approximation later.108As discussed earlier the amplitude for flipping the bath spins is much less than thecoupling to the central spin bâ¥i Ïâi so that Î²i 1. This allows for a large variety oftransitions which still satisfy the orthogonality blocking condition Î¾Î·`â1s`â1 = Î¾Î·`s`.In this case we will see the orthogonality blocking approximation will give us theleading term in an expansion for the dynamics in the small parameter Î²i.With the above assumptions and initial conditions in mind we find that theorthogonality blocking approximation just means that between each central spinflip the bath is in the zero polarisation group. So that the only relevant part of theoff diagonal Hamiltonian isHâ¥ â 12â0Ï+P0eiâiâB Î²iÏziP0 +12â0ÏâP0eiââiâB Î²iÏziP0 (7.35)where P0 is the projection operator that projects the bath onto the zero magenti-sation polarisation group. P0 can be writtenP0 =â« 2pi0dÎ¾2piexp(iâiâBÎ¾Ïzi). (7.36)Now one can employ the transition expansion (7) to obtain an expression for thereduced density matrix of the central system, which can be specified with the cor-relatorsãÏ+(t)ã =âân=0{(âiâ0t)2n(2n)![ãÏâãA(n)+â + ãÏ+ãA(n)++]+(âiâ0t)2n+1(2n+ 1)![12(1 + ãÏ zã)A(n)+â +12(1â ãÏ zã)ãA(n)+â]}(7.37)ãÏâ(t)ã =âân=0{(âiâ0t)2n(2n)![ãÏâãA(n)ââ + ãÏ+ãA(n)â+]+(âiâ0t)2n+1(2n+ 1)![12(1 + ãÏ zã)A(n)ââ +12(1â ãÏ zã)ãA(n)ââ]}(7.38)ãÏ z(t)ã =âân=0{(âiâ0t)2n(2n)!12[(1 + ãÏ z(0)ã)A(n)zâ + (1â ãÏ z(0)ã)A(n)zâ]+(âiâ0t)2n+1(2n+ 1)![ãÏ+(0)ãA(n)z+ + ãÏâ(0)ãA(n)zâ]}â 12. (7.39)Here the quantities A(n)ÂµÎ½ are sums of influence functionals for the spin bath with thesame numbers of flips of the central spin where the central spins path has the same109end points, mathematicallyA(n)+â â¡122nnâ`=0(2n2`)â©(P0Uâ P0U)` (P0UP0Uâ )nâ`âª(7.40)A(n)++ â¡â122nnâ1â`=0(2n2`+ 1)â©P0U(P0Uâ P0U)` (P0UP0Uâ )nâ`P0Uâª(7.41)A(n)+â â¡122n+1nâ`=0(2n+ 12`)â©(P0Uâ P0U)` (P0UP0Uâ )nâ`P0Uâª(7.42)A(n)+â â¡â122n+1nâ`=0(2n+ 12`+ 1)â©P0U(P0Uâ P0U)` (P0UP0Uâ )nâ`âª(7.43)A(n)ââ â¡122nnâ`=0(2n2`)â©(P0UP0Uâ )` (P0Uâ P0U)nâ`âª(7.44)A(n)â+ â¡â122nnâ1â`=0(2n2`+ 1)â©P0Uâ (P0UP0Uâ )` (P0Uâ P0U)nâ`P0Uâ âª(7.45)A(n)ââ â¡122n+1nâ`=0(2n+ 12`)â©(P0UP0Uâ )` (P0Uâ P0U)nâ`P0Uâ âª(7.46)A(n)ââ â¡â122n+1nâ`=0(2n+ 12`+ 1)â©P0Uâ (P0UP0Uâ )` (P0Uâ P0U)nâ`âª(7.47)A(n)zâ â¡12â©(P0Uâ P0U)nâª(7.48)A(n)zâ â¡â 12â©(P0UP0Uâ )nâª(7.49)A(n)z+ â¡12â©(P0Uâ P0U)nP0Uâ âª(7.50)A(n)zâ â¡â 12â©P0U(P0UP0Uâ )nâª. (7.51)Here(n`)= n!`!(nâ`)! is the binomial coefficient, U = U({Î²i}) is the unitary rotationU({Î²i}) â¡ eiâj Î²jÏxj , (7.52)and all the averages are traces over the bath, ã. . .ã = trB(. . . ÏÂ¯B(0)). Now the problemof the calculation of the reduced density matrix is reduced to averaging factors thatdepend on expectations of strings of operators acting on the bath space. Prokefâevand Stamp [86] used this approximation to compute the return probability prâ(t) forthe central spin when it is initially polarised in the up direction. We will present thiscalculation here and then show that the results obtained by Prokefâev and Stamp110can be generalised to arbitrary initial conditions.From equation (7.39) we haveprâ(t) = 12(1 + ãÏz(t)ã) =âân=0(âiâ0t)2n(2n)!12â©(P0Uâ P0U)nâª. (7.53)The expectation appearing in this expression can be factorised using the represen-tation (7.36) for the projection operators,â©(P0Uâ P0U)nâª(7.54)=â« 2pi0d2nÎ¾(2pi)2nâiâBâ©(eiÎ¾1Ïzi eâiÎ²iÏxi eiÎ¾2Ïzi eiÎ²iÏxi)(eiÎ¾3Ïzi eâiÎ²iÏxi eiÎ¾4Ïzi eiÎ²iÏxi). . .(eiÎ¾2nâ1Ïzi eâiÎ²iÏxi eiÎ¾2nÏzi eiÎ²iÏxi)âª.So we need to approximate each of these factors. If i is a bath spin which is initiallyup, then one hasâ©(eiÎ¾1Ïzi eâiÎ²iÏxi eiÎ¾2Ïzi eiÎ²iÏxi)(eiÎ¾3Ïzi eâiÎ²iÏxi eiÎ¾4Ïzi eiÎ²iÏxi). . .(eiÎ¾2nâ1Ïzi eâiÎ²iÏxi eiÎ¾2nÏzi eiÎ²iÏxi)âª=eiâ` Î¾`{1â nÎ²2i â Î²2i2nâ1â`=12nâ`â²=`+1(â1)`+`â²eâ2iâ`â²Ë`=`+1Î¾Ë`}+O (Î²3i )= exp{iâ`Î¾` â nÎ²2i â Î²2i2nâ1â`=12nâ`â²=`+1(â1)`+`â²eâ2iâ`â²Ë`=`+1Î¾Ë`}+O (Î²3i ) . (7.55)Similarly if j is a bath spin which is initially down, then one hasâ©(eiÎ¾1Ïzj eâiÎ²jÏxj eiÎ¾2Ïzj eiÎ²jÏxj)(eiÎ¾3Ïzj eâiÎ²jÏxj eiÎ¾4Ïzj eiÎ²jÏxj). . .(eiÎ¾2nâ1Ïzj eâiÎ²jÏxj eiÎ¾2nÏzj eiÎ²jÏxj)âª= exp{âiâ`Î¾` â nÎ²2j â Î²2j2nâ1â`=12nâ`â²=`(â1)`+`â²e2iâ`â²Ë`=`+1Î¾Ë`}+O (Î²3j ) . (7.56)111Thereforeâ©(P0Uâ P0U)nâª(7.57)ââ« 2pi0d2nÎ¾(2pi)2nexp{ânâiâBÎ²2i ââiâBÎ²2i2nâ1â`=12nâ`â²=`(â1)`+`â²e2iÎ·iâ`â²Ë`=`+1Î¾Ë`}=â« 2pi0d2nÎ¾(2pi)2nexpï£±ï£²ï£³ânNÎ²2 âNÎ²22nâ1â`=12nâ`â²=`+1(â1)`+`â² cosï£«ï£­2 `â²âË`=`+1Î¾Ë`ï£¶ï£¸ï£¼ï£½ï£¾ .(7.58)Where we have assumed that the Î²i variables are sharply peaked around their meanÎ²i â Î² â¡âi Î²i/N . Now with the substitution,Ï` = 2â``â²=1Î¾`â² + `pi (7.59)we getâ©(P0Uâ P0U)nâª=â« 2pi0d2nÏ(2pi)2nexp{ânNÎ²2 âN2Î²22nâ1â`=12nâ`â²=`+1cos (Ï` â Ï`â²)}.(7.60)Which can be thought of as a partition function of 2n âclassical pseudo-spinsâ,~s` = (cosÏ`, sinÏ`), defining ~S =â` ~s`, we haveâ2nâ1`=1â2n`â²=`+1 2 cos (Ï` â Ï`â²) +2n = ||~S||2 = S2(Ï), so,â©(P0Uâ P0U)nâª=â« 2pi0d2nÏ(2pi)2nexp{âNÎ²2S2(Ï)} (7.61)=â«d~SeâNÎ²~S2â« 2pi0d2nÏ(2pi)2nÎ´(~S ââ`~S`)(7.62)=â«d ~X2piâ«d~SeâNÎ²2 ~S2+i ~XÂ·~S[â«dÏ2pieâi ~XÂ·~s`]2n(7.63)=â« â0dXX2Îºeâ X24NÎ² J0(X)2n. (7.64)We denote an nâth order Bessel function of the first kind [26] by Jn(X). Inserting(7.64) into (7.53) we haveprâ(t) =â« â0dXX2Îºeâ X24NÎ²2 12âân=0(âiâ0J0(X)t)2n(2n)!. (7.65)112The expression (7.65) is of the form of an average over the variable X, which entersinto the motion through a renormalised transition element â(X) = â0J0(X). Theaveraged function (7.65) is exactly the value for the same probability Pââ, which wewould calculate if the Hamiltonian was just H0(â0J0(X)) =12â0J0(X)Ïx. X has aprobability distribution functionP (X) =X2ÎºeâX24NÎº , (7.66)the parameter Îº which controls the probability density of the average isÎº =12âjÎ²2j . (7.67)In order to generalise the calculation given above to the case where the centralspin has arbitrary initial conditions and to calculate all the components of ãÏ (t)ãone needs to: (i) derive expressions analogous to (7.58) for all of the expectationsappearing in equations (7.40-7.51). (ii) Transform the {Î¾q} variables in the resultingexpression to get an exponent which is manageable. This is done in appendix E.1.The result in the lowest order orthogonality blocking approximation is that in orderto calculate ãÏ (t)ã we need to perform the same averaging procedure, that isãÏ (t)ã =â« â0dXXeâX22Îº ãÏ (t; â0J0(X))ã0, (7.68)where ãÏ (t; â0J0(X))ã0 is the expectation of the central qubit polarisation calculatedusing the bare Hamiltonian H0(â0J0(X)) =12â0J0(X)Ïx.The Decay of ãÏx(t)ãNow the equation (7.68) is unsatisfactory, as it it predicts that a central spin ini-tialised in the state ÏÂ¯S(0) = 12(1 + Ïx) should remain static. The reason for thiscan be understood physically. The most secular terms conserve the âlongitudinalenergyââj ÏjÏzÏzj , which is initially zero. Thus the longitudinal energy term in theHamiltonian, which changes the x component of the central spin has no effect whenonly these terms are kept. Because of this we expect, something like we had in theexample in section 6.3.2, one component of the central spin (in this case ãÏxã and inthe example, ããÏ zããÎ¾) decays at a much slower rate that the other components anda higher order approximation is needed to calculate this decay.In order to find any dynamics for ãÏx(t)ã for the Hamiltonian (7.26), one needsto go to a higher order in the orthogonality blocking approximation. The next lo-west order approximation can be obtained by allowing two of the transitions whichchange the bathâs polarisation group, first jumping from its initial value and then onthe second central spin flip, jumping back to its original value (transition sequences113with only one change in the polarisation group do not contribute to the dynamicsof ãÏ ã, as they give zero when we trace out the bath). We show in the appendix E.2that this ânext to leading orderâ orthogonality blocking approximation to ãÏx(t)ã,gives zero (when we have the same parameters and initial conditions discussed inthe previous section). This is because of a cancellation of the next to leading orderterms, which relies on the initial state of the bath having zero z magnetisation aswell as zero external bias and so represents a somewhat special case. Rather thancomputing higher order terms in the transition expansion, we will briefly describeother effects which we have not included in the transition calculations so far, thatcan also cause ãÏxã to evolve.Small residual bias on the central qubit. Suppose the central qubit is im-perfectly tuned to near its âsweet spotâ and there is a small bias Î¾0 6= 0 remainingon the central spin. Suppose that this is much less than the coupling between thebath and central spin (Î¾0 Ï0). In this case a single transition which flips thecentral spin, will not be able to satisfy the resonance condition (7.33) exactly, butwe will still be able to obtain an approximation by satisfying these conditions ap-proximately. In this approximation the mathematics used to evaluate the strings ofoperators appearing in equations (7.40-7.51) is unaffected as an average, howeverbetween each central spin flip the density matrix is accumulating phase at a rate ofÂ±Î¾0 from the mismatched bias. Thus at time t the central system polarisation willbeãÏ (t)ã =â« â0dXXeâ X24NÎ² ãÏ (t; â0J0(X), Î¾0)ã0, (7.69)where ãÏ (t; â0, Î¾0)ã0 is the polarisation computed, from the âbare HamiltonianâH0 â¡ 12(â0Ïx + Î¾0Ï z). The averaging in equation (7.69) causes the decay of ãÏxã,because the small bias can rotate this component of the central polarisation intothe y â z plane, then the averaging over different transition amplitudes causes thedecay.Disorder in the coupling strengths, {Ïj}. We relax the condition that allcoupling strengths Ïj are the same by allowing them to each have a small deviationÎ´Ïj from the mean (C.F. equation (7.34)). In this case each polarisation groupof the bath spins will have a range of energies. This will affect the dynamics intwo ways, (i) there will be states with zero bath z polarisation where the centralspin still feels a non-zero bias, (ii) transitions which still approximately satisfy theresonance condition (7.33) can change the bias the central spin feels. If we assumefor simplicity that the distribution of of the couplings is approximately Gaussianwith a small standard deviation Î´Ï |Ï0|, then distribution of possible biases feltby the central spin is as shown in figure 1.3, each polarisation group is spread out.In this case a typical bath state with zero polarisation has a bias on the central spin114of order Î´Ï/âN and an transition where this does not change is approximately onresonance. Thus when the variables Î²i are small the dominant transitions do notchange the bias, so we expect the dynamics to be similar to those obtained with asmall residual bias, described in the previous paragraph.Additional averaging over initial states. One can expect the previous casesto cover a realistic experiment on a system described by the Hamiltonian (7.26).The bath starts in a definite pure state which only exerts a small bias on the centralsystem. We will discuss in more detail the model with parameters suitable forrealistic system of a single molecule magnet in chapter 8. However most experimentson such systems are performed on an ensemble of such systems. Even when studyinga single system in order to reconstruct the full central system reduced density matrix,one needs to perform multiple experiments and in each of these experiments one doesnot have full control of the bath. In both of the cases the results of experimentswill be an average over a set of initial bath configurations, much like in section 6.1.One can imagine a realistic case where the system is prepared with the bath in azero polarisation group state for each run of the experiment, the central systempolarisation vector recovered from such experiments will be an average of dynamicsdescribed in equation (7.69) over the probability distribution P0(Î¾0) for biases inthis polarisation group,ãÏ (t)ã =â« âââdÎ¾0P0(Î¾0)â« â0dXXeâ X24NÎ² ãÏ (t; â0J0(X), Î¾0)ã0, (7.70)where ãÏ (t; â0, Î¾0)ã0 is the polarisation computed, from the âbare HamiltonianâH0 â¡ 12(â0Ïx + Î¾0Ï z).7.2.4 Dynamics of Decoherence in the Canonical VariablesAs in section 6.1.3 it is interesting to ask where the information lost by the centralqubit goes. In particular we are interested whether the picture described in section6.1.3, where information cascades from the central qubit to higher and higher ordercorrelators with the bath, holds up when the bath spins can flip. In this section weexplore this question.When written in terms of the canonical variables the Hamiltonian (7.26) is nolonger only contains local fields and pairwise interactions so the structure of thehierarchy of equations of motion will be much more complicated. In these canonical115variables the equation of motion for the central qubit polarization isddtãÏ ã =â02{iã(U â U â )Ï zãxË+ ã(U + U â )Ï zãyË (7.71)â[iã(U â U â )Ïxã+ ã(U + U â )Ïyã]zË}+âjÏj zË Ã ãÏÏzj ã.It follows from the orthogonality blocking approximation that the key terms in thedecay of the central spin polarisation are those which involve correlations of U({Î²i})with the central spin. We have seen in equation (7.30) how terms involving U({Î²i})can be expressed as sums over clusters of bath spins of terms which depend on thex components of the bath spins in the cluster. So we expect the information lost bythe central spin to be transferred into correlators between central spin componentsand operators likeâC Ïxi for clusters C of bath spins. Now we move on to calculatingsome of these correlators.We begin by considering all possible correlators between two or fewer bath spinsand the central spin. We restrict ourselves to a central system intialised in thestate ÏÂ¯S(0) = | âããâ |. The details of all calculations in this section are containedin appendix E.3, here we describe how the calculations are done and the results.In the leading order orthogonality blocking approximation transition expansion thebath must end in the same polarisation group as it begins. This means that in thisapproximation any calculation of the expectation of an operator OË, will result in anexpression of the formãOË(t)ã = tr{â(. . .)P0OËP0(. . .)Ï(0)}, (7.72)where (. . .) represents terms omitted for clarity. Thus only operators whose actionpreserves the bath polarisation group contribute at the lowest order. Therefore thecorrelators ãÏÂµÏxi ã = ãÏÂµÏyi ã = 0. In appendix E.3.2 we show that to leading orderour expansion the correlators ãÏÂµÏzi ã and ãÏÂµÏzi Ïzj ã, which involve the z componentsof bath spins are,ãÏÂµÏzi ã(t) =Î·iãÏÂµã(t) (7.73)ãÏÂµÏzi Ïzj ã(t) =Î·iÎ·jãÏÂµã(t). (7.74)Therefore these correlators decay at the same rate as the associated central spinpolarisation components. For the correlators of the form ãÏÂµÏxi Ïxj ã, ãÏÂµÏyi Ïxj ã, andãÏÂµÏyi Ïyj ã, which contain the transverse component of pairs of bath spins, howeverwe get non-trivial results. This is important as these were precisely the correlatorsthat we argued in the previous paragraph should contribute to the decay of the116central spin polarisation. The calculation of these correlators is in appendix E.3.1,it is done as follows. Because Ïxj and Ïyj are off diagonal, bath spins i and j mustflip along with one of the central spins flips to contribute to these correlators. Theleading order calculation consists of summing a contribution from every central spinflip that these co-flips could coincide with. Naturally the result of this calculationdepends on the initial state of the spins i and j, as they can either be both initiallyup, down, or one may be up and the other down. In the case where both spins iand j are initially up (or down) we have the results,ãÏxi Ïxj ã =ãÏyi Ïyj ã = Î²iÎ²j(â0t)24â«dXP (X)[J1(X)]2 (7.75)ãÏxi Ïyj ã =ãÏyi Ïxj ã = 0 (7.76)ãÏxÏxi Ïxj ã =ãÏxÏyi Ïyj ã = ãÏxÏxi Ïyj ã = ãÏxÏyi Ïxj ã = 0 (7.77)ãÏyÏxi Ïxj ã =ãÏyÏyi Ïyj ã (7.78)=Î²iÎ²j(â0t)22â«dXP (X)[J1(X)]2 sin[â0J0(X)t]ãÏyÏyi Ïxj ã =ãÏyÏxi Ïyj ã = 0 (7.79)ãÏ zÏxi Ïxj ã =ãÏ zÏyi Ïyj ã (7.80)=â Î²iÎ²j (â0t)24â«dXP (X)[J1(X)]2 cos[â0J0(X)t]ãÏ zÏyi Ïxj ã =ãÏ zÏxi Ïyj ã = 0. (7.81)117While if spin i is initially up and spin j is initially down we then have,ãÏxi Ïxj ã =ãÏyi Ïyj ã = âÎ²iÎ²j(â0t)24â«dXP (X)[J1(X)]2 (7.82)ãÏxi Ïyj ã =ãÏyi Ïxj ã = 0 (7.83)ãÏxÏxi Ïxj ã =ãÏxÏyi Ïyj ã = 0 (7.84)ãÏxÏxi Ïyj ã =â ãÏxÏyi Ïxj ã = Î²iÎ²jâ0tâ«dXP (X)J0(X) (7.85)ãÏyÏxi Ïxj ã =ãÏyÏyi Ïyj ã (7.86)=â Î²iÎ²jâ«dXP (X){[â0J0(X)t] cos[â0J0(X)t]+ 12 [â0J1(X)t]2 sin[â0J0(X)t]}ãÏyÏyi Ïxj ã =ãÏyÏxi Ïyj ã = 0 (7.87)ãÏ zÏxi Ïxj ã =ãÏ zÏyi Ïyj ã (7.88)=Î²iÎ²j2â«dXP (X){[â0J0(X)t] sin[â0J0(X)t]â 12 [â0J1(X)t]2 cos[â0J0(X)t]}ãÏ zÏyi Ïxj ã =ãÏ zÏxi Ïyj ã = 0. (7.89)The first thing one should note about the results in equations (7.75-7.89), is that inboth cases there are correlators which appear to diverge as t â â. However thisdivergence is not a problem because our expansion is in small (Î²iÎ²j), is not validfor large Î²iÎ²jâ0t or largeâÎ²iÎ²jâ0t.The expressions for the non-zero correlators are all of the form, of an averagedexpressions which grow over time a slow time scale depending on Î²iÎ²j . In thecase where both i and j are initially up (or down) the non-zero correlators (7.78)and (7.80), obtained when the two bath spins are aligned oscillate with amplitudeswhich grow quadratically with time. While in the case where the two bath spins areinitially anti-aligned the correlators (7.86) and (7.88), have oscillating componentsthat grow linearly with time as well as the quadratically growing components, whilethe correlators (7.85) grow linearly with time. This difference in scaling comes fromthe fact that, in the situation where the spins i and j are initially different, thereis the possibility that both spins i and j may co-flip along with the same centralspin flip. We also note that, as expected correlators of the form ãÏÂµÏxi Ïxj ã, whichare important in the decay of the central spin, grow.Interestingly equations (7.82) and (7.75) show, in both cases the two bath spinsbecome correlated over time. This correlation increases quadratically and in theopposite sense in the two cases, initially aligned spins develop positive correlati-118ons between their x components, while initially anti-aligned spins develop negativecorrelations between their x components.We have also calculated the correlators ãÏyÏxi Ïxj ã and ãÏ zÏxi Ïxj ã in the lowestorder orthogonality blocking approximation for the case where there is an additionalweak bias on the central spin, the results are given in appendix E.3.3. In this casewe get a more complicated set of results. We still see quadratic divergences in thevarious correlators at long times. We will plot these solutions for a specific case inthe next chapter. Next we discuss the scaling of the growth rate of larger correlatorsbetween the central spin and more bath spins.Now we consider larger correlators between the central spin and many bath spins.In particular we consider correlators of the formâ©ÏÂµâjâC Ïxjâªfor some subset ofthe bath C â B , which are important in the decay of the central qubit polarisation.Because the only terms which contribute to our leading order expansion are thosewhich contain operators which preserve the bath polarisation, at this order thecorrelator is zero unless |C| is even. Thus it suffices to consider correlators of theformâ©ÏÂµâjâG Ï+jâjâK Ïâjâª, where G âª K = C and |G| = |K|. We have seen theexpressions will in general depend on the initial states of the spins in G and K, wetake U (D) to be the subsets of the bath which are initially up (down). Then wewill consider the scaling of the correlatorâ©ÏÂµâjâGÏ+jâjâKÏâjâª(t) (7.90)= tr{U â E(t)ÏÂµâjâGâ©UÏ+jâjâKâ©UÏâjâjâGâ©DÏ+jâjâKâ©DÏâj UE(t)Ï(0)}.Every spin in C must flip so the leading order expression will have a prefactor ofâjâC Î²j . Each bath spin in C will flip alongside one of the central spin flips containedin the transition expansion for either UE(t) or Uâ E(t). Those bath spins in (G â© U)or (K â© D) must co-flip with one of the central spin flips in the expansion ofUE(t).While bath spins in (G â©D) or (Kâ©U) must co-flip with one of the central spin flipsin the expansion of U â E(t). Define the indices nL and nR,nL â¡ |G â© D|+ |K â© U| (7.91)nR â¡ |G â© U|+ |K â© D|. (7.92)The leading order transition expansion for the correlator (7.90) will then be of theform, â©ÏÂµâjâGÏ+jâjâKÏâjâª(t) (7.93)=(âjâCÎ²j)â0tnL+nRâm=|nLânR|(â0t)mAÂµm,C(â0t).119Where AÂµm,C(â0t) is an average over some oscillating term. We see from the ex-pression (7.93), that in the regime of validity of our approximations, correlatorsinvolving more bath spins grow slower, while the exact form of the growth dependson the specific correlator.7.3 ConclusionIn 7.1 we derived a transition expansion for the influence functional for a systemof qubits coupled to a spin bath. We have seen that this expansion is most usefulwhen a small number different types of transition are important to understand thedynamics.Then, in section 7.2 we used this expansion to derive the dynamics of the centralspin in the model for precessional decoherence. We saw that the transition Hamil-tonian could be simplified by working in basis where the bath spin states depend onthe central spin. Then using the approximation of Prokefâev and Stamp [86] we sawthat the important transitions preserved the polarisation group of the bath spin,and we recovered their result for the return probability. Then we showed explicitly(in appendix E.1) how this generalises to give the full central qubit reduced den-sity matrices. We then discussed limitations and corrections to this approximation.We then turned to the build up of correlations between the central qubit and thebath during decoherence. We argued the decoherence in this model via a cascade ofcorrelations, like in the degeneracy blocking model discussed in section 6.1. Exceptin the precessional decoherence case the information is transferred into correlati-ons with the transverse components of the bath spins, rather than the longitudinalcomponents as seen in degeneracy blocking.In the next chapter we will apply our results for precessional decoherence, to theFe8 magnetic molecule qubit.120Chapter 8Application: The Fe8 QubitAs discussed in the section 1.10.2 there are magnetic molecule qubits which posseswell understood interactions with their local environment. In this chapter we willexplore the dynamics of decoherence in a real magnetic molecule system, the Fe8qubit, which is well described by the precessional decoherence Hamiltonian. Exa-mine the results of the previous chapter for the dynamics of the entanglement ofthe central qubit with its spin environment in the context of this system. In pre-vious work by Stamp, Tupitsyn, and collaborators [73, 98, 101, 103], the relevantparameters have been calculated for the Fe8 molecule. We will now use these resultsto calculate the dynamics of the central spin and its correlators with pairs of bathspins in these systems.8.1 The âFe8â QubitHere we describe a single such qubit design based on the so called âFe8â molecule[40],shown in figure 8.1, the molecule consists of a core of eight Fe3+ ions linked by oxoand hydroxo ions and surrounded in a cage of organic ligands. In this sectionwe discuss a qubit made from one of these molecules, ignoring the effect of theenvironment and in the next section we will discuss its coupling to the environment.Low temperatures (T 4K), the low energy dynamics of the electronic spinsof the Fe3+ ions can be described by the following Hamiltonian H0Fe8(S), written interms of the total electronic spin S, which is S = 10 [107, 108, 117]H0Fe8(S) = âDS2z + ES2x +Kâ¥4 (S4+ + S4â)â geÂµBÂµ0S Â·Hâ¥. (8.1)Here Hâ¥ is an applied transverse field (in the y direction), the energies D, E, Kâ¥4arise from the details of the molecule, ge is the electronic g-factor, ÂµB is theBohr magneton, and Âµ0 is the permeability. Without the applied field, S feelsan anisotropic potential set by the parameters D/kB = 0.23 K, E/kB = 0.094 K,Kâ¥4 = â3.28 Ã 10â5 K [107, 108, 117]. The Hamiltonian (8.1) has been calculatedusing an ultraviolet high frequency cut-off of â¦0/kB = 4.6 K (â¦0 = 6Ã 1011Hz).The semi-classical potential obtained from the Hamiltonian (8.1) is plotted infigure 8.2, for the experimentally achievable case where the applied transverse fieldis Âµ0Hâ¥ = 2.5 TyË. From this figure we see there are two minima of energy onthe Bloch sphere with opposite Sz components, states in these two minima can be121Figure 8.1: Structure of the Fe8 molecule. The crosshatched circles are the Fe3+ions, the hatched circles are the oxygen atoms, and the empty circlesrepresent, in order of decreasing size, nitrogen and carbon atoms. [Re-printed figure with permission, from D Gatteschi, A. Caneschi, L., R.Sessoli, Science, 265, 1054 (1994), [40]. Copyright 1994 by the AmericanAssociation for the Advancement of Science.].122taken as the two qubit levels, call these states | âã, and | âã. These states are strictlydefined by| âã â¡ 1â2(|sã+ |aã) (8.2)| âã â¡ 1â2(|sã â |aã), (8.3)where |sã and |aã are the two lowest energy states of the Hamiltonian (8.1) (|sã, and|aã have symmetric and antisymmetric wave functions respectively ). From thesestates one can build the usual Pauli operators (Ï z â¡ | âããâ | â | âããâ | etc.) andthen the effective Hamiltonian for a single isolated Fe8 qubit is of the form [73],H0effFe8 = â12â0(Hâ¥)Ïx. (8.4)Where â0(Hâ¥) is as shown in figure 8.3 and has a strong dependence on the appliedfield. Physically this dependence arises from the fact that as |Hâ¥| is increased, thetwo energy minima move closer to Sy axis (and one another), therefore the tunnelingamplitude increases.8.1.1 The Coupling of the Fe8 Qubit to the Spin BathAt low temperatures, Fe8 is crystalline and the central spin is coupled to an envi-ronment primarily consisting of nuclear spins and phonons[85, 98, 101, 103, 118].The typical time scale Ïph associated with the phonon decoherence is [85]Ïâ1ph = ââ10(4Sâ¦0â0(kBÎ¸D)2)2coth(4â0kBT). (8.5)Where Î¸D is the Debye temperature for the crystal, Î¸D â 33K. So Ïph can be madelarge by using a low field Hâ¥ to tune â0. Ïph is plotted as a function of the appliedfield in figure 8.4. This time scale is very long when Hâ¥ is small, we will 8 that, atlow fields, the spin bath causes much faster decoherence. So we turn to a discussionof the nuclear spin bath.In terms of the electronic spin operators sa and the nuclear spin operators Ijthe Hamiltonian HFe8SB, which governs the dynamics of the nuclear spins in the Fe8molecule is of the form,HFe8SB =âjaAajÂµÎ±sÂµaIÎ±j ââjgNj ÂµNj Âµ0H0 Â· Ij , (8.6)where AajÂµÎ± are the various hyperfine couplings, H0 is the applied magnetic field,gNi and ÂµNj are the g-factor and magnetic moment of the jâth nucleus. The preciseset of nuclear spins and hyperfine couplings depend on the isotopic make-up of themolecule, but these can be calculated for specific cases [99, 101]. A histogram ofthe different hyperfine coupling strengths for the 1H nuclei and 57Fe nuclei (for the123Figure 8.2: The energy of the electronic spin vector in the Fe8 molecule. Both themain plot and the inset plot shaded colour shows the semicassical energyof the electronic spin in the Fe8 molecule as a function of its direction onthe Bloch sphere, when the applied field is Âµ0Hâ¥ = 2.5 TyË. The scaleshown on the right gives the energy/kB in Kelvins. The black bullets(â¢) mark the positions of the energy minima where the qubit states arelocalised. The dotted black line marks the semiclassical tunneling route.The inset plot is the same as the main plot but viewed from a differentangle.124Figure 8.3: The field dependence of the Fe8 qubitâs tunneling amplitude â0(Hâ¥),when the field Hâ¥ is pointing in the yË direction. Data for this plot isobtained from [73].125Figure 8.4: The time scale associated with decoherence due to phonons in the Fe8qubit. The field dependence of Ïph as a function of the applied field,when the field Hâ¥ is pointing in the yË direction. Data for this plot isobtained from [73], using equation 8.5 and assuming a temperature ofT = 0.5K.126Figure 8.5: A histogram of the different hyperfine coupling strengths. The blueboxes show the hyperfine coupling strengths for the H nuclei and thered histrogram shows those for 57Fe nuclei (in the case where all Fe3+ions are 57Fe). All data is binned into 0.5 MHz intervals. Data for thisplot is taken from [99]case where all the hydrogens in the molecule are 1H and all the iron ions are 57Fe,both of which are spinâ12) is shown in figure 8.5. A comparison of figures 8.5 and8.3, shows that if |Hâ¥| . 1T the hyperfine coupling may be quite large comparedto the tunneling amplitude.Now one can proceed truncating the Hamiltonian HFe8 = HFe8(S) +HFe8SB, toget an effective qubit and spin bath Hamiltonian like (1.48) for the molecule. Inthis case one finds the Î±i terms dressing the qubit flipping term are negligible [98]and one is left with the effective Hamiltonian (neglecting the dipolar interactionsbetween nuclear spins, which can be calculated from the molecular structure andare â¼ 10â4 â 0.1 MHz [99]),Heff Fe8 =â0(Hâ¥0 )2Ïx +Ï z2âiÏâi Ïzi +12âiÏâ¥i Ïxi . (8.7)Here â0(Hâ¥0 ) is the effective central spin flip amplitude discussed in the previoussection, Ïâi depends on how much the effective field on the iâth nucleus changes when127the central qubit flips,Ïâi =1Iiâ£â£â£â£â£âaAaiÂµÎ±IÎ±(ãsÂµaãâ â ãsÂµaãâ)â£â£â£â£â£ (8.8)(ãsÂµaãâ â¡ ãâ |sÂµa | âã and like wise for ãsÂµaãâ), Ïâ¥i is,Ïâi =1Iiâ£â£â£â£â£âaAaiÂµÎ±IÎ±(ãsÂµaãâ + ãsÂµaãâ)â gNi ÂµNi Âµ0Hâ¥0 Iiâ£â£â£â£â£ . (8.9)In the previous two sections we have seen that we can find a well defined effectiveHamiltonian for an Fe8 single molecule magnet and its spin bath we will use thisin chapter 8 where we study the dynamics of the central spin and the bath in thissystem.8.2 The Fe8 Molecule in Real SystemsSo far experiments on Fe8 magnetic molecules, have been on crystalline samples[41].This means the qubits in the crystal suffer from some effects which are undesira-ble for the purpose of this study. Only an ensemble of qubits in the crystal maybe addressed experimentally and because of the long ranged dipole fields at play,different qubits can be feeling quite different fields, so we will have to average theresults for single over a distribution of different fields. Decoherence also occurs dueto the coupling between the average qubit magnetisation in the sample with mag-nons [73, 103], the decoherence rate due to these interactions, is quite complicatedto calculate, and in general depends on the sample geometry. But if one is able tocreate a small enough crystal the magnon decoherence could be minimised.8.3 Dynamics of Dechoerence Due to 1H NucleiWe are interested in the details of the loss of information in the central qubit andhow correlations build up with the bath spins. As we saw in the previous chapterthe relevant parameter determining for the âstrengthâ of the spin bath is Îº. Themost important nuclei for the spin decoherence at low fields (where the effect ofphonons is small), are the 1H nuclei [99]. This Îº parameter has been calculated[99] from the various 1H couplings and is shown in figure 8.6 as a function of fieldstrength Hâ¥.We now consider a definite case, when the applied field is Âµ0Hâ¥ = (0.025 T)yË,then the tunneling element is â0 â 2.47 Ã 104 Hz. First we give results for ahypothetical experiment where we prepare a single Fe8 qubit, so that the bath of1H nuclei is in a definite state in the zero polarisation group and we have tunedthe z component of the magnetic field so the central qubit experiences no bias. If128Figure 8.6: A plot of the variable Îº parametrising the bath of 1H nuclei in the Fe8.The ploted data is obtained from [99] and it corrisponds to the casewhere all of all of the hydrogen atoms in the molecule are 1H.then the central system is prepared so that it is fully polarised in the z direction,we are be able to use the results of the previous section and the time evolution ofthe central spin is as given in figure 8.7. The results we would obtain were we ableto measure the time dependence of correlators between the central spins given inequations (7.75-7.81) or (7.82-7.89), (depending on the initial states of the specificbath spins) are also shown in figure 8.7. We see that correlator ãÏxi Ïxi ã increasesand amplitude amplitude of correlators like ãÏ zÏxi Ïxi ã increase as the central qubitloses information to the bath. If we were not able to measure these correlators forspecific pairs of bath spins, but instead we had to measure the average of thesecorrelators over all choices for pairs of bath spins then the only correlators withpairs of off diagonal bath components, which would not average out to zero, wouldbe ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã and ãÏyÏxj Ïxj ã and they would be as shown in figure 8.8. Inthis case we would still see the growth of the important correlators as the centralspin losses information.Unfortunately such experiments on a single Fe8 molecule have not yet beenachieved. Experiments have been done on crystals of Fe8, but in these samples theindividual molecules feel a range of biases Î´Î¾, which has been measured Î´Î¾ â¼ 5Ã107Hz [103], which will completely wash out the details of the dynamics seen here. Evenif a single molecule could be isolated and experimented on the range of differentcouplings due to the dipolar hyperfine coupling between the 1H molecules is quite129Figure 8.7: Plots of the correlators for the case of a single Fe8 molecule coupled toits 1H nuclear spin bath, as decribed in section 8.3. The top graph showsthe time dependence of the components of the central spin polarisation,ãÏ ã, the solid red curve is ãÏxã, the dashed green curve is ãÏyã, and thedotted blue curve is ãÏ zã. The bottom left graph shows components ofcorrelators involving two bath spins, i and j, where spin i is intialy upand j is intialy down. The bottom right graph shows components of thesame correlators (when they are non-zero) in the case that spins i andj are intialy aligned. In the main plots on the bottom, the dotted bluecurve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã, the dashed green curve is ãÏyÏxj Ïxj ã, thesolid red curve is ãÏyÏxi Ïxj ã = ãÏyÏyi Ïyj ã, and in the inset the thick solidred curve is ãÏxÏxi Ïyj ã, the dashed red curve is ãÏxÏyi Ïxj ã, and the thinblack curve is ãÏxi Ïxj ã.130Figure 8.8: A plot of correlators between the bath spins and the central spin, forthe case of a single Fe8 molecule coupled to its1H nuclear spin bath,when we have averaged over the different choices for sites i and j. Thedotted blue curve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã and the dashed green curveis ãÏyÏxj Ïxj ã.large, we estimate Î´Ï â¼ 2 Ã 105 Hz (from data in [99]), which will have a similareffect destroying the results we see here. As such in the next section we will considera molecule where all of the hydrogen atoms are substituted by deuterium, and thedecoherence is caused by the other nuclei.8.4 Dynamics of Decoherence Due to 57Fe NucleiWe saw in the previous section that a realistic test of the precessional decoherencemodel due to the 1H spin bath is unfeasible, even on a single molecule, due to thespread of different couplings between the hydrogen nuclei and the central qubit.We saw in section 8.1.1 that in the case where the Fe3+ ions are the 57Fe isotope,then the spread of the distribution of the contact hyperfine couplings between theseions and the electronic spins is smaller. We calculate Î´Ï â¼ 2 Ã 105 Hz (fromdata in [99]), which is much smaller, than that associated with the hydrogen. Alsowith the hydrogen spin bath eliminated by isotopic substitution, the coherence timewill be longer and higher a field Hâ¥ can be used resulting in a bigger â0, so thedimensionless ratio Î´Ï/â0 can be made much smaller.We consider a case where all the nuclei except the iron nuclei are spin zeroisotopes and all the iron nuclei are 57Fe. If the applied field is Hâ¥ = 1 TyË, thenwe have a transition amplitude â0 = 81.0 MHz (see figure 8.3), at T = 0.5K)and the timescale associated with phonon decoherence is Ïph = 25 s (from equation(8.5), at T = 0.5K). By assuming the contact hyperfine interaction between thenuclear spins and the electronic spins is isotropic, we can calculate the Î²i â 0.05and Îº â 0.01 using equations (8.8) and (8.8). Then we plot results for the central131spin polarisation, and the non-zero correlators with pairs of bath spins in figure 8.9,for the initial period of decoherence. In these results we have averaged the resultsin appendix E.3.3, for precessional decoherence with a small bias, over a Gaussiandistribution of biases with standard deviation Î´Ï. This average should accountfor experiments either perform an ensemble of different measurements, or study anensemble of different systems with slightly different biases. We also assume we cannot probe particular pairs iron nuclei, so we have averaged over the different choicesfor bath spins i and j. We see that as the central qubit decays correlations build upwith pairs of bath spins.8.5 ConclusionThe important result of this section is that, if one can perform measurements ona single Fe8 molecule, in which all the nuclei are isotopes with spin zero, exceptthe iron ions which are the 57Fe isotope, then we can predict the correlators ofcomponents of the central spin with the transverse components of pairs of bathspins. We think these could be measured using NMR techniques, the details of suchan experiment are left for future work.132Figure 8.9: The initial stages of decoherence in an Fe8 qubit, when we have averagedover the different choices for bath spins i and j. The top figure showscomponents of the central qubit ãÏ ã, the solid red curve is ãÏxã, thedashed green curve is ãÏyã, and the dotted blue curve is ãÏ zã. Thebottom figure is plot of correlators between the pairs of bath spins andthe central spin, the dotted blue curve is ãÏ zÏxi Ïxj ã = ãÏ zÏyi Ïyj ã and thedashed green curve is ãÏyÏxj Ïxj ã.133Chapter 9Evolution of a Large âCatStateâHere we will consider the time evolution of a set S containing N central spins wherethe initial state is a âcat-stateâ. These states have a wave function which is acoherent superposition of two states where all the spins have opposite componentsin the z (quantisation) direction. We can specify these states in terms of the ket|sã with a vector s = (s1, s2, . . . , sN ) where each element sa = Â±1 indicates whetherthe aâth spin in S is up or down and |sã = âaâS |saã. We consider initial states thatcan be described by the wave function,|Ïs,Ïã = 1â2(|sã+ eiÏ| â sã), (9.1)where we have allowed for an arbitrary phase angle Ï between the two parts of thesuperposition.The initial reduced density matrix for S (we allow for the possibility that S ispart of a large set of qubits the others forming an environment) is thenÏÂ¯S(0) =12(|sããs|+ | â sããâs|+ eiÏ| â sããs|+ eâiÏ|sããâs|). (9.2)In the following sections we will discuss the specifics of the time evolution of thesecat states in a variety of cases. First in section 9.1 we will discuss some quantitieswhich will be useful to characterise the information contained in the reduced densitymatrix ÏÂ¯S(t) as time evolves. In section 9.2 we consider the case where each spinonly feels the its own longitudinal field in that case the phase angle the evolutionis equivalent to the phase angle Ï evolving with time at a frequency determined bythe vector s. In section 9.3 we study the case where each spin feels a general localfield and find that when the fields have a transverse component for large N many ofthe quantities we are interested are zero most of the time except at given recurrencetimes determined by the mean strength of the field and that the effect of these localfields being disordered is to modulate these recurrences by a function that decays ona time scale determined by the spread of the frequencies. In section 9.4 we discussthe case where each spin interacts with its own static environment so that the fieldeach spin feels is effectively averaged over a range of values, we find that thereare several regimes for the decay of the recurrences seen in this model in various134quantities and that the importance of these regimes is determined by how largeN is. In section 9.5 we will consider the case where the effect of the environmentcauses more general averaging of the central spin dynamics and elucidate which ofthe results in the degeneracy blocking case can be generalised.9.1 Quantities of InterestWe will assume that the system S and the environmental bath B are initially ina product state Ï(0) = ÏS(0)ÏB(0). First we can split the initial reduced densitymatrix ÏÂ¯S(0) = DË(0) + CË(0) into a diagonal DË(0) and off diagonal parts CË(0),DË(0) =12(|sããs|+ | â sããâs|)=12N+1âaâS(1 + saÏza ) +12N+1âaâS(1â saÏ za ) (9.3)CË(0) =12(eiÏ| â sããs|+ eâiÏ|sããâs|)=12(eiÏ2NâaâS(Ïxa â isaÏya ) + h.c.). (9.4)Where h.c. denotes the Hermitian conjugate of the previous term. From the expres-sions (9.3) and (9.4) we can describe all correlations which specify the initial state.The diagonal part (9.3) contains correlators between z components of the spins only.DË(0) is a valid (impure) density matrix on its own and would describe state whichcould be described by classical a probability distribution where for the out come ofthe z component vector sË occurs with probability PsË =12(Î´s,sË + Î´âs,sË). A correlatorbetween the z components of a cluster C â S of the spins is only non-zero when thecluster contains an even number of spins (|C| is even) in which case it isâ©âaâCÏ za (0)âª=âaâCsa. (9.5)The off diagonal part (9.3) contains only correlators between x and y componentsof spins furthermore it contains only correlators containing all N qubits, no smallerclusters. This means that if any spin is traced out of ÏÂ¯S(0) then all the off-diagonalinformation is lost and there are no non-zero correlators involving Ïxa or Ïya in theresulting density matrix, so we see that CË(0) contains information about the N -partite quantum entanglement of the state. Now as the density matrix evolveswith time we can define time dependent operators DË(t) = trBU(t)DË(0)ÏÂ¯B(0)U â (t)and CË(t) = trBU(t)CË(0)ÏÂ¯B(0)U â (t) which give us the parts of the density matrixÏÂ¯S(t) = trBU(t)Ï(0)U â (t) = DË(t) + CË(t) sourced by the initial diagonal and offdiagonal parts respectively. In general DË(t) and CË(t) will contribute to all possible135correlators at finite times and we can expand them in terms of these contributions,DË(t) =12NâCâSDCÂµ1,...,Âµ|C|(t)âaiâCÏÂµaa (9.6)CË(t) =12NâCâSCCÂµ1,...,Âµ|C|(t)âaiâCÏÂµaa (9.7)with : (9.8)DCÂµ1,...,Âµ|C|(t) =trSDË(t)âaiâCÏÂµaa (9.9)CCÂµ1,...,Âµ|C|(t) =trSCË(t)âaiâCÏÂµaa . (9.10)Note that the above implies Dâ (t) = 1, Câ (t) = 0 and that a correlator between acluster C of spins has a contribution from both DË(t) and CË(t),â©âaâCÏÂµaa (t)âª= DCÂµ1,...,Âµ|C|(t) + CCÂµ1,...,Âµ|C|(t). (9.11)One could imagine an experiment to measure the components of the tensors {CCÂµ1,...,Âµ|C|(t), DCÂµ1,...,Âµ|C|(t)}suppose one has the means to prepare two copies of central system, (a) in the catstate (9.2) but also (b) in the impure state where ÏS(0) = DË(0) then one couldmeasure the correlatorâ©âaâC ÏÂµaaâªin both systems after some time then one hasDCÂµ1,...,Âµ|C|(t) =â©âaâC ÏÂµaaâª(b)and CCÂµ1,...,Âµ|C|(t) =â©âaâC ÏÂµaaâª(a)âDCÂµ1,...,Âµ|C|(t). Ini-tially when the central spins are in a cat state the tensors {CC ,DC} areCCÂµ1...Âµ|C|(0) =Î´CSRe eiÏâaâS(xËÂµa â iyËa) (9.12)DCÂµ1...Âµ|C|(0) ={âaâC sazËÂµa for |C| even0 for |C| odd . (9.13)One quantity of interest is the probability pr(t) for the state at t to return to itsinitial state. When written in terms of the operators CË and DË the return probabilityispr(t) = trSÏÂ¯S(t)ÏÂ¯S(0) = trSDË(t)DË(0) + trSDË(t)CË(0) + trSCË(t)DË(0) + trSCË(t)CË(0) (9.14)the last term trSCË(t)CË(0) is of particular interest because it tells us how much of theoff-diagonal part of the density matrix returns to its original value we will refer tothis as the off-diagonal part of the return probability pc(t)pc(t) â¡ trS CË(t)CË(0). (9.15)136Note that pc(t) is not a valid probability as in may be negative so it is probablybest thought of as the inner-product of CË(t) with its initial value. We can write theterms appearing in the above expression in terms of the tensors {DCÂµ1...(t), CCÂµ1...(t)},trSDË(t)DË(0) =âCâS|C| even12NDCzz...z(t)âaâCsa (9.16)trSCË(t)DË(0) =âCâS|C| even12NCCzz...z(t)âaâCsa (9.17)trSDË(t)CË(0) =ReeiÏ2NDSÂµ1...ÂµN (t)âaâS(xËÂµa â isayËÂµa) (9.18)trCDË(t)CË(0) =ReeiÏ2NCSÂµ1...ÂµN (t)âaâS(xËÂµa â isayËÂµa). (9.19)Also note the return probability can be written directly in terms of the correlatorsat the initial and final timespr(t) =âCâS12Nâ©âaâCÏÂµaa (t)âªâ©âaâCÏÂµaa (0)âª. (9.20)The return probability diagnoses how likely the system is to return to its initialstate. In general a large isolated system could take an effectively infinite amount oftime to have a good probability of returning to its initial state thus we could expectpr(t) and related metrics to decay on time scales less than this recurrence time eventhough no information would be lost to an environment. A quantity that tells usabout information loss to the environment is the trace of the reduced density matrixsquaredp(t) â¡ trSÏÂ¯S(t)2 (9.21)which is sometimes called the purity[78] and is related to the linear entropy discussedin section 1.4.1. The purity is one for a pure state and 12Nâ¤ p < 1 for a mixedstate. The purity can be expressed in terms of CË(t) and DË(t)p(t) â¡ trSÏÂ¯S(t)2 = trSDË(t)DË(t) + 2trSCË(t)DË(t) + trSCË(t)CË(t). (9.22)Note that expressions like trSCË(t)DË(t) can be written in terms of the tensor compo-nents for exampletrSCË(t)DË(t) =âCâS12NCCÂµ1...Âµ|C|(t)DCÂµ1...Âµ|C|(t). (9.23)137If our central system is isolated then the purity is constant and we have the followingtrSDË(t)DË(t) = trSU(t)DË(0)U â (t)U(t)DË(0)U â (t) =14trS(|sããs|+ | â sããâs|) = 12(9.24)trSCË(t)DË(t) = 0 (9.25)trSCË(t)CË(t) =12(9.26)so that the purity can be divided into equal parts belonging to the diagonal andoff-diagonal parts which are constant over time. The purity can also be expandedin terms of correlatorsp(t) =12NâCâSâ©âaâCÏÂµaa (t)âªãâaâCÏÂµaa (t)âªâ¡âCâSpC(t). (9.27)There are contributions to the purity from each of the correlators, of particularinterest is pS which describes the contribution to the purity from the Nth ordercorrelators. We can get an understanding of the entanglement properties by lookingat the correlators. Of particular interest is the off-diagonal N point correlators forthe central system that is the correlators of the formâ©âaâC ÏxaâbâSC ÏybâªwhereC â S. Really we should look at the irreducible version of this correlator (initially theirreducible correlator is identical to the correlator). Initially the N point correlatorsin the xâ y plane are,â©âaâCÏxa (0)âbâSCÏyb (0)âª= cos(Ïâ pi2 |C|). (9.28)We can get an idea of how much information is contained in these correlators bylooking at the contribution to the purity from these correlators alonepâ¥S(t) =12NâCâSâ©âaâCÏxaâbâSCÏybâª2(9.29)initially pâ¥S(0) = 12 (so it contains the entire purity of the off-diagonal part of thedensity matrix). As time goes by pâ¥S may be lost to the environment or othercorrelators in the system. An alternate formula pâ¥S(t) which may be simpler tocalculate ispâ¥S(t) =12Nâ©ÏÂµ11 . . . ÏÂµNNâªâaâSPâ¥ÂµaÎ½aâ©ÏÎ½11 . . . ÏÎ½NNâª(9.30)where Pâ¥ÂµÎ½ = Î´ÂµÎ½ â zËÂµzËÎ½ is the projection operator that projects every polarisationinto the xâ y plane.1389.2 Motion in Applied Longditudional FieldsThe first case we will consider is where the central Hamiltonian contains a locallongitudinal field for each central spin and there is no interaction with the bath sothat the central system is isolated with a HamiltonianH =12âaÎ¾aÏza . (9.31)The off diagonal part of Ï evolvesCË(t) =12U(t)(eâiÏ|sããâs|+ eiÏ| â sããs|)U â (t) (9.32)=12(eâiÏââa isaÎ¾at|sããâs|+ eiÏ|+ eiÏ+âa isaÎ¾at| â sããs|). (9.33)Thus the Hamiltonian effectively causes the phase Ï to accumulate over time. Thusthe contribution from the off diagonal part to the return probability oscillates withtime and does not decaytrCË(t)CË(0) = cos(Ï+âasaÎ¾at). (9.34)The diagonal part of the density matrix commutes with the Hamiltonian and isconserved. The return probability in this case is thenpr(t) = trÏ(t)Ï(0) =12(1 + trCË(t)CË(0)). (9.35)Only the off-diagonal correlators have a time dependence and because there are onlylocal fields. A correlator on set C only depends on correlators containing all the spinsfrom that set initially so that the only non-zero off diagonal correlators are thoseencompassing the whole system, so for an arbitrary subset C â S we haveâ© âaâSCÏxa (t)âbâCÏyb (t)âª= cos(Ï+âcâSscÎ¾ct+pi2|C|). (9.36)All of the off diagonal correlators oscillate at the same frequency as the returnprobability. As the Hamiltonian (9.31) contains no coupling to the environment theoff diagonal and diagonal purities are conserved and the off diagonal part of CË(t)remains off diagonal, so the purity contained in the N order off diagonal correlatorsis pâ¥S(t) = trCË(t)2 = 12 constant. Intuitively we can understand this evolution asall of the components contained in the correlators freely rotating around the zË axis,only the N âth order correlators have any components in the xây plane so only theyevolve but their magnitude remains the same.1399.3 Motion in General fieldsConsider the case where the Hamiltonian consists of local fields for every site whichare each individually at an angle Î¸a to the axis of quantisation and have strengthÏaH =12âaÏa(cos Î¸axË + sin Î¸azË) Â· Ï a =âa(âaxË + Î¾azË) Â· Ï a. (9.37)Because the Hamiltonian is local the different order correlators evolve independentlyand so that the only tensor components of CË(t) that is non-zero are those contai-ned in CSÂµ1...ÂµN (i.e. those containing components of all the spins) which we willabbreviate as CÂµ1...ÂµN in this section. At time t we haveCÂµ1...ÂµN (t) =12Re eiÏâa(gÂµaxa (t)â isagÂµaya (t)). (9.38)Where gÂµÎ½a (t) is the single spin Green function (cf.(4.24)) for the qubit agÂµÎ½a (t) = hËÂµa hËÎ½a + cos(Ïat)(Î´ÂµÎ½ â hËÂµa hËÎ½a)+ sin(Ïat)ÎµÂµÎ½Î»hËÎ»a (9.39)where hË = cos Î¸azË + sin Î¸axË is the direction of the field on spin a. Thus the off-diagonal contribution to the return probability istrCË(t)CË0 =12Re{e2iÏâa12[gxxa â sagyya â i(sagxya + gyxa )] (9.40)+âa12[gxxa + sagyya â i(sagxya â gyxa )]}=12Re{e2iÏâaâK[12cos2 Î¸a(1â cosÏat)]Â·âa6âK[cos2 Î¸a + (2â cos2 Î¸a) cosÏat+ 2i sin Î¸a sinÏat]+âaâK12[cos2 Î¸a + (2â cos2 Î¸a) cosÏatâ 2i sin Î¸a sinÏat]Â·âa6âK[12cos2 Î¸a(1â cosÏat)]}. (9.41)Where K is the set of all spins such that sa = +1. To simplify the preceding analysiswe shall assume that all sa = 1. First consider the case where the field on each spin140is the same so that Î¸a = Î¸0 and Ïa = Ï0 in which casetrCË(t)CË0 =12Re{e2iÏ[12cos2 Î¸0(1â cosÏ0t)]N(9.42)+12N[cos2 Î¸0 + (2â cos2 Î¸0) cosÏ0tâ 2i sin Î¸0 sinÏ0t]N}.Each of the terms inside the curly brackets is proportional to some number whichis less than one raised to a large power N as such they are exponentially small inN unless the number has a magnitude close to one. So terms are not exponentiallysmall in N when, â£â£â£â£12 cos2 Î¸0(1â cosÏ0t)â£â£â£â£ =1â Î¸0 = pin and Ï0t = pi(2m+ 1)(9.43)12â£â£cos2 Î¸0 + (2â cos2 Î¸0) cosÏ0tâ 2i sin Î¸0 sinÏ0tâ£â£ =1â Ï0t = 2pi` or Î¸0 = pi2(2k + 1)(9.44)for n,m, `, k â Z. We can expand around these points to get the large N asymptoticbehaviour at these points, for instance let Ï0t = pi+ Ï0Î´t and Î¸0 = pi+ Î´Î¸0 then forsmall Î´t and Î´Î¸0 [12cos2 Î¸0(1â cosÏ0t)]Nâ¼ eâNÎ´Î¸20â14N(Ï0Î´t)2 (9.45)replacing the small terms with trigonometric approximations with the correct peri-odicity gives, [12cos2 Î¸0(1â cosÏ0t)]Nâ¼ eâN sin2 Î¸0âN cos2(Ï0t2 ). (9.46)Similarly near t = 2pi/Ï0` for ` â Z the magnitude of the second product in (9.42)is122â£â£cos2 Î¸0 + (2â cos2 Î¸0) cosÏ0tâ 2i sin Î¸0 sinÏ0tâ£â£N â¼ eâN4 cos2(Î¸0)Ï20(tâ 2piÏ0 `)2(9.47)so the product will be eâN4cos2(Î¸0)(tâ 2piÏ0`)2eiÎ¦(t) where Î¦(t) is the phase angle closeto each of the t = 2pi/Ï0` we findÎ¦(t) â¼ âNÏ0 sin Î¸0(tâ 2piÏ0`)= âNÎ¾0(tâ 2piÏ0`). (9.48)141So that as N ââ12N[cos2 Î¸0 + (2â cos2 Î¸0) cosÏ0tâ 2i sin Î¸0 sinÏ0t]N(9.49)â¼ââ`=0eâN4Ï20 cos2(Î¸0)(tâ 2piÏ0`)2âiNÎ¾0(tâ 2piÏ0 `)or equivalently12N[cos2 Î¸0 + (2â cos2 Î¸0) cosÏ0tâ 2i sin Î¸0 sinÏ0t]N â¼ eâN cos2 Î¸0 sin2(Ï0t2 )âiNÎ¾0t+iÏ(t)(9.50)here Ï(t) is a slowly varying function of time that ensures the phase is zero whenever t = 2pi`/Ï0. So that for large N we have the off-diagonal part of the returnprobabilitypc(t) = trCË(t)CË0 â¼12cos(2Ï)eâN sin2 Î¸0âN cos2(Ï0t2 ) (9.51)+12cos(NÎ¾0t+ Ï(t))eâN cos2 Î¸0 sin2(Ï0t2 ).Now consider the case where N is even and the effect of the diagonal part of theinitial density matrix, we have (when all sa = +1)DSÂµ1...ÂµN (t) =âagÂµaza (t). (9.52)So the contribution to the return probability from the diagonal N point correlatoralone is1212NDS Âµ1...ÂµN (t)DSÂµ1...ÂµN (0) =1212Nâagzza (t) (9.53)=1212Nâa[sin2 Î¸a + cosÏat(1â sin2 Î¸a)](9.54)all terms in the product (9.54) are less than 12 so this is exponentially small for largeN compared to trCË(t)CË0. Similarly the cross terms which contribute to the returnprobability,trDËCË0 =12NReeiÏâacos Î¸a [sin Î¸a(1â cosÏat)â i sinÏat] (9.55)trCËDË0 =12NReeiÏâacos Î¸a [sin Î¸a(1â cosÏat) + i sinÏat] (9.56)142are exponentially small as well. So we see that the leading order contribution fromthe N point correlators to the return probability is (9.51). The total diagonal partof the return probability istrDË(t)DË(0) =14tr(U(t)|sããs|U â (t)|sããs|+ U(t)| â sããâs|U â (t)| â sããâs| (9.57)+ U(t)| â sããâs|U â (t)|sããs|+ U(t)|sããs|U â (t)| â sããâs|)=12âaâS(1 + sagzza (t)2)+12âaâS(1â sagzza (t)2)(9.58)=12âÎ·=Â±1âaâS(1 + saÎ· sin2 Î¸a + saÎ· cos2 Î¸a cosÏat2)(9.59)â¼12[eâN cos2 Î¸0 sin2(Ï0t2 ) + eâN sin2 Î¸0âN cos2(Ï0t2 )]. (9.60)Note that in the above we only assumed Ïa = Ï0, Î¸a = Î¸0 and sa = +1 in the laststep. Plots of pc(t) = trCË(t)CË0 are shown in figure 9.1. We see that when Î¸0 issignificantly less that pi/2 so that there is a significant transverse field on the spinsthere are peaks of pc at multiples of the half the period of each qubitâs rotation as thetransverse components of all the spins are tipped in and out of the z = 0 plane. Inbetween these peaks the return probability is vanishingly small for large N . WhenÎ¸0 is close to zero there is a second set of peaks whose height is determined by cos 2Ïthis arises as when the field is entirely transverse. The second set of peaks comesfrom the qubitsâ transverse polerisation components anti aligning after rotation ofan angle of pi.Correlators of the formâ©âaâSC ÏxaâbâC Ïybâªare interesting for diagnosing theoff diagonal elements of the density matrix. These correlators have the followingtime dependenceâ© âaâSCÏxaâbâCÏybâª(t) =ReeiÏâaâSC(gxxa â igyxa )âbâC(gxyb â igyyb)+âaâSCgxzaâbâCgyzb(9.61)=ReeiÏâaâSC[cos2 Î¸a + cosÏat(1â cos2 Î¸a)â i sin Î¸a sinÏat]Â·âbâC[â sin Î¸b sinÏbtâ i cosÏbt] (9.62)+âaâSCcos Î¸a sin Î¸a [1â cosÏat]âbâCcos Î¸b sinÏbt.Now if N 1 it follows that at least one of C and SC are large. For |C| 1 and1430 4 800.50 4 800.50 4 8-0.500.50 4 8-0.500.50 4 8-0.500.50 4 8-0.500.5Figure 9.1: Some plots of pc(t) for cat states in systems where all qubits have thesame field. The solid (blue) line shows the exact expression (9.42) andthe dotted (red) line shows the approximation (9.51), in (a) N = 10,Î¸0 = 0, in (b) N = 10, Î¸0 =pi4 , in (c) N = 10, Î¸0 =38pi, in (d) N = 20,Î¸0 = 0, in (e) N = 20, Î¸0 =pi4 , and in (f) N = 20, Î¸0 =38pi. In allcases we have set Ï = pi8 . Note that the approximate formula works bestclose to the peaks but fails in between in particular the phase of theoscillating function is not well described by the approximate formula inbetween the peaks. Not shown is the case where Î¸0 =pi2 where we get apure sinusoidal waveform.144|SC| 1 we haveâaâSC[cos2 Î¸0 + cosÏ0t(1â cos2 Î¸0)â i sin Î¸0 sinÏ0t](9.63)â¼ââ`=0ei(Nâ|C|)Î¾0(tâ 2pi`Ï0)â 132 (Nâ|C|) sin2 2Î¸0(Ï0tâ2pi`)4âbâC[â sin Î¸0 sinÏ0tâ i cosÏ0t] â¼ââ`=0eâ |C|2cos2 Î¸0(Ï0tâpi`)2+i|C|(Î¾0[tâ pi`Ï0]âpi2|C|)(9.64)âaâSCcos Î¸0 sin Î¸0 [1â cosÏat] â¼eâNâ|C|2 cos2 2Î¸0â(Nâ|C|) cos2(Ï0t/2) (9.65)âbâCcos Î¸0 sinÏ0t â¼ sgn(sin|C| Ï0t)eâ|C|2 sin2 Î¸0â|C|2 cos2 Ï0t. (9.66)Thus if both |C| 1 and |SC| 1 we have,â© âaâSCÏxaâbâCÏybâª(t) (9.67)â¼ââ`=0cos[N sin Î¸0 (Ï0tâ 2pi`) + Ïâ pi2 |C|]Â· eâ 132 (Nâ|C|) sin2 2Î¸0(Ï0tâ2pi`)4â |C|2 cos2 Î¸0(Ï0tâ2pi`)2+ sgn(sin|C| Ï0t)Â· exp{â |C|2 sin2 Î¸0 â |C|2 cos2 Ï0tâ Nâ|C|2 cos2 2Î¸0 â (N â |C|) cos2(Ï0t/2))}.(9.68)So each of the correlators oscillates in a similar way to the return probabilities. Plotsof the off diagonal correlators in this case are shown in figure 9.2.9.3.1 The Effect of DisorderNow we can add the effect of the disorder in the Hamiltonian. We will start byconsidering the case where the field strength each qubit feels {Ïa} are different. Ingeneral there are a large set of different parameter regimes that one can considereven in this simple case. We will specialise to the case where the histogram of fieldstrengths {Ïa} is sharply peaked around a mean value Ï0. This has the effect ofspreading out each of the recurrences that occur in various quantities in the uniformcase. For instance consider the expression for the off diagonal part of the returnprobability in the uniform case as long as the recurrences occur the same argument145that lead to equation (9.51) leads topc(t) = trCË(t)CË(0) â¼12cos(2Ï)eâN sin2 Î¸0ââa cos2(Ïat2 ) (9.69)+12cos(âÎ¾at+ Ï(t))eâ cos2 Î¸0âa sin2(Ïat2 )now unlike the uniform field case all of the cos2(Ïat2)= 0 are not equal to zero at thesame time. If we calculate this term at the nâth recurrence time tn =(1+2n)piÏ0(puttingÏa = Ï0 + Î´Ïa and expanding in tnÎ´Ïa) we get cos2(Ïatn2) â¼ 14Î´Ï2at2n thus the firstterm in (9.51) can be corrected by putting eââa cos2(Ïat2 ) â eâN cos2(Ï0t2 )âN4 Î´Ï2t2with Î´Ï2 â¡ âa Î´Ï2a/N . Using this kind of approximation all terms in equation(9.51) one gets,pc(t) = trCË(t)CË(0) â¼12cos(2Ï)eâN sin2 Î¸0âN cos2(Ï0t2 )âN4 Î´Ï2t2+12cos(âÎ¾at+ Ï(t))eâN cos2 Î¸0[sin2(Ï0t2 )+Î´Ï2t24]âN2 sin2 Î¸0Î´Ï2t2 .(9.70)Some plots of pc(t) with disordered local field are shown in figure 9.3. As one wouldexpect there is no decay when Î¸0 = pi and the field is truly longitudinal we just seecoherent oscillations at the frequencyâa Î¾a.The analogous expansion approximation for the off-diagonal correlators in thepresence of disorder gives,â© âaâSCÏxaâbâCÏybâª(t) (9.71)â¼ sgn(sin|C| Ï0t)eâ|C|2 sin2 Î¸0â|C|2 cos2 Ï0tâNâ|C|2 cos2 2Î¸0â(Nâ|C|) cos2(Ï0t/2))+ eâ 132 (Nâ|C|)Î´Ï4SCt4â|C|2 cos2(Î¸0)Î´Ï2Ct2Â·ââ`=0cos[N sin Î¸0 (Ï0tâ 2pi`) + Ïâ pi2 |C|]Â· eâ 132 (Nâ|C|) sin2 2Î¸0(Ï0tâ2pi`)4â |C|2 cos2 Î¸0(Ï0tâ2pi`)2where we have defined the âmomentsâ of Î´Ïa over a set C byÎ´Ï2C =âaâCÎ´Ï2a/|C| (9.72)Î´Ï4C =âaâCÎ´Ï4a/|C|. (9.73)146We see from equation (9.71) that when |C| = O(1) (and Î¸0 6 âpi4 (2n+1) for n â Z) thecomponent ofâ©âaâSC ÏxaâbâC Ïybâª(t) which comes from the initial off-diagonal co-relations dies down much slower with the disorder when compared to the componentfrom the diagonal correlators. This does not contribute to the return probability asthe phases of these different correlators mean their contributions to pc(t) all cancelout. Plots of the off diagonal correlators in this case are shown in figure 9.2.We have found that both the return probability and the amplitude of the differentoff-diagonal correlators all die down over time when we have disordered local fieldseven in the absensence of an environment (this is of course on a time scale such thatthe recurrence times for the whole system are infinite). We know that the puritymust be constant in an isolated system. What about the part of the purity containedin the off diagonal correlators? We find up to an exponentially small termpâ¥S(t) =12âaâS{cos2 Ïat+ sin2 Î¸a sin2 Ïat+ 2 sin2(Ïat2)cos2 Î¸a[sin2(Ïat2)+ 1]}(9.74)â¼12(eâN2 cos2 Î¸0Î´Ï2t2â2N cos2 Î¸0 sin2(Ï0t/2) + eâ8N sin2(Î¸0/2)âN(1+cosÏ0t)âN2 Î´Ï2t2)(9.75)which also decays over time unless the field is purely longitudinal. The total puritycontained in the Nth order correlators must be constant over time as there are nointeraction terms in the Hamiltonian (9.37) so the decay of pâ¥S occurs when thereis a transverse component to the local fields as the purity âcontainedâ in the 2NN âth order correlators with components in the xâ y plane is spread amongst all ofthe 3N N âth order correlators.So far we have examined the effect of disorder in the local field strength felt byeach qubit. Now we consider disorder in the orientation of the local field. First wewill consider the case where all the fields have the same strength Ïa = Ï0 and thereis only a disorder in the angular variables. The effect of very weak disorder will bemost dramatic when Î¸0 =pi2 and the local fields are almost longitudinal, so thatwithout disorder we would expect coherent oscillations at a frequency NÏ0 for thereturn probability. When we have disordered angles the generalisation of equation(9.51) for the large N off-diagonal return probability is,pc(t) = trCË(t)CË(0) â¼ 12cos(2Ï)eââa sin2 Î¸aâN cos2(Ï0t2 ) (9.76)+12cos(âÎ¾at+ Ï(t))eââa cos2 Î¸a sin2(Ï0t2 ).Thus we can write pc(t) in terms of the averaged quantitiesâa sin2 Î¸a = Nsin2 Î¸,1470 2 4-1010 2 4-1010 2 4-1010 2 4-101Figure 9.2: Some plots of off diagonal correlators for cat states in system with N =20 central qubits where all qubits feel a central field. The solid (blue)line shows the exact expression (9.42) and the dotted (red) line showsthe approximation (9.71). Plot (a) showsâ©âNa Ïxaâªwithout disorder, (b)showsâ©âNa Ïxaâªwhere the central field strength has gaussian disordercharacterised by the mean Î´Ï2 = 0.05Ï0, (c) showsâ©âNa Ïyaâªwithoutdisorder and (d) showsâ©âNa Ïyaâªwhere the central field strength hasgaussian disorder characterised by the mean Î´Ï2 = 0.05Ï0 . In all caseswe have set Ï = 0 and all Î¸a = Î¸0 =pi4 .1480 4 800.50 4 800.50 4 8-0.500.50 4 8-0.500.50 4 8-0.500.50 4 8-0.500.5Figure 9.3: Some plots of pc(t) for cat states in systems when there is a disorderedcentral field strength. The solid line shows the exact expression (9.62)and the dashed line shows the approximations (9.70), in (a)âÎ´Ï2=0.01Ï0,Î¸0 = 0, in (b)âÎ´Ï2= 0.01Ï0, Î¸0 =pi4 , in (c)âÎ´Ï2= 0.01Ï0,Î¸0 =38pi, in (d)âÎ´Ï2= 0.05Ï0, Î¸0 = 0, in (e)âÎ´Ï2= 0.05Ï0, Î¸0 =pi4 ,and in (f)âÎ´Ï2= 0.05Ï0, Î¸0 =38pi. In all cases we have set Ï =pi8and N = 20. The approximate formula works best from times suchthat Î´Ï2t2 1, increasing N makes the approximation better at longertimes as it kills off the smaller oscillations. Not shown is the case whereÎ¸0 =pi2 where we get a pure sinusoidal waveform.149Ï1Ï15Ï14Ï13Ï12Ï11Ï2Ï25Ï24Ï23Ï22Ï21 ...Figure 9.4: The configuration of the bath discused in section 9.4, each central systemqubit {Ï a for i = 1 . . . N} is coupled to its own set of of bath spinsBa = {Ïai for i = 1 . . . |Ba|}. We have illustrated the case where thereare two central spins..âa cos2 Î¸a = Ncos2 Î¸ andâa Î¾a = NÎ¾,pc(t) = trCË(t)CË(0) â¼12cos(2Ï)eâNsin2 Î¸âN cos2(Ï0t2 )+12cos(NÎ¾t+ Ï(t))eâNcos2 Î¸ sin2(Ï0t2 ). (9.77)The first term in the equation (9.77) is exponentially small so long as sin2 Î¸ â¼ O(1)and peaks when Ï0t = pi(2n + 1) for n â Z. The second term in equation (9.77) isexponentially small for cos2 Î¸ â¼ O(1) unless Ï0t = 2npi for n â Z. So we see thatdisorder in the direction of the local fields changes the structure of the recurrencesbut in order to get decay of the amplitudes of these regular recurrences we needdisorder in the strengths of the local fields so that we have a spread of the oscillationfrequencies.9.4 Degeneracy Blocking for a Large Cat StateConsider now a multi-qubit generalisation of the degeneracy blocking Hamiltonianfrom section 6.1,H =âaâSâa2Ïxi +âaâSâiâBa12ÏaiÏzaiÏza . (9.78)We will consider a case where the interactions between bath and central qubits islike that shown in figure 9.4. Each central spin Ï a is coupled to its own bath of bathspins Ba so that the total bath is B = B1âªB2âª. . .âªBN . For simplicity we assume thatthe bath Ba belonging to each central spin Ï a is the same size and the interactionsbetween each central spin and its bath spins are the same, that is Ïai = Ïi. Withan initial density matrix that is a product state Ï(0) = ÏÂ¯S(0)ÏÂ¯B(0) and the initial150reduced density matrix the central system ÏÂ¯S(0) as the cat state (3.41). We takethe initial reduced density for the bath separates ÏÂ¯B(0) =âa ÏÂ¯Ba(0) where eachÏÂ¯Ba(0) = ÏÂ¯B1(0) is the same.As we saw in section 6.1 the only relevant part of ÏB1 for determining the dyna-mics of the central spin corralators is the part that contains information about theoperator Î¾Ëa â¡âiâBa ÏaiÏzai which commutes with the Hamiltonian. Thus if we arenot interested in the time dependence of the bath correlators we can with out lossof generality consider a spectral decomposition of ÏÂ¯Ba(0) of the form,ÏÂ¯Ba(t) =â« âââdÎ¾P aDB(Î¾)Î´Ë(Î¾ â Î¾Ëa). (9.79)Here P aDB(Î¾) acts as a time independent probability density for the bias field. Thusin this case the Green function for the single qubit correlations of qubit a isgÂµÎ½a (t) =â« âââdÎ¾P aDB(Î¾)gÂµÎ½a (t; Î¾) (9.80)wheregÂµÎ½a (t; Î¾) = cos2 Î¸a(Î¾)xËÂµxËÎ½ + sin2 Î¸a(Î¾)zËÂµzËÎ½ + cos Î¸a(Î¾) sin Î¸(Î¾) (xËÂµzËÎ½ + zËÂµxËÎ½)+ cosÏÎ¾t[(1â cos2 Î¸(Î¾))xËÂµxËÎ½ + (1â sin2 Î¸(Î¾))zËÂµzËÎ½ + yËÂµyËÎ½â 2 cos Î¸(Î¾) sin(Î¾)Î¸ (xËÂµzËÎ½ + zËÂµxËÎ½)]+ sinÏÎ¾t(2 cos Î¸(Î¾) (yËÂµzËÎ½ â zËÂµyËÎ½) + 2 sin Î¸(Î¾) (xËÂµyËÎ½ â yËÂµxËÎ½))(9.81)with: cos Î¸(Î¾) â¡ Î¾ÏÎ¾(9.82)ÏÎ¾ â¡âÎ¾2 + â2a. (9.83)The time evolution of the N point correlator CÂµ1...ÂµN defined in the previous sectionis justCÂµ1...ÂµN =âaâSgÂµaÎ½aa CNÎ½1...Î½N(9.84)where gÂµaÎ½aa is the degeneracy averaged single spin Green function (6.15). So that151the pc(t) istrCË(t)CË0 =12Re{e2iÏâa12[gxxa â gyya â i(gxya + gyxa )] (9.85)+âa12[gxxa + gyya â i(gxya â gyxa )]}=12Re{e2iÏâa12â« âââdÎ¾aPaDB(Î¾a)[cos2 Î¸a(1â cosÏat)]+12âa12â« âââdÎ¾aPaDB(Î¾a)[cos2 Î¸a + (2â cos2 Î¸a) cosÏatâ 2i sin Î¸a sinÏat]}.(9.86)Examine the two integrals appearing in the above expression with the assumptionthat the spectral density is an even function P aDB(Î¾) = PaDB(âÎ¾) then we can writethe above integrals as integrals over the frequency as follows,12â« âââdÎ¾aPaDB(Î¾a) cos2 Î¸a(1â cosÏat) = Reâ« ââadÏâ2aPaDB(âÏ2 ââ2a)ÏâÏ2 ââ2a(1â eiÏt)(9.87)12â« âââdÎ¾aPaDB(Î¾a)[cos2 Î¸a + (2â cos2 Î¸a) cosÏatâ 2i sin Î¸a sinÏat]=â« ââadÏP aDB(âÏ2 ââ2a)ÏâÏ2 ââ2a[â2a + (2Ï2 ââ2a) cosÏtâ 2iÏâÏ2 ââ2a sinÏt].(9.88)Which can be written as constant terms added to various Fourier integrals with anamplitude that is smooth except at Ï = âa. Theorems in Fourier analysis [66] implythat the long time behaviour can be obtained by expanding the Fourier amplitudesaround any points where they fail to be smooth so that the long time t â âbehaviour of the integrals appearing in (9.86) is12â« âââdÎ¾aPaDB(Î¾a) cos2 Î¸a(1â cosÏat) â¼ 12â« âââdÎ¾aPa(Î¾a)â2aÎ¾2a + â2a(9.89)â P aDB(0)âpiâa2tcos(âat+pi4)12â« âââdÎ¾aPaDB(Î¾a)[cos2 Î¸a + (2â cos2 Î¸a) cosÏatâ 2i sin Î¸a sinÏat]â¼12â« âââdÎ¾aP aDB(Î¾a)â2aÎ¾2a + â2a+ P aDB(0)âpiâa2tcos(âat+pi4). (9.90)152Assuming all âa = â0, PaDB(Î¾a) = PDB(Î¾a) are the same, and defining gxxâ â¡12â«âââ dÎ¾aPDB(Î¾a)â20Î¾2a+â20. Then we get an approximate formula for the off-diagonal returnprobability in the long time limit.pc(t) â¼ cos 2Ï[gxxâ â PDB(0)âpiâ02tcos(â0t+pi4)]N(9.91)+[gxxâ + PDB(0)âpiâ02tcos(â0t+pi4)]N(9.92)Now consider the case where we have a Gaussian density of bias fields with standarddeviation Î´Î¾,PDB(Î¾a) =eâ Î¾2a2Î´Î¾2â2piÎ´Î¾. (9.93)in which case equation (9.91) becomespc(t) â¼ cos 2Ï[gxxâ â12ââ0Î´Î¾2tcos(ât+ pi4)]N+[gxxâ +12ââ0Î´Î¾2tcos(â0t+pi4)]N.(9.94)withgxxâ =âpi2|â0|2Î´Î¾eâ202Î´Î¾2 erfc( |â0|â2Î´Î¾0). (9.95)The approximation contained in equation (9.94) is tested in figure 9.5 where we seethat when the standard deviation of the bath bias is small the approximation takesa long time to become accurate and even with a modestly large number of spins thereturn probability can be very small by the time the approximation is accurate. Thisis because in the expansion (9.94) one is assuming that all dimensionless parametersproportional to t are large in particular one is assuming Î´Î¾t is large. This alsoexplains why one does not obtain the Î´Î¾ = 0 result from sending Î´Î¾ â 0 in equation(9.94). To remedy this one can calculate an approximation to the integrals (9.87-9.88) using the steepest descent method [7] in a way that takes into account thefact that Î´Î¾ can be small. This approximation is derived in appendix F.1 insteadof treating t as a large parameter it treatsât2Î´Î¾2 +â20Î´Î¾2as a large parameter and153gives the following results for the integrals (9.87-9.88)12â« âââdÎ¾aeâ Î¾2a2Î´Î¾2â2piÎ´Î¾cos2 Î¸a(1â cosÏat) â¼ gxxâ â12cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14(9.96)12â« âââdÎ¾aeâ Î¾2a2Î´Î¾2â2piÎ´Î¾[cos2 Î¸a + (2â cos2 Î¸a) cosÏatâ 2i sin Î¸a sinÏat]â¼gxxâ +12cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14(9.97)with: tanÏa(t) =Î´Î¾2tâa(9.98)gxxâ â¼12âân=0(â2)nÎ(12 + n)Î(12) ( Î´Î¾â0)2n. (9.99)Therefore in this limit we have,pc(t) â¼ cos 2Ïï£®ï£¯ï£¯ï£°gxxâ â 12 cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14ï£¹ï£ºï£ºï£»N+ï£®ï£¯ï£¯ï£°gxxâ + 12 cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14ï£¹ï£ºï£ºï£»N.(9.100)Figures 9.5 and 9.6 show the how the approximation works (9.100) we see it workswell for longer times and that the errors present at shorter times are increased whenN is increased. If N is large enough we see a regime where there are still coherentrecurrences but the approximation (9.100) does not work so well as by the time theapproximation is accurate pc(t) is already negligible so we will have to work a bitharder to describe this region.When N is very large we have seen that the approximation (9.100) will fallsapart because the large exponents mean that the large product in equation (9.86)dies away much faster than each of the terms. We can get the amplitude of theâearlyâ recurrences by noting that after a small number of periods the phase shiftis negligible thus we can get the amplitude of the early peaks by evaluating theintegrals in (9.86) at the times tn = 2pin/â0 for integer n then expand in small Î´Î¾yielding the following result for then height, pcn of the nâth peak in the plot of pc(t)(this requires Î´Î¾â 1 so that there are well defined peaks)pcn â¼ 12eâ 3Nt2nÎ´Î¾416â20 for:tÎ´Î¾2â0 1. (9.101)1540 5000.50 10000.50 5000.50 1000.50 10000.50 1000.5Figure 9.5: Some plots of pc(t) for cat states in systems with a simple degeneracyblocking spin bath. The solid line shows the exact expression (9.86),the dotted red line shows the long time approximation (9.94), and thedashed black line shows the long time small Î´Î¾ approximation (9.100).In (a) Î´Î¾ = 0.2â0 and N = 1, in (b)Î´Î¾ = 0.2â0,N = 5, and the inset isthe same plot magnified at earlier times , in (c) Î´Î¾ = 0.1â0 and N = 1,in (d) Î´Î¾ = 0.1â0,N = 5, and the inset is the same plot magnifiedat earler times. In all cases we have set Ï = pi4 . Note the differencein scale on the time axes between (a-b) and (c-d). We see in generalthe approximation (9.100) does a much better job when Î´Î¾ is small forsmaller times.155Which is plotted in figure 9.6 where we can see the cross over between the differentregimes.There is another limit which is worth considering, when either N or Î´Î¾ are largeenough we only see the initial peak in the plot of pc(t), in which case the pc(t) caneasily be calculated by expanding expressions in the small variable â0t from whichone findspc(t) â¼ 12eâN2t2(Î´Î¾2+12 â20). (9.102)So we see that the time dependence of pc(t) can be divided into three regimes:(i) In all cases there is a peak at t = 0 which will die away like 12eâN2t2(Î´2+12 â20)fortimes t ââ10 and Î´Î¾â1, (ii) the first few recurrences will occur at t = 2pin/Ï0 forn â Z and have an amplitude â¼ 12 exp(â3pi2NnÎ´Î¾44â40)provided tÎ´Î¾2â 1 and will bevisible if N is not to large compared to4â403pi2Î´Î¾2and (iii) so long as Î´Î¾ is small enoughthere will also be a long time algebraic decay where pc(t) is well approximated by(9.100). Note that in general for a given t > 0, pc(t) scales with N like pc â¼ A(t)Nfor some A < 1 so that for any given time there is an N that will make pc negligibleat that time.Now we look at the correlators in the limit where we can make the steepestdescent approximation discussed in the appendix F.1. We have,â© âaâSCÏxaâbâCÏybâª(t) =ReeiÏâbâCâ«dÎ¾bP (Î¾b) [â sin Î¸b sinÏbtâ i cosÏbt] .âaâSCâ«dÎ¾aP (Î¾a)[cos2 Î¸a + cosÏat(1â cos2 Î¸a)â i sin Î¸a sinÏat](9.103)+âaâSCâ«dÎ¾aP (Î¾a) cos Î¸a sin Î¸a [1â cosÏat]Â·âbâCâ«dÎ¾bP (Î¾b) cos Î¸b sinÏbt (9.104)â© âaâSCÏxaâbâCÏybâª(t) â¼ Î´C,S2N(1 +t2Î´Î¾4â2)âN4sinN(â0t+12Ï)+(1 +t2Î´Î¾4â2)â |C|4cos(Ïâ pi2|C|)2|C| cos|C|(â0t+12Ï)(9.105)Â·[gxxâ +Î´Î¾2ââ0(t2Î´Î¾4 + â20)â 34 cos(â0t+34Ï)]Nâ|C|. (9.106)1560 10 2000.50 20 4000.50 10 2000.50 20 4000.50 10 2000.50 20 4000.5Figure 9.6: Some plots of pc(t) for cat states in systems with a simple degene-racy blocking spin bath. The solid line shows the exact expression(9.86) and the dotted red line shows the long time small Î´Î¾ approxi-mation (9.100) and the dashed black line shows the approximation tothe recurrence peak height (9.101). In (a) Î´Î¾ = 0.1â0 and N = 25,in (b)Î´Î¾ = 0.1â0,N = 50, in (c) Î´Î¾ = 0.1â0 and N = 100, in(d) Î´Î¾ = 0.05â0,N = 25, in (e) Î´Î¾ = 0.05â0,N = 50, and in (f)Î´Î¾ = 0.05â0,N = 100. In all cases we have set Ï =pi4 . We see ingeneral the approximation (9.100) does a much better job when Î´Î¾ issmall for smaller times.157The Î´C,S term here is the contribution from the initial correlatorâ©âS Ïzaâªand isonly present when N is even in which case the z correlations can evolve into ycorrelations only (this is a consequence of having a bias density which is symmetricaround zero). So for long enough times correlators containing more x componentsdie off slower than those containing more y. In the case of the correlators the longtime small Î´Î¾ approximation works a lot better for large N . Plots of the off diagonalcorrelators in the degeneracy blocking case are shown in figure 9.7.The off-diagonal purity in this degeracy blocking case simplifies (the simple formis again due to the symmetry of the bias distribution PDB(Î¾)),pâ¥S(t) =12{cos 2Ïâa[(gxxa (t))2 â (gyya (t))2]+âa[(gxxa (t))2 + (gyya (t))2]}(9.107)â¼12ï£±ï£´ï£´ï£²ï£´ï£´ï£³cos 2Ï(gxxâ )Nï£®ï£¯ï£¯ï£°gxxâ + cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14ï£¹ï£ºï£ºï£»N(9.108)+ï£®ï£¯ï£¯ï£°ï£«ï£¬ï£¬ï£­gxxâ + 12 cos(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)14ï£¶ï£·ï£·ï£¸2+14cos2(âat+12Ïa(t))(1 + t2Î´Î¾4â2a)12ï£¹ï£ºï£ºï£»Nï£¼ï£´ï£´ï£´ï£½ï£´ï£´ï£´ï£¾ .One can also calculate the N -partite purity. I have omitted the expressions butproduced plots in figure (9.8), where we see that the purity has decaying recurrenceson top of a decaying âbaselineâ both of which decay in a similar way to the off-diagonal pc(t) (exponentially for small times or large enough N and algebraicallyfor long times).9.5 General averagingNow consider the case where each spin is coupled to its own bath so we still havea situation like that in figure 9.4 but with a more general bath Hamiltonian. Astracing out the bath acts like a form of averaging it is natural to consider the casewhere the effect of the bath is to average the Green function in a way which isanalogous to (9.80). Indeed many central spin models (for example the precessionaldecoherence model discussed in chapter 7) reduce to finding a Green function forcentral spin a of the form,gÂµÎ½a (t) =â«dnÎ± gÂµÎ½a (t;Î±)P (Î±) (9.109)158Figure 9.7: Some plots of off diagonal correlators for cat states in system. The solid(blue) line shows the exact expression and the dotted (red) line shows theapproximation (9.106). Plot (a) showsâ©âNa Ïxaâªwith N = 25 centralqubits at Î´Î¾ = 0.1â0, (b) showsâ©âNa Ïyaâªwith N = 25 central qubitsat Î´Î¾ = 0.1â0, (c) showsâ©âNa Ïxaâªwith N = 101 central qubits atÎ´Î¾ = 0.1â0, and (d) showsâ©âNa Ïyaâªwith N = 101 central qubits atÎ´Î¾ = 0.1â0. In all cases we have set Ï =pi4 .1590 10 2000.50 10 2000.50 10 2000.50 10 2000.5Figure 9.8: Some plots of pS(t) for cat states in systems with a simple degeneracyblocking spin bath. In (a) Î´Î¾ = 0.2â0 and N = 5, in (b)Î´Î¾ = 0.2â0,N =25, and the inset is the same plot magnified at earlier times , in (c)Î´Î¾ = 0.1â0 and N = 25, in (d) Î´Î¾ = 0.1â0,N = 100.160where gÂµÎ½a (t;Î±) is the Green function for central spin a defined in terms of an ndimensional set of parameters Î± = (Î±1, Î±2 . . . Î±n) which are then averaged over withsome weight function P (Î±). In some cases (including all those listed above) gÂµÎ½a (t;Î±)will be expressible in terms of the Green function for a central spin in a simple fieldHa =12â¦(Î±)nË(Î±) (we can always chose nË(Î±) so that â¦(Î±) > 0) so thatgÂµÎ½a (t;Î±) = nËÂµ(Î±)nËÎ½(Î±) + [Î´ÂµÎ½ â nËÂµ(Î±)nËÎ½(Î±)] cos[â¦(Î±)t] + sin[â¦(Î±)t]ÎµÂµÎ½Î»nËÎ»(Î±).(9.110)Then the average (9.109) takes can be split into two different parts: (i) an evenfunction of timegÂµÎ½a1 (t) =â«dnÎ±P (Î±)nËÂµ(Î±)nËÎ½(Î±) +â«dnÎ±P (Î±) [Î´ÂµÎ½ â nËÂµ(Î±)nËÎ½(Î±)] cos[â¦(Î±)t](9.111)=gÂµÎ½a0 +â«dÏ cos(Ït)gÂµÎ½a2 (Ï) (9.112)gÂµÎ½a1 (Ï) â¡â«dnÎ±P (Î±) [Î´ÂµÎ½ â nËÂµ(Î±)nËÎ½(Î±)] Î´(â¦(Î±)â Ï) (9.113)gÂµÎ½a0 â¡â«dnÎ±P (Î±)nËÂµ(Î±)nËÎ½(Î±) (9.114)this also has eigenvalues which are less than one and (ii) an odd function of timeÎµÂµÎ½Î»gÎ»a2(t),gÂµa2(t) =â«dnÎ±P (Î±) sin[â¦(Î±)t]nËÂµ(Î±) =â« â0dÏgÂµa2(Ï) sin(Ït) (9.115)gÂµa2(Ï) =â«dnÎ±P (Î±)nËÂµ(Î±)Î´(â¦(Î±)â Ï) (9.116)which is the alternating tensor multiplied by a vector with length less than or equalto one. In the long time limit the behaviour of gÂµÎ½a (t) will be dominated by anyparts of the tensors gÂµa1(Ï) and gÂµÎ½a2 (Ï) which fail to be analytic [66] and the Greenfunction will decay to a constant symmetric tensor gÂµÎ½aâ = limtââ gÂµÎ½a1 (t) which willhave positive eigenvalues less than one. In the orthogonality blocking case above thegÂµa2 = 0 and gÂµÎ½a1 (Ï) has an isolated singularity at Ï = â0 at that singularity differentcomponents of gÂµÎ½a2 (Ï) go likeâ¼ (Ïââ0)â12 Î(Ïââ0), Î(Ïââ0),âÏ ââ0Î(Ïââ0)and (Ïââ0)Î(Ïââ0) which leads to the long time algebraic decay in the oscillationsseen in the previous section. The probability density function in the orthogonalityblocking example above P â PDB(Î¾) = eâÎ¾2/2Î´Î¾2/â2piÎ´Î¾2 is well behaved enoughthat we can find an approximation that combines the long time behaviour using asteepest descent approximation. As we will see probability densities which do nothave an exponential decay will require a different approach. In general for the abovetype of averaging without interactions between the central spins we will have for our161generalised cat states,CË(t) =12NCSÂµ1...Âµn(t)âaâSÏÂµaa (9.117)CSÂµ1...Âµn(t) =12ReeiÏâaâS[gÂµaxa (t)â isagÂµaya (t)] (9.118)pc(t) =12Ree2iÏâa[gxxa â gyya â isagxya1 ] +12Reâa[gxxa + gyya â isagza2] (9.119)â©âaâSÏÂµaaâª=12ReeiÏâaâS[gÂµaxa (t)â isagÂµaya (t)] +âagÂµaza (9.120)pc =12Ree2iÏâa[gÂµxa gÂµxa â gÂµya gÂµya â 2isagÂµxa gÎ½xa ] +12âa[gÂµxa gÂµxa â gÂµya gÂµya ](9.121)pâ¥c =12Ree2iÏâa[(gxxa )2 â (gyya )2 + 4gxya1gza2 â 2isa (gxxa gxya + gyya gyxa )]+12âa[(gxxa )2 + (gxya )2 + (gyxa )2 + (gyya )2]. (9.122)Where we have assumed N is even. When N is odd the last term in equation (9.120)should be omitted.9.5.1 Spherically Symmetric Degeneracy BlockingConsider the slightly generalised version of the degeneracy blocking spin bath wherethe z components of the bath spins are allowed to couple to any component of thecentral spin,H =âaâSâa2Ïxi +âaâSâiâBaÏaiÏzaimËi Â· Ï a. (9.123)in the same limit discussed above the field on each central spin is of the formâaxË + Î¾a = â¦(Î¾a)nË(Î¾a) where Î¾a is a three dimensional vector drawn from somedistribution P (Î¾a). â¦(Î¾a) and nË(Î¾a) are in this case,â¦(Î¾a) =â(âa + Î¾x)2 + Î¾2y + Î¾2z (9.124)nË(Î¾a) =âaxË+ Î¾aâ¦(Î¾a)(9.125)A simple assumption is that P (Î¾a) is a spherically symmetric Gaussian distribution,P (Î¾a) =eâÎ¾2a/2Î´Î¾2(2piÎ´Î¾2)3/2. (9.126)162In this case the tensors gÂµa1(Ï) and gÂµÎ½a2 (Ï) are easly resolved in the âspherical polarco-ordinatesâ Î¾x = â¦ cos Î¸ ââa, Î¾y = â¦ sin Î¸ cosÏ, Î¾z = â¦Î¾y = â¦ sin Î¸ cosÏ,ga1(Ï) =Ï2 2pieâ(Ï2+â2)/2Î´Î¾2(2piÎ´Î¾2)3/2â« pi0dÎ¸ sin Î¸eÏâ cos Î¸/Î´Î¾2ï£«ï£­1â cos2 Î¸ 0 00 1â 12 sin2 Î¸ 00 0 1â 12 sin2 Î¸ï£¶ï£¸(9.127)=Ï22pieâ(Ï2+â2)/2Î´Î¾2(2piÎ´Î¾2)3/2â« 1â1dueÏâÎ´Î¾2uï£«ï£­1â u2 0 00 12(1 + u2) 00 0 1â 12(1 + u2)ï£¶ï£¸(9.128)ga2(Ï) =xËÏ2 2pieâ(Ï2+â2)/2Î´Î¾2(2piÎ´Î¾2)3/2â« pi0dÎ¸ sin Î¸ cos Î¸eÏâ cos Î¸/Î´Î¾2(9.129)=xËÏ22pieâ(Ï2+â2)/2Î´Î¾2(2piÎ´Î¾2)3/2â« 1â1duueÏâÎ´Î¾2u(9.130)of which the only non-zero matrix elements are,gxxa1 (Ï) =Î´Î¾â30Ïâ2pi[eâ (Ïââ0)22Î´Î¾2 (Î´Î¾2 + â0Ï) + eâ (Ï+â0)22Î´Î¾2 (Î´Î¾2 ââ0Ï)](9.131)gyya1(Ï) =gzza1(Ï) (9.132)=1â2piâ30ÏÎ´Î¾[eâ (Ï+â0)22Î´Î¾2 (Î´Î¾4 + â20Ï2 ââ0ÏÎ´Î¾2) (9.133)â eâ(Ïââ0)22Î´Î¾2 (Î´Î¾4 + â20Ï2 + â0ÏÎ´Î¾2)]gxa2(Ï) =1â2piÎ´Î¾â2[eâ (Ï+â)22Î´Î¾2(1â â0ÏÎ´Î¾2)â eâ(Ïââ)22Î´Î¾2(1 +â0ÏÎ´Î¾2)]. (9.134)these are smooth functions for all real frequencies so that their Fourier transformsare exponentially decaying oscillations with an underlying frequency of â0 (thepeaks of the Fourier amplitudes),gxxa1 (t) =gxxa0 +Ï2eâ12 Î´Î¾2tâ20cos â0t (9.135)gyya1(t) =gyya0 +eâ12 Î´Î¾2tâ20[(â20 + Î´Î¾2) cos â0t+ tâ0Î´Î¾2 sin â0t](9.136)gxa2(t) =eâ12 Î´Î¾2tâ20[â0tÎ´Î¾2 cos â0t+ (â20 â Î´Î¾2) sin â0t]. (9.137)1630.10.212=Figure 9.9: A plot of gzza1(Ï) precessional decoherence with various values of Îº, asindicated in the legend.9.5.2 Precessional DecoherenceNow we discuss the many qubit version of the precessional decoherence model dis-cussed in section 7.2, we make the same assumptions about the initial conditions ofthe bath which we did in that section. In this case the dynamics of the bath on eachspin are accounted for by performing an average of over an auxiliary variable X withâ¦(x) = â0|J0(2XâÎº)|, nË(x) = xË and P (X) = Xeâ X24NÎº where Îº is the precessionaldecoherence variable defined in equation 7.67. The tensors, gÂµa1(Ï) and gÂµÎ½a2 (Ï), inthis case aregÂµÎ½a1 (Ï) =2(Î´ÂµÎ½ â xËÂµxËÎ½)â« â0dX4ÎºXeâX24Îº Î´(â0J0(X)â Ï) (9.138)gÂµa2(Ï) =2xËÂµâ« â0dX4ÎºXeâX24Îº Î´(â0J0(X)â Ï). (9.139)With the aid of the delta function identity Î´(f(X)) = Î´(X)/|f â²(X)|, we can concludethat gÂµÎ½a1 (Ï) and gÂµÎ½a2 (Ï) fail to be smooth whenever Ï = â0J0(jâ²0,2m), (jâ²0,m is themth zero of the derivative of the J0(X) and m = 0, 1, 2 . . .). gzza1(Ï) is shown in figure9.9, from which we can see that when Îº 1, the only non-analytic point that issignificant is Ï = â0J0(0) = â0. This is because the amplitude of the contributionto gzza1(Ï) from at jâ²0,m is â¼ eâ(jâ²0,m)2/4Îº for small Îº and in that limit only the firstzero jâ²00 contributes.164So for small Îº we getâ« â0dX4ÎºXeâX24Îº Î´(â0J0(X)â Ï) (9.140)=â« j010dX4ÎºXeâX24Îº Î´(â0J0(X)â Ï) +O(eâj201/4Îº)=Jâ100(Ïâ)Î(â0 â Ï)Î(Ï)â0ÎºJ1[Jâ100(Ïâ)] expï£±ï£´ï£²ï£´ï£³â[Jâ100(Ïâ0)]24Îºï£¼ï£´ï£½ï£´ï£¾+O(eâj201/4Îº). (9.141)Here Jâ100 (X) is the inverse function of the J0(X), defined by Jâ100 (J0(X)) = X,on the domain 0 < X < j01, and j01 â 2.4048 is the first zero of J0(x). Thenperforming a steepest descent calculation like that in appendix F.1 gives us,gÂµÎ½a1 (t) â¼xËÂµxËÎ½ + (Î´ÂµÎ½ â xËÂµxËÎ½)cos â0t+ â0tÎº sin â0tâ1 + Îº2â20t2(9.142)gÂµa2(t) â¼xËÂµsin â0tââ0tÎº cos â0tâ1 + Îº2â20t2. (9.143)One then finds the correlators for cat state initial conditions,â©âa6âCÏxaâbâCÏybâª= cos(Ïâ pi2|C|) cos|C| (â0t+ ÏÎº(t))(1 + Îº2â20t2) |C|2(9.144)tanÏÎº(t) = Îºâ0t (9.145)so we see in the lowest order orthogonality blocking approximation, Ïxa is conserved(as discussed in section 7.2.3). The off diagonal return probabilitypc(t) =12Ncos 2Ï(1â cos (â0t+ ÏÎº(t))â1 + Îº2â20t2)N+12N(1 +cos (â0t+ ÏÎº(t))â1 + Îº2â20t2)N.(9.146)We see that this is similar to the behaviour in the degeneracy blocking case, in thelimit we have looked the key piece of information that determines the long timebehaviour in the small Îº limit is the exponent close to Ï = â0 and the discontinuityof the Fourier tensors gÂµÎ½a1 (Ï) and gÂµa2(Ï) at this point.9.5.3 Degeneracy Blocking with a Lorentzian Field DistributionConsider the example in section 9.4 but with the bias weighting function,PDB(Î¾) =Î³pi(Î³2 + Î¾2)(9.147)1650 5 1000.5Figure 9.10: A plot of pS(t) for cat states in systems with a simple degeneracyblocking spin bath with a Lorentzian field distribution. The solid (blue)line shows the exact value and the dotted (red) line shows an approx-imation derived as in the appendix F.2. We have Î³ = 0.01â0 andN = 100.which has ill defined moments. We can still obtain the long time behaviour byexpanding the Fourier tensors gÂµÎ½a1 (Ï) and gÂµa2(Ï) around their Ï = â0 values butthis gives an expression which is only valid for large Î³t so if we want to include thepossibility of Î³ being small we will have to derive another expression. The steepestdescents approach we used in section 9.4 does not work well here as we cannot treatthe probability distribution as a term which varies exponentially fast with small Î³and modify the contour of integration appropriately. The problem is that as Î³ â 0the two poles in PDB(Î¾) at Î¾ = Â±iÎ³ come closer and closer to the real axis. Soany approximations which will be valid for small Î³t need to have the poles treatedproperly, in practice as the Green function has branch points at Î¾ = Â±â0 thismeans an approximation derived for small Î³ will break down when Î³ â¼ â0. Suchan approximation is derived in the appendix F.2. The long time decay of variousquantities is algebraic still but the short time behaviour is now non-analytic, a plotof pc(t) is shown in figure 9.10.9.6 ConclusionIn this chapter we have presented some results for the dynamics of entangled statescontaining a large number of central qubits, which may be coupled to a bath. Wehave limited our discussion to cases where the central qubits do not interact either166directly or through a bath. We hope to tackle the interacting problem in futurework.First in section 9.1, we discussed some simple ways to characterise these states,based on their correlators, as they evolve from their initial values. Then in section9.2 and 9.3 we studied their dynamics under the influence of only local fields. Wesaw that if the fields were uniform then the motion could be characterised by dif-ferent recurrences, and disorder could cause various recurence amplitudes to decay,even without interactions which change the amount of information stored in thedifferent orders of correlators. Then in sections 9.4 and 9.5, we considered someexamples where spin baths had to be averaged out. We found we could get resultsby investigating the averaged single qubit Green functionâs spectral properties andthat in general, when there are many central qubits, care needed to be taken toobtain the correct asymptotic behaviour for high order correlators.167Chapter 10ConclusionsHere we discuss the new work results presented in this thesis, and how we it couldbe extended.In chapters 2 and 3 showed how we can understand multipartite entanglmentas correlations. In this picure of entanglement, the different types of entanglmentare be specified by different correlated parts of the reduced density matrices or bycorrelations between different clusters of qubits. We saw that the use of these de-compositions is in many ways a more transparent way of characterising multipartiteentanglement than the entanglement measures that have been discussed in the lite-rature.In particular in chapter we 2 we identified multiple different ways of splitting themany body density matrix into âcorrelated partsâ, as an expansion over differententangled sets of the system, using ÏÂ¯CA and, over different partitions of the systeminto entangled sets using ÏÂ¯CCA . It would be interesting to see whether more preciseconnections between ÏÂ¯CA, ÏÂ¯CCA and the formal measures of entanglement can be made.Then we focused on gaining an understanding the time evolution of multipartiteentanglements, an important problem for the construction of a quantum computer.First we considered a formal approach to quite general problems involving pairwiseinteractions. In chapter 4 we showed correlated parts of the reduced density matrixÏÂ¯CA, can be used to study the dynamics of the reduced density matrix, in cases whenthere are general pairwise interactions and we derived a hierarchy of equations ofmotion for many-body density matrices. We then saw that in the qubit case withonly pairwise interactions and local fields included, there is a hierarchy of equationsof motion for the qubit correlators, which links correlators containing a cluster ofspins to correlators containing that cluster with one spin added or removed. Whenthings are rewritten in terms of supervectors of qubit correlators, we found thatthat the resulting matrix equations of motion involve sparse matrices, making thempractically useful. It remains to be seen under what circumstances one could derivesimilar hierarchies of equations for the correlated parts of reduced density matricesor connected correlators between spin components, and whether they can be put ina useful form.This work showed that the structure of the hierarchy of equations of motion imp-lied that the information lost by a central qubit into a bath of spins is transferred tolarger and larger correlators between the central qubit and bath spins, in a cascade168of coherence. We saw cascade exists this both in the simple degeneracy blockingmodel studied in chapter 6 and in the more realistic precessional decoherence modelstudied in chapter 7. This presents an intuitive way of understanding the importantprocess of decoherence. It would be interesting to see how this cascade works inother situations, it is clear that while the details of this cascade might be differentin other models, this picture is somewhat generic.We showed how the cascade works out for the specific example of an Fe8 magneticmolecule qubit in chapter 8. We investigated under what conditions one could expectto see signs of this cascade, if we could measure the correlators between pairs of bathspins and the central spin.Finally in chapter 9, we studied the decay of entanglement in non-interactingmodels with many central spins, and saw the effect of large numbers in this case.We saw that the correlators responsible for Nâpartite enanglement decayed at arate N times greater than a single central qubit. This places limitations on theconstruction of quantum computers which require the information in multipartiteenanglment to function. An obvious avenue for future work is to extend this analysisto interacting systems with many qubits. 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B, 53:13682, 1996.179Appendix AProperties of EntanglementDensity MatricesIn this Appendix we prove two properties of the entanglement density matrices thatwere quoted without proof in section 2.0.3. We use same notation as that definedin this section.A.1 Proof of Eqtn. (2.23)We wish here to prove the result given in eqtn. (2.22) (or, equivalently eq. (2.23))for the entanglement correlated density matrices.We do this by induction. The n = 2 case comes from tracing out all of S excepti and j from the equation for the density matrix (2.2), so thatÏÂ¯ij = ÏÂ¯iÏÂ¯j + ÏÂ¯Cij â ÏÂ¯Cij = ÏÂ¯ij â ÏÂ¯iÏÂ¯j (A.1)as required. Now we make the inductive assumption that for all k < n and Bk â Anwe haveÏÂ¯CBk =kâm=2(â1)(kâm)âCmâBkï£«ï£­ÏÂ¯Cm âjâBkCmÏÂ¯jï£¶ï£¸â (â1)k(k â 1)âjâBkÏÂ¯j . (A.2)Substituting equation (A.2) intoÏÂ¯An =âjâAnÏÂ¯j + ÏÂ¯CAn +nâ1âk=2âCkâAnï£«ï£­ âjâAnCkÏÂ¯jï£¶ï£¸ ÏÂ¯CCk (A.3)then gives an expression of the formÏÂ¯CAn = ÏÂ¯An +nâ1â`=2âF`âAnÎ¾`ÏÂ¯F`âiâAnF`ÏÂ¯i + Î¾0âiAnÏÂ¯i. (A.4)This is because terms in (A.3) contain one ÏÂ¯CCk multiplied by the single cellreduced density matrices for the rest of the cells, and terms in (A.2) contain one180reduced density matrix over a larger set multiplied by single cell reduced densitymatrices, and all subsets of the same size appear symmetrically in (A.3) and (A.2).Thus the final expression is a sum over terms which are the product of a singlereduced density matrix over a set F` â An multiplied by single cell reduced densitymatrices with a coefficient depending only on the size ` of the set F`. Now we needto find Î¾` and Î¾0.To find Î¾` we note that every Bk â F` (Bk â An) gives a contribution â(â1)kâ`to Î¾`, so there arenâ`Ckâ` such Bkâs for a given k; thus the coefficient isÎ¾` = ânâ1âk=`(â1)kâ`(nâ`Ckâ`)= ânâ`â1âp=0(â1)p(nâ`Cp)= ânââ`p=0(â1)p(nâ`Cp)+ (â1)nâ`(nâ`Cnâ`)= â (1â 1)nâ` + (â1)nâ`= (â1)nâ` (A.5)as required.To find Î¾0 we note that there is a contribution â1 from the first term in equation(A.3) as well as a contribution (â1)k(k â 1) from every Bk with n â 1 â¥ k â¥ 2.There are nCk different Bks for each k, so thatÎ¾0 = â 1 +nâ1âk=2(â1)k(k â 1) (nCk)= (â1)n+1(nâ 1) (A.6)as required; this completes the proof.A.2 Proof that any partial trace of ÏÂ¯CAn is zeroIn the main text we took the result in eqtn. (2.9) to be a defining property of thepartial trace. However, one can also derive the result explicitly from the expression(2.23). We now show this.Let us begin with (2.23) of the main text, viz.,triÏÂ¯CAn =nâm=2(â1)(nâm)âCmâAntriï£«ï£­ÏÂ¯Cm âjâAnCmÏÂ¯jï£¶ï£¸â (â1)n(nâ 1)âjâAniÏÂ¯j . (A.7)with the notation as before.181We start by noting thattriï£«ï£­ÏÂ¯Cm âjâAnCmÏÂ¯jï£¶ï£¸ =ï£±ï£´ï£´ï£²ï£´ï£´ï£³ÏÂ¯Cmâjâ(Ani)CmÏÂ¯j i 6â CmÏÂ¯CmiâjâAnCmÏÂ¯j i â Cm(A.8)It then follows that we can writenâm=2(â1)(nâm)âCmâAntriï£«ï£­ÏÂ¯Cm âjâAnCmÏÂ¯jï£¶ï£¸ = â`âAnitriÏÂ¯i`(â1)nâ2âjâAn{i,`}ÏÂ¯j+nâ2âm=2âCmâ(Ani)(â1)mtri(ÏÂ¯Cm ÏÂ¯i â ÏÂ¯Cmâª{i}) âjâ(Ani)CmÏÂ¯j=â`âAni(â1)nâ2âjâAn{i}ÏÂ¯j= (nâ 1)(â1)nâ2âjâAn{i}ÏÂ¯j (A.9)so thattriÏÂ¯CAn = (nâ 1)(â1)nâ2âjâAn{i}ÏÂ¯j â (â1)n(nâ 1)âjâAniÏÂ¯j= 0 (A.10)which is the result we wanted.182Appendix BDerivation of Equations ofMotion hierarchiesIn the main text we simply quoted the results for the equations of motion, for botha general multipartite system, and also for an N -qubit system. Here we give thederivations of these results.B.1 Equation of Motion for N-Partite systemWrite begin by writing the Hamiltonian as a âfreeâ single-system part, plus a pair-wise interaction term, viz.,H = H0 +HI =âjï£«ï£­H0j + 12 âi 6=jHIijï£¶ï£¸ (B.1)The equation of motion is theniâtÏS = [H, ÏS ]=âAâS[H ,ï£«ï£­âj 6âAÏÂ¯jï£¶ï£¸ ÏÂ¯CA]=âjâSï£±ï£²ï£³ï£®ï£°H0j + âj 6=iâS12H0ij ,ï£«ï£­âj 6âAÏÂ¯jï£¶ï£¸ ÏÂ¯CAï£¹ï£»ï£¼ï£½ï£¾ (B.2)for the part of the above containing the non-interacting part of the Hamiltonian eachj is either in A or not A, for the interacting part there are three possible situations(see figure B.1): Both i, j â A, only one of i or j in A, and both i, j /â A. We can183Figure B.1: The different classes of interaction involving A. In (i) we have interacti-ons entirely between cells inside A; in (ii) we have interactions betweenclls inside A and cells outside; and in (ii) the interactions are entirelybetween cells outside A. The interactions are denoted by the wavy line.split the sums up accordingly; one hasâAï£®ï£°H,ï£«ï£­âj /âAÏÂ¯jï£¶ï£¸ ÏÂ¯CAï£¹ï£» = âAâS{âjâA[H0j , ÏÂ¯CA]âi/âAÏÂ¯i + ÏÂ¯CAâj /âA[H0j , ÏÂ¯j] âi/âAâª{j}ÏÂ¯i+âjâAâiâA{j}[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k +âjâAâi/âA[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k+ ÏÂ¯CAâj /âAâi/âAâª{j}[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k}. (B.3)Now we trace out a set C of cells. This givestrC[H,âAÏÂ¯CAâi/âAÏÂ¯i]=âAtrCâjâA[H0j , ÏÂ¯CA]âi/âAÏÂ¯i +âAtrCÏÂ¯CAâj /âA[H0j , ÏÂ¯j] âi/âAâª{j}ÏÂ¯i+âAâjâAâiâA{j}trC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k +âAâjâAtrCâi/âA[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k+âAtrCÏÂ¯CAâj /âAâi/âAâª{j}[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k. (B.4)Let us simplify each term in the above equation separately:1. First consider the terms involving H0j :(a) Consider the first sum in eqtn. (B.4), viz.,âAtrCâjâA[H0j , ÏÂ¯CA]âi/âAÏÂ¯i (B.5)â¢ The terms are trivially zero when the overlap C â© A 6= â or {j}.â¢ If the overlap contains exactly one cell C â© A = {j}, then we havethe following identitytrj[H0j , ÏÂ¯CA]= (H0j )mn(ÏÂ¯CA)MMmn â (H0j )nm(ÏÂ¯CA)MMmn = 0.184whereM is an index on the Hilbert space of states on the set of cellsA{j}and m,n are indices on the Hilbert space at j, and repeated indices aresummed so (ÏÂ¯CA)MMmn =âM ãmM |(ÏÂ¯CA)MMmn |nMã.We see therefore that only terms with no overlap C â© A = â contributeto the first sum in eqtn. (B.4):âAâStrCâjâA[H0j , ÏÂ¯CA]âi/âAÏÂ¯i =âAâ(SC)âjâA[H0j , ÏÂ¯CA]trCâi/âAÏÂ¯i (B.6)=âAâSCâjâA[H0j , ÏÂ¯CA] âiâS(AâªC)ÏÂ¯i(b) Consider now the second sum in (B.4), viz.,trCÏÂ¯CANâj /âA[H0j , ÏÂ¯j] âi/âAâª{j}ÏÂ¯iThe terms are zero when A â© C 6= â and when j â C, so thatâAâStrCÏÂ¯CAâi/âA[H0j , ÏÂ¯j] âj /âAâª{j}ÏÂ¯i =âAâ(SC)ÏÂ¯CAnâj /â(AâªC)[H0j , ÏÂ¯j]trCâi/âAâª{j}ÏÂ¯i=âAâ(SC)âj /â(AâªC)ÏÂ¯CA[H0j , ÏÂ¯j] âiâS(AâªCâª{j})ÏÂ¯i=âAâ(SC)âjâAÏÂ¯CA{j}[H0j , ÏÂ¯j] âiâS(AâªC)ÏÂ¯i(B.7)The last line here requires a bit of thought; it reflects the fact that sum-ming over all possible A â (SC), then over j â (S(C âªA), is equivalentto summing over all possible A â (SC) and all possible j in A.2. Now consider the terms involving the interaction Hamiltonian HIij .(a) Consider first the third sum in equation (B.4), viz.,âAâjâAâiâA{j}trC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k (B.8)which contains all the terms where cells inside A are interacting witheach other, ie., case (i) in figure B.1.â¢ If the intersection C â© A contains cells other than i or j, thentrC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k = 0.185â¢ If the intersection C â© A = {i, j} thentrC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k = 0.so that there are only nonzero terms in the sum when the intersectionA â© C contains exactly one or zero elements.â¢ If the intersection is one of C â© A = {i} or {j} thentrC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k = triorj([12HIij , ÏÂ¯CA]) âk/â(AâªC)ÏÂ¯kwhich is not necessarily zero.â¢ When both i, j are in AC thentrC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k =[12HIij , ÏÂ¯CA] âk/â(AâªC)ÏÂ¯k.Thus there only two kinds of term in the sum (B.8) that matter. Thefirst are those where both i, j 6â C and C â©A. The second are those whereonly one of i or j are in C (say i â C) and A â© C = {i}. ThusâAâjâAâiâA{j}[12HIij , ÏÂ¯CA]trCâk/âAÏÂ¯k =âAâ(SC)âjâAâiâA{j}trC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k+âAâSAâ©C={j}âjâAtrC[12HIij , ÏÂ¯CA] âk/âAÏÂ¯k=âAâ(SC)âjâAâiâA{j}[12HIij , ÏÂ¯CA] âk/âAâªCÏÂ¯k+âAâ(SC)âiâCâjâA[HIij , ÏÂ¯CAâª{i}] âk/âAâªCÏÂ¯k (B.9)(b) The fourth sum in eqtn. (B.4), viz.,âAâjâAtrCâi/âA[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k (B.10)is a sum over terms involving interactions between j in A and i not in A(ie., terms like (ii) in figure B.1).â¢ When neither i nor j are in C, thentrC[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k =[12HIij , ÏÂ¯CAÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k.186â¢ When there is an overlap AâªC which contains an element other thanj, thentrC[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k = 0.â¢ When i is in C and A â© C = â , thentrC[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k =[12tri(HIij ÏÂ¯i), ÏÂ¯CA] âk/â(AâªCâª{i})ÏÂ¯k.â¢ When j is in C but i is not, thentrC[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k = trj[12HIij , ÏÂ¯CAÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k.â¢ If i and j are in C, thentrC[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k = 0.Thus the sum (B.10) isâAâjâAtrCâi/âA[12HIij , ÏÂ¯CAÏÂ¯i] âk/âAâª{i}ÏÂ¯k =âAâ(SC)âjâAâi/â(AâªC)[12HIij , ÏÂ¯CAÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k+âAâ(SC)âjâAâiâC[12tri(HIij ÏÂ¯i), ÏÂ¯CA] âk/â(AâªC)ÏÂ¯k+âAâSAâ©C={j}âi/â(AâªC)trj[12HIij , ÏÂ¯CAÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k (B.11)=âAâ(SC)âjâAâi/â(AâªC)[12HIij , ÏÂ¯CAÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k+âAâ(SC)âjâAâiâC[12tri(HIij ÏÂ¯i), ÏÂ¯CA] âk/â(AâªC)ÏÂ¯k+âAâ(SC)âjâCtrj[12HIij , ÏÂ¯CAâª{j}ÏÂ¯i] âk/â(AâªCâª{i})ÏÂ¯k (B.12)(c) The fifth sum in eqtn. (B.4), viz.,âAtrCÏÂ¯CAâj /âAâi/âAâª{j}[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k (B.13)is a sum over the interactions shown in figure B.1 (iii), where both i andj are not in C. Then187â¢ When i, j â C, we havetrCÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k = 0.â¢ When A â© C 6= â , we havetrCÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k = 0.â¢ When one of i and j (say i) is in C and the other is not, then (andA â© C = â )trCÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k = ÏÂ¯CA[12tri(HIij ÏÂ¯i), ÏÂ¯j] âk/â(Aâª{j}âªC)ÏÂ¯k.â¢ When nether i nor j are in C and A â© C = â , we havetrCÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k = ÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}âªCÏÂ¯k.Thus the sum (B.13) is given byâAtrCÏÂ¯CAâj /âAâi/âAâª{j}[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}ÏÂ¯k =âAâCâiâCâj /âAâªCÏÂ¯CA[12tri(HIij ÏÂ¯i), ÏÂ¯j] âk/â(Aâª{j}âªC)ÏÂ¯k+âAâCâiâCâjâA{i}ÏÂ¯CA[12HIij , ÏÂ¯j ÏÂ¯i] âk/âAâª{i,j}âªCÏÂ¯k. (B.14)Thus finally, inserting equations (B.6),(B.7),(B.9),(B.12), and (B.14) into (B.4), we188havetrC[H, ÏS ] =âAâSC{âjâA[H0j , ÏÂ¯CA] âiâS(AâªC)ÏÂ¯i+ ÏÂ¯CAâiâS(AâªC)[H0i , ÏÂ¯i] âjâS(AâªCâª{i})ÏÂ¯j+âjâAâiâA{j}[12HIij , ÏÂ¯CAn] âkâS(AâªC)ÏÂ¯k +âiâCâjâAtri[HIij , ÏÂ¯CAâª{i}] âkâS(AâªC)ÏÂ¯k+ ÏÂ¯CAâkâS(AâªC)âlâS(AâªCâª{k})[HIkl, ÏÂ¯k] âjâS(AâªCâª{k,l})ÏÂ¯j+ ÏÂ¯CAâkâS(AâªC)âlâC[trl(HIklÏÂ¯l), ÏÂ¯k] âjâS(AâªCâª{k})ÏÂ¯j+âjâAâkâS(AâªC)[HIjk, ÏÂ¯CAÏÂ¯k] âiâS(AâªCâª{k})ÏÂ¯i+âlâCâkâS(AâªC)trl[HIlk, ÏÂ¯CAâª{l}ÏÂ¯k] âjâS(AâªCâª{k})ÏÂ¯j+âlâCâkâAtrl[HIjlÏÂ¯l, ÏÂ¯CA] âjâS(AâªC)ÏÂ¯j}. (B.15)Comparing this with equation (B.3), we see that all of those terms above which donot contain an explicit trace can be collected to give[HSC , ÏÂ¯SC], withHSC =âjâSCï£«ï£­H0j + 12 âjâS(Câª{j})HIijï£¶ï£¸ (B.16)so thatiâtÏÂ¯SC =[HSC , ÏÂ¯SC]+ TT (B.17)189The extra âtrace termâ TT isTT = iâlâCtrl{ âAâ(SC)âiâA([HIil, ÏÂ¯CA]ÏÂ¯l +[HIil , ÏCAâª{l}]+[HIil , ÏÂ¯i]ÏC(A{i})âª{l} +[HIil , ÏÂ¯i]ÏC(A{i})ÏÂ¯l) âjâ(SC)AÏÂ¯j}(B.18)= iâlâCtrlï£®ï£° âiâSCHIil,âAâ(SC)âª{l}ÏCSËâjâ((SC)âª{l})AÏÂ¯jï£¹ï£» (B.19)=âlâCï£®ï£° âiâSCHIil, ÏÂ¯(SC)âª{l}ï£¹ï£» (B.20)Thus, finally, we have the resultiâtÏÂ¯SC =[HSC , ÏÂ¯SC]+ iâlâCï£®ï£° âiâSCHIil, ÏÂ¯(SC)âª{l}ï£¹ï£» . (B.21)If we now relabel the set (SC)â A, we get the result (4.3) in the text.B.2 Equation of Motion for N-Qubit systemWe now want to derive the equations of motion (4.7) for N qubits. The HamiltonianisH =âi12hi Â· Ïi +Nâi=1âj<i12VÂµÎ½ij ÏÂµi ÏÎ½j (B.22)which we write as H = H0 +HV .We wish to calculateddtâ©âjâCÏÂµjjâª= âiâ©[H,âjâCÏÂµjj]âª. (B.23)We thus need the commutatorsï£®ï£°H0,âjâCÏÂµjjï£¹ï£» = âiâC12hÎ»i [ÏÎ»i , ÏÂµii ]âjâC{i}ÏÂµjj= iâiâCÎµÂµiÎ»Î½ihÎ»i ÏÎ½iiâjâC{i}ÏÂµjj (B.24)190and ï£®ï£°HV ,âjâCÏÂµjjï£¹ï£» = Nâi=1âk<i12VÎ±Î²ikï£®ï£°ÏÎ±i ÏÎ²k ,âjâCÏÂµjjï£¹ï£» (B.25)The commutator on the right of the previous expression is non zero when eitherone of i, k or both i and k are in C. Consider the case when i is in C but k is not;then we have ï£®ï£°ÏÎ±i ÏÎ²k ,âjâCÏÂµjjï£¹ï£» = 2iÎµÂµiÎ±Î½iÏÎ½ii ÏÎ²k âjâCiÏÂµjj (B.26)On the other hand if both i and k are in C, then we haveï£®ï£°ÏÎ±i ÏÎ²k ,âjâCÏÂµjjï£¹ï£» = [ÏÎ±i ÏÎ²k , ÏÂµii ÏÂµkk ] âjâC{i,k}ÏÂµjj= 2i(ÎµÂµiÎ±Î½iÎ´ÂµkÎ²ÏÎ½jj + ÎµÂµkÎ²Î½kÏÎ½kk Î´ÂµjÎ±) âjâC{i,k}ÏÂµjj (B.27)putting equations (B.26) and (B.26) into the equation of motion for the correlator(B.23) one gets the hierarchy of equations of motion,ddtâ©âiâAÏÂµiiâª=âiâAÎµÂµiÎ±Î½hÎ±iâ©ÏÎ½iâjâA{i}ÏÂµjjâª+âiâAâ` 6âAÎµÂµiÎ±Î½V Î±Î»i`â©ÏÎ»` ÏÎ½iâjâA{i}ÏÂµjjâª+âiâAâjâA{i}ÎµÂµiÎ±Î½VÎ±Âµjijâ©ÏÎ½iâkâA{i,j}ÏÂµkkâª.(B.28)which is the hierarchy of equations of motion for the spin correlators that we wishedto derive (cf. eqtn. (4.7)).191Appendix CMatrix Propagator for 2-SpinSystemIn the main text we worked out explicitly the equation of motion for the entangle-ment correlators of a simple 2-spin system, with the HamiltonianH = 12 [â1Ïx + â2Ïx + ÏÏ zÏz] (C.1)and eigenvalues 1, 2 (compare eqtn. (4.40) et seq.).Here we write out explicitly the propagators which appear in the block matrixG(z) (the result for G(t) then being given by Fourier transformation). We haveg11(z) =(Ï2z2Ï30(z2 + Ï230)+Ï2z2Ï21(z2 + Ï221)+(1â Ï4Ï230Ï221)1z)xËxË+z2Ï30Ï21{(Ï220 ââ21z2 + Ï210+Ï220 ââ22z2 + Ï220)(yËyË + zËzË)â 2Ï2(yËyËz2 + Ï210+zËzËz2 + Ï220)}+yËzË â zËyË4Ï30Ï20{[Ï21(â1 + â2)â Ï30(â1 ââ2)] Ï210z2 + Ï210+ [Ï21(â1 ââ2) + Ï30(â1 + â2)] Ï220z2 + Ï220}(C.2)g12(z) = xËxË{2â1â2Ï2Ï230Ï221z+zÏ22(1Ï30(z2 + Ï230)â 1Ï21(z2 + Ï221))}= g21(z)(C.3)(C.4)192for the âsmallâ matrix propagators, andgÂµ1pÎ½Î² =[z(z2 + Ï221)(z2 + Ï230)]â1â1Ï(2â2zxËÂµzËÎ½ yËÎ² + [z2 + â21 ââ22 + Ï2]xËÂµzËÎ½ yËÎ²â [z2 + â21 + â22 + Ï2]xËÂµyËÎ½ zËÎ² â â2z [z2 ââ21 + â22 + Ï2]xËÂµyËÎ½ yËÎ²)+ Ïzâ2yËÂµxËÎ½ yËÎ² + z2yËÂµxËÎ½ zËÎ² + â1â2zËÂµxËÎ½ yËÎ² + zâ1zËÂµxËÎ½ zËÎ²z2Ï2 + (z2 + â21)(z2 + â22)(C.5)gÂµÎ±pp Î½Î² =1zxËÂµxËÎ±xËÎ½ xËÎ² +[(z2 + â21)(z2 + â22) + z2Ï2]â1{z(z2 + â21 + Ï2)xËÂµyËÎ±xËÎ½ yËÎ²+ â2(z2 + â21) [xËÂµzËÎ±xËÎ½ yËÎ² â xËÂµyËÎ±xËÎ½ zËÎ²] + z(z2 + â21)xËÂµzËÎ±xËÎ½ zËÎ²+ z(z2 + â22 + Ï2)yËÂµxËÎ±yËÎ½ xËÎ² + â1(z2 + â22) [zËÂµxËÎ±yËÎ½ xËÎ² â yËÂµxËÎ±zËÎ½ xËÎ²]+ z(z2 + â22)zËÂµxËÎ±zËÎ½ xËÎ²}+ (z2 + Ï221)â1(z2 + Ï230)â1{(z2 + â21 + â22 + Ï2)Â·[(z + Ï2z)(yËÂµyËÎ±yËÎ½ yËÎ² + zËÂµzËÎ±zËÎ½ zËÎ²) + z (yËÂµzËÎ±yËÎ½ zËÎ² + zËÂµyËÎ±zËÎ½ yËÎ²)]+ â2(z2 ââ21 + â22 + Ï2) [yËÂµzËÎ±yËÎ½ yËÎ² â yËÂµyËÎ±yËÎ½ zËÎ² + zËÂµzËÎ±zËÎ½ yËÎ² â zËÂµyËÎ±zËÎ½ zËÎ²]+ â1(z2 + â21 ââ22 + Ï2) [zËÂµyËÎ±yËÎ½ yËÎ² + zËÂµzËÎ±yËÎ½ zËÎ² â yËÂµyËÎ±zËÎ½ yËÎ² â yËÂµzËÎ±zËÎ½ zËÎ²]+ 2â1â2z (z2 + Ï2) [yËÂµyËÎ±zËÎ½ zËÎ² â zËÂµzËÎ±yËÎ½ yËÎ²] + 2zâ1â2 [yËÂµzËÎ±zËÎ½ yËÎ² â zËÂµyËÎ±yËÎ½ zËÎ²]}.(C.6)for the âlargeâ matrix propagators. In these equations xË, yË, and zË are unit Cartesianvectors, and zË should not be confused with the complex frequency z.Formulae for g22, g21, and g2p, can be obtained from the expressions for g11, g21,and g1p, if we make the replacements â1 â â2 and â2 â â1 and adjust the tensorindices accordingly (Âµ â Î±, Î½ â Î², Î± â Âµ, Î² â Î½). gp1(z) and gp2(z) can beobtained from g1p,g2p using the identities gp1(z) = gT1p(âz) and gp2(z) = gT2p(âz)(we have obtained these identities by examining the full solution).193Appendix DNumerical Calculation of SingleSpin Evolution in a TimeDependent Field.Numerically integrating the equation of motion with a time dependent bias (6.42)can cause troubles. Using an ineffective time integration scheme (eg Runge-Kutta)one finds that the length of the qubit polarisation |ãÏ (t)ã| drifts over time eventhough it is an exact constant of motion. To fix this one must use an integratorthat explicitly conserves the polarisation. The simplest such integrator based of thetrotter product formula. We derive this integrator as follows, begin by writing theformal solution to the equations of motion as a at time tn+1 in terms of ãÏ (tn)ã atan earlier time t1 using a time ordered exponential,ãÏ (tn+1)ã = T exp{â« tn+1tndtâ²[â + Î¾(tâ²)]} ãÏ (tn)ã. (D.1)Where T is the time ordering operator and the matrices Î¾(t) and â areâ = â0ï£«ï£­0 0 00 0 â10 1 0ï£¶ï£¸ (D.2)Î¾(t) = Î¾(t)ï£«ï£­0 â1 01 0 00 0 0ï£¶ï£¸ . (D.3)Write Î´t = tn+1 â tn then we can expand the time ordered exponential using theTrotter product formulaT exp{â« ttndtâ²[â + Î¾(tâ²)]}= exp (Î´tâ) exp(â« Î´ttndtâ²Î¾(tâ²))+O(Î´t2â2, Î´t2Î¾(tn)2, Î´t3Î¾â²â²(tn))(D.4)= exp (Î´tâ) exp (Î´tÎ¾(tn)) +O(Î´t2â2, Î´t2Î¾(tn)2, Î´t3Î¾â²â²(tn)).(D.5)Note the error estimates above assume that Î¾(t) is sufficiently smooth, in the parti-cular case where Î¾(t) is white noise with standard deviation Î» the last error estimate194becomes O(Î»Î¾(tn)Î´t3/2). Thus we can numerically integrate the equations of motionby setting Î´t sufficiently small and iterating the following,ãÏ (tn+1)ã = exp (Î´tâ) exp (Î´tÎ¾(tn)) ãÏ (tn)ã (D.6)with tn = nÎ´t. Because the matrices Î¾(tn) and â are antisymmetric both of thematrix exponentials in (D.6) are orthogonal and each time step explicitly conservesthe length of the polarisation vector.D.1 Treatment of Stochastic ForcesOne can treat stochastic terms in the Hamiltonian numerically using the well develo-ped theory of stochastic differential equations [39]. The diffusing bias Î¾(t) discussedin section 6.3.2 can be approximated, for a single realisation of the noise as follows.A set of random variables {ri} are drawn from a Gaussian distribution with meanzero and variance of one, one for each time step. Then the integral of the bias overthe nâ²th time step can be approximated by (Î¾0 is the initial value for the bias)â« t+Î´tntndsÎ¾(s) â Î¾0Î´t+ Î´t32 Î32nâj=0rj â¡ Î´Î¸n. (D.7)With this the solution to the equation of motion can then be obtained iterativelyusing a modified version of equation (D.6)ãÏ (tn+1)ã = exp (Î´tâ) expï£®ï£°Î´Î¸nï£«ï£­0 â1 01 0 00 0 0ï£¶ï£¸ï£¹ï£» ãÏ (tn)ã. (D.8)Then averages over Î¾ can be computed by averaging over many different realisationsfor the {ri} variables.195Appendix EDetails of precessionaldecoherence calculations:In this appendix we show the details omitted in section 7.2.In section E.1 show how to generalise the result derived in 7.2.3 for the returnprobability of the central spin, to get the full reduced density matrix for the centralspin. In section E.2 we calculate the next-to leading order correction to ãÏ+ã dis-cussed in section 7.68. Then in section E.3 we calculate correlators of the centralspin with the spin bath, deriving the results presented in section 7.2.4.196E.1 Calculating the full reduced density matrix for thecentral spin.Calculating the full reduced density matrix from section 7.2.3. Write the equations(7.40-7.51) as followsA(n)+â â¡122nnâ`=0(2n2`)a(n`)+â (E.1)A(n)++ â¡â122nnâ1â`=0(2n2`+ 1)a(n`)++ (E.2)A(n)+â â¡122n+1nâ`=0(2n+ 12`)a(n`)+â (E.3)A(n)+â â¡â122n+1nâ`=0(2n+ 12`+ 1)a(n`)+â (E.4)A(n)ââ â¡122nnâ`=0(2n2`)a(n`)ââ (E.5)A(n)â+ â¡â122nnâ1â`=0(2n2`+ 1)a(n`)â+ (E.6)A(n)ââ â¡122n+1nâ`=0(2n+ 12`)a(n`)ââ (E.7)A(n)ââ â¡â122n+1nâ`=0(2n+ 12`+ 1)a(n`)ââ (E.8)A(n)zâ â¡12a(n0)zâ (E.9)A(n)zâ â¡â 12a(n0)zâ (E.10)A(n)z+ â¡12a(n0)z+ (E.11)A(n)zâ â¡â 12a(n0)zâ . (E.12)So the variables a(n`)ab for appropreate a, b â {a,+,â, â, â} appearing in equations(E.1-E.12) are expectations of strings of operators appearing in equations (7.40-7.51). Using the same manipulations leading up to equation (7.58) the a(n`)ab coeffi-cients can be written in the following forma(n`)ab =â« 2pi0dnabÎ¾(2pi)nabexpï£±ï£²ï£³ânabNÎ²2 âNÎ²2nabâ1âq=1nabâqâ²=q(cab)n`qqâ² cosï£«ï£­2 qâ²âqË=q+1Î¾qËï£¶ï£¸ï£¼ï£½ï£¾ .(E.13)197a b Î³ab Ë`ab nab+ â 1 ` 2n+ + â1 `+ 1 2n+ â 1 ` 2n+ 1+ â â1 `+ 1 2n+ 1â â â1 ` 2nâ + 1 `+ 1 2nâ â â1 ` 2n+ 1â â 1 `+ 1 2n+ 1z â 1 0 2nz â â1 0 2nz + 1 0 2n+ 1z â â1 0 2n+ 1Table E.1: Parameters used in the transition expansion calculation of the reduceddensity matrix.The variables (cab)n`qqâ² are of the form (note qâ² > q in all terms)(cab)n`qqâ² ={Î³ab(â1)q+qâ² for q > Ë`ab or qâ² â¤ Ë`abâÎ³ab(â1)q+qâ² for q â¤ Ë`ab and qâ² > Ë`ab(E.14)and the parameters Î³ab, Ë`ab, and nab which appear in these expressions are given inthe table E.1. Then by substitutingÏq = 2qâqâ²=1Î¾qâ² + qpi + pi [Î(q â `+ ) + Î(âÎ³ab)] (E.15)we geta(n`)ab =â« 2pi0dnabÏ(2pi)nabexpï£±ï£²ï£³ânabNÎ²2 âN2Î²22nabâ1âq=1nabâqâ²=qcos(Ïq â Ïqâ²)ï£¼ï£½ï£¾ (E.16)and the same set of steps that leads to (7.64) givesa(n`)ab =â« â0dXXeâ X24NÎ² J0(X)nab . (E.17)198Plugging this into equations (E.1-eq:lastAap) , givesA(n)+â â¡â«dXP (X)122nnâ`=0(2n2`)[J0(X)]2n (E.18)A(n)++ â¡ââ«dXP (X)122nnâ1â`=0(2n2`+ 1)[J0(X)]2n+2 (E.19)A(n)+â â¡â«dXP (X)122n+1nâ`=0(2n+ 12`)[J0(X)]2n+1 (E.20)A(n)+â â¡ââ«dXP (X)122n+1nâ`=0(2n+ 12`+ 1)[J0(X)]2n+1 (E.21)A(n)ââ â¡â«dXP (X)122nnâ`=0(2n2`)[J0(X)]2n (E.22)A(n)â+ â¡ââ«dXP (X)122nnâ1â`=0(2n2`+ 1)[J0(X)]2n+2 (E.23)A(n)ââ â¡â«dXP (X)122n+1nâ`=0(2n+ 12`)[J0(X)]2n+1 (E.24)A(n)ââ â¡ââ«dXP (X)122n+1nâ`=0(2n+ 12`+ 1)[J0(X)]2n (E.25)A(n)zâ â¡â«dXP (X)12 [J0(X)]2n (E.26)A(n)zâ â¡ââ«dXP (X)12 [J0(X)]2n (E.27)A(n)z+ â¡â«dXP (X)12 [J0(X)]2n+1 (E.28)A(n)zâ â¡ââ«dXP (X)12 [J0(X)]2n+1. (E.29)Plugging equations (E.18-E.29) into equations (7.37-7.37), gives the desired result,the same expansion one would have for an isolated qubit under the influence of theHamiltonian H0 =12â0J0(X)Ïx averaged over the variable X.E.2 Next to leading order contributions to ãÏx(t)ãAs discussed in section 7.68 we show the ext to leading order contributions to ãÏx(t)ãis zero. We compute the next to leading order contribution to ãÏ+(t)ã from an initialdensity from the initial central spin density matrix | âããâ | and show that this is zero.Then we will explain why this generalises to all next to leading order contributions199to ãÏx(t)ã.Î´ãÏ+(t)ã (E.30)=âân=0(âiâ0)2nnâ`=1(â1)`â« t0{DÎ·Ë(tË)}2` â« t0{DÎ·(t)}2nâ2`NâM=âN 6=0{2`â1âp=1â©(pâ1âq=1P0U(â1)q+1)[P0U(â1)p+1PMU (â1)p+2]Â·( 2â`q=p+2P0U(â1)q+1)( 2nâq=2`+1P0U(â1)q)âªei(â1)p2 Ï0M(tËpâtËp+1)+2nâ1âp=2`+1â©( 2â`q=1P0U(â1)q+1)( pâ1âq=2`+1P0U(â1)q)[P0U(â1)pPMU (â1)p+1]Â·( 2nâq=p+2P0U(â1)q)âªei(â1)p+12 Ï0M(tpâtp+1)}here â« t0{DÎ·(t)}2` =â« t0dt2`â« t2`0dt2`â1 . . .â« t20dt1 (E.31)and we have used the following convention for non-commutative productsnâi=1AËi â¡ AË1AË2AË3 . . . AËn. (E.32)Equation (E.30) consists of sums over the contributions from different paths fore thecentral spin where the pâth tunneling event for the central spin is accompanied bya jump of bath z magnetisation from zero to any possible value M then back in thefollowing central spin tunneling event. Now we calculate time integrations occuringin (E.30) using the identityâ« tp0dtpâ1 . . .â« t20dt1 =tpâ1p(pâ 1)! (E.33)200so that â« t0dt2` . . .â« tp+10dtpâ« tp0dtpâ1 . . .â« t20dt1ei(â1)p2 Ï0M(tpâtp+1) (E.34)=â« t0dtn . . .â« tp+10dtptpâ1p(pâ 1)!ei(â1)p2 Ï0M(tpâtp+1)=â« t0dt2` . . .â« tp+20dtp+11(pâ 1)!eâi(â1)p2 Ï0Mtp+1Â·((â1)p+1iMââÏ0)pâ1 â« tp+10dtpei(â1)p2 Ï0Mtp=â« t0dt2` . . .â« tp+20dtp+11(pâ 1)!eâi(â1)p2 Ï0Mtp+1Â·((â1)p+1iMââÏ0)pâ1(â1)p+1iMÏ0[ei(â1)p2 Ï0Mtp+1 â 1]â¼â« t0dt2` . . .â« tp+20dtp+11(pâ 1)!eâi(â1)p2 Ï0Mtp+1Â· (â1)p+1iMÏ0((â1)p+1iMââÏ0)pâ1ei(â1)p2 Ï0Mtp+1â¼ (â1)p+1iMÏ0â« t0dt2` . . .â« tp+20dtp+11(pâ 1)! tpâ1p+1â¼ (â1)p+1iMÏ0t2`â1(2`â 1)!where we have kept the leading order secular terms. Thus the contribution (E.30)becomesÎ´ãÏ+(t)ã (E.35)=âân=0(âiâ0)2nnâ`=1(â1)`NâM=âN 6=0it2nâ1(2`)!(2nâ 2`)!MÏ0{2`â1âp=1(â1)p+12`â©(pâ1âq=1P0U(â1)q+1)[P0U(â1)p+1PMU (â1)p+2]( 2â`q=p+2P0U(â1)q+1)Â·( 2nâq=2`+1P0U(â1)q)âª+2nâ1âp=2`+1(â1)p(2nâ 2`)â©( 2â`q=1P0U(â1)q+1)( pâ1âq=2`+1P0U(â1)q)[P0U(â1)pPMU (â1)p+1]Â·( 2nâq=p+2P0U(â1)q)âª}.201Now the terms in curly brackets in equation (E.35) only depend on the magnitude ofM , this is because the terms containing M are expectations of strings of projectionoperators sandwiching rotation matrices U , the unitary can be writtenU (â1)q=âCâB(âiâC(â1)qi sinÎ²i(Ï+i + Ïâi ))âj 6âCcosÎ²i. (E.36)Which depends on the raising operator and the lowering operator in the same wayso that the terms where this string of operators specifies a jump to the manifold ofstates with magnetisation +M will have the same amplitude as that where thereis a jump to the manifold of states with magnetisation âM . Therefore the +Mand âM terms in the sum (E.35) cancel exactly. This same set of statements isgoing to be true for any string of operators generated by the next to leading orderorthogonality blocking approximation to the transition expansion of ÏÂ¯S .E.3 Calculating correlators between bath and centralspins.Our goal here is to calculate various correlators given in section7.2.4, between thecentral spin and upto two other spins. As discussed in the text only correlatives ofoperators which preserve the bath magnetisation are non zero in the orthogonalityblocking approximation. Therefore the only applicable bath operators are,Ïxi Ïxj = Ï+i Ï+j + Ï+i Ïâj + Ïâi Ï+j + Ïâi Ïâj (E.37)Ïxi Ïyj = âiÏ+i Ï+j + iÏ+i Ïâj â iÏâi Ï+j + iÏâi Ïâj (E.38)Ïyi Ïyj = âÏ+i Ï+j + Ï+i Ïâj + Ïâi Ï+j â Ïâi Ïâj (E.39)Ïzi (E.40)Ïzi Ïzj . (E.41)We will show later in section E.3.2 that, to leading order the correlators of the formãÏÂµÏzi ã and ãÏÂµÏzi Ïzj ã are to leading order,ãÏÂµÏzi ã(t) =Î·iãÏÂµã (E.42)ãÏÂµÏzi Ïzj ã(t) =Î·iÎ·jãÏÂµã, (E.43)where ÏÂ¯i(0) = |Î·iããÎ·j |. First we focus on the correlators between the central spinand the operators (E.37-E.37), which involve transverse components of the bathspins.E.3.1 Correlators Involving Transverse Components of the BathSpinsWe begin by computing the contribution toâ©| âããâ |Ï+i Ïâjâª(t) from the initial cen-tral spin density matrix ÏÂ¯S(0) = | âããâ |.202In the leading order orthogonality blocking approximation the operatorâ©| âããâ |Ï+i Ïâjâª(t)ends up sandwiched between two P0 operators, so we haveâ©| âããâ |Ï+i Ïâjâª(t)ââân=0(âiâ02)2n nâ`=0â«{DÎ·Ë}2`â«{DÎ·}2nâ2`Â·â© 2â`q=1(P0U(â1)q+1)(Ï+i Ïâj )2nâq=2`+1(P0U(â1)q+1)âª. (E.44)(E.45)Then using the usual maneuvers we have, the most secular terms in the transitionexpansionâ©| âããâ |Ï+i Ïâjâª(t)ââân=0(âiâ0t2)2n nâ`=01(2`)!(2nâ 2`)!â«d2nÎ¾(2pi)2nÂ· exp{ânÎ²2(N â 2)â (N â 1)Î²22nâ1â`â²=12nâ`â²â²=`â²(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)}Â·â© 2â`q=1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))(Ï+i Ïâj )Â·2nâq=2`+1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))âª. (E.46)Obviously this expression depends on whether spins i and j are intially up or down,now we consider the different possibilities. Firstly We only need to evaluate thecorrelators involving bath spins i and j (the rest we have done in the previous203sections). These take the following possible valuesãâ |2â`q=1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)Ï+i2nâq=2`+1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)| âã (E.47)â Î²ieiâ2nq=1 Î¾q2nâ`â²=2`+1i(â1)`â²+1eâ2iâ`â²q=2`+1 Î¾qãâ |2â`q=1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)Ï+i2nâq=2`+1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)| âã (E.48)â Î²ieâiâ2nq=1 Î¾q2â``â²=1i(â1)`â²+1e2iâ2`q=`â²+1 Î¾q (E.49)ãâ |2â`q=1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)Ïâi2nâq=2`+1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)| âã (E.50)â Î²ieiâ2nq=1 Î¾q2â``â²=1i(â1)`â²+1eâ2iâ2`q=`â²+1 Î¾qãâ |2â`q=1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)Ïâi2nâq=2`+1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)| âã (E.51)â Î²ieâiâ2nq=1 Î¾q2nâ`â²=2`+1i(â1)`â²+1eâ2iâ`â²q=2`+1 Î¾qNow we consider the different cases.204Only One of the Spins i and j Initially UpTherefore considering the case where the spin i is initially up and the spin j isinitially down, we have,â© 2â`q=1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))Ï+i ÏâjÂ·2nâq=2`+1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))âª(E.52)â âÎ²iÎ²j[2nâ 2`+ 22nâ`â²=2`+2`â²â1â`â²â²=2`+1(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)]â© 2â`q=1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))Ïâi Ï+jÂ·2nâq=2`+1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))âª(E.53)â âÎ²iÎ²j[2`+ 22â``â²=2`â²â1â`â²â²=1(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)]205Thusâ©| âããâ |Ï+i Ïâjâª(t)â âÎ²iÎ²jâân=0(âiâ0t2)2n nâ`=01(2`)!(2nâ 2`)!â«d2nÎ¾(2pi)2n(E.54)Â· exp{ânÎ²2(N â 2)â (N â 1)Î²22nâ1â`â²=12nâ`â²â²=`â²(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)}Â·[2nâ 2`+22nâp=2`+2pâ1âq=2`+1(â1)p+q cos(2pâk=q+1Î¾k)]â âÎ²iÎ²jâân=0(âiâ0t2)2n nâ`=01(2`)!(2nâ 2`)!â«d2nÏ(2pi)2n(E.55)Â· exp{ânÎ²2(N â 2)â (N â 1)Î²22nâ1â`â²=12nâ`â²â²=`â²cos(Ï`â² â Ï`â²â²)}Â·[2nâ 2`+ 22nâ1âp=2`+2pâ1âq=2`+1cos(Ïp â Ïq)]= âÎ²iÎ²jâân=0(âiâ0t2)2n [ nâ`=02nâ 2`(2`)!(2nâ 2`)!â«dXP (X)[J0(X)]2n (E.56)+nâ1â`=0(2nâ 1â 2`)(2nâ 2`)(2`)!(2nâ 2`)!â«d ~X(2pi)eâ X28NÎ²22NÎ²[J0(X)]2nâ2â«d2Ï(2pi)2cos(Ï1 â Ï2) exp{âiX1(cosÏ1 + cosÏ2)â iX2(sinÏ1 + sinÏ2)}].In equation (E.55) we have introduced Ï variables just like in the appendix E.1. Thenin equation (E.56) we use same steps leading equation (7.64), as well as making useof the fact that each of the terms over the sums over p and q in equation (E.55) areidentical. The integral in equation (E.56) can be evaluated:â«d2Ï(2pi)2cos(Ï1 â Ï2) exp{âiX1(cosÏ1 + cosÏ2)â iX2(sinÏ1 + sinÏ2)}=â«d2Ï(2pi)2cos(Ï1 â Ï2)eâiX(cosÏ1+cosÏ2) (E.57)=(â«dÏ2picosÏ1eâiX cosÏ1)2+(â«dÏ2pisinÏ1eâiX cosÏ1)2(E.58)=â [J1(X)]2. (E.59)206Therefore we haveâ©| âããâ |Ï+i Ïâjâª(t)â âÎ²iÎ²jâân=0(âiâ0t2)2n [ nâ`=02nâ 2`(2`)!(2nâ 2`)!â«dXP (X)[J0(X)]2n (E.60)+nâ1â`=0(2nâ 1â 2`)(2nâ 2`)(2`)!(2nâ 2`)!â«d ~X(2pi)eâ X28NÎ²22NÎ²[J0(X)]2nâ2â«d2Ï(2pi)2cos(Ï1 â Ï2) exp{âiX1(cosÏ1 + cosÏ2)â iX2(sinÏ1 + sinÏ2)}].â Î²iÎ²j4â«dXP (X){[â0J0(X)t] sin[â0J0(X)t] (E.61)â 12 [â0J1(X)t]2 (cos[â0J0(X)t] + 1)}.207Similar calculations give:â©| âããâ |Ïâi Ï+jâª(t)â Î²iÎ²j4â«dXP (X){[â0J0(X)t] sin[â0J0(X)t] (E.62)â 12 [â0J1(X)t]2 (cos[â0J0(X)t] + 1)}â©| âããâ |Ï+i Ïâjâª(t)â âÎ²iÎ²j4â«dXP (X){[â0J0(X)t] sin[â0J0(X)t] (E.63)â 12 [â0J1(X)t]2 (cos[â0J0(X)t]â 1)}â©| âããâ |Ïâi Ï+jâª(t)â âÎ²iÎ²j4â«dXP (X){[â0J0(X)t] sin[â0J0(X)t] (E.64)â 12 [â0J1(X)t]2 (cos[â0J0(X)t]â 1)}â©| âããâ |Ï+i Ïâjâª(t)â iÎ²iÎ²j4â«dXP (X){[â0J0(X)t] (cos[â0J0(X)t]â 1) (E.65)+ 12 [â0J1(X)t]2 sin[â0J0(X)t]}â©| âããâ |Ï+i Ïâjâª(t)â âiÎ²iÎ²j4â«dXP (X){[â0J0(X)t] (cos[â0J0(X)t] + 1) (E.66)+ 12 [â0J1(X)t]2 sin[â0J0(X)t]}.208Combining the expressions (E.61-E.67) we get the following expressions for the de-sired correlators, at leading order in our approximationsãÏxi Ïxj ã =ãÏyi Ïyj ã = â(â0t)24â«dXP (X)[J1(X)]2 (E.67)ãÏxi Ïyj ã =ãÏyi Ïxj ã = 0 (E.68)ãÏxÏxi Ïxj ã =ãÏxÏyi Ïyj ã = 0 (E.69)ãÏxÏxi Ïyj ã =â ãÏxÏyi Ïxj ã = â0tâ«dXP (X)J0(X) (E.70)ãÏyÏxi Ïxj ã =ãÏyÏyi Ïyj ã (E.71)=â Î²iÎ²jâ«dXP (X){[â0J0(X)t] cos[â0J0(X)t]+ 12 [â0J1(X)t]2 sin[â0J0(X)t]}ãÏyÏyi Ïxj ã =ãÏyÏxi Ïyj ã = 0 (E.72)ãÏ zÏxi Ïxj ã =ãÏ zÏyi Ïyj ã (E.73)=Î²iÎ²j2â«dXP (X){[â0J0(X)t] sin[â0J0(X)t]â 12 [â0J1(X)t]2 cos[â0J0(X)t]}ãÏ zÏyi Ïxj ã =ãÏ zÏxi Ïyj ã = 0. (E.74)When Both Spins i and j are Initially Up (or Down)In this case similar steps to those in section E.3.1 lead toâ©| âããâ |Ï+i Ïâjâª(t)= Î²iÎ²jâân=2(âiâ0t2)2n â«dXP (X)[J1(X)]2[J0(X)]2nâ2 (E.75)Â·nâ`=1(2nâ 2`)(2`)(2`)!(2nâ 2`)!= Î²iÎ²jâân=2(âiâ0t2)2n â«dXP (X)[J1(X)]2[J0(X)]2nâ2 (E.76)Â· 22nâ3(2nâ 2)!= Î²iÎ²j(â0t)28â«dXP (X)[J1(X)]2 [1â cos(â0J0(X)t)] .209Similarly â©| âããâ |Ï+i Ïâjâª(t)= Î²iÎ²j(â0t)28â«dXP (X)[J1(X)]2 (1 + cos(â0J0(X)t)) (E.77)â©| âããâ |Ï+i Ïâjâª(t)= iÎ²iÎ²j(â0t)28â«dXP (X)[J1(X)]2 sin[â0J0(X)t] (E.78)â©| âããâ |Ï+i Ïâjâª(t)= âiÎ²iÎ²j (â0t)28â«dXP (X)[J1(X)]2 sin[â0J0(X)t]. (E.79)So we haveãÏ zÏxi Ïxj ã =â ãÏ zÏyi Ïyj ã (E.80)=Î²iÎ²j(â0t)22â«dXP (X)[J1(X)]2 [cos(â0J0(X)t)â 1]ãÏxÏyi Ïxj ã =ãÏxÏxi Ïyj ã = 0 (E.81)ãÏxÏxi Ïxj ã =ãÏxÏxi Ïyj ã = ãÏxÏyi Ïxj ã = ãÏxÏyi Ïyj ã = 0 (E.82)ãÏyÏxi Ïxj ã =â ãÏyÏyi Ïyj ã (E.83)=Î²iÎ²j(â0t)22â«dXP (X)[J1(X)]2 sin[â0J0(X)t]ãÏyÏxi Ïyj ã =ãÏyÏyi Ïxj ã = 0. (E.84)E.3.2 Correlators Involving Longditudional Components of theBath SpinsWe begin by computing the contribution to ã| âããâ |Ïzi ã (t) from the initial centralspin density matrix ÏÂ¯S(0) = | âããâ |.In the leading order orthogonality blocking approximationã| âããâ |Ïzi ã (t)ââân=0(âiâ02)2n nâ`=0â«{DÎ·Ë}2`â«{DÎ·}2nâ2`Â·â© 2â`q=1(P0U(â1)q+1)(Ïzi )2nâq=2`+1(P0U(â1)q+1)âª. (E.85)210NowãÎ·i|2â`q=1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi)Ïzi2nâq=2`+1(eiÎ¾qÏzi ei(â1)q+1Î²Ïxi +)|Î·iã (E.86)â Î·ieiÎ·iâk Î¾k exp{ânÎ²2i â Î²2i2`â1âp=12â`q=p+1n(â1)p+qeâ2iÎ·iâ`â²kË=p+1Î¾Ë`+ Î²2i2`â1âp=12nâq=`(â1)p+qeâ2iÎ·iâqk=p+1 Î¾kâ Î²2i2nâ1âp=`2nâq=p+1(â1)q+peâ2iÎ·iâqk=p+1 Î¾k}.Thusã| âããâ |Ïzi ã (t)â Î·iâân=0(âiâ0t2)2n nâ`=01(2`)!(2nâ 2`)!â« 2pi0d2nÎ¾(2pi)2neâÎ´K(n`)i ({Î¾k})Â· expï£±ï£²ï£³ânNÎ²2 âNÎ²22nâ1â`=12nâ`â²=`+1(â1)`+`â² cosï£«ï£­2 `â²âË`=`+1Î¾Ë`ï£¶ï£¸ï£¼ï£½ï£¾ (E.87)where Î´K(n`)i ({Î¾k}) is a small change in the âactionâ,Î´K(n`)i ({Î¾k}) = Î²2i2`â1âp=12nâq=`(â1)p+qeâ2iÎ·iâqk=p+1 Î¾k . (E.88)If, as we have been assuming in the rest of the text Î²i â Î² andN 1 then the changein the action is dominated by the rest of the âactionâ in equation E.87. In this casewe recover the result stated in section (E.3) (that is, in this case ã| âããâ |Ïzi ã (t) âÎ·i ã| âããâ |ã (t)) . This argument is general to all correlators between central spinoperators and Ïzi .E.3.3 Calculations of Correlators When There is a Small Bias onthe Central QubitHere we show how the results described in section 8.1.1 were calculated. We caculatea set of correlators that are needed to calculate ãÏxÏxÏxã and ãÏyÏxÏxã, in theprecessional decoherence case, when there is a small additional bias Î¾0 Ï0. In thiscase the orthogonality blocking approximation is still valid, but the wave functionsaccumulate phase according to the bias on the central qubit. We calculated the211needed correlators for the cases where i and j are initially both spin up (or down)and when one is initially spin up.In all cases we have to deal with a phase factor of the form(â« t0dtn . . .â« t20dt1â« t0dtËn . . .â« tË20dtË1)(E.89)Â· expï£±ï£²ï£³âi s02nâ`=0(â1)`(t`+1 â t`)Î¾0 + i sË02nËâË`=0(â1)Ë`(tËË`+1 â tËË`)Î¾0ï£¼ï£½ï£¾ ,every term in the transition expansion. Here s0 and sË0 specify the central qubitâsinitial matrix element (in all cases we consider in this section s0 = sË0 = 1), and itis understood that we will set tn = tËnË = t. Expressions like (E.89) can be takencare of by taking Laplace transforms with respect to t and tËnË, after inverting thistransform one gets(â« t0dtn . . .â« t20dt1â« t0dtËn . . .â« tË20dtË1)(E.90)Â· expï£±ï£²ï£³âi s2nâ`=0(â1)`(t`+1 â t`)Î¾0 + i sË2nËâË`=0(â1)Ë`(tËË`+1 â tËË`)Î¾0ï£¼ï£½ï£¾ ,=â« iâ+Î´âiâ+Î´dz1â« iâ+Î´âiâ+Î´dz2e(z1+z2)tnâ`=01z1 + is02 (â1)`Î¾0nËâ`=01z2 + isË02 (â1)Ë`Î¾0.Only One of the Spins i and j is Initially UpWe begin by computing the contribution toâ©| âããâ |Ï+i Ïâjâª(t) from the initial cen-tral spin density matrix ÏÂ¯S(0) = | âããâ |. To leading order in the orthogonalityblocking approximation, useing the Laplace transform identity (E.90), we getâ©| âããâ |Ï+i Ïâjâª(t)ââ« iââiâd2ze(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)âân=0ââm=0(âiâ02)2(n+m)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)nÂ·â© 2mâq=1(P0U(â1)q+1)(Ï+i Ïâj )2nâq=1(P0U(â1)q+1)âª. (E.91)(E.92)212Then using the usual maneuvers we have, the most secular terms in the transitionexpansionâ©| âããâ |Ï+i Ïâjâª(t)ââân=0ââm=0(âiâ02)2(n+m) â« iââiâd2zÂ· e(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n â« d2(n+m)Î¾(2pi)2(n+m)Â· exp{â(n+m)Î²2(N â 2)â (N â 1)Î²22(n+m)â1â`â²=12(n+m)â`â²â²=`â²(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)}Â·â© 2mâq=1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))(Ï+i Ïâj )Â·2(n+m)âq=2m+1(eiÎ¾q(Ïzi +Ïzj )ei(â1)q+1Î²(Ïxi +Ïxj ))âª. (E.93)â âÎ²iÎ²jâ« iââiâd2ze(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)âân=0ââm=0(âiâ02)2(n+m)(E.94)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n â« d2(n+m)Î¾(2pi)2(n+m)(E.95)Â· exp{â(n+m)Î²2(N â 2)â (N â 1)Î²22(n+m)â1â`â²=12(n+m)â`â²â²=`â²(â1)`â²+`â²â² cos(2`â²âq=`â²â²+1Î¾q)}Â·[2n+22(n+m)âp=2m+2pâ1âq=2m+1(â1)p+q cos(2pâk=q+1Î¾k)]â âÎ²iÎ²jâ« iââiâd2ze(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)âân=0ââm=0(âiâ02)2(n+m)(E.96)Â·nâ`=0(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n â« d2(n+m)Ï(2pi)2(n+m)(E.97)Â· exp{â(n+m)Î²2(N â 2)â (N â 1)Î²22(n+m)â1â`â²=12(n+m)â`â²â²=`â²cos(Ï`â² â Ï`â²â²)}Â·[2n+ 22(n+m)â1âp=2m+2pâ1âq=2m+1cos(Ïp â Ïq)]= âÎ²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)âân=0ââm=0(âiâ02)2(n+m)(E.98)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n[J0(X)]2(n+m)â2[2n[J0(X)]2 â 2n(2nâ 1)[J1(X)]2]213So inverting the Laplace transformation (we have used Mathematica software forthis),â©| âããâ |Ï+i Ïâjâª(t)= âÎ²iÎ²jâ«dXP (X)â2016â¦6[Î¾0 sin(tâ¦2)+ iâ¦ cos(tâ¦2)](E.99)Â·{sin(tâ¦2)[J0(X)2(3â20J1(X)2(â12Î¾0 + tâ¦2(Î¾0t+ 2i))â 16Î¾0â¦2 + 8itâ¦4)+ J1(X)2(12Î¾30 + tâ¦4(Î¾0tâ 6i)â Î¾0â¦2(â4 + Î¾0t(Î¾0t+ 2i)))]+ tâ¦ cos(tâ¦2)(J0(X)2[8Î¾0â¦2 + 3â20J1(X)2(6Î¾0 â itâ¦2))â J1(X)2(6Î¾30 + Î¾0â¦2(2â iÎ¾0t) + itâ¦4)]}withâ¦ = â¦(X) â¡â[â0J0(X)]2 + Î¾20 . (E.100)Expressions such as (E.99) are difficult to understand and for our purposes will beused to generate values on a graph anyway. So for the remainder of this section wewill give expressions for the other correlators we require in the form of a series like214(E.98), which can be resolved using Mathematica.â©| âããâ |Ï+i Ïâjâª(t)â Î²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tâân=0ââm=0(âiâ02)2(n+m)+2Â·nâ`=0(1z21 + Î¾20/4)m+1( 1z22 + Î¾20/4)n+1(E.101)Â· [J0(X)]2(n+m)[(2n+ 1)[J0(X)]2 â (2n+ 1)2n[J1(X)]2]â©| âããâ |Ï+i Ïâjâª(t)â âÎ²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tz2 â iÎ¾0/2âân=0ââm=0(âiâ02)2(n+m)+1(E.102)Â·(1z21 + Î¾20/4)m+1( 1z22 + Î¾20/4)nÂ· [J0(X)]2(n+m)â1[2n[J0(X)]2 â 2n(2nâ 1)[J1(X)]2]â©| âããâ |Ï+i Ïâjâª(t)â Î²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tz1 + iÎ¾0/2âân=0ââm=0(âiâ02)2(n+m)+1(E.103)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n+1Â· [J0(X)]2(n+m)â1[(2n+ 1)[J0(X)]2 â (2n+ 1)2n[J1(X)]2].215Both i and j Initially Up (or Down)Considering the case where both the spins i and j are initially up (or down), wehave,â©| âããâ |Ï+i Ïâjâª(t)= Î²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)t(z1 + iÎ¾0/2)(z2 â iÎ¾0/2)âân=0ââm=0(âiâ02)2(n+m)(E.104)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n[J0(X)]2(n+m)â2(2n)(2m)[J1(X)]2â©| âããâ |Ï+i Ïâjâª(t)= âÎ²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tâân=0ââm=0(âiâ02)2(n+m)+2(E.105)Â·nâ`=0(1z21 + Î¾20/4)m+1( 1z22 + Î¾20/4)n+1(E.106)Â· [J0(X)]2(n+m)(2n+ 1)(2m+ 1)[J1(X)]2â©| âããâ |Ï+i Ïâjâª(t)= Î²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tz2 â iÎ¾0/2âân=0ââm=0(âiâ02)2(n+m)+1(E.107)Â·(1z21 + Î¾20/4)m+1( 1z22 + Î¾20/4)nÂ· [J0(X)]2(n+m)â1(2n)(2m+ 1)[J1(X)]2â©| âããâ |Ï+i Ïâjâª(t)= âÎ²iÎ²jâ«dXP (X)â« iââiâd2ze(z1+z2)tz1 + iÎ¾0/2âân=0ââm=0(âiâ02)2(n+m)+1(E.108)Â·(1z21 + Î¾20/4)m( 1z22 + Î¾20/4)n+1Â· [J0(X)]2(n+m)â1(2n+ 1)(2m)[J1(X)]2216Appendix FSome Steepest DescentsIntegralsF.1 Gaussian Average of a Green functionConsider the integralI1 â¡ 12â« âââdÎ¾eâÎ¾22Ï2â2piÏcos(tââ02 + Î¾2)(F.1)for â0t 1. Begin by substituting Ï =âÎ¾2 ââ020I1 = Reâ« ââ0dÏÏâÏ2 ââ02eâÏ2ââ022Ï2â2piÏeiÏt (F.2)=1â2piÏReeiâ0tâ« â0dÏÏ + â0âÏ(Ï + 2â0)exp{â Ï22Ï2+ (itââ0Ïâ2)Ï}. (F.3)In the last step we have made the substitution Ï â Ï + â0. The above workinggives the following useful formula which can be used to evaluate integrals of theformâ« âââdÎ¾â2piÏ2eâÎ¾22Ï2+iâÎ¾2+â02tf(Î¾) (F.4)=â2piÏ2eiâ0tâ« â0dÏ(Ï + â0âÏ(Ï + 2â0))S[f ](â(Ï + â0)2 ââ02)eâÏ22Ï2+(itââ0Ïâ2)Ïwhere S[f ](Î¾) = 12 [f(Î¾) + f(âÎ¾)] is the symmetrised version of the function f(Î¾).As this is a Fourier one could obtain the long time behavior for I1 by expandingÏ+â0âÏ(Ï+2â0)eâÏ22Ï2 around Ï = 0 where the Fourier transform is not smooth [66] thiswould give use I1 â¼ ReCeiâ0t/Ïât which does not recover the behavior of theintegral as Ï â 0 so a more sophisticated approach must be taken. To obtain anapproximation to I1 for tâ0 1 which holds no matter how small Ï we use themethod of steepest decent. We deform the contour of integration in the complexplane so that the real part of the exponent â Ï22Ï2+ (it â â0Ïâ2)Ï decreases most217Re(Ï)Im(Ï) 000Figure F.1: The contour used in the steepest descents integral. The shaded colourshows the real part of the exponent in equation (F.3). The dashed lineis deformed to the filled line along which the real part of the exponentdecreases most rapidly. The contribution from the vertical line vanishesas it is moved to positive infinity along the real axisrapidly (see figure F.1). This will be along the curve where the imaginary part ofthe integral exponent is zero. Along this contour we may expand about the pointat which the real part of the exponent is maximised this is the origin. Evaluatingthe gradient of the real part of the exponent:ââRe[â(x+ iy)22Ï2+ (itââ0Ïâ2)(x+ iy)]x=y=0=(â0Ï2t)(F.5)so that we should substitute Ï = reiÏ with tanÏ = tÏ2â0and then expand aboutÏ = 0 which givesI1 =ââ0piReeiâ0t+iÏ/22Ïâ« â0drârexp(âât2Ï4 + â02Ï2r)(F.6)=12(t2Ï4â02 + 1)â 14cos(â0t+12Ï)(F.7)218F.2 Loretzian AverageConsider the integralI2 =â« âââdÎ¾Î³pif(Î¾)eiâÎ¾2+â02t(Î¾2 + Î³2)(F.8)we are interested in a regime for which â0t 1 with no particular restriction onÎ³t but with â0 > Î³. f(Î¾) is a function which is well behaved. The long time partof the Fourier transform can still be calculated by a steepest descent analysis, thepoint Î¾ = 0 is a saddle point for the exponent so integral will be dominated by theÎ¾ = 0 the contribution from neighbourhood Î¾ = 0. In order to account for the factthat the poles of the loretzian may be close too Î¾ = 0 we expand the prefactor tothe exponential functionÎ³pif(Î¾)eiâÎ¾2+â02t(Î¾2 + Î³2)=Ar(Î¾) +A+(Î¾) +Aâ(Î¾) (F.9)A+(Î¾) =âif(iÎ³)2pi(Î¾ â iÎ³) (F.10)Aâ(Î¾) =if(âiÎ³)2pi(Î¾ â iÎ³) (F.11)Ar(Î¾) =Î³pif(Î¾)eiâÎ¾2+â02t(Î¾2 + Î³2)âA+(Î¾)âAâ(Î¾). (F.12)Here we have split off the poles of the âamplitude function to leave a regularisedpart the integral I2 can then be split upI2 =I2+ + I2â + I2r (F.13)I2+ =â« âââdÎ¾A+(Î¾)eiâÎ¾2+â02t (F.14)I2â =â« âââdÎ¾Aâ(Î¾)eiâÎ¾2+â02t (F.15)I2r =â« âââdÎ¾Ar(Î¾)eiâÎ¾2+â02t. (F.16)Then expanding around the saddle point we find that along the steepest descentcontour Î¾ â¼ reipi4 and that the exponent is iâ0tâ r2t2â0 so that we findI2+ + I2â â¼12exp(â itÎ³22â0+ itâ0)ï£®ï£°1 + ierfiï£«ï£­1 + i2âtÎ³2â0ï£¶ï£¸ï£¹ï£» [f(âiÎ³) + f(iÎ³)](F.17)I2r =eipi4â« âââdrAr(reipi4)eiâ0tâ r2t2â0 (F.18)219where Ar(reipi4)can be expanded around zero to give an approximation to I2r.
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Aspects of decoherence in qubit systems
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