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Quantum mechanics of composite objects with internal entanglement Suzuki, Fumika 2017

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Quantum Mechanics of CompositeObjects with Internal EntanglementbyFumika SuzukiB.Sc., The University of Leeds, 2010M.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Fumika Suzuki 2017AbstractAlthough many quantum mechanics problems of a point particle have beenwell understood, the realistic physical model is often a composite objectconsisting of particles bound together, and its quantum mechanics is an im-portant problem with applications in many areas of physics. Unlike a pointparticle, a composite object possesses internal structure described by somedegrees of freedom which are often entangled with each other. We call theentanglement among these degrees of freedom “internal entanglement”, todistinguish it from any other entanglement they may share with an externalobject. Examples of internal entanglement include the entanglement be-tween a vibrational and a rotational mode of a molecule, and that betweenits position and its internal clock-state etc.In this thesis, we study quantum mechanics of composite objects by fo-cusing on the effects of internal entanglement. We first look at the tunnellingof a diatomic molecule type composite object through a half-silvered mirrorin continuum space and observe that its spatial superposition state madeby the mirror can decohere by emission of radiation due to the fact thatits internal degrees of freedom are entangled with radiation fields, and thatthere exists entanglement between its position and its internal (i.e., relativeposition) degrees of freedom. Secondly, we study its lattice analog in molec-ular crystals, namely we replace the diatomic molecule by a biexicton, andthe mirror by an impurity. We find that discreteness of a lattice makes thewave vector of a biexicton and the relative distance between two excitons ofit entangled with each other. We investigate how this inseparability affectsthe creation of the biexciton-impurity bound states and the entanglementdynamics. Finally we propose a possible application of our study of internalentanglement to the Anderson model of a composite quasiparticle.iiLay SummaryQuantum mechanics is a branch of physics that successfully explains thebehaviours of microscopic systems and has become one of the most funda-mental theories in science. In quantum mechanics, particles are described interms of waves that can interfere and show non-classical behaviours. How-ever it is still less clear how classical mechanics of the macroscopic worldmay emerge from underlying quantum mechanics. One possible explanationfor the transition from quantum mechanics to classical mechanics is deco-herence, which is the idea that quantum systems interact with the environ-ment and lose interference phenomena, consequently macroscopic systemsappear classical. Instead of the interaction with the external environment,we study the effects of the interactions among degrees of freedom of quantumcomposite objects on their quantum mechanical behaviours in this thesis,with possible applications to the study of quantum-to-classical transitions ofmacroscopic objects, quantum gravity, quantum chemistry, and condensedmatter physics.iiiPrefaceParts of the contents of Chapter 2 were published in the article: F. Suzukiand F. Queisser, Environmental gravitational decoherence and a tensor noisemodel. J. Phys.: Conf. Ser. 626 012039 (2015) [1]. The project wasdesigned by me and F. Queisser. I performed the analytical and numericalcalculations with F. Queisser. The chapter is also based on my extendedwork F. Suzuki, arXiv:1705.05426 (2017) [2]. This chapter has benefittedfrom the helpful comments of G. W. Semenoff and W. G. Unruh.A version of Chapter 3 has been published: F. Suzuki, M. Litinskaya andW. G. Unruh, Scattering of a composite quasiparticle by an impurity on alattice. Phys. Rev. B. 96, 054307 (2017) [3]. I performed all the analyticaland numerical calculations and wrote the original manuscript. The Hamil-tonian of the problem was introduced by M. Litinskaya, and the methodsto solve eigenstates of exciton with an impurity and biexciton eigenstatesexplicitly were found by W. G. Unruh. M. Litinskaya and W. G. Unruh alsoworked on my manuscript to improve explanations of content and figures. Ialso would like to thank R. V. Krems and T. Momose who held the meetingthat initiated the project.Some of the contents of Chapter 4 are based on work by the authorimproved by discussions with W. G. Unruh during the project above. Thesection about the Anderson model is based on an idea and the guidance ofR. V. Krems. Although this chapter only shows analytical and numericalcalculations that I performed myself, some of them have benefitted fromdiscussions with T. Chattaraj as well as from his numerical studies.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantum Mechanics of Composite Objects . . . . . . . . . . 11.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 What Is Decoherence? . . . . . . . . . . . . . . . . . 41.3.2 Various Models of Decoherence . . . . . . . . . . . . . 131.4 Composite Objects . . . . . . . . . . . . . . . . . . . . . . . 161.5 Applications of Entanglement and Decoherence Studies . . . 181.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Decoherence of a Composite Object in QED and Weak Grav-ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Spatial Superposition of a Composite Object in ContinuumSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24v2.2 Decoherence by Emission of Radiation . . . . . . . . . . . . . 292.3 Decoherence by Quantized Static Fields . . . . . . . . . . . . 432.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Scattering of a Composite Quasiparticle by an Impurity ona Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Biexciton States . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Interaction of Excitons and Biexcitons with Impurity . . . . 613.2.1 Exciton-Impurity Interaction . . . . . . . . . . . . . . 613.2.2 Biexciton-Impurity Interaction . . . . . . . . . . . . . 673.2.3 Poles of the Reflection Amplitude . . . . . . . . . . . 723.2.4 Bound States in the Continuum . . . . . . . . . . . . 763.3 Decoherence by Internal Degrees of Freedom . . . . . . . . . 803.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.5 Supplemental Material: From Discrete Model to ContinuumModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 Anderson Model of a Composite Quasiparticle . . . . . . . 904.1 Biexciton States and Continuum Two-Exciton States . . . . 914.2 Quantum Walk and Group Velocity of a Biexciton . . . . . . 944.3 Interaction with a Single Impurity . . . . . . . . . . . . . . . 1004.3.1 Impurity-Induced Bound States . . . . . . . . . . . . 1004.3.2 Transitions between Biexciton States and ContinuumTwo-Exciton States by a Scattering . . . . . . . . . . 1014.4 Anderson Model of a Composite Quasiparticle . . . . . . . . 1064.4.1 Anderson Localization . . . . . . . . . . . . . . . . . 1064.4.2 Anderson Model of Single Exciton and Biexciton . . . 1085 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122viAppendicesA Path Integral for a Forced Harmonic Oscillator . . . . . . . 129B Influence Functional from Radiation Fields . . . . . . . . . 132C Exciton Scattering by an Impurity . . . . . . . . . . . . . . . 135D Solving Lippmann-Schwinger Equation with Method of Con-tinued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 139viiList of Tables4.1 1/IPRbiexc and the von Neumann entanglement entropy Sof the entanglement between the CM and relative positiondegrees of freedom of the wave packet in random potentials att = 6000. 1/IPRexc is obtained from a single exciton with asimilar group velocity to that of the corresponding biexciton. 113viiiList of Figures1.1 The absolute value of the initial density matrix ρsys(x0, x′0; 0)of a quantum particle in a spatial superposition state aftera double-slit. The off-diagonal element of the density ma-trix represents quantum coherence (i). Without the interac-tion with an environment, the off-diagonal elements of thereduced density matrix remain, while they decay if the in-teraction with the environment is turned on (ii). Withoutthe environment, we have the interference pattern, while it iswashed out when the environmental decoherence occurs (iii). 142.1 Particle 1 and particle 2 of a composite object scatter by deltapotentials V1δ(x1) and V2δ(x2) respectively. . . . . . . . . . . 242.2 The wave packet after the scattering by a mirror potentialwhen V1 is zero and V2 is nonzero where the initial wavepacket before the scattering is in the form Ψ0(X,x) (above).Probability of finding the internal states φn in the transmittedand in the reflected part. The initial energy of CM degrees offreedom is twice larger than than that of the relative positiondegrees of freedom. State number n = 0 corresponds to theground state, and n = 1 is for the first excited state etc. Thesimilar result was also found in [60] (below). . . . . . . . . . . 282.3 One can partially tell whether it is the transmitted part orthe reflected part by looking at which-way information fromelectromagnetic or gravitational radiation. . . . . . . . . . . . 31ix2.4 The absolute value of p0(n1, n2) has off-diagonal elementsrepresenting superposition of the ground and excited state(above). After decoherence due to radiations, those superpo-sition states get suppressed in pt(n1, n2) (below). . . . . . . . 422.5 The time-evolution of the von Neumann entanglement en-tropy which measures the entanglement between x1 and x2according to Hˆ and Hˆ ′. . . . . . . . . . . . . . . . . . . . . . 462.6 Interference pattern produced by the reduced density matrixin x1 at the moment of maximal overlap. The interference islargely suppressed when the system evolves according to Hˆ,whereas the clear interference pattern can be produced whenthe system evolves according to Hˆ ′. . . . . . . . . . . . . . . . 472.7 External and internal observer with superposed gravity andspacetime generated by the superposed matter. We have anexternal observer in quantum mechanics, while measurementsare made from the inside of the system by an internal observerin general relativity. . . . . . . . . . . . . . . . . . . . . . . . 483.1 The diagram of the lattice with N = 4. The red points areall equivalent. Note that r, s + 4 is not the same as r, s, butr, s+ 8 is. r + 4, s+ 4 is the same as r, s. . . . . . . . . . . . 563.2 Single exciton interacting with impurity: Single bound state.(a) Energy spectrum for various combinations of J and V0.J > 0: Continuum states (red solid line) and bound states(i) or (ii); J < 0: Continuum states (blue dashed line), andbound states (iii) or (iv). (b) Wave-function correspondingto bound states (i-iv). . . . . . . . . . . . . . . . . . . . . . . 64x3.3 Energy spectrum of an exciton with an impurity given bynumerical diagonalization of Hamiltonian. E0/|J | = 1000,V0/J = 2.5 N = 40. We see that antisymmetric states (firstthree examples are surrounded by blue square) do not in-teract with an impurity, while symmetric states (first threeexamples are surrounded by red square) interact with an im-purity. Their energies agree perfectly with energies given byanalytical expressions (3.29) and (3.34). . . . . . . . . . . . . 653.4 Biexciton interacting with an impurity: Multiple bound states.(a) Biexciton scattering states for D > 0 (red solid line) andD < 0 (blue dashed line), bound biexciton-impurity states(dots (i,ii) for D > 0 and (iii,iv) for D < 0), and two-excitonunbound states (grey shaded region). Right panels zoom theregions (i) and (iii) with multiple states. (b) The probabilitydistributions of the bound states (i,iii), and (c) of the boundstates (ii,iv). . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.5 Examples of scattering states of a single exciton and those ofa biexciton with an impurity. Amplitudes of biexciton scat-tering states are populated around an impurity, unlike singleexciton scattering states with an impurity. . . . . . . . . . . 703.6 Number of bound states as function of parameters. All ener-gies are in units of |J |. The number of states depends on thewidth of φK(|xrel|). The bottom plots show the wavefunctionsof relative coordinate for three values of parameters; “case (i)”and “case (ii)” refer to the notations of Figure 3.4 (a). Firstand last bottom plots illustrate the difference in the width ofφK(|s|) at K = 0 and K = pi/2 for same values of all otherparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.7 Scattering amplitude of a single exciton (a) and biexciton inthe perturbative limit (b) as function of complex momentum.The insets show the continuum spectrum (red stripe) and thesingle bound state (red dot) calculated numerically, and thearrow indicates the analytical estimates (see text). . . . . . . 75xi3.8 Bound states in the continuum. (a) D = 4.1J , V0 = 8J . (b)D = 4.1J , V0 = J . . . . . . . . . . . . . . . . . . . . . . . . . 803.9 Time evolution of the biexciton wave packet (3.49) interact-ing with the impurity. Shown is the probability distributionρ(r, s; r, s; t), with N = 40, r ∈ (−N,N), s ∈ (−N,N). . . . . 813.10 (a) Comparison of diagonal and off-diagonal matrix elementsfor the reduced density matrix after scattering, C(r, r′) at t =35. (b) Interference produced by the reduced density matrixin the CM-coordinate at the moment of maximal overlap,ρcm(r, r; t = 54). (c) Similar calculation for single exciton. (d)Mode distribution before and after scattering. Expectation ofenergy 〈Eb〉 = −4.7 is the same in both. . . . . . . . . . . . . 844.1 Comparison between eigenvalues obtained analytically andthose obtained by numerical diagonalization of H ′0 or H0 withparameters N = 20, D/J = 4.1 and D < 0. . . . . . . . . . . . 954.2 Example of eigenstates obtained by numerical diagonalizationof H ′0 or H0 on physical space (even r+s space) with D/J = 4.1. 964.3 The propagation of the initial wave packet (left) moving to-wards the opposite edge of the lattice. At t = 25, the wavepacket consisting of modes around K ∼ pi/4 (above) arrivesat the other side slightly earlier than that consisting of modesaround K ∼ 0,±pi/2 does (below). D/J = 4.25 and N = 26. . 984.4 (a) The dependence of the group velocity vb of a biexciton onK andD/J . (b) The probability distributions of the biexcitonwave packets at t = 25 that consist of modes K ∼ pi/4 withD/J = 3.25 and D/J = 9.25 respectively. They propagatefrom the left end to the right end, similarly to the wave packetin Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99xii4.5 Examples of the eigenstates obtained by numerical diagonal-ization of H ′0 + V ′ or H0 + V with parameters D/J = 2.25,V0/J = 5. The biexciton-impurity bound state (left) and thestate with one free exciton, and one exciton bound to theimpurity (right). . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6 The increase of the entanglement entropy between the CMand relative position degrees of freedom by a single scatter-ing due to an impurity, with different values of D/J (above).Comparison between the wave packets with D/J = 2.25 andD/J = 4.25 after the scattering. We have the nonzero prob-abilities of finding states with large s, showing the existenceof the continuum two-exciton states after the scattering whenD/J = 2.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7 The backward and the forward path become identical to eachother when xB = xA and they interfere constructively witheach other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.8 Anderson localization of a single exciton (above) and thatof a biexciton with D/J = 4.25 (below). The comparisonbetween the time-evolutions of the wave packet with (right)and without (left) random potentials, W = 0.45|J |. Here weplotted the probability distribution of a biexciton wave packetby rescaled r → r/2 so that r = (m+ n)/2. . . . . . . . . . . 1094.9 (a) The increase of the entanglement entropy between the CMand relative position degrees of freedom in random potentialsof the wave packet, with different values of D/J . (b)The en-ergy spectrum of the continuum two-exciton states and thebiexciton states with D/J = 2.25. Black line indicates thesame energy. When the initial mode distribution is centredaround K0 = pi/2, there exist many continuum two-excitonstates that the initial biexciton states can decay into, in con-trast to the case of K0 = 0 where there exist only a fewcontinuum two-exciton states that have the same energy asthe initial biexciton states. . . . . . . . . . . . . . . . . . . . . 111xiii4.10 Comparison between the probability distribution for the wavepacket in random potentials after t = 6000, with the ini-tial mode distribution centred at K0 = 0 and at K0 = pi/2(above). The corresponding reduced density matrix ρcm(r, r; t)(below). D/J = 2.25. . . . . . . . . . . . . . . . . . . . . . . . 1154.11 The probability distribution for the wave packet in randompotentials after t = 6000, with the initial wave packet consist-ing of the continuum two-exciton states of the same energyas biexciton states with K0 = pi/2 when a two-particle inter-action D is turned off (above). The corresponding reduceddensity matrix ρcm(r, r; t) (below). . . . . . . . . . . . . . . . 116C.1 Integration contour for various n and k. . . . . . . . . . . . . 136xivAcknowledgementsFirstly I would like to thank Takamasa Momose for his kindness and con-tinuous support during my PhD studies and for teaching me about variousexperiments in physics by showing me a lot of the interesting work conductedin his lab. I also would like to thank Roman Krems and Kirk Madison fortheir efforts to connect my theoretical interest with realistic quantum opticalsystems and for providing fruitful physics problems.In my study of theoretical physics, I have been benefitted greatly fromdiscussions with Bill Unruh and Gordon Semenoff. I would like to thank BillUnruh for always providing me with sincere comments and teaching me howto think about physics problems by asking me important questions. Thanksto him, I have also learned how to conduct collaborative research throughdiscussions with other researchers. I would like to thank Gordon Semenofffor teaching me path integral formalisms in quantum field theories, whichhas helped inform my work in Chapter 2.I also would like to thank a number of researchers who have kindly dis-cussed various issues with me, especially R. Froese, F. Queisser, M. Litin-skaya, I. Averbukh, Y. Shikano, B. L. Hu, C. Anastopoulos, and M. Hotta.Thanks also to D. Stephen for precious suggestions on the form and contentof the final draft. Another word of thanks goes to the staff, faculty, postdocsand graduate students at UBC.Thanks to the co-operation and understanding of my family withoutwhich I would not have been able to continue my study. I am grateful to myparents, my sisters, my brother-in-law and our dogs for their support andencouragement.xvChapter 1Introduction1.1 Quantum Mechanics of Composite ObjectsAlthough common examples in quantum physics textbooks are often re-stricted to simplified models in which the system is considered to be a singlepoint particle, many realistic physical models are in fact composite objectsconsisting of at least two constituent particles. By composite object, wemean a system whose constituent particles are bound together in their rel-ative distance by the particle-particle interaction, e.g., a diatomic moleculein Chapter 2 and a biexciton in Chapter 3. The quantum mechanics ofcomposite objects provides important problems with applications in manyareas of physics. These problems include tunnelling of composite objectsthrough barriers [4], with applications in nuclear fusion [5], induced decayof false vacuum [6], and tunnelling of Cooper pairs in superconductors andWannier-Mott excitons in semiconductor heterostructures [7]. It had beenshown that the probability of tunnelling through a barrier for an objectpossessing an internal degree of freedom – for example, a diatomic molecule– may greatly exceed that of a structureless object with similar propertiesdue to the appearance of quasi-bound states in the combined scatteringand molecular binding potentials [4–6]. Furthermore, the interaction of amolecule with an external potential can induce transitions between molec-ular states due to coupling between the relative and centre of mass (CM)position degrees of freedom [8, 9]. Two- or many-photon transport throughan impurity [10, 11] or impurities [12] with potential applications in quan-tum information processing has also been studied. It is also expected thatthe interference and superposition of a composite object may provide aninsight into the quantum-to-classical transition [13, 14] and the effects of1relativity and gravity on quantum mechanics [15].Unlike a point particle, a composite object possesses internal structuredescribed by some degrees of freedom which are often entangled with eachother. This entanglement may act as a source of decoherence in one of thecoordinates. We call the entanglement among these degrees of freedom “in-ternal entanglement”, to distinguish it from any other entanglement theymay share with an external object. In this thesis, we study quantum me-chanics of composite objects by focusing on the effects of internal entangle-ment. The famous example of internal entanglement was discussed in [16].They considered the situation where an internal clock is entangled with theexternal spatial superposition of a molecule. For example, an internal clockcan be a marker on the surface of the molecule. If the marker rotates with afixed angular frequency, the number of turns it makes can be considered asa measure of time of an internal clock. If the molecule passes through thedouble-slit, the part of the wave packet that passes the left-hand slit arrivesat the left-hand first-order interference maximum earlier than that whichpasses the right-hand slit, in terms of time measured by an internal clock.Therefore, an internal clock gives which-way information telling whether thewave packet comes from the left-hand or the right-hand slit. In this way,the internal clock gets entangled with the position of a molecule. Howeverthe time difference of an internal clock is normally so small that it does notcause observable decoherence. Note that there is also an attempt to observethe decoherence phenomena by the time difference of an internal clock en-tangled with a spatial superposition state of a composite object due to thetime dilation by gravitational fields [15].As a simple example of internal entanglement, in this thesis, we lookat the entanglement between the relative separation of constituent particlesand the position of the composite object where the position of the compositeobject is represented by the CM coordinate. One of the important objec-tives of this thesis is to investigate the effects of internal entanglement onexperiments which test the quantum spatial superposition of a compositeobject.Before going into details, let us briefly review the concept of entangle-2ment and decoherence in the following few sections.1.2 EntanglementIn this section, we review the basic idea of quantum entanglement.Let us consider two quantum mechanical systems A and B whose statesare represented by Hilbert spaces HA and HB respectively. Then the generalstate of the bipartite composite system of A and B represented by Hilbertspace HA ⊗HB can be written as|Ψ〉AB =∑i,jcij |i〉A ⊗ |j〉B (1.1)where {|i〉A} and {|j〉B} are bases for HA and for HB respectively.If cij can be written as cij = cAi cBj such that |ψ〉A =∑i cAi |i〉A and |φ〉B =∑j cBj |j〉B, we can write |Ψ〉AB as a product state |Ψ〉AB = |ψ〉A ⊗ |φ〉B.Then it is said that the subsystems A and B are in a pure state. Howeverif there exist i, j such that cij 6= cAi cBj , then the state |Ψ〉AB can not bewritten as a product state, and it is said that the subsystems A and B areentangled.The density matrix of the composite system can be written asρ = |Ψ〉ABAB〈Ψ| =∑i,j,i′,j′cijc∗i′j′(|i〉A ⊗ |j〉B)(A〈i′| ⊗B 〈j′|). (1.2)Then the reduced density matrices of a system A and a system B arerespectively,ρA = trBρ =∑j′′B〈j′′|ρ|j′′〉B =∑i,i′,jcijc∗i′j |i〉AA〈i′|,ρB = trAρ =∑i′′A〈i′′|ρ|i′′〉A =∑j,j′,icijc∗ij′ |j〉BB〈j′|. (1.3)When cij = cAi cBj (i.e., a system A and B are in a pure state), we haveρ2A = ρA and ρ2B = ρB. In this case, a systemB does not affect measurements3of a system A and vice versa.When a system A and B are in an entangled state, however, we can findan observable OA for a system A, and an observable OB for a system B suchthat [17]AB〈Ψ|OA ⊗OB|Ψ〉AB 6= AB〈Ψ|OA|Ψ〉ABAB〈Ψ|OB|Ψ〉AB. (1.4)The amount of entanglement in a bipartite quantum state can be mea-sured by the von Neumann entanglement entropy S defined asS = −trρA log2 ρA. (1.5)Note that S = −trρA log2 ρA = −trρB log2 ρB in the case of a bipartitepure state where we can write the density matrix as ρ = |Ψ〉ABAB〈Ψ| andthe state is completely known for the composite system of A and B.If we use the eigenvalues λi of ρA, we haveS = −∑iλi log2 λi. (1.6)For a pure state, we have the von Neumann entanglement entropy S = 0,while S = log2N for a maximally mixed state where N is the dimension ofthe Hilbert space.In this thesis, it is often considered that a quantum mechanical systemA and B represent the degrees of freedom of a single composite object, andwe call the entanglement among them internal entanglement.1.3 Decoherence1.3.1 What Is Decoherence?Although quantum coherence is a fundamental property of quantum me-chanics, a quantum system appears to lose its coherence when it is entangledwith the other quantum systems. When one traces out quantum systemswhich one does not look at but are entangled with the quantum system con-4cerned (sometimes called a coarse-grained description of the system [18]),it appears as if the behaviour of the system given by quantum mechanicalprobability is suppressed but is described by classical probability theory in-stead [19]. The phenomena is called decoherence. The quantum systemswhich are traced out can be external/environmental degrees of freedom orinternal degrees of freedom of a single composite object as long as one doesnot have access to these degrees of freedom.As an example, let us consider a system concerned is in a superpositionof states |Ψ1〉 and |Ψ2〉:|Ψ〉 = 1√2(|Ψ1〉+ |Ψ2〉). (1.7)If the system interacts with an external environment |E〉, then we have|Ψ〉 ⊗ |E〉 = 1√2(|Ψ1〉+ |Ψ2〉)⊗ |E〉 interaction−−−−−−→ 1√2(|Ψ1〉|E1〉+ |Ψ2〉|E2〉).(1.8)This indicates that if the system is in a state |Ψ1〉 (or |Ψ2〉) then theenvironment becomes a state |E1〉 (or |E2〉) after the interaction with it. Forexample, if the system has a superposed momentum, then an environmen-tal particle gets scattered by it due to the interaction with it and the finalmomentum of an environmental particle depends on the momentum of thesystem concerned. Therefore, by measuring the momentum of an environ-mental particle, one can obtain the information about the momentum of thesystem.Now let us look at the reduced density matrix of the system ρsys. Bytracing out the environmental degrees of freedom, we haveρsys =12(|Ψ1〉〈Ψ1|+ |Ψ2〉〈Ψ2|+ |Ψ1〉〈Ψ2|〈E2|E1〉+ |Ψ2〉〈Ψ1|〈E1|E2〉). (1.9)If |E1〉 and |E2〉 are almost orthogonal states, i.e., 〈E2|E1〉 ≈ δ12, theoff-diagonal element of the reduced density matrix, |Ψ1〉〈Ψ2| and |Ψ2〉〈Ψ1|representing quantum coherence get suppressed, which is called environmen-tal decoherence. In the process of decoherence, the reduced density matrix5of the system obtained by tracing out environmental degrees of freedommakes a transition from a pure state to a mixed state such that the vonNeumann entanglement entropy is non-zero, and there exists the basis inwhich the off-diagonal elements of the reduced density matrix of the systemget suppressed.Note that emergence of decoherence can depend on which basis the sys-tem has a superposition state in and whether the environment can obtainwhich-way information, so that it can get entangled with the state of thesystem by interacting with it. As an example in [13] shows, if we considerthe superposition|Ψ′〉 = 1√2(|Ψ+〉+ |Ψ−〉) (1.10)where|Ψ±〉 = 1√2(|Ψ1〉 ± |Ψ2〉), (1.11)then the above environment fails to obtain which-way information,|Ψ′〉|E〉 interaction−−−−−−→ |Ψ′〉|E1〉 (1.12)and the system and the environment do not get entangled, meaning we donot have decoherence on such a superposition state of the system by thisenvironment. Therefore, it is important to consider what kind of which-wayinformation the environment carries away, as well as the initial state of thesystem and its time-evolution when we study decoherence.The time-evolution of the density matrix of the system coupled to theenvironment is often conveniently studied in the path integrals formalism.One of the common examples is a system x coupled to an environment Xmodelled as the large bath consisting of a harmonic oscillator [20, 21]:Ssys[x] + Senv[X] + Sint[x,X]=∫ t0Lsys(x, x˙)dt+∫ t0Lenv(X, X˙)dt+∫ t0Lint(x,X)dt, (1.13)6whereLenv =∑k12MX˙2k −∑k12Mω2kX2k ,Lint = −x∑kCkXk. (1.14)The time-evolution of the density matrix ρ(x, x′;X,X ′; t) of the wholesystem is given byρ(x, x′;X,X ′; t) =∫∫∫∫dx0dx′0dX0dX′0J(x, x′;X,X ′; t|x0, x′0;X0, X ′0; 0)×ρsys(x0, x′0; 0)ρenv(X0, X ′0; 0)(1.15)whereJ =∫∫∫∫DxDx′DXDX ′ exp i(Ssys[x]− Ssys[x′] + Senv[X]− Senv[X ′]+Sint[x,X]− Sint[x′, X ′]). (1.16)However, if we are only interested in the reduced density matrix ρred(x, x′; t)of the system x, we haveρred(x, x′; t) =∫∫dx0dx′0Jred(x, x′; t|x0, x′0; 0)ρsys(x0, x′0) (1.17)whereJred(x, x′; t|x0, x′0; 0) =∫∫DxDx′ei(Ssys[x]−Ssys[x′])F [x, x′] (1.18)and the influence functional F [x, x′] is given byF [x, x′] =∫∫∫dXdX ′0dX0ρenv(X0, X′0; 0)∫ XX0∫ XX′0DXDX ′ei(Senv[X]−Senv[X′]+Sint[x,X]−Sint[x′,X′])(1.19)7where [22] ∫ XX0∫ XX′0DX(s)DX ′(s)ei(Senv[X]−Senv[X′]+Sint[x,X]−Sint[x′,X′])=∏kKk(Xk, Xk0)K∗k(Xk, X′k0) (1.20)and [21] (Appendix A)Kk(Xk, Xk0) =√Mkωk2pii~ sinωktexpi~S(k)cl (Xk, Xk0),Scl(Xk, Xk0)=Mkωk2 sinωkt[(X2k +X2k0) cosωkt− 2XkXk0+2ckXkMkωk∫ t0x(s) sinωksds+2ckXk0Mkωk∫ t0x(s) sinωk(t− s)ds− 2c2kM2kω2k∫ t0ds∫ s0ds′x(s)x(s′) sinωk(t− s) sinωks′]. (1.21)Here the subscript 0 indicates the initial state.ThenKk(Xk, Xk0)K∗k(Xk, X′k0) =Mkωk2pi~ sinωktexpi~S′(k)cl ,S′(k)cl =Mkωk2 sinωkt[(X2k0 −X ′2k0) cosωkt− 2Xk(Xk0 −X ′k0) +2ckXkMkωkI1+2ckXk0MkωkI2 − 2ckX′k0MkωkI3 − 2c2kM2kω2kI4](1.22)8whereI1 =∫ t0(x(s)− x′(s)) sinωksds, I2 =∫ t0x(s) sinωk(t− s)ds,I3 =∫ t0x′(s) sinωk(t− s)ds,I4 =∫ t0ds∫ s0ds′[x(s)x(s′)− x′(s)x′(s′)] sinωk(t− s) sinωks′.(1.23)If the environmental oscillator modes are assumed to be in the groundstate initially, we haveρenv(X0, X′0; 0) =∏kρ(k)env(Xk0, X′k0, 0)=∏k(Mkωkpi~)1/2exp[−Mkωk2~(X2k0 +X′2k0)].(1.24)By recalling Gaussian integrals with a complex phase:∫ ∞−∞exp i(Ax2 +Bx)dx =(piiA)1/2exp(− iB24A)(1.25)we perform the integrals in (1.19), leading toF [x, x′]=∏kexp[− 14~Mkωk sin2 ωkt(c2k(I21 + I22 − 2I2I3 + I23+2I1(I2 − I3) cosωkt− 2i(I1(I2 + I3)− 2I4) sinωkt))]. (1.26)9The complex part of the phase givesic2kI1(I2 + I3)− 2I42~Mkωk sinωkt=i2~Mkωk sinωkt∫ t0∫ s0(x(s)− x′(s))(x(s′) + x′(s′))×[sinωks sinωk(t− s′)− sinωks′ sinωk(t− s)]dsds′= i∫ t0∫ s0dsds′(x(s)− x′(s))c2k sinωk(s− s′)2~Mkωk(x(s′) + x′(s′)) (1.27)while its real part gives−c2k(I21 + I22 − 2I2I3 + I23 + 2I1(I2 − I3) cosωkt)4~Mkωk sin2 ωkt= − c2k4~Mkωk sin2 ωkt∫ t0∫ s0(x(s)− x′(s))(x(s′)− x′(s′))×[cosωk(s′ − s)− cosωk(s′ − s− 2t)2− cosωk(s′ − s+ 2t)2]= −∫ t0∫ s0(x(s)− x′(s))c2k cosωk(s− s′)2~Mkωk(x(s′)− x′(s′)) (1.28)where we used the property of the double integral:∫ t0∫ t0f(s, s′)dsds′ =∫ t0∫ s0(f(s, s′) + f(s′, s))dsds′ (1.29)for a function f(s, s′).By combing the two terms, we obtainF(x, x′)= exp[i∫ t0∫ s0dsds′(x(s)− x′(s))∑kc2k sinωk(s− s′)2~Mkωk(x(s′) + x′(s′))−∫ t0∫ s0(x(s)− x′(s))∑kc2k cosωk(s− s′)2~Mkωk(x(s′)− x′(s′))]. (1.30)Note that if we choose the initial density matrix for a harmonic oscillator10to be in thermal equilibrium instead of (1.24):ρenv(X0, X′0; 0) =∏kρ(k)env(Xk0, X′k0, 0)=∏kMkωk2pi~ sinhβ~ωk× exp{− Mkωk2~ sinh(β~ωk)[(X2k0 +X′2k0) cosh(β~ωk)− 2Xk0X ′k0]}(1.31)where β = 1/kBT , we obtainF(x, x′)= exp[i∫ t0∫ s0dsds′(x(s)− x′(s))∑kc2k sinωk(s− s′)2~Mkωk(x(s′) + x′(s′))−∫ t0∫ s0(x(s)− x′(s))∑kc2k cosωk(s− s′) coth ~ωkβ/22~Mkωk(x(s′)− x′(s′))].(1.32)Indeed (1.30) and (1.32) are identical in the zero temperature limit,T → 0. In Chapter 2, we use this type of the influence functional fromradiation fields.As an example, let us consider the situation where the action Ssys of thesystem concerned is a free particle:Ssys =∫ t0m2x˙2dt (1.33)and its kernel can be written asKsys(x, x0) =( m2pii~t)1/2exp[im(x− x0)22~t]. (1.34)This kernel gives the free-particle propagation with dispersion. If we act(1.34) on the initial wave packet ψ0(x0, 0) in a spatial superposition state11(e.g., an electron after passing a double-slit, Figure 1.1 (i)),ψ(x, t) =∫ ∞−∞dx0Ksys(x, x0)ψ(x0, 0),ψ(x0, 0) =1√21(2piσ2)1/4[exp(− 14σ2(x0 − d)2)+ exp(− 14σ2(x0 + d)2)](1.35)where d and −d are the centre of two Gaussian distribution with widthσ, then the off-diagonal element of the density matrix remains (Figure 1.1(ii)) and we obtain the interference pattern after some time, as the plot ofψ(x, t)∗ψ(x, t) plotted in Figure 1.1 (iii) shows.Let us now consider the effect of the environmental decoherence on thisinterference pattern. The limit of (1.32) when 2kBT >∼~Ω  ~ωk where Ωis the cutoff frequency of the environment can be written as [20]F(x, x′) = exp(−a∫ t0(x(s)− x′(s))2)ds(1.36)if we ignore the imaginary part of the phase. Here a = 2mγkBT~2 and γ is therelaxation constant.Then the reduced density matrix propagator (1.18) is given byJred(x, x′; t|x0, x′0; 0)=∫∫DxDx′ exp(im2~∫ds(x˙2 − x˙′2)− a∫ds(x− x′)2). (1.37)Substituting the classical path xc(s) of a free particle with the boundarycondition x(0) = x0, x(t) = x:xcl(s) = x0 +x− x0ts (1.38)12into (1.37), we obtain [23]Jred(x, x′; t|x0, x′0; 0)=m2pi~texp[im2~t((x− x0)2 − (x′ − x′0)2)−13at((x− x′)2 + (x− x′)(x0 − x′0) + (x0 − x′0)2)]. (1.39)Then (1.17) with ρsys(x0, x′0; 0) = ψ(x0, 0)ψ∗0(x′0, 0) gives that the off-diagonal element of the reduced density matrix ρred(x, x′; t) is suppressedafter some time (Figure 1.1 (ii)), and the interference is washed out (Figure1.1 (iii)). Here the environment monitors a position of a system, and de-coheres its spatial superposition state [24]. Since the quantum interferenceis a consequence of the coherent superposition obtained by the quantummechanical probability, we see that the decoherence effect suppresses in-terference phases by causing transitions from the behaviour of the systemobeying quantum mechanical probability to that obeying classical probabil-ity theory.1.3.2 Various Models of DecoherenceIn the previous section, we discussed one simple model for decoherence wherea single particle is coupled to the surrounding environment described by abath consisting of harmonic oscillators. Although the model is often used[25], decoherence is caused by the entanglement between the system con-cerned and the extra degrees of freedom (e.g., environment) which makesthe state of the system a mixed state with non-zero von Neumann entan-glement entropy, and therefore there are a huge number of other decoher-ence models studied [26–28]. Early study of decoherence was done by H.D. Zeh [29] and decoherence due to scatterings by environmental particleswas discussed in [26]. Although decoherence sometimes happens along withdissipation, it is shown that decoherence can also occur in the presence ofenergy conservation [30].13Figure 1.1: The absolute value of the initial density matrix ρsys(x0, x′0; 0) ofa quantum particle in a spatial superposition state after a double-slit. Theoff-diagonal element of the density matrix represents quantum coherence(i). Without the interaction with an environment, the off-diagonal elementsof the reduced density matrix remain, while they decay if the interactionwith the environment is turned on (ii). Without the environment, we havethe interference pattern, while it is washed out when the environmentaldecoherence occurs (iii).14Under the assumption that quantum theory is universally valid, the en-tanglement dynamics between the quantum system concerned and the otherextra quantum systems that one does not look at (e.g., environment) is inprinciple reversible since quantum evolution according to the whole Hamil-tonian (e.g., Hsys + Hint + Henv) is unitary. However, if there exist a largenumber of degrees of freedom, it is often technically difficult to reverse theentanglement dynamics by controlling those degrees of freedom involved indynamics. When Hamiltonian Hint + Henv can be assumed to be time-independent smooth bounded function, one may use Poincare´ recurrencetime given by the inverse of its level spacing to estimate the time when thetotal system returns to the state arbitrary close to the initial state. For afinite number of reservoir oscillators, we have a finite recurrence time whenwe have the trivial entanglement dynamics. The recurrence time can beinfinitely long if it is assumed that we have an infinitely large environmentand the dynamics given by the Hamiltonian is trivial [31].There also exists a semiclassical process called dephasing where the phaseof each of the energy eigenstate gets altered by the interaction with the otherdegrees of freedom, or due to unknown attributes of the environment. Theenvironment may cause constant unknown changes in the energy eigenstateswhich leads to loss of coherence, however the effect can be cancelled bysuitable manipulation of the system. For example, the loss of coherence inspin echo [32] can be called dephasing.The entanglement dynamics can become more complicated when Hamil-tonian is nontrivial (e.g., Hint is time-dependent). It was discussed in[33] that the system gradually couples to an environment equivalent to theCaldeira-Leggett model can give maximally incoherent density matrix, how-ever its density matrix becomes coherent again when the system gets slowlydecoupled from the environment. In this thesis, we discuss the entangle-ment dynamics of finite degrees of freedom, but with nontrivial Hamiltonian,namely the CM and relative position degrees of freedom of a composite ob-ject do not interact with each other constantly, but they interact at themoment when the composite object scatters by a half-silvered mirror or animpurity. With nontrivial dynamics, often one can not predict the behaviour15of decoherence by just looking at the Hamiltonian of the environment (inour case, the internal degrees of freedom of the composite object acts likethe environment), but needs to keep careful track of dynamics given by thewhole Hamiltonian.Many prior works in decoherence have been done with a single particleinteracting with the external environment. Instead here we look at a singlecomposite object with entangled internal degrees of freedom. Unlike theexternal environment such as particles scattering by the system concerned,internal degrees of freedom of a composite object are always there in princi-ple when one studies quantum mechanics of composite objects. For example,rotational and vibrational modes can be considered as internal degrees offreedom of a molecule. The internal entanglement (entanglement amonginternal degrees of freedom of a composite object) can cause decoherencein one of the degrees of freedom of the object when the other degrees offreedom are traced out, and can affect how the composite object interactswith the other external objects. In Chapter 2, we discuss the spatial super-position of the composite object can get entangled with radiation fields dueto the internal entanglement between the CM and relative position degreesof freedom, and in Chapter 3, we find the inseparability of these two degreesof freedom of biexciton can affect how it creates bound states with an im-purity, and causes decoherence of CM degrees of freedom when the relativeposition degrees of freedom is traced out.1.4 Composite ObjectsIn this thesis, we deal with two types of composite objects. In Chapter 2,we use a diatomic molecule type composite object in continuum space wheretwo particles interact with each other through the harmonic potential andthe Hamiltonian is given byHˆ0 =Pˆ 22M+pˆ22µ+12µω2xˆ2. (1.40)16Here Pˆ , pˆ are the CM coordinate and the relative coordinate momentumoperator respectively, M is the sum of two masses, µ is the reduced massand ω is the natural frequency of the oscillator.It is well known that the harmonic potential creates multiple boundstates in the relative degrees of freedom whose wave-functions can be writtenasφn(x) =1√2nn!(µωpi)1/4e−µωx2/2Hn(√µωx) (1.41)where Hn represents the Hermite polynomial of degree n, and the energyeigenvalues En = (n+ 1/2)ω.The objects described by wave-functions with these bound states in therelative degrees of freedom are treated as composite objects in this thesis.In Chapter 3, we use a biexciton as a composite quasiparticle on a lattice.Let us consider N interacting particles with the Hamiltonian,Hˆ =N∑n=1∇2n2m+∑m 6=nD(xm − xn) (1.42)where the first part is the kinetic energy of particles and the second part isthe two-particle interaction potential.We now introduce the second quantization and write the Hamiltonian as[34]Hˆ =∫dxΨˆ†(x)(−∇22m)Ψˆ(x)+∫dxdx′Ψˆ†(x′)Ψˆ†(x)D(x− x′)Ψˆ(x)Ψˆ(x′)=∑m,nJmnaˆ†maˆn +∑m1,m2,m3,m4Dm1m2m3m4 aˆ†m1 aˆ†m2 aˆm3 aˆm4 (1.43)where the field operators are expanded in terms of Wannier statesΨˆ†(x) =∑mw(x− xm)aˆ†m (1.44)17and aˆ†m is the creation operator which creates a particle at site m. Wannierfunction w(x− xm) is well localized in each lattice site and the particles inan optical lattice can be described by the Hubbard model (1.43).Since the wave-functions are well localized, we may keep only the largestcontributions from the nearest neighbours (the nearest-neighbour approxi-mation) and write the Hamiltonian asHˆ =∑n(J(aˆ†n+1aˆn + aˆ†n−1aˆn) +Daˆ†n+1aˆn+1aˆ†naˆn). (1.45)A two-particle interaction D creates biexciton which is a bound state inthe relative degrees of freedom, and we derive its wave-function in Chapter3.1.5 Applications of Entanglement andDecoherence StudiesEntanglement and decoherence appear everywhere in quantum mechanics,and recently they have been studied in diverse fields including quantuminformation, condensed matter physics, quantum cosmology and quantumgravity. Although this thesis does not go into details about those topics, wepresent an incomplete list of their applications to several areas of physics.Firstly, they are one of the central topics in quantum information. The useof quantum entanglement between a number of qubits in order to obtain anexponential speed-up of a quantum computation over classical computationwas discussed in [35]. However it is also known that decoherence is oneof the biggest obstacles to build a quantum computer in realistic physicalsystems [36]. Quantum entanglement also plays a key role in understand-ing phenomena in condensed matter physics and many-body physics. Forexample, it was shown that topological entanglement entropy [37], whichmeasures the long-range quantum entanglements in a many-body quantumsystem, is a useful quantity to characterize topological orders. The role ofentanglement in quantum phase transitions has also been investigated re-18cently [38]. The area law [39], which states that the entanglement entropyof the ground states between two partitions scales as the boundary betweenthem, is now used to develop powerful analytical and numerical methods toanalyze correlated many-body systems [40–42].The area law, however first appeared in the context of the study of blackhole entropies [43, 44]. In quantum cosmology and black hole physics, manydiscussions of entanglement and decoherence appear in the so-called blackhole information loss problem [17], that is a problem about the final quantumstate resulting from black holes that evaporate by thermal radiation. A purestate evolves to a mixed state due to radiation and information loss mayoccur. Decoherence of black holes by assuming Hawking radiation as thesource of decoherence is also studied in [45]. There also exist studies whichinvestigate the emergence of classical behaviour in quantum cosmology [46,47].As it can been seen, the study of entanglement and decoherence attractsinterest from a wide range of research areas.In this thesis, we will study decoherence of a composite object possess-ing internal entanglement due to emission of radiation in the framework ofquantum field theories (Chapter 2), the behaviour of internal entanglementwhen the composite object is defined on a discrete lattice (Chapter 3), anda possible application of the study of internal entanglement to the Andersonmodel of a composite quasiparticle (Chapter 4).1.6 OutlineThis thesis is organized as follows.In Chapter 1, we provided a brief review of the fundamental backgroundsof our study in order to help readers to understand the contents of thefollowing chapters.In Chapter 2, we study the quantum spatial superposition of a diatomicmolecule type composite object and its decoherence due to emission of elec-tromagnetic and gravitational radiation. The time-evolution of the reduceddensity matrix of the system with the influence functional from radiation19fields is solved using the path integral formalism. It is shown that the spa-tial superposition state of the composite object can decohere by emissionof radiation due to the facts that radiation is entangled with the internaldegrees of freedom of the composite object that describe the relative sepa-ration between its constituent particles, and that the composite object canhave internal entanglement between its position and its internal degrees offreedom. We also briefly discuss decoherence due to the quantization of thestatic Coulomb field generated by a superposed particle and the problemswhich arise when the electromagnetic field is replaced by gravity.In Chapter 3, we study scattering of a composite quasiparticle, whichpossesses a degree of freedom corresponding to relative separation betweentwo bound excitations, by a delta-like impurity potential on a one-dimensionaldiscrete lattice. Firstly, we show that, due to specific properties of theirdispersion, lattice excitations bind to impurities with both negative andpositive potentials. We demonstrate that the finite size of the compositeexcitation leads to formation of multiple excitation-impurity bound states.The number and the degree of localization of these bound states depend onthe signs and relative magnitudes of the impurity potential and the bindingstrength of two quasiparticles. We also report the existence of excitation-impurity bound states whose energies are located in the continuum band.Secondly, we study a change in the entanglement between the centre of massand relative coordinate degrees of freedom of a biexciton wave packet duringsingle impurity scattering and decoherence caused by it. For a compositequasiparticle on a lattice, the entanglement between its relative and centreof mass coordinate degrees of freedom arises naturally due to inseparabilityof the two-particle Hamiltonian. One of the main focuses of our study isto investigate how this inseparability affects the creation of the biexciton-impurity bound states and the entanglement dynamics.In Chapter 4, we extend our work in Chapter 3, by including continuumtwo-exciton states in a basis. We investigate how the transitions betweenbiexciton states and continuum two-exciton states due to a scattering byan impurity change the entanglement between the CM and relative positiondegrees of freedom of the wave packet prepared as a biexciton wave packet20initially. We also propose a possible application of our study to the Andersonmodel of a composite quasiparticle. The purpose of this chapter is not topresent the complete results, but to propose a possible extension of ourresults obtained in Chapter 3.In Chapter 5, we present our conclusions.21Chapter 2Decoherence of a CompositeObject in QED and WeakGravityIn this chapter, we consider a spatial superposition of a diatomic moleculetype composite object in continuum space prepared by a half-silvered mir-ror. We observe that the process of creating a spatial superposition of thecomposite object by the mirror makes its position and its internal (i.e., rel-ative position) degrees of freedom get entangled with each other. We studydecoherence of the spatial superposition state of the composite object withthe above internal entanglement by emission of radiation, due to the factthat the internal degrees of freedom of the composite object are entangledwith radiation fields.Recently, the studies of gravitational decoherence [48–50], decoherencestudies in the framework of field theories [51–55] or in ADM formalism [56],and the effects of gravity or relativity on quantum mechanics [15, 57] haveattracted considerable interest. An essential observation, which we shalldevelop further in this chapter, is the fact that decoherence, even thoughdriven by interaction with an environment, can depend on the details of theinitial state of a system and on the time evolution of that state. Derivation ofmaster equations and a reduced density matrix propagator with the influencefunctional from an environmental bath is an important step, but it does notgive the complete picture in many cases [33].The focus of this chapter is to study decoherence of the composite objectdue to emission of quantized radiation fields. Discussions of environmental22decoherence often involve a single particle interacting with an environmentalbath. Our purpose in studying the composite object here is to have a simplemechanism whereby the system can emit radiation.We assume there are no ambient radiation fields in the initial state andstudy decoherence of the spatial superposition of the composite object dueto its own emission of radiation. The starting point of the problem of de-coherence by the quantized fields generated by a quantum system itself wasdiscussed in [58, 59]. When a system is in a quantum superposition, a fieldgenerated by it should be also in a quantum superposition, and thereforeit is necessary to quantize fields including a gravitational field if they aregenerated by a quantum system. If the system concerned is only a singlequantum particle, we do not have emission of radiation generally, howeverwhen a quantum superposition of a composite object possessing dipole orquadrupole moment is considered, it would be possible to have decoherencecaused by radiation fields generated by it.Recent studies of [60] showed that the process of making a spatial super-position state of a composite object using a half-silvered mirror causes itsposition and its internal (relative position) degrees of freedom get entangledwith each other, namely it was observed that the reflected part has moreexcited internal modes than the transmitted part has. This indicates thatone is able to tell whether the wave packet is the reflected or the transmittedpart by looking at the internal degree of freedom. The states of the inter-nal degrees of freedom can be measured indirectly by observing radiationemitted from it. Following [60], we prepare a spatial superposition state ofa composite object consisting of two particles interacting with each othervia a harmonic potential in Section 2.1. We review how the position of thecomposite object gets entangled with its internal degrees of freedom, anduse such spatial superposition state of the composite object possessing in-ternal entanglement between its position and its internal degrees of freedomto study decoherence by emission of radiation in Section 2.2. In the section,we use the density matrix propagator with the influence functional fromradiation fields to study the time-evolution of the reduced density matrix ofthe composite object. It is observed that the superposition of the ground23Figure 2.1: Particle 1 and particle 2 of a composite object scatter by deltapotentials V1δ(x1) and V2δ(x2) respectively.and excited states of the internal degrees of freedom, and the spatial super-position of the transmitted and the reflected part associated with it, can besuppressed by tracing out radiation fields. Radiation fields consist of on-shellparticles that can freely propagate following their dynamical equations. Onthe other hand, in Section 2.3, we look at off-shell photons that do not havetheir own dynamical equations to obey, and appear only as an interactionbetween two charges by a static Coulomb field. In the section, we discussdecoherence caused by a static Coulomb field and the problem which ariseswhen the field is replaced by gravity. Units are chosen throughout such thatc = ~ = 1.2.1 Spatial Superposition of a Composite Objectin Continuum SpaceIn this section, we create a spatial superposition state of a diatomic moleculetype composite object by using the procedure of [60], which is used in the24following section in order to study decoherence of the spatial superpositionof the composite object by its own emission of radiation.Let us consider the Hamiltonian for two particles interacting with eachother by a harmonic potential written asHˆ0 =Pˆ 22M+pˆ22µ+12µω2xˆ2 (2.1)where Pˆ , pˆ are the CM coordinate and the relative coordinate momentumoperator respectively, M is the sum of two masses M = m1 + m2, µ is thereduced mass and ω is the natural frequency of the oscillator.We prepare the initial wave packet using eigenstates of (2.1) in the formΨ0(X,x) = f(X)φ0(x) wheref(X) =(2piσ2)1/4exp(iP0X − (X −X0)2σ2),φn(x) =1√2nn!(µωpi)1/4e−µωx2/2Hn(√µωx). (2.2)Here X is the CM position of two particles, x is the relative distance betweenthem, P0 is the initial CM momentum, σ is the width of the initial wavepacket with its centre being X0. We have the internal (relative position)degrees of freedom of the wave packet is in the ground state φ0(x) initially.In order to make a spatial superposition state of this composite object,[60] introduced a delta potential at the origin, Vˆ = V1δ(xˆ1) + V2δ(xˆ2) =V1δ(Xˆ + m2M xˆ)+ V2δ(Xˆ − m1M xˆ)to Hˆ0. They assumed the wave packet isinitially centred to the left of the potential (X0 < 0), and then it scattersby the potential to be separated into the transmitted and reflected parts(Figure 2.1). They observed that this process makes the CM position degreesof freedom and the internal (relative position) degrees of freedom of thecomposite object get entangled with each other even if the wave packetΨ0(X,x) is in the pure state initially. Since the excitations of the internaldegrees of freedom are sensitive to the energy transfer from the CM motionto the relative motion, their behaviour largely depends on the initial energyof the CM motion ECM and that of the relative motion Erel. Here, we25assumed that one particle is much heavier than the other, i.e., m1  m2(e.g., a particle 1 is a proton and a particle 2 is an electron) and only aparticle m2 interacts with the delta potential, i.e., V1 = 0. Then, when onechooses ECM = 2Erel and the other parameters such that the initial wavepacket can be half transmitted and half reflected, one obtains the reflectedpart has more excited modes than the transmitted part has as shown inFigure 2.2 where the wave packet which is moved far away from the potentialafter the scattering is expressed as Ψ(X,x) =∑m,nCm,ngm(X)φn(x) =∑n un(X)φn(x) with gm(X), φn(x) being eigenstates in the CM and in therelative coordinate respectively and un(X) =∑mCm,ngm(X). Following[60],∫ −−∞ |un(X)|2dX and∫∞ |un(X)|2dX were plotted as probabilities offinding the reflected part and transmitted part having its internal degreesof freedom in the state φn(x) respectively.As we see, the wave packet after the scattering Ψ(X,x) can be roughlywritten asΨ(X,x) ≈ uT0 (X)φ0(x) + uR0 (X)φ0(x) + uR1 (X)φ1(x) + · · ·(2.3)where superscripts T and R represent “transmitted” and “reflected” respec-tively.If we look at the reduced density matrix for the CM position ρCM (X,X′),26we findρCM(X,X′) =∫Ψ(X,x)Ψ∗(X ′, x)dx= (uT0 (X)uT∗0 (X′) + uT0 (X)uR∗0 (X′)+uR0 (X)uT∗0 (X′) + uR0 (X)uR∗0 (X′))∫φ0(x)φ∗0(x)dx+(uT0 (X)uR∗1 (X′) + uR0 (X)uR∗1 (X′))∫φ0(x)φ∗1(x)dx+(uR1 (X)uT∗0 (X′) + uR1 (X)uR∗0 (X′))∫φ1(x)φ∗0(x)dx+uR1 (X)uR∗1 (X′)∫φ1(x)φ∗1(x)dx+ · · ·= uT0 (X)uT∗0 (X′) + uT0 (X)uR∗0 (X′)+uR0 (X)uT∗0 (X′) + uR0 (X)uR∗0 (X′) + uR1 (X)uR∗1 (X′) · · ·(2.4)by using∫φn(x)φ∗m(x)dx = δnm.This indicates that the coherence terms such as uT0 (X)uR∗1 (X′) etc. getsuppressed if one traces out the internal degrees of freedom φn(x). Indeedthe von Neumann entanglement entropy SCM−rel = −tr(ρCM log2 ρCM) thatmeasures the entanglement between the CM position and the relative sepa-ration of two particles increased to 1.3 after a scattering while it was zero be-fore the scattering. However since both of the reflected and the transmittedpart has the nonzero amplitude for having the internal states in the groundstate, we still have the partial coherence terms representing superposition ofthe reflected and the transmitted parts, uT0 (X)uR∗0 (X′)+uR0 (X)uT∗0 (X ′) arepresent in the system. In the case of Figure 2.2, about 24% of coherence re-mains even if the states of the internal (relative position) degrees of freedomare completely measured since about 24% of states are in the ground statein the reflected part, as well as almost all states are in the ground state inthe transmitted part.In the following section, we use this spatial superposition of the compos-ite object to study its decoherence by its own emission of radiation.27Figure 2.2: The wave packet after the scattering by a mirror potential whenV1 is zero and V2 is nonzero where the initial wave packet before the scat-tering is in the form Ψ0(X,x) (above). Probability of finding the internalstates φn in the transmitted and in the reflected part. The initial energy ofCM degrees of freedom is twice larger than than that of the relative positiondegrees of freedom. State number n = 0 corresponds to the ground state,and n = 1 is for the first excited state etc. The similar result was also foundin [60] (below).282.2 Decoherence by Emission of RadiationWhen decoherence is discussed, it is important to clarify the property ofthe interaction between the environment and the system concerned. Theenvironment discussed here is consisted of radiation fields generated by thecomposite object, while we do not have any ambient radiation fields presentinitially. When the composite object has the oscillating dipole moment orexcited internal states, we have radiation emitted from the system, whileradiation is generally not emitted if its internal states are in the groundstate. The similar example was also discussed in [53, 54]. Here the emittedradiation as the environment can monitor whether the internal states of thesystem are in the excited state or in the ground state. Therefore if we havethe composite object with its internal states being in a superposition of thosestates, then radiation fields generated by it give which-way information andcause decoherence on the superposition. This kind of superposition state isalready prepared in the previous section. Here, we study its decoherencedue to emission of radiation using the path integrals formalism.As it was shown in the previous section, since we can roughly knowwhether the wave packet is the reflected part or the transmitted part bylooking at the internal (relative position) degrees of freedom x, we nowfocus on x and simplify the wave packet of the composite object concernedby writingΨ(x) ∼ 1√2φT/R0 (x) +1√2φR2 (x). (2.5)The wave packet describes that the internal degrees of freedom is in asuperposition of the ground state and the second excited state, and the spa-tial superposition state (reflected/transmitted) is partially entangled withit, i.e., φT/R0 (x) = (uR0 + uT0 )φ0(x) and φR2 (x) = uR2 φ2(x). It was seen inFigure 2.2 that the first excited state was also populated in the reflectedpart, and the ground state is more populated than the excited states are,however we use (2.5) to simplify calculations below.29Then the density matrix isρ0(x, x′) = Ψ(x)Ψ∗(x′)=12φT/R0 (x)φT/R∗0 (x′) +12φR2 (x)φR∗2 (x′)+12(φT/R0 (x)φR∗2 (x′) + φR2 (x)φT/R∗0 (x′)).(2.6)Here one knows the wave packet is either the reflected part or the transmittedpart if the ground state φT/R0 (x) is measured, while it is the reflected partwhen the excited state φR2 (x) is measured.Assuming that two particles of the composite object have opposite chargesq and −q, the superposition of the ground state and excited state in the in-ternal degrees of freedom suggests that there exists a superposition of adipole qx or quadrupole moment Qij associated with it. Superposed dipolemoment generates superposed electromagnetic radiation, while superposedquadrupole moment might generate superposed gravitational radiation ifEinstein’s equations are valid even in the quantum regime. Therefore, radi-ation is entangled with the internal states of the object (Figure 2.3):Ψ(x)Φ(a)interaction−−−−−−→ 1√2φT/R0 (x)Φ0(a) +1√2φR2 (x)Φ2(a)(2.7)where Φn(a) represents radiation fields associated with φn(x).We now derive the influence functional from radiation fields for the re-duced density matrix propagator of the internal degrees of freedom. Here wework in the Coulomb gauge, ∇ ·A = 0. The action for the internal degreesof freedom interacting with radiation fields can be written asS = S0 + Sint + Srad (2.8)30Figure 2.3: One can partially tell whether it is the transmitted part or thereflected part by looking at which-way information from electromagnetic orgravitational radiation.whereS0 =∫ (µ2x˙2 − 12µω2x2)dt,Sint =∫∫d3rdtj(r, t) ·A(r, t)= −q∫ ∑kx˙ · a(k)eik·x(t)dt= −q∫ ∑kx˙λaλ(k)eik·x(t)dt,Srad = −14∫d4rFµνFµν=12∫ ∑k(a˙∗(k) · a˙(k)− k2a∗(k) · a(k))dt=12∫ ∑k(a˙λ∗(k)a˙λ(k)− k2aλ∗(k)aλ(k))dt (2.9)31in the case of electromagnetic radiation fields. Note that Coulomb inter-actions and the other interactions between two particles are assumed to beincluded in the effective potential modelled as a harmonic potential for sim-plicity. Here λ = 1, 2 represents polarizations, j(r, t) = −qx˙(t)δ3(r − x(t))and xλ are the components of x in the direction transverse to k [21]. In theprevious section, we defined x as the relative distance between two particlesx = x2 − x1. Here we use the assumption that x˙2  x˙1 (with m1  m2)and x˙ ≈ x˙2, i.e., the term qx˙1 is negligible compared to −qx˙2 ≈ −qx˙.When two particles do not have charges, however they are still coupledto the gravitational radiation fields by their energy-momentum tensor T ijinstead, Sint and Srad can be replaced by [61]Sint =∫d4rhij(r, t)Tij(r, t),Srad =14piG∫d4rhµν,αhµν,α=14piG∫d4r(∂thij∂thij − hij,khij,k) (2.10)in the weak gravity regime.As in the case of the electromagnetic radiation, we expand the transverse-traceless parts hij of the gravitational fields and Tij into plane waves,hij(r, t) =∑kh(λ)ij (k)eik·r, T ij(r, t) =∑kT ij(λ)(k, t)eik·r(2.11)where h(λ)ij (k) = (λ)ij (k)s(λ)k (t) with the transverse-traceless polarization ten-sors (λ)ij (k).Note that here we take the general assumption of quantum mechanicsthat we have an external observer who is in the source-free region and per-forms a measurement from the outside of the system. With this assumption,the transverse-traceless gauge, i.e., h0µ = hij,i = hii = 0, can be introducedas in (2.10). In general relativity, however, an observer is normally locatedinside of the system and this fact causes problems as explained in [48, 49],32which will be briefly discussed in the next section.Here we follow the quantum mechanical view point and write the time-evolution of the density matrix asρ(x, x′, aλ, aλ′; t)=∫dx0dx′0daλ0daλ′0 Ψ(x0)Φ(aλ0)Ψ∗(x′0)Φ∗(aλ′0 )×W (x, x′, aλ, aλ′ , t|x0, x′0, aλ0 , aλ′0 , 0) (2.12)where aλ → h(λ)ij for gravitational radiation and the density matrix propa-gator W isW = K(x, aλ|x0, aλ0)K∗(x′, aλ′ |x′0, aλ′0 ),K =∫∫exp i(S0 + Sint + Srad)Dx∏λ=1,2Daλ. (2.13)Then the reduced density matrix ρred(x, x′; t) for x can be written asρred(x, x′; t) =∫ρ(x, x′, aλ, aλ; t)daλ=∫dx0dx′0∫ xx0∫ x′x′0DxDx′ei(S[x]−S[x′])F(j, j′)ρ0(x0, x′0) (2.14)where j and j′ represent current associated with x and x′ respectively.Since Srad + Sint gives the action for the forced harmonic oscillator, wecan calculate the influence functional F as (Appendix B)F(j, j′) =∫∫∫ ∏λ=1,2daλdaλ0daλ′0 Krad(aλ, aλ0)K∗rad(aλ, aλ′0 )Φ(aλ0)Φ∗(aλ′0 )= exp Θphoton (2.15)33whereΘphoton = iq2∫ t0∫ s0dsds′(x˙λ(s)eik·x(s) − x˙′λ(s)eik·x′(s))×γ(s,s′)λρ (x˙ρ(s′)e−ik·x(s′) + x˙′ρ(s′)e−ik·x′(s′))−q2∫ t0∫ s0dsds′(x˙λ(s)eik·x(s) − x˙′λ(s)eik·x′(s))×η(s,s′)λρ (x˙ρ(s′)e−ik·x(s′) − x˙′ρ(s′)e−ik·x′(s′)) (2.16)andKrad =∫exp i(Sint + Srad)∏λ=1,2Daλ,Φ(aλ0) = exp[−∑kk2aλ∗0 (−k)aλ0(k)](2.17)assuming there are no ambient radiation fields except those emitted by thecomposite object concerned. The correlation functions are given by summingover photon polarizations,γ(s,s′)λρ = Pλρ∑k12ksin k(s− s′),η(s,s′)λρ = Pλρ∑k12kcos k(s− s′) (2.18)with Pλρ = δλρ − kλkρk2 . The similar expression of the influence functionalwas also obtained in [53, 54].In the case of gravitational radiation fields, the influence functional canbe obtained in the similar way [52, 56] asF(T ij , T ij′) = exp Θgrav (2.19)34whereΘgrav= i∫ t0∫ s0dsds′(T ij(k, s)− T ij′(k, s))γ(s,s′)ij,kl (T kl(−k, s′) + T kl′(−k, s′))−∫ t0∫ s0dsds′(T ij(k, s)− T ij′(k, s))η(s,s′)ij,kl (T kl(−k, s′)− T kl′(−k, s′))(2.20)with the correlation functionsγ(s,s′)ij,kl = 2piG∑kΠij,klksin k(s− s′),η(s,s′)λρ = 2piG∑kΠij,klkcos k(s− s′). (2.21)The sum over polarizations of two on-shell graviton states gives the po-larization tensor2∑λλij(k)λ∗kl (k) = Πij,kl =12(PikPjl+PilPjk−PijPkl) wherethe projection operators are Pij = δij − kikj|k|2 .Here T ij and T ij′are associated with x and x′ respectively. If we intro-duce the energy-momentum tensor,T ij(k, s) = µx˙i(s)x˙j(s)eik·x(s) (2.22)35we obtain the expressions similar to (2.15) and that obtained in [52]:F(T ij , T ij′) = exp Θgrav= exp[iµ2∫ t0∫ s0dsds′(x˙i(s)x˙j(s)eik·x(s) − x˙′i(s)x˙′j(s)eik·x′(s))×γ(s,s′)ij,kl (x˙k(s′)x˙l(s′)e−ik·x(s′) + x˙′k(s′)x˙′l(s′)e−ik·x′(s′))−µ2∫ t0∫ s0dsds′(x˙i(s)x˙j(s)eik·x(s) − x˙′i(s)x˙′j(s)eik·x′(s))×η(s,s′)ij,kl (x˙k(s′)x˙l(s′)e−ik·x(s′) − x˙′k(s′)x˙′l(s′)e−ik·x′(s′))].(2.23)We now evaluate decoherence of the internal states (and spatial superpo-sition of the transmitted part and the reflected part associated with them) byradiation fields. We look at the reduced density matrix ρred(x, x′; t) (2.14).The imaginary and real parts of the phase Θphoton/grav in the influence func-tional (2.15), (2.23) lead to dissipation and decoherence respectively. Sincethe internal degrees of freedom follows motion of the harmonic oscillatorS0 (2.9), its classical path xc(s) with the boundary condition x(0) = x0,x(t) = x can be written asxc(s) = x0sinω(t− s)sinωt+ xsinωssinωt. (2.24)For simplicity, we assume that the oscillator moves in one-dimension.36Ignoring damping, we substitute (2.24) into (2.15) and (2.23) to obtainIm(Θphoton)=q2ω4pi2∫ t0∫ s0dsds′(x˙c(s)− x˙′c(s)) sinω(s− s′)(x˙c(s′) + x˙′c(s′))=αω8pi sinωt(x′(x0 − 2ω2t2x0 + 3x′0)− x(x′0 − 2ω2t2x′0 + 3x0)+ cos 2ωt(x0 + x′0)(x− x′) +2ωttanωt(xx0 − x′x′0)+(x2 + x20 − x′20 − x′2)(cosωt− ωt cos 2ωtsinωt))t>1/ω−→ αω2t8pi sinωt(2ωt(xx′0 − x0x′) + 2(xx0 − x0x′)/ tanωt+(x′2 + x′20 − x2 − x20) cos 2ωt/ sinωt), (2.25)andRe(Θphoton)= −q2ω4pi2∫ t0∫ s0dsds′(x˙c(s)− x˙′c(s)) cosω(s− s′)(x˙c(s′)− x˙′c(s′))= − αω16pi sin2 ωt×((1 + 2ω2t2 − cos 2ωt+ 2ωt sin 2ωt)((x− x′)2 + (x0 − x′0)2)+(x− x′)(x0 − x′0)(cos 3ωt− 8ωt sinωt− (1 + 4ω2t2) cosωt))t>1/ω−→ − αω3t28pi sin2 ωt((x− x′)2 + (x0 − x′0)2−2(x− x′)(x0 − x′0) cosωt)(2.26)for electromagnetic radiation.37For gravitational radiation, we haveIm(Θgrav)= −Gµ2ω2pi∫ t0∫ s0dsds′(x˙c(s)x˙c(s)− x˙′c(s)x˙′c(s))× sinω(s− s′)(x˙c(s′)x˙c(s′) + x˙′c(s′)x˙′c(s′))= − Gµ2ω3576pi sin4 ωt(60ωt(x′4 + 4x′20 x′2 + x′40 − x4 − 4x2x20 − x40)+240ωt(x3x0 + xx30 − x′0x′(x′20 + x′2)) cosωt+120ωt(x′20 x′2 − x2x20) cos 2ωt+4(18x4 + 27x3x0 + 18x40 − 18x′40 + x2(41x20 + x′0(25x′0 − 4x′))+x20x′(4x′0 − 25x′) + xx0(27x20 − 4x′20 + 4x′2)−x′(27x′30 + 41x′20 x′ + 27x′0x′2 + 18x′3))sinωt−4(6x4 + 40x3x0 + 6x40 − 6x′40 − 8x20x′(x′0 + 2x′)+8xx0(5x20 + x′20 − x′2)− x′(40x′30 + 27x′20 x′ + 40x′0x′2 + 6x′3)+x2(27x20 + 8x′0(2x′0 + x′)))sin 2ωt+2(4x4 + 6x3x0 + 4x40 − 4x′40 − 3x20x′(8x′0 + x′)−x′(6x′30 + 67x′20 x′ + 6x′0x′2 + 4x′3)+6xx0(x20 + 4(x′0 − x′)(x′0 + x′)) + x2(67x20 + 3x′0(x′0 + 8x′)))sin 3ωt+(3x4 − 16x3x0 + 3x40 − 3x′40 − 16x2x′0x′ + 16x′0x′(x20 + x′20 )+16x′0x′3 − 3x′4 − 16xx0(x20 + x′20 − x′2))sin 4ωt+2(x20 + x′20 )(x− x′)(x+ x′) sin 5ωt)t>1/ω−→ 5Gµ2ω3t48pi sin4 ωt(x4 + 4x2x20 + x40 − x′40 − 4x′20 x′2 − x′4+4(x′0x′3 + x′30 x′ − xx0(x2 + x20)) cosωt+2(xx0 − x′0x′)(xx0 + x′0x′) cos 2ωt), (2.27)38andRe(Θgrav)= −Gµ2ω2pi∫ t0∫ s0dsds′(x˙c(s)x˙c(s)− x˙′c(s)x˙′c(s))× cosω(s− s′)(x˙c(s′)x˙c(s′)− x˙′c(s′)x˙′c(s′))= −Gµ2ω3144pi(4(8xx0(x20 − x′20 − x′2) + 2x2x′(3x′ − 4x′0)−3x′4 + 8x′0x3 + 8x′0x′(x′20 − x20)− 3(x20 − x′20 )2 + 8x3x0 − 3x4)−8(x0 − x′0)(x0 + x′0)(x− x′)(x+ x′) cosωt+9(x− x0 + x′0 − x′)2(x− x0 − x′0 + x′)21sin2 ωt2+(5x4 − 12x3x0 + 5x40 + 2x2(3x′20 + 6x′0x′ − 17x20 − 5x′2)+2x20(3x′2 − 5x′20 + 6x0x′)+4xx0(3x′2 + 3x′20 − 3x20 + 14x′0x′)+(x′0 + x′)2(5x′20 − 22x′0x′ + 5x′2)) 1cos2 ωt/2+(x+ x0 − x′0 − x′)2(x+ x0 + x′0 + x′)21cos4 ωt/2)t>1/ω−→ −Gµ2ω3144pi(4(8xx0(x20 − x′20 − x′2) + 2x2x′(3x′ − 4x′0)−3x′4 + 8x′0x3 + 8x′0x′(x′20 − x20)− 3(x20 − x′20 )2 + 8x3x0 − 3x4)+9(x− x0 + x′0 − x′)2(x− x0 − x′0 + x′)21sin2 ωt2+(x+ x0 − x′0 − x′)2(x+ x0 + x′0 + x′)21cos4 ωt2)(2.28)where α = q24pi ≈ 1/137 is the fine-structure constant. Here we used thefact that the frequency of the emitted electromagnetic or gravitational waveis similar to the natural frequency ω of the oscillator, in contrast to thesituation where one needs to introduce a cutoff frequency of the ambientenvironmental bath. We also used k is mostly perpendicular to x, i.e., the39wave vector of the emitted radiation is mostly perpendicular to a dipole orquadrupole moment, and assumed t > 1/ω to simplify the expressions.The averaged power emitted in the form of the electromagnetic radiationby the classical path xc is given by the Larmor’s formula,Pphoton =〈23αx¨c2〉=αω3(4xx0 sinωt− 4xx0ωt cosωt− (x2 + x20)(sin 2ωt− 2ωt))6t sin2 ωtt>1/ω−→ αω4(x2 + x20 − 2xx0 cosωt)3 sin2 ωt(2.29)and the averaged power emitted in the form of the gravitational radiation isPgrav =G45〈...Qij...Qij〉=12Gµ2ω545t sin4 ωt(4ωtx4 + 16ωtx2x20 + 4ωtx40−16ωtxx0(x2 + x20) cosωt+ 8ωtx2x20 cos 2ωt+4x3x0 sinωt+ 4xx30 sinωt− 12x2x20 sin 2ωt+4x3x0 sin 3ωt+ 4xx30 sin 3ωt−x4 sin 4ωt− x40 sin 4ωt)t>1/ω−→ 48Gµ2ω645 sin4 ωt(x2 + x20 − 2xx0 cosωt)2 (2.30)where [62]Qij =∫d3xρ(3xixj − r2δij),ρ = µδ(x− xc)δ(y)δ(z), r2 = x2c ,Q11 = 2µx2c , Q22 = Q33 = −µx2c . (2.31)We find similarities between (2.29, 2.30) and (2.25, 2.26, 2.27, 2.28).Indeed Θphoton/grav → 0 and Pphoton/grav → 0 as ω → 0, i.e., if there are40no radiations emitted, we do not have decoherence.We now evaluateρred(x, x′; t) =∫dx0dx′0K0(x, x0)K∗0 (x′, x′0)eΘphoton/gravρ0(x0, x′0)(2.32)where ρ0(x0, x′0) is given in (2.6).We plotp0(n1, n2) =∫∫dx0dx′0φ∗n1(x0)ρ0(x0, x′0)φn2(x′0) (2.33)in Figure 2.4. The diagonal elements p0(n1, n2 = n1) represents probabili-ties of finding the wave-function in the state n1 = n2 where n1 = n2 = 0,n1 = n2 = 2 correspond to the ground state and the second excited staterespectively. The off-diagonal elements p0(n1, n2 6= n1) represents a quan-tum superposition of state n1 and state n2. For example, n1 = 0, n2 = 2corresponds to a superposition of the ground and the second excited state.Since S0 is the action for harmonic oscillator, we have [21]K0(x, x0)=( µω2pii sinωt)1/2exp(iµω2 sinωt[(x2 + x20) cosωt− 2xx0]).(2.34)To simplify the calculations, we replace Θphoton/grav by Re(Θphoton/grav)so that we focus on the decoherence effects caused by radiation emission.We plot the absolute value of pt(n1, n2) =∫∫dxdx′φ∗n1(x)ρred(x, x′; t)φn2(x′)numerically in Figure 2.4 for the case of Θphoton. It is observed that after t =5.3(2pi/ω), we see that the off-diagonal elements of pt(n1, n2) representingsuperposition of the ground state and excited state are largely suppressed.We see that the von Neumann entropy of the entanglement between theinternal degrees of freedom and radiation fields increased from Sint−rad =−tr(pt log2 pt) = 0 (at t = 0) to 1.7 (at t = 5.3(2pi/ω)). Since the CMposition degrees of freedom of the composite object is also entangled with41Figure 2.4: The absolute value of p0(n1, n2) has off-diagonal elements rep-resenting superposition of the ground and excited state (above). After de-coherence due to radiations, those superposition states get suppressed inpt(n1, n2) (below).42the internal degrees of freedom that are entangled with radiation fields,superposition of the reflected part and transmitted part of the wave packetshould be partially suppressed as well due to emission of radiation, i.e., ifradiation is emitted, we know that the wave packet is the reflected part. Itis also shown that the second excited state is less populated and the firstexcited state gets populated instead after time t. The effect of Re(Θgrav) issimilar, although it is much weaker than that of Re(Θphoton) by a factor ofGµ2/α even if it exists.2.3 Decoherence by Quantized Static FieldsSo far we discussed decoherence caused by the superposed radiation fieldsgenerated by the superposed four-current or energy-momentum tensor ofa composite object. The decoherence is caused by on-shell particles rep-resenting the electromagnetic waves or gravitational waves that propagateaccording to their own dynamical equations of motions. On the other hand,off-shell photons generally can only appear as an interaction between par-ticles. Note that the situation can become more complicated in the case ofgravity where spacetime becomes dynamical. Off-shell particles often ap-pear in the action of the particles, and they were assumed to be includedin the effective potential modelled as a harmonic potential in S0 (2.9) forsimplicity in the previous section.Of course, without the simplification, Hamiltonian of two particles in-teracting with each other by a Coulomb interaction isHˆ =pˆ212m1+pˆ222m2+q24pi0|xˆ1 − xˆ2| (2.35)where the third term is the contribution from the off-shell photons.Normally the Hamiltonian of two particles interacting via the potentialdepending only on their relative distance is separable in the relative andCM coordinates, therefore its eigenstates are also separated in those twocoordinates, and the entanglement between them does not change when thesystem evolves according to Hamiltonian (2.35). However since (2.35) is not43separable in (x1, x2)-coordinates, its eigenstates generally have entangledcoordinate x1 and x2, and the time-evolution according to the Hamiltoniancan change the entanglement between those two coordinates. In other words,off-shell photons make x1 and x2 entangled with each other, unlike on-shellphotons that monitor the internal excitations of the composite object in theprevious section.For example, let us consider we have an electron in a spatial superpo-sition state of two locations, x1 = d > 0 and x1 = −d. We may write itswave-function asψ(x1) = ψ1(x1) + ψ2(x1)=1√21(2piσ21)1/4(e− 14σ21(x1−d)2+ e− 14σ21(x1+d)2)(2.36)where σ1 is the width of Gaussian distribution. We assume that the twoGaussians have essentially no overlap.Then we have the superposed static Coulomb field generated by thesuperposed charge density of an electron. Therefore, if we could measurethe static field, one can tell whether an electron is at position d or −d.However, since off-shell photons can only appear in terms of the interactionbetween particles, we put a particle 2 with a negative charge at the originin order to measure the Coulomb field:φ(x2) =1(2piσ22)1/4e− 14σ22x22. (2.37)Even if we start with the pure state, ψ(x1)φ(x2), the time-evolution bythe Hamiltonian (2.35) makes x1 and x2 get entangled after some time, i.e.,if a particle 2 is repelled in the negative direction, we know a particle 1 is atd and if a particle 2 is repelled in the positive direction, we know a particle441 is at −d respectively:ψ(x1)φ(x2)interaction−−−−−−→ ψ1(x1)φ1(x2) + ψ2(x1)φ2(x2).(2.38)Then if we trace out the position x2 of a particle 2, we obtain the decoher-ence of the spatial superposition of a particle 1. Note that the decoherence ismore easily controlled and sometimes coherence can be restored, in contrastto the decoherence due to radiation fields discussed in the previous section,where the environment was a large bath of radiation fields, while here theexternal object traced out is only a single particle x2. For example, if onemakes the mass of a particle 2 heavier or the width σ2 of φ(x2) larger so thatthe particle becomes insensitive to the Coulomb field of a particle 1, thenthe change in the entanglement between x1 and x2 becomes much smallergenerally. Moreover if we attach a particle 2 to a harmonic oscillator:Hˆ ′ = Hˆ +12cx22 (2.39)so that it tries to pull a particle 2 back toward the origin after some time,we see that the von Neumann entanglement entropy which measures theentanglement between x1 and x2 decreases and coherence can be restoredagain after some time (Figure 2.5). In the figure, we chose σ1 = σ2, d/σ1 = 5,m1 = m2,cσ1q2/(4pi0σ21)= 1/200, and looked at the time-evolution of theentropy of the entanglement between x1 and x2 when the system evolvesaccording to the Hamiltonian Hˆ (2.35) and Hˆ ′ (2.39) respectively with theinitial wave-function ψ(x1)φ(x2) being in the pure state. In both cases, theentropy increases initially, however when a particle 2 gets far away from theorigin, the spring tries to pull it back and the entropy decreases in the caseof Hˆ ′. We see that the interference pattern produced by the reduced densitymatrix in x1 when two packets overlap maximally is largely suppressed whenthe system evolves according to Hˆ, whereas the clear interference patterncan be produced when the system evolves according to Hˆ ′ (Figure 2.6). Ofcourse, if we have many particles interact with a particle 1 via the Coulomb45Figure 2.5: The time-evolution of the von Neumann entanglement entropywhich measures the entanglement between x1 and x2 according to Hˆ andHˆ ′.interaction and then fly away like radiation, it becomes equally difficult torestore the coherence again.The discussions above would be valid even if the Coulomb field is re-placed by the static Newtonian gravitational field, as long as it is assumedthat we have an external observer, which is the general assumption in theconventional quantum mechanics. On the other hand, the observer is con-sidered to be located inside of the system in general relativity, and is affectedby spacetime structure made by the system concerned and himself and hisbackreaction on it. It was considered in [48, 49] that the superposition ofmatter causes the superposition of time of the internal observer which maylead to gravitational decoherence as the quantum gravity effect. Here itseems that gravity makes not only spatial coordinates but also space andtime get entangled with each other (Figure 2.7). However it is still an openproblem how the quantum mechanics of an internal observer [63, 64] orquantum reference frame [65, 66] can be related to those studies of gravi-tational decoherence, what is the Schro¨dinger equations with the entangled46Figure 2.6: Interference pattern produced by the reduced density matrix inx1 at the moment of maximal overlap. The interference is largely suppressedwhen the system evolves according to Hˆ, whereas the clear interferencepattern can be produced when the system evolves according to Hˆ ′.space and time, and the relations between observables of an internal observerand those of an external observer or the relations between the Schro¨dingerequation in ADM formalism derived by one coordinate/gauge fixing and bythe other coordinate/gauge fixings in the strong gravity regime [50] are stillunclear. Those questions are beyond the scope of this chapter, and we per-formed all calculations of Section 2.2 with the assumption that we have anexternal observer who is in the source-free region, in flat spacetime, in theweak-gravity regime.2.4 ConclusionWe have studied decoherence of superposed internal degrees of freedom andspatial superposition of a composite object entangled with them due to itsown emission of radiation. We used the fact that the reflected part hasmore excited internal modes than the transmitted part has when the spa-tial superposition of a composite object is made by a half-silvered mirror.It is possible to study the time-evolution of the reduced density matrix ofthe internal degrees of freedom using the density matrix propagator withthe influence functional from radiation fields. The influence functional canbe evaluated for the electromagnetic radiation and for the gravitational ra-47Figure 2.7: External and internal observer with superposed gravity andspacetime generated by the superposed matter. We have an external ob-server in quantum mechanics, while measurements are made from the insideof the system by an internal observer in general relativity.48diation assuming that Einstein’s equations are valid even in the quantumregime and we have an external observer. We found that a superpositionof the ground and excited state of the internal degrees of freedom can besuppressed by emission of radiation. Since there exists the entanglementbetween the position and the internal degrees of freedom of the compositeobject, it is also possible that radiation emission can cause the partial de-coherence on the spatial superposition (i.e., reflected or transmitted) of thecomposite object. On-shell particles representing electromagnetic or grav-itational waves are responsible for the decoherence. On the other hand,off-shell photons can only appear as the interactions between the chargesvia the static Coulomb field and play a different role in decoherence. Thestatic field makes positions of particles entangled and causes decoherence ofa spatial superposition of one particle when the external particles interactedby the particle concerned due to the field are traced out. However, it is easyto recover coherence in some cases where the number of the external parti-cles involved is a few, compared to decoherence via radiation fields where itis much more difficult to restore coherence again from a large bath of radi-ation. Note that decoherence discussed above is not directly related to thegravitational decoherence discussed in [48, 49]. Although it is expected thatthe static Newtonian gravitational field acts in a similar way as the staticCoulomb field does, the situation can become more complicated in the caseof gravity, as it seems that it may make spatial and time coordinates getentangled, and we have an internal observer instead in general relativity. Wedid not address this problem here, and evaluated everything using the as-sumption that we have an external observer who is in the source-free region,in the weak-gravity regime.49Chapter 3Scattering of a CompositeQuasiparticle by an Impurityon a LatticeIn the previous chapter, we studied the spatial superposition of a diatomicmolecule type composite object and its decoherence due to emission of ra-diation in continuum space. We prepared the initial spatial superpositionstate of the composite object using the mirror potential. However, realizingan interaction between a diatomic molecule and a semi-transparent infinites-imally thin mirror is technically demanding in continuum space. Therefore,in this chapter, we study its lattice analog in molecular crystals, where animpurity acts as a delta potential, which scatters collective many-atom exci-tations – Frenkel excitons [67]. Scattering of a single exciton by an impurityhas been studied in literature [68]. For a two-exciton bound state (biexciton)the problem is more complicated, as recent numerical studies have shown[69]. Scattering of composite objects in the lattice configuration can be stud-ied also with the help of cold optical systems. Major success in trappingultracold atoms [70] and molecules [71] in optical lattices allows for creationof controllable periodic ensembles in many ways similar to natural crystals,which support rotational Frenkel excitons [72, 73]. The exciton-exciton in-teractions can be controlled by applying external electric field, and undercertain conditions a biexciton is formed [74]. Perturbing the ideal transla-tional invariance of the lattice ensemble by replacing one of the moleculesby a molecule of a different kind simulates an impurity in a natural crystal[72].50Here we study scattering of a Frenkel biexciton by an impurity in aone-dimensional (1D) lattice. In our model the on-site interaction for exci-tations is not a free parameter but corresponds to an infinite repulsion, asone molecule cannot be excited twice. Some of the continuum models andtheir results can be obtained from the corresponding lattice model by takinga limiting procedure (e.g., the lattice constant a→ 0, the number of latticesites N →∞) with the excitation hopping strength J ∝ 1/a2 (Section 3.5).However, although many continuum models of two particles interacting viathe potential depending only on their relative distance are separable in therelative and CM coordinates, it is often the case that the correspondinglattice Hamiltonian and its eigenstates are no longer separable in those co-ordinates due to discreteness (Section 3.1). As a result, the entanglementbetween relative and CM coordinate degrees of freedom can naturally arisefor a composite quasiparticle on a lattice, and indeed we will observe thatthe width of a biexciton wave-function in the relative coordinate dependson the CM wave vector K. Moreover, the lattice models have energy bandswhich are bound both from above and below, that allows the creation ofbound states with both attractive and repulsive interactions (Section 3.2).In the continuum models the energy is bound from below but has no upperbound, and the bound states are only associated with attractive interactions.An important objective of this chapter is to investigate how inseparabilityof relative and CM coordinates and finiteness of the energy band of a com-posite quasiparticle on a lattice affects creation of the biexciton-impuritybound states and the internal entanglement dynamics.The chapter is organized as follows: In Section 3.1, we derive biexcitonstates analytically in an ideal 1D lattice with periodic boundary conditions.We use them as a basis for the following discussion. In Section 3.2 we studyand contrast the eigenstates of an exciton and a biexciton in a 1D latticewith an impurity. We find that the free-space intuition can not be directlyapplied to a lattice setup: In particular, binding between the impurity andlattice excitations occurs at both signs of the impurity potential. For anexciton the exact solution is reported. For a biexciton we show numericallythat the extra (relative) degree of freedom results in formation of multiple51biexciton-impurity bound states, – in contrast to one exciton, which alwayshas one bound state near a delta-like potential. The number of boundstates and the degree of their localization are determined by the signs andrelative values of the exciton-exciton and biexciton-impurity interactions.The bound states are also studied analytically by looking at the poles ofthe scattering amplitude for exciton and biexciton. Furthermore, we reportthat our model with the impurity can be approximately solved and thereexist bound states in the continuum [75] in which two excitons are mutuallybound and bound to the impurity and the energies of the states are locatedin the continuum band. In Section 3.3, we study scattering of a biexcitonwave packet by an impurity, and a change in the entanglement between itsrelative and CM coordinate degrees of freedom. In Section 3.4 we presentour conclusions and discuss further applications of the obtained results.3.1 Biexciton StatesWe consider a 1D lattice of molecules or any other two-level objects withperiodic boundary conditions, with the lattice constant equal to 1, andstudy the excitation transfer between the molecules. The Hamiltonian inthe nearest-neighbour approximation isHˆ0 =N/2∑n=−N/2+1(E0aˆ†naˆn + J(aˆ†n+1aˆn + aˆ†n−1aˆn)+Daˆ†n+1aˆn+1aˆ†naˆn + Laˆ†naˆnaˆ†naˆn) (3.1)where n labels the sites of a 1D lattice, N is the total number of lattice sites,and n + N is taken to be just a different label for the site n for arbitraryn. While N could be arbitrary, we take N to be even. The analysis forodd N is possible, but more complicated than for even N . Operators aˆ†n,aˆn describe excitation and de-excitation of n-th molecule, J describes theexcitation hopping strength between molecules in sites n and n ± 1, whileD denotes a two-particle interaction strength between the excitations, andE0 is the one-particle excitation energy. L→∞ accounts for the hard-core52constraint, i.e., to the fact that one molecule can accommodate at mostone excitation. The number operator Nˆ = ∑n aˆ†naˆn commutes with theHamiltonian and the number of the excitations is conserved.A single exciton [67] can be represented by the eigenstates of the firsttwo terms of Hamiltonian (3.1) when the total number of excitations inthe lattice is one. Here we consider Hamiltonian acting on two-excitonsubspace, then the basis can be written as |m,n〉 = aˆ†maˆ†n|0〉 ∈ Hm ⊗ Hn,meaning that m-th and n-th sites are excited. Then |m,n〉 ≡ |m,n + N〉,|m,n〉 ≡ |m+N,n〉. In addition we have indistinguishability of excitations|n,m〉 ≡ |m,n〉 and |n, n〉 does not exist by the hard-core constraint. Then,in the nearest-neighbour approximation (3.1) can be written asHˆ0 = 2E0Iˆm ⊗ Iˆn + JN/2∑m=−N/2+1(|m+ 1〉〈m|+ |m− 1〉〈m|)⊗ Iˆn+JIˆm ⊗N/2∑n=−N/2+1(|n+ 1〉〈n|+ |n− 1〉〈n|)+N/2∑m,n 6=mD(δ(m− n+ 1) + δ(m− n− 1))|m,n〉〈m,n|+L∑m,nδ(m− n)|m,n〉〈m,n|. (3.2)We now define r = m + n (r/2 is the centre of mass coordinate) andrelative coordinate of two excitations s = m− n on a lattice. The variablesr and s are not independent: they must be both even or odd. This indicatesr + s should be even for physical states. However we extend the space ofstates to all r and s for simplicity, i.e., Hilbert space Hm⊗Hn is a subspaceof Hr ⊗ Hs. If we take −N/2 + 1 ≤ m,n ≤ N/2, then −N + 1 ≤ r ≤ Nand −N + 1 ≤ s ≤ N . We define |r, s〉 ∈ Hr ⊗ Hs and introduce theunitary transformations Rˆ =∑r,s |r+ 1, s〉〈r, s| and Sˆ =∑r,s |r, s+ 1〉〈r, s|.The symmetries of m,n translate to |r, s〉 ≡ |r + 2N, s〉 ≡ |r, s + 2N〉 ≡|r+N, s+N〉 for arbitrary r, s. The constraint that |n,m〉 ≡ |m,n〉 becomesthat |r, s〉 ≡ |r,−s〉 so everything can be taken as defined only for positive53s. Now, a hopping m → m + 1 gives r → r + 1, s → s + 1. A hoppingm→ m−1 gives r → r−1, s→ s−1. A hopping n→ n+1 gives r → r+1,s→ s− 1. A hopping n→ n− 1 gives r → r − 1, s→ s+ 1, therefore∑m|m+ 1〉〈m| ⊗ Iˆn =∑r,s(|r + 1, s〉〈r, s|)(|r, s+ 1〉〈r, s|) = RˆSˆ,∑m|m− 1〉〈m| ⊗ Iˆn =∑r,s(|r − 1, s〉〈r, s|)(|r, s− 1〉〈r, s|) = Rˆ†Sˆ†,Iˆm ⊗∑n|n+ 1〉〈n| =∑r,s(|r + 1, s〉〈r, s|)(|r, s− 1〉〈r, s|) = RˆSˆ†,Iˆm ⊗∑n|n− 1〉〈n| =∑r,s(|r − 1, s〉〈r, s|)(|r, s+ 1〉〈r, s|) = Rˆ†Sˆ.(3.3)Then, (3.1) acting on two-exciton subspace is equivalent toHˆ0 = 2E0∑r,s|r, s〉〈r, s|+ J(Rˆ+ Rˆ†)(Sˆ + Sˆ†)+D∑r,sδ(|s| − 1)|r, s〉〈r, s|+ L∑r,sδ(s)|r, s〉〈r, s|. (3.4)Note that the Hamiltonian only couples even r+s states with each other,and odd r+s states with each other. So one can always project the solutionsback onto the |m,n〉 set of states.We consider the wave-function Φ(r, s) such that the state is|Φ〉 =∑r,sΦ(r, s)|r, s〉 (3.5)and Φ(r, s = 0) = 0 arises from the limit of finite energy when L→∞.The periodic boundary conditions and indistinguishability of excitations54impose the following symmetry requirements:1 (Figure 3.1)Φ(r, s) ≡ Φ(r + 2N, s) ≡ Φ(r, s+ 2N)≡ Φ(r +N, s±N) ≡ Φ(r,−s).(3.6)The Hamiltonian commutes with Rˆ and we can simultaneously diagonal-ize Rˆ and Hˆ0. The eigenvalues of Rˆ must be pure phases. Since Rˆ2N = Rˆ,they can be written as eiK with the eigenstatesΦ(r, s) = eiKrφK(s), K =2pilK2N, (3.7)where lK ∈ [−N + 1, N ] is an integer. The last two symmetry requirementsin (3.6) indicateφK(s) = (−1)lKφK(N − |s|) (3.8)which means that the relative coordinate wave-function φK(s) is even orodd about s = ±N/2 (as we assume N is even, N/2 is an integer) accordingto the parity of lK :φK(N/2 + |s|) = (−1)lKφK(N/2− |s|). (3.9)Projecting the eigenvalue equation Hˆ0|Φ〉 = E|Φ〉 onto 〈r, s| away froms = 0,±1, we get2J cosK(φK(s− 1) + φK(s+ 1)) = (E − 2E0)φK(s)(3.10)1We have (2N)2 states inHr⊗Hs. Φ(r+N, s±N) ≡ Φ(r, s) divides it by 2, eliminatings = 0 subtracts 2N , Φ(r, s) ≡ Φ(r,−s) divides it by 2 and even r+ s divides it by 2. Thisprocess gives N(N − 1)/2 states, which is the number of states in Hm ⊗Hn.55Figure 3.1: The diagram of the lattice with N = 4. The red points are allequivalent. Note that r, s+4 is not the same as r, s, but r, s+8 is. r+4, s+4is the same as r, s.56which yieldsφK(s) =cos k(N/2− |s|), if lK is even,sin k(N/2− |s|), if lK is odd (3.11)up to normalization. The corresponding energy eigenvalue is given byE = 2E0 + 4J cosK cos k. (3.12)The eigenvalue equation at s = 1 gives2J cosKφK(2) +DφK(1) = 4J cosK cos kφK(1). (3.13)Then substituting (3.11) into (3.13), we haveD2J cosK=cos kN/2cos k(N/2− 1) , if lK is even,D2J cosK=sin kN/2sin k(N/2− 1) , if lK is odd. (3.14)In this chapter, we are interested in the bound two-exciton complex,biexciton [74, 76, 77], which appears as a result of the exciton-exciton in-teractions given by the D and by the L terms in the Hamiltonian. Forbiexciton, the wave-function of the relative coordinate decays exponentiallywith the growth of separation and k is complex, i.e., k = kr + iki. As thetwo-particle energies are proportional to cos k, we conclude that kr = 0 orpi to keep the biexciton energy real. Now a transformation K → K ± pi andk → k ± pi giveseiKr cos k(N/2− |s|)→ ei(K±pi)r cos(k ± pi)(N/2− |s|)= (−1)reiKr cos(k(N/2− |s|)± pi(N/2− |s|))= (−1)reiKr(cos k(N/2− |s|) cospi(N/2− |s|)∓ sin k(N/2− |s|) sinpi(N/2− |s|))= (−1)r+seiKr cos k(N/2− |s|) (3.15)57when lK and N are even.eiKr cos k(N/2− |s|)→ ei(K±pi)r cos(k ± pi)(N/2− |s|)= (−1)reiKr cos(k(N/2− |s| ± pi/2± pi(N/2− |s| − 1/2))= (−1)reiKr(cos k(N/2− |s| ± pi/2) cos(±pi(N/2− |s| − 1/2))− sin k(N/2− |s|) sin(±pi(N/2− |s| − 1/2)))= (−1)r−|s|(−1)(N−1)/2eiKr cos k(N/2− |s| ± pi/2)= (−1)r−|s|(−1)(N±1)/2eiKr sin k(N/2− |s|) (3.16)when N is odd and lK is even.eiKr sin k(N/2− |s|)→ ei(K±pi)r sin(k ± pi)(N/2− |s|)= (−1)reiKr sin(k(N/2− |s| ± pi(N/2− |s|))= (−1)reiKr(sin k(N/2− |s|) cos(±pi(N/2− |s|))+ cos k(N/2− |s|) sin(±pi(N/2− |s|)))= (−1)r−|s|(−1)N/2eiKr sin k(N/2− |s|)= (−1)r−|s|eiKr sin k(N/2− |s|) (3.17)when N is even and lK is odd.eiKr sin k(N/2− |s|)→ ei(K±pi)r sin(k ± pi)(N/2− |s|)= (−1)reiKr sin(k(N/2− |s| ± pi/2± pi(N/2− |s| − 1/2))= (−1)reiKr(sin k(N/2− |s| ± pi/2) cos(±pi(N/2− |s| − 1/2))+ cos k(N/2− |s| ± pi/2) sin(±pi(N/2− |s| − 1/2)))= (−1)r−|s|(−1)(N−1)/2eiKr sin k(N/2− |s| ± pi/2)= (−1)r−|s|(−1)(N∓1)/2eiKr cos k(N/2− |s|) (3.18)when N and lK are odd.58Therefore,eiKr cos k(N/2− |s|) (lK is even)→(−1)r+seiKr cos k(N/2− |s|), N even,(−1)r+s(−1)N±12 eiKr sin k(N/2− |s|), N odd.eiKr sin k(N/2− |s|) (lK is odd)→(−1)r+seiKr sin k(N/2− |s|), N even,(−1)r+s(−1)N∓12 eiKr cos k(N/2− |s|), N odd. (3.19)Furthermore, it can be confirmed that if K, k obey the equation (3.14),then K ± pi, k ± pi also do. Note that these arguments are valid for bothreal and complex k, i.e., they are valid for the biexciton states as well asfor continuum two-exciton states. For even N , the wave-function for K±pi,k ± pi is the same as that for K, k up to the factor (−1)r+s which is equalto 1 on the even r + s sublattice. When N is odd, these transformationsbecome more complicated, and it is for this reason that we chose N to beeven throughout this chapter. Eq.(3.14) with complex k shows that thecondition for biexciton to appear is |D| ≥ |2J cosK|. When |D/2J | > 1,one has a biexciton solution for each value of K. If we choose |D/2J | > 1and large N , then the equation can be simplified toD2J cosK≈ e−i(kr+i|ki|), kr = 0 or pi. (3.20)Here the choice of kr correlates with the sign of D/2J cosK. In par-ticular, if sgn(J) = sgn(D), the solution of (3.20) exists if: (1a) kr = 0,−pi/2 ≤ K ≤ pi/2, or (1b) kr = pi, pi/2 ≤ |K| ≤ pi. Similarly, if sgn(J) 6=sgn(D), the solution exists if: (2a) kr = 0, pi/2 ≤ |K| ≤ pi, or (2b) kr = pi,−pi/2 ≤ K ≤ pi/2. However, as discussed above, we have ΦK±pi(r, s) =(−1)r+sΦK(r, s) for even N and linearly dependent states can be summedas 12(|ΦK〉 + |ΦK±pi〉) = 12(1 + (−1)r+s)|ΦK〉, which is zero for odd r + sstates and |ΦK〉 for even r+ s states. In the following sections we use these59states setting sgn(J) = sgn(D), kr = 0, and K ∈ (−pi/2, pi/2] 2.Eq.(3.20) giveski ≈ |ln |αK || with αK = 2J cosKD. (3.21)Substituting ki into (3.12), we can write the biexciton energy asEb(K) ≈ 2E0 +D(1 + α2K). (3.22)The biexciton wave-function can now be written as 3|ΦK〉 =√2N∑r,seiKrφK(s)|r, s〉 (3.23)andφK(s) =1Ne cosh ki(N/2− |s|), lK even,φK(s) =1No sinh ki(N/2− |s|), lK odd,(3.24)where the normalization constants areNe,o =√N − 1± sinh ki(N − 1)/ sinh ki(upper sign for Ne, lower for No). Since ki is related to K by (3.21), thebiexciton wave-function can be expressed just in terms of K.We note that ki goes to infinity as K → pi/2. Since (3.24) is maximumat |s| = 1, N − 1, all values except those are infinitely smaller. Then afternormalization, biexciton wave-function is represented by delta functions atK = pi/2:φK=pi/2(s) =12(δ(|s| − 1) + (−1)lKδ(N − |s| − 1)). (3.25)Note that φK(s) is defined on 0 < s < N and extended to other valuesby applying the symmetries (3.6). In the above expressions and the corre-sponding normalizations, we already extended φK(s) to −N < s < N with2For sgn(J) 6= sgn(D), either the K-domain or the value of kr should be modified.3For even r + s sublattice, the sum can be taken over {r, s} having the same parity,and s 6= 0 and positive: ∑{r,s} =∑N−1s=−N+1∑N−|s|r=−N+2+|s| with step 2 in the sum over r.60the symmetry φK(s) = φK(−s). Also we have φK(s = 0) = 0.We use the biexciton wave-function derived here as a basis in the follow-ing sections.3.2 Interaction of Excitons and Biexcitons withImpurityWe now assume that an impurity is located at the origin of the lattice. Inthis section we study the biexciton-impurity bound state(s), and Section 3.3discusses scattering of a biexciton wave packet by an impurity potential.Note that, in both sections, we assume a biexciton is tightly bound in itsrelative coordinate with |D/2J |  1, and transitions from biexciton statesto continuum two-exciton states and vice versa are not considered. Howeverwe will study those transitions in Chapter 4.3.2.1 Exciton-Impurity InteractionAs a benchmark, let us consider interaction of a single exciton with an im-purity. This problem was addressed in [68] with a parabolic approximationfor the exciton dispersion. Here we show that the account of the convex-concave dispersion of the exciton leads to qualitatively new behaviour ofexciton-impurity binding.In the nearest-neighbour approximation, the Hamiltonian isHˆ = J∑n(|n〉〈n− 1|+ |n− 1〉〈n|)+ V0∑nδ(n)|n〉〈n|+E0∑n|n〉〈n| (3.26)where |n〉 = aˆ†n|0〉 and V0 is the impurity strength, which is equal to thedifference in the excitation energies of the impurity and the host molecules.For the periodic boundary conditions, exciton states without an impurity61can be written as|ϕ(k)〉 = 1√N∑nϕk(n)|n〉, ϕk(n) = eikn (3.27)where k = 2piν/N is the wave vector, and ν ∈ [−N/2 + 1, N/2] is an integer.The eigenstates for an exciton interacting with an impurity in a 1Dlattice can be found exactly. Using (3.27), we divide the eigenstates intoantisymmetric and symmetric ones:|ϕa(ka)〉 =√2N∑nsin kan|n〉,|ϕs(ks)〉 = 1√N∑n(cos ksn+ α sin ks|n|)|n〉,(3.28)where α is yet unknown constant to be determined from the boundary con-ditions.The projection of the eigenvalue equation Hˆ|ϕa/s〉 = E|ϕa/s〉 onto a state〈n| gives for arbitrary n 6= 0Ee(k) = E0 + 2J cos k. (3.29)The antisymmetric states |φa(ka)〉 vanish at the impurity location n = 0,so they are impurity-free states; ka = 2piνa/N , νa is an integer in the interval[−N/2+1, N/2]. The states |φs(ks)〉, in contrast, interact with the impurity.The projection of the eigenvalue equation onto the state 〈n = N/2| withthe account of the periodicity requirement N/2 + 1 → −N/2 + 1 gives the62equation, which connects α and ks:〈n = N/2 + 1|ϕs(ks)〉 = 1√N(cos ks(N/2 + 1) + α sin ks|N/2 + 1|)=1√N(cos ksN/2 cos ks − sin ksN/2 sin ks+α(sin ksN/2 cos ks + cos ksN/2 sin ks))= 〈n = −N/2 + 1|φs(ks)〉=1√N(cos ks(−N/2 + 1) + α sin ks| −N/2 + 1|)=1√N(cos ksN/2 cos ks + sin ksN/2 sin ks+α(sin ksN/2 cos ks − cos ksN/2 sin ks)) (3.30)and 〈n = N/2 + 1|ϕs(ks)〉 = 〈n = −N/2 + 1|ϕs(ks)〉 givesα = tanksN2. (3.31)The projection of the eigenvalue equation onto the state 〈n = 0| givesthe remaining equation relating α and ks with V0:〈n = 0|Hˆ|ϕs(ks)〉=J√N(cos ks(1) + α sin ks|1|+ cos ks(−1) + α sin ks| − 1|)+V0√N=2J cos ks√N= 〈n = 0|E|ϕs(ks)〉 (3.32)which yieldsα sin ks = −V0/2J. (3.33)From (3.31) and (3.33) we conclude that the wave vectors ks obey thefollowing equation:tanksN2sin ks = − V02J. (3.34)Its solutions tend to ks = 2pins/N when V0 → 0, and the parameterV0/2J determines their shifts from the impurity-free values. All solutionsbut one are real values corresponding to exciton-impurity scattering states.63Figure 3.2: Single exciton interacting with impurity: Single bound state.(a) Energy spectrum for various combinations of J and V0. J > 0: Contin-uum states (red solid line) and bound states (i) or (ii); J < 0: Continuumstates (blue dashed line), and bound states (iii) or (iv). (b) Wave-functioncorresponding to bound states (i-iv).64Figure 3.3: Energy spectrum of an exciton with an impurity given by nu-merical diagonalization of Hamiltonian. E0/|J | = 1000, V0/J = 2.5 N = 40.We see that antisymmetric states (first three examples are surrounded byblue square) do not interact with an impurity, while symmetric states (firstthree examples are surrounded by red square) interact with an impurity.Their energies agree perfectly with energies given by analytical expressions(3.29) and (3.34).65The energies of the scattering states Ee(ks,a) are plotted in Figure 3.2 (a)as function of the wave vector for J > 0 (red solid line) and J < 0 (bluedashed line); E0/|J | = 1000, |V0|/|J | = 2.5. Also eigenvalues obtainedby numerical diagonalizations of Hamiltonian agree perfectly with energiesobtained by analytical expressions (3.29) and (3.34) (Figure 3.3). Indeed,we see that antisymmetric states do not interact with an impurity, whilesymmetric states interact with an impurity.One solution of (3.34), k = kb ≡ k′ + ik′′ with Re(k) = k′, Im(k) = k′′is complex, and describes exciton bound to the impurity. Interestingly, thestructure of kb is determined by the relative signs of V0 and J . As the energyis real, cos kb = cos k′ cosh k′′ − i sin k′ sinh k′′ must be real as well, whichmeans that k′ equals to either 0, or pi. As follows from (3.34), the first caseis realized when V0/2J > 0, the second – when and V0/2J < 0. For large Nwe can write the wave vector for the bound state as the simplified form:sgn(V0) = sgn(J) : k′ = 0, k′′ = arcsinhV02J;sgn(V0) 6= sgn(J) : k′ = pi, k′′ = −arcsinh V02J.(3.35)Therefore, the spectrum always possesses a single bound state with theenergy Ee(kb). For a given sign of J , it forms both for negative and positiveimpurity potentials. In the first case the bound state splits downwards,in the second – upwards from the continuum band. If the signs of J andV0 coincide, the state is described by purely imaginary wave vector (cases(i) and (iii) in Figure 3.2 (a)). If the signs of J and V0 are different, itswave vector has a real part pi (cases (ii) and (iv)). The wave-function ofthe bound sates looks the same for all four situations (i-iv) (Figure 3.2(b)), as if a bound state forms under repulsive forces exactly as it doesfor attractive forces. In fact, bound complexes “forming under repulsiveforces” in a lattice geometry have recently been in the focus of attention[78, 79]. As we show below, this equivalence between attractive and repulsivepotentials allows for a simple interpretation in terms of the effective massm−1eff = (∂2Ee(k)/∂k2)/~2 of the exciton.66Indeed, the effective masses defined in the centre (Re(k) ∼ 0) and at theedge (Re(k) ∼ pi) of the exciton energy band have different signs, owing tothe convex-concave dispersion of the exciton. Due to the structure of theSchro¨dinger equation, Hˆ → −~2∆/(2meff) + Vˆ (n), a particle with negativeeffective mass “sees” an attractive potential as repulsive, and repulsive asattractive [80]. For concreteness of the following discussion, assume J > 0.Then mcentreeff < 0, and medgeeff > 0 (for J < 0 it will be the other way around).Then V0 > 0 will be felt as an attractive potential by the states with k ≈ 0,while V0 < 0 will be felt as an attractive potential by the states with k ≈ pi,see Figure 3.2 (a). This interpretation is confirmed by the location of thestates at the (k,E)-plane obtained analytically, see (3.35). Similar logicworks for J < 0, with the flipping of the sign of V0. We conclude thatstates (ii) and (iii) correspond to attraction of a positive-mass particle by anegative potential, the states (i) and (iv) – to attraction of a negative-massparticle by a positive potential.3.2.2 Biexciton-Impurity InteractionHere we turn to biexciton scattering by an impurity, and find numerically theeigenstates of time-independent Schro¨dinger equation Hˆ|Ψ〉 = E|Ψ〉, whereHˆ = Hˆ0 + Vˆ with Vˆ = V0∑r,s(δ(r + s) + δ(r − s))|r, s〉〈r, s|4. Note thatthe impurity potential Vˆ also obeys periodic boundary conditions of |r, s〉.We expand |Ψ〉 in the basis of the free-biexciton wave-functions as |Ψµ〉 =∑Kuµ(K)|ΦK〉, where µ is the state index and the biexciton wave-function|ΦK〉 is given in (3.23). By choosing just the biexciton states as a basis, ourresults are approximate, which should be good as long as |D/2J |  1 (i.e.,the energy of the biexciton states is much larger than any of the unboundstates). We have:∑K′〈ΦK |Vˆ |ΦK′〉uµ(K ′) = (Eµ − Eb(K))uµ(K), (3.36)4Vˆ = V0∑m,n(δ(n) + δ(m))|m,n〉〈m,n| = V0∑r,s(δ(r+ s) + δ(r− s))|r, s〉〈r, s| showsthat only even r + s states interact with an impurity.67where 〈ΦK |Vˆ |ΦK′〉 = VKK′ is written asVKK′ =4V0N∑s 6=0(r+s even)φ∗K(s)φK′(s)ei(K′−K)s. (3.37)The eigenvalues of biexciton-impurity bound states can be obtained bynumerical diagonalization of the matrix MKK′ = Eb(K)δKK′ + VKK′ andthey are shown in Figure 3.4 (a) for N = 40, E0/|J | = 1000, |V0|/|J | = 4,|D|/|J | = 4.1 as blue and red dots. All combinations of signs of D and V0 areconsidered. As follows from (3.22), the effective mass of biexciton is definedby the sign of D, and the sign of J is irrelevant, so we set sgn(J) = sgn(D).For reference, thick solid lines show the impurity-free biexciton dispersion,the grey shade shows two-exciton unbound states, and the isolated points– the biexciton-impurity bound states. Their positions on the Re(K)-axis(near Re(K) = 0 for cases (i) and (iii) and near Re(K) = pi/2 for cases (ii)and (iv)) is justified by the analogy with the exciton case, and will be provedanalytically below, see (3.41). Note that although K is not a good quantumnumber in the interaction region, we can use it as a quantum number in theasymptotic region, i.e., it is a good quantum number in the region awayfrom the scattering centre.Near Re(K) ∼ 0 we see four – marked as a, b, c, d – isolated eigenvaluesof the type (i) for D > 0, V0 > 0 (attraction of a negative-mass quasiparticleby a positive potential), and of type (iii) for D < 0, V0 < 0 (attraction of apositive-mass quasiparticle by a negative potential). The bound character ofthese states is confirmed by the decaying shape of their probability distribu-tion in r-coordinate at s = 1 (Figure 3.4 (b)); At larger s the wave-functionshave same behaviour. States a and b are well-split from the continuum andare strongly localized near the impurity. The states c and d lie very close tothe continuum and are loosely bound to the impurity. In turn, near Re(K)∼ pi/2 we see two isolated eigenvalues – they are marked by e and f – ofthe type (ii) for D > 0, V0 < 0 (attraction of a positive-mass quasiparticleby a negative potential), and of type (iv) for D < 0, V0 > 0 (attractionof a negative-mass quasiparticle by a positive potential). Their probabil-68Figure 3.4: Biexciton interacting with an impurity: Multiple bound states.(a) Biexciton scattering states for D > 0 (red solid line) and D < 0 (bluedashed line), bound biexciton-impurity states (dots (i,ii) for D > 0 and(iii,iv) for D < 0), and two-exciton unbound states (grey shaded region).Right panels zoom the regions (i) and (iii) with multiple states. (b) Theprobability distributions of the bound states (i,iii), and (c) of the boundstates (ii,iv).69Figure 3.5: Examples of scattering states of a single exciton and those ofa biexciton with an impurity. Amplitudes of biexciton scattering states arepopulated around an impurity, unlike single exciton scattering states withan impurity.70ity distribution in r-coordinate at s = 1 is shown in (Figure 3.4 (c)). Weconclude that the interplay between the biexciton binding and impurity po-tential leads to formation of multiple bound states with various degree ofexcitation localization. We also see that the scattering states of a biexci-ton with an impurity behave differently from those of a single exciton withan impurity, especially amplitudes of the scattering states of a biexcitonare populated around an impurity, unlike those of a single exciton with animpurity (Figure 3.5).The appearance of additional bound states and the variation in theirnumber for different combinations of sgn(D/V0) can be explained by av-eraging of the scattering potential by the relative coordinate of two boundexcitons. We draw the analogy with the work [81], which examines the aver-aging of the interface roughness potential in semiconductor heterostructuresby electron-hole relative coordinate in a Wannier-Mott exciton. As a resultof a composite structure of this exciton, the correlation length of the effectivepotential acting on the exciton centre of mass greatly exceeds a typical scaleof the initial disorder potential, being of the order of the electron-hole meanseparation. In a similar way, (3.37) shows that the relative coordinate oftwo excitations, which is described by φK(s), acts onto the CM coordinateof the biexciton as an effective potential V effKK′(s) = V0φ∗K(s)φK′(s). Thiscan be seen by analogy with the problem of one structureless particle inan extended potential Vˆ =∑m V∗(m)|m〉〈m|, whose matrix element in thewave vector space is Vkk′ =1N∑m V∗(m)ei(k−k′)m. The spatial extent of theeffective potential is determined by the spread of involved φK(s) and φK′(s).We checked that for small values of D the wave-function φK∼pi/2(s) is muchnarrower than φK∼0(s). Then near Re(K) ∼ 0, the averaging is effectiveand the potential is similar to a well of finite width, while near Re(K) ∼ pi/2the effective potential is close to the underlying delta function. For large Dthe wave-function φK∼0(s) narrows due to stronger exciton-exciton binding.Accordingly, for parameters (D,V0) and (D,−V0) we expect strong asym-metry in the number of bound states for small D, and the same number ofbound states when D is large.This is confirmed by examining the number of bound states as function71of (D/J, V0/J) summarized in Figure 3.6. The obtained numbers of boundstates are marked as red italic numbers near each plateau in (V0, D)-plane.The state was considered as bound if its energy fell out of the impurity-free biexciton energy band, and its amplitude was a decaying function ofr. The averaging of the potential by the relative coordinate is illustratedby showing the profiles of three wave-functions: φK=pi/2(s) for D = 2.1J ,φK=0(s) for D = 4.1J , φK=0(s) for D = 2.1J . Each plot is associated withthe corresponding point in (D,V0)-plane by an arrow. The left graph isfor V0 = −5|J |, the middle and the right – for V0 = 5|J |. Indeed, largenumbers of bound states are achieved only with |D| ≤ 2.1|J | and sgn(D)= sgn(V0), when the biexciton is weakly bound and the bound state formsnear Re(K) = 0. However, at small values of D biexciton is resonant withthe continuum two-exciton states, and the scattering of biexciton into two-exciton continuum and into the states with one free exciton, and one excitonbound to the impurity may become important. We will address this problemin Chapter 4.3.2.3 Poles of the Reflection AmplitudeThe bound states can be examined analytically as the poles of the scat-tering amplitude with complex momentum which can be derived from theLippmann-Schwinger equation [82]. In this section, we compare the poles ofthe scattering amplitude derived from Lippmann-Schwinger equation withthe assumption N → ∞, and numerical results for (3.36). Treating k =k′ + ik′′, we find that the scattering amplitude of an exciton with the dis-persion Ee(k) = E0 + 2J cos k scattered by an impurity can be calculated(Appendix D) asRe(k) = V0[(2J cos k′ sinh |k′′| − V0)−2iJ sin k′ cosh |k′′|]−1. (3.38)Note that the exact solution of the problem without the assumption of720V123456 7892 3456Number of bound states00.000.06N/2s00.000.14N/2sφ      (  )sK= 0φ      (  )sK= 0φ        (  )sK= π/2case (i), D = 2.1case (i), D = 4.1case (ii), D = 2.100.00.8N/2sDFigure 3.6: Number of bound states as function of parameters. All energiesare in units of |J |. The number of states depends on the width of φK(|xrel|).The bottom plots show the wavefunctions of relative coordinate for threevalues of parameters; “case (i)” and “case (ii)” refer to the notations of Fig-ure 3.4 (a). First and last bottom plots illustrate the difference in the widthof φK(|s|) at K = 0 and K = pi/2 for same values of all other parameters.73N → ∞ is already given in Section 3.2.1 and the poles of (3.38) appear atspecific values of k given by (3.35).For biexciton the Lippmann-Schwinger equation |Ψ〉 = |ΦK〉+Gˆ0Vˆ |Ψ〉 ismore complicated, as the potential for the interaction between the impurityand biexciton is non-separable. We solve it approximately using method ofcontinued fractions [83] in the first-order (see Appendix D). The scatteringamplitude with complex K = K ′ + iK ′′ is found as (see Appendix D):Rb(K) = 2DV0S(K ′, |K ′′|)[(J2 cos(2K ′) sinh(2|K ′′|)−2DV0S(K′, |K ′′|))− iJ2 sin(2K ′) cosh(2|K ′′|)]−1,(3.39)whereS(K) =N/2∑s=−N/2+1e−2|K′′|sφK′−i|K′′|(s)φK′+i|K′′|(s). (3.40)The factor V0S(K) accounts for averaging of the potential by the wave-function of relative coordinate, φK(s). The scattering amplitude (3.39) haspoles for K ′ = 0 and pi/2, with the corresponding equation for K ′′ beingsinh(2|K ′′pole|) =2DV0S(K′pole, |K ′′|pole)J2 cos(2K ′pole),K ′pole = 0,pi2.(3.41)This shows that the small parameter of the perturbative expansion is2DV0/J2. The first-order approximation allows us to confirm the abovepicture of potential averaged by the wave-function of relative coordinate. Italso reproduces the same effect as for a single exciton: when D and V0 havesame signs, the bound state appears near K ′ = 0, while when D and V0have opposite signs, the bound state appears near K ′ = pi/2. The physicalmeaning of this effect, as can be seen from Figure 3.4 (a), again lays in the740 Im(k) πR   (k) [arb. units]eE E(ii, iii) (i, iv)(a)V  =    2.5 J0 +−k “pole(b)V  =    0.25 J+−K “pole(iii) (iv)0 Im(K) π/2R   (K) [arb. units]b E EFigure 3.7: Scattering amplitude of a single exciton (a) and biexciton in theperturbative limit (b) as function of complex momentum. The insets showthe continuum spectrum (red stripe) and the single bound state (red dot)calculated numerically, and the arrow indicates the analytical estimates (seetext).different signs of the effective mass of the biexciton at Re(K) ∼ 0 and atRe(K) ∼ pi/2. We remark that the numerical studies of Ref. [69] reportanomalously high transmission at Re(K) = pi/2 (in our notations, whichcorresponds to Re(K) = pi in the notations of Ref. [69]), and attribute itto existence of a two-body resonant localized state. The authors considerednegative J and D and positive V0. Our results confirm this interpretation.Comparison of (3.41) and (3.35) reveals that the bound states at Re(K)= pi/2 are a general property of a restricted (cosine) energy band, ratherthan being due to the composite character of the scattered biexciton.In Figure 3.7 we show the scattering amplitudes: for an exciton as afunction of k′′ (3.38), and for a biexciton as a function of K ′′ (3.39) with thefollowing parameters: V0 = ±2.5J for panel (a), D = 4J and V0 = ±0.25J(so that we remain in the perturbative limit) for panel (b). Both scattering75amplitudes show a single pole, Re at k′′ given by (3.35), and Rb at K ′′given by (3.41). The insets show numerically calculated energy bands, withthe bound states shown as dots above or below the continuum. The arrowsindicate, respectively, the position of Ee(k′pole+ik′′pole) and Eb(K′pole+iK′′pole).For exciton the agreement is perfect, for biexciton - approximate.3.2.4 Bound States in the ContinuumIn the previous section, we focused on biexciton-impurity bound states whoseenergies are located outside the continuum band. In this section, we showthat our model can also have bound states in the continuum in which twoexcitons are mutually bound and bound to the impurity and the energies ofthe states are located in the continuum band. Although an impurity destroysintegrability of the model [84], it was observed that the Bose-Hubbard modelwith a particle-particle interaction and an impurity potential in the two-particle sector is semi-integrable in some cases [85]. Our model has L→∞that accounts for the hard-core constraint, therefore m, n translations donot commute with the Hamiltonian even if a two-particle interaction D andan impurity potential V0 are zero. However, the model can be approximatelysolved as follows.We assume that the strength of a two-particle interaction D and that ofan impurity potential V0 are not similar, i.e., either |V0|  |D|, or |V0|  |D|(the condition |D| > 2|J | for biexciton formation is always assumed). Undereach of these conditions, potentials in the CM and relative coordinate decou-ple, and the two-particle wave function looks as if D and V0 independentlycreate bound states in the relative and the CM coordinate, respectively.When they are similar, i.e., |V0| ∼ |D|, often each exciton individually formsa bound state with a combined potential of D and V0 in m and n coor-dinates respectively rather than in the relative and the CM coordinates.For simplicity, we do not consider this situation and derive only an ap-proximate solution under the assumption that the strong exciton-excitoninteraction (D) creates a bound state in the relative coordinate, and weakexciton-impurity interaction (V0  D) perturbatively bounds biexciton to76the impurity. Alternatively, each of two excitons strongly bounds to theimpurity, and weak exciton-exciton interaction (D  V0) forms a biexciton.Then an ansatz for a two-exciton state with a two-particle interaction Dand an impurity potential V0 can be approximately written as|ΦaK〉 ≈2√N∑r,ssin(Kar)φKa(s)|r, s〉. (3.42)Here we write the ansatz as a product of wave-functions in the CM andin the relative coordinate since we assume that these two degrees of freedomdecouple because of incomparable scales for V0 and D. The wave vector Kbears index a as we will consider states which are antisymmetric under thereplacement r → −r. The problem involving the states which are symmetricunder this transformation becomes more complicated as in the case of [85].Given that φK(s) is symmetric under s→ −s, |ΦaK〉 is antisymmetric underthe transformation (r, s)→ (−r,−s).Now we consider the eigenvalue equations. We have four cases: (i) for|r| 6= |s| and s 6= 0,±1 where neither V0 nor D appears, (ii) for |r| 6= |s| ands = ±1 where D appears but V0 does not, (iii) for |r| = |s| and s 6= 0,±1where V0 appears but D does not, and (iv) for |r| = |s| and s = ±1 whereboth D and V0 appear. The eigenvalue equations corresponding to (i) and(ii) are already given in (3.10) and (3.13) respectively.For (iv), we project the eigenvalue equation Hˆ|ΦaK〉 ≈ E|ΦaK〉 onto 〈r, s|at r = s = 1:2J cosKaφK(2) + (D + V0)φKa(1) ≈ 4J cosKa cos kφKa(1). (3.43)Ideally we look for the state that satisfies all equations correspondingto cases (i), (ii), (iii) and (iv). However, (3.43) reduces to (3.13) when|D|  |V0|, and it reduces to the eigenvalue equation corresponding to (iii)which is shown below when |V0|  |D|. Therefore, in either of these twolimits, we only derive the state that satisfies three equations correspondingto (i), (ii) and (iii). Eqs. (3.10) and (3.13) are already solved in Section3.1, and it was found that k satisfies (3.21) when the state is bound in the77relative coordinate due to a two-particle interaction D. Then only thing leftis to evaluate complex Ka that gives the bound state in the CM coordinatedue to the impurity potential V0.For (iii), we project the eigenvalue equation Hˆ|ΦaK〉 ≈ E|ΦaK〉 onto 〈r, s|at r = s (here we choose r = s = N/2− 1), which leads toV0 ≈ 2J(cosKa − sinKatanKa(N/2− 1))cos k (3.44)where Ka is complex for the bound state in the CM coordinate.When we have Ka = iK′′a or Ka = pi/2+iK′′a and large N , (3.44) reducestoV0/2J ≈ ±e−K′′a cosh ki, Ka = iK ′′a ,−V0/2J ≈ ±eK′′a cosh ki, Ka = pi2+ iK ′′a , (3.45)where we have + sign and − sign on the right hand side when k = iki andk = pi + iki respectively. Here we substituted complex k given by (3.21)obtained from (3.13) so that the state is bound in both CM coordinatewith complex K and in relative coordinate with complex k respectively.When Ka = iK′′a , the equation with + sign corresponds to the case wheresgn(J) = sgn(D) = sgn(V0), and that with − sign corresponds to the casewhere sgn(J) 6= sgn(D) = sgn(V0). On the other hand, when Ka = pi/2 +iK ′′a , the equation with + sign corresponds to the case where sgn(J) =sgn(D) 6= sgn(V0), while that with − sign corresponds to the case wheresgn(J) = sgn(V0) 6= sgn(D). Substituting Ka obtained by (3.45) into (3.12)givesEb1 =DV0(D + V0 −√4J2 + (D − V0)2)2(DV0 − J2) ,Eb2 =DV0(D + V0 +√4J2 + (D − V0)2)2(DV0 − J2)(3.46)where both equations in (3.45) give the same result and 2E0 should be added78if E0 is not set to equal to zero.We find that Eb1 can fall into the continuum band that covers the interval[2E0−4J, 2E0 +4J ] while Eb2 falls outside of the band when D,V0 > 0. Onthe other hand, Eb2 can fall into the continuum band and Eb1 falls outsideof it when D,V0 < 0. Therefore, one of them corresponds to the type ofbound states which we discussed in the previous section, while the other isa bound state in the continuum similar to that obtained in [85].We have also studied numerical solutions of the Schro¨dinger equationsin the full basis, which included all continuum states as well biexciton statesdiscussed in previous sections. We find four distinct types of bound states:the first type is the biexciton state where two particles are bound in the rel-ative coordinate with complex k but the biexciton complex is not bound bythe impurity (free biexciton). The second type corresponds to two particleswhose CM position is bound by the impurity with complex K but they arenot bound in the relative position (two excitons whose CM position is boundto the impurity). The third type of bound states corresponds to the casewhere one particle is bound by the impurity (the wave vector of one particleis complex) but the other is not. Finally, the forth type of bound statescorresponds to two particles mutually bound and bound to the impurity. Inthis forth case, the energies of the bound states are approximately given byEq. (3.46), and one of them falls into the continuum. Figure 3.8 shows anexample of a bound state in the continuum of the forth type. In the limitD  V0, we found almost exact agreement between Eb1 in (3.46) and theeigenvalue of the state obtained by numerical diagonalization. Indeed wesee the probability distribution of the state demonstrates clear decouplingbetween r and s (Figure 3.8 (a)), which indicates that the ansatz (3.42) writ-ten as a product of wave-functions in the CM and in the relative coordinateworks well. Meanwhile, in the limit D  V0, we found that the formula forEb1 works only approximately, and the probability distribution shows thatfull decoupling between r and s is not achieved (Figure 3.8 (b)).79Figure 3.8: Bound states in the continuum. (a) D = 4.1J , V0 = 8J . (b)D = 4.1J , V0 = J .3.3 Decoherence by Internal Degrees of FreedomIn this section we study the change in internal entanglement between theCM motion and the internal (relative coordinate) degree of freedom s viascattering by an impurity. We consider a single scattering of a narrow biex-citon wave packet∑K uK |ΦK〉 with the expansion coefficients uK by animpurity.Let W (r, s) =∑K uKfK(r)φK(s) where fK(r) = eiKr and φK(s) isgiven in (3.24) denote its real-space projection onto a state |r, s〉. The re-duced density matrix of ρ(r, s; r′, s′) = W (r, s)W ∗(r′, s′) isρcm(r, r′) =∑sρ(r, s; r′, s)=∑K,K′ u∗K′uKf∗K′(r′)fK(r)∑sφ∗K′(s)φK(s).(3.47)Therefore, tracing over the relative coordinate suppresses the contribu-tion of the pairs of components with different |K|; The less similar φK andφK′ functions are, the stronger the suppression. In a sense, the relativecoordinate acts as a source of decoherence. To quantify this decoherence,we consider a thought experiment, in which the two ends of a lattice are80Figure 3.9: Time evolution of the biexciton wave packet (3.49) interactingwith the impurity. Shown is the probability distribution ρ(r, s; r, s; t), withN = 40, r ∈ (−N,N), s ∈ (−N,N).81connected (ring geometry). A wave packet propagates towards the impu-rity and is split by it into transmitted and reflected parts. The two partspropagate away from the impurity, meet at the opposite side of the ring andinterfere. The off-diagonal elements of ρcm describing this interference ofthe wave packet with itself quantify the degree of decoherence.We solve the time-dependent Schro¨dinger equation, i~ ∂∂t |Ψ(t)〉 = Hˆ|Ψ(t)〉,where Hˆ = Hˆ0 + Vˆ as in the previous section. In the basis set of staticimpurity-free biexciton wave-functions (3.23), |Ψ(t)〉 = ∑K uK(t)|ΦK〉, thetime-dependent expansion coefficients uK(t) of the wave vectorK obey equa-tions of motions given byi~∂uK(t)∂t=∑K′〈ΦK |Hˆ|ΦK′〉uK′(t) (3.48)where 〈ΦK |Hˆ|ΦK′〉 = MKK′ = Eb(K)δKK′ + VKK′ is given in (3.37).Here we project the Hamiltonian onto the biexciton set of states, butignore two-exciton continuum states. This simplification is only possiblewhen the biexciton state is well split from the two-exciton continuum with|D| > 4|J | and the impurity potential V0 is not large compared to D, i.e.,not |V0|  |D|. However, when |D| ∼ 2|J |, a scattering by an impurity candestroy biexciton states and make them decay into two-exciton continuumstates. In this section, we do not study such physical process, but show onlyone simple example in which a biexciton wave packet scatters by an impuritywithout the transitions between biexciton and two-exciton continuum states.We will look at those transitions with weak |D| ∼ 2|J | in Chapter 4.We consider an initial wave packet of the form|Ψ0〉 = 1N∑KuK(0)|ΦK〉, (3.49)where N is the normalization factor, anduK(0) = e− 12(K−K0)2/(∆K0)2 . (3.50)We solve a matrix differential equation (3.48) numerically with an initial82condition (3.50). We choose the parameters as K0 = 3pi/8, ∆K0 = pi/24,D/J = 4.5, |V0/J | ∼ |D/J | such that half of the wave packet is transmittedand half of it is reflected in the CM coordinate. Here D/J and V0/J haveopposite signs. The calculations are done with the number of moleculesN = 40, and 2E0 chosen as a reference point for energy. We measure timein units of 1/|J |. The wave packet shown in Figure. 3.9 (a) starts at t = −30and is split by the impurity at around t = 0 into reflected and transmittedparts (Figure. 3.9 (b)). Finally, two parts meet at the opposite side of thering and interfere with each other (t = 54, Figure. 3.9 (c), shows the momentof maximal overlap).The von Neumann entanglement entropy [13] of the entanglement be-tween CM and relative position degrees of freedom, is defined as S(t) =− tr[ρcm(r, r′; t) log2 ρcm(r, r′; t)], and ρcm(r, r′; t) =∑s Ψ(r, s; t)Ψ∗(r′, s; t)where Ψ(r, s; t) =∑K uK(t)fK(r)φK(s). We diagonalize ρcm(r, r′; t) asρcm =∑r ηr|r〉〈r| and evaluate S = −∑r ηr log2 ηr. The entropy for theinitial biexciton wave packet [(3.49) and Figure 3.9 (a)] is calculated asS = 0.18. In contrast to the case of the free-space composite object [33], thelattice Hamiltonian (3.1) is not separable into relative and CM coordinatesparts. As a result, the r − s entanglement of a state changes when it prop-agates. Careful choice of the wave packet parameters can make this changenegligibly small for long enough propagation times. For our parameters∆S ∼ 10−2 from t = −30 to around t = 0. However, when the wave packetis scattered by the impurity at t ∼ 0, the interplay between the exciton-exciton interaction and the impurity potential changes the entropy rapidlyto S = 0.38. Note that it is likely that a larger change in the entropy can beobserved when the transitions between biexciton and two-exciton continuumstates occur.The off-diagonal elements of the reduced density matrix ρcm(r, r′) areindicators of the degree of decoherence present in the system. To quantifythe contrast between diagonal and off-diagonal elements, we introduce thefunctionC(r, r′) =|ρcm(r, r′)|12 |ρcm(r, r) + ρcm(r′, r′)|(3.51)83Figure 3.10: (a) Comparison of diagonal and off-diagonal matrix elementsfor the reduced density matrix after scattering, C(r, r′) at t = 35. (b) In-terference produced by the reduced density matrix in the CM-coordinateat the moment of maximal overlap, ρcm(r, r; t = 54). (c) Similar calcula-tion for single exciton. (d) Mode distribution before and after scattering.Expectation of energy 〈Eb〉 = −4.7 is the same in both.84and plot it after scattering (t = 35) in Figure 3.10 (a). The diagonal elementsare equal to 1, and many of off-diagonal elements are only 10− 20% smallerthan the diagonal ones, indicating large degree of coherence still presentin the system. Finally, coherence is manifested in the interference patternwhen the reflected and transmitted wave packets meet at the opposite sideof the ring. Figure 3.10 (b) shows the interference pattern at the momentof maximal overlap, t = 54; the relative coordinate is traced out. Themiddle of the plot (r = ±40) corresponds to the point on the ring oppositeto the impurity, where the reflected and transmitted wave packets meet.The bottom panel shows the same data as a density plot, simulating theinterference picture observed in experiment. The interference fringe loses∼15% of its maximal visibility. For comparison, Figure 3.10 (c) shows asimilar calculation done for a single exciton scattered by the impurity. Weconsider an exciton wave packet with the same shape function uk(0) as in(3.49) scattered by an impurity, which provides half-to-half splitting of thewave packet. Plotted is the density matrix ρe(x, x) of exciton at the momentof the maximal overlap. The amplitude of fringe oscillations vanishes in theinterference minima at panel (c), confirming that the flattening of contrastin panel (b) is due to the entanglement between relative and CM coordinatesof the biexciton wave packet.Finally, Figure 3.10 (d) shows the mode distribution over K-states inthe initial wave packet (at t = −30), and after scattering (at t = 35). In[33], decoherence due to the excitation of internal degrees of freedom wasobserved. In our approximation where we only look at the biexciton states,there is only one internal state for each K, namely one biexciton state. If onestarts out with a wave packet with narrow distribution in K then the internalstates for different K have a large overlap. However, due to scattering by animpurity, the overlap gets smaller as the states spread out and decoherenceoccurs. In the example of Figure 3.9 and 3.10, the transmitted wave packethas more modes excited near K ∼ pi/2 than the reflected one has, andtherefore there is some measurement of K or s that has some possibility ofdifferentiating them.853.4 DiscussionWe have discussed two phenomena relevant to interaction of a compositeparticle on a lattice with a point defect. Firstly, in contrast with the caseof a single exciton having only one bound state near an impurity, for biex-citons the interplay between exciton-exciton interaction and the impuritypotential leads to formation of additional bound states. This can be viewedas if these two interactions introduce a finite-scale effective potential forthe CM coordinate. At the same impurity strength, the number of boundstates is larger when sgn(V0) = sgn(D), and the bound states form out ofthe scattering states near K ∼ 0. We have also demonstrated, both ana-lytically and numerically, the existence of bound states in the continuumfor our model. Secondly, we have studied a change of the entanglement ina biexciton wave packet via one scattering event. We have observed thatthe entropy increases as a result of scattering. Still, we expect that the de-crease of the entropy can in principle also be observed. When an initial wavepacket is nearly a pure state, generally the increase of the entanglement dueto a scattering event would be observed, as we saw in the previous section.However, if the initial wave packet had high entanglement of the CM motionwith the internal degrees of freedom, it may drop as a result of scattering.For a free-space particle of Ref. [33] with a quadratic dispersion, Hamilto-nian can be separated into the CM and relative coordinates. In contrast,for a biexciton on a lattice possessing cosine dispersion and subject to pe-riodic boundary conditions, the CM and relative coordinate wave-functionsdepend on one and the same wave number K, as can be seen from (3.23).As a result, a wave packet constructed as a sum of eigenmodes is normallyentangled in r − s-coordinate basis. We observed that while for the stateswith K ∼ pi/2 the relative coordinate wave-functions φK(s) are all alike, forK ∼ 0 and small D their width strongly depends on K (see Figure. 3.6 foran example). If a wave packet is formed from the states near K ∼ pi/2, wecan approximately factorize it as∑K fK(r)φK(s) ≈ φK=pi/2(s)∑K fK(r),which is nearly a pure state. In contrast, when K ∼ 0 and |D| ∼ 2.1|J |, thestates do not disentangle. This suggest a possibility to compose a highly86entangled wave packet out of these low-K small-D states, which could pu-rify upon propagation, by scattering with the impurity, or by other physicalprocesses.Our work demonstrates important differences between lattice and free-space models. In particular, the change of the sign of the effective mass of thequasiparticle, and the resulting equivalence between the effects produced by“attractive” and “repulsive” potentials, have no free-space analogs. Further-more, the property of the biexciton wave-function derived from the latticeHamiltonian (3.1) which is inseparable in r − s-coordinate with periodicboundary conditions plays an important role: The dependence of the rela-tive coordinate wave-function φK(s) on the centre of mass wave vector K isresponsible (i) for different number of bound states for sgn(V0) = ±sgn(D),and (ii) for the entanglement dynamics. Yet another difference involves res-onance states. In the framework of the free-space model [33], long-livedresonances for a composite particle at a delta-like mirror were found usingcomplex scaling method. This method, however, becomes ineffective on alattice, where the periodic boundary conditions rule out the very possibil-ity of complex eigenenergies associated with Hermitian Hamiltonians [86].Roughly speaking, an ideal discrete periodic system is doomed to experi-ence periodic evolution with revivals instead of exponential decay. On theother hand, it is clear that analogs of continuum resonances should exist innon-ideal lattice models, as soon as the lifetimes associated with resonancesare smaller than the lifetime of the system, or when the decay length of theexcitation is smaller than the system physical size.3.5 Supplemental Material: From DiscreteModel to Continuum ModelIn the discrete model, we had (3.4):87Hˆ0 = 2E0Iˆm ⊗ Iˆn + JN/2∑m=−N/2+1(|m+ 1〉〈m|+ |m− 1〉〈m|)⊗ Iˆn+JIˆm ⊗N/2∑n=−N/2+1(|n+ 1〉〈n|+ |n− 1〉〈n|)+N/2∑m,n 6=mD(δ(m− n+ 1) + δ(m− n− 1))|m,n〉〈m,n|+L∑m,nδ(m− n)|m,n〉〈m,n|. (3.52)If we letJ = − 12ma2, E0 = −2J (3.53)where a = ∆x, we obtainHˆ0ψ = − 12mψi−1,j + ψi+1,j − 2ψi,j∆x2− 12mψi,j−1 + ψi,j+1 − 2ψi,j∆x2+D∆x2(ψi,i−1 + ψi,i+1 − 2ψi,i)∆x2+ (L+ 2D)ψi,i(3.54)where ψi,j = ψ(x1i, x2j) assuming that a square grid is used so that a =∆x1 = ∆x2, and x1, x2 are coordinates of particle 1 and 2 respectively. Wehave L+ 2D →∞ and so ψi,i = 0.This is a finite difference approximation of the derivatives of the contin-uum model. When a = ∆x→ 0, we recoverHˆ0 = − 12m∂2∂x21− 12m∂2∂x22+D∆x2∂2∂x22∣∣∣∣∣x1=x2(3.55)where the first and second term are completely separable in the CM andrelative coordinates.88The energy of a single exciton on a lattice isEe(k) = E0 + 2J cos(ak) = −2J(1− cos(ak))→ −Ja2k2 = k22m(3.56)as a→ 0.Therefore, in the continuum limit, the energy spectrum is bounded be-low, i.e., ~k2/2m = 0 when k = 0, but is unbounded above ~k2/2m  0when k  0. This indicates that we can not have a bound state withrepulsive potentials in the continuum model.The energy of two excitons was written asE = 2E0 + 4J cos aK cos ak = −4J(1− cos aK cos ak)→ −4J(1− (1− a2K2/2)(1− a2k2/2)) = −2Ja2K2 − 2Ja2k2 +O(an)(3.57)which shows the inseparability of K and k vanishes in the limit a→ 0.89Chapter 4Anderson Model of aComposite QuasiparticleIn Chapter 3, we studied the biexciton wave-function interacting with animpurity where we only had the biexciton states as a basis by assuming|D/2J |  1. In this chapter, we perform similar calculations by includingthe continuum two-exciton states in a basis. In Section 4.1, we evaluate thewhole eigenvalues and corresponding eigenstates of the Hamiltonian for twoexcitons interacting with each other, and confirm that analytical expressionsfor the eigenvalues of the biexciton states and the continuum two-excitonstates derived in Chapter 3 agree with those obtained by numerical diago-nalization of the Hamiltonian. In Section 4.2, we study the quantum walk ofa biexciton on a lattice by using the time-dependent Schro¨dinger equation.Here we look at the dependence of the group velocity of a biexciton on awave vector K and a two-particle interaction strength D. In Section 4.3,we study the eigenstates of the biexciton and the continuum two-excitonstates with an impurity. Then we investigate a scattering of a biexcitonwave packet by an impurity. We find that the significant change in the en-tanglement entropy which measures the entanglement between the CM andrelative position degrees of freedom can be observed when the transitionsfrom the biexciton states into the continuum two-exciton states by a scat-tering occur. Finally we discuss a possible application of these studies tothe Anderson model of a composite quasiparticle in Section 4.4. Currently alot of effort is devoted to understand the role of interactions in many-bodyAnderson localization [87, 88]. It might be possible to characterize localiza-tion phenomena of a composite quasiparticle in terms of entanglement and90decoherence caused by it. The purpose of this chapter is not to present thecomplete results, but to propose a possible extension of our results to thestudies of Anderson localization of interacting particles.4.1 Biexciton States and ContinuumTwo-Exciton StatesIn this section, we derive the matrix of graph that represents the Hamil-tonian for two quasiparticles interacting with each other in the nearest-neighbour approximation, and compare the eigenvalues and eigenstates ob-tained by numerical diagonalization of the matrix and their analytical ex-pressions derived in Chapter 3. We again consider a 1D lattice with periodicboundary conditions.Let us recall the Hamiltonian introduced in Chapter 3:Hˆ ′0 = JN/2∑n=−N/2+1(|m+ 1〉〈m|+ |m− 1〉〈m|)⊗ Iˆn+JIˆm ⊗N/2∑n=−N/2+1(|n+ 1〉〈n|+ |n− 1〉〈n|)+DN/2∑m,n 6=m(δ(m− n+ 1) + δ(m− n− 1))|m,n〉〈m,n|+L∑m,nδ(m− n)|m,n〉〈m,n| (4.1)or in (r, s)-coordinates,Hˆ0 = L∑r,sδ(s)|r, s〉〈r, s|+ J(Rˆ+ Rˆ†)(Sˆ + Sˆ†) +D∑r,sδ(|s| − 1)|r, s〉〈r, s|(4.2)where E0 is omitted for simplicity.We consider the wave-function Φ′(m,n) if we work in (m,n)-space or91Φ(r, s) if we work in (r, s)-space such that the state is given by|Φ′〉 =∑m,nΦ′(m,n)|m,n〉, |Φ〉 =∑r,sΦ(r, s)|r, s〉. (4.3)Then the time-independent Schro¨dinger equation can be written asE〈m,n|Φ′〉 =∑m′n′〈m,n|Hˆ ′0|m′, n′〉〈m′, n′|Φ′〉,E〈r, s|Φ〉 =∑r′s′〈r, s|Hˆ0|r′, s′〉〈r′, s′|Φ〉 (4.4)where〈m,n|Hˆ ′0|m′, n′〉 = J(δm,m′+1 + δm,m′−1)δn,n′ + Jδm,m′(δn,n′+1 + δn,n′−1)+Lδm,nδm′,n′ +D(δm,n−1δn,n′ + δm,n+1δn,n′),〈r, s|Hˆ0|r′, s′〉 = J(δr,r′+1 + δr,r′−1)(δs,s′+1 + δs,s′−1) + Lδr,r′δs,s′=0+Dδr,r′δs,s′=±1. (4.5)Therefore Hˆ ′0 and Hˆ0 are represented by the matrix of the graph suchthatH ′0 = J(KN ⊗ IN + IN ⊗KN ) + LN/2∑i=−N/2+1(uiuTi )⊗ (uiuTi )+DN/2∑i=−N/2+1((ui−1uTi−1)⊗ (uiuTi ) + (ui+1uTi+1)⊗ (uiuTi )),H0 = JK2N ⊗K2N + LI2N ⊗ON2N +DI2N ⊗ON±12N (4.6)where ui is a N×1 column vector with its ith entry being one and all the restof the entries being zero, uTi is its transpose, i.e., uiuTi is an outer productof ui with itself. IN is the N × N identity matrix, KN is the N × N shift92matrix with the periodic boundary condition,KN =0 1 0 . . . 0 11 0 1 0 . . . 00 1 0 1 . . . 0. . . . . . . . . . . . . . . . . . . . . . .0 . . . 1 0 1 00 . . . 0 1 0 11 0 . . . 0 1 0. (4.7)ON2N and ON±12N are the 2N × 2N matrix where all their entries are zero,except their Nth and N ± 1th diagonal elements are one respectively.In the previous chapter, we extended the space of states from −N/2+1 ≤m,n ≤ N/2 to −N + 1 ≤ r, s ≤ N in order to simplify our analyticaldiscussion. In numerical study here, both H ′0 and H0 work. When onechooses to use H0, it is useful to add the termLN∑i,j=−N+1(uiuTi )⊗ (ujuTj ), where i+ j is odd, (4.8)to H0 so that odd r+ s states are unphysical, i.e., eigenvalue of those statesare so high that no physical states can reach the energy.Then numerical diagonalization of H ′0 or H0 gives the biexciton eigen-states and the continuum two-exciton states whose corresponding eigenval-ues agree with those given by equations derived in Chapter 3:E = 4J cosK cos k (4.9)where k satisfiesD2J cosK=cos kN/2cos k(N/2− 1) , if lK is even,D2J cosK=sin kN/2sin k(N/2− 1) , if lK is odd. (4.10)93As it was discussed, real ks satisfying (4.10) correspond to eigenvaluesof the continuum two-exciton states, and complex ks give eigenvalues of thebiexciton states. Figure 4.1 compares the eignevalues obtained analyticallyusing (4.9), (4.10) and those obtained by numerical diagonalization of H0or H ′0 (both give the same results) by choosing the parameters N = 20,D/J = 4.1 and D < 0. We see the agreement between the eigenvaluesobtained analytically and those obtained numerically. Examples of corre-sponding eigenstates are shown in Figure 4.2. We now have both biexcitoneigenstates and continuum two-exciton states as a basis, and use this basisin the following sections.4.2 Quantum Walk and Group Velocity of aBiexcitonFirstly let us look at the quantum walk of a biexciton on an ideal 1D lat-tice with periodic boundary conditions. The time-dependent Schro¨dingerequation gives the dynamics of a wave packet:iddt〈m,n|Ψ′(t)〉 =∑m′n′〈m,n|Hˆ ′0|m′, n′〉〈m′, n′|Ψ′(t)〉,iddt〈r, s|Ψ(t)〉 =∑r′s′〈r, s|Hˆ0|r′, s′〉〈r′, s′|Ψ(t)〉. (4.11)where 〈m,n|Hˆ ′0|m′, n′〉 and 〈r, s|Hˆ0|r′, s′〉 are given in (4.5).The solution to (4.11) can be expressed as a matrix exponential,Ψ′(m,n, t) = e−iH′0tΨ′(m,n, 0), Ψ(r, s, t) = e−iH0tΨ(r, s, 0) (4.12)with the initial wave packet Ψ′(m,n, 0) and Ψ(r, s, 0) respectively.(4.12) gives the quantum walk of two excitons interacting with eachother. When we start with a biexciton wave packet initially, it gives thequantum walk of a biexciton until the biexciton states break up into thecontinuum two-exciton states by the external potential etc. The group ve-94(a) Eigenvalues obtained by (4.9) and (4.10).(b) Eigenvalues obtained by numerical diagonzalization.Figure 4.1: Comparison between eigenvalues obtained analytically and thoseobtained by numerical diagonalization of H ′0 or H0 with parameters N =20, D/J = 4.1 and D < 0.95(a) Biexciton eigenstates.(b) Continuum two-exciton states. (c) Continuum two-exciton states.Figure 4.2: Example of eigenstates obtained by numerical diagonalizationof H ′0 or H0 on physical space (even r + s space) with D/J = 4.1.96locity of the quantum walk can be determined as follows.Recall that the energy of a single exciton Ee(k) and that of a biexcitonEb(K) with |D/2J | > 1 and large N can be respectively written asEe(k) = 2J cos k, Eb(K) ≈ D + 4J2 cos2KD. (4.13)Then the group velocity ve of a single exciton is given byve =∂∂kEe(k) = −2J sin k. (4.14)Therefore the absolute value of the group velocity of the exciton wavepacket is large when k ∼ ±pi/2 and is small when k ∼ 0.Similarly the group velocity vb of a biexciton in the CM degrees of free-dom is given byvb =∂∂KEb(K) = −4J2 sin 2KD. (4.15)Note that the biexciton wave-function is bound in the relative coordinate,and so we generally do not write its group velocity in the coordinate.Then the absolute value of the group velocity of the biexciton wavepacket is large when K ∼ ±pi/4 and is small when K ∼ 0,±pi/2. Figure4.3 shows the propagation of the initial wave packet spatially localized atthe edge of the lattice moving towards the opposite edge. At t = 25, thewave packet consisting of modes around K ∼ pi/4 arrives at the other sideslightly earlier than that consisting of modes around K ∼ 0,±pi/2 does.Furthermore the group velocity of a biexciton depends on D/J . As atwo-particle interaction strength D gets larger, the absolute value of thegroup velocity decreases (Figure 4.4 (a)). The dependence of the groupvelocity on D/J can be confirmed by looking at the time-evolution of thebiexciton wave packet directly. In the same condition as Figure 4.3, thebiexciton wave packet with D/J = 3.25 already has arrived at the oppositeend of the lattice, while the biexciton wave packet with D/J = 9.25 is stillin the middle of the lattice at t = 25 (Figure 4.4 (b)).97Figure 4.3: The propagation of the initial wave packet (left) moving towardsthe opposite edge of the lattice. At t = 25, the wave packet consisting ofmodes around K ∼ pi/4 (above) arrives at the other side slightly earlier thanthat consisting of modes around K ∼ 0,±pi/2 does (below). D/J = 4.25and N = 26.98Figure 4.4: (a) The dependence of the group velocity vb of a biexciton on Kand D/J . (b) The probability distributions of the biexciton wave packetsat t = 25 that consist of modes K ∼ pi/4 with D/J = 3.25 and D/J = 9.25respectively. They propagate from the left end to the right end, similarly tothe wave packet in Figure 4.3.99Figure 4.5: Examples of the eigenstates obtained by numerical diagonaliza-tion of H ′0 + V ′ or H0 + V with parameters D/J = 2.25, V0/J = 5. Thebiexciton-impurity bound state (left) and the state with one free exciton,and one exciton bound to the impurity (right).4.3 Interaction with a Single Impurity4.3.1 Impurity-Induced Bound StatesIn Chapter 3, we studied the interaction of a biexciton with a single impurityby assuming |D/2J |  1. Here we use a basis including the continuumtwo-exciton states derived in Section 4.1 and study the interaction of thebiexciton and the continuum two-exciton states with an impurity.Firstly, we add the impurity potential Vˆ ′ = V0∑m,n(δ(m)+δ(n))|m,n〉〈m,n|and Vˆ = V0∑r,s(δ(r + s) + δ(r − s))|r, s〉〈r, s| into Hˆ ′0 and Hˆ0 respectively.Then〈m,n|Vˆ ′|m′, n′〉 = V0(δm,m′=0δn,n′ + δm,m′δn,n′=0),〈r, s|Vˆ |r′, s′〉 = V0(δr,r′δs,s′=r + δr,r′δs,s′=−r). (4.16)Therefore Vˆ ′ and Vˆ can be represented by the matrix of the graph such100thatV ′ = V0(ON/2N ⊗ IN + IN ⊗ON/2N),V = V0N∑i=−N+1((uiuTi )⊗ (uiuTi ) + (uiuTi )⊗ (u−iuT−i)). (4.17)If we diagonalize H ′0 +V ′ or H0 +V numerically, we obtain the eigenstatecorresponding to the biexciton-impurity bound state (Figure 4.5, left) as wellas that corresponding to the situation where one exciton is free, while theother exciton is bound to the impurity (Figure 4.5, right). Figure 4.5 showseigenstates obtained by choosing parameters D/J = 2.25, V0/J = −5. InChapter 3, we projected the Hamiltonian onto the biexciton set of states,but ignored the continuum two-exciton states and focused on the biexciton-impurity bound states. However the above example clearly shows that thereexist states with one free exciton, and one exciton bound to the impurity,that would become important in the time-dependent dynamics as well.The study of the entanglement dynamics during a single scattering inChapter 3 also ignored the continuum two-exciton states by using the initialbiexciton wavepacket with |D| > 4|J | such that it scatters by an impuritywithout the transitions between the biexciton states and the continuum two-exciton states. However, when |D| ∼ 2|J |, a scattering by an impurity candestroy the biexciton states and make them decay into the continuum two-exciton states. In the next section, we study the entanglement dynamicsduring a single scattering by an impurity, including the transitions betweenthe biexciton states and the continuum two-exciton states by choosing |D| ∼2|J |.4.3.2 Transitions between Biexciton States and ContinuumTwo-Exciton States by a ScatteringWe now study the change in the entanglement between the CM and therelative position degrees of freedom during a single scattering by an impu-rity. One example of the entanglement dynamics of a biexciton wave packet101during a single impurity scattering was already studied in Chapter 3. How-ever here we include the cases of weak D, i.e., |D/2J | ∼ 1, and the possibletransitions from the biexciton states to the continuum two-exciton statesand vice versa.Recall that the range of the energy spectrum of the continuum two-exciton states is given by|E| ≤ 4|J |. (4.18)while the range of the energy spectrum of the biexciton states is approxi-mately,D + 4J2/D ≤ Eb(K) ≤ D, if D < 0,D ≤ Eb(K) ≤ D + 4J2/D, if D > 0. (4.19)Therefore the range of the energy spectrum of the biexciton states andthat of the continuum two-exciton states partially overlap when |D| ≤ 4|J |,and the transitions between these states obeying the energy conservationcan occur.As in Chapter 3, we prepare the initial biexciton wave packet of the form|Ψ0〉 = 1N∑KuK(0)|ΦbiexcK 〉 (4.20)where the biexciton wave-function |ΦbiexcK 〉 = |ΦK〉 is given in (3.23), anduK(0) = e− 12(K−K0)2/(∆K0)2 . (4.21)Here we choose the parameters as K0 = pi/4, ∆K0 = pi/4 and N = 26.Then the initial biexciton wave packet here has the wider distribution inmomentum space, and the narrower distribution in position space comparedto that used in Chapter 3. We also prepare the initial wave packet such thatit is spatially localized to the left of the impurity and moves toward the rightto hit it as time evolves.102Then the wave packet at time t is given byΨ(r, s, t) = e−i(H0+V )tΨ(r, s, 0) (4.22)where Ψ(r, s, 0) = 〈r, s|Ψ0〉. Note that we can equally use (m,n)-coordinatesfollowing (4.12).We look at the increase of the von Neumann entanglement entropy thatmeasures the entanglement between CM and relative position degrees offreedom during a single scattering by an impurity such that the wave packetis half transmitted, and is half reflected. In Figure 4.6, we compare theresults with different values of D/J . Here the impurity potential V0 and Dhave the opposite signs. Generally, the impurity potential strength neededto separate the wave packet in a 50:50 superposition of reflected and trans-mitted waves should be larger as D gets smaller and the group velocity of thewave packet increases. Since the group velocity is large when D is small, thewave packet with D/J = 2.25 hits the impurity around t = 6, earlier thanthe wave packets with D/J = 3.25 and D/J = 4.25 do, and the increase ofthe entropy is also significant. This large increase of the entropy is causedby the breakup of the biexicton states into the continuum two-exciton statesdue to the scattering by an impurity. If we have the wave packet Ψ(r, s, t)that moved far away from the impurity potential after the scattering, wecan definepbiexc =∑K∣∣∣∣∣∑r,sΦbiexc∗K (r, s)Ψ(r, s, t)∣∣∣∣∣2(4.23)as the probability to find the biexciton states remained in the wave packetafter the scattering, where ΦbiexcK (r, s) = 〈r, s|ΦbiexcK 〉.For the wave packet with D/J = 4.25, we have pbiexc = 1, indicating thatthe transitions from the biexciton states to the continuum two-exciton statesdo not occur by the scattering. As discussed earlier, the range of the energyspectrum of the biexciton states does not overlap with that of the continuumtwo-exciton states when |D| > 4|J |, therefore the transitions between themare unlikely to occur as long as the scattering by an impurity is nearly elastic.103Indeed we do not observe a considerable change in the entropy (Figure 4.6(above)) and the wave packet is still bound in the relative coordinate aftera scattering (Figure 4.6 (below)).On the other hand, for the wave packet with D/J = 3.25 and D/J =2.25, the range of the energy spectrum of the biexciton states partially over-laps with that of the continuum two-exciton states and the transitions be-tween two types of states can occur by the scattering. When D/J = 3.25,we have pbiexc ≈ 0.93 indicating that around 7% of the biexciton states havedecayed into the continuum two-exciton states. When D/J = 2.25, we havepbiexc ≈ 0.6 indicating the break up of around 40% of the biexciton statesinto the continuum two-exciton states, which resulted in the significant in-crease of the entropy as shown in Figure 4.6. In the figure (below), we alsosee that the wave packet with D/J = 2.25 after the scattering has nonzeroprobability of finding the states with large relative coordinate s showingthere exist the continuum two-exciton states after the scattering by an im-purity, which is in contrast to the fact that the wave packet with D/J = 4.25after the scattering has no probability of finding states with large s and thewave packet is still tightly-bound in s, showing that it is still consisted ofonly the biexciton states after the scattering.Furthermore we can compare the amount of the biexciton states re-mained in the reflected part and in the transmitted part by definingpbiexcR =∑K∣∣∣∣∣∑s∑r<0Φbiexc∗K (r, s)Ψ(r, s, t)∣∣∣∣∣2,pbiexcT =∑K∣∣∣∣∣∑s∑r>0Φbiexc∗K (r, s)Ψ(r, s, t)∣∣∣∣∣2. (4.24)When D/J = 2.25, we have pbiexcR = 0.35 and pbiexcT = 0.25, which showsthat more biexicton states have decayed into the continuum two-excitonstates in the transmitted part. In the situation similar to that the diatomicmolecule obtained different excitations of the internal degrees of freedomin the reflected part and in the transmitted part after the scattering withthe mirror in Chapter 2, it may be possible to obtain which-way informa-104Figure 4.6: The increase of the entanglement entropy between the CM andrelative position degrees of freedom by a single scattering due to an impurity,with different values of D/J (above). Comparison between the wave packetswith D/J = 2.25 and D/J = 4.25 after the scattering. We have the nonzeroprobabilities of finding states with large s, showing the existence of thecontinuum two-exciton states after the scattering when D/J = 2.25.tion that tells whether the wave packet is the reflected or the transmittedpart by looking at the amount of the biexciton states that decayed into thecontinuum two-exciton states in the wave packet.1054.4 Anderson Model of a CompositeQuasiparticle4.4.1 Anderson LocalizationIn this section, we present a potential application of the model studied sofar to the Anderson model of a composite quasiparticle. Anderson local-ization is the absence of diffusion of waves in disordered materials such asalloys and amorphous media, that makes them become insulators, in con-trast with the ideal crystals which have ordered structure and are alwaysconductors. Mathematically one is able to study the Anderson localizationby connecting the Green’s function and the transport of electrons [89, 90].It is also possible to understand the Anderson localization in terms of thepath integral formalism as follows [91, 92].Let us consider the action for a particle moving in random potentialsVR,S =∫12mx˙2 − VR(x)dt. (4.25)Then the propagation amplitude of a particle from a point xA to xB canbe given by a kernel K(xB, tB;xA, tA),K(xB, tB;xA, tA) =∫ xBxAexp(i~S[x(t)])Dx. (4.26)In the semi-classical approximation, the path integrals can be written asthe sum over all classical paths that obey the equations of motion from theaction,K(xB, tB;xA, tA) =∑classical paths iAi exp(iS[xi(t)]/~) (4.27)where the prefactor Ai comes from the Gaussian fluctuations around theclassical path.Then the probability W for a particle to arrive at a point xB from xA is106given by the forward-backward path integral:W (xB, tB;xA, tA) =∑i=j|Ai|2 +∑i 6=jAiA∗j exp(i(S[xi(t)]− S[xj(t)])/~).(4.28)The second term represents the quantum coherence due to the inter-ference of different paths. The randomness of VR leads to the absence ofthe correlation between the phases S[xi(t)] and S[xj(t)], therefore we havethe destructive interference as a result, and the second term vanishes whenxA 6= xB. However, when we consider a path such that the initial pointand the final point are identical, i.e., xA = xB, and the particle comes backto the initial point (Figure 4.7), then the forward path and the backwardpath become identical to each other and we obtain coherence relation be-tween S[xi(t)] and S[xj(t)] which leads to the constructive interference. Inthis case, the second term representing coherence remains. Therefore, theprobability that a particle comes back to its initial point is higher than theprobability that it goes far away from the initial point in random potentials.In this way, Anderson localization can be understood as the interferencephenomenon.The question then arises as to whether the emergence of classicalitycaused by decoherence due to internal degrees of freedom affects Andersonlocalization of a composite quasiparticle. Similar questions were addressedto the quantum kicked rotor model in quantum chaos [93]. The model ex-hibits localization of the wave-function in momentum space due to quantumcoherence, which is analogous to Anderson localization of electrons in ran-dom potentials [94]. The effects of noise or the heat bath on the quantumkicked rotor were investigated in [95, 96] where it was found that they dis-rupt the localization of a wave-function in the model. In the following, wepropose a setup that may give an insight into the similar problem usingAnderson model of a composite quasiparticle where the source of noise canbe internal degrees of freedom of itself.107Figure 4.7: The backward and the forward path become identical to eachother when xB = xA and they interfere constructively with each other.4.4.2 Anderson Model of Single Exciton and BiexcitonThe Anderson model Hamiltonian for a single exciton can be written asHˆexc = J∑n(|n〉〈n− 1|+ |n− 1〉〈n|) +∑nVn|n〉〈n| (4.29)where Vn are random potentials that are characterized by a Gaussian normaldistributions P (Vn) of width W and mean zero,P (Vn) =1√2piW exp(−V2n /2W2). (4.30)Similarly the Anderson model Hamiltonian for a biexciton or for twoexcitons interacting with each other by the interaction strength D can begiven by Hˆ ′0 + Vˆ ′R or Hˆ0 + VˆR whereVˆ ′R =∑m,n(Vm + Vn)|m,n〉〈m,n|,VˆR =∑m,n∑r,s(Vmδ(r + s− 2m) + Vnδ(r − s− 2n))|r, s〉〈r, s| (4.31)wth −N/2 + 1 ≤ m,n ≤ N/2.108Figure 4.8: Anderson localization of a single exciton (above) and that ofa biexciton with D/J = 4.25 (below). The comparison between the time-evolutions of the wave packet with (right) and without (left) random po-tentials, W = 0.45|J |. Here we plotted the probability distribution of abiexciton wave packet by rescaled r → r/2 so that r = (m+ n)/2.We prepare the biexciton wave packet of the form (4.20) with D/J =4.25, K0 = 0, ∆K0 = pi/21, N = 26, and the exciton wave packet in thesimilar condition, such that both are located at the origin of the latticeinitially.Without the random potentials, we see that both biexciton wave packetand exciton wave packet spread diffusively on the lattice after a long timet = 6000 (Figure 4.8 (left)). However, when the random potentials withW = 0.45|J | are turned on, then both wave packets are spatially localizedafter the same amount of time (Figure 4.8 (right)), indicating that Andersonlocalization takes place. In the figure, we used rescaled r → r/2 so thatr = (n + m)/2 in order to make comparison easy between the probability109distribution of a biexciton and that of a single exciton. In what follows, weuse rescaled r → r/2.In the study of Anderson localization, it is useful to evaluate the inverseparticipation ratio (IPR) defined asIPRexc =∑n|ψ(n; t)|4 (4.32)where ψ(n; t) is the wave packet of a single exciton at time t.1/IPRexc is approximately the number of sites where the wave packet isnonzero, and we have IPRexc → 1 for localized states and IPRexc → 0 forextended states.Since the biexicton wave packet is already bound in the relative coordi-nate, we are generally interested in Anderson localization in the CM degreesof freedom. Therefore we defineIPRbiexc =∑r|ρcm(r, r, t)|2 (4.33)where ρcm(r, r′; t) =∑s Ψ(r, s; t)Ψ∗(r′, s; t) is the reduced density matrixof a biexciton in the CM degrees of freedom at time t. Then 1/IPRbiexcgives approximately the number of r-sites where the biexciton wave packetis nonzero.The important difference between the exciton wave packet and the biex-citon wave packet is that the latter has the entanglement between the CMand relative position degrees of freedom, even when we do not have the ex-ternal environment around. Since Anderson localization is the interferencephenomena as discussed in the previous section, one may ask whether de-coherence caused by the entanglement affects Anderson localization of thebiexciton in the CM degrees of freedom.Figure 4.9 (a) shows the increase of the entanglement entropy betweenthe CM and relative position degrees of freedom of the wave packet in ran-dom potentials with different values of D/J where the initial wave packettakes the form of (4.20) with ∆K0 = pi/21, N = 26. It is initially located atthe origin of the lattice. The plot compares the entropy increase where the110Figure 4.9: (a) The increase of the entanglement entropy between the CMand relative position degrees of freedom in random potentials of the wavepacket, with different values of D/J . (b)The energy spectrum of the con-tinuum two-exciton states and the biexciton states with D/J = 2.25. Blackline indicates the same energy. When the initial mode distribution is cen-tred around K0 = pi/2, there exist many continuum two-exciton states thatthe initial biexciton states can decay into, in contrast to the case of K0 = 0where there exist only a few continuum two-exciton states that have thesame energy as the initial biexciton states.111centre of mode distribution of the initial wave packet is chosen as K0 = 0 andD/J = 4.25, 3.25 and 2.25, while one plot shows the entropy increase wherethe centre of mode distribution is shifted to K0 = pi/2 and D/J = 2.25.In all cases, the entropy increases initially and then becomes stable aftersome time. This is due to the fact that the initial energy of the wave packetshould be conserved at later times, and the transition from one state to theother state obeys energy conservation. Initially the entropy increases sincethe initial states make transitions to the other states, however once all pos-sible states that give the same energy as the initial states are occupied, theentropy stops increasing and becomes stable. This is shown by comparingthe entropy increase of the initial wave packet with K0 = 0 and that withK0 = pi/2. When we start with the initial wave packet with D/J = 2.25and K0 = 0, there exist only a few continuum two-exciton states that theinitial states can make transitions to by the energy conservation (Figure 4.9(b)). However the initial wave packet with D/J = 2.25 and K0 = pi/2 hasmany continuum two-exciton states that the initial states can make transi-tions to (Figure 4.9 (b)), and the entropy increase is also large comparedto the case of K0 = 0 (Figure 4.9 (a)). We have the time scale differencein the growth between two cases since the initial entanglement entropy issmaller for the case of K0 = pi/2 as discussed in the previous chapter, i.e.,for the states with K0 ∼ pi/2 the relative coordinate wave-functions are allalike, and multiple scatterings by impurities are needed until all possiblecontinuum two-exciton states get occupied for the case of K0 = pi/2.Now let us look at localization of the wave packet in random potentials.Table 4.1 shows 1/IPRbiexc and the von Neumann entanglement entropy Sof the entanglement between the CM and relative position degrees of freedomafter a long time t = 6000 with different values of D/J . We chose W =0.45|J |. As D/J gets smaller, the group velocity of a biexciton increases.In order to see the effect of the increase of the group velocity, we included1/IPRexc given by a similar calculation done for a single exciton whosegroup velocity is chosen to be similar to that of the corresponding biexcitonon a same lattice with same random potentials, i.e., we chose Jexc = 2J/Dso that ve ≈ vb where Jexc and J are the hopping strength of a single exciton112D/J 1/IPRexc 1/IPRbiexc S6.25 (K0 = 0) 3.0 6.3 0.45.25 (K0 = 0) 4.8 3.5 0.44.25 (K0 = 0) 2.8 4.9 0.73.25 (K0 = 0) 4.4 12 1.52.25 (K0 = 0) 3.8 14 2.22.25 (K0 = pi/2) 3.8 35 3.6Table 4.1: 1/IPRbiexc and the von Neumann entanglement entropy S of theentanglement between the CM and relative position degrees of freedom of thewave packet in random potentials at t = 6000. 1/IPRexc is obtained froma single exciton with a similar group velocity to that of the correspondingbiexciton.and of a biexciton respectively.Since the decrease of D/J changes the group velocity only slightly, we seethat the similar localizations of a single exciton (characterized by 1/IPRexc)are observed with different group velocities, i.e., for each D/J we have1/IPRexc is around 3− 5.When D/J is large, such that D/J = 4.25, 5.25, 6.25, we do not seethe correlation between 1/IPRbiexc and the entropy S. Also 1/IPRbiexc issimilar to 1/IPRexc, which indicates the localization of the CM coordinateof a biexciton is similar to that of a single exciton. However, when D/J getssmaller, such that D/J < 4, then 1/IPRbiexc and the entropy S increasedramatically. In this regime, the transitions between the biexciton statesand the continuum two-exciton states can occur. When K0 = 0, only asmall number of states can make transitions to the continuum two-excitonstates. However, as Table 4.1 shows, this small number of transitions to thecontinuum two-exciton states increase the entropy significantly and weakenlocalization phenomena when D/J = 3.25 and D/J = 2.25. When K0 =pi/2, the large amount of states can decay into the continuum two-excitonstates. In such a situation, the localization effect in the CM degrees offreedom is largely weakened as shown in Figure 4.10. Note that if we start113with the wave packet consisting of the continuum two-exciton states of thesame energy as biexciton states with K0 = pi/2, the localization effect isstronger when a two-particle interaction D is turned off (Figure 4.11). Thesimilar result was discussed in [88] where it was found that inter-particleinteractions decrease the localization.The enhancement of entanglement between the CM and relative positiondegrees of freedom due to impurity potentials and a two-particle interactionmay make the relative position degrees of freedom acts like the noise thatcauses suppression of effects of quantum interference in the CM degrees offreedom as was the case with [95, 96] where the noise disrupts the local-ization phenomena in the quantum kicked rotor model. However one needsto take into account of other possible factors that can affect the localiza-tion phenomena such as different number of biexicton-impurity bound statescreated for K ∼ 0 and for K ∼ pi/2 as discussed in the previous chapter,and the probability of tunnelling through impurities etc. Furthermore, wedid not observe the significant change in the entanglement entropy and thelocalization phenomena until we reach the regime where the biexciton statescan break up into the continuum two-exciton states. Since our model onlyallows one bound state (one biexciton state) for each K, it may be prefer-able to use the model with the long-range inter-particle interactions that cancreate multiple bound states in the relative position degrees of freedom (i.e.,multiple internal states) to investigate the effect of the internal entangle-ment on the localization phenomena of composite object without breakingit by scatterings.As discussed above, further studies are clearly needed and currently un-derway. However the above investigation suggests that it might be possibleto use our model in order to study the effect of the entanglement and deco-herence caused by the particle-particle interaction and the impurity poten-tials on Anderson localization of a composite quasiparticle or that with theparticle-particle interactions.114Figure 4.10: Comparison between the probability distribution for the wavepacket in random potentials after t = 6000, with the initial mode distributioncentred at K0 = 0 and at K0 = pi/2 (above). The corresponding reduceddensity matrix ρcm(r, r; t) (below). D/J = 2.25.115Figure 4.11: The probability distribution for the wave packet in randompotentials after t = 6000, with the initial wave packet consisting of thecontinuum two-exciton states of the same energy as biexciton states withK0 = pi/2 when a two-particle interaction D is turned off (above). Thecorresponding reduced density matrix ρcm(r, r; t) (below).116Chapter 5ConclusionsIn this thesis, we studied two types of models that describe a compositeobject and its scattering by a potential. The purposes of the study were tounderstand the effects of internal entanglement between the position of acomposite object and its internal (relative position) degrees of freedom onexperiments which test its quantum spatial superposition state, on the waythe composite object interacts with the external objects such as radiation(Chapter 2) and an impurity (Chapter 3) and on quantum interference phe-nomena in one degree of freedom of the composite object as the cause of theemergence of classicality (Chapter 4).In Chapter 2, we studied a diatomic molecule-type composite objectscattered by a half-silvered mirror modelled as a delta potential in contin-uum space. We found that the creation of a spatial superposition state ofthe composite object by means of the mirror can result in entanglement be-tween its position and its internal (relative position) degrees of freedom, andsuch superposition can decohere by emission of radiation since the internaldegrees of freedom are entangled with radiation fields. When there exists theinternal entanglement between the CM and relative position degrees of free-dom, tracing out the relative position degrees of freedom causes decoherencein the CM degrees of freedom. However if one allows experiments which lookat the whole degrees of freedom (CM+relative) of the composite object, wedo not have decoherence as a pure state remains a pure state. If one looksat only part of the system, then there can be decoherence. In the study, wehad radiation fields as the additional external degrees of freedom that areentangled with the relative position degrees of freedom. In such a situation,even if one looks at the whole degrees of freedom of the composite object,117tracing out radiation fields causes decoherence on superposition of the com-posite object. Therefore, one can understand the internal entanglement asa source of decoherence when one only looks at the one degree of freedom ofthe composite object, or it makes the superposition state of the compositeobject get entangled with the external objects in a different way from theway that of a single point particle does. We evaluated decoherence effects byemission of electromagnetic radiation and gravitational radiation using thepath integral formalism under the assumption that Einstein’s equations arevalid even in the quantum regime, and that we have an external observer.In the case of gravity, the problem can become complicated since generalrelativity has an internal observer rather than an external one. In such asituation, the spatial superposition of matter may get entangled with thetime measured by the internal observer and the inconsistencies between gen-eral relativity and quantum mechanics become crucial, as it was discussedin [48, 49]. We did not address this problem and only dealt with the possibledecoherence phenomena of the spatial superposition of the composite objectdue to emission of gravitational radiation in the weak gravity regime.In Chapter 3, we studied a biexciton scattering by an impurity, which isthe lattice analog of the model studied in Chapter 2. Although the Hamil-tonian of the diatomic molecule-type composite object in continuum spacewhich appeared in Chapter 2 was separable in the CM and relative coor-dinates, and therefore the corresponding eigenstates did not have entangle-ment among these degrees of freedom, we observed that its lattice analogueHamiltonian and eigenstates are no longer separable in these coordinatesdue to the discreteness of space, and found that the CM and relative posi-tion degrees of freedom are entangled in biexciton states. The inseparabilityof these degrees of freedom leads to a dependence of the width of a biex-citon wave-function in the relative coordinate on the CM wave vector K,and a difference in the number of biexciton-impurity bound states formedwhen K ∼ 0 (the impurity potential V0 and D have same signs) and whenK ∼ pi/2 (the impurity potential V0 and D have opposite signs). Further-more we found that the biexciton wave packet consisting of modes aroundK ∼ 0 has stronger internal entanglement than that consisting of modes118around K ∼ pi/2. Chapter 3 demonstrated an example where the choice ofspace in which a composite object is defined can affect the behaviour of itsinternal entanglement.Finally, in Chapter 4, we proposed a possible application of our studyto the Anderson localization of a composite quasiparticle or that with theparticle-particle interaction. In our model, the significant increase of the en-tanglement entropy and weakening of the localization phenomena in the CMdegrees of freedom were observed only when the transitions from biexcitonstates to continuum two-exciton states can occur. It might be possible thatthe relative position degrees of freedom behaves like noise that affects theAnderson localization as the quantum interference phenomenon in the CMdegrees of freedom. However, in order to study the pure effect of internalentanglement on the localization phenomena of a composite quasiparticlewithout these transitions, one may need to introduce the model with morebound states (internal states) in the relative position degrees of freedomso that the composite object does not easily break up by scatterings, andfurther studies are underway.The results in this thesis suggest that the study of internal entanglementof composite objects may give insight into experiments involving their spatialsuperposition, quantum tunnelling of a molecule, decoherence studies in theframework of quantum field theories and gravity, and complex phenomena incondensed matter systems. A composite object whose constituent particlesare bound to each other in their relative distance appears to be a singleobject, but its behaviour is largely different from that of a point particledue to the existence of internal entanglement. When the internal degreesof freedom of a composite object are separable and are not entangled witheach other, then one is able to study the reduced density matrix of interestby tracing out nonessential degrees of freedom without any effect. Howeverwhen these degrees of freedom of a composite object are entangled witheach other, then one is no longer able to perform a measurement of quantummechanical phenomena occurring in one degree of freedom without the effectof tracing out the other degrees of freedom that are entangled with it.119When we study a physical process of a composite object that changesits internal entanglement, then techniques used to study the quantum me-chanics of a point particle generally can not be applied. For example, it isexperimentally observed that the quantum tunnelling rate of the chemicalreactions between a methyl radical and a hydrogen molecule is higher thanthat given by the theoretical prediction which does not take the internalstructure of the molecule into consideration [97, 98]. In this thesis, we haveobserved that the tunnelling of the composite object normally changes itsinternal entanglement due to the interplay between its internal structure anda potential barrier. It may be possible to connect the change in its internalentanglement with its behaviour in quantum tunnelling.We also found that the spatial superposition of a composite object caneasily radiate and decohere since the excitations of the internal degrees offreedom can be entangled with its position, whereas we do not normallyhave such a mechanism in a point particle without any dipole moment. Acomposite object can have an internal structure that makes it interact withthe environment (e.g., quantized radiation fields in our study) in a differentway from how a point particle does. Therefore, in order to evaluate thelifetime of a superposition state of a composite object explicitly, it becomesimportant to understand how the internal degrees of freedom are entangledwith the environment and with the degrees of freedom of interest.Internal entanglement may also be used in quantum computing. [99] pro-posed to use entangled rotation and vibration states of molecules to encodeinformation. In the study, the control of internal entanglement becomes akey issue. Compared to the entanglement of the object with the externalenvironmental bath, internal entanglement of the composite object may becontrolled easier in some cases since the degrees of freedom entangled arealways present in the object concerned unlike environmental particles thatscatter with the object and fly away with information. For example, inChapter 3, we observed that the biexciton states near K ∼ pi/2 have weakerinternal entanglement than states near K ∼ 0. Therefore one may be ableto make internal entanglement weaker when one introduces the interactionsuch that it causes transitions of states from those near K ∼ 0 to K ∼ pi/2.120Finally, it may be possible to apply our study of entanglement to classifi-cations of complex phenomena in condensed matter systems. 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(A.2)129By substituting (A.2) into (A.1), we have the classical action isScl =∫ t0(12mX˙2cl −12mω2X2cl(s) + f(s)Xcl(s))ds=12m[Xcl(s)X˙cl(s)]t0 +12∫ t0f(s)Xcl(s)ds=mω2 sinωt[(X2 +X20 ) cosωt− 2XX0 +2Xmω∫ t0f(s) sinωsds+2X0mω∫ t0f(s) sinω(t− s)ds− 1m2ω2∫ t0f(s) sinωsds∫ t0f(s′) sinω(t− s′)ds′+sinωtm2ω2∫ t0f(s)∫ s0f(s′) sinω(s− s′)ds′ds](A.3)where the second step used the integration by parts.The forth line can be written assinωtm2ω2∫ t0f(s)∫ s0f(s′) sinω(s− s′)ds′ds=1m2ω2∫ t0∫ s0f(s)f(s′)[sinω(t− s′) sinωs− sinω(t− s) sinωs′]ds′ds(A.4)by usingsinωt sinω(s− s′) = sinω(t− s′) sinωs− sinω(t− s) sinωs′,(A.5)while the third line is− 1m2ω2∫ t0∫ t0f(s)f(s′) sinωs sinω(t− s′)dsds′= − 1m2ω2∫ t0ds∫ s0ds′f(s)f(s′)[sinω(t− s′) sinωs+ sinω(t− s) sinωs′]dsds′ (A.6)130since for a function f(s, s′) we have a double integral can be written as∫ t0∫ t0f(s, s′)dsds′ =∫ t0∫ s0(f(s, s′) + f(s′, s))dsds′. (A.7)Therefore, the kernel for a forced harmonic oscillator is given byK =( mω2pii~ sinωt)1/2eiScl/~ (A.8)whereScl =mω2 sinωt[(X2 +X20 ) cosωt− 2XX0 +2Xmω∫ t0f(s) sinωsds+2X0mω∫ t0f(s) sinω(t− s)ds− 2m2ω2∫ t0∫ s0f(s)f(s′) sinω(t− s) sinωs′dsds′]. (A.9)131Appendix BInfluence Functional fromRadiation FieldsWe evaluate the influence functional F(j, j′) from radiation fields:F(j, j′) =∫∫∫ ∏λ=1,2daλdaλ0daλ′0 Krad(aλ, aλ0)K∗rad(aλ, aλ′0 )Φ(aλ0)Φ∗(aλ′0 ).(B.1)It is assumed that there are no ambient radiation fields initially. Thenthe initial wavefunctional for radiation fields is in the ground state,Φ(aλ0) = exp[−∑kk2aλ∗0 (−k)aλ0(k)](B.2)up to normalization.The influence functional can be written asF(j, j′) =∫∫∫da1da10da1′0 Krad(a1, a10)K∗rad(a1, a1′0 )Φ(a10)Φ∗(a1′0 )×∫∫∫da2da20da2′0 Krad(a2, a20)K∗rad(a2, a2′0 )Φ(a20)Φ∗(a2′0 ).(B.3)132For each polarization λ = 1, 2, we haveKrad(aλ, aλ0) =∫exp i(Srad + Sint)Daλ=∫Daλ exp i∫ (∑k12(a˙λ∗(k)a˙λ(k)− k2aλ∗(k)aλ(k))+x˙λ(t)aλ(k)eik·x(t))dt. (B.4)Since (B.4) is the path integral for the forced harmonic oscillator, wehave Krad ∝ eiScl where [21] (Appendix A)Scl =∑kk2 sin kt[(aλ(k)aλ(−k) + aλ0(k)aλ0(−k)) cos kt− 2aλ(k)aλ0(−k)+2aλ(−k)k∫ t0x˙λeik·x(s) sin ksds+2aλ0(−k)k∫ t0x˙λ(s)eik·x(s) sin k(t− s)ds− 2k2∫ t0ds∫ s0ds′x˙λ(s)eik·x(s)x˙λ(s′)e−ik·x(s′) sin k(t− s) sin ks′].(B.5)By substituting (B.2) and (B.5) into (B.3), we obtain (2.15).For the case of gravitational radiation, we again has the action for theforced harmonic oscillator in the weak gravity regime (2.10).By introducing the initial ground state wavefunctional [100–102],Φ(h(λ)0 ) = exp[−∑kk2h(λ)ij (k)h(λ)ij(−k)](B.6)with rescaled hij →√2piGhij , and Krad ∝ eiScl (subscript 0 indicates the133initial state) whereScl =∑kk2 sin kt[(h(λ)ij (k)h(λ)ij(−k) + h(λ)0ij (k)h(λ)ij0 (−k)) cos kt−2h(λ)ij (k)h(λ)ij0 (−k) +2√2piGh(λ)ij (−k)k∫ t0T ij(λ)(k, s) sin ksds+2√2piGh(λ)0ij (−k)k∫ t0T ij(λ)(k, s) sin k(t− s)ds−4piGk2∫ t0ds∫ s0ds′T ij(λ)(k, s)T (λ)ij (−k, s′) sin k(t− s) sin ks′](B.7)intoF(T, T ′) =∫∫∫ ∏λ=1,2dh(λ)dh(λ)0 dh(λ)′0 Krad(h(λ), h(λ)0 )K∗rad(h(λ), h(λ)′0 )×Φ(h(λ)0 )Φ∗(h(λ)′0 ) (B.8)we obtain (2.23).134Appendix CExciton Scattering by anImpurityFor an exciton scattered by an impurity in a 1D lattice, the Hamiltonian is(3.26) with free-exciton states (3.27) and energy (3.29).The Lippmann-Schwinger equation represents the total wave-function|ψ〉 as a sum of the incident state |ϕ〉 and the scattered state, i.e., |ψ〉 =|ϕ〉 + Gˆ0Vˆ |ψ〉, where Gˆ0 = [Ee(k) − Hˆ0 + i]−1 is the Green’s functionof the exciton and  stands for any positive infinitessimal. Projecting theLippmann-Schwinger equation onto 〈n| givesψ(n) =1√Nϕk(n) + V0 ψ(0)∑qeiqnE(k)− E(q) + i . (C.1)We find ψ(0) by letting n = 0 in this equation, and reduce it toψ(n) =1√N[eikn +V0I11− V0I2], (C.2)whereI1 =1N∑qeiqnEe(k)− Ee(q) + i ,I2 =1N∑q1Ee(k)− Ee(q) + i . (C.3)In [68] this expression was examined with parabolic approximation forthe dispersion of exciton, which is valid at small k. Here we go beyond this135π−πRe[k]Im[k]a) n > 0, Im[z  ] > 0π−πz0−z00π−πIm[k]b) n > 0, Im[z  ] < 0z0−z00Re[k]Im[k]z0−z0c) n < 0, Im[z  ] > 0π−πRe[k]Im[k]z0−z0d) n < 0, Im[z  ] < 00 0Re[k]k −kk k−k −kqq qqk kk −k−k kqq qqFigure C.1: Integration contour for various n and k.136approximation. Replacing the sum by the integral assuming N →∞:∑q→ N2pi∫ pi−pidq, (C.4)and writing Ee(k) = (E0 + 2J)− 4J sin2(k/2), we getI1 =18piJ∫ pi−pidq eiqnsin2(q/2)− sin2(k/2)− iJ . (C.5)We are interested in the complex values of k = k′ + ik′′ (Re(k) = k′,Im(k) = k′′), as they provide poles of the scattering amplitude correspondingto bound states. We choose k′ = 0 or pi to keep the exciton energy real.Then the integral has no poles on the real axis, and iJ can be omitted.The function has two series of poles, q = ±k, one laying in the upper, onein the lower half-plane. In each series the poles are shifted with respectto each other by 2pi. We choose the integration contour as a rectangle,with one horizontal side along the real axis segment [−pi, pi], two verticalsides and the second horizontal line approaching +i∞ if n > 0, and −i∞ ifn < 0. These contours are shown in Figure C.1, along with a pair of poles±k with −pi < k′ < pi. If k′ = pi, the contour is shifted by pi to the right,using the periodicity of the integrand. The integrals along the right and leftsides of the rectangles cancel each other, and the integral along the top sidevanishes, and only one of the poles q = ±k always lays within the contour.Using Taylor series, we can writesin2(q/2)− sin2(k/2) = (q − k) sin(k/2) cos(k/2) +O(q − k)2 (C.6)which givesI1 =18piJ∫ pi−pidq eiqnsin2(q/2)− sin2(k/2) =2pii8piJeik|n|sin k/2 cos k/2. (C.7)Considering four combinations of sgn(n) and sgn(k′′), as shown in Figure137C.1, we find that the result of integration is:I1 =e−|k′′||n|2J sinh |k′′| if k′ = 0,−(−1)ne−|k′′||n|2J sinh |k′′| if k′ = pi.(C.8)and I2 = I1 with n = 0.Then (C.2) can be written asψ(n) =1√N[eikn +Re(k)eik|n|](C.9)where the scattering amplitude Re(k) isRe(k) =V02J sinh |k′′| − V0 if k′ = 0,− V02J sinh |k′′|+ V0 if k′ = pi.(C.10)The poles of this scattering amplitude coincide with those determinedby (3.34) in the limit of large N . Again, as in Section 3.2, we note that itis the combination of signs of J and V0, that determines which of these twocases is realized, as sinh |k′′| is always positive. When J > 0 and V0 > 0,the only pole in the reflection amplitude appears at k = i sinh−1(V0/2J). Itcorresponds to a single bound state marked by (i) in Figure 3.2 (a) abovean upper bound of the continuum spectrum with V0 > 0. The pole atk = pi + i sinh−1(−V0/2J) is realized when V0 < 0, and corresponds to abound state (ii) in Figure 3.2 (a) below the lower bound. For J < 0, on theother hand, poles at k = i sinh−1(V0/2J) (iii) and k = pi+ i sinh−1(−V0/2J)(iv) correspond, respectively, to a bound state below the lower bound withV0 < 0, and above the upper bound with V0 > 0.138Appendix DSolving Lippmann-SchwingerEquation with Method ofContinued FractionsIn the method of continued fractions [83], the Lippmann-Schwinger equationis solved by means of iterations. In the first order,|ΨK〉 ≈ |ΦK〉+ β(0)K Gˆ0Vˆβ(0)K − γ(1)K|ΦK〉, (D.1)where |ΦK〉 is a biexciton wave-function without the impurity (3.23), β(0)K =〈ΦK |Vˆ |ΦK〉, γ(1)K = 〈ΦK |Vˆ Gˆ0Vˆ |ΦK〉.We calculate γ(1)K asγ(1)K =∑Q|〈ΦK |Vˆ |ΦQ〉|2Eb(K)− Eb(Q) + i=ND4piJ2pi/2∫−pi/2dQ |〈ΦK |Vˆ |ΦQ〉|2sin2Q− sin2K + iD . (D.2)For complex K, this integral can be carried out by the same method asin Appendix C for single exciton, with the result (K˜ = K ′ + iK ′′):γ(1)K˜=iND|〈ΦK˜ |Vˆ |ΦK˜〉|22J2 sin 2K˜. (D.3)139and the matrix element of the potential is〈ΦK˜ |Vˆ |ΦK˜〉 =4V0NS(K ′, |K ′′|), (D.4)whereS(K ′, |K ′′|) =N/2∑s=−N/2+1e−2|K′′|sφK′−i|K′′|(s)φK′+i|K′′|(s) (D.5)is a real-valued function of K ′ and |K ′′|. The function S(K ′, |K ′′|) multipliesV0, and reflects the averaging of the delta-potential by the wave function ofbiexciton’s relative coordinate. Then the scattered state in (D.1) has polesif1 =2iV0DS(K′, |K ′′|)/J2sin(2K ′) cosh(2|K ′′|) + i cos(2K ′) sinh(2|K ′′|) . (D.6)This equality with real biexciton energy can be satisfied when (a) K ′ = 0,or (b) K ′ = pi/2. These two cases lead to two types of poles, defined,respectively, by the following equations:K ′ = 0, S(0, |K ′′|) = J2 sinh(2|K ′′|)2V0D,K ′ =pi2, S(pi/2, |K ′′|) = −J2 sinh(2|K ′′|)2V0D.(D.7)The first type of poles correspond to the poles marked as (i),(iii) inFigure 3.4 (a), the second – to those marked as (ii) and (iv). Again, whichof the two poles can be realized is determined by the signs of V0 and D (thelatter determines the signs of the effective mass of biexciton near K ∼ 0 andnear K ∼ pi/2).Then the projection of (D.1) onto 〈r, s| can be written asΨK(r, s) ≈ ΦK(r, s) +Rb(K)ΦK(r, s), (D.8)140where the scattering amplitude Rb(K) isRb(K) = 2iDV0S(K′, |K ′′|)J2 sin 2(K ′ + i|K ′′|)− 2iDV0S(K ′, |K ′′|) . (D.9)141

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