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Excitonic modes and phonons in biological molecules Zhu, Zhen 2018

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Excitonic Modes and Phonons inBiological MoleculesbyZhen ZhuB.Sc., Peking University, 2006M.Sc., The University of British Columbia, 2009A 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)March 2018c© Zhen Zhu 2018AbstractThere are two kinds of environmental modes in open quantum systems: thedelocalized modes which can typically be modeled by “oscillator bath” mod-els and the localized modes which can be mapped to “spin bath” modes. Tounderstand the quantum phenomena in photosynthetic energy transfer, weat first conduct thorough studies of the proper modeling of light harvestingmolecules as well as their interactions with the central system. These modescan couple to the system by either modulating the on-site energy (Holsteincoupling) or modulating the hopping amplitude (Peierls coupling). Only theHolstein couplings of delocalized modes have been extensively studied. Theimportance of other types of couplings is rarely discussed in the literature.For the spin bath, we study a particle hopping around a general lattice,coupled to a spin bath. Analytical results are found for the dynamics of theinfluence functional and for the reduced density matrix of the particle invarious parameter regimes. Spin baths behave qualitatively differently fromoscillator baths and dissipation and decoherence happen independently indifferent parameter regimes.For the Peierls couplings, we start with a dimer model for light harvest-ing molecules, which contains a reaction center and both types of phononcouplings. We find that the effect of Peierls type coupling on the transferrate can be significant even when it is not noticeable in the spectrum. Ourstudy suggests that Peierls couplings cannot be easily neglected in light har-vesting molecules in which the energy difference between the sites is usuallymuch larger than the hopping amplitude. We apply our method to a reallight harvesting model. Although we do not have much detailed informationof the Peierls couplings in vivo, we find that vibrational phonons can affectthe path-selecting of the central particles as well as increasing the transferrate.iiLay SummaryPhotosynthesis may be the single most important biochemical process onEarth, since all higher life depends on it. The underlying mechanism ofphotosynthesis has fascinated scientists for centuries and yet still remainscontroversial. Recent experiments suggest that the photosynthetic mech-anism may involve the coherent quantum-mechanical transport of energyinside ”light-harvesting molecules” (in a way similar to quantum coherencein, eg., lasers). We study this subject from a theoretical physics perspec-tive, and propose a new theoretical approach. This research should helpus to understand how Nature captures solar energy with such astonishingefficiency.iiiPrefaceThe original work presented in Section 2.4 was carried out by Zhen Zhu whoalso developed the conception, scope and methods of this research.The original work presented in Chapter 3 is a generalization of the workdone by N.V. Prokof’ev and P. C. E. Stamp[1]. Section 3.2 was publishedin “Pure phase decoherence in a ring geometry”[2] in Phys. Rev. A at2010. Zhen Zhu conducted most of calculations in the manuscript. Theother original work in this chapter was carried out by Zhen Zhu, with variousdegrees of conception, consultation and editing support from P. C. E. Stamp.The original work presented in Chapter 4-5 was carried out by Zhen Zhuwho also developed the methods of this research, with various degrees ofconception, methods, consultation and editing support from P. C. E. Stampand Mona Berciu.Original work reported from Chapter 3-5 are under preparation for pub-lication in the near future.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Biological Systems with Quantum Phenomena . . . . . . . . 31.1.1 Fenna-Matthew-Olson Complexes . . . . . . . . . . . 41.1.2 Light Harvesting System I,II . . . . . . . . . . . . . . 61.1.3 Avian Compass . . . . . . . . . . . . . . . . . . . . . 61.2 Quantum Model for Photosynthetic Energy Transfer . . . . 81.2.1 Single Chlorophyll Excitations and Diagonal couplings 101.2.2 Exciton Propagation and Off-Diagonal Couplings . . 121.2.3 Vibrational Modes . . . . . . . . . . . . . . . . . . . . 141.2.4 Two-Phonon Interaction . . . . . . . . . . . . . . . . 161.3 The Proposed Hamiltonian for Energy Transfer . . . . . . . 171.3.1 Coherence in light harvesting molecules . . . . . . . . 191.3.2 Acoustic Modes and Vibrational Modes . . . . . . . . 201.3.3 Diagonal Coupling and Off-Diagonal Coupling . . . . 211.3.4 Two-Phonon Terms . . . . . . . . . . . . . . . . . . . 222 Limitations of Master Equation Approaches to Open Quan-tum Systems and Decoherence . . . . . . . . . . . . . . . . . 232.1 A Brief History of Studies on Open Quantum Systems . . . . 24vTable of Contents2.2 Quantum Master Equation Approaches . . . . . . . . . . . . 272.2.1 Redfield Master Equation . . . . . . . . . . . . . . . . 292.2.2 Example: Central Spin Model . . . . . . . . . . . . . 332.3 Path Integral Formalism . . . . . . . . . . . . . . . . . . . . 362.4 Decoherence and False Decoherence . . . . . . . . . . . . . . 402.4.1 Central Spin Couples to a Spin Chain . . . . . . . . . 403 Spin Bath Model . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 The Model of a Discretized System Coupled to a Spin Bath . 513.1.1 Parameter Regions . . . . . . . . . . . . . . . . . . . 533.2 Pure Phase Decoherence Region . . . . . . . . . . . . . . . . 553.2.1 Bare System Behaviour . . . . . . . . . . . . . . . . . 563.2.2 System Plus the Bath . . . . . . . . . . . . . . . . . . 623.2.3 Wave Package Interference . . . . . . . . . . . . . . . 663.3 High Field Region . . . . . . . . . . . . . . . . . . . . . . . . 693.3.1 Orthogonal Blocking Approximation . . . . . . . . . . 713.3.2 Perpendicular Coupling . . . . . . . . . . . . . . . . . 733.3.3 Spectral Function . . . . . . . . . . . . . . . . . . . . 793.3.4 General Coupling . . . . . . . . . . . . . . . . . . . . 833.3.5 General Initial State of Spin Bath . . . . . . . . . . . 843.3.6 Finite hk Expansions . . . . . . . . . . . . . . . . . . 854 Oscillatior Bath Model: Role of Vibrational Modes in En-ergy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1 Model of Resonant Energy Transfer into a Reaction Center . 914.2 Holstein Coupling . . . . . . . . . . . . . . . . . . . . . . . . 944.3 Peierls Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Jaynes-Cummings Model and Beyond . . . . . . . . . 984.4 The Full Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 1024.5 Role of Off-Diagonal Couplings in Resonant Energy Transfer 1044.6 Two-Phonon Interactions for Optical Phonons . . . . . . . . 1054.6.1 Two-Phonon Interaction in the Holstein model . . . . 1064.6.2 A General Path Integral Formalism . . . . . . . . . . 1105 Multi-Pathway Light Harvesting Molecules . . . . . . . . . 1175.1 Approach to the Complete Hamiltonian for Energy Transfer 1185.2 Application to the FMO Complex . . . . . . . . . . . . . . . 1215.2.1 Back to Two Level Model . . . . . . . . . . . . . . . . 1245.2.2 Reduced 3-Site Model for FMO complexes . . . . . . 1255.3 Pathway Selecting Mechanism . . . . . . . . . . . . . . . . . 129viTable of Contents5.3.1 Ladder Model . . . . . . . . . . . . . . . . . . . . . . 1315.3.2 Symmetric Model . . . . . . . . . . . . . . . . . . . . 1355.3.3 Symmetric Pathways with a Single Entry . . . . . . . 1375.4 Summary and Future Researches . . . . . . . . . . . . . . . . 139Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141AppendicesA Spin Chain Decoherence . . . . . . . . . . . . . . . . . . . . . 156A.1 Neglecting Commutator . . . . . . . . . . . . . . . . . . . . . 156A.2 Adiabatic Decoupling . . . . . . . . . . . . . . . . . . . . . . 157B Pure Phase Decoherence . . . . . . . . . . . . . . . . . . . . . 160B.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.2 Including Phase Decoherence . . . . . . . . . . . . . . . . . 163C Orthogonal Blocking Approximation . . . . . . . . . . . . . . 168C.1 A Useful Integral . . . . . . . . . . . . . . . . . . . . . . . . 168C.2 OBA Expansion of the System . . . . . . . . . . . . . . . . . 169C.3 Zero Polarization . . . . . . . . . . . . . . . . . . . . . . . . 174C.4 M = 1 Polarization . . . . . . . . . . . . . . . . . . . . . . . 175C.5 General Coupling . . . . . . . . . . . . . . . . . . . . . . . . 178C.6 General Initial State in the High Field Limit . . . . . . . . . 180D Oscillator Bath model: Role of Peierls Coupling in EnergyTransfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183D.1 Green Function . . . . . . . . . . . . . . . . . . . . . . . . . 183D.2 Bethe Lattice Reaction Center . . . . . . . . . . . . . . . . . 185D.3 Logarithm Fitting . . . . . . . . . . . . . . . . . . . . . . . . 187viiList of Figures1.1 The illustration of the molecular structure of FMO complexes.The left panel shows the three identical subunits of the FMOtrimer. The right panel is the structure of each subunit. Itis composed by 8 bacteriachlorophylls. Reprinted (adapted)with permission from [3]. Copyright (2011) American Chem-ical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The Molecular structure of light harvesting molecules LH1and LH2 based on [4]. The look is from the top so the B800ring and the B850 ring of LH2 overlap with each other. Theright panel is the B875 ring of LH1. Both of them have acircular symmetry. . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Molecular structure of the magnetoreception molecules FADH*based on [5]. The blue arrows suggest the electron pathways,though there is no experimental evidence of this. . . . . . . . 81.4 The structure and the absorption/fluorescence spectrum of asingle Bchl-a. The structure of a single Bchl-a is illustrated inthe left panel(without the hydrocarbon tail). The left panelis the absorption spectrum of Bchl-a and the left panel is itsfluorescence spectrum. We can see that there are two majorpeaks in the absorption: the Soret Bands and the Only theQy “band” transition is included in the current calculationsof excitonic energy transfer. Reprinted with permission from[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 The overlap integral of two 2pz orbitals displaced in x direc-tion by a distance d. We can see that the integral is stillnoticeable around 7a0. a0 is the Bohr Radius which equals to0.53A˚. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 The spectral function J(ω) for each Bchls in FMO complex.The result is acquired by fitting the auto-correlation functionof the absorption spectrum. Reprinted with permission from[7]. Copyright (2011) American Chemical Society. . . . . . . . 16viiiList of Figures2.1 Illustration of our spin chain model. A central spin couplesto a spin chain. . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 Decoherence factor κ(t) (2.83) of model 1 as a function oftime. There is no real decoherence and κ goes back to 1periodically. . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 This is the decoherence factor of model 2 as a function oftime. The loss of cohcerence is clear over time. . . . . . . . . 462.4 Figure of the coupling J(t) over time. These lines representk = 0.1, 0.5, 1, 10 respectively. . . . . . . . . . . . . . . . . . . 473.1 Illustration of truncation to discretized system: At left, an8-site ring with nearest-neighbour hopping between sites. Atright a potential U(R) with 8 potential wells (shown heresymmetric under rotations by pi/4), depicted as a contourmap (with lower potential shown darker). When truncatedto the 8 lowest eigenstates, this is equivalent to the 8-site model. 563.2 A particular path in a path integral for the particle, shownhere for an N = 3 ring. This path, from site 0 to site 1, haswinding number p = 1. . . . . . . . . . . . . . . . . . . . . . . 583.3 Results for the free particle for N = 3 and for a particleinitially on site 1. Left: The probabilities to occupy site 1(full line), 2 (large dashes), and 3 (small dashes). Right: thecurrent from site 1 to site 2. Top: Φ = 0. Bottom: Φ = pi/2(i.e. φ = pi/6). . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Plot of Pj1(t) for a 3-site ring, for a particle initially on site1, in the strong decoherence limit. Left: The probability tooccupy site 1 (full line), 2 (large dashes), and 3 (small dashes).Right: the current from site 1 to site 2 (compare Fig. 3.3).The results do not depend on Φ. . . . . . . . . . . . . . . . . 623.5 Interference between 2 wavepackets in the strong decoherencelimit. The packets start at site 0 and site jo = 50 at t = 0,and their relative velocity is pi2 , in phase units. . . . . . . . . . 683.6 Figures of ρ00(t) in (3.86) in high field limit. Different colorsfor different coupling strength parameter κ = κ01 = κ12 = κ02(3.84). Blue:κ = 10−3; Red:κ = 10−1; Green κ = 102; . . . . . 78ixList of Figures3.7 Figures of spectral function for both the zero field limit (3.94)and the hight field limit (3.98). Blue: zero field limit(thepure phase decoherence region); Red: High field limit. Eachfigure has different coupling strengths. The first row, left:κ, λ = 0.001; right: κ, λ = 0.01. The second row: left 0.05;right 0.1. The third row: left 0.5; right 1.0. The last row: 10.0. 824.1 An illustration of our whole system. In the top panel weshow a schematic picture for our model. c1 and c2 are thetwo pigments in a molecule. The reaction center model inthe hexagon is a Bethe lattice. Phonons couple to the systemthrough gH (Holstein coupling) and gP (Peierls coupling). Inthe bottom panel, we show the energy difference in the model.∆ is the energy differences between c1 and c2. 2D is thebandwidth of the RC. t0 is the hopping amplitude betweenc1 and c2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Plots for the pure Holstein coupling case. For all plots, wechoose the energy of site c2 to be ε2 = 2 and the energy ofsite c1 to be ε = 0 (∆ = 2). The intra-molecules hoppingamplitude is t0 = 0.1. The RC has a bandwidth D = 1 andcouples to the molecule by t1 = 0.2. (a): Plot of the transferrate when the phonon frequency is around the resonant pointΩ ≈ ∆ = 2. From (b) to (d) the phonon frequency is set tobe on resonance Ω = ∆ = 2. (b): The transfer rate in theintermediate gH region. (c): The reduced density matrix ele-ments ρ11(the population on site c2) and ρ00(the populationon site c1) when gH = 1.5; the sub-panel is the transfer ratein the large gH region. (d): Local spectra at site c1. . . . . . 954.3 Plots for the pure Peierls coupling case. The values of the pa-rameters in the Hamiltonian are the same as Fig.4.2. Phononscouple to the hopping amplitude rather than the on-site en-ergies. (a): Plot of transfer rate when the phonon frequencyaround the resonant point Ω ≈ ∆ = 2. From (b) to (d) thephonon frequency is set on resonance: Ω = ∆ = 2. (b): thetransfer rate in small gP region. (c): Plot of the reduceddensity matrix elements ρ11 and ρ00 when gP = 0.5; the sub-panel is the transfer rate in intermediate gP region. (d): localspectra at site c1. . . . . . . . . . . . . . . . . . . . . . . . . . 96xList of Figures4.4 The plots obtained from fitting (1 − ρRC) as an exponentialfunction of t for the pure Holstein case and the pure Peierlscase. The parameters of the bare model are the same as thevalues in previous figures. The range of t is taken from 0 to200. (a) The transfer rate γ as a function of gH (blue circle)and gP (red star); (b) The transfer rate γ as a function oft˜H0 (blue circle) and t˜P0 (red star);(c) The linear correlationcoefficient between log(1−ρRC) and t; (d) the log(1−ρRC) v.s.t plot in two transition regions: the lines within the dashedcircle represent the region which the exponential model startsto fail; the line within the solid circle represent where theexponential behavior is restored. . . . . . . . . . . . . . . . . 974.5 The plot of the transfer rate γ as a function of Peierls couplinggP for the JCM and the full Hamiltonian. . . . . . . . . . . . 994.6 The plots of the transfer rate γ as a function of Peierls cou-pling gP . Different colors stand for the JCM(green) , the Rabimodel (red) and the full Hamiltonian (blue). . . . . . . . . . 1004.7 The first 11 energy levels as a function of Peierls couplinggP for the JCM (top), the Rabi Model (middle), and the fullHamiltonian with t0 = 0.1(bottom). . . . . . . . . . . . . . . 1014.8 Plots when both types of phonon couplings exist. The val-ues of the parameters in the Hamiltonian are the same as forFig.4.2. (a): The local spectra at site c1 with various cou-plings. (b): The transfer rate when t˜0 is fixed at 0.05, withgP determined by (4.20). (c): The transfer rate when gP isfixed at 0.06. (d) the transfer rate when gH is fixed at 0.7. . 1034.9 The plots of transfer rate γ for the full Hamiltonian (3.1) withboth the Peierls coupling and the Holstein coupling. Thevalues of the parameters in the Hamiltonian are the sameas Fig.4.2. (a) gH = 0.1, 0.6, γ as a function of gP . (b)gP = 0.1, 0.05, γ as a function of gH . . . . . . . . . . . . . . 1045.1 This is the energy path-way of a 8-site model from with on-siteenergies, as well as the scheme to reduce to a 3-site model.Site-3 is the site that connecting to the RC. Site-8 has thehighest on-site energy and receives excitons from the base-plate directly. Reprinted with permission from [8]. Copyright(2011) American Chemical Society. . . . . . . . . . . . . . . . 121xiList of Figures5.2 the plot of the transfer rate of the original two level systemwith different deviations d and phonon frequencies Ω, D =1, t0 = 0.2 a) the transfer rate with Deviation d = 0, differentlines stands for different phono frequencies, b) the transferrate with Deviation d = 1.0, different lines stands for differentphono frequencies. . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 a) The plot of transfer rate of Hamiltonian (5.12) with differ-ent types of phonon couplings, the reaction center parametersare chosen as D = 1, t0 = 0.2, d(deviation) = 1; b) the trans-fer rate of off-diagonal coupling with different t0, D = 1, d = 1c) the transfer rate of off-diagonal coupling with different d,D = 1, t0 = 0.2; d) the transfer rate of off-diagonal couplingwith different bandwidth D,t0 = 0.2, d = 1. . . . . . . . . . . 1265.4 The plots of Hamiltonian (5.12) with different Peierls cou-plings, the reaction center parameters are chosen as D =1, t0 = 0.2, d(deviation) = 1 a) the local spectrum at site-1,b), c), d): the time evolution of each pigments with differentgP s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.5 This is the cofactors-only structure of the reaction center ofRhodobacter sphaeroides . DA DB and BA BB are four Bchlpigments with absorption peak at 870nm (P870). HA and HBare two bacteriapheophytins(BPh, basically the Bchl withoutthe central Mg2+), and QA and QB are two quinones. At thebottom there is a metal ion Fe2+. Reprinted with permissionfrom [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.6 Illustration of the two-pathway models. (a)The ladder model:site-2 serves as the ladder between site-3 and site-1. (b)Thesymmetric Model: site-2 has the same energy as site-1. Inboth cases, the exicitation is initially created at site-1 andeventually transferred into RC.(c) The symmetric 4-site model.1315.7 The transfer rate of the Hamiltonian (5.14) . The phononenergy Ω = 2. The parameters of the RC couplings are thesame as Fig. 4.2. Different lines stand for different types ofphonon couplings. For simplicity we assume the same cou-pling strength for all coupled phonon modes. . . . . . . . . . 132xiiList of Figures5.8 The time evolution of the Hamiltonian (5.14) for the weakcoupling case gP = 0.16(top panel) and strong coupling gP =0.72(bottom panel). The parameters are the same as Fig. 5.7.Different colors indicate different site populations(ρ11, ρ22,ρ33). a) phonons couple to all three off-diagonal hoppings,b) phonons couple to 1-2 and 2-3 hopping, c) phonons onlycouple to 1-3, d) no phonon coupling at all. . . . . . . . . . . 1335.9 The time evolution of the Hamiltonian (5.14). Different col-ors indicate different off-diagonal components (ρ12, ρ13, ρ23).Here we use the normalized density matrix element|ρij |trρ . . . 1345.10 The time evolution of the Hamiltonian (5.15). Different colorsindicate different site populations(ρ11, ρ22, ρ33). The top fourfigures are for the weak interaction gP = 0.16; and the bot-tom four figures are for the strong interaction gP = 0.72. a)phonons couple to all three off-diagonal hoppings, b) phononscouple to 1-2 and 2-3 hopping, c) phonons only couple to 1-3,d) no phonon coupling at all. Top panels are small couplingsgP = 0.16. Bottom panels are large couplings gP = 0.72 . . . 1365.11 The transfer rate of the Hamiltonian (5.15) . The phononenergy Ω = 2. The parameters of the RC couplings are thesame as Fig. 4.2. Different lines stand for different types ofphonon couplings. For simplicity we assume the same cou-pling strength for all coupled phonon modes. . . . . . . . . . 1375.12 The transfer rate of the 4 site model (5.16). The transfer ratewhen phonons couples to different hopping amplitude. Allcouplings are either the same or zero. . . . . . . . . . . . . . 1385.13 The time evolution of a 4 site model (5.16). Different colorsindicate different site populations(ρ11, ρ22, ρ33, ρ44). a, b)Peierls phonons couple to the left branch of the molecules;c,d) Peierls phonons couple to both branches. . . . . . . . . 138D.1 Illustration of phase diagram of single site system. The greyzone indicates there is no real root of (D.15). In this zone theexciton would be transferred into the sink completely. In thegreen zone there is one real root of (D.15) . There are tworeal roots of (D.15) in the rest of the area. . . . . . . . . . . 186xiiiList of FiguresD.2 The plots of fitting (1 − ρRC) as a exponential function of twhen Ω ≈ ∆. The parameters of the bare model is same asthe values in previous figures. The range of t is taken from0 to 200. (a) red star :gH = 0, gP = 0.05, the decay rate γas a function of Ω/∆; blue circle star :gP = 0, gH = 0.5, thedecay rate γ as a function of Ω/∆. (b) the linear correlationcoefficient between log(1− ρRC) and t. . . . . . . . . . . . . 187D.3 The plots of fitting (1 − ρRC) as a exponential function oft for the pure Holstein case and the pure Peierls case. Theparameters of the bare model is same as the values in previousfigures. The range of t is taken from 0 to 200. (a) the decayrate F as a function of gH(blue circle) and gP (red star); (b)the decay rate F as a function of t˜H0 (blue circle) and t˜P0 (redstar);(c) the linear correlation coefficient between log(1−ρRC)and t; (d) the log(1−ρRC) v.s. t plot in two transition regions:the lines within the dashed circle represent the region wherethe exponential model start to fail; the line within the solidcircle represent where the exponential behavior is restored. . 188xivAcknowledgementsIt is a pleasure to thank those who made this thesis possible.I owe my deepest gratitude to my supervisor, Prof. Philip Stamp. Thisthesis would not have been possible without his great efforts to educate andto help me in so many ways. He taught me not only physics, but also howto write up and explain things clearly and simply. Throughout my years ofgraduate study at UBC, he has always been a good adviser and a friend tome.It is an honor for me to thank the many people with whom I discussed therelated work and have been benefited greatly from such discussion: MonaBerciu, Amnon Aharony, Ora Entin-Wohlman, Hans Briegel, Klaus Schul-ten, Peter Hore, Igor S. Tupitsyn, Birgitta Whaley, Ilia Solov’yov and BillUnruh.I am indebted to my many colleagues for the stimulating discussion andsometimes even excited and heated argument: Ryan McKenzie, Tim Cox,Leon Ruocco, Si Chen, Jinshan Wu, Daoyan Wang, Jianyang He and QingdiWang.Most importantly, none of this would have been possible without thelove and patience of my family. Especially I want to mention my parentsYuqin Hu and Miaolong Zhu, my grandmother Guihua Hu, my aunts anduncles Yudi, Yuhua, Yujun, Qi and Gang. My family has been a constantsource of love, concern, support and strength all these years. I would like toexpress my heart-felt gratitude to my family.xvDedicationTo my late father Miaolong ZhuxviChapter 1IntroductionOne of the most exciting experiments in the last decade is probably the dis-covery of coherent quantum beating behaviors in photosynthetic systems[9–13]. People used to believe that quantum mechanics is generally unnecessaryto understand biological systems. Due to the enormous number of environ-mental degrees of freedom and the high physiological temperature, quantumcoherence is believed to fade away rapidly and a classical description of bothdynamics and states suffices. This discovery upsets scientists. At both lowtemperature(77K)[9, 10] and room temperature (277K)[13], quantum co-herent beating is observed in various types of light harvesting molecule ata time scale comparable to the photosynthetic process. Controversies havethus arisen about the actual functional role quantum coherence plays in thisphotosynthesis. There is no widely accepted conclusion that has been drawnyet. To address this problem, we have to thoroughly understand the behav-ior of open quantum systems coupling to various types of environmentalmodes.There are usually two kinds of environmental modes in mesoscopic ormacroscopic systems. One can categorize the environment oscillators ininsulating systems by their origins: i) delocalized modes: such as the collec-tive movements of whole molecules; ii) localized modes: such as the intra-molecule vibration modes.Delocalized modes are usually weakly-coupled low energy modes. Thenumber of such modes is usually huge due to the enormous number of degreesof freedom of biological molecules. They usually have a delocalized origin(representing acoustic phonons, collective movements of electrons, phonons,photons, spin waves, etc.) so they can typically be modeled by ”oscillatorbath” models [14–17]. The oscillator bath models have been extensivelystudied in the literature. It is one of the fundamental topics in solid statephysics, as well as many other aspects of physics. In the language of spec-troscopy, acoustic phonons are usually associated with the low energy con-tinuum in spectra . In these models, decoherence always goes hand-in-handwith dissipation because of to the fluctuation-relaxation theorem[18, 19].There are other modes that have localized microscopic origins(optical1Chapter 1. Introductionphonons, defects, dislocations, dangling bonds, nuclear and paramagneticimpurity spins, vibrations of chemical bonds, charged amino acid residues,etc.). In spectroscopy, these modes are related to discrete peaks in spectra.Their numbers are usually fewer compared with delocalized modes (whichnumbers are proportional to the enormous degrees of freedom in huge bio-logical molecules), but their energies can be much higher, sometimes evencomparable to the exciton energy light harvesting molecules. Under cer-tain circumstance, these localized modes can be mapped into “spin bath”models[1, 20]. They behave quite differently from delocalized modes, so onecannot establish a general mapping between them[20]. In spin baths models,decoherence does not necessarily come together with dissipation. These lo-calized modes are often far from equilibrium while in oscillator bath modelsenvironmental modes are always assumed to be in thermal states.The importance of these localized modes, espeically these vibrationalmodes in biological molecules, has already been realized in resonant energytransfer processes in light harvesting molecules (see for example [7, 21, 22]).However, in most of these studies, only the coupling between the vibra-tional modes and the on-site energies are considered. This type of cou-pling is known as the Holstein coupling in solid state physics[23]. However,phonons could also couple to hopping amplitudes between the different sitesof pigments. The phonon-modulated transitions were first mentioned byPeierls with regard to the Peierls transition[24].They were then studied inpolyacetylene[25, 26] in the form of Su-Schrieffer-Heeger(SSH) model. Peo-ple found that the Peierls(and SSH) model show qualitatively different be-haviors compared to the Holstein model[27]. There are already plenty ofexperiments suggesting that vibration modes can directly couple to the exci-ton creation process in photosynthetic pigments[28]. Therefore, in principle,these modes should also couple to the hopping amplitude of excitons betweenpigments. Unfortunately, this type of coupling has never been discussed inbiological molecules. Although the role of oscillator baths in biological sys-tems has already been extensively studied[29–32], most studies only includethe diagonal couplings , a.k.a , the modulation of the on-site energy.On the other hand, although oscillator baths have been studied in-depthfor various systems during the last half century, research on spin baths isscattered in the literature. Many previous works focus on either mappingspin baths into oscillator baths [17] or studying some simplified spin toymodels such as the ”central spin model” (a central 2-level system (TLS)coupled to a background spin bath; for examples, see [1, 20, 33, 34]).In this thesis, we are going to address the following four objectives:(i) Understand the limitation of existing approaches in this newly found21.1. Biological Systems with Quantum Phenomenaquantum biology field; (ii) Extend the existing theoretical and numericaltechniques of spin bath models; (iii) Study the role of vibrational modes inresonant energy transfer, with both diagonal and non-diagonal couplings;(iv) Apply these studies in biological systems and understand the role oflarge scale quantum coherence in these systems.This thesis is organized as follows: the rest of this section serves as an in-troduction to quantum phenomena in biological systems. The proper quan-tum modeling of biological systems is also reviewed in this chapter. We givethe outline of this thesis and point out our contributions. In Chapter 2, wereview the current popular approaches for quantum open systems and theirlimitations in application to biological systems. We also introduce the pathintegral formalism which is used later in this thesis. In Chapter 3, we studythe spin bath model coupled to a general lattice model. We develop methodsto solve the model in different parameter regions systematically. In Chapter4, we study the role of vibrational modes in resonant energy transfer. Weinclude both the diagonal (Holstein) couplings, the non-diagonal (Peierls)couplings as well as the two-phonon couplings. In Chapter 5, we propose acomplete quantum Hamiltonian for excitons coupled to phonons in Fenna-Matthew-Olson complexes(FMO). We apply the theoretical and numericaltechniques developed in the previous chapters to address the various termsin this Hamiltonian. Although the lack of knowledge of the detailed phononspectrum in FMO prevent us from doing a thorough research, we still findsome interesting mechanisms introduced by the coupling to environmentalmodes.1.1 Biological Systems with QuantumPhenomenaIn principle, quantum mechanics ultimately dictates the atomic and molec-ular details of biological systems, as the correct underlying theory. Butsuch quantum effects are ”trivial” at the biological level. In this sectionwe review three specific biological systems which shows interesting quantumphenomena at time scales which are relevant to their biological functional-ity. In these cases, a proper quantum description of biological molecules isimportant to study the biological functionality. Such ”non-trivial” quantumeffects are the main motivation of this thesis. Although in this thesis, wemainly focus on the first system, i.e. the Fenna-Matthew-Olson complexes,a brief review of this new field of quantum biology could help understandingthe overall context of the following research.31.1. Biological Systems with Quantum Phenomena1.1.1 Fenna-Matthew-Olson ComplexesPhotosynthesis happens in various stages. The first stage is the absorptionof light. Excitations are created in antennae called light harvesting com-plexes(LHC), usually in form of excitons[35]. The excitions are then trans-ferred to reaction centers(RC) where the photosynthetic chemical reactionseventually happen. The transfer efficiency of LHC, i.e. the ratio betweenthe amount of photons transported to RC and the amount of photons ab-sorbed by the antennae, is fantastically high, almost 100% efficient[4]. Thetypical time for the whole exciton transfer process in LHCs is at the orderof several pico seconds [36].The Fenna-Matthew-Olson (FMO) complex is an LHC which exists inthe purple bacteria Chlorobium tepidum. It was the first pigment-proteincomplex whose structure was solved by X-ray crystallography[37, 38]. Therecently refined structure of FMO can be found in [3]. It has a global C3symmetry with an arrangement of three identical subunits. People usedto believe that each subunit contains seven bacteriachlorophylls (BChls)with the nearest neighbor distance of 11.3 to 14.4 A˚, while the distancebetween nearest neighbors in different subunits of the trimer is about 24 A˚[39, 40]. An eighth BChls has been found recently[41]. They are almost notinteracting with Bchls in two different subunits[37]. FMO complexes act as”antennae” in photosynthetic process. Their main function is to transportphoton energies into the RC for photosynthesis chemical reactions.In 2007 , Engel et al observed the quantum beating in FMO complexesat 77K by using a 2D femtosecond nonlinear spectroscopy. They foundthat the quantum coherence time of energy transfer is unexpectedly longerthan any previous theoretical predictions(> 660fs[9]). This coherence timeis comparable to the previous mentioned transfer time, which makes it inel-igible to fully understand the functionality of LHCs. Shortly after, similarexperiments were repeated under room temperature, which also showed longcoherence time[13]. Furthermore, people found that this quantum coherenttransfer is stable against the change of the structures of biological moleculesby cultivating mutations of light harvesting molecules[42].The existing theory of photosynthesis is based on the Fo¨rster theory[43].The Fo¨rster is a semi-classical theory which only assumes classic energytransfer. Therefore it is insufficient to study the newly-found quantum beat-ing phenomenon. Due to the amorphous nature of the biological molecules,the couplings to delocalized modes are also not negligible[30] (e.g., to lo-cal distortion of the molecule around pigments). The fact that thetotaltransfer process takes less than several pico-seconds suggests that localized41.1. Biological Systems with Quantum Phenomenamodes probably play an important role here, although the incoherent os-cillator bath-mediated transition would take over at longer times[1]. Tocompare with experiments, both localized and delocalized modes have to beincluded to study this macroscopic quantum coherent phenomenon. Fur-thermore, biological systems mostly operate away from equilibrium, whichalso renders most existing oscillator bath approaches questionable[44], sincethese approaches all rely on the the assumption that baths are always inthe thermal equilibrium state. The Markovian approximation also becomesquestionable since the hopping amplitude of excitons in this central systemis typically 10 ∼ 102cm−1 which is the same order as the coupling to theenvironment modes. Many theories have been proposed to describe the dy-namics of light harvesting molecules. We are going to review them in detaillater.Trimetric FMO complexes Bchls in each monomerFigure 1.1: The illustration of the molecular structure of FMO complexes.The left panel shows the three identical subunits of the FMO trimer. Theright panel is the structure of each subunit. It is composed by 8 bacte-riachlorophylls. Reprinted (adapted) with permission from [3]. Copyright(2011) American Chemical Society.51.1. Biological Systems with Quantum Phenomena1.1.2 Light Harvesting System I,IIAnother interesting light harvesting system besides FMO is the light har-vesting complex I (LH1) and light-harvesting complex II (LH2) found inthe photosynthetic bacterium Rhodopseudomonas acidophila[35, 36, 45–48].The LH2 complex serves as an antenna , which transfer photon energiestowards LH1 . Their molecular structures are shown in Fig.1.2. Each LH2consists of 24 bacteriochlorophylls(BChls). These BChls have a C8 globalsymmetry as illustrated in Fig. 1.2 [49]. Sixteen of the BChls in a com-plex form a ring structure that is responsible for the strong absorption peakaround 850nm (B850) at room temperature, and the remaining eight BChlsare bound near the cytoplasmic surface, and are responsible for anotherabsorption peak around 800nm (B800).[35]. LH1 surrounds a reaction cen-ter (RC) which is the destination of incoming excitons. LH1 contains 32Bchls and forms a larger ring with C16 symmetry. The overall diameter ofLH1 is 118 A˚[36]. The large LH1 ring lies in the center of the process ofphotosynthesis. It absorbs excitations from LH2 networks and send theminto the RCs. Similar quantum coherent beating phenomena were recentlyfound in both LH1[50] and LH2[11]. Compared to FMO, these types oflight harvesting complex has more BChls and therefore more computation-ally challenging. However their ring-like symmetric property can let us avoidthe localization problem in FMO complexes. The calculation of this kindof central systems is also useful in other condensed matter system such assuperconductors[51], which may also contain a flux threading the ring, whichgives an interesting Aharonov-Bohm syle interference[2].1.1.3 Avian CompassThe mechanism of avian navigation and bacteria guidance has been studiedfor hundreds of years. The once popular explanation is that certain mag-netite particles exist in bacteria and birds. However it cannot explain theobservation that the primary navigation mechanism is light activated[52, 53].A novel magnetoreception explanation has arisen based on the radical-pair-based mechanism in cryptochrome[54]. Similarly to photosynthetic process,the chromophore flavin adenine dinucleotide (FAD) absorbs a photon andis excited to FAD* state[55]. After protonation to form FAD*, an electron-hole paired excitation is formed and the electron is allowed to hop withinthe cryptochrome. The spin configuration of this pair is initially singlet(or triplet in a different kind of cryptochrome[56]). The hopping electron’sspin is constantly altered by the external magnetic field, i.e. the earth’s61.1. Biological Systems with Quantum PhenomenaLH2LH1Figure 1.2: The Molecular structure of light harvesting molecules LH1 andLH2 based on [4]. The look is from the top so the B800 ring and the B850ring of LH2 overlap with each other. The right panel is the B875 ring ofLH1. Both of them have a circular symmetry.magnetic field, while the spin of the hole is fixed. Meanwhile, the electronspin is also interacting with surrounding nucleus spins through hyperfinecouplings. The strength of hyperfine couplings in FAD molecules is from101 ∼ 102µT while the earth magnetic field is from 23µT to 68µT [57].When the spin configuration of this electron-hole pair is singlet(or triplet)again, they might recombine to produce FAD again and thus terminate thereaction. Therefore, the yield of the final product (FADH) of this reac-tion is controlled by the angles between the earth’s magnetic field and thehyperfine fields. Since the direction of hyperfine fields is fixed in the cryp-tochrome in birds’ eyes, birds are allowed to sense the angle between theirlines of sight and the earth’s magnetic field, and know the right directionto go for. The structure of this cryptochrome is illustrated in Fig. 1.3.There is already some convincing experimental evidence which support thisradical-pair explanation[58].The electrons are interacting with a large number of nucleus spins dur-71.2. Quantum Model for Photosynthetic Energy Transfering the process. How these bath spins make the avian compass extremelyaccurate is still not clear. Experiments suggest that this system should havea long coherence time (at least 10−4s compared to the 660fs in LHC)[55].The functional role of this nucleus bath has already become another hottopic in quantum biology in the last few years[5, 56, 59].Structure of the favin radical FADH*HydrogenOxygenNitrogenCarbonEarth Magnetic FieldsFigure 1.3: Molecular structure of the magnetoreception molecules FADH*based on [5]. The blue arrows suggest the electron pathways, though thereis no experimental evidence of this.1.2 Quantum Model for Photosynthetic EnergyTransferIn light harvesting molecules, the energy is usually transferred as Frenkelexcitons, which are localized around each chromophore pigment. Therefore,the system is usually depicted as a particle (exciton) hopping between dif-ferent sites (chlorophylls) while coupling to the surrounding environment.This phenomenon of resonant energy transfer (RET) was first observed atthe beginning of last century. Before the breakthrough of the recent quan-tum biology experiments, the most successful theory about RET was the81.2. Quantum Model for Photosynthetic Energy TransferFo¨rster theory, which was proposed by Fo¨rster in the late 1940’s. He de-rived an equation that relates the interchromophore distances to the spec-troscopic properties of chromophores[43]. The Fo¨rster theory then becamethe standard way of spectroscopic distance determination in biology[36, 60–62]. The Fo¨rster theory is a semi-classical theory which involves incoherenthopping between different sites. Fo¨rster assumed weak inter-site couplingsand assumed that one can use the equilibrium Fermi Golden Rule to treatthe electron coupling between site to site. In Fo¨rster theory, the probabilityPi to find a molecule i excited at time t can be determined byddtPi =∑j(WijPj −WjiPi) (1.1)Here Wij is the Fo¨rster transfer rate from molecule j to molecule i. Usingthe Fermi Golden rule, we can get Pi as a function of the electronic couplingVij , i.e.,Wij =pih∫ ∞0dε|Vij |2Jij(ε). (1.2)Here Jij(ω) ∝∫dλf¯i(λ)f¯j(λ)λ4 is the overlap integral integral between thenormalized emission spectrum f¯i(λ) and the normalized emission spectrumf¯j(λ). This theory was rather successful in the past. The sixth-power de-pendence of Pi on the inter-molecular distance was verified experimentallyin 1978 [63]. People are still working on this model and trying to improveits performance in photosynthetic systems (e.g. [64]). Incoherent hoppingbetween sites is a central assumption in Fo¨rster theory, so it is not applicablefor the study of the newly observed quantum effects.Fortunately, there are already studies on the exciton transport, sinceit is a well-known phenomenon which is important in solid state physics.But there are many difference between these two fields of science. Thebiological systems usually contain a huge number of degrees of freedom andare much ”dirtier” (a huge number of environmental modes). In contrastwith low temperature physics experiments, quantum biological experimentsare usually conducted at either 77K or room temperature. The environmentin biological experiments is also more complex and less controllable. It iscrucial for us to establish proper models for these biological systems withtheir environments before we study the actual behaviour of this excitontransfer process with the tools in quantum physics.Thanks to the recent development of refined experiments, we are pro-vided with new tools to look into this fragile and sensitive quantum en-vironment. In this section, we review the microscopic origins of various91.2. Quantum Model for Photosynthetic Energy Transferenvironmental modes and their coupling to either the on-site energy of theexciton (diagonal couplings ) or to the hopping amplitude (off-diagonal cou-plings). We take the FMO complexes and the associated Bacteriochloro-phylls (BChls) as a typical example. We can see that both delocalized andlocalized modes are important in biological systems and they couple to thecentral system through both diagonal and off-diagonal couplings. We alsobelieve that the two-phonon interaction terms beyond the linear approxima-tion might also be relevant.1.2.1 Single Chlorophyll Excitations and DiagonalcouplingsMost chlorophylls are planar molecules which have a Magnesium coordinateatom as their centres. The illustration for Bacteriochlorophyll a (Bchl-a)in purple photosynthetic bacteria is shown in Fig.1.4. It is a quasi-planarmolecule with a big hydrocarbon tail. There are fours pyrrole rings surroundthe central Magnesium atom. These pyrrole rings form a chlorin macrocyclearound the coordinate atom, and form a multi-band pi-bonded system alongthe molecular plane, in the language of the molecular orbital theory[6].The transition in the Bchl-a which is relevant to the biological processis believed to be the pi → pi∗ transitions in the conjugated pi system. Thereare many different types of such transitions among these bands, which canbe seen from their absorption spectra. A simplified theoretical model isthe“four orbits” model[6]. Only the two lowest unoccupied molecular or-bitals and the two highest occupied molecular orbitals are principally in-volved in these transitions. The two lower energy transitions are called “Qbands” and the two higher energy transitions are called “B bands ” or “SoretBands” (see Fig. 1.4). These “bands”, which are actually transitions, havedifferent electronic transition dipole moments µij = e〈i|r|j〉 associated witheach of them. For the lowest energy Qy band, the polarization is along thedirection of the solid Mg-N bond in Fig. 1.4 with a slight deviation; the Qxband polarization is not as big as the Qy band polarization and theoreticalcalculations also suggest that it is not perpendicular to the Qy polarization.The Soret bands have mixed polarizations which are scarcely discussed. Thetransition dipole moment of the Qy transition is 6.1D in vacuo[29]. Thoughthere is more than one excitation in the system, in almost every previous cal-culation, the Qy “band” is the only transition to be included in the excitontransfer.After being embedded into the protein, the exact excitation energy ofa single Bchl changes due to the effect of the surrounding charges and sol-101.2. Quantum Model for Photosynthetic Energy TransferFigure 1.4: The structure and the absorption/fluorescence spectrum ofa single Bchl-a. The structure of a single Bchl-a is illustrated in the leftpanel(without the hydrocarbon tail). The left panel is the absorption spec-trum of Bchl-a and the left panel is its fluorescence spectrum. We can seethat there are two major peaks in the absorption: the Soret Bands and theOnly the Qy “band” transition is included in the current calculations ofexcitonic energy transfer. Reprinted with permission from [6].111.2. Quantum Model for Photosynthetic Energy Transfervent dielectricities. In experiments, the excitation energy is determined byfitting its absorption spectrum. It is generally believed that the major in-fluence on the excitation energy i of each single Bchl a comes from thethe surrounding charged amino acid residues of the protein. The calcula-tion based this assumption is quite accurate compared to the experimentaldata[29]. By applying the dipole approximation, the electric field createdby the surrounding charges would give an extra term to the single electronHamiltonian of the Qy orbital in the form of∆H =e∑jqj~ri · ~RijR3ij, (1.3)where j are the indices of all different amino residues, qj is the charge ofthe amino residues,  is the optical dielectric constant in the protein andRij is the distance between the central chlorophyll and the jth residue. Infirst order approximation, this term gives an energy shift to the single elec-tron state∑j1qj~µ· ~RijR3ij. Here ~µ =∫d3rφ(r)~rφ∗(r) is the permanent dipolemoment of a particular single electron state. Therefore, in the Hartree-Forkapproximation, the shift to the exciton energy is∆i =1∑jqj ~∆µ · ~RijR3ij, (1.4)where ~∆µ = ~µ1 − ~µ0 is the difference of the permanent dipole moments be-tween the ground state and the excited state(Qy transition). The absolutevalue of ~∆µ is around 1.6D to 2.4D. There are more than 100 amino acidresidues which have significant effects on the excitation energy (> 20cm−1).These residues are mainly from the protein frame in which the Bchls areembedded. The net energy shift ranges from −105cm−1 to 195cm−1. Sincethe energy shift depends on the relative distance between the Bchls and theamino acid residues, it couples to the phonons in the system, i.e. the quan-tized movements of atoms in biological molecules. This is the microscopicorigin of the diagonal couplings of the phonon modes in the Hamiltonian.1.2.2 Exciton Propagation and Off-Diagonal CouplingsIn the photosynthetic energy transfer process, the excitation is transferredfrom one chlorophyll to another. The system in which the coherent quantumbeating phenomenon is found is actually quite compact. For example, theminimum distance between two BChls in one FMO complex is around 11A˚121.2. Quantum Model for Photosynthetic Energy Transfer. This is the centre to centre distance. The distance measured between theclosest atoms is much smaller, which ranges from 3.8A˚ (Bchl 1 to Bchl 2)to 11.3A˚ (Bchl 2 to Bchl 7)[65]. For comparison, the C − C single bondlength is around 1.5A˚. For the long range Coulomb interaction, we can usethe point-dipole approximation and assume the inter-Bchl couplingVmn = fµ2vacr3ij(~ei · ~ej − 3(~ei · ~eij)(~ej · ~eij)), (1.5)where f = 1µ()2µ2vacis the term representing the screening effect of dielectricmolecular environment; and µ() is the transition dipole moment of Qytransition after screening. The empirical value of f is 0.8 and we can simplyuse this to calculate the the dipole interaction between Bchl-1 and Bchl-2as 111cm−1. This is also called the Fo¨rster picture in which there is no realparticle exchange in this type of interaction. The statistical property of theelectron is irrelevant here.However, there are several drawbacks to the model. At first, the Mg-N distance is around 2.0A˚ to 2.2A˚[65], which is quite comparable to theclosest atomic distance between two different pyrrole rings. This fact leadsto several problems for this simple picture: i) the effect of higher ordermultipole interaction is not negligible.[66]. ii) the screening factor f becomesa function of distance rather than a constant[29]; iii) electrons can be directlyexchanged between pi orbitals in different Bchls [67].This can be understood in the Hartree-Fock picture. The interactionbetween the two electrons on two neighboring Bchls can be written asV = Vijc†j,ec†i,gci,ecj,g. (1.6)Here i, j are indices for different Bchls; g represents the ground state of eachBchl and e represents the first excited state. The interaction term Vij canbe written asVij =∫d~r1d~r2ψ∗j,e(r1)ψj,g(r1)e2|r1 − r2|ψ∗i,g(r2)ψi,e(r2)−∫d~r1d~r2ψ∗j,e(r1)ψi,e(r1)e2|r1 − r2|ψ∗i,g(r2)ψj,g(r2)= (ψj,eψj,g|ψi,gψi,e)− (ψj,eψi,e|ψi,gψj,g)= V cij − V exij ,(1.7)where ψi,e(r) is the single electron wave function for the e state of the ith131.2. Quantum Model for Photosynthetic Energy TransferBchl. Here we denote(ψaψb|ψcψd) ≡∫d~r1d~r2ψa(r1)ψb(r1)e2|r1 − r2|ψ∗c (r2)ψd(r2). (1.8)The first term V cij = (ψj,eψj,g|ψi,gψi,e) can lead to the usual dipole-dipoleinteraction form by expanding ri,2 around Ri,j . The wave functions ψi,e(g)are centred near Ri which is the position of the center of the Bchl, so the firstnonzero order gives the dipole-dipole interaction in (1.5) with the transitiondipole moments ~µ =∫d~rψ∗i,e(r)~rψi,g(r). The second part Vexij is called theexchange interaction. Its value depends on the overlap of wave functions atdifferent Bchls. By applying the Mulliken’s approximation of multicentreintegrals, we can write it asV exij ≈〈i, e|j, e〉〈i, g|j, g〉2·[(ψj,eψj,e|ψi,gψi,g) + (ψi,eψi,e|ψi,gψi,g)](1.9)The pi orbitals are composed of pz orbitals, therefore the overlap of wavefunctions 〈i, e|j, e〉 decays exponentially by distance. As we mentioned theclosest atom-atom distance between different Bchls is around 3.8A˚. At thisscale, the integral actually is not necessarily small. For example, the overlapintegral between two pz orbitals which is displaced in the x direction by dis illustrated in Fig. 1.5.The exchange interaction V ex has a different dependence on the distancecompared to the dipole interaction V c (V c ∝ 1R3and V ex ∝ e−R/Rc , inwhich Rc is the extension of the molecular orbital wavefunction). Whetherthe extra exchange term V ex is important or not still remains unknown.But both terms couple to the distances between Bchls, and thus to thephonon modes in the system. Since V ex modulates the hopping amplitudeof excitations, this is the origin of the off-diagonal phonon interaction in theHamiltonian.1.2.3 Vibrational ModesNow we look at the details of the wave functions and their dependence on theatomic distances. The actual state wave function ψe(g) can be approximatelycalculated in the Pariser-Parr-Pople Hamiltonian given by [68],H =∑iσ(−Ii −∑j 6=iRij)niσ +∑i 6=j,σtijc†i,σcj,σ+12∑i 6=j,σ,σ′Rijni,σnj,σ′ .(1.10)141.2. Quantum Model for Photosynthetic Energy Transfer5 10 15 20da00. 1.5: The overlap integral of two 2pz orbitals displaced in x directionby a distance d. We can see that the integral is still noticeable around 7a0.a0 is the Bohr Radius which equals to 0.53A˚.Here i, j are the indices for different pi orbitals. σ, σ′ are spin indices, Iiis the ionization energy of orbital i, Rij is the effective Coulomb interac-tion integral, and tij is the hopping integral between different orbitals. TheHamiltonian (1.10) is a semi-classical empirical Hamiltonian. For a singleBchl-a pigment, it takes into account for 24 pi orbitals (18 C atoms, 2 Oatoms and 4 N atoms) and 26 electrons (two extra electrons from the co-ordinate atoms Mg in the center). Solving this Hamiltonian would give thewave function ψe(g) and the associated eigen-energy. In (1.10), both Rij andtij depend on the relative distance rij between the different atoms. In con-trast with the collective movement of the molecules, the distances betweenatoms are affected by the vibration movements between Bchls, i.e. the vibra-tional modes. These modes are localized intra-chlorophyll phonon modes,which can have high frequencies. Vibrational modes are significant in thespectrum (see Fig. 1.6). We can see the sharp peaks ranging from 200cm−1to 1700cm−1 in addition to the low energy continuum. The strongest peakaround 1500cm−1 is associated with the C=O and C=C double bonds in theBchl structure[7]. As a result, these modes also change the wave functionψ(r) in equation (1.7) and hence couple to the off-diagonal terms Vij . Thatis the microscopic origin of off-diagonal couplings to the vibrational modes.These intra-chlorophyll vibrational modes are fundamentally differentfrom the collective modes which have been extensively studied in the litera-ture continuum modes described in the previous two sections. They have dif-151.2. Quantum Model for Photosynthetic Energy TransferFigure 1.6: The spectral function J(ω) for each Bchls in FMO complex. Theresult is acquired by fitting the auto-correlation function of the absorptionspectrum. Reprinted with permission from [7]. Copyright (2011) AmericanChemical Society.ferent dispersion relations and their numbers are small. Furthermore, thesemodes also couple to the transition dipole moments ~d =∫d~rψ∗e(~r)~rψg(~r).Therefore, when a single chlorophyll is excited by incoming photons, thevibrational modes are also excited by the phonons. In experiment, peopleusually use coherent light signals to excite the Bchls and they could con-struct a particularly structured vibrational states, which could make thedecoherence being significantly slower due to the existence of vibrationalmodes[69]. Vibrational modes could introduce new physics into the system.It is important to study this type of environmental modes individually.1.2.4 Two-Phonon InteractionAnother comment we have to make is about the possible effect from thehigher order phonon couplings. Most of the time we only keep the linearterms for the interactions between the bath and the system for the followingthree reasons: i) The weak interaction: the interaction between the systemand a single mode in the environment is usually small. ii) The n > 2 multi-phonon processes do not introduce new physics: they only give a weakrenormalization to the parameters in the influence function kernel[17] foracoustic phonons at low temperature. iii) The Gaussian assumption: sincethe number of modes couples to the central system is huge, one can utilizethe central limit theorem to justify the elimination of any cumulants higher161.3. The Proposed Hamiltonian for Energy Transferthan second order. The effect of a two-phonon interaction can be absorbedinto linear terms if it is pure Gaussian.However, these reasons are not applicable in these biological systems,espeically with the addition of vibrational modes. Although Fig. 1.6 doesnot include the off-diagonal couplings, we can still see that the couplingsto theses vibrational modes could be much stronger than the couplings ofdelocalized modes (for example g ≈ 0.1eV for the 0.21eV peak, comparedwith those g ≈ 0.02 to the continuum), which are also strong compared withthe net shift of the on-site energy from −105cm−1 to 195cm−1. Furthermore,the number of such mode is also much fewer than the collective movementof the environment. The coupling strength for a single vibrational modescould be even stronger than the coupling to the continuum, which has ahuge number of modes due to the enormous degrees of freedom. Therefore,we cannot assume weak interactions and a huge number of modes.In addition, we cannot assume the low temperature condition for biologi-cal molecules either: the system is either at 77 K (in experiments) and 300K(in real life). Meanwhile, the coupling strength and hopping amplitude ofexcitons are both ∼ 102cm−1 ∼ 100K which are comparable to the ambienttemperature. Also the vibrational modes in photosynthetic molecules havea completely different spectrum compared to the acoustic phonons in solidstate physics.Therefore, if we include the couplings to the vibrational modes, we can-not use the usual argument to neglect the effect the two phonon interactions.We shall see in this thesis that the two-phonon process can introduce newphysics into the system.1.3 The Proposed Hamiltonian for EnergyTransferIn this thesis, we are going to focus on the exciton energy transfer in lightharvesting molecules. Based on the previous review, we propose the appro-priate complete Hamiltonian for an exciton coupling to vibrational modes171.3. The Proposed Hamiltonian for Energy Transferof the environment asH = H0 +H1 +H2 (1.11)H0 =∑iεic†ici +∑〈ij〉(tijc†icj + h.c.) + t0(c†1d0 + d†0c1) +HB (1.12)HB =∑〈ij〉V (d†idj + h.c.) (1.13)H2 =∑kΩka†kak. (1.14)H1 =∑i∑kg(1)H,k,i(a†k + ak) +∑k,k′g(2)H,k,k′,i(a†k + ak)(a†k′ + ak′) c†ici+∑〈ij〉∑kg(1)P,k,〈ij〉(a†k + ak) +∑k,k′g(2)P,k,k′,〈ij〉(a†k + ak)(a†k′ + ak′) c†icj+ h.c.(1.15)where H0 is the bare molecular Hamiltonian without coupling to phonons,and {c†i , ci} are the creation and annihilation operators of local excitons onthe i-th chromophore. Here HB is the phenomenological Hamiltonian of thereaction center (RC) and t0 is the transfer amplitude to RC. The detailsof this reaction center model will be discussed in details in Chapter 4 andAppendix D.2.The “g”s are the couplings to vibrational phonon modes. The couplingswith subscript H stand for the diagonal couplings (which is known as theHolstein coupling in solid state physics[23]) and the couplings with subscriptP stand for the off-diagonal couplings (which was first mentioned by Peierls[24] with regard to the Peierls transition). The superscripts (1) and (2)stand for the first order phonon coupling and for the second order phononcoupling.The environmental modes are represented by operators {a†k, ak}, includ-ing both the delocalized modes (acoustic phonons) and localized modes (e.g.vibrational modes). Together we have a complete description of the energytransfer model in light harvesting molecules in (5.1).These equations are difficult to solve. Phonon couplings with differentcoupling strengths and frequencies require different theoretical and numeri-cal techniques to address. Different parameters can have a huge impact onthe qualitative behaviour of the model. The implication of exciton-phonon181.3. The Proposed Hamiltonian for Energy Transfercoupling has been examined by many authors using many methods over thepast few years. In the latter part of this sub-section, we will give a reviewof the prior work, and outline our approaches in this thesis.1.3.1 Coherence in light harvesting moleculesIn experiments, coherence is detected by specialized measurements. Shortlaser pulses are applied to the sample consecutively, creating superpositionsof excitations, which can be detected using two-dimensional spectroscopy[70].In principle, the cross-peak in the 2-D spectrum is the Fourier transform overt and τ in the following expression:(i/~)3(~µij · ~E1)(~µji · ~E2)(~µkl · ~E3)(~µlk · ~E4)× (Gij(τ)Gjk(T )Gkl(t)). (1.16)Here the ~µij are the absorption dipole moments of the eigenstate |i〉; the~E1,2,3,4 are polarization vectors of laser pulses; and Gij(t) is the free propa-gator of the density matrix element ρij . If there is no decoherence, this latteris simply exp (−iωijt) with ωij = ωi−ωj . When one takes the Fourier trans-form over τ and t, this term gives a cross-peak at (ωij , ωkl) and oscillates ina time T . In the real world, the oscillation does not last forever because ofdecoherence. In FMO complex experiments, the observed oscillation time iscomparable with the time of the relevant biological process, i.e. the excitontransport time.However, whether or not these results indicate coherent energy transportin vivo remains controversial. All of the experiments are done in laborato-ries. The sample FMO complexes are extracted from the membranes of cells.The light signals to create excitons are not incoherent solar-like light, butultrashort coherent laser pulses. Some have argued that the dynamics ob-served from ultra-short coherent excitation does not reflect what happens inprocesses induced by solar-like radiation[69]. It is suggested that electroniccoherence decays significantly faster for incoherent light than for coherentultrafast excitation. Furthermore, simply describing decoherence based onthe damping of oscillations of off-diagonal elements of density matrix can bemisleading, since such damping can also describe “False decoherence”[71].In “False decoherence”, although one still observes damping of the reduceddensity matrix, the coherence information is actually not lost but preservedin the surrounding environmental modes. If one restores the interaction withthe environment, one can restore the coherence as well. In Section 2.4, wewill have a detailed discussion of False decoherence.Although whether or not the beating phenomena indicates quantum co-herence remains controversial, various experiments confirm the failure of191.3. The Proposed Hamiltonian for Energy Transferthe old semi-classical picture. Some work suggested that the observed beat-ing has properties of quantum mechanics rather than a classical coherencepicture[72]. Experimental approaches that certify the non-classical natureof coherence in chemical and biophysical systems require us to employ aquantum model to understand its functions.1.3.2 Acoustic Modes and Vibrational ModesMany attempts have been made to understand the role of coherence in lightharvesting molecules, despite the controversies. The traditional Fo¨rster the-ory [43] and Markovian Redfield theory [73] cannot predict the newly foundlong coherence time. New methods have been developed for the so called“non-Markovian” region for intermediate phonon-exciton couplings. TheHierarchy Equation of Motion(HEOM) is an exact method to deal with aspectral density of Drude type[30, 74–76]. The Non-Markovian quantumstate diffusion(NMQSD) method is also applied to address more structuredspectra (those with weakly interacted high frequency modes)[77]. Thesemethods are mainly employed to deal with the phonon modes in the lowenergy continuum.Later on, people started to notice the importance of vibrational modes[21].The coherent beating amplitude is amplified by the resonance between theexciton energy difference and vibrational frequencies[78]. The effects of thesevibrational modes on the 2-D spectroscopy has been extensively examined[79–81]. The results suggest that these resonant vibrational modes greatly in-crease the coherence time of the energy transfer.Regarding the functionality of the biological molecules, it is generally be-lieved that a balance of coherent and incoherent energy transfer is requiredto achieve the fastest transfer rate. Many works show that there is an op-timal decoherence rate, which leads to an optimal transfer rate(e.g.[82]).In the pure coherent limit, one can have constructive interference betweenmultiple pathways for back reactions, therefore decreasing the transfer rate.In the pure classical limit, the exciton can be easily trapped in energy land-scapes. These two competing mechanisms yield an optimal transfer ratewith an intermediate decoherence rate. The transition between these twolimits is smooth and seems robust against small parameter changes.There are certain drawbacks to the work already appearing in the lit-erature. First, we note that most of this work is based on the quantummaster equation approach. The quantum master equation, based on theBorn approximation and a Markov approximation, is successful for weaklyinteracted phonon modes, but not appropriate for strong couplings. This is201.3. The Proposed Hamiltonian for Energy Transferespecially important if we are going to deal with the strongly coupled vi-brational modes. Mapping these modes into a spin bath model can greatlysimplify the calculation in certain parameter regimes. In Chapter 2, wecritically review the limitations of the popular master equation approaches,and in Chapter 3, we study the spin bath model which is useful in relatedcalculations.Second, in the context of biological functionality, we note that the quan-tity most often studied is the transfer rate, i.e., the exciton population trans-ferred into the reaction center. In most of the literature, the population ofthe lowest energy site in FMO complexes is used as an indication of thetransfer rate instead. However, as the exciton can be trapped by the strongphonon-exciton coupling, the population on the lowest energy site is not aproper indication of the transfer rate. In our thesis, we are going to use aquantum model for the reaction center, which allows possible back reactionprocesses.Last but not least, only the diagonal phonon coupling is included in allprevious studies. In the next paragraph, we will give a brief review of bothdiagonal and off-diagonal phonon couplings in physics.1.3.3 Diagonal Coupling and Off-Diagonal CouplingThe polaron problem, i.e., the problem of a particle dressed by an environ-ment, is an old problem in physics. It was firstly introduced by Landau in1933 and later by Pekar in 1946[83–85].The diagonal coupling models (such as the Holstein model[23], the Fro¨hlichmodel[86] and the breathing-mode model[87]) show qualitatively similar re-sults: the stronger coupling leads to a bigger dressing cloud, and hence createa heavier polaron. It involves a smooth crossover between weak and strongcouplings , with no sharp transition in between. This is called the “polaronparadigm[88]. This is expected for diagonal couplings, where phonons onlymodulate the potential energy of the exciton. Therefore, a stronger couplingmeans it is harder for the exciton to ”climb out” the potential well createdby the polarization cloud. It actually can be shown that for diagonal modelshaving gapped bosonic modes, sharp transitions in the polaronic propertiesare impossible; all physical quantities must vary smoothly with couplingstrength [89]. Therefore, we can see that the robustness and optimal trans-fer rate in light harvesting molecules is actually a common feature for allthe diagonal phonon coupling models.The off-diagonal coupling to phonons was first mentioned by Peierls [24],with regard to the Peierls transition. It was then studied in polyacetylene[25,211.3. The Proposed Hamiltonian for Energy Transfer26] using the Su-Schrieffer-Heeger (SSH) model. Unlike diagonal models,off-diagonal models do allow sharp transitions between different regimes[90].The polarons can be very light even at very strong coupling because hoppingintegrals can be larger in the presence of phonons. The polaron dispersionin these models can be quite different from that of the bare particle. In thisthesis, we are going to compare the effects of both diagonal couplings and off-diagonal couplings in biological models. We will show that the local spectracan be changed qualitatively with the presence of off-diagonal couplings.This property allows us to engineer various macroscopic properties such asthe transfer pathways selecting mechanism (see Chapter 5).1.3.4 Two-Phonon TermsMost theories about polaron physics only contain linear terms. Studies ofquadratic phonon coupling are rarely found in the literature. Due to theparticular symmetry of certain systems, pure quadratic phonon couplingis expected to occur in certain intramolecular modes of vibration, such asthe out-of-plane bending modes in aromatic hydrocarbons. These couplingsare estimated to be of importance in exciton transport but probably not incharge-carrier transport[91]. For the large polaron case(weak interaction),we can use second order perturbation theory to show that the effective massof the polaron is barely changed by quadratic phonon couplings [92]. In thesecases, the quadratic phonon interaction term can be captured by a linearHamiltonian with renormalized parameters, although the effective mass andtransition rate have different dependence on the temperature[93, 94]. How-ever, for medium and strong linear coupling, even small quadratic electron-phonon coupling terms are found to lead to very significant quantitativechanges in the properties of the Holstein polaron[95]. These results suggestthat the linear approximation is likely to be inappropriate to model sys-tems with strong electron-phonon coupling. Actually, going back to 1989,it was already realized that the most important quadratic phonon couplingsare the couplings between the modes with close frequencies[96]. In Section4.6, we show that the only non-trivial effects from the two-phonon inter-actions come from the cross-interaction terms g(2)k,k′,〈ij〉(ak + a†k)(ak′ + a†k′)when k 6= k′ and Ωk ≈ Ωk′ . These can be mapped into the spin bath inter-action (Ωk −Ωk′)σz + g(2)k,k′,〈ij〉σx, in which the σks are the operators of spin12 systems.22Chapter 2Limitations of MasterEquation Approaches toOpen Quantum Systems andDecoherenceAn open quantum system is a quantum system S (the central system) cou-ples to another quantum system B (the environment/bath). They form aclosed quantum system S +B with S as the subsystem. Although the totalsystem still obeys Hamiltonian dynamics, the evolution of the subsystemcan no longer be represented as unitary Hamiltonian dynamics. The the-ories of open quantum system aim to derive the dynamics of the reducedsystem S. The quantum dynamics of such mesoscopic or macroscopic sys-tems is always complicated due to their coupling to environments. In lightharvesting system, the central system is a Frenkel exciton propagating inthe light harvesting molecules and the environment is all the surroundingphonon modes with the aforementioned coupling to the transfer of such aexciton. Despite the biological significance, the understanding of the dy-namics of quantum systems in contact with different kind of environmentmodes is also an important topic in physics. It has already been noticedthat there are fundamental differences between a system coupled to a spinbath or an oscillator bath[97]. A thorough study is needed to compare thesetwo models, each of which has its own theoretical importance.The studies of biological systems focus on the their biological function-ality, namely the energy transfer rate, efficiency, robustness, etc. After thediscovery of the quantum beating phenomenon, quantum decoherence be-came another aspect to study in this field. Quantum coherence is a measureof how much a quantum state differs from a classic state and decoherence isthe loss of quantum coherence. The concept of decoherence was introducedsince 1953 [98–103] and has been a subject of active research since 1980s[104].Many kinds of decoherence mechanisms exist, for example, environment-232.1. A Brief History of Studies on Open Quantum Systemsinduced decoherence; 3rd-party decoherence; intrinsic decoherence due togravity, etc.[105]. Here we are only interested in the environment-induceddecoherence, in which the quantum coherence is lost by interacting with en-vironmental modes. People usually refer to the decoherence as the reductionof the off-diagonal elements of the density matrix, after “averaging over” theenvironment modes. However the definition of decoherence is not quite sostraightforward that there are subtleties involved.In this chapter, we would like to review the two most commonly usedapproaches for open quantum systems: the method of master equations andthe path integral formalism. In the first part of this chapter, we reviewthe key steps in deriving the famous Redfield equation and point out thelimitation of the master equation approach. Although historically it hasbeen successful in various fields, it could be questionable if we blindly applyit into any open quantum system. In latter part of this chapter, we usethe central spin model as an example to show the danger when not usingthe appropriate method for the system we are studying. After that weshow the path integral formalism and the influence functional method, whichwill be used in the later chapter on the spin bath model. We then use aslightly different central spin model to show the subtleties in the definitionof quantum decoherence.2.1 A Brief History of Studies on Open QuantumSystemsThe traditional way to deal with open quantum systems started from An-derson and Kubo’s pioneering works on the spectrum of nuclear magneticresonance in 1954 [106, 107]. They treated the environment as a sourceof random forces which act on the central system. They introduced atime-dependent effective hamiltonian H(t) = H0 + HI(Ω(t)) in which H0is the bare Hamiltonian and HI(Ω(t)) is the effective influence of a bathwith stochastic variables Ω(t). By making a Gaussian-Markov assumption(i.e.〈Ω(t)Ω(0)〉 ∝ e−γt) on the stochastic evolution, they got a stochastic Li-ouville equation for the central system. This method is successful in diversetopics such as NMR, ESR, muon spin rotation, Mo¨ssbauer spectroscopy,dielectric relaxation, and linear and nonlinear spectroscopies[108].However, this phenomenological model ignores the microscopic nature ofthe bath modes and the back reaction of the central system on the baths.It cannot describe a system with non-Markovian dynamics. The researchon a dynamic bath model was started near the end of 1950s [109–113],242.1. A Brief History of Studies on Open Quantum Systemsincluding the important work by Feynman & Vernon in 1963[14]. However,this work did not attract major attention until Caldeira & Leggett’s famouswork on quantum tunneling in dissipative systems[15]. They modeled theenvironment as N uncoupled harmonic oscillators, which was originated byFeynman & Vernon. By employing the path integral method to take intoaccount the intrinsic dynamics of the bath, they got effective interactionsbetween classical paths, and demonstrated the quantum microscopic natureof friction.Tanimura and Kubo then included this path integral formalism of anoscillator bath to overcome the drawbacks of the stochastic theory[114].They derived a set of hierarchy equations of motion (HEOM) for the evo-lution of the reduced density matrix. By introducing auxiliary stochasticvariables, they showed that the HEOM is actually a generalization of theprevious quantum stochastic Liouville equation[115]. The HEOM has re-cently been widely used in quantum chemistry, especially in treating thespectrum of a biological molecule with strong interactions to the environ-ment [30, 75, 76, 116, 117].However, in these models, quantum decoherence always happens to-gether with the relaxation, due to the fluctuation-dissipation theorem. Theloss of phase is associated with friction from the environment which makesit impossible to study dissipation and decoherence separately. Furthermore,the whole work is based on the oscillator bath model. As Feynman showed,this is only valid when the coupling is weak [14, 118]: only under the weakcoupling limit can we linearize the couplings and neglect high order cumu-lants. The derivation of the HEOM is based on the Gaussian-Markovianproperty of the spectral function, i.e.〈q(t)q(0)〉 ∝ e−γt . The delocalizedmodes couple to the central system with couplings ∼ N−1/2 (N as the num-ber of environmental modes). Their couplings are automatically weak forlarge N . The couplings to the localized modes, on the other hand, are oftenindependent of N and not necessarily small. The localized modes can havevarious microscopic origins such as nuclear spins, topological defects, andvarious more subtle modes associated with frustration, boundaries and in-trinsic disorder. These modes are usually discrete and can be mapped intoa spin system with spin S such that 2S+ 1 = M , where M is the number ofenergy levels of each discrete mode. The decoherence in spin bath modelsis due to the precession of spins in time-variant fields coming from the cen-tral system, which allows us to study the pure phase decoherence withoutdissipation.Unfortunately, spin baths have not been studied as completely as oscil-lator baths. One usual approach is to omit any cumulant higher than two252.1. A Brief History of Studies on Open Quantum Systems[17]. Such treatment eliminates the non-Gaussian behavior and is thereforeunable to capture the strong nonlinearity of spin baths. Even if we includethe higher level cumulants, it is still far from an exact method since thecommutators between spin operators Sx,y,z are spin operators themselves,not the c-number commutators like the Fermion or Boson cases.Another approach is the cluster expansion of spin correlators[119–122].Though varying in details, cluster expansion methods are aiming at deal-ing with the expectation value of T exp(∫ t0 dt′H(t′)) for an interactive bathHamiltion H(t′) = V αβij σαi σβj . By expanding the exponent we can get acluster expansions based on the series of the commutators between spinoperators. This model is computationally friendly and has been applied tospin decoherence experiments[123]. In these works, the interactions betweenbath spins and the central systems are always assumed not to flip the cen-tral spin. Therefore, there is no “real” interaction between the bath and thecentral systems; the decoherence is due to the mismatch of phases betweenspins under the different effective spin Hamiltonians which are associatedwith different eigenstates of the central system. Obviously, this assumptionis not always correct for any real system.A more general form of spin-environment interaction is considered in spinstar models[33, 34]. In order to compensate for the complexity of generalinteractions, the intrinsic dynamics of the spin itself are usually omitted, insome cases even the dynamics of the central system is not included. Sinceexact results can be usually acquired in these models, people already foundsome peculiar properties of spin baths which have never been found amongoscillator ones, such as ballistic long time behavior and incomplete decoher-ence. However the problems considered still lack generality.Prokof’ev and Stamp have done a thorough study on a central spinmodel coupled to a spin bath [1, 124] and their work has been successfullysupported by various experiments [125–135]. Their technique is based on apath integral and instanton operator method. However many experimentalsystems do not have such a simple central system (a two level system),and the path counting is generally not as straightforward as the TLS. Itis even more different if we study the decoherence of quantum walks inan infinite lattice. The peculiarities found in a study of a spin bath on alattice system have already been noticed by them but have yet been studiedsystematically[97].262.2. Quantum Master Equation Approaches2.2 Quantum Master Equation ApproachesAs we reviewed in last section, the quantum master equation has been thecenter of the studies in open quantum systems. The master equation ap-proach is widely used in quantum optics and quantum chemistry, as wellas in quantum biology, with both phenomenological and microscopic mod-els (e.g.[32, 136, 137]). In this chapter we are going to derive the masterequation and show the limitation of this approach.The master equation is a probability equation to describe the time evolu-tion of a system that can be modeled as being in a probabilistic combinationof states at any given time, and in which the switching between states is de-termined by a transition rate matrix,ddtPij(t) =∑kAik(t, t′)Pkj(x′, t′). (2.1)In mathematics it is called the Kolmogorov equation for his contribution toMarkov processes. The master equation is usually used with a discretizedphase space. By replacing discrete indices i, j with continuous variables x, y, we can obtain its counterpart for the continuous phase space, namely theFokker-Planck equation. A quantum master equation is a generalization ofthe master equation, involving a differential equation for the entire densitymatrix. Its continuous counterpart, a quantum Fokker-Planck equation, is adifferential equation for the Wigner quasi-distribution function [138] of thequantum system.The common derivation of a quantum master equation is usually basedon the Liouville equation of the whole quantum systemddtρ(t) = Lρ(t) = [H, ρ]. (2.2)We can start by writing the Hamiltonian asH = H0 +HB +HI , (2.3)where H0 is the Hamiltonian of the central system, HB is the bath Hamil-tonian and HI is the interaction between them. In most cases we can writeHI asHI =∑iFiLi. (2.4)Here Fi are the operators acting on the bath and Li are the operators actingon the central system. We can define the Hamiltonian in the interaction272.2. Quantum Master Equation Approachespicture asHI(t) = ei(H0+HB)tHIe−i(H0+HB)t. (2.5)The ultimate goal is to calculate the reduced density matrix of the centralsystemρS(t) = trB[ρ(t)], (2.6)where ρ(t) is the density matrix of the whole system. We start with the mostgeneral master equation: the Nakajim-Zwanzig Master Equation[139, 140](NZME). The core idea of the NZME is to using the following projectionoperators to divide the Hamiltonian into “relevant” part(central system plusa fixed bath) and “irrelevant” part (the bath variation part):Pρ = trB[ρ]⊗ ρB, (2.7)Qρ = (1−Q)ρ. (2.8)Then solving the reduced density matrix ρS(t) is equivalent to finding Pρ(t).Here ρB is a constant density matrix for the bath. In principle it can bechosen arbitrarily, but usually one chooses ρB to be a state which is closeto the actual bath state. In some cases, the choice for ρB is obvious: forexample, a thermal state for a thermal bath. The dynamics of the totalsystem can be written asdρdt= L(t)ρ ≡ [HI(t), ρ], (2.9)where L(t) is the time evolution operator in the interaction picture. Wethen getddtPρ = PLPρ+ PLQρ, (2.10)ddtQρ = QLPρ+QLQρ. (2.11)The solution of (2.11) can be formally written asQρ(t) = G(t, 0)Qρ(0) +∫ t0dsG(t, s)QLPρ(s) (2.12)with G(t, 0) as the propagator of the “irrelevant” part of the system:G(t, 0) = T+[ei∫ t0 dsQL(s)]. (2.13)282.2. Quantum Master Equation ApproachesHere T+ is the time ordered operator. Plugging (2.12) into (2.10), we canwrite the exact master equation asddtPρ(t) = PL(t)Pρ(t)+PL(t)G(t, 0)Qρ(0)+∫ t0dsPL(t)G(t, s)QL(s)Pρ(s).(2.14)This equation is the Nakajima-Zwanzig Master Equation (NZME). It is anexact equation. The right hand side contains an inhomogeneous term de-pending on the initial state of the bath and a time convolution integral.The mixture of time evolution operator and projections actually makes itextremely difficult to solve analytically. The first term on the right handside of (2.14) is a dynamic term. To simplify the equation, we can assumethatPL(t)Pρ(t) = 0. (2.15)This is equivalent to assuming that for any eigenstate in the bath’s Hilbertspace, the expectation value of the bath operators involved in the bath-system coupling vanishes :〈b|Fi|b〉 = 0, (2.16)where |b〉 are the states of the bath. However, this condition is not necessarysince even if it is not zero, we can just add a renormalization term∑itrB[ρBFi] [Li, . . . ]to the time evolution super operator L (see for example [141]). In the restof this chapter, we always assume (2.15) holds.2.2.1 Redfield Master EquationThe NZME (2.14) is exact but difficult to solve. In practice people employseveral approximations to mitigate the difficulty. The Redfield equationis among one of the most popular quantum Markovian master equations.The derivation of the Redfield equation involves multiple approximationprocedures which are popular in many master equation approaches. In thissection we would like to review the derivation of the Redfield master equationand review the critical steps. Although the Redfield equation has beensuccessful in history, there are several drawbacks which makes it applicableto many real world systems.To begin with, we assume that the initial state of the system is a productstate, i.e.ρ(0) = ρS(0)⊗ ρB. (2.17)292.2. Quantum Master Equation ApproachesTherefore Qρ(0) = 0, i.e. the inhomogeneous term which depends on theinitial state is gone. This assumption is quite common when people aredealing with open quantum systems. However, in reality, validating thisassumption requires either the central system and the environment to beprepared separately, or the relaxation time of the bath to be much shorterthan the other time scales of the system. Actually in most cases the bathand the central system are initially entangled due to interactions.Secondly, we introduce the weak coupling assumption: HI(s)t 1. Weassume that there is no evolution for the“irrelevant” part of the Hamiltonian,i.e.G(t, 0) = T+[ei∫ t0 dsQL(s)]≈ 1. (2.18)This is the zeroth order perturbation of the superoperator QL, which means(2.18) is only valid in the weak interaction regime and for a short time scale.This weak interaction assumption is sometimes called the Redfield limit[30].We can see later that this is also a Gaussian approximation: you eliminateany cumulant higher than two for the bath correlation function. The masterequation then becomesddtPρ(t) =∫ t0dsPL(t)L(s)Pρ(s). (2.19)We can write out the operators explicitly to get the familiar master equation:ddtρS(t) = −∫ t0dstrB[HI(t), [HI(s), ρS(s)⊗ ρB]]. (2.20)In the literature, (2.20) is usually referred as the second-order time-convolu-tion equation(TC2). It still contains memory effect of the past in the con-volution integral explicitly.The next step is to introduce the Markov approximation to make themaster equation local in time:ρS(s) ≈ ρS(t). (2.21)We can then write the master equation in the form of the commonly usedRedfield equation:ddtρS(t) = −∫ t0dτ([Li(t), Lj(t− τ)ρS(t)]Gij(τ)− [Li(t), ρS(t)Lj(t− τ)]Gji(−τ)).(2.22)302.2. Quantum Master Equation ApproachesIn the literature, (2.22) is sometimes called the second-order perturbativetime-convolutionless equation (TL2). If the bath energy level is continuousand infinite, one can expect the correlation function G(τ) to decay in time.We denote the bath correlation time τB as the decay time scale. The Markovapproximation is usually justified by assuming that t τB. The TL2 seemslike a further approximation of TC2. However, one can show that (2.22) isactually the second order correction to the time convolutionless form of theNZME. They should be at the same level of accuracy since they are boththe second-order perturbations of the original master equation. The detailscan be found in [142].Although the TL2 does not include the time convolution integral, itis still not a completely Markovian equation since the integral is over aparticular time span τ ∈ {0, t}. To transform it to a completely Markovianform, we put the upper limit of the integral from t to ∞. The reasoning isstill based on the assumption that G(τ) decays in time and t  τB. Theresult is the Redfield equation:ddtρS(t) = −∫ ∞0dτ([Li(t), Lj(t− τ)ρS(t)]Gij(τ)− [Li(t), ρS(t)Lj(t− τ)]Gji(−τ)).(2.23)One thing we need to comment on here is that (2.23) is a time local Markovmaster equation but not in Lindblad form, which means the resulting densitymatrix can be non-physical. One can easily verify that the Redfield equationpreserves the trace as well as the Hermicity of the reduced density matrix.But it does not preserve the positivity. One has to be careful about thepossible resulting negative density matrix.To transform the equation into the Lindblad form, we need to perform asecular approximation. We project the equation onto the eigenstates of H0denoted as |s〉 such that H0|s〉 = Es|s〉. In this case,ddt[ρS(t)]ss′ =∑pp′Kss′pp′(t)[ρS(t)]pp′ (2.24)withKss′pp′(t) = e−i∆Ess′pp′ t[∑k(δs′pΓ+skkp + δspΓ−p′kks′)− Γ+p′s′sp − Γ−p′ss′p](2.25)312.2. Quantum Master Equation ApproacheswhereΓ+ss′pp′ =∑ij〈s|Li|s′〉〈p|Lj |p′〉∫ ∞0dτei(Ep−Ep′ )τGij(τ), (2.26)∆Ess′pp′ = Es − Es′ + Ep − Ep′ (2.27)and Γ−ss′pp′ = (Γ+p′ps′s)∗. We define the intrinsic dynamic time scale of thecentral system as τS ∝ |Es − Es′ |−1. If τS  t, we may neglect the fastoscillatory terms when Es − Es′ 6= 0 since these terms oscillate rapidlyduring an appreciable time scale. In quantum optics this is also called theRotating Wave approximation. If we only include terms where ∆Ess′pp′ = 0,the right hand side does not depend on time explicitly and it can be easilyrewritten into a form which is consistent with Fermi’s Golden Rule:ddt[ρS(t)] =∑ω∑ijΓij(ω) (Li(ω)ρS(t)Lj(ω)− (Li(ω)Lj(ω)ρS(t)) + h.c,(2.28)where the operator Li is defined asLi(ω)ss′ =∑Es−Es′=ω〈s|Li|s′〉 (2.29)andΓij(ω) =∫ ∞0dτeiωτGij(τ) (2.30). Now (2.28) is a Markov master equation. We can diagonalize the matrixΓij to transform this equation into the standard Lindblad form [142].Summary : We have three typical time scales in the system: the timescale of the intrinsic dynamics of the central system τS , the bath correlationtime τB and the appreciable time span t. t is usually replaced by the so-called bath relaxation time τR, which represents the time scale of the bathsystem interactions.To simplify the NZME (2.14) to get a time-local Markov master equation(2.28), we assume: i) The product initial state ρ(0) = ρS(0) ⊗ ρB; ii) Aweak interaction between the bath and the system so as to make a 2ndorder perturbation to get both TL2(2.22) and TC2 (2.20); iii) The Born-Markov approximation by sending the the integral limit to ∞; iv) A secularapproximation to eliminate the fast oscillatory terms.The first and second steps both depend on the weak interaction assump-tion and the first term also requires that the bath and the system be prepared322.2. Quantum Master Equation Approachesseparately as well as τS  τB. The third step rely on the assumption thatGij(τ) decays in time, and that the decay rate τB  τR. However, it isonly well-defined if the bath is infinitely large and its energy levels containsa continuum. However, in an ideal non-interacting spin bath model, thebath energy levels are discretized and their numbers are finite. As a result,the bath is peusdo-periodic in time and the Poincare´ recurrence time is al-ways finite which means the definition of τB can be pathological. In the laststep, the secular approximation requires τS  τR to allow us to ignore fastoscillatory components in the master equation.We notice that this whole process imposes a strong restriction on theapplicable systems: a infinitely large bath with continuous energy levels, ahierarchy of time scales τB  τS  τR, as well as the weak interactions.There are tons of literature working on different forms of Redfield equations(2.20), (2.22), (2.28), trying to compare the accuracy of the aforementionedapproximations (for example [30] [143]). However, most of them are onlytweaking the relevant size between these three time scales but not touchinganything else in the model. For example, the hierarchy equation of mo-tion technique developed by Ishizaki and Tanimura [30][76][75] is an exactmethod for such systems. But it has a even stronger restriction on the bathmodel: not only has it to obey all previous restrictions, but the bath spec-tral function also has to be the particular Drude form J(ω) ∝ γ2ωω2+γ2. Thisbath itself exhibits Markov properties and therefore it is not a surprise thatone can write the final hierarchy equation of motion in time local fashion.Unfortunately, most system in the real world do not falls into this exact cat-egory. Just as we reviewed in the first chapter, there are all kinds of differentenvironmental modes, which requires us to employ different techniques. Inmost systems, the three time scales τB, τS , τR might not be easily defined.For example, in spin systems, although the spin-lattice relaxation time t1 isanalogous to the τR in damped oscillators, the spin-spin relaxation time t2has a different microscopic origin and does not have any analogous in oscil-lator baths. Writing every model into the Redfield master equation actuallyhomogenizes the difference between systems and it might lead to artificialresults. In the next subsection, we would like to show how disastrous thiscan be when we blindly apply the master equation to every open quantumsystems.2.2.2 Example: Central Spin ModelTo see the limitation of the approaches of master equations, we are goingto use the central spin model as a simple example. It will be shown how332.2. Quantum Master Equation Approachesmaster equation approaches lead to unphysical results.The simplest case of the central spin model can be written asH =∑kgkσzkτz, (2.31)where τ is the central spin and the σks are the operators of environmentalspins. There is no dynamics either for the bath or for the central spin. Thismodel can be solved exactly, therefore it is useful for us to compare resultsfrom different methods.Exact Result Assume that the initial state of the system is|ψ(0)〉 = (a| ↑〉+ b| ↓〉)⊗Nk=1 (αk| ↑〉k + βk| ↓〉k). (2.32)It is easy to show that the reduced density matrix is thenρ(t) =( |a|2 a∗bκ(t)ab∗κ∗(t) |b|2). (2.33)We can see that the system has no dissipation– the diagonal elements donot change in time. The off-diagonal elements of the reduced density matrixare tuned by a decoherence factorκ(t) =N∏k=1(|αk|2e−igkt + |βk|2eigkt) . (2.34)Since |αk|2 + |βk|2 = 1, we can writeκ(t) =N∏k=1(cos gkt+ i sin gkt(|αk|2 − |βk|2)). (2.35)It can be shown that in the large N limit, this factor can be approximatedas a Gaussian decay, i.e.κ(t) ≈ eiB¯te−s2N t2/2, (2.36)whereB¯ =∑kgk(|αk|2 − |βk|2) (2.37)s2N =∑k4g2k|αk|2|βk|2. (2.38)This derivation can be found in [144]. It can also be easily seen by using thecentral limit theorem: without dynamics, the spins now act on the centralspin as a random field, and if the number is large enough the distributionof the field is always Gaussian .342.2. Quantum Master Equation ApproachesMaster equation: Born Approximation Actually in this simple case,TL2 and TC2 are the same. We can simply write the equation of motion asddtρ = −∫ t0ds∑k 6=k′2gkgk′(|αk|2 − |βk|2)(|αk′ |2 − |βk′ |2)(ρ− τ zρτ z). (2.39)Sinceρ− τ zρτ z =(0 2ρ102ρ01 0), (2.40)the diagonal matrix elements do not change. But the evolution of the off-diagonal element satisfiesddtρ10 = −∑k 6=k′4gkgk′(|αk|2 − |βk|2)(|αk′ |2 − |βk′ |2)tρ10. (2.41)Its solution has an exponential decay:ρ10(t) = ρ10(0)e−λRt2 , (2.42)with the decoherence factorλR =∑k 6=k′2gkgk′(|αk|2 − |βk|2)(|αk′ |2 − |βk′ |2). (2.43)Although this correctly recovers the Gaussian decay of the exact result(2.36), the decay rate λR is not correct. Actually, the master equation ap-proach only captures this lowest order information and discards the rest: thefirst order perturbation over coupling strength gk for the decay rate r(t) =∏Nk=1(cos gkt+ i sin gkt(|αk|2 − |βk|2))is proportional to gk(|αk|2 − |βk|2).One might argue that since there is no dynamics of the central spin, theRedfield equation, which is only valid in the weak interaction region, cancertainly fail in this region. However, even when we add the dynamics tothe central spin, i.e. set the field h non-zero, the factor λR does not change.The Born approximation always leads to the following coefficienttrB[∑kσzk∑k′σzk′ρB] =∑k 6=k′2gkgk′(|αk|2 − |βk|2)(|αk′ |2 − |βk′ |2). (2.44)We can also see that for finite N , the exact result is a pseudo-periodicfunction in time. But in the master equation approach, in this case, the TCformalism of Redfield equations always gives Gaussian decay and the time-local Lindblad form always gives exponential decay. These are the intrinsic352.3. Path Integral Formalismproperties embedded in these methods: as long as your central system isa two-level system one always gets such behavior, regardless of the natureof the problem. Therefore, we cannot apply the master equation approachto any open quantum systems without looking at the detailed structures oftheir environments.In the next part, we would like to introduce another approach to theopen quantum system: the path integral formalism, which we are goingto use later in this thesis. In contrast to the Markovian master equationapproaches, it preserves the memory effect explicitly and allows us to dothe perturbation with different parameters. We can see later that it can beuseful in certain models.2.3 Path Integral FormalismThe path integral was developed by Feynman in 1948 [145]. The idea is togeneralize the action principle in classical mechanics and write the evolu-tion operator of a quantum state as a functional integral over all possiblepath for the classic actions. The path-integral approach has been proved tobe equivalent to the other forms of quantum mechanics and quantum fieldtheory.In this section, we are going to introduce path integral method using theFeynman and Vernon formalism[14]. We again assume the Hamiltonian fora central system couples to the bath asH = H0(x) +HB(q) +HI(x,q). (2.45)Here x and q are the generalized coordinates for the central system and thebath respectively. They are classical variables. Based on the path integralform, the time evolution of the whole system’s density matrix ρ can bewritten asρ(xf ,qf ; x′f ,q′f ) =∫dqidq′idxidx′iρ(xi,qi; x′i,q′i)× J(xf ,qf ; x′f ,q′f t; xi,qix′i,q′i, 0),(2.46)where the subscripts f, i indicate final and initial states respectively. The362.3. Path Integral Formalismintegration kernel can be written asJ(xf ,qf ; x′f ,q′f t; xi,qix′i,q′i, 0)=∫ q(t)=qfq(0)=qiDq∫ q′(t)=q′fq′(0)=q′iDq′∫ x(t)=xfx(0)=xiDx∫ x′(t)=x′fx′(0)=x′iDx′· exp{ i~(S[x,q]− S[x′,q′])},(2.47)where Dx means functional integration for all possible paths x from time 0to t. S is the action corresponds to the total Hamiltonian H. Similarly wecould write it asS[x,q] = S0[x] + SB[q] + SI [x,q]. (2.48)Suppose that the density operator of the global system at the initial timet = 0 is in product form, i.e.ρ(0) = ρS(0)⊗ ρB(0). (2.49)We can take a partial trace over B in equation (2.46) and getρS(xf ,x′f ; t) =∫dxidx′i∫ x(t)=xfx(0)=xiDx∫ x′(t)=x′fx′(0)=x′iDx′exp{ i~(S0[x]− S0[x′])}F [x(t),x′(t)]ρS(xi,x′i; 0),(2.50)with the kernelF [x(t),x′(t)] ≡∫dqfdqidq′i∫ q(t)=qfq(0)=qiDq∫ q′(t)=qfq′(0)=q′iDq′exp{ i~(SB[q] + SI [q,x]− SB[q′]− SI [q′,x′])}ρB(q,q′; 0).(2.51)Since we take a trace over system B, the final state of B should be the sameqf for both q(t) and q′(t). This method is called the influence functionalmethod and F [x(t),x′(t)] is the so-called “influence functional”. Noticethat since q′(t) = q(t) = qf , the integral over qf is actually equivalent to thepartial trace over all bath states. Although we still use the product initialstate condition ρ(0) = ρS(0) ⊗ ρB(0), it is actually not necessary. We useit mainly for simplicity. One can write down the influence functional for ageneral initial state: absorb the term ρS(xi,x′i; 0) into F [x(t),x′(t)]in (2.50)and replace the ρB(q,q′; 0) in (2.51) with ρ(xi,x′i; q,q′; 0).372.3. Path Integral FormalismPath integral and Master equation approaches The influence func-tional F [x(t),x′(t)] is a functional over the path x(t),x′(t). It does not onlydepend on the initial and the end point of the path x(t) but also on thepositions at any time between them. For a superoperator K(t, t′) whichis defined as ρ(t) = K(t, t′)ρ(t′), this situation means that the quantumsemi-group assumption does not hold:K(t, 0) 6= K(t, t′)K(t′, 0); 0 < t′ < t. (2.52)In other words, we cannot write it as a Markovian time local master equa-tion. But one can still derive master equations (or quantum Fokker-Planckequations) from the path integral formalism under certain conditions. Wecan take a simple example to see how this is achieved. Consider a 1-D freeparticle with a friction term caused by the environment. We have its action:S0[x] =∫ t0dτ(12Mx˙2(τ)), (2.53)and the influence functional with pure friction:F [x(t),x′(t)] = 1Aexp[−∫ t0dτγ(x(τ)− x′(τ))2] . (2.54)For a short interval of time (t, t+), we can assume that∫ t+t f(τ)dτ ≈ f(t)and x˙ = x(t+)−x(t) , and also eliminate the functional integral∫d∆x′∫ x(t)=xx(t)=x−∆xDx→∫d∆x. (2.55)Then we can write the simplified path integral in this period of time asρS(x, x′; t+ ) ≈ 1A∫d∆xd∆x′ exp[i12M(∆x)2− i12M(∆x′)2]· e−γ(x−x′)2ρS(x−∆x, x′ −∆x′; t)≈ 1A∫d∆xd∆x′ exp[iM2∆x2 − iM2∆x′2] [1− γ(x− x′)2]·[ρS(t)− ∂ρ∂x∆x− ∂ρ∂x′∆x′ +∂2ρ∂x2∆x2 +∂2ρ∂x′2∆x′2 +∂2ρ∂x∂x′∆x∆x′].(2.56)382.3. Path Integral FormalismThe integral is of the Fresnel type and the dominant term comes fromexp[iM2 ∆x2 − iM2 ∆x′2], which shows that ∆x is on the order of 1/2. There-fore , we can expand the path integral in order of . The zeroth order termsare only exp[iM2 ∆x2 − iM2 ∆x′2], which gives the renormalization factorA =2piM. (2.57)The first order of  gives the equationdρSdt= ρS(t+ )− ρS(t) = − i2M∂2ρS∂x2+i2M∂2ρS∂x′2− γ(x− x′)2ρS , (2.58)which can be written in the operator form of (2.28):dρSdt= −i[H0, ρS ]− γ[x, [x, ρS ]]. (2.59)We can introduce the Wigner distributionW (x, p) =12pi∫dyeipy〈x+ y/2|ρS |x− y/2〉 (2.60)to write (2.59) as the quantum Fokker Planck equationdWdt=ddxpW + γd2Wdp2. (2.61)By assuming a proper type of influence functional F , we can actually writethe path integral as a Gaussian time local master equation. However, thisprocedure is only valid for a Ohmic oscillator bath at high temperature[146],in which case one can get the Markov influence functional kernel F . TheHEOM mentioned in the previous chapter can also be derived from the pathintegral formalism by assuming a Drude-Lorentz form spectrum, which canbe used to acquire a Markov influence functional kernel which is proportionalto e−γt [147]. In the path integral approach, the Markov approximation canbe applied by choosing a specific form of bath spectrum.Summary Formally, equation (2.50) gives the exact time evolution of thereduced density matrix of the central system. Unfortunately, just as in theexact NZME, in most cases it is very difficult to calculate. We can only dothe functional integration analytically for a limited class of Hamiltonians.One of the most extensively studied Hamiltonian is a harmonic oscillatorbath coupled linearly to the central system. In this case the integrals are392.4. Decoherence and False Decoherenceall Gaussian and can be solved analytically. It can be mapped into a mas-ter equation or a Fokker-Planck equation by assuming a Markov spectralfunction for the bath, which is equivalent to doing the Born-Markov ap-proximation in the previous section. The influence functional method isoften used with a continuous Hilbert space to take advantage of continuousdynamical variables. The discretized model,both for the central system andthe bath, are barely touched in the literature . In this dissertation, we aregoing to use the path integral method for discretized models extensively inChapter 3. Although we cannot get a closed form for a single spin if itis exposed to a time-dependent field, we can still make approximations forthe influence functional under certain limits. In Section 4, we are going touse the general continuous path integral formalism to deal with two-phononinteractions.2.4 Decoherence and False DecoherenceThe standard technique to determine the decoherence is to look at the re-duced density matrix, in which one has traced out the environmental vari-ables in an open quantum system. If the state changes from a pure state toa mixed state one argues that the system has lost quantum coherence, andquantum interference effects are suppressed. In most experiments, there isno good measure to quantify the coherence for a general mixed state. De-coherence is usually referred as the decreasing of the off-diagonal elementsHowever this criterion is vague and sometimes too strong.In this section we again use a central spin model coupled to differentbaths as an example. By choosing different bath structures and adiabaticallycoupling/decoupling the system from the bath, we study the“decoherence”phenomenon in detail. We can see that in some instances the informationof the quantum coherence is not truly lost with apparent decreasing of off-diagonal elements. We claim that the decoherence is a more subtle subjectand that people might be misled by the so-called “False Decoherence”.2.4.1 Central Spin Couples to a Spin ChainThe environment is modeled as a 1-D spin chain {σi} and our central systemis a single spin τ402.4. Decoherence and False DecoherenceFigure 2.1: Illustration of our spin chain model. A central spin couples toa spin chain.Model 1The first case we are considering for the environment is a spin chain withHamiltonianH = Jτyσy0 + VN−1∑i=0σzi σzi+1, (2.62)where τy is the central spin operator and the {σi} are operators for N bathspins; J is the coupling and NV is the band width of the bath spins. We doNOT use the periodic boundary condition here. Similar to the central spinmodel in 2.2.2, we have the reduced density matrix( |a|2 a∗bκ(t)ab∗κ∗(t) |b|2). (2.63)Again, we can see that since the total Hamiltonian commutes with the cen-tral spin operator τy, the diagonal component (represented in τy eigenstates)of the reduced density matrix does not change in time. Only the off-diagonalcomponents ρ12(t) and ρ21(t) might decay in time. Also, we haveρ12(t) = κ(t)ρ12(0), (2.64)with κ(t) being the “decoherence factor”κ = trb(e−iH+tρbe−iH−t), (2.65)412.4. Decoherence and False Decoherencewhere H± are the block Hamiltonians in the τy = ±1 subspaces respectively.H+ = Jσy0 + VN−1∑i=0σzi σzi+1 (2.66)H− = −Jσy0 + VN−1∑i=0σzi σzi+1. (2.67)This model can be solved exactly. The result isκ(t) =V 2 + J2 cos 2√1 + J2/V 2tV 2 + J2= 1− 2J2 sin2√1 + J2/V 2tV 2 + J2. (2.68)But for convenience for the next model, we introduce the Jordan-Wignertransformation to diagonalize this Hamiltonian:σzi = −∏j<i(1− 2c†jcj)(c†i + ci), (2.69)σxi = 1− 2c†ici, (2.70)σyi = i∏j<i(1− 2c†jcj)(c†i − ci). (2.71)Substituting this back to the Hamiltonian (2.62) givesH = iJτy(c†0 − c0) + VN−1∑i=0(c†ic†i+1 − cic†i+1 + c†ici+1 − cici+1). (2.72)We can make Fourier transforms:ckn =1√N∑jcjeijkn (2.73)c†kn =1√N∑jc†je−ijkn ; kn =2pinN(2.74)kn =2pinN. (2.75)as well as a Bogoliubov transformation:γkn = e− ikn2(coskn2ckn − i sinkn2c†−kn), (2.76)ckn = coskn2eikn2 γkn + i sinkn2eikn2 γ†−kn . (2.77)422.4. Decoherence and False DecoherenceThe resulting Hamiltonian isH =∑n(V (2γ†knγkn − 1) +iJ√Nτy(γ†kn − γkn))=∑nHn,(2.78)with Hn = V (2γ†knγkn − 1) + iJ√N τy(γ†kn− γkn). It looks block-diagonal butit actually is not. The problem here is that the individual Hamiltonians Hndo not commute with each other, i.e.[Hn, Hn′ ] = −2J2N(γ†kn − γkn)(γ†kn′− γkn′ ). (2.79)It can be shown that for the weak interaction limit J/√NV  1, the commu-tators between Hn can be neglected as they are higher order in O(J2/NV 2).The details can be found in Appendix A. If we ignore the commutators in(2.79), the Hamiltonian (2.78) becomes block diagonal. In each {γ†kn , γkn}subspace, we have thatHn(τy = +1) =(1 iJ√NV− iJ√NV−1)(2.80)Hn(τy = −1) =(1 − iJ√NViJ√NV−1). (2.81)We assume the bath is in a fully mixed state ρb =12I. It is then straightfor-ward to calculate the decoherence factor κ(t) asκ = 1− 2 sin2(√1 +J2NV 2t)J2NV 2 + J2≈ 1− 2 sin2 t J2NV 2.(2.82)For the N →∞ limit, we have the total decoherence factor asκ = limN→∞(1− 2 sin2 t J2NV 2)N= e−2J2 sin2 t/V 2 . (2.83)We can see that κ gives the correct result compared with the exact result(2.68) up to order J2/V 2.432.4. Decoherence and False DecoherenceAs we can see from both results, κ is periodic in time, which means it willcome back to its initial state without real loss of coherence in time (see Fig.2.2). There is no real decoherence happening in this model if you are givensufficient time for observation. Although in practice, you might observe theinitial drop of the off-diagonal elements and apparently, “decoherence”.Figure 2.2: Decoherence factor κ(t) (2.83) of model 1 as a function of time.There is no real decoherence and κ goes back to 1 periodically.Model 2The last model is rather simple and in fact, equivalent to a two-spin model.But the method we develop is useful. In this part, we change the structureof the bath spins toH = Jτyσy0 + VN−1∑i=0(σ+i σ−i+1 + h.c.). (2.84)The trick we introduced in the last subsection is used here to transform theHamiltonian intoH =∑n(2V cos kn(γ†knγkn −12)− iJ√Nτy(γ†kn − γkn)). (2.85)It has the same structure as (2.78), except that now the spectrum is cos knVinstead of a constant. Unfortunately, there is no exact result for this model.But we can follow the approximate procedure in the previous section to get442.4. Decoherence and False Decoherencethe decoherence factor up to order J2:12tr(e−iH−n te−iH†nt)= 1−2 sin2√V 2 cos2 kn +J2N tcos2 kn +J2NV 2· J2NV 2≈ 1− 2 sin2(V t cos kn)J2NV 2 cos2 kn; (N →∞)≈ e−2 sin2(V t cos kn)J2NV 2 cos2 kn .(2.86)After including all the Hns, we haveκ(t) = exp(−∑n2 sin2(V t cos kn)J2NV 2 cos2 kn)≈ exp(− J2piV 2∫ pi−pidksin2(V t cos k)cos2 k); (N →∞)= exp(−2J2t2 1F2(12;32, 2;−V 2t2)).(2.87)The behaviour of κ(t) is illustrated in Fig. 2.3. Its long time asymptoticbehaviour can still be studied. When t→∞, we can expand the generalizedhypergeometric function as(V t)2 1F2(12;32, 2;−V 2t2)= V t+12√1piV tcos(2V t− pi4)+O(1(V t)32).(2.88)Therefore, κ → e−J2t/V → 0 as t → ∞. We can see now in this model thesystem is completely decohered and different from our previous case; it nevergoes back to its initial condition. In model 2, the quantum information istruely lost into the environment. We can see the main difference now isthat the eigenstates of the bath spins are spin waves. The “information” ispropagating through the whole spin chain and if the chain is long enough,it never feeds back to the central system.452.4. Decoherence and False DecoherenceFigure 2.3: This is the decoherence factor of model 2 as a function of time.The loss of cohcerence is clear over time.Adiabatic DecouplingNow we still consider the same system, but make the coupling between thecentral system and the bath vary in time:H = J(t)τyσy0 +Hbath, (2.89)where J(t) is the time-varying coupling with J → J0 as t→ −∞ and J → 0as t→∞. We choose our J(t) asJ(t) =J0ekt + 1. (2.90)462.4. Decoherence and False DecoherenceFigure 2.4: Figure of the coupling J(t) over time. These lines representk = 0.1, 0.5, 1, 10 respectively.Therefore, ddtJ(t) = −J0kekt/(ekt + 1)2. When k → 0, ddtH0 ∝ ddtJ(t)will go to zero; when k → ∞, it will become a Heaviside step functionwith a jump at t = 0 (see Fig.2.4 ). Therefore, the k = 0 limit meansthat we adiabatically decouple the system from the bath. Meanwhile thek → ∞ limit refers to the sudden turn-off of the coupling. For this suddendecoupling case, the result should be the same as what we get from previoussections since the central system stops evolving after the turn-off. But forthe adiabatic decoupling case, it is more complicated.With time dependent coupling J(t), we can still follow the same proce-dure to transform the Hamiltonian toH =∑n2En(γ†nγn −12) +∑ iJ(t)√N(γ†n − γn), (2.91)where En = V for our first model and En = V cos kn for our second model.ThenH±n =(2En ∓ iJ(t)√N± iJ(t)√N0). (2.92)Our goal is to solve this Hamiltonian in each {γ†n|0〉, |0〉} subspace. A vector(a(t), b(t)) in this subspace evolves as− i(ddta(t)ddtb(t))=(2En ∓ iJ(t)√N± iJ(t)√N0)(a(t)b(t)). (2.93)472.4. Decoherence and False DecoherenceIt can be rewritten asa(t) = ∓√NJ(t) b˙(t), (2.94)¨b(t)− (iEn + J˙(t)J(t))b˙(t) + J2(t)N b(t) = 0. (2.95)The derivation is lengthy, and can be found in Appendix A. At the end, wecan find for the long time asymptotic limit t→∞ of the decoherence factor:κ(t) =∏n(1− 2J2NE2n). (2.96)For our first bath model, Hbath = V∑N−1i=0 σzi σzi+1, En = V for all subspaces.Then the decoherence factor isκ1(t) = (1− 2J2NE2n)N = e−2J20 . (2.97)For our second bath structure Hbath = V∑N−1i=0 (σ+i σ−i+1 + h.c.), En =V cos kn for each subspaces. The decoherence factor isκ2(t) = e− 12pi∫ 2pi0 dk2J2V 2 cos2 k → 0. (2.98)So we find “partial decoherence” for model 1 (κ → e−2J2)and “completedecoherence” for model 2 (κ → 0) in the adiabatically decoupled limit.Apparently, the system would end up in a mixed state and the coherence iseventually lost. But we can still ask the question: is the information trulylost? Or equivalently, is this real decoherence?To see this, we should reverse the system back to its original condition,by further looking at a general slow-varying J(t). In this case (2.95) stillholds.i)For our first model, En = V . If the coupling is varying slowly, ˙J(t)J(t)En, we can safely neglect the J˙(t) term in (2.95) and haveb(t) =C1e−iV t/2−iV t/2√1+4J2(t)NV 2 + C2e−iV t/2+iV t/2√1+4J2(t)NV 2 ,a(t) =− i2C1√NJ(t)(V + V√1 +4J2(t)NV 2)e−iV t/2−iV t/2√1+4J2(t)NV 2− i2C2√NJ(t)(V − V√1 +4J2(t)NV 2)e−iV t/2+iV t/2√1+4J2(t)NV 2 .(2.99)482.4. Decoherence and False DecoherenceWe can see that in this adiabatic limit, the result has no dependence on J˙(t).The decoherence factor κ1(t) should be solely dependent on J(t). This meansthat if one slowly changes the coupling strength back to its original value,κ1(t) would go back to its original value regardless of the apparent loss ofdecoherence we obtained from (2.97). One can recover the full coherenceif by decoupling the model 1 to its environment and recoupling it backadiabatically. Roughly speaking, in this case, the coherence is not truly lostfrom the environment. It is encoded in the bath and can be transferred backto the central system as long as the interaction recovers.ii)For model 2, on the other hand, we cannot neglect the J˙(t) term sinceEn = V cos kn can be zero. The slow varying condition ˙J(t)  J(t)Enbreaks down for the modes in the center of the band. Actually it is easyto prove that κ(t) would stay at 0 after adiabatically decoupling from theenvironment and never come back to its initial state even if one recover theinteraction. In this case, the coherence is truly lost through the modes nearkn ≈ pi2 , 3pi2 .Discussion From the study of these two central-spin models, we can seethat decoherence is a subtle subject and cannot be easily judged by the be-haviour of the off-diagonal elements during a certain period of time. For truedecoherence to happen, the “information” has to be carried away by the en-vironment, separated from the system by some mechanism so that it cannotcome back into the system. In model 2, the spin wave modes with kn ≈ pi2 , 3pi2serve this purpose. In model 1, the apparent loss of decoherence is actuallyencoded in the environment and if one adiabatically restores the interaction,coherence is fully restored. This usually referred as “False Decoherence” inthe literature [71], which was initially discussed for high frequency modes inthe environment. These high frequency modes do not affect the dynamics oflow energy physics, but formally can still introduce false decoherence. Thissuggests the importance of not making quick conclusions about the deco-herence of the system, especially for these systems which strongly couple tocertain localized modes in the environment.49Chapter 3Spin Bath ModelBoth oscillator and spin bath models map to many important models inphysics. For example, the spin-boson model mentioned before can be mappedinto the Kondo model, Sine-Gordon models, and various other 2-dimensionalfield theories for specific parameter values. Although for the spin bathmodel, similar mappings have yet to be exploited in great detail, they willobviously be very useful for, e.g. lattice spin models. Actually, as mentionedin the introduction, lattice spin models are important in understanding therole of quantum phenomena in various biological systems. Physicists needto understand the mechanisms of decoherence and quantum dissipation innature, and the crossover to (or the “emergence of) classical behaviour fromquantum physics as one increases either size, temperature T, external fields,or couplings to the environment. Such quantum-classical crossover phenom-ena are not only relevant to the traditional low temperature microscopicsystems, but also observed in high temperature macroscopic biological sys-tems.The oscillator bath model as an environment has been studied exten-sively in the literature. Meanwhile, spin lattice models has not been studiedin depth, although people have already noticed the peculiarity of spin bathdynamics (e.g.[97]). We shall see in this chapter that it is even more differentif we study the decoherence of quantum walks in a general lattice.The general model for a particle moving between a set of sites througha spin bath can be written asH = Hband +HSB, (3.1)whereHband =∑ij[∆ijc†icjeiAoij+i∑k(φijk +αijk ·σk) +H.c.] (3.2)+∑j(j +∑kλkj · σk)c†jcj , (3.3)503.1. The Model of a Discretized System Coupled to a Spin BathandHSB =∑khk · σk +∑k,k′V αβkk′ σαk σβk′ . (3.4)Here i, j are the site indices. The operator c†j creates a particle at site j;in this chapter we assume a single particle only, which means the statisticalproperty of the particle is irrelevant. σk is the spin operator. k is the bathspin index. In the following section, the microscopic origin of this model isdiscussed.The dynamics of the spin bath depends on the size of the system param-eters: the central particle parameters ∆ij , i, and the spin bath parametersV αβkk′ , hk, as well as both the diagonal couplings λkj and off-diagonal couplingαijk to the spin bath. In this chapter, we are going to explore the dynamicsof the central particle in various parameter regimes. Some analytical resultsare acquired by taking the system into certain asymptotic limits. These re-sults already show the extraordinary behavior of spin baths compared withoscillator ones. General techniques are given afterwards, with some numeri-cal results. Although the most general Hamiltonian for a spin bath has notyet been solved, we are trying to cover most parameter regions.3.1 The Model of a Discretized System Coupledto a Spin BathIn this section, we discuss the microscopic origin of spin bath models. Thebehavior of the model depends on the ratios between parameters. We willoutline the various dynamics of the model in different parameter regimes.At first we look at the bare system Hamiltonian. We can truncate thefollowing high-energy Hamiltonian of form:HV =12M(P−A(R))2 + U(R), (3.5)into the discetized effective Hamiltonian:Ho = −∑<ij>[∆ijc†icj eiA0ij +H.c.]+∑jεjc†jcj . (3.6)This “1-band” Hamiltonian is the result of truncating to low energies.The Hamiltonian (3.5) describes a particle of mass M moves in a potentialU(R) characterized by N potential wells in a certain configuration. Thenεj is the energy of the lowest state in the j-th well, and ∆ij is the tunneling513.1. The Model of a Discretized System Coupled to a Spin Bathamplitude between the i-th and j-th wells. In path integral language, thistunneling is over a semiclassical “instanton” trajectory Rins(τ), and thisoccurs over a timescale τB ∼ 1/Ω0 (the “bounce time” [148]), where Ω0is roughly the small oscillation frequency of the particle in the potentialwells. In a semiclassical calculation, the phase Aoij is that incurred along thesemiclassical trajectory by the particle, moving in the gauge field A(R).The second step is to add the interaction to the environment. The mostobvious interaction between the particle and a set of bath spins has the localform [124]:Hint(R) =∑kF (R− rk) · σk =∑kHkint(R) (3.7)where F (r) is some vector function, and rk is the position of the k-th bathspin. The diagonal coupling F kj , or its linearized form ωjk, is then easilyobtained from (3.7) when we truncate to the single band form. But theterm (3.7) must also generate a non-diagonal term, which is more subtle.We can see this by defining the operatorTˆ kij = exp [−i/~∫ τf (Rj)τin(Ri)dτ Hkint(R, σk)], (3.8)where the particle is assumed to start in the i-th potential well centered atposition Ri, at the initial time τin, and finish at position Rj in the adjacentj-th well at time τf ; the intervening trajectory is the instanton trajectory(which in general is modified somewhat by the coupling to the spin bath).Now we operate on σk with Tˆkij , to get|σfk〉 = Tˆ kij |σink 〉 = ei(φijk +αijk ·σk)|σink 〉, (3.9)where we note that both the phase φijk multiplying the unit spin operator σ0k,and the vector αijk multiplying the other 3 spin operators σxk , σyk , σzk, are ingeneral complex. In this way the instanton trajectory of the particle acts asan operator in the Hilbert space of the k-th bath spin [1, 149]. Note that oneimportant implication of this derivation is that typically |αijk |  1, in factexponentially small, since the interaction energy scale set by |F (R − rk)|is usually much smaller than the “bounce energy” scale ~/|Ωo| set by thepotential U(R), i.e. the tunneling of the particle between wells is a suddenperturbation on the bath spins [1] .Therefore, the most general coupling terms between the particle and thespin bath has the formHint =∑Nsk [∑jF kj (σk)cˆ†j cˆj +∑<ij>(Gkij(σk)cˆ†i cˆj +H.c.)] (3.10)523.1. The Model of a Discretized System Coupled to a Spin Bathin which both the diagonal coupling F kj and the non-diagonal coupling Gkijare vectors in the Hilbert space of the k-th bath spin.Starting with the particle-bath interaction given in (3.7), we will endup with an effective Hamiltonian for the interaction terms in form given in(3.10), in which the non-diagonal interaction Gkij(σk) has assumed a ratherspecial form.One can in fact have a more general form for Gkij(σk) in the lowest-bandapproximation, provided one also introduces in the microscopic Hamiltoniana couplingHint(P) =∑kG(P ,σk) =∑kHkint(P) (3.11)to the momentum of the particle. This can include various terms, includingfunctions of P × σk and P · σk; a detailed analysis is fairly lengthy. Themain new effect of these is to generate terms in the band Hamiltonian whichcouple the spins to the amplitude of tij as well as to its phase; these do notappear in (3.15).The spin bath itself, independent of the particle, has the HamiltonianHSB =∑khk · σk +∑k,k′V αβkk′ σαk σβk′ . (3.12)Each environment spin has some local field hk acting on it, and the in-teractions V αβkk′ are typically rather small because the spin bath representslocalized modes in the environment.Considering the origin of these terms, we can combine the bare Hamil-tonian (3.6) and the interaction terms (3.10) to get the band HamiltonianasHband = Ho +Hint (3.13)=∑ij[∆ijc†icjeiAoij+i∑k(φijk +αijk ·σk) +H.c.] (3.14)+∑j(j +∑kλkj · σk)c†jcj (3.15)in which the diagonal couplings to the spin bath assume a “Zeeman” form,of strength |λjk|, linear in the {σk}, and the non-diagonal couplings appearin the form of extra phase factors in the hopping amplitude between sites.3.1.1 Parameter RegionsSolving the general Hamiltonian (3.1) and (3.15) analytically is difficult.But we can study the behaviour of the central system in different parameter533.1. The Model of a Discretized System Coupled to a Spin Bathregions. We are going to outline and discuss them briefly in this section. Thedetailed calculations are provided in the next two sections of this chapter.I. Off-diagonal Coupling: Pure phase decoherence region If hk ·σkcommutes with the off-diagonal coupling αijk ·σk , the interactions with thecentral system will not flip the bath spins between the eigenstates of thebath Hamiltonian HSB = hk · σk . Therefore there is no dissipation in thisregion since there is no energy transfer between the bath and the system.The interaction terms only add time dependent phase factors to bath spinstates which cause the decoherence. This is one of the major differencesfrom the oscillator bath case where decoherence always happens togetherwith dissipation due to the fluctuation-dissipation theorem. Therefore, wecall this region the pure phase decoherence region. It has been studied fora ring system [2] and there is no difficulty to generalize it to an arbitrarylattice by using the influence functional technique. We can see that theinfluence functional kernel in this case smears the phase correlation betweendifferent paths and eventually leads to a non-dissipative decoherence of thereduced density matrix.II. Off-diagonal Coupling: High field region If hk ·σk does not com-mute with αijk ·σk and the commutators are much larger than |αijk ∆ij |, bathspins are completely driven by this effective “external field”. The techniquewe are going to use for this region is the orthongonal blocking approxi-mation which was originally discussed for the TLS central system[1]. Wegeneralize the method here to deal with general lattices. We divide thetotal Hilbert space of bath spins into different polarization groups. Thosepolarization groups are the sets of bath states which have similar eigen-values of∑k hk · σk. Since |hk|s are large, the energy difference betweendifferent polarization groups is usually large too. Due to such large energygaps, the transitions between different polarization groups rarely happen.We can do a perturbation expansion on the maximum number of transitionsallowed. Since these transitions are naturally correlated in nature, we canfind effective interactions between different paths and derive an effective in-fluence functional kernel. This method can be systematically generalized tothe intermediate parameter region. We can see that the bath spins behavedifferently compared with those in the pure phase decoherence region. Com-bining them together give a complete description of a spin bath with onlyoff-diagonal coupling.543.2. Pure Phase Decoherence RegionIII.Diagonal Coupling: Anderson localization and beyond The di-agonal coupling∑j(j +∑k λkj · σk)c†jcj is more complicated to deal with.The site energy {j} is generally not homogeneous and therefore we expectthe localization phenomenon in the bare Hamiltonian. In fact, this problemcan be converted into the famous Anderson model[150]. In mathematics ,it is also called the random Jacobi problem[151]. The wave function of thecentral particle can be localized by a randomly distributed set of {j} dueto the interference between multiple paths. This localization mechanismimpedes the energy transfer of the central system.If the diagonal coupling term λkj · σk commutes with hk · σk, againthere is no energy transfer into the bath. If the spin bath is initially in ahigh temperature state, this interaction basically gives a decoherence kernel∝ exp(−λ2) to each path in the path integral. This decoherence kernel candestroy the mentioned interference between different paths and thereforeeventually destroy the localization. In biophysics, this destruction of weaklocalization is also called an “environment-assisted” transition [32, 136].Anderson localization with quantum environments is a huge topic whichhas tons of literature about it. If the commutator between λkj ·σk and hk ·σkis not negligible, this problem becomes even more complicated. If hk →∞,we can imagine that the “orthogonality blocking approximation” method isstill valid. For large but finite hk, a perturbation expansion method canbe made based on the path integral, but the convergence of the expansionstrongly depends on the geometry of the lattice which is beyond the scope ofthis thesis. Therefore, in this thesis we are not going to follow this directionin any depth. ,which could be useful for various lattice models.3.2 Pure Phase Decoherence RegionIf only off-diagonal couplings are considered, and hk · σk commutes withαijk ·σk, we get the “Pure Phase Decoherence Region”. It is useful to look atmodels in this region since the environment causes pure phase decoherence,with no dissipation.Such models become particularly interesting when the decoherence isacting on systems propagating in ‘closed loops’. Models of rings coupledto oscillator baths have already been studied [152]. However such models,where decoherence is inextricably linked to dissipation, do not capture thelargely non-dissipative decoherence processes that dominate many solids atlow T .In this section we are going to illustrate the character of spin bath in553.2. Pure Phase Decoherence Regionthis region on a ring structure (see Fig. 3.1). As we reviewed in Section 1.1,the light harvesting systems I and II also have these ring-like structures. Intheory, by making i identical at each site, we can eliminate the unnecessarylocalization effect to focus on the effect of the pure phase decohrence. Andthe 1-D ring is the smallest possible structure we can study to showcase thisphenomenon. This calculation can be easily generalized to other lattices.Figure 3.1: Illustration of truncation to discretized system: At left, an 8-site ring with nearest-neighbour hopping between sites. At right a potentialU(R) with 8 potential wells (shown here symmetric under rotations by pi/4),depicted as a contour map (with lower potential shown darker). Whentruncated to the 8 lowest eigenstates, this is equivalent to the 8-site model.3.2.1 Bare System BehaviourWe first consider the dynamics of a particle in some initial state moving onthe N -site ring without bath. In (3.6), we assume a symmetric ring so that∆ij is only nonzero for neighboring sites and identical for every such pair.The gauge potential is given byAoij =e2H ·Ri ×Rj = Φ/N (3.16)(We use MKS units, in which ~ = 1.) Here, H is the magnetic field, and Riis the radius-vector to the ith site; in cylindrical coordinatesRj = (Ro,Θj)Θj = 2pij/N (3.17)563.2. Pure Phase Decoherence Regionfor a ring of radius Ro. We now define operatorsc†j =√1N∑kneiknjc†kn ,c†kn =√1N∑`e−ikn`c†` ,kn =2pinN, n = 0, 1, . . . , N − 1 , (3.18)(we have slightly switched notation from the last section, now denotingmomenta by kn instead of pn). The bare Hamiltonian is thenHo =∑kn2∆o cos(kn − Φ/N)c†knckn . (3.19)For this free particle the dynamics is entirely described in terms of thebare 1-particle Green functionGojj′(t) ≡ 〈j|Go(t)|j′〉 ≡ 〈j|e−iHot|j′〉=1N∑ne−i2∆0t cos(kn−Φ/N)eikn(j′−j) . (3.20)which gives the amplitude for the particle to propagate from site j′ at timezero to site j at time t. This can be evaluated in various ways (see AppendixB.1); the result can be usefully written asGojj′(t) =+∞∑p=−∞JNp+j′−j(2∆ot)e−i(Np+j′−j)(Φ/N+pi/2) (3.21)where∑p is a sum over ‘winding numbers’ around the ring. The ”returnamplitude” Go00(t) is then given byG00(t) =∑peipΦ(−i)|Np|J|Np|(2∆ot)=∑peipΦINp(−2i∆ot) (3.22)where in the last form we use the hyperbolic Bessel function.It is often more useful to have expressions for the density matrix; eventhough these depend trivially for a free particle on the Green function, they573.2. Pure Phase Decoherence Regionare essential when we come to compare with the reduced density matrix forthe particle coupled to the bath. One has, for the ‘bare’ density matrix ofthe system at time t,ρo(t) = e−iHotρo(0)eiHot. (3.23)Thus, suppose we have an initial density matrix ρol,l′ = 〈l|ρ(t = 0)|l′〉 at timet = 0 (where l and l′ are site indices), then at a later time t we haveρojj′(t) ≡ 〈j|ρo(t)|j′〉 = 〈j|e−iHot|l〉ρl,l′〈l′|eiHot|j′〉= ρl,l′Gojl(t)Goj′l′(t)†. (3.24)where we use the Einstein summation convention (summing over l, l′). Inwhat follows we will often choose the special case where the particle beginsat t = 0 on site 0, so that ρl,l′ = δ0lδl′0, and then we have〈j|ρo(t)|j′〉 = Goj0(t)Goj′0(t)†. (3.25)tj30021321Figure 3.2: A particular path in a path integral for the particle, shown herefor an N = 3 ring. This path, from site 0 to site 1, has winding numberp = 1.The evaluation of the time-dependent density matrix for the free par-ticle turns out to be quite interesting mathematically. As discussed in theAppendix B.2, one can evaluate ρojj′(t) as a sum over winding numbers, toproduce either a sum over pairs of paths in a path integral, to give a doublesum over winding numbers, or as a single sum over winding numbers (seeFig. 3.2). Consider first the double sum form; for the special case where583.2. Pure Phase Decoherence Regionρl,l′ = δ0lδl′0 (the particle starts at the origin), this can be written asρojj′(t) =∑pp′ei(p−p′)ΦeiΦ(j−j′)/N (−i)Np+j(i)Np′+j′× JNp+j(2∆ot)JNp′+j′(2∆ot),(3.26)where p, p′ are winding numbers (see Appendix B.2 for the derivation fora general initial density matrix). This form has a simple physical interpre-tation - the particle propagates along pairs of paths in the density matrix,one finishing at site j and the other at site j′, and the order of each Besselfunction simply gives the total number of sites traversed in each path, withappropriate Aharonov-Bohm phase multipliers for each path.Consider now the answer written as a single sum over winding numbers;again assuming ρl,l′ = δ0lδl′0, we get:ρojj′(t) =1NN−1∑m=0∞∑p′=−∞JNp′+j′−j [4∆ot sin(km/2)]· eiΦ[p′+(j′−j)/N ]−ikm(j+j′−Np′)/2 .(3.27)The physical interpretation of this form is less obvious, but the sums aremuch easier to evaluate since they only contain single Bessel functions in-stead of pairs of them. Thus wherever possible we reduce double sum formsto single sums. Notice that for these finite rings, the bare density matrix isof course strictly periodic in time. Notice also that the diagonal elements ofρ(t) are generally periodic with Φ. However, the off-diagonal elements areonly periodic in Φ/N . In contrast, eiφ(j−j′)〈j|ρ(t)|j′〉 is periodic in Φ, withperiod 2pi. This latter is the quantity needed for calculating the currents,as we will see below.From either Gojj′(t) or ρojj′(t) we may immediately compute two usefulphysical quantities. First, the probability P oj0(t) to find the particle at timet at site j, assuming it starts at the origin; and second, the current Ij,j+1(t)between adjacent links as a function of time. This probability P oj0(t) is givenbyP oj0(t) = 〈j|ρo(t)|j〉 = |Goj0(t)|2. (3.28)which from above can be written in double sum form asP oj0(t) =∑pp′JNp+j(2∆ot)JNp′+j(2∆ot)× e−iN(p′−p)(Φ/N+pi2 )593.2. Pure Phase Decoherence Regionor in single sum form asP oj0(t) =1NN−1∑m=0∞∑p=−∞eip(Φ+Nkm/2)× JNp[4∆ot sin(km/2)] .5 10 15D0t13231P j5 10 15D0t-0.40.4IH1®2LD05 10 15D0t13231P j5 10 15D0t-®2LD0Figure 3.3: Results for the free particle for N = 3 and for a particle initiallyon site 1. Left: The probabilities to occupy site 1 (full line), 2 (large dashes),and 3 (small dashes). Right: the current from site 1 to site 2. Top: Φ = 0.Bottom: Φ = pi/2 (i.e. φ = pi/6).One may also compute moments of these probabilities (eg., the 2nd mo-ment∑j j2P oj0(t) tells us the rate at which a density matrix spreads in time),by a simple generalisation of these formulas.To give some idea of how for the free particle behaves, it is useful to lookat these results for a small 3-site ring, where the oscillation periods are quite603.2. Pure Phase Decoherence Regionshort. One then has, for the case where the particle starts at the origin, thatP oj0(t) =13(1 + (3δj,0 − 1)[J0(2∆o√3t)+ 2∞∑p=1J6p(2∆o√3t) cos(2pΦ)]+ (δj,1 − δj,2)2√3∞∑p=1J6p−3(2∆o√3t) sin((2p− 1)Φ)) . (3.29)In Fig. 3.3 the return probability P o00(t) is plotted for N = 3, using (3.29);we see that the periodic behaviour depends strongly on the flux Φ.Turning now to the current Ioj,j+1(t) between site j and site j + 1, thisis given byIoj,j+1(t) = 2 Im [∆oe−iΦ/Nρoj+1,j(t)]= −i∆o∑pp′ei(p−p′)Φ(INp+j+1(x)INp′+j(x∗)− INp+j(x)INp′+j+1(x∗))(3.30)where we define x = −2i∆ot. Again, one can write the current as either adouble sum over pairs of winding numbers, or as a single sum (see AppendixB.2 for the general results and derivation). For the case where the particlestarts from the origin, these expressions reduce toIj+1,j = 2∆o∑pp′JNp+j(2∆ot)JNp′+j+1(2∆ot) cos[(pi2N + Φ)(p′ − p)]=2∆oNN−1∑m=0∑pJNp+1(4∆ot sinkm2)e−ikm(Np+12+j)iNp+1· cos[(pi2N + Φ)p] (3.31)for the double and single sums respectively.Again, the currents across any links must be strictly periodic in time;and again, it is useful to show the results for a 3-site system. For this caseN = 3 , and assuming that the particle begins at the origin, we findI0,1 =2∆o32∑m=1∑pJ3p+1(4∆ot sinmpi3)× e−impi(3p+1)/3i3p+1 cos[(3pi2+ Φ)p](3.32)613.2. Pure Phase Decoherence Regionwhich we can also write in the formI0,1 =2∆o3∑pJ3p+1(2√3∆ot)i3p+1 cos[(3pi2+ Φ)p]×2∑m=1(e−ipi(3p+1)/3 + e−i2pi(3p+1)/3)(3.33)Now let us write e−ipi(3p+1)/3 + e−i2pi(3p+1)/3 = (−)pe−ipi/3 + e−2ipi/3. If p iseven, this becomes −i√3 and cos[(3pi2 + Φ)p] = (−)3p/2 cos(Φp); If p is odd,it becomes−1 and cos[(3pi2 + Φ)p] = (−)3(p−1)/2 sin(Φp). Therefore, we haveI0,1 =23∆o∞∑p=−∞J3p+1(2∆o√3t)K(p,Φ) ,K(p,Φ) = sin(pΦ) if p = odd ,K(p,Φ) =√3 cos(pΦ) if p = even . (3.34)These results are shown in Fig. 3.4. Notice that in this special case theresult is periodic in Φ; this is not however true for a general initial densitymatrix ρl,l′ , when the periodicity is in Φ/N .5 10 15D0t13231P j5 10 15D0t-0.40.4IH1®2LD0Figure 3.4: Plot of Pj1(t) for a 3-site ring, for a particle initially on site 1,in the strong decoherence limit. Left: The probability to occupy site 1 (fullline), 2 (large dashes), and 3 (small dashes). Right: the current from site 1to site 2 (compare Fig. 3.3). The results do not depend on Φ.3.2.2 System Plus the BathWe now wish to solve for the dynamics of the particle once it is coupled tothe bath, via the HamiltonianHφ = −∆0∑<ij>[c†icjei(A0ij+∑kαijk ·σk) +H.c.]. (3.35)623.2. Pure Phase Decoherence RegionHere we neglect the σk term since it commutes with the full Hamiltonian.Before going to the actual calculation, it is useful to note what are theimportant parameters in this problem. Consider first the simplest com-pletely symmetric case where αmnk → αk for all links {mn}. Assuming that|αk|  1 for all k, as discussed in section II, then it has been usual to definea parameter[1, 149]λ =12∑k|αk|2 (3.36)which is intended to measure the strength of the pure phase decoherence(this parameter has been referred to as the ‘topological decoherence strength’in the literature[1]. If the number N of bath spins is large, then we canhave λ  1; this is the limit of strong phase decoherence. However weshall see in what follows that under certain circumstances the decoherencecharacteristics depend on the functionF0(p) =∏kcos(Np|αk|) (3.37)which, depending on the values of the |αk|, can show very interesting prop-erties.In the more general case where the couplings {αmnk } differ from one linkto another, one can in principle define a set of decoherence parameters λmnfor each link, but this turns out to be not very useful.We now wish to solve for the reduced density matrix of the particle onceit is coupled to the spin bath, assuming the system to be described by Hφin (3.35). This is most easily done in a path integral framework, becausefor the tight-binding model of the ring we are using, the particle paths arevery simple (see Fig. 3.2).As shown in Appendix B.2, the reduced density matrix for the particlecan be written as follows. We begin by writing the “bare” free particledensity matrix as a double sum over winding numbers:ρojj′(t) =∑pp′ρojj′(p, p′; t) (3.38)Then one finds that in the presence of phase coupling to the spin bath, thereduced density matrix takes the formρjj′(t) =∑pp′∑ll′ρoj−l,j′−l′(p, p′; t)ρll′Fl,l′j,j′(p, p′) (3.39)633.2. Pure Phase Decoherence Regionwhere the influence functional has now reduced to a much simpler weightingfunction F l,l′j,j′(p, p′), which we call the ‘influence function’. In the same wayas the original influence functional, it depends in general on the initial stateρl,l′ of the density matrix at time t = 0. In the appendix the full expressionfor F l,l′j,j′(p, p′) is given; but here we will only use it for the usual case whereρl,l′ = δ0lδl′0, ie., the particle starts at the origin. We will also assume thepurely symmetric case where αijk → αk for every link. In this case theinfluence function reduces to (see Appendix B.2):Fj,j′(p, p′) = 〈e−iN [(p−p′)+(j−j′)]∑kαk·σk〉 (3.40)Notice that Fj,j′(p, p′) is a function only of the distance j − j′ betweeninitial and final sites. We may easily evaluate Fj,j′(p, p′) by assuming theusual thermal initial bath spin distribution, with equally populated states;we then get:Fj,j′(p, p′) =∏kcos((N [p− p′] + j − j′)|αk|) (3.41)Other initial distributions for the spin bath are easily evaluated from (B.23).From expressions like (3.41) one can then write down expectation valuesof physical quantities as a function of time. The simplest example is theprobability for the particle to end up at some site after a time t, havingstarted at another. Thus, eg., the probability Pj0(t) to move to site j fromthe origin in time t is now given byPj0(t) = ρjj(t)=∑pp′JNp+j(2∆ot)JNp′+j(2∆ot)× e−iN(p′−p)(Φ/N+pi2 ) F0(p, p′)which is a simple generalization of the free particle result in (3.29); we notethat only the termF0(p, p′) =∏kcos(N(p− p′)|αk|) (3.42)in the influence function survives in this expression. Note that since thisfunction depends only on the difference p− p′, it is identical to the functionF0(p) defined in (3.37) above (letting p′ = 0). We shall see below that thering current is also controlled by this function, and that it is therefore of643.2. Pure Phase Decoherence Regionquite general use in discussing the decoherence in this system. Note that ithas a complex multiperiodicity, as a function of the Ns different parameterspλk = Np|αk|; we do not have space here to examine the rich variety ofbehaviour found in the system dynamics as we vary these parameters.To give something of the flavour of this behaviour, suppose we have aGaussian distribution for the |αk|, given byP (|αk|) = e−|αk|2/2λo/√2piλo (3.43)so thatF0(p) = e−λp2/2, λ = Nλo (3.44)The limit λ → ∞ is the “strong decoherence” limit for this distribution,where we have F0(p)→ δp,0. In this limit the behaviour does not depend onflux at all.Now consider the results away from this limit - to be specific we take thecase where N = 3 again. For this 3-site ring one hasP11(t) =13(1 + 2[J0(2∆o√3t) + 4∞∑p=1J6p cos(2pΦ)F0(6p)]). (3.45)To analyse this result, note that for x  (6p)2, we can use J6p(x) ≈(−1)p√2/(pix) cos(x − pi/4). If the function F0(6p) truncates terms withp > pmax then for 2∆o√3t (6pmax)2 we have e.g.P11(t) ≈ 13[1 +2A√pi∆o√3tcos(2∆o√3t− pi/4)] ,A = 1 + 2∞∑p=1(−1)p cos(2pΦ)F0(6p) . (3.46)For Φ = 0 (or pi/2), the sum in the amplitude A reduces to∑(−1)pF0(6p)[or to∑F0(6p)]. Clearly, switching from Φ = 0 to Φ = pi/2 causes alarge increase in A. As λ increases, pmax decreases, and Eq. (3.46) appliesat shorter times. However, if λ > 0.1 the whole sum becomes negligible,and we are left with the Φ−independent asymptotic result. In fact, theinverse Fourier transform of the amplitude A(φ) can be used to measure thedecoherence function F0(6p)!Turning now to the current through the ring, we generalize the freeparticle results in the same way as above. Quite generally one hasIj,j+1 = −i〈∆˜j,j+1ρj+1,j − ∆˜j+1,jρj,j+1〉 (3.47)653.2. Pure Phase Decoherence Regionwhere one averages over the operator∆˜j,j+1 = ∆oeiΦ/Nei∑kαj,j+1k ·σk (3.48)This expression is evaluated in detail in the Appendix B.2. Here we consideronly the special case where the particle starts from the origin, and αijk → αk.Then one hasIj,j+1(t) =2∆oNN−1∑m=0∑pJNp+1(4∆ot sinkm2)e−ikm(Np+12+j)iNp+1· F0(p) cos[(pi2N + Φ)p] (3.49)One can also analyse these results as a function of the decoherencestrength and of the flux. Here we only quote the result in the strong deco-herence limit - then one hasI(j, j + 1)→ 2√33∆o(ρj,j − ρj+1,j+1)J1(2∆o√3t) . (3.50)for some general initial density matrix ρl,l′ . Again we see that the result iscompletely independent of the flux.3.2.3 Wave Package InterferenceIt is interesting to now turn to the situation where two signals are launchedat t = 0 from 2 different points in the ring. The idea is to see how the spinbath affects their mutual interference, and how, by effectively coupling tothe momentum of the particle, it destroys the coherence between states withdifferent momenta. We do not give complete results here, but only enoughto show how things work.We therefore start with two-wave-packets which will initially be in a purestate, and will then gradually be dephased by the bath. In the absence ofa bath, we will assume the wave function of this state to be the symmetricsuperpositionΨ(t) =1√2(ψ1(t) + ψ2(t)) (3.51)where the two wave-packets are assumed to have Gaussian form:|ψ1(t)〉 =N−1∑n=0e−(kn−pi/2)2D/2× e−ix0kn−i2∆0t cos(kn−Φ/N)|kn〉 (3.52)663.2. Pure Phase Decoherence Region|ψ2(t)〉 =N−1∑n=0e−(kn−pi/2)2D/2|2pi − kn〉 (3.53)where we assume the usual symmetric ring with flux Φ. At t = 0, one ofthe packets is centred at the origin, and the other at site jo. Note that thevelocity of each wave-packet is conserved. At times such that ∆ot = 2n,the two wave packets cross each other. From (3.52) we see that the maineffect of the flux is to shift the relative momentum of the wave-packets. Italso affects the rate at which the wave-packets disperse in real space - thisdispersion rate is at a minimum when φ = pi2 .The free-particle wave function in real space is then|Ψj(t)〉 =N−1∑n=0e−(kn−pi/2)2(ei(j−j0)kne−2i∆ot cos (kn+Φ/N)+ e−ijkne−2i∆ot cos (kn−Φ/N))|j〉(3.54)so that the probability to find a particle at time t on site j is P (j) = |Ψj(t)|2.Let us now consider the effect of phase decoherence from the spin bath.Using the results for Pjj′(t) from the last section, with an initial reduceddensity matrix given byρ(j, j′; t = 0) = |Ψj(t = 0)〉〈Ψj′(t = 0)| (3.55)we find a rather lengthy result for the probability that the site j is occupiedat time t:Pj(t) =N−1∑n,n′=0+∞∑m=−∞e−((kn−pi/2)2+(kn′−pi/2)2)D/2F0(m)×{ei(j−j0)(kn−kn′ )Jm(4∆ot sin ((kn − kn′)/2))eim((kn+kn′ )/2+Φ/N)++ e−i(kn−kn′ )jJm(4∆ot sin ((kn − kn′)/2))eim((kn+kn′ )/2−Φ/N)++[ei((j−j0)kn+jkn′ )Jm(4∆ot sin ((kn + kn′)/2))eim((kn−kn′ )−Φ/N)+ h.c.]}(3.56)Here we have used the generating series for Bessel functions, viz.,eix sin θ =+∞∑m=−∞Jm(x)eimθ (3.57)to separate the parts depending on flux in the final expression.673.2. Pure Phase Decoherence RegionM ∆0W3Figure 3.5: Interference between 2 wavepackets in the strong decoherencelimit. The packets start at site 0 and site jo = 50 at t = 0, and their relativevelocity is pi2 , in phase units.683.3. High Field RegionOne can also, in the same way, derive results for the current in the situa-tion where we start with 2 wave-packets. We see that expressions like (3.56)are too unwieldy for simple analysis. However in the strong decoherencelimit (3.56) simplifies to:P (j) =N−1∑n,n′=0e−((kn−pi/2)2+(kn′−pi/2)2)D/2· {ei(j−j0)(kn−kn′ )J0(4∆t sin ((kn − kn′)/2))+ e−ij(kn−kn′ )J0(4∆t sin ((kn − kn′)/2))+ [ei((j−j0)kn+jkn′ )J0(4∆t sin ((kn + kn′)/2)) + h.c.]}(3.58)and again we see that the flux has disappeared from this equation. Thisresult is shown in Fig 3.5.We notice 2 interesting things here. First, the interference between thetwo wave-packets is completely washed out, as one might expect. Howevernotice also that each wave packet splits into parts which move in oppositedirections. This is because the interaction with the fluctuating bath fluxcan actually change the direction of parts of each wave-packet (note thatthe transformation Φ→ Φ + pi reverses the momentum).One can also derive results for the current dynamics in the situationwhere we start with 2 wave-packets.3.3 High Field RegionIn this section, we are going to study the effect of the ‘transverse field’ term∑k hk · σk, if hk · σk does not commute with αijk · σk. If the commutatorsare much larger than |αijk ∆ij |, bath spins are significantly driven by thefields acting on them. The large fields divide surrounding bath spins intocertain polarization groups, which are the sets of bath states which havesimilar eigenvalues of∑k hk · σk, and make the hoppings between differentpolarization groups forbidden. That is the reason we call it the “High FieldRegion” in the first section of this chapter.In this section, we mainly use the “Orthogonal Blocking Approxima-tion”(OBA) to explore this problem. We start with the lowest order ofOBA with only perpendicular couplings before we generalize the methodinto general couplings as well as general initial states. At the end we discussthe technique of finite hk expansion which allow us systematically general-izing the OBA to a wide parameter region. Together with the last section,693.3. High Field Regionwe can obtain a complete description of the effect of off-diagonal couplingsin the spin bath model.The Hamiltonian we are considering in this section can be written asH =∑<ij>∆oei(φij+∑kαij ·σk)c†icj +∑khk · σk. (3.59)For simplicity we are assuming ∆ij = ∆0 for all non-zero 〈ij〉 pairs, althoughthe result can be generalized to arbitrary ∆ij without difficulty. The fieldshk act on each spin in bath may not always point in the z-direction sincethis effective “magnetic field” has various origins. But we can rotate everyσk to let hk acting along its “z-direction”. Now the dynamics of the spinbath is entangled with the central system. We have a rough argument onthis issue. When αk is small then eiαijk ·σk ≈ 1+ iαijk ·σk. The dynamic termsof the kth surrounding spin are(hk + i∆o∑ijαijk c†icj) · σk. (3.60)We can see that the external field acting on a bath spin is affected by the pathchosen by the central particle. By each hopping of the central particle, theenvironmental spins precess along different axes. Let’s write the interactionvector αij = αij‖ +αij⊥. Here αij⊥ is the component perpendicular to hk andαij‖ is the component parallel to hk.Parallel Coupling The parallel component αij‖ does not change the di-rection of the “effective field” (3.60). If there is only a parallel component,all the bath spin related terms commute with each other. As a result, theeffect of the spin bath should be the same as the pure phase decoherencecase in the previous chapter: it only changes the relative phase of differentpaths of the central spin. The trajectory of the central particle does not mixbath spins in different polarization groups. In the latter part of this section,we are going to show in detail that this is also true if both components arenonzero.Perpendicular Coupling The αijk,⊥ term, however, changes the preces-sion axis of the bath spins while the particle is hopping within the lattice.Therefore, the Berry phase of the environmental spin is entangled with thepath which the central particle chooses. This kind of entangled precession isa unique source of decoherence in the spin bath which cannot be reproduced703.3. High Field Regionin oscillator bath models. In this section we are going to focus more on theαijk,⊥ terms.3.3.1 Orthogonal Blocking ApproximationIn this high field limit, we start with the the “Orthogonal Blocking Approx-imation” (OBA), which was originally discussed in [1] with a TLS as thecentral system. In this section , we are going to generalize this approxima-tion to a general lattice in a formal way.Consider a time-independent Hamiltonian which can be separated intotwo parts, i.e. H = H0 +H1, with H1 as the perturbation term. We expandthe time evolution operator U(t) = e−iHt in orders of H0,e−iHt = e−i(H0+H1)t =1 + ...∑{ni}(−it)L(L+∑Li=0 ni)!(−iH1t)n0H0(−iH1t)n1·H0...(−iH1t)nL−1H0(−iH1t)nL + ....(3.61)Noticing that∑nxn(n+m)!=ex(m− 1)!xm∫ x0dt tm−1e−t =1(m− 1)!∫ 10da am−1e(1−a)x,(3.62)we can write each term (which is referred as a “path” in path-integral lan-guage) on the LHS of (3.61) as∑ni(−it)L(L+∑Li=0 ni)!(−iH1t)n0H0(−iH1t)n1H0...(−iH1t)nL−1H0(−iH1t)nL=(−it)L(L− 1 +∑Li=1 ni)!∫ 1a1=0da1aL−11 e−ia1H1tH0∑ni(−ia1H1t)n1H0...· (−ia1H1t)nL−1H0(−ia1H1t)nL= .......= (−it)L(L+1∏i=1∫ 1ai=0dai)δ(∑ai − 1)e−ia1H1tH0e−ia2H1tH0...· e−iaLH1tH0e−iaL+1H1t= (−it)L∫ +∞−∞dξ2pie−iξP˜ (ξ)H0P˜ (ξ)...H0P˜ (ξ),(3.63)713.3. High Field RegionHere we rewrite the Delta function as δ(∑ai − 1) =∫ +∞−∞dξ2pie−iξ(∑ ai−1)so that the ai integrals are independent with each other. We define theprojection operator P˜ (ξ) in the last line of (3.63) asP˜ (ξ) =∑nf(Mn, ξ, t)|φn〉〈φn|, (3.64)In our spin bath model, we take H1 =∑k hk ·σk; its eigenstates |φn〉s satisfyH1|φn〉 = Mnh|φn〉, where h represents the value of the field strength |hk|.The function f(M, ξ) is defined asf(M, ξ, t) =ei(ξ−Mht) − 1i(ξ −Mht) . (3.65)The function f(Mn, ξ) is centered at the point ξ = Mnt. If Mn 6=Mm, the integral∫ dξ2pie−iξ∏n f(Mn, ξ) is inversely proportional to (Mn −Mm) (see Appendix C). If (Mm −Mn)ht  1 , we can neglect all pathsin this summation except the ones with the same polarization all alongthe path, i.e. Mn = M, ∀n in the path. This implies that the pathswhich involve bath spins hopping between different polarization groups donot contribute significantly to the total time evolution. The “OrthogonalBlocking Approximation” is obtained by ignoring these terms in the path-integral by utilizing the following projection operator:PM ≡ δ(∑khk · σk −Mh) =∫ 2pi0dξ2pieiξ(∑k hk·σk−Mh). (3.66)In OBA, we insert such a projection operator in place of the P˜ (ξ) terms inthe last line of (3.63) to enforce the restriction on bath spin flipping. Theresult ise−iHt =∑Me−iMht∞∑L=0(−it)LL!(PMH0PM )L . (3.67)The summation over M is over all possible polarizations of bath spins. It isequivalent to replacing the Hamiltonian withH = H0 +H1 →∑MPMH0PM +H1 (3.68)If h→∞ this method is exact. This approximation is usually used whenthe external field strength h, which determines the energy difference betweendifferent polarization groups, is much larger than other energy scales in thesystem, i.e. the high field limit. In our case, the high field limit is validwhen |αijk ∆ij |  |hk| in (3.59).723.3. High Field Region3.3.2 Perpendicular CouplingWe start with a Hamiltonian which only contains the perpendicular off-diagonal coupling. We will study the general case with both perpendicularand parallel couplings in section 3.3.4.If we only allow the spins to couple to a single hopping amplitude, wecan rotate all the αkij to x-direction as well as keeping all hk at z-direction.The Hamiltonian can be written asH = ∆oei∑〈ij〉(φij+∑k αijk σxk )c†icj + h.c.+∑khkσzk. (3.69)To shorten the expressions, we again choose ∆〈ij〉 = ∆o. We can seelater that this choice does not change the main result in this chapter.The goal here is to calculate the reduced density matrixρij(t) = 〈∑k,lG(i, k; t)G†(j, l; t)ρkl(t = 0)〉 . (3.70)Here 〈...〉 is the partial trace over the bath variables andG(i, k; t) = 〈i|eiHt|k〉is the propagator from site i to site k of the central particle.We can write the Green’s functions in the path integral language:G(i, k; t)ρkl(0)G†(j, l; t) = 〈φi|e−iHt|φk〉〈φk|ρ(0)|φl〉〈φl|eiHt|φj〉= 〈φi|∞∑L=0(−it)LL!HL|φk〉〈φk|ρ(0)|φl〉〈φl|∞∑L=0(it)LL!HL|φj〉=∑L,L′∑i1...iL−1,i′1,i′L′−1tL+L′iL−L′L!L′!〈φiL |H|φiL−1〉〈φiL−1 |H|φiL−2〉...〈φi1 |H|φi0〉· ρkl(0)〈φi′0 |H|φi′1〉...〈φi′L′−1 |H|φi′L′ 〉.(3.71)Here L (L′) is the length of a path (conjugate path). The sequence {i0, ....iL}({i′0, ....i′L−1}) represents the sites passing by the central particle in this path,with i0.....L being the indices of lattice sites. i0 = k, iL = i (i′0 = l, i′L′ = j)are the starting and ending sites of this path.The summation over L,L′ and i1...iL−1, i′1, i′L′−1 is equivalent to the sum-mation over all possible paths. So we can define the contribution function733.3. High Field RegionW of a pair of paths asWi0...iLi′L′ ....i′0≡ tL+L′iL−L′L!L′!〈φiL |H|φiL−1〉〈φiL−1 |H|φiL−2〉...〈φi1 |H|φi0〉· 〈φi′0 |H|φi′1〉...〈φi′L′−1 |H|φi′L′ 〉,(3.72)andG(i, k; t)ρkl(0)G†(j, l; t) = ρkl(0)∑all possible pathsWi0...iLi′L′ ....i′0. (3.73)The problem is now converted to calculating the path contribution functionW .At the first glance, we might want to treat the terms∑k αijk σxk as aperturbation to the hopping amplitude ∆o. However, this picture is wrongsince it ignores the dynamics of bath spin variables. The quantum spin isrotating around the field acting on it. In (3.69), the “effective” field actingon a bath spin is heff = (∑〈ij〉∆oc†icji sinαijk +h.c.)xˆ+hkzˆ. Because of theαijk term, this “effective” field keeps changing its amplitude and directionwhile the central particle is hopping from site to site. This cause the pre-cession of bath spins. Since the interaction is through σxk terms, quantummechanically, this precession of bath spins is equivalent to saying that somebath spins are flipped when the central particles hop from site to site.Unfortunately, this behavior is not well described by the usual perturba-tive approach towards the central system. The dynamic of the bath variablesare smeared out through the premature partial trace process and eventuallylost in the Markovian approximation. On the other hand, OBA preservesthe behavior and allows us to see this point clearly.743.3. High Field RegionWe can write down the contribution function W using the OBA asWi0...iLi′L′ ....i′0=(∆0t)L+L′iL−L′L!L′!〈δ(∑khkσzk −M)ei∑k αiLiL−1k σxk ...· δ(∑khkσzk −M)ei∑k αi1i0k σxk δ(∑khkσzk −M)· δ(∑khkσzk −M)ei∑k αi′0i′1k σxk ...δ(∑khkσzk −M)· ei∑k αi′L′−1i′L′k σxk δ(∑kσzk −M)〉=(∆0t)L+L′iL−L′L!L′!∫ ∞−∞L∏j=1dξj2piL′∏j=1dξ′j2pi〈∏kT kLT†kL′ 〉.. (3.74)Here 〈...〉 is the average over the spin bath variables. The operatorei∑k αiLiL−1k σxk flips the bath spins. Suppose the initial polarization of bathspins is M ; in this limit the bath spins cannot flip at all unless the totalpolarization M is preserved. This is usually referred as a resonant transitionin the literature. In the last line we again use (3.66) to expand the deltafunction, so that the initial and final state of a particular bath spin arerelated by the following transition operators:T kL|σink 〉 = |σoutk 〉. (3.75)T kL = eiξLhkσzkeiαiLiL−1k σxk ......eiξ1hkσzkeiαi1i0k σxkeiξ0hkσzk . (3.76)These transition operators describe the dynamics of the bath spins andpreserve information about the precession of the bath spins.The result will depend on the initial polarization M of the bath spins.To make things as simple as possible we drop all degeneracy blocking effects,i.e., we make all hk the same. This is generally true if all bath spins havethe same origin (e.g. nuclear spins of the same atoms, the same type ofamino acid residue, etc.) In this case the bath can be separated into 2K+ 1different polarization groups with the degeneracyp(M) =K!(K+M2 )!(K−M2 )!2K. (3.77)Here M =∑k σzk and K is the total number of bath spins. One importantcomment is that the small M ≈ 0 configurations have way larger phase space753.3. High Field Regionthan the configurations with large M . We could expand (3.77) aslogp(M)p(0)≈ e−M22K+O(M4). (3.78)The leading term away from the identity is proportional to exp(−M2/2K)for small M . Henceforth, we can write the result as an expansion of thetotal polarization M asWi0...iLi′L′ ....i′0=(∆0t)L+L′iL−L′L!L′!(∏n∫ 2pi0dξn2pi)(∏n′∫ 2pi0dξ′n′2pi)·A0(L,L′)K∑M=−Ke−M22Kθ(K)AM (L,L′).(3.79)Here θ(K) =∑KM=−K e−M22K is the normalization factor for this summation.The A0(L,L′) term represents the contribution from the M = 0 polarizationgroup which is a major part of the total bath Hilbert space if the numberof bath spins is large. The AM (L,L′) terms represent the contribution fromother polarization groups. Actually, detailed calculations show that thecontribution from the M = 0 polarization group is of O(α2) order while thecontribution of the M = 1 polarization group is of order O(α4). Combinedwith the e−M22K factor, we can safely neglect all the M 6= 0 terms in (3.79).The detailed calculations and expressions for the derivation of (3.79) canbe found in Appendix C, as well as the calculations of both A0(L,L′) andA1(L,L′).Zero PolarizationThe term which contributes most is from the zero polarization group M = 0.If we ignore the non zero polarizations of the bath spins, we have the result(for details again check Appendix C):W 0i0...iLi′L′ ....i′0=(i∆ot)L+L′iL−L′L!L′!∫ ∞0z dz2pie−12z2· (∏nJ0(∑kαinin+1k z))(∏n′J0(∑kαi′n′ i′n′+1k z)).(3.80)(J0(x) is the Bessel function of the first kind). Compared to the path inte-gral of the bare system, we can see that J0(∑k αi′n′ i′n′+1k z) can be absorbed763.3. High Field Regioninto ∆0. Actually, even if the system is not symmetric, this term can stillbe absorbed into ∆i′n′ i′n′+1. This justifies the simplification we make by as-suming all ∆ij = ∆0. As a result, we get a new Hamiltonian for the centralparticle asH0(z) =∑〈ij〉∆〈ij〉J0(∑kαijk z)c†icj . (3.81)And the density matrix evolution under such a Hamiltonian isρ˜0(z; t) = e−iH(z)tρ0eiH(z)t. (3.82)We can then write the reduced density matrix ρij(t) in the high field limitasρ(t) =∫ ∞0z dz e−z22 ρ0(z; t). (3.83)Since |J0(∑k αijk z)| ≤ 1, the renormalized Hamiltonian H0(z) is alwaysmore “localized” than the original one. The quenching factorκij = |∑kαijk |2, (3.84)makes the hopping terms small and the transfer between sites is suppressed.Therefore the excitations are more likely to be localized to their initial state.We can use κij as the control parameter of the decoherence in this high fieldregion.As a simple example, we consider the central system as a triangularlattice with three identical sites. The hopping amplitudes between threesites are ∆ij = ∆0. Without spin bath, the probability for the centralparticle to return to the original point after time t isP00(t) =19(5 + 4 cos(3∆ot)). (3.85)Then when we turn on the spin bath interactions, we haveρ00(t) =∫ ∞0dz e−z22z9[5 + 4 cos(3∆otJ0(∑kαkz))]. (3.86)Here we have assumed the symmetric case so κij = κWe plot (3.86) for different κ in Fig 3.6. Since the OBA is mainlyestablished to study the long time behavior of the system, it is interestingto check the asymptotic values as t → ∞ for the reduced density matrixelements.773.3. High Field Region20 40 60 80 100Dt0.450.500.550.600.650.700.75PooHtLFigure 3.6: Figures of ρ00(t) in (3.86) in high field limit. Different colors fordifferent coupling strength parameter κ = κ01 = κ12 = κ02 (3.84). Blue:κ =10−3; Red:κ = 10−1; Green κ = 102;There are two interesting things that need to be pointed out here. Thefirst point concerns the asymptotic value of the diagonal elements. Onemight expect that as t→∞, the system will go to the thermal equilibriumstate since spins are coupled to each site equally. For a 3-site symmetricsystem, this means ρ00 → 13 . However,ρ00(t)→ 59+4 cos(3∆ot+ θ(κ∆ot))9√1 + 9κ2∆2ot2→ 59, (3.87)where θ(κ∆ot) is a real integral function of κ∆ot. This shows that in the highfield limit, since the environmental spins are frozen in certain polarizationsubspaces, the dynamics of the central system is also “slaved” and is notsusceptible to thermalization.The second point concerns the off-diagonal terms in the reduced densitymatrix. The time evolution of off-diagonal elements gives the decoherenceof the system. In the oscillator bath, these elements usually go to zero aftera long time. But for this spin bath case, we haveρ01(1)→ 19(−1 + e3i∆ot3i∆otκ+ 1− e−3i∆ot−3i∆otκ+ 1)→ −19. (3.88)783.3. High Field RegionWe can see that the coherence in the central system is “protected” by thebath spins. But this coherence protection phenomenon is not universal inall kinds of lattices. If we look at the general symmetric ring lattice with Nidentical sites we studies in the previous chapter, the density matrix aftertime t will beρojj′(t) =1N2N−1∑n,n′=0e−i(knj−kn′j′)+4i∆ot sin[φ−(kn+kn′ )/2] sin[(kn−kn′ )/2], (3.89)where kn =2npiN and φ are the flux threading the ring. If there are no knand kn′ which satisfy sin[φ − (kn + kn′)/2] = 0, then ρjj′(t) → 0 for alloff-diagonal terms j 6= j′ as t → ∞. The long time asymptotic values foroff-diagonal elements are non-zero only if we can find such kn and kn′ .3.3.3 Spectral FunctionThe Fourier transform of the reduced density matrix element ρ00(t) givesthe spectral function. For a bare latticeSo(ω) =12pi∫ +∞−∞dtρ00(t)eiωt =1N2∑i,jδ(ω − εi + εj), (3.90)where εi are the energy eigenvalues. The spectrum is composed of severalδ-function peaks residing at all the resonant frequencies.Again we take the N = 3 symmetric ring as an example; one then hasSo(ω) =19[5δ(ω) + 2δ(ω − 3∆o) + 2δ(ω + 3∆o)]. (3.91)There are three peaks at ω = 0,±3∆. This basically reflects the bandstructure of this model.In the previous sections, we studied both the pure phase decoherencelimit with hk = 0 and the high field limit with hk →∞. In this section weare going to study the spectrum of both cases. It helps us understanding thefundamental difference between these two types of environmental couplings.Pure phase decoherence In the pure phase decoherence limit, we quotethe result in the previous section for a 3-site systems, viz.,ρ00(t) =13(1 + 2J0(2√3∆ot) + 4∞∑p=1J6p(2√3∆ot)F (6p)). (3.92)793.3. High Field RegionIf we have a Gaussian distribution for the coupling αk, we can take F (6p) =e−36λp2 , with λ = 12∑k α2k. The Fourier transform of the Bessel functions is12pi∫ +∞−∞Jn(at)eiωtdt=e−inθ + (−1)neinθ2pi√a2 − ω2 Θ(a2 − ω2),(3.93)where Θ(x) is the Heviside step function and θ = arcsin ωa . Notice that thisexpression is only nonzero when ω ∈ (−|a|, |a|). If ω2 = a2, this function willdiverge. However, this divergence is killed when λ = 0, i.e., for the bare ringcase. The poles move to ω = 3∆ due to the summation∑+∞n=−∞ ein arcsin aω .As a result, we have the spectral function in the pure phase decoherencelimit,Sp(ω) =13[δ(ω) +2pi√12∆2o − ω2Θ(12∆2o − ω2)+ 4∞∑p=1cos(6pθ)pi√12∆2o − ω2Θ(12∆2o − ω2)F (6p)].(3.94)Compared to the bare ring results, we can see that since we have a dampingterm in the series, the sum over p converges. As a result, the poles moveto −2√3∆ and +2√3∆. But there are still three divergent peaks in thespectrum, which means the system still preserve the band structure of thebare lattice qualitatively. And it is always true even when the interactiongoes to infinity. For the limit when λ → ∞, i.e.where there is an infinitynumber of bath spins, we still have three divergent peaks in the spectrum:Sp(ω)→ 13[δ(ω) +2pi√12∆2 − ω2 η(12∆2o − ω2)], λ→∞. (3.95)High Field Limit In the high field limit case, we haveρ00(t) =∫ ∞0z dz e−z221N2∑i,jei(εi−εj)J0(∑k αkz)t. (3.96)Its Fourier transform isSh(ω) =1N2∫ ∞0z dz e−z22∑i,jδ(ω + (εi − εj)J0(∑kαkz)). (3.97)803.3. High Field RegionWe still consider the same three-site symmetric model. The spectral functionis thenSh(ω) =59δ(ω) +29∫ ∞0z dz e−z22·(δ(ω + 3∆oJ0(∑kαkz)) + δ(ω − 3∆o∏kJ0(αkz))).(3.98)The first term is identical to the bare ring case, which indicates a δ-peakat the origin. However, the two non-zero peaks are no longer divergent.Actually, for a frequency ω which is close to the 3∆o, the spectral functionisSh(ω) ≈ 227∆oκe3∆o−ω3∆oκ , ω → 3∆o. (3.99)In the weak coupling limit κ → 0, this function becomes a δ function likethe bare ring case. But it is no longer divergent for any nonzero κ. In thestrong coupling limit κ → +∞ (for infinite numbers of bath spins), Sh(ω)actually goes to zero for any nonzero ω. The whole spectrum becomes asingle δ function at the origin.This means that unlike the pure phase decoherence case, the basic bandstructure of the central system is not preserved in the high field limit. Thedisappearance of non-zero peaks means these energy levels are no longerstable. Fig. 3.7 gives the plot of spectral functions in both cases.813.3. High Field Region-4 -2 2 4ΩD0.ΩLD- -4 -2 2 4ΩD0.ΩLD-4 -2 2 4ΩD0.ΩLD-4 -2 2 4ΩD0.ΩLD-4 -2 2 4ΩD0.10.20.3SHΩLD-4 -2 2 4ΩD0.10.20.3SHΩLD-4 -2 2 4ΩD0.10.20.3SHΩLDFigure 3.7: Figures of spectral function for both the zero field limit (3.94)and the hight field limit (3.98). Blue: zero field limit(the pure phase deco-herence region); Red: High field limit. Each figure has different couplingstrengths. The first row, left: κ, λ = 0.001; right: κ, λ = 0.01. The secondrow: left 0.05; right 0.1. The third row: left 0.5; right 1.0. The last row:10.0.823.3. High Field Region3.3.4 General CouplingUp to this point we have ignored the parallel coupling term αij‖ . To simplifythe Hamiltonian, in the previous sections we also assumed that by rotatingall of the hk to the z direction, we can simultaneously align almost all thecouplings αijk to the x direction. We have established the method to dealwith such systems and show the unique properties of this high field spin bathmodel. However, the restrictions on the directions of αijk are not necessaryand can be easily generalized to the most general case.We start from equation (3.59) with hk rotated to the z-direction:H =∑<ij>∆oei(φij+∑kαij ·σk)c†icj +∑khzkσzk. (3.100)Now αijk contains all three component αijk,x, αijk,y, αijk,z. As long as hzk is stillmuch larger than the other energy scales in this model, we can still use theOBA to study the result. The derivation is similar to the perpendicularcoupling case but with a more complicated transition operator T kL. Detailscan be found in Appendix C. The result can be written asW 0i0...iLi′L′ ....i′0=(i∆ot)L+L′iL−L′L!L′!∫ ∞0z dz2pie−12z2(∏nJ0(κginin+1z))·(∏n′J0(κgi′ni′n+1z))exp−12∑k(∑nαinin+1kz −∑nαi′ni′n+1kz)2 .(3.101)Here κgij is a function associated with the perpendicular coupling αijk,⊥, whichis a direct generalization of (3.84), i.e.,κgij =√(∑kαijkx)2 + (∑kαijky)2 = |∑kαijk,⊥|2. (3.102)We also have an extra decoherence term, which depends on the parallelcouplings αijkz. If the lattice is a symmetric ring, this term gives us a deco-herence factor e−(N(m−m′)+j−j′)2λ with λ = 12(∑k αkz)2. We immediatelyrealize that this is exactly the decoherence factor we get in the pure phasedecoherence case. Only the parallel coupling αijk,‖ enters this decoherencefactor, which is directly associated with the λ we introduced in (3.36).Therefore, the parallel couplings αijk,‖ and perpendicular couplings αijk,⊥have different effects. The former ones cause pure phase decoherence and the833.3. High Field Regionlatter ones cause energy dissipation. We have already shown that these twohave fundamentally different effects on the dynamics of the central systemthrough their spectral. And this only relies on the angles between hk and αijk .Therefore, it allows us to manipulate the decoherence/dissipation propertiesof the system by changing the direction of fields hk. If we know that αijkare mostly lying in on plane, we could change the polarization field to thedirection perpendicular to this plane, in order to protect the central systemfrom phase decoherence and effectively localize the excitation hopping. Onthe other hand, if we want to protect the system from dissipation, we couldrotate hk to reach a minimum κgij .3.3.5 General Initial State of Spin BathUntil now we have only considered the initial density matrix of the bath asan identity matrix. If the system is ergodic, it will eventually “forget” itsinitial state and the time average of the dynamics of the system will convergeto a universal value. Therefore we could define an equilibrium state of thebath. But in previous sections, we have seen that the long term behaviourof the central system strongly depends on its initial state. It is worthwhileto explore the dependence of the bath behavior on its initial state.In this section, we set the initial state to be ρo = ⊗∏k ρk(nˆ), in whichρk(nˆ) ≡ 12I + 12 nˆ · σ, where ~n is the so called “Bloch vector” in 3-D spacewith |nˆ| ≤ 1.Pure phase decoherence Recall that the kernel of the influence func-tional in the pure phase decoherence region can be written asF0(p¯) =∏kcos(Np¯|αk|), (3.103)where p¯ is the winding number difference of the two path. This is derivedbased on an identity for the bath initial state. It is straight forward toevaluate the influence functional kernel in this new initial state:F ′0(p¯) =tr(ρoe−i∑k Np¯αk·σk)=∏k〈nˆk|e−iNp¯αk·σk |nˆk〉=∏kcos(Np¯αk)(1 + iaˆk · nˆ tan(Np¯αk))=F0(p¯)∏k(1 + iaˆk · nˆ tan(Np¯αk)).(3.104)843.3. High Field RegionThe first term is just the original influence functional kernel. The correctionterm can be written as eiNp¯∑k ak·nˆ at the first order correction level. Thisbasically gives a flux renormalization with the value ∆Φ =∑k αk · nˆ. Itwill not increase or reduce the decoherence time scale of the system, butchange the Aharonov-Borm flux by a factor which is proportional to theinner product between the initial state vector nˆ and the sum of the couplingvectors αˆk. Therefore, this initial state does not change the behavior of thissystem qualitatively.For the high field limit, the expansion is more complicated. But we canstill get some analytical results in some cases.If nˆk only has a z-component, which means the initial state is polarizedin the same direction as the external field hk, the probability to find thesystem at a∑k σzk = M state is no longer proportional toK!( k+M2)!(K−M2)!.The effect of this term will be discussed next in the high order correctionsection. We can see the leading order is O(α3k). If nzk = ±1, then the systemis completely frozen at an extreme state in which there is no freedom forthe bath spins to flip. In this case, there is no decoherence of the centralsystem. This bath only renormalizes the total flux of the central system asit does in the low field limit.If nˆk only has a y-component, which means the initial states are allpolarized along the y-component, then we have similarly thatρij(t) =∫ ∞0z dz J0(nyz)e− z2−n2y2 ρ0ij(t; z). (3.105)The derivation of (3.105) can be found in Appendix C.6. There is a addi-tional term J0(nyz)en2y2 . This is a quenching term since it is always smallerthan 1. In this case, dissipation effects are enhanced and the central systemapproaches its final states more quickly than the origin case.We can see that the baths will have different decoherence effects dependson their different initial states. This is because when the hk field is applied,the intrinsic dynamics of the bath itself matters. The memory effect of thebath can fundamentally change the decoherence or the dissipation rates ofthe central system, leading to various behaviors: from a pure phase deco-herence (nx = ny = 0, nz 6= 0) to a strong dissipation (nx 6= 0, ny = nz = 0).3.3.6 Finite hk ExpansionsThe orthogonal blocking approximation is only exact when the field h→∞.If h is not sent to infinity but a finite large value, then the transitions between853.3. High Field Regiondifferent polarization group can happen and the result will be a series over∆o/h. The goal in this subsection is to calculate the next non-trivial orderwith a finite but large h and to show that we can systematically extend theOBA to any order of ∆o/h.To start, we go back to the original derivation of the OBA in 3.3.1. Fora time-independent Hamiltonian H = H0 + H1, since I =∑M PM , we caninsert this identity operator and rewrite the original Hamiltonian asH =∑M,M ′PMH0PM ′ +H1. (3.106)From (3.63) we can write the propagator of this Hamiltoinan aseiHt =∑L,{Mj}(−it)Lg[t, {Mj}]PM0H0PM1H0...PML−1H0PML . (3.107)The function g is defined asg[t, {Mj}] =∫ ∞∞eiξ2pidξL∏j=0f(Mj , ξ, t), (3.108)where f is defined in (3.65). In the OBA, we do not allow any transitionsbetween different polarization groups. We only include the terms whereM0 = M1 = ... = ML = M , and in this case g(t, {Mj}) = e−iMhtL! . This isequivalent to taking the following effective Hamiltonian:H =∑MPMH0PM +H1, (3.109)where PM is the projection to polarization group M . In the next leadingorder, we loosen the restrictions and allow the system to hop into anotherpolarization group M ′ only once. That is we include paths which containonly one Mk = M′ and another Mj (j 6= k) still equal to M . The contribu-tion to the propagator of such a path is(−it)Lg[t, {Mj}]PMH0PM ...H0PM ′H0....PMH0PM , (3.110)whereg[t, {Mj}] =L∑a=1−1(L− a)!e−iMht(i(M −M ′)ht)a +(−1)Le−iM ′ht(i(M ′ −M)ht)L . (3.111)863.3. High Field RegionDetails of the calculation of g can be found in Appendix C.1. We can seethat now g is a polynomial of (i(M −M ′)ht)−1. If we are only interested inthe long time behavior of the system, we only need to keep the lowest orderterms, which are proportional to (i(M −M ′)ht)−1, giving usg[t, {Mj}] ≈ −1(L− 1)!e−iMhti(M −M ′)ht +O(1(i(M −M ′)ht)2)(3.112)Then we can write the contribution of this path as−1(L− 1)!(−it)L e−iMht(i(M −M ′)ht)PMH0...H0PM ′H0...H0PM=(−it)L−1(L− 1)! e−iMhtPMH0...H0PM ′H0(M −M ′)h...H0PM .(3.113)This is the same as the first order perturbation expansion of the followingeffective Hamiltonian:H =∑MPMH0 + ∑M ′ 6=MH0PM ′H0(M −M ′)hPM +H1 (3.114)Then we can follow the same procedure as in the OBA (Section 3.3.1) andwrite the path contribution function asW 0i0...iLi′L′ ....i′0=iL−L′(∆0t)L+L′L!L′!∫ ∞−∞∏nξn2pi∏n′ξ′n′2pi∑M ′〈eiξ0(∑k σzk−M)ei∑k αiLiL−1k σxk ..eiξm(∑k σzk−M ′)..eiξ′0′ (∑k σzk−M)ei∑k αi′L′−1i′L′k σxk 〉(3.115)We can go through all the procedures in 3.3.2, before we summed over allpossible M ′ from −K to +K. If the number of bath spins K is very large,we can make the approximation that K →∞ and use the identity:∞∑M ′=1sin(M ′θ)M=12(pi − θ), if θ ∈ (0, 2pi). (3.116)Then the result of the next nontrivial order to the OBA can be written as:ρ(t) =∫ ∞0z dz e−z22 e−iH˜(z)tρ(0)eiH˜(z)t. (3.117)873.3. High Field RegionBut H˜0(z) now becomesH˜0(z) = H0(z)− piH20 (z)h, (3.118)with H0(z) defined by (3.81) as in the zeroth order OBA term.We can systematically extend this method to higher orders by consid-ering two or more different Mj 6= M for each pair of paths. The standardprocedure would be:(i) calculate the integral g[t, {Mj}] for the given number of allowed tran-sitions;(ii) discard terms higher than the given order of ((M −M ′)ht)−1. Thecoefficients in the expansion give the coefficients in the effective Hamiltonian;(iii)follow the OBA procedure from Section 3.3.1 and sum over all pos-sible polarization groups allowed.At the end, we always get an expression similar to (3.117) but with amore complicated H˜0(z).88Chapter 4Oscillatior Bath Model: Roleof Vibrational Modes inEnergy TransferThe oscillator bath has been extensively studied, as reviewed in previouschapters. This problem is one of the fundamental topics in condensed matterphysics as well as in other aspects of science. One can categorize differentoscillator environments by their origins: i)acoustic phonons, the collectivemovements of the whole molecule: the coupling to the central system is long-range and weak for each single mode. ii) optical phonons: in the language ofbiology these are usually referred to the intra-molecular vibrational modes.The former modes are usually weakly-coupled low energy modes whosenumbers are enormous due to the huge number of degrees of freedom inbiological complexes. They are associated with the low energy continuumin the spectrum. With these modes, decoherence goes hand-in-hand withdissipation[18, 19]. They have been extensively studied in the literature(see [17, 142] for a good summary). The latter ones, on the other hand,have localized microscopic origins (vibrations of chemical bonds, chargedresidues, etc.). These phonon modes can be observed as discrete peaks inthe spectrum. Their numbers are fewer and their energy scale is higher,usually comparable to other relevant energy scales in the system. Thesemodes also couples to the hopping amplitude of the excitons: excitons canhop down to other sites with lower site energies by exchanging energies withthese vibrational phonons. The importance of these modes has been realizedin resonant energy transfer processes in light harvesting molecules (see forexample [21]).These vibrational modes couple not only to the on-site energies (whichis known as the Holstein coupling in solid state physics[23]), but also to thehopping amplitudes between the different pigments. The phonon-modulatedtransition was first mentioned by Peierls [24] with regard to the Peierls tran-sition. It was then studied in polyacetylene[25, 26] using the Su-Schrieffer-894.1. Model of Resonant Energy Transfer into a Reaction CenterHeeger (SSH) model. It had already been realized that the SSH model cancause qualitatively different behaviors when compared with the Holsteinmodel[27]. The Holstein couplings have been extensively studied in biologi-cal molecules, but these Peierls-type couplings have never been discussed inbiological molecules. Experiments have already suggested that vibrationalmodes can directly couple to photon absorption[28]. It is natural to spec-ulate that these modes also couple to the resonant energy transfer withinmolecules.In this chapter, we investigate a dimer model for light harvesting mole-cules, which keeps the essential properties of a reaction-center-connectedlight harvesting complex. We try to address the role of phonons in resonantenergy transfer by including phonon couplings of both Holstein type andPeierls type. We find that the two types of coupling affect the energy transferrate in qualitatively different ways.In this chapter we also study two-phonon interactions, i.e. the inter-action terms which are proportional to gkl(a†k + ak)(a†l + al). Two-phononinteractions are usually negligible for acoustic phonon modes in solid statephysics because the coupling strength is usually much smaller than the lin-ear interactions. There are arguments that for acoustic phonons interactingwith electrons in a crystal, the lowest order in coupling strengths one can ac-quire from two-phonon couplings is an Ohmic spectral function which takesthe form J(ω) → αω[17]. This term can be absorbed into the one-phononterms and is negligible since in general α ∝ 1N for acoustic phonon modes[17].However, this calculation only takes into account the second order cumu-lant and ignores higher order ones (which are not zero for the second ordercouplings). For optical phonons, on the other hand, there is little literaturewhich considers the effect of the higher order couplings. In this chapter, westudy two-phonon interactions between optical phonon modes with frequen-cies close to each other. We show that the two-phonon interaction can bemapped into a spin bath through path integral techniques.904.1. Model of Resonant Energy Transfer into a Reaction CenterFigure 4.1: An illustration of our whole system. In the top panel weshow a schematic picture for our model. c1 and c2 are the two pigmentsin a molecule. The reaction center model in the hexagon is a Bethe lat-tice. Phonons couple to the system through gH (Holstein coupling) and gP(Peierls coupling). In the bottom panel, we show the energy difference in themodel. ∆ is the energy differences between c1 and c2. 2D is the bandwidthof the RC. t0 is the hopping amplitude between c1 and c2.4.1 Model of Resonant Energy Transfer into aReaction CenterThe model we will use to study the energy transfer into a reaction center isH = H0 +H1 +H2 (4.1)H0 = ε2c†2c2 + ε1c†1c1 + t0(c†2c1 + c†1c2)+t1(c†1d0 + d†0c1) +HB (4.2)HB =∑〈ij〉V (d†idj + h.c.) + ε (4.3)H1 =∑i=0,1(gH(a†i + ai) +g2HΩi)c†ici+(gP (a†2 + a2) +g2PΩ2)(c†2c1 + h.c.) (4.4)H2 =∑i=0,1,2Ωia†iai (4.5)914.1. Model of Resonant Energy Transfer into a Reaction Centerwhere H0 is the bare molecular Hamiltonian without coupling to phonons,and {c†i , ci} are the creation and annihilation operators of local excitons. Wechoose a dimer model in which there are only two pigments in a molecule.The exciton is first created at site c2, which has the highest on-site energyε2 in the molecule. Let us define ∆ ≡ ε2 − ε1 is the energy differencebetween site c1 and site c2. Site c1 is the pigment connected to the reactioncenter(RC). It couples to the RC through the term t1(c†1d0 + d†0c1). Thecenter of the absorption spectrum of the RC is set at ε = 0 (an illustrationof this model is shown in Fig. 4.1).The interplay between RCs and photosynthetic antennae is complicatedin vivo. Although the time scale for an RC to complete a chemical reactioncircle is much longer than the exciton transfer time scale, the transfer timeto the RC is actually close to the order of the intra-molecule transfer time(e.g. in purple bacteria, 20ps for LH1-LH1 transfer, 56.8 − 130ps for LH1-RC transfer, and 11.9ps for RC-LH1 back transfer [153][67]; for smallerphotosynthetic molecules the intra-molecule transfer rate is faster). The RCshuts down after receiving an exciton to complete the chemical reaction, butthe transfer to the RC is still relevant under the relevant biological time scale.However, in most theoretical studies of the intra-molecule phonon-assistedtransportation, the existence of RCs is either ignored or simply treated asa non-hermitian term in master equations for the dynamics. In the lattercase, the effect of an RC is usually simplified into a single predeterminedrelaxation rate and the quantum-mechanical back transfer from the RC tothe photosynthetic antennae is ignored. In this work we are going to studythe effect of environmental phonons, not only on the transfer rate betweentwo pigments within a single molecule, but also on the transfer rate tothe RC. The transfer rate to the RC, as the most important measure ofthe biological functionality of this system, should not be predetermined. Toavoid this, our phenomenological RC model should have a finite band width,and should be incorporated into the quantum model of the whole system.In principle, many forms of lattice model which have finite band widthcould do the work. Here we choose a Bethe lattice (or Cayley tree) asour RC Hamiltonian HB in (4.1). A Bethe lattice is a lattice which hasno closed loops and is completely characterized by the number of nearestneighbors Z. In Fig. 4.1 we show a portion of a Bethe lattice with Z =3. In our calculations, we choose a lattice with infinite branching numberZ →∞ but fixed bandwidth D = 2V√Z − 1 to model the RC. In this casethe propagator within the RC is known to be the Hubbard Green functionG0B(z) =2z+√z2−D2 [154]. In this chapter, we always set D = 1. There are924.1. Model of Resonant Energy Transfer into a Reaction Centerseveral advantages to using this model: i) It does not contain any singularitywithin the spectrum, unlike the case of crystalline lattices. ii) The treestructure eliminates the possible emergent resonance between loop pathswhich would cause unnecessary complications. iii) It maintains analyticbehavior near the band edges: the density of states is simply a semicircle.iv) It is exactly solvable and easy to study.Without any phonon coupling, this RC model is equivalent to the Ander-son-Fano model of impurity scattering. The exact solution is easily found[155]. The properties of this RC model are determined by both the energydifference ε1 and coupling strength t1. In the region ε21 + 4t21 < D2, nolocalized state is formed and all excitons are eventually transferred to theRC(see D.2). This is the property we need for an RC , therefore we alwayswork at ε1 = 0 and t1 = 0.2D in this entire paper. A detailed description ofour model for the RC can be found in Appendix D.2.In (4.1), H2 is the Hamiltonian of environmental vibrational modes.These modes couple to the central system through H1. In (4.4), gH is theHolstein coupling which modulates the on-site energy of the system ε2, ε1;and gP is the Peierls coupling which modulates the hopping amplitude t0(see Fig.(4.1)). In this chapter, we only include the phonon modes whosefrequency Ω is on resonance with the intra-molecular energy difference ∆.The reason is that we are interested in the high energy vibrational modes,and various studies have already shown that these modes have the biggesteffect in the resonant energy transfer process[21, 22, 156].One comment we want to make is that Holstein couplings and Peierlscouplings cannot be transformed into each other in our model, due to theexistence of the RC. Without the RC, gH and gP can be transformedinto each other by employing the canonical transformation H → eSHe−Swith S = pi/4(c†0c1 − c†1c0) . This transformation also changes the value∆ → t0 and t0 → −∆. If there is a big difference between ∆ and t0, thistransformation would change the basic physical features of the bare sys-tem. Among these light harvesting molecules which have been studied most(e.g. FMO, PE545), ∆ is usually larger than t0 by at least one order ofmagnitude[29, 157]. The addition of the RC also makes this transformationinvalid, since the RC couples to the latticeThe formal analytical solution of the bare system is easy to establish.The Green function of the whole Hamiltonian can be acquired by diagonal-izing the bare Hamiltonian without the RC (for details see Appendix D.1).The numerical results we show in this thesis include 10 phonons in the wholesystem. The results already converge with fewer phonons. The transfer rateis defined as the probability of the exciton being transferred to the RC from934.2. Holstein Couplingthe central system, i.e.ρRC = 1− ρ22 − ρ11, (4.6). where ρ22 and ρ11 are the elements of the reduced density matrix whichrepresent the probability to find the exciton at site c2 and site c1 respectively.We also define the logarithmic fitting parameter γ(1− ρRC) ∝ e−γt. (4.7)The validity of this exponential relaxation approximation is discussed inAppendix D.3.4.2 Holstein CouplingIn the pure Holstein coupling case gP = 0, we start by confirming that theresonance condition Ω0,1 = ∆ is important. We can see from Fig. 4.2(a) thateven a little deviation from the resonant frequency has a significant impacton the transfer rate. We can see that there are multiple double peaks in thespectrum around ω = n∆ under the resonance condition (see Fig. 4.2(d)).The degenerate states associated with these peaks are the states in whicheither the exciton is at site c1 with n phonons in the cloud, or the excitonis at site c2 with n− 1 phonons in the cloud. These resonance states createextra channels for the exciton to move down to the lower energy sites. If wetreat the t0 as a perturbation, the energy splitting of the lowest resonancestates with energy E = Ω = ∆ without the RC ist˜H0 =√2〈0|c2Hc†1a†0,1|0〉 =√2gHt0∆e−g2H/Ω2. (4.8)In the weak interaction limit gH  Ω, this resonant transfer is the dominantmechanism for energy transfer, and t˜H0 increases as gH increases. This isthe well known phonon-assisted transfer mechanism: the exciton leaves aphonon in the cloud to assist hopping to a low energy site[158]. We see thatt˜H0 starts to fall off when gH is larger thanΩ√2. Fig.4.4(a) is the illustrationfor the transfer rate γ as a function of coupling strength. We can see fromthat this property qualitatively holds for the pure Holstein case(the blueline in Fig.4.4(a)): the coupling has an optimal value around gH = 0.6for the maximum transfer rate. This turning point has gH smaller thanΩ√2. The reason is that distorted phonon clouds in the large coupling regionalso impede the transfer from site c1 to the RC. The overlap between thedistorted phonon cloud and free phonon cloud is ∝ e−g2H/2Ω2 . In the strong944.3. Peierls Coupling0 50 100 150t0. Rate Ω/∆=0.90 Ω/∆=0.95 Ω/∆=1.00 Ω/∆=1.05 Ω/∆=1.100 50 100 150t0. RategH= 0.0gH= 0.1gH= 0.3gH= 0.50 50 100 150t0.00.51.0Transfer RategH=0.7gH=1.0gH=1.50 50 100 150 200t0. ρρ11ρ000 1 2 3 4 ω0123456A(ω)gH=0.0gH=0.3gH=0.7gH=1.0(a) (b)(c)(d)Figure 4.2: Plots for the pure Holstein coupling case. For all plots, wechoose the energy of site c2 to be ε2 = 2 and the energy of site c1 to beε = 0 (∆ = 2). The intra-molecules hopping amplitude is t0 = 0.1. The RChas a bandwidth D = 1 and couples to the molecule by t1 = 0.2. (a): Plotof the transfer rate when the phonon frequency is around the resonant pointΩ ≈ ∆ = 2. From (b) to (d) the phonon frequency is set to be on resonanceΩ = ∆ = 2. (b): The transfer rate in the intermediate gH region. (c): Thereduced density matrix elements ρ11(the population on site c2) and ρ00(thepopulation on site c1) when gH = 1.5; the sub-panel is the transfer rate inthe large gH region. (d): Local spectra at site c1.interaction region, the exciton starts to oscillate within the molecule ratherthan propagating to the RC. The frequency of the oscillation is very close tothe value of t˜H0 . We cannot observe such oscillation in the small gH region(even though t˜H0 could have the same value as in the strong interactionregion), since the exciton will be promptly transferred to the reaction centeronce it reaches the c1 site.4.3 Peierls CouplingIn this section we study the Peierls type only coupling, i.e. gH = 0. In asimilar way to the previous case, the resonance condition Ω ≈ ∆ is again954.3. Peierls Coupling0 50 100 150t0. RateΩ/∆=0.90Ω/∆=0.95Ω/∆=1.00Ω/∆=1.05Ω/∆=1.100 50 100 150t0. RategP=0.00gP=0.03gP=0.06gP=0.100 25 50 75t0.00.51.0Transfer RategP=0.1gP=0.3gP=0.5gP=0.70 25 50 75t0. ρρ11ρ00-1 0 1 2 3 4 ω0246A(ω)gP=0.00gP=0.06gP=0.10gP=0.70(a) (b)(c)(d)Figure 4.3: Plots for the pure Peierls coupling case. The values of theparameters in the Hamiltonian are the same as Fig.4.2. Phonons couple tothe hopping amplitude rather than the on-site energies. (a): Plot of transferrate when the phonon frequency around the resonant point Ω ≈ ∆ = 2.From (b) to (d) the phonon frequency is set on resonance: Ω = ∆ = 2. (b):the transfer rate in small gP region. (c): Plot of the reduced density matrixelements ρ11 and ρ00 when gP = 0.5; the sub-panel is the transfer rate inintermediate gP region. (d): local spectra at site c1.important to the transfer rate (see in Fig. 4.3(a)). Under the resonancecondition, the transfer rate is more sensitive to the size of the Peierls couplinggP . We find that even when the Peierls coupling is so small that the changeof the spectrum is almost invisible (see the gP = 0.06, gP = 0.10 spectrain Fig. 4.3(d)), the increase of the total transfer rate is already significant(Fig. 4.3(b)).To understand this case, we again treat t0 as a perturbation. The unper-turbed Hamiltonian without the RC is a Jaynes Cummings Hamiltonian ifwe apply the Rotating Wave Approximation [159]. The unperturbed eigen-values areEn = Ω(n+12)± gP√n+ 1. (4.9)When gP is small, the energy states with the same n are nearly degeneratewith each other. The splitting of the lowest degenerate states with n = 0 is964.3. Peierls Coupling0.0 0.5 1.0 1.5g0.γgHgP0.0 0.1 0.2t00.γgHgP0.0 0.5 1.0 1.5g0.940.960.981.00Correlation CoefficientgHgP0 50 100 150 200t-10.0-5.00.0Log(1-ρRC )gP=0.50gP=0.53gP=0.56gP=0.72gP=0.77gP=0.80(a) (b)(c)(d)Figure 4.4: The plots obtained from fitting (1 − ρRC) as an exponentialfunction of t for the pure Holstein case and the pure Peierls case. Theparameters of the bare model are the same as the values in previous figures.The range of t is taken from 0 to 200. (a) The transfer rate γ as a functionof gH (blue circle) and gP (red star); (b) The transfer rate γ as a functionof t˜H0 (blue circle) and t˜P0 (red star);(c) The linear correlation coefficientbetween log(1−ρRC) and t; (d) the log(1−ρRC) v.s. t plot in two transitionregions: the lines within the dashed circle represent the region which theexponential model starts to fail; the line within the solid circle representwhere the exponential behavior is restored.simplyt˜P0 = gP . (4.10)If we compare this with (4.8), we notice that the factor t0/∆ is missing in(4.10). This factor suppresses the energy splitting in the pure Holstein case.Under the condition t0/∆  1 , Peierls couplings are not negligible evenwhen they are small compared with Holstein couplings. They have similareffects to the transfer rate as the Holstein couplings with much larger values.We can check whether the transfer rate agrees with each other when gP andgH are carefully chosen so that gP =√2gHt0∆e−g2H/Ω2 . Actually we can seethat both coupling agrees pretty well in Fig. 4.4(b) for the weak interactionregion.Fig. 4.4 is an illustration of the transfer rate γ. In the weak interaction974.3. Peierls Couplingregion, the transfer rate increases as gP increases, similar to the Holsteincase (Fig. 4.4(a)). However, without the factor t0/∆ in (4.10), the transferrate is more sensitive to the Peierls coupling. It increases much faster forthe pure Peierls coupling case under the condition t0/∆  1, which is thecommon situation in photosynthetic molecules. This indicates that Peierlscouplings cannot be ignored even when they are much smaller than Holsteincouplings. In this region, the perturbation theory works well for both casesand they agree with each other when we plot γ as a function of t˜0 in bothcases (Fig. 4.4(b)). Even after the Holstein transfer rate starts to drop, thehigh transfer rate is still maintained in the Peierls case.4.3.1 Jaynes-Cummings Model and BeyondWe can see from Fig. 4.4 that as we continue increasing the interactions(gP and gH), the Peierls coupling shows a more interesting behavior. Thespectrum is no longer qualitatively similar to the small interaction case,as shown in Fig. 4.3(d). The transfer speed γ drops through differentplateaus before it eventually reaches its final plateau when gP ≥ 0.82. Inthis subsection, we study the details of the transfer rate in the pure Peierlscoupling case.We first look at a simplified model, i.e. the Jaynes-Cummings Model(JCM) mentioned in the previous section. In this model, we omit the hop-ping term t0 and the non-energy conserving term a†c†2c1 +ac†1c2. The Hamil-tonian becomesH = H0 +H1 +H2 (4.11)H0 = ε2c†2c2 + ε1c†1c1+t1(c†1d0 + d†0c1) +HB (4.12)H1 = gP (a†c†1c2 + ac†2c1) (4.13)H2 = Ωa†a (4.14)with HB remaining the same (a Bethe Lattice with band width D). Underthis approximation, the Hamiltonian can be solved analytically without thereaction center. In this section, we only consider the resonant case, i.e.Ω = ε2 and ε1 = 0.The energy levels of the JCM without the RC areE±n = Ω(n+ 1)± gP√n+ 1. (4.15)984.3. Peierls Coupling0 0.5 1gP00.1Transfer Ratet0=0.01t0=0.1t0=0.2JCMFigure 4.5: The plot of the transfer rate γ as a function of Peierls couplinggP for the JCM and the full Hamiltonian.The eigenvectors associated with E±n are|n,+〉 =√22(|1, n+ 1〉+ |2, n〉)|n,−〉 =√22(|1, n+ 1〉 − |2, n〉).(4.16)The state |1, n+1〉 means the exciton is at site 1 and there are n+1 phononsin the system. Using the equations in the Appendix D.1, the resulting Greenfunction G0(i, j; z) of the total system can be written as[G0(1, 1; z)]pq= δp,q12(1z − pΩ− gP√p +1z − pΩ + gP√p). (4.17)Here p and q are the indices of phonon states and we are only interested inthe case when i = j = 1. The poles of G0(1, 1; z) are determined by theequationt21(z − pΩ) +√(z − pΩ)2 −D2×(1z − pΩ− gP√p +1z − pΩ + gP√p)= 1.(4.18)This has two real solutions when gP ≥√(1− t21)/p. For the lowest excita-tion space |1,±〉, we have gP ≥√1− t2. If we start with the exciton sitting994.3. Peierls Couplingat site 2 and no phonon in the system intially, the two states associated withE±1 are actually the only relevant states in the system (since in the JCM wedo not have terms which connect different En subspaces). In Fig. 4.5, wecan see that the JCM has a clear change of dynamics near gP = 1. Beyondthis point, there is a trapped region when two bound states are formed andthe transfer rate towards the RC becomes small. On the other hand, in theweak interaction region gP < 0.1, the JCM is also quite accurate comparedwith the full interaction cases. This shows that our perturbative descriptionin the previous section is a good approximation in the weak gP region.0 0.5 1gP00.050.1Transfer Ratet0=0.1JCMRabiFigure 4.6: The plots of the transfer rate γ as a function of Peierls couplinggP . Different colors stand for the JCM(green) , the Rabi model (red) andthe full Hamiltonian (blue).However, the pure JCM cannot recover the behavior between 0.1 < gP <1. In particular, the decrease in the transfer rate near gP ≈ 0.4 is notcaptured. To recover this behavior, we add back the non-energy conservingterm a†c†2c1 + ac†1c2 so that the phonon interaction term becomesH1 = gP (a† + a)(c†1c2 + c†2c1) (4.19)Now the system effectively becomes a Rabi Model without the RC. Wecan obtain the eigenstates and eigenvalues of the Rabi Model analytically,but this involves solving transcendental equations and there is no closedform for the results[160]. Therefore we use only the numerical results here.In Fig. 4.6, we see the comparison between the results from JCM, Rabiand full Hamiltonians. We can see that Rabi Model agrees with the fullHamiltonian in the intermediate region 0.3 < gP < 0.8. In this region, we1004.3. Peierls Coupling0 0.5 1 1.5 2gP0246810E0 0.5 1 1.5 2gP0246810E0 0.5 1 1.5 2gP0246810EFigure 4.7: The first 11 energy levels as a function of Peierls coupling gPfor the JCM (top), the Rabi Model (middle), and the full Hamiltonian witht0 = 0.1(bottom).notice that in the Rabi Mode, the transfer rate actually has several plateausas it drops from 0.08 to 0.01. The most significant of these are the dropnear gP = 0.42 (first drop), gP = 0.54 (second drop), and gP = 0.82 (finaldrop). The final drop occurs earlier than in the JCM case (gP = 1). Thiscan be understood by comparing the eigenvalues of the Rabi Model and theJCM model (see Fig. 4.7). As we start with the state |2, 0〉 (the particleat site 2, zero phonon) , we expect that the eigenstates in this subspace arestill dominant for the transfer even in the Rabi model. These states startas the green and red lines in Fig. 4.7(a). In the JCM, only these two statesmatter, and the final drop (which represents the formation of bound states)happens around the point where the energy level drops within the bandrange of the RC (−1 < E < 1), which is around gP = 1 (the intersectionbetween the black dashed line E = 1 and the red line in Fig. 4.7(b)). In theRabi case, however, the intersection point is at gP = 0.833. This value isquite close to the final drop of the Rabi case at around gP = 0.82. The extraterms in the Rabi Model introduce the interaction between the |1,±〉 stateswith other |2n+ 1,±〉 states, so other states can also contribute to the total1014.4. The Full Hamiltoniantransfer rate. We can do a similar thing for the higher state subspace. Thenext subspaces are the two |3,±〉 states which starts at E = 6. We find theintersection point between the E = 5 line and the brown line in Fig. 4.7(b).This point turns out to be at gP = 0.53, which is close to the value of gP atthe second drop. We follow the same procedure for the |5,±〉 states and theresult is gP = 0.43 which is close to the value of the first drop. Based onthis observation, we suggest that the multiple drops between 0.3 < gP < 0.8can be understood as the shutdown of different transfer channels associatedwith the |2n + 1,±〉 subspaces. For higher states, the plateau is too smallto be tracked in the calculation and we see a continuous decrease.The last step is to restore the t0(c†2c1 +c†1c2) term in the Hamiltonian. Inthe weak interaction region, the parameter t0 in the original Hamiltonian isnot that important for the qualitative behavior. We can see in Fig. 4.5 thatthe transfer rate γ is very robust against changes in t0. The t0(c†2c1 + c†1c2)term does introduce an interaction between the n = 1 states and arbitrary|n,±〉, so there are more dips in the pattern. The overall behavior does notchange qualitatively from t0 = 0.01 to t0 = 0.2, as shown in Fig. The Full HamiltonianIn this section we study the full Hamiltonian (4.1) with both the Peierlscoupling and the Holstein coupling. Due to the large parameter space weare not going to do a thorough exploration. We mainly focus on findingwhether the previous results hold in this case. We find in Fig. 4.8(a) thatthe local spectrum is mainly determined by the Holstein coupling, but thetransfer rate to the RC is very sensitive to changes in gH even when gP  gH( Fig. 4.8(c,d)).In a similar way to the Holstein coupling only case, there is an optimalvalue of the coupling gH when the transfer rate achieves the fastest transferrate. (in Fig. (4.8c), the optimal value is around gH = 0.6 if gP is fixed).If gH is fixed, the system also behaves similarly to the Peierls-only case, asthe transfer rate increases as gP increases (Fig. 4.8(d)). Comparing thesetwo figures we confirm that the transfer rate to the RC is still very sensitiveto gP even when a much larger gH exists, although Fig. 4.8(a) shows thatthe local spectrum is mainly determined by the Holstein coupling.In the mixed case, if we follow the perturbation argument, the energysplitting of the lowest degenerate states ist˜0 =√A2 +B2, (4.20)1024.4. The Full Hamiltonian0 1 2 3 4ω012345A( ω)0 25 50 75t0. RategH=0.1gH=0.4gH=0.7gH=1.0gH=1.220 30 40 50 60 70t0.80.91.0Transfer RategH=0.2gH=0.4gH=0.6gH=0.8gH=1.00 10 20 30 40 50t0. RategP=0.01gP=0.04gP=0.07gP=0.10(a) (b)(d)(c)gH=0.8 gP=0.06gH=0.7 gP=0.10gH=0.7 gP=0.04gH=0.2 gP=0.06Figure 4.8: Plots when both types of phonon couplings exist. The valuesof the parameters in the Hamiltonian are the same as for Fig.4.2. (a): Thelocal spectra at site c1 with various couplings. (b): The transfer rate whent˜0 is fixed at 0.05, with gP determined by (4.20). (c): The transfer ratewhen gP is fixed at 0.06. (d) the transfer rate when gH is fixed at 0.7.with A =√2gHt0∆e−g2H/Ω2 and B = gP e−g2H/Ω2. In Fig (4.8b), we keep t˜0fixed and vary gH . We see that in the small gH region (gH = 0.1, 0.4, 0.7),the curves agree with each other well. In the large gH case(gH = 1.0, 1.2),the curves start to deviate since the phonon cloud coupled to site c1 is toodistorted to allow the exciton to move to the RC.In Fig.4.9 (a), we fix gH , and the behavior of γ is qualitatively thesame as for the pure Peierls case shown in Fig.4.3 (a). The plateau effectis barely noticeable due to the accuracy of the numeric calculation. Theinitial increase of the transfer rate to the RC is still rapid and sensitiveto gP , even when a relatively large gH exists. The result with gP fixed isshown in Fig.4.9 (b). The choice of gP is already large enough (gP = 0.05 isalready larger than the turning point of t˜H0 in the pure Holstein case, see Fig.4.4(b)) for the transfer rate γ to decrease monotonically as gH increases. Inboth cases, the existence of the Peirels coupling changes the behavior of thesystem qualitatively, even when its value is much smaller than the Holsteincoupling.1034.5. Role of Off-Diagonal Couplings in Resonant Energy Transfer0.0 0.5 1.0 1.5g0.γgH=0.1gH=0.60.0 0.5 1.0 1.5 2.0g0.γgP=0.10gP=0.05(a)(b)Figure 4.9: The plots of transfer rate γ for the full Hamiltonian (3.1) withboth the Peierls coupling and the Holstein coupling. The values of theparameters in the Hamiltonian are the same as Fig.4.2. (a) gH = 0.1, 0.6, γas a function of gP . (b) gP = 0.1, 0.05, γ as a function of gH .4.5 Role of Off-Diagonal Couplings in ResonantEnergy TransferIn the previous sections, we investigated a dimer model of light harvestingmolecules coupled to an RC and resonating environmental phonon modes.We included two different types of phonon couplings and we studied theireffects on both the transfer rate to the RC and the local spectra. In thecase of the pure Holstein coupling, we demonstrated the resonant phonon-assisted transfer mechanism: the coupling to phonons increases the transferrate when it is small but eventually makes the exciton too heavy to propa-gate. We also showed that it can be described by degenerate perturbationcalculations, in both the weak and strong coupling regions. In the case of thepure Peierls coupling, we found that in the weak coupling region gP  Ω, if1044.6. Two-Phonon Interactions for Optical Phononst0/∆ is small, a large Holstein coupling gH is equivalent to a small Peierlscoupling gP =√2gHt0∆e−g2H/Ω2 . A small gP will have almost no effect on thelocal spectra, but still affects the transfer rate significantly. In the intermedi-ate region gS ∼ Ω, the Peierls couplings start to cause qualitatively differentbehavior compared with the Holstein couplings. The transfer rate is stillhigh in this region. The spectrum contains complicated peaks located awayfrom the resonant points, and the perturbation calculation starts to fail. Weconfirmed that our results in the two individual cases are still significant.The interesting properties of the Peierls couplings are not overshadowed bythe existence of large Holstein couplings.The excitation energy of the photosynthetic molecules (Bchls in the FMOcase) changes due to the effect of the surrounding charges and solvent di-electricity, which is the major source of the Holstein coupling. Meanwhile,the hopping amplitude between Bchls is also modulated by the vibrationalmodes, since these modes change the wave function as well as the over-lap integral between sites. This leads to the Peierls coupling term in theHamiltonian. Most works only consider the former term and ignore the ef-fect of the latter, since it is usually small and negligible in the spectrum.Our studies suggested that we cannot ignore the phonon-modulated hop-ping amplitude in the resonant energy transfer mechanism within biologicalmolecules. The large t0/∆ ratio in various light harvesting molecules makesPeierls couplings important even if they are negligible in the spectra.In the next chapter, we are going to use a more realistic model of the lightharvesting molecules, which contains more sites and more phonon modescompared with (3.1), to study the role of vibrational modes in these biologi-cal systems. It will be interesting to know whether or not the Peierls couplingplays a functionally important role in real-life photosynthetic processes. Ourfindings also show that Peierls couplings have qualitatively different dynam-ics beyond the small coupling region which requires new theoretical toolsto describe. This might not be the case in photosynthetic molecules but isstill interesting on its own from theoretical stand point, which could lead tointeresting further research.4.6 Two-Phonon Interactions for OpticalPhononsIn this section we study the second-order phonon couplings for optical pho-nons. We start the section by studying two-phonon interaction terms addedto the energy transfer model we have already studied in 4.2. In this simple1054.6. Two-Phonon Interactions for Optical Phononsmodel, we neglect the linear couplings and find out that the cross interactionterms can be mapped into a spin bath if the frequency of the two interactingphonon modes are close to each other. The effect of this interaction is similarto the pure phase decoherence region in Section 3.2. In the second partof this section, we employ the path integral method to show that such amapping is general for optical phonon modes when the linear terms are notzero.In this section, we assume that the coupling is always to the spatialcoordinate of the bath oscillators (i.e. xˆ = a† + a) for simplicity.4.6.1 Two-Phonon Interaction in the Holstein modelFirst we consider two-phonon interactions added to the energy transfermodel which has already been studied in previous sections. The Hamil-tonian can now be written asH = H0 +H1 +H2 (4.21)H0 = ε2c†2c2 + ε1c†1c1 + t0(c†2c1 + c†1c2)+t1(c†1d0 + d†0c1) +HB (4.22)HB =∑〈ij〉V (d†idj + h.c.) (4.23)H1 =∑i=0,1(g(1)i (a†i + ai) + g(2)i (a†i + ai)2 +g(1)2iΩi)c†ici (4.24)+g(2)ij (a†i + ai)(a†j + aj)(c†icj + c†jci)H2 =∑i=0,1,2Ωia†iai. (4.25)We see that there are two types of interaction being added to the system:the self interaction terms,g(2)i (a†i + ai)2; (4.26)and the cross terms,g(2)ij (a†i + ai)(a†j + aj). (4.27)We still assume that Ωi = ε2 − ε1 = Ω, which is the resonance condition.We also assume that there are only two phonon modes in the system; onecouples to site-1 and another couples to site-2.1064.6. Two-Phonon Interactions for Optical PhononsSelf Interaction TermsIf there are no cross interaction terms, i.e. g(2)ij = 0 ∀i 6= j, then there areonly self-interacting terms g(2)i (a†i + ai)2. We can follow the same pertur-bative argument as we did in Section 4.2. Without loss of generality, weassume g(2)i = g(2),∀i and g(1)i = g(1),∀i.The truncated Hamiltonian in the lowest degenerate energy subspace isH = 0 A BA 0 0B 0 ∆ω (4.28)whereA = g(1)2Ω(Ω + Ω′)2e−( g(1)Ω′ )2 2ΩΩ+Ω′B = −g(1)√ΩΩ′2Ω(Ω + Ω′)2e−( g(1)Ω′ )2 2ΩΩ+Ω′ .Here ∆ω = Ω′ − Ω. Ω′ =√Ω + g(2) is the detuned frequency caused bythe second order coupling term g(2). When g(2) = 0, we have Ω = Ω′, and(4.28) reduces to the result of the first order phonon couplings in Section4.2. Compared to the previous result, we can see that the dependence onthe first order coupling g(1)H is similar to what we found in previous sectionsand there is nothing qualitatively changed by the second order couplings.The self-interacting terms only reduce the effective transfer amplitude t˜H0 bya certain amount in the weak interaction region. We find that the transferrate decreases as we increases the second order coupling g(2), although theamount it changes by is not very significant unless one increases the g(2)beyond 0.1Ω, which is quite large compared to the first order couplings.This is what we expected since the self interacting terms effectively changethe phonon frequency a little bit, and cause the system to deviate from theresonance condition, which results in the decrease of the energy transfer.Cross Interaction TermsOn the other hand, the cross interaction terms g(2)ij (a†i +ai)(a†j+aj) are moreinteresting than the self interaction terms. We consider the case in whichthe two coupled modes Ωk and Ωl have similar energy Ωk ≈ Ωl. Includingthe cross interaction terms, we find the effective Hamiltonian matrix in the1074.6. Two-Phonon Interactions for Optical Phononsresonance energy level subspace with N phonons to beH1 = ε+N∆ω√Ng(2)ij 0 · · · 0√Ng(2)ij (N − 1)∆ω√2(N − 1)g(2)ij · · · 00√2(N − 1)g(2)ij (N − 2)∆ω · · · 0.... . .. . ....· · · ∆ω √Ng(2)ij0 · · · √Ng(2)ij 0,(4.29)where ∆ω = Ωi + 2g(2)i − Ωj − 2g(2)j . This can be directly mapped into aspin N/2 systemH1 = εN + ∆ωSzN/2 + 2g(2)ij SxN/2, (4.30)where the Sx,y,zN/2 are spin operators of N/2 spins, and εN = ε+N2 ∆ω+g(2)i +g(2)j . The linear coupling couples the Hilbert space between different spinspaces which cannot be represented in the form of these spin operators.This mapping can be easily verified in some simple models. As an exam-ple, we consider two identical phonon modes that couple to the off-diagonalterms of (4.21) with only the second order coupling. This problem, withoutthe RC, is equivalent to the spin boson model:H = tσx + g(2)12 (b†1 + b1)(b†2 + b2)σz + Ω0(b†1b1 + b†2b2). (4.31)This can be further transformed intoH = tσx +12(g(2)12 (b†1 + b1)2 − g(2)12 (b†2 + b2)2)σz + Ω0(b†1b1 + b†2b2). (4.32)In the path integral picture, we marked the path of the central system by{tn}, which satisfies 0 = t0 < t1 < ...tN−1 < tN < tN+1 = T . Any pathof the central spin from t = 0 to t = T which starts at σz = −1 can bedescribed by a set of {tn} such that the path can be represented by{tn} := −12+n=N∑n=1η(t− tn)(−1)n+1. (4.33)The propagator of a phonon mode which has a time dependent frequencyΩ(t) can be written as〈x|Te−i∫ to H(t′)dt′ |y〉 = 1√2pii sin Θ(t)ei2 sin Θ(t)((x2+y2) cos Θ(t)−2xy), (4.34)1084.6. Two-Phonon Interactions for Optical Phononswhere Θ(t) =∫ t0 Ω(t)dt. If the initial state is a thermal state with kBT =β−1, the influence functional of a single phonon associated with these twopaths {tn}, {t′m} can be represented byF1({tn}, {t′m}) =1Z1√2pii sin(Θ(t)−Θ(t′)− iβΩ0)·∫ ∞−∞dx exp(− i(1− cos(Θ(t)−Θ(t′)− iβΩ0))sin(Θ(t)−Θ(t′)− iβΩ0) x2)=sinh 12βΩ0i sin 12(Θ(t)−Θ(t′)− iβΩ0).(4.35)If one considers both phonons, then the total influence functional is theproduct of themF ({tn}, {t′m}) = F1(({tn}, {t′m}))F2(({tn}, {t′m}))=sinh2 12βΩ0− sin 12(Θ(t)−Θ(t′)− iβΩ0) sin 12(Θ′(t)−Θ′(t′)− iβΩ0).(4.36)Notice that Θ(t) and Θ′(t) are generated by an identical central path withthe opposite sign of coupling ±B12. Therefore, we have the relation thatΘ(t) + Θ′(t) = T2 (Ω+ + Ω−),12(Θ(t) − Θ′(t)) = 12(Ω+ − Ω−)∑n(tn+1 −tn)(−1)n := Θ0(t). Based on this we can rewrite the denominator as12(cos(Θ0(t) − Θ0(t′)) − cos(−iβΩ0)). Then for this spin boson model wehaveF ({tn}, {t′m}) =4 sinh2 12βΩ0eβΩ0 + e−βΩ0 − 2 cos(Θ0(t)−Θ0(t′))=11 +sin2 12(Θ0(t)−Θ0(t′))sinh2 12βΩ0.(4.37)On the other hand, if we start with the spin Hamiltonian as in (4.30), we1094.6. Two-Phonon Interactions for Optical Phononshave the influence functionalF ({tn}, {t′m})=∞∑N=0e−βNΩ0ZN/2∑SzN=−N/2e2ig(2)12 SzN(∑n(tn+1−tn)(−1)n−∑m(t′m+1−t′m)(−1)m)=∞∑N=0(e−βNΩ0Zsin(N + 1)(Θ˜(t)− Θ˜′(t))sin(Θ˜(t)− Θ˜′(t)))=11 +sin2 12(Θ˜(t)−Θ˜′(t))sinh2( 12βΩ0),(4.38)where Z =∑∞N=0(N + 1)e−βNΩ0 = e−βΩ0(1−e−βΩ0 )2 , and Θ˜(t) = g(2)12∑n(tn+1 −tn)(−1)n. In the small coupling region, Ω± =√Ω0(Ω0 ± 2g(2)12 ) ≈ Ω0± g(2)12 ,and Θ˜(t) ≈ Θ(t). Therefore these two expressions (4.37) and (4.38) agreewith each other in this weak interaction regime.For the light harvesting molecules problem, we realize that the opticalphonon (the intra-molecule vibronic modes) has a very high excitation en-ergy compared to room temperature (from 500cm−1 to 1500cm−1), and alsohigh compared to the hopping amplitude of the excitons (around 101cm−1).Therefore, in the case when 2ωk,l  T , we can replace these two resonatingenergy levels as a two level system, ie. an environmental spin. In (4.29),let us take N = 1. Then if ∆ω  |g(2)12 |, this is exactly the pure phase de-coherence term described in the Section 3.2. The phase decoherence rate isgoverned by the parameter κ =∑i Re(g(2)12 )2, which is not necessarily smallafter summation of all correlated energy levels.4.6.2 A General Path Integral FormalismIn the previous subsection, we showed that the cross interaction terms ofthe two-phonon interactions can be mapped into a spin bath of the simplestspin boson model without first order phonon couplings. In this section wetry to show that this mapping is still valid when we include the first ordercouplings, making it applicable to the most general models.In this section, we use the path-integral formalism and the Lagrangian1104.6. Two-Phonon Interactions for Optical Phononsof the system takes the formL = L0 + LI + LB (4.39)L0 = L0(t, q(t), q˙(t)) (4.40)LB =∑α12mαx˙2α −12mαω2αx2α (4.41)LI =∑αg(1)α xαq(t) +∑α,βg(2)αβxαxβG[q(t)]. (4.42)where L0 is the Lagrangian of the central system and q(t) is the general-ized coordinate. The Lagrangian of the phonon bath is LB, and LB is theinteraction term. Without loss of generality we assume the phonons coupledirectly to q(t) to first order. The first order linear coupling is g(1)α , and g(2)αβis the second order coupling.For a product initial state, we can write the influence functional kernel(Feynman-Vernon kernel) asF [q, q′] =∫dxidx′idxfρB(xi,x′i)F (q; xi,xf )F∗(q′; x′i,xf ). (4.43)In which ρB(xi,x′i) is the initial density matrix of the phonon modes andF (q; xi,xf ) =∫ x(t)=xfx(0)=xiDx exp(i∫ T0dt(LB(x) + LI(q(t),x))). (4.44)Up to this point, these are all standard procedures. The next step is toevaluate the functional integral F (q; xi,xf ). If one has only the first ordercoupling, the calculations are straightforward. Unfortunately, we now havesecond order terms in LI . To reduce the second order terms to the first orderterms, we introduce a set of auxiliary variables y(t) and use the identityexpi∫ T0dt q(t)∑α,βg(2)αβxαxβ=1Zy∫Dy exp−14∑α,β∫ T0dt yα[A−1(t)]αβyβ + i∑α∫ T0dt yα(t)xα(t)(4.45)where the matrix A(t) contains all the information about the two-phononinteractions:[A(t)]αβ = −iG[q(t)]g(2)αβ . (4.46)1114.6. Two-Phonon Interactions for Optical PhononsHere Zy is the normalization factor, and logZy ∝ −∫dt log detA(t). Nowthe second order couplings are reduced to first order couplings: g(1)α xαq(t)→(g(1)α q(t) + y(t))xα. Therefore, we can use the result of the linear couplingsfor the influence functional kernel.Influence Functional KernelIf the initial phonon state is a thermal state, the influence functional kernelfor the real time path integral can be written asF [q, q′] =∫Dy∫Dy′ 1ZyZy′exp[−Φ(y, y′, q, q′)]exp−14 ∑α,β∫ T0dt([A−1(t)]αβyαyβ + [A′−1(t)]αβy′αy′β) .(4.47)Here we have[A′(t)]αβ = iG[q′(t)]g(2)αβ , (4.48)Φ(y,y′, q, q′) = Φ0(q, q′) + Φ1(y,y′, q, q′) + Φ2(y,y′), (4.49)Φ0(q, q′) =∑α∫ T0dt∫ t0dt′{ [q(t)− q′(t)]Lα1 (t− t′) [q(t′)− q′(t′)]−i [q(t)− q′(t)]Lα2 (t− t′) [q(t′) + q′(t′)] }+iµα∫ T0dt[q2(t)− q′2(t)] , (4.50)Φ2(y,y′) =∑α∫ T0dt∫ t0dt′{[yα(t)− y′α(t)] Lα1 (t− t′)(g(1)α )2[yα(t′)− y′α(t′)]−i [yα(t)− y′α(t)] Lα2 (t− t′)(g(1)α )2[yα(t′) + y′α(t′)]}+iµα(g(1)α )2∫ T0dt[y2α(t)− y′α2(t)], (4.51)1124.6. Two-Phonon Interactions for Optical PhononsΦ1(y,y′, q, q′)=∑α∫ T0dt∫ T0dt′{[yα(t)− y′α(t)] Lα1 (t− t′)g(1)α[q(t′)− q′(t′)]]−i [yα(t)− y′α(t)] η(t− t′)Lα2 (t− t′)g(1)α[q(t′) + q′(t′)]+i[yα(t) + y′α(t)]η(t′ − t)Lα2 (t− t′)g(1)α[q(t′)− q′(t′)]}+i2µαg(1)α∫ T0dt[yα(t)q(t)− y′α(t)q′(t)]. (4.52)In (4.50), Φ0(q, q′) is the influence funtional when there is no secondorder couplings andLα1 (t) =(g(1)α )2mαωαcoth(βωα2) cosωαt (4.53)Lα2 (t) =(g(1)α )2mαωαcoth(βωα2) sinωαt (4.54)µα1 =(g(1)α )2mαω2α(4.55)are the standard Feynman-Vernon kernels, which are quadratic functionsof the first order couplings. The two-phonon interactions introduced theadditional term Φ1(y,y′, q, q′). Its effects can be calculated by calculatingthe functional integral over y and y′ in (4.47).For a multidimentional Gaussian Integral, we have∫dxe−12∑ij xiAijxj+∑iBixi =√(2pi)ndetAe−12BTA−1B. (4.56)To use this formalism, we write all terms in Φ1(y,y′, q, q′) and Φ2(y,y′)in matrix form . The basis of vectors y is chosen asy =y1(t)y′1(t)...yα(t)y′α(t).... (4.57)1134.6. Two-Phonon Interactions for Optical PhononsAs a result, Φ2(y,y′) is quadratic over the variables y and can be writtenasΦ2(y,y′) =∫ T0dt∫ t0dt′yT(t)Q1(t− t′) 0 · · ·0 Q2(t− t′) · · ·....... . .y(t′)=∫ T0dt∫ t0dt′yT(t)∆A(t− t′)y(t′).(4.58)This defines the kernel matrix ∆A(t − t′). Here Qα(t − t′) is a two by twomatrix which is independent of the first order coupling g(1), i.e.,Qα(t) =1mαωα×(coth(βωα2 )e−iωαt + 2iδ(t)/ωα − coth(βωα2 )eiωαt− coth(βωα2 )e−iωαt coth(βωα2 )eiωαt − 2iδ(t)/ωα).(4.59)On the other hand, Φ1(y,y′, q, q′) is linear in the first order coupling g(1)α .This term can be written asΦ1(y,y′, q, q′) =∫ T0dtB1,α(t)yα(t) +B2,α(t)y′α(t)=∫ T0dtB(t) · y(t),(4.60)whereB1,α(t) =∫ T0dt′g(1)αmαωαcoth(βωα2)(e−iωα|t−t′|q(t′)− eiωα(t−t′)q′(t′))+ i2g(1)αmαω2αq(t)(4.61)B2,α(t) =∫ T0dt′g(1)αmαωαcoth(βωα2)(eiωα|t−t′|q′(t′)− e−iωα(t−t′)q(t′))− i 2g(1)αmαω2αq′(t).(4.62)We also rewrite the non-interacting integral kernel in (4.47) asexp(−12∫ T0dt∫ T0dt′yT(t)A−10 (t− t′)y(t′))(4.63)1144.6. Two-Phonon Interactions for Optical PhononswithA0(t) = 2δ(t− t′)×−iG[q(t)]g(2)11 0 −iG[q(t)]g(2)12 0 · · ·0 iG[q′(t)]g(2)11 0 iG[q′(t)]g(2)12 · · ·−iG[q(t)]g(2)21 0 −iG[q(t)]g(2)22 0 · · ·0 iG[q′(t)]g(2)21 0 iG[q′(t)]g(2)22 · · ·............. . . .(4.64)Therefore, the influence functional kernel can be formally written asF [q, q′] =e−Φ0(q,q′)∫Dy 1ZyZy′exp{− 12∫ T0dt∫ T0dt′yT(t)· (A−10 (t− t′)− 2∆A(t− t′))y(t′) + ∫ T0dtB(t) · y(t)}=e−Φ0(q,q′)√detA−10det(A−10 − 2∆A) exp [−12BT(A−10 − 2∆A)−1B],(4.65)where e−Φ0(q,q′) is the influence functional kernel of the linear coupling terms.The rest of the terms are the effects of the two-phonon couplings. The termsin exponents, i.e. BT(A−10 − 2∆A)−1B are quadratic functions of q and q′,which can be absorbed into the linear coupling kernel Φ0(q, q′) as shownin (4.50). Considering that A0 ∝ g(2)αβ and ∆A ∝ 1mαωα , if the two-phononinteractions are small compared to the phonon frequency (which is usuallytrue for optical phonons), this kernel is dominated by A−10 . Since B ∝ g(1)αmαωα,the exponents are proportional to (g(1)α )2mαωαg(2)αmαωαto lowest order in g(2). Weexpect that in this situation, the extra dynamics introduced by the two-phonon interactions can be safely treated as a perturbation to the linearcouplings without any intrinsic difference.On the other hand, the factor√detA−10 / det(A−10 − 2∆A)cannot beabsorbed into the first order dynamics. The factor ∆A is completely inde-pendent of g(1). To actually calculate this factor requires us to calculate thelogarithm of the product of matrices. We can use the Baker - Campbell -1154.6. Two-Phonon Interactions for Optical PhononsHausdorff formula to write it formally as√detA−10det(A−10 − 2∆A) = exp [−tr log (A−10 − 2∆A)− tr logA0]= exp tr{−Y + 12[X,Y ]− 112([X, [X,Y ]]− [Y, [Y,X]]) + ...} (4.66)in which X = logA0 and Y = log(1 − 2A0∆A). Unfortunately this pro-cedure is too complicated to get a closed form for the general case. How-ever, we notice that this term does not depend on the first order couplings.In the last section, we already calculated the pure two-phonon interac-tion kernel (4.37), which is the g(1)αβ = 0 case in (4.65). Since the kernel(4.37) is analogous to the coupling to the spin bath kernel (4.38), so is the√detA−10 /det(A−10 − 2∆A)factor. Therefore, the mapping to a spin bathin the previous section is still valid after we turn on the linear couplingterms.116Chapter 5Multi-Pathway LightHarvesting MoleculesIn the last chapter we have explored the functional importance of Peierlspolarons in energy transfer in two-site models. In this chapter we are goingto expand the same technique to study the more complex light harvestingcomplexes. Due to the lack of knowledge of the actual parameters in realworld biological molecules, we are not going to try to reproduce the details ofexperimental observations. Instead, we focus on exploring new mechanismsintroduced by the Peierls polarons and the spin baths.In the first section of this chapter, we write down the full Hamiltonianof vibrational modes coupled to energy transfer again. We summarize themethods we are using from the previous chapters to approach this Hamil-tonian. In the next section, we apply the model to the FMO complex.To reduce complexity, our calculations are mainly based on a reduced 3-site model of FMO. Site-energies and hopping amplitudes are taken from[8, 29, 161]. The phonon frequencies are no longer in resonance with theenergy gaps, in this simplified model we still find that the Peierls couplingsintroduce resonant peaks in the weak interaction region. The sensitivity tothe coupling strength is qualitatively different from the Holstein couplingcase. In the last subsection, we will study the path-selecting mechanismin several multiple-pathway models. We find that the effect of off-diagonalcouplings strongly depends on the energy difference between the two sites itcouples to. It can enhance the transfer rate as well as the coherence betweenthe two sites. Although the two branches of biological molecules have thesame chemical structure, the transfer rate through each of them can be dueto different environmental phonons.1175.1. Approach to the Complete Hamiltonian for Energy Transfer5.1 Approach to the Complete Hamiltonian forEnergy TransferWe again write down the complete Hamiltonian for an exciton coupling tovibrational modes of the environment in Chapter 1 as:H = H0 +H1 +H2 (5.1)H0 =∑iεic†ici +∑〈ij〉(tijc†icj + h.c.) + t0(c†1d0 + d†0c1) +HB (5.2)HB =∑〈ij〉V (d†idj + h.c.) (5.3)H2 =∑kΩka†kak. (5.4)H1 =∑i∑kg(1)H,k,i(a†k + ak) +∑k,k′g(2)H,k,k′,i(a†k + ak)(a†k′ + ak′) c†ici+∑〈ij〉∑kg(1)P,k,〈ij〉(a†k + ak) +∑k,k′g(2)P,k,k′,〈ij〉(a†k + ak)(a†k′ + ak′) c†icj+ h.c.(5.5)Based on our studies in previous sections, we can at first map severalterms into spin bath models to simplify the expression, and help to under-stand the dynamics of the system.There are two major sources of spin bath interactions in the system.One is from the direct truncation of the high energy phonon modes. For thephonons with energy Ωk  kBT ( at room temperature 300K ≈ 208cm−1),considering the weak interaction assumption g2/Ω  tij , |i − j | (if thephonon energy is around 200cm−1, the criterion is g  100cm−1 which isquite reasonable in the FMO complex), we can safely assume that only theirground states and first excitation states are involved in the energy transfer.Therefore, we can truncate such phonon modes into two-level systems, a.k.a.,spin 12 modes. Replacing the {ak, a†k} with {σ−k , σ†k} and (a†kak − 12) with σzk1185.1. Approach to the Complete Hamiltonian for Energy Transfer, we can write down the mapped spin bath coupling terms asHhf =∑k,〈ij〉(g(1)P,k,〈ij〉(ak + a†k)c†icj + h.c.) +∑kΩka†kak→∑k,〈ij〉(g(1)k,〈ij〉 · σxkc†icj + h.c.) +∑kΩkσzk.(5.6)Due to the big energy difference Ωk, this interaction falls into the high fieldregime which we have already studied in 3.3. The effects from the associatedphonon modes dissipate the system and we can introduce the parameter κijto describe the dissipation rate:κij =∑k(g(1)ktij)2. (5.7)The resulting reduced density matrix can then be approximated asρ(tij , t) =∫ ∞0zdze−z2/2ρ0(z, t), (5.8)where ρ0(z, t) is the density matrix under the modified Halmiltonian tij →tijeκijz.Another source of spin bath couplings is from the two-phonon cross-interactions g(2)H,P,k,k′ when Ωk ≈ Ωk′ . In Section 4.6, we have already shownthat the only non-trivial effects from the two-phonon interactions come fromthe cross-interaction terms g(2)k,k′,〈ij〉(ak+a†k)(ak′+a†k′) when k 6= k′. They canbe mapped into the spin bath interaction (Ωk−Ωk′)σz+g(2)k,k′,〈ij〉σx, in whichthe σks are the operators of spin12 systems. As a result, the cross-interactionterms can be written asHpf =∑k,〈ij〉(g(2)P,kk′,〈ij〉(ak + a†k)(ak + a†k)c†icj + h.c.) +∑kΩka†kak→∑〈kk′〉,〈ij〉(g(2)k,k′,〈ij〉 · σx〈kk′〉c†icj + h.c.) +∑〈kk′〉(Ωk − Ωk′)σz〈kk′〉.(5.9)This mapping is valid when i) the phonon frequency is much larger than thecoupling strength , i.e., |Ωk|, |Ωk′ |  g(2)k,k′,〈ij〉; and ii) the phonon frequencydifference is much smaller than the other relevant energy scales phononscouple to, i.e., |Ωk − Ωk′ |  tij , |i − j |, g(2)k,k′,〈ij〉. Noticing that the terms|Ωk −Ωk′ | are equivalent to the local fields hk acting on bath spins, we can1195.1. Approach to the Complete Hamiltonian for Energy Transferimmediately recognize that the second criterion is exactly the condition ofpure phase decoherence regime of spin baths we discussed in Section 3.2.From that section, we show that if Ωk ≈ Ωk′ , the couplings to bath spins inthis regime would only cause phase decoherence , with no energy dissipation.The actual decoherence rate is determined by the topology of the centrallattice. In general, for a particular path of the exciton, the factor thatgoverns the decoherence rate is given byλ =12∑k,〈ij〉g(2)kk′,ijtij2 . (5.10)Here the summation indices 〈ij〉 are taken along a particular path. Forthis particular path, the decoherence rate is proportional to e−λt (see 3.2).Summing over all possible paths, we can get the dynamics of the off-diagonalelements of the reduced density matrix.If we know enough details of the phonon couplings to calculate κ and λ,we can use the aforementioned mappings to spin baths to isolate the effectsintroduced by the high energy modes as well as the two-phonon interac-tions. This procedure leaves us the coupling to the modes with energy Ωkclose to the energy scales of the central system, which are more likely to beon resonance with certain energy gaps |i − j | between different Bchls inlight harvesting molecules. In Chapter 4, we have already studied the casewhen the phonon energy is on resonance exactly. Similar techniques willbe employed in this chapter to study the case when the phonon energy isnot on resonance. We shall see that most of our results in Chapter 4 arestill applicable with new parameters. Combining these results with the spinbath mapping techniques, we can acquire a thorough understanding of theHamiltonian (5.1) to (5.5).One may notice that we omit the diagonal couplings to the vibrationalphonon modes in the mapping to spin baths. Our argument is based on theproperties of Holstein couplings. In the parameter region of our problem,Holstein couplings always show similar behavior: increasing the transfer ratein the weak interaction region but decreasing the transfer rate when thecoupling strength is beyond a certain point(the optimal coupling strength).The overall dependence on the coupling strength varies smoothly. Thereis no rapid raise nor dropping for the energy transfer rate. We also findthat such dynamics are quite robust regardless of the details of the models.This robustness can also be verified in later sections of this chapter whenthe resonant condition is not satisfied. Therefore, including multiple sourcesof Holstein couplings would only result in a renormalization of the optimal1205.2. Application to the FMO ComplexFigure 5.1: This is the energy path-way of a 8-site model from with on-siteenergies, as well as the scheme to reduce to a 3-site model. Site-3 is thesite that connecting to the RC. Site-8 has the highest on-site energy andreceives excitons from the baseplate directly. Reprinted with permissionfrom [8]. Copyright (2011) American Chemical Society.coupling strength as well as the enery spectrum. Practically, if we want tostudy the effect of multiple sources of Holstein couplings qualitatively, wecould simply study them as a whole, as a single bath with a renormalizedspectrum. Actually in experiments, we cannot distinguish the effects fromdifferent sources either. Obviously, this argument is not universal and onlyvalid if the bare system is not in the localized regime. If the system is in alocalized phase, the increasing of Holstein couplings could eventually destroythe localization and cause a phase transition. But in FMO complexes, wecan clearly see a hierarchy of on-site energies so localization is not the casehere. As a result , we do not have to map certain Holstein couplings intothe spin bath model separately in the problems we are insterested in thisthesis.5.2 Application to the FMO Complex.Usually the energy transfer in the FMO complexes is considered as atwo-pathway transfer. It was generally believed that there were two majorpathways for FMO complexes : 1-2-3 and 5-6-7-4-3 (see for example [8, 30,162] ). The site numbers and on-site energies are illustrated in 5.1. Site3 is the site which is connected to the RC. Being the two pigments whichare closest to the baseplate, site 1 and site 5 are believed to accept excitons1215.2. Application to the FMO Complexfrom the baseplate and start a two-pathway transfer towards site 3. Lateron, however, the discovery of the 8th pigment changed this picture[3, 8, 161,163]. The crystal structure indicates that the 8th pigment resides roughlymidway between the baseplate and the Bchl at site 1. Based on the crystalstructure and photoemission spectrum of FMO complexes, it was shownthat the 8th pigment provides the most efficient pathway due to its positionand orientation with respect to the surrounding chlorosome[8, 161, 164].One of the most widely-used Hamiltonians for the 8-site FMO complexwas worked out by Renger and his co-workers ([161]); in a site representationon e hasH0 =310 −98 6 −6 7 −12 −10 38−98 230 30 7 2 12 5 86 30 0 −60 −2 −10 5 2−6 7 −60 180 −65 −17 −64 −27 2 −2 −65 405 89 −6 −2−12 11 −10 −18 89 320 32 −10−10 5 5 −64 −6 32 270 −1138 8 2 −2 5 −10 −11 505cm−1. (5.11)The parameters in (5.11) were calculated based on the traditional masterequation method with Holstein couplings to acoustic phonons. Obviouslythese numbers can be problematic when we include Peierls couplings. How-ever, since we expect that the Peierls coupling is smaller than the Holsteincoupling and not significant in spectra, this is still among the most accurateHamiltonians of the FMO complex we can have. If the exciton is initiallycreated at site 8, numerical calculations show that the left path-way 8-1-2-3becomes the the dominant pathway in this model [161, 164]. The large en-ergy gaps between the eighth pigment and the rest of the pigments, as well asthe weak inter-path interactions, suppress the oscillations between the twoprimary pathways. The energy transfer process changes from an oscillationbetween two pathways into a ladder diffusion from site 8 to site 3. Consid-ering the fact that site 1 and site 2 have close on-site energy and stronglycoupled to each other, one can also treat them as a single site. In this way,we acquire a reduced 3-site effective model for the light harvesting molecules.Renger and his co-workers show that there is no qualitative difference forsite-populations between the 8-site model and reduced 3-site model[161]. Itis not clear that only taking into account of a subset of sites from the wholesystem would affect other functional properties of the whole system besidesthe site-populations. In this thesis, we mainly focus on the site-populations,as the energy (exciton) transfer is the major function of FMO complexes.1225.2. Application to the FMO ComplexTherefore, we use the simplified 3-site model as our starting point.The three site effective model based on the work of Schmidt et al[161] is(again in the site representation):H0 =505(3.70) 38(0.282) 038 270(2.00) 30(0.222)0 30 0 cm−1 (5.12)The numbers in the brackets in (5.12) are the dimensionless renormalizedsite-energies for numerical purposes. The energy difference between thelowest two sites are renormalized to 2. In the following sections, the energyvaue of any dimensionless number a is a × 135cm−1(e.g. , the numbers inthe bracket in (5.12)).The next step is to determine the vibrational phonon frequencies andtheir couplings. The possible frequencies for the vibrational phonon modesrange from 20cm−1 to > 1600cm−1. Based on the previous studies, wecan argue that only the phonons with frequencies close to the energy dif-ferences between sites are important, which are at the order of 102cm−1.In [29], it was estimated that the typical vibrational phonon frequencyΩ = 180cm−1 = 1.33 in our renormalized unit. The associated Huang-Rhys factor S ≡ g22Ω = 0.22. The corresponding coupling strength g = 0.77.Noticing that this is the Holstein coupling of the most dominant vibrationalphonons; we cannot use the value for the Peierls couplings. In this section,we assume that such phonons also couple to the central system throughPeierls couplings. Not knowing the actual size of Peierls couplings, we varytheir coupling strength from 0 to 2 to see their effects on the overall dynam-ics.The last step is to determine the parameters of the RC. Unfortunately, wedo not have detailed knowledge of the interaction between FMO complexesand RCs in vivo. In most experiments on the FMO complexes, these com-plexes are separated from their natural biological environment and studiedin laboratory solvents. All we know about the RC is based on spectroscopy.From the photoemission spectrum of the RC in purple bacteria, we can seethat the energy difference of absorption peaks of the closes pigments be-tween the FMO and the RC is about 580cm−1[6]. This is a huge energygap which is larger than any on-site energy differences within the FMO.From its absorption spectrum, the bandwidth of RC is apparently 150cm−1,which apparently does not overlap with the absorption spectrum of FMO[4].However, it is not guaranteed that there is no intermediate Bchl pigmentlost during distilling procedures, as with the 8th pigment of the FMO com-plexes. It is difficult to determine the actual parameter of the RC and the1235.2. Application to the FMO Complexdynamic of the system actually heavily depends on the position of the RCabsorption bands. Therefore, we also have to vary the parameters of RC tosee the dynamics of the central system in different regimes. In most cases,we are going to face situations different from our toy models in the lastchapter: the phonon frequency Ω is no longer on resonance with the siteenergy differences and the site-energies do not necessarily overlap with theabsorption band of RC.5.2.1 Back to Two Level ModelBefore we actually start calculating the reduced 3-site model, we go backto the two-site model, and tune the phonon frequency Ω off resonance andchange the position of the absorption band of RC. From this simple toymodel we can obtain a qualitative understanding of the behavior of thesystem in this general case. We follow the same notations for variables inthis section as in Chapter 4.The model Hamiltonian can be written asH0 =∑〈ij〉(∆ij +∑kbkij(c†ij + cij))(|ai〉〈aj |+ h.c.)+∑ii|ai〉〈ai|+∑〈ij〉Ωc†ijcij .(5.13)We use the same site subscript 〈ij〉 for phonon modes since we assume thereis only one vibrational phonon coupling to each hopping amplitude.At first, we set d = 0, which is the case we studied in 4: the bottomsite sits at the center of the RC absorption spectrum. The dependence oftransfer rate on the phonon frequency is illustrated in Fig. 5.2(a). We seethat the overall transfer rate decreases as the phonon frequency is tunedaway from resonance. The high-transfer-rate plateau gradually turns intoa sharp transfer peak. The position of the peak can be understood byeigenvalue calculations: the maximum transfer happens when the lowesteigen-energy of the polaron drops into the band of the RC. The smaller thephonon frequencies are, the larger the coupling strength of the transfer peakis. For small phonon frequency Ω = 1.0, we can even see another peak in thelarge coupling regime which is associated with the second lowest eigenenergyof the polaron.In Fig. 5.2(b), we set the energy deviation d = 1 − 0 (the energydifference between site-1 of the lattice and site-0 in the RC model) to 1.Therefore, the bottom site is right outside the absorption spectrum of the1245.2. Application to the FMO ComplexRC. There is no energy overlap with the site and the absorption spectrumof the RC in this case. The dynamics is qualitatively different from theprevious case in Fig. 5.2(a). First we can see that Ω ≈ 2 − 1 = 2 is nolonger the resonance condition. The transfer rate drops rapidly when Ω = 2compared to the previous case. The energy of site 2 does not overlap withthe RC band so the lowest polaron state  ≈ d ± gP is also outside theRC bandwidth. The actual resonance happens when the phonon freqencyΩ comes close to the energy difference d = 1. Beyond Ω = 1, if the phononfrequency increases, the height of the peak actually decreases.From both cases, we can see that instead of the high-transfer plateau,transfer peaks become more common when the parameters are not fullyon resonance. The location of these peaks are determined by whether aparticular energy eigenstate of the polaron is on resonance with the RCband or not. Therefore, we can see a sudden increase or decrease of thetransfer rate around these critical couplings. The transfer rate is sensitiveto the change of coupling strength near these critical point couplings butremains almost constant in the rest of the parameter regime.0 0.5 1 1.5gP0. RateΩ=2.0Ω=1.8Ω=1.6Ω=1.4Ω=1.2Ω=1.0d=00 0.5 1 1.5gP0. RateΩ=2.0Ω=1.8Ω=1.6Ω=1.4Ω=1.2Ω=1.0d=1Figure 5.2: the plot of the transfer rate of the original two level systemwith different deviations d and phonon frequencies Ω, D = 1, t0 = 0.2 a)the transfer rate with Deviation d = 0, different lines stands for differentphono frequencies, b) the transfer rate with Deviation d = 1.0, differentlines stands for different phono frequencies.5.2.2 Reduced 3-Site Model for FMO complexesNow we turn to the reduced 3-site model for FMO complexes (5.12) withthe phonon frequency set to be Ω = 1.33(180cm−1). The parameters wevary during the studies are : the phonon coupling strength g, the transfer1255.2. Application to the FMO Complex0.0 0.5 1.0 1.5g0.020.04γRabigPgH0.0 0.5 1.0 1.5gP0.γt0=0.1t0=0.2t0=0.3t0=0.60.0 0.5 1.0 1.5gP0.γd=0d=1d=20 0.5 1 1.5gP0.γD=0.5D=1.0D=1.5(a)(b)(c) (d)Figure 5.3: a) The plot of transfer rate of Hamiltonian (5.12) with differenttypes of phonon couplings, the reaction center parameters are chosen as D =1, t0 = 0.2, d(deviation) = 1; b) the transfer rate of off-diagonal couplingwith different t0, D = 1, d = 1 c) the transfer rate of off-diagonal couplingwith different d, D = 1, t0 = 0.2; d) the transfer rate of off-diagonal couplingwith different bandwidth D,t0 = 0.2, d = 1.amplitude to the RC t0 , the energy deviation of the RC d, and the halfband width D of the RC.In Fig. 5.3(a), we examine the effects of three different types of phononcouplings: the Holstein coupling, the full Peierls coupling and the Rabi-type Peierls coupling. We set the band width D of the RC as 1, t0 as0.2 and the deviation d as 1. For the Holstein coupling case, we again seethe phonon-assisted transfer which we have discussed multiple times before.This robustness against the change of parameters is a typical quality ofHolstein couplings in this type of model. In contrast, there is a sharp peakin the transfer rate for both Peierls coupling cases. The peak is locatedaround gP = 0.5. The associated Huang-Rhys factorg2P2Ω2= 0.07, which is1265.2. Application to the FMO Complexnoticeably smaller than the reported value for the Holstein coupling model in[29]. The spectra and the time evolution of the site populations are plotted inFig. 5.4. We can see that in the small interaction gP = 0.2 case, the excitonis mainly oscillating between site-1 and site-2. The oscillation frequency isapproximately the site energy difference 2− 1. Near the transfer rate peak(gP = 0.5), although there are still oscillations between 1-2, the transfer tosite 3 is significantly increased. Actually it is similar to the transfer peak inthe two-site case we just discussed in the previous part in Fig. 5.2(b). Theresonance between the polaron state and the RC band significantly enhancesthe transfer between site-3 and the RC. For the large coupling gP = 0.8, theexciton populations on site-1 and site-2 do not change compared to thegP = 0.5 case. However, we can see a significant increase of the populationson site-3. Due to the big phonon couplings, the distorted phonon cloudprevents the further transfer into the RC. The back reaction from the RCbecomes significant.In the rest of the panels in Fig.5.3, we change the parameters of theRC to see their effects on the exciton dynamics. Fig.5.3 (b) shows thedependence on the coupling strength t0 between the system and the RC. The result is straightforward: increasing the coupling strength increasesthe transfer rate and the change of t0 does not change the position of thetransfer peak. Fig.5.3 (d) shows the dependence on the bandwidth D ofthe RC. We can see that as long as the bandwidth D does not exceed thephonon energy Ω, i.e., only the zero-phonon polaron state on site-3 beingin the range of the RC band, the change of bandwidth does not change thetransfer rate peak. When the band width exceeds Ω = 1.33 (the D = 1.5line), another transfer peak appears, which is the result of the one-phononpolaron state is in the range of RC band. Fig.5.3 (c) shows the dependenceon the energy difference d. Unlike the other cases, the position of the transferpeak changes for different d. This shows that the transfer peak is almostexclusively dependent on the energy difference between the RC and site-3.Summary We find that the reduced 3-site model (5.12) with Peierls cou-pling qualitatively shows the same behavior as the two level system we stud-ied in 5.2.1. The high transfer plateau is reduced to a transfer peak at acoupling strength which is still much smaller than the Holstein couplings.The sensitivity of the transfer rate near the transfer peak suggests that wecannot ignore Peierls couplings even if they are small. We show that only theenergy difference between the RC band and the lowest energy site changesthe position of such transfer peaks. Increasing the RC bandwidth would1275.2. Application to the FMO Complex-2 0 2 4 6ω02040A( ω)0 50t0.ρ(t)ρ22ρ11ρ000 50t0.ρ(t)ρ22ρ11ρ000 50t0.ρ(t)ρ22ρ11ρ00(a) (b)(d) gP=0.8gP=0.6gP=0.5gP=0.4gP=0.1gP=0.8gP=1.0gP=0.5gP=0.2(c)Figure 5.4: The plots of Hamiltonian (5.12) with different Peierls couplings,the reaction center parameters are chosen as D = 1, t0 = 0.2, d(deviation) =1 a) the local spectrum at site-1, b), c), d): the time evolution of eachpigments with different gP scause multiple transfer peaks. For the Holstein coupling case, however, theresult is similar to what we studied before: increasing the Holstein phononcoupling gradually increases the transfer rate until it reaches the optimalvalue; further increasing the coupling will reduce the transfer rate due tothe heavy phonon cloud.The major obstacle preventing us from calculating the actual 8-sitemodel (5.11) is the lack of knowledge of the actual coupling details of thelight harvesting molecules. Most experiments on light harvesting moleculesare photon-based. Photons create electronic excitations in the form of exci-tons but photons do not directly interact with phonons. It is the excitonsthat are directly observed. To get the phonon structure, we have to assumea particular theory of the phonon-exciton interactions and fit the parame-ters based on this pre-assumed theory[7, 29, 39, 40]. This has great successin low temperature physics since both the environment and the system are1285.3. Pathway Selecting Mechanismrelatively “simple” and “clean”. Biological processes happen at room tem-perature and are usually more “messy”. This is the intrinsic problem of thecurrent theoretical treatments in this field: the pre-assumed theory deter-mines the values inferred for the parameters so there is no way to justify thetheory itself. To overcome this difficulty, we need either i) a direct probe tothe actual phonon structure such as neutron scattering experiments, ratherthan the current indirect photon probe, or ii) a first-principle calculation forthe phonon couplings based on the structure of light harvesting moleculesand the protein they are embedded in. Unfortunately, both methods arebeyond the expertise of the authors. We cannot find similar researches onthe detailed phonon structures in light harvesting molecules in the literaturesince quantum biology is a relatively new field for both physics and biology.In the reduced 3-site model, although we do not know the details of thephonon spectra and RC parameters, we can vary these parameters to ex-plore the dynamics in different parameter regions. However, if we go fromthe 3-site model to the full 8-site model, the number of hopping amplitudesbetween the 8 different sites is dramatically increased. There are 28 inde-pendent hopping amplitudes and associated phonon coupling spectra. Dueto the lack of symmetry of their spatial distribution, it is not valid to as-sume the same coupling strength for each hopping amplitude. This leadsto a much larger parameter space to explore. It is almost impossible forus to control variables in the way we do in the 3-site model. Also for the8-site Hamiltonian, the dynamics are more sensitive to the hopping ampli-tudes as well as the coupling strengths in (5.11), due to its mutiple-pathwaynature. Any inaccuracy of the hopping amplitudes could give a completelydifferent pathway scheme. The addition of the acoustic phonon continuumand Holstein couplings might also change the fitting results of the hoppingamplitudes. Therefore, at this stage, we are mainly working on the reducedfew-site models and trying to explore the new mechanism introduced byPeierls couplings, rather than actually fitting the experiment data.5.3 Pathway Selecting MechanismThe reduced 3-site model in the previous section is a straightforward lad-der model: there is no direct interaction between site-2 and site-0. Butas we mentioned before, it is not clear whether another pathway in FMOhas functional importance. And the two pathway structure is not uniqueto the FMO complex; it is actually common among the light harvestingmolecules. Another typical example is the reaction center of the purple bac-1295.3. Pathway Selecting Mechanismterium Rhodobacter sphaeroides (see Fig. 5.5). It has a pesudo two-foldsymmetry, labelled by paths A and B in the Fig.5.5.Figure 5.5: This is the cofactors-only structure of the reaction center ofRhodobacter sphaeroides . DA DB and BA BB are four Bchl pigmentswith absorption peak at 870nm (P870). HA and HB are two bacteriapheo-phytins(BPh, basically the Bchl without the central Mg2+), and QA andQB are two quinones. At the bottom there is a metal ion Fe2+. Reprintedwith permission from [6].In the RC of Rhodobacter sphaeroides, the top part (the part which isclosest to FMO complex) is composed by two cofactors, each of which hasa BChl-a P870 pigment, which is named after the aborption peak of its Qyband. Note that 870nm is approximately 11630cm−1. The lowest excitationenergy of the BChl in FMO complex is 12195cm−1, which is 568cm−1 higherthan the absorption peak. The top P870 pigments absorbs an exciton toform an excited state P870∗. This state is unstable and it decays in 3ps by donating an electron to the neighbour Bchl to form a radical pairP870+BChl−A . This state decays to P870+BPh−A radical pairs rapidly in1ps. Eventually this electron is transferred to quinone QA which makesit P870+Q−A. This step takes a much longer time, which is around 200ps.Q−A, Q−B and other cytocrhomes form another chemical reaction circle whichtakes a much longer time to complete.Although it has an apparent two-fold symmetry in the structure, theenergy path-way is not symmetric. Estimates of the ratio for the electrontransfer probability down the A branch compared with the B branch areabout 100 : 1[6]. Neither the function of the other structure nor the reason1305.3. Pathway Selecting MechanismFigure 5.6: Illustration of the two-pathway models. (a)The ladder model:site-2 serves as the ladder between site-3 and site-1. (b)The symmetricModel: site-2 has the same energy as site-1. In both cases, the exicita-tion is initially created at site-1 and eventually transferred into RC.(c) Thesymmetric 4-site model.for this path-selecting phenomenon is clear at the present time.It would be interesting to study how phonon couplings affect the transferpattern in different pathways. In this section, we are trying to answer thesequestions and to gain some insight of the real biological world, by studyinga couple of two-pathway models.5.3.1 Ladder ModelThe major difference between the two-site and three-site systems is that forthe three-site system we actually have multiple pathways from the startingsite to the sink. Here we will start with a three-site ladder model but withdirect couplings between the top and the bottom site (see Fig.5.6(a)). Inthis case, we can really talk about the choice between multiple pathwaysand the interaction between them. The bare Hamiltonian of our first modelis, in site representation:H0 = 4 0.1 0.20.1 2 0.10.2 0.1 0 (5.14)Site-1 has the highest energy and the excitation is initially created on thissite. Site-3 has the lowest energy and resides in the center of the RC band,1315.3. Pathway Selecting Mechanism0 0.5 1 1.5 2gP0. RateHolsteinPeierls 1-3 onlyPeierls 1-2, 2-3Full Peierls Peierls 1-2 onlyPeierls 2-3 onlyFigure 5.7: The transfer rate of the Hamiltonian (5.14) . The phonon energyΩ = 2. The parameters of the RC couplings are the same as Fig. 4.2.Different lines stand for different types of phonon couplings. For simplicitywe assume the same coupling strength for all coupled phonon modes.which has a width of 1. Site-2 serves as a “ladder” in between the othertwo sites (see Fig. 5.6(a)). In this model the excitation is created at site1, which is similar to the eighth pigment of FMO complexes. To reach theRC, there are two major path ways, i.e. site-1→site-3, and site-1→site-2→site-3. This is intended to be an analogue to the FMO complexes withtwo different path ways. The bare system evolution can be found in Fig.5.8(d). We can find that ρ22 is almost zero throughout the time so that theexciton is directly transferred into site-3. The first pathway 1→3 dominatethe excitation transfer.Then we have phonons coupled to this model. We use the same numericalin Chapter 4. The phonon energy Ω = 2 which matches the energy gapbetween the site 1-2 and 2-3. Fig. 5.7 is the general result of the transferrate in this model with various types of phonon couplings. Fig 5.8 showsthe time evolution for site populations for some particular choice of phononcouplings.We can see that for the Holstein coupling case, the the result does notchange much from the two-site case in Chapter 4. There is an optimaltransfer rate when gH ∼ 1. And we can see that Holstein coupling does notchange the site population qualitatively either.On the other hand, for the Peierls coupling case, the highest transferrate is achieved when phonons only couple to the hopping amplitude 1-2and 2-3. Similar to the two-site case studied in the last chapter, the transfer1325.3. Pathway Selecting Mechanism0 50t0.ρ 1,2,3ρ11ρ22ρ330 50t0.ρ 1,2,3ρ11ρ22ρ330 50t0.ρ 1,2,3ρ11ρ22ρ330 50t0.00.51.0ρ 1,2,3ρ11ρ22ρ33(a) (b)(c) (d)gP=0.16 gP=0.161-2, 2-3 onlygP=0.161-3 onlygP=00 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50t0.ρ 1,2,3ρ11ρ22ρ330 50t0.00.51.0ρ 1,2,3ρ11ρ22ρ33(a) (b)(c) (d)gP=0.72 gP=0.721-2, 2-3 onlygP=0.721-3 onlygP=0Figure 5.8: The time evolution of the Hamiltonian (5.14) for the weak cou-pling case gP = 0.16(top panel) and strong coupling gP = 0.72(bottompanel). The parameters are the same as Fig. 5.7. Different colors indicatedifferent site populations(ρ11, ρ22, ρ33). a) phonons couple to all three off-diagonal hoppings, b) phonons couple to 1-2 and 2-3 hopping, c) phononsonly couple to 1-3, d) no phonon coupling at all.1335.3. Pathway Selecting Mechanism0 50t0.00.2ρρ12ρ13ρ230 50t0.00.2ρ 1,2,3ρ12ρ13ρ230 50t0.00.2ρρ12ρ13ρ230 50t0.0ρρ12ρ13ρ23(a) (b)(c) (d)gP=0.16gP=0.161-2, 2-3 onlygP=0.161-3 onlygP=0Figure 5.9: The time evolution of the Hamiltonian (5.14). Different colorsindicate different off-diagonal components (ρ12, ρ13, ρ23). Here we use thenormalized density matrix element|ρij |trρrate increases rapidly in the weak interaction region and falls off when boundstates are formed in the large interaction region. This is what we expectsince the phonon energy is resonant with the energy difference between 1-2 and 2-3 so the phonon coupling to these branches has more impact onthe total transfer rate. The full Peierls coupling shows the same behaviorqualitatively but with a smaller transfer rate. This shows that in this case1→2→3 becomes the dominant energy transfer pathway. We can confirmthis by comparing the time evolution of ρ22 and ρ33 in both cases in Fig. 5.8.We can see that the behavior of the density matrix element is qualitativelythe same. The population on site-3 follows the trend of the population onsite-2 in both cases.For other cases, we can see from Fig.5.7 that the coupling between 1-2or 2-3 alone has almost no effect on the transfer rate. The coupling on onlyone of the hopping amplitudes in the second path is not enough to improvethe overall energy transfer.For the couplings between site 1-3 only, we have effectively a phonon-assisted transfer but with no resonant condition. Overall, this does notchange the transfer rate significantly. We can also see from Fig. 5.8(c) that1345.3. Pathway Selecting Mechanismthe sole coupling to 1-3 qualitatively does not change compared to the baresystem even in the large interaction region. On the other hand, we noticethat there is a sudden change of transfer rate around the intermediate regiongP ∼ 1. We can understand this behavior by looking at the eigenstates ofthe system without the RC. When gP < 1, the drop is caused by the formingof bound states as we explained in 4.3.1. The reason for the sudden increasein the transfer rate afterwards is because around gP = 1, the eigenenergyof the two-phonon ploaron state starts to drop into the range of the RCspectrum. Since the energy difference between 1-3 is exactly two times thephonon energy, this state actually satisfies the resonant condition and affectsthe transfer rate more than the non-resonant one-phonon state.We can also see the time evolution of the off-diagonal elements in Fig.5.9.The trace of the reduced density matrix is not conserved here so we use thenormalized density matrix element|ρij |trρ . Again we can confirm that the 1-2-3 pathway is dominant for the full Peierls coupling case since the behavior ofthe off-diagonal elements is almost the same for the full Peierls coupling caseand the 1-2 2-3 coupling case. The decay of the oscillation of the off-diagonalelements is usually a signal of decoherence in spectroscopic experiments. Wecan see that for the gP = 0 case we already have the decoherence due to thecoupling to the bath. After we turn on the coupling, one of the interestingthings here is that, even though the effect of the 1-3 only coupling does noteffect the diagonal elements significantly, it does cause decoherence betweensite-1 and site-2. This suggest that the off-diagonal coupling can affect thedecoherence rate, but at the same time does not affect the energy transferrate.5.3.2 Symmetric ModelIn this section we study a symmetric model. The bare Hamiltonian in siterepresentation of our second model isH0 = 2 0.2 0.10.2 2 0.10.1 0.1 0 (5.15)Here we change the energy of site-1 to the same energy as site-2 (see Fig.5.6).We also set the hopping amplitude between site 1-2 to be the biggest am-plitude. The time evolution of the bare Hamiltonian can be found in Fig.5.10(d). The excitation is mainly oscillating between site 1 and site 2 asexpected.1355.3. Pathway Selecting Mechanism0 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50 100t0.00.51.0ρ 1,2,3ρ11ρ22ρ33(a) (b)(c) (d)gP=0.16 gP=0.161-3, 2-3 onlygP=0.161-2 onlygP=00 50 100t0.00.51.0ρ 1,2,3ρ11ρ22ρ330 50 100t0.ρ 1,2,3ρ11ρ22ρ330 50 100t0.00.51.0ρ 1,2,3ρ11ρ22ρ330 50 100t0.00.51.0ρ 1,2,3ρ11ρ22ρ33(a) (b)(c) (d)gP=0.72 gP=0.721-3, 2-3 onlygP=0.721-2 onlygP=0Figure 5.10: The time evolution of the Hamiltonian (5.15). Different colorsindicate different site populations(ρ11, ρ22, ρ33). The top four figures arefor the weak interaction gP = 0.16; and the bottom four figures are forthe strong interaction gP = 0.72. a) phonons couple to all three off-diagonalhoppings, b) phonons couple to 1-2 and 2-3 hopping, c) phonons only coupleto 1-3, d) no phonon coupling at all. Top panels are small couplings gP =0.16. Bottom panels are large couplings gP = 0.721365.3. Pathway Selecting Mechanism0 0.5 1 1.5 2gP0. RateFull PeierlsPeierls 1-2Peierls 1-3, 2-3Figure 5.11: The transfer rate of the Hamiltonian (5.15) . The phononenergy Ω = 2. The parameters of the RC couplings are the same as Fig. 4.2.Different lines stand for different types of phonon couplings. For simplicitywe assume the same coupling strength for all coupled phonon modes.As a function of phonon couplings, the transfer rate to the RC is il-lustrated in Fig. 5.11. If phonons only couple to the hopping amplitude1-2 (the brown line in Fig. 5.11), it does not affect the transfer rate sig-nificantly. We can see from the time evolution in Fig. 5.10 that the effectof 1-2 coupling is mainly to change the oscillation frequency between site1-2. Since there is no energy difference between 1-2, there is no emergentresonant effect either.On the other hand, the coupling to the amplitude 1-3 and 2-3 has a moresignificant effect on the transfer rate, just the same as what we saw in theprevious model.5.3.3 Symmetric Pathways with a Single EntryTo better illustrate the path-selecting mechanism, instead of the 3-site model,we study a symmetric 4-site model in this section. The Model Hamiltonianis, in site representation:H0 =4 0.2 0.2 00.2 2 0 0.20.2 0 2 0.20 0.2 0.2 0 . (5.16)The system has a single entry at site-1 and two symmetric paths: 1→3→4and 1→2→4 (see Fig.5.6(c)). .1375.3. Pathway Selecting Mechanism0 0.5 1 1.5 2gP,H0.010.0150.020.0250.030.0350.04Transfer RateFull Peierls CouplingHolstein CouplingPeierls Coupling1-3,3-4 onlyFigure 5.12: The transfer rate of the 4 site model (5.16). The transfer ratewhen phonons couples to different hopping amplitude. All couplings areeither the same or zero.0 50 100 150t0.ρ 1,2,3,4ρ11ρ22ρ33ρ440 50 100 150t0.ρ 1,2,3,4ρ11ρ22ρ33ρ440 50 100 150t0.ρ 1,2,3,4ρ11ρ22,33ρ440 50 100 150t0.ρ 1,2,3,4ρ11ρ22,33ρ44(a) (b)(c) (d)gP=0.11-3,3-4 onlygP=1.01-3,3-4 onlygP=0.1full interactiongP=1.0full interactionFigure 5.13: The time evolution of a 4 site model (5.16). Different colorsindicate different site populations(ρ11, ρ22, ρ33, ρ44). a, b) Peierls phononscouple to the left branch of the molecules; c,d) Peierls phonons couple toboth branches.1385.4. Summary and Future ResearchesWe study two cases: i) Peierls phonons coupled to both paths and 2)Peierls Phonon coupled to only one of the path(1→3→4). The time evolutionof site-populations is illustrated in the dependence of transfer rate in Fig.5.13. We can see the clear path selecting mechanism here. As for the boththe weak interaction case gP = 0.1 and the strong interaction case gP = 1.0,the population on site-3 is significantly larger than the population on site-2. The pathway which couples to phonons (→3→4) has the dominant rolein energy transfer. If the phonon couples to both pathways, however, thetransfer rate is suppressed. The backwards transfers from site-4 to both site-2 and site-3 are both enhanced by phonon couplings and therefore impedethe transfer to the RC. The excitons are more likely to be trapped comparedwith the single pathway case.This can be confirmed with the dependence of transfer rate on couplingstrength plots in Fig.5.12. The transfer rate when the phonons only coupleto a single branch is actually higher than the transfer rate when the phononscouple to both branches. We have already found similar behaviors in Fig.5.7and Fig. 5.11. This shows that for a multiple pathway model, it is actuallymore efficient to enhance the transfer rate along a particular pathway ratherthan several of them. It is interesting to see that the full Holstein couplingnow has the higher transfer rate with the optimal value gH ≈ 0.75. However,the Peierls coupling stil has a more significant effect on the transfer rate inthe small interaction region.5.4 Summary and Future ResearchesIn this chapter, we at first gave a complete description for the vibrationalmodes coupling in light harvesting molecules. The two-phonon interactionscan be mapped into a spin bath model in the pure decoherence region, whichintroduce decoherence without dissipation. The high energy phonon modescan be mapped into a spin bath model in the high field region which hasa Non-Markovian dissipative behavior. The intermediate vibrational modescan be treated with the method we developed in Chapter 4.However, the lack of knowledge of the actual phonon couplings in thereal system prevents us from studying the model thoroughly. Instead, westudied the reduced 3-site model of FMO complexes and explored differentparameter regions of the unknown variables. In this case, the phonon energyis no longer on resonance. We find that the Holstein coupling still givesqualitatively the same effects as in Chapter 4. The Peierls coupling, onthe other hand, gives a sharp transfer peak in the intermediate coupling1395.4. Summary and Future Researchesregion, instead of the high-transfer-rate plateau we found in Chapter 4. Theoptimal value of the coupling strength g is robust against several parametersin the model, such as the coupling strength to the RC t0, and to the RCbandwidthD. Only the energy deviation of the RC d and phonon frequenciesΩ can change the positions of the transfer peaks. The typical values of theHuang-Rhys factor associated with transfer-rate peaks (∼ 0.07) are stillmuch smaller than the reported value for Holstein phonons (∼ 0.22).In the last part of this chapter, we studied the path selecting mechanismfor multiple-pathway models. Our models have two pathways to reach theRC. A Peierls coupling to phonons greatly enhances the site populationon the coupled pathway . The transfer rate is also increased significantlyif the phonon energy is close to the energy different between sites. Wealso find that it is more efficient for energy transfer if Peierls phonons onlycouple to a particular pathway in the system rather than couple to bothof them. The competition between different pathways actually impedes thetransfer into the RC.Therefore, beside the resonant transfer mechanism weintroduced in Chapter 4, vibrational phonons can also help to increase theoverall transfer rate by enhancing a particular path and suppressing theoscillation between pathways. This might explain the Nature’s preferencefor a particular pathway in an apparently symmetric structure.In future researches, one can combine the numerical calculations of theacoustic phonon modes together with the vibrational modes discussed here.Master equation approaches are generally better for these acoustic phononmodes so one can combine such techniques with the methods we developedin this thesis. We have already shown the sensitivity of the transfer rate tothe Peierls coupling and the robustness of Holstein couplings. It could beinteresting to see how the acoustic phonon modes change the transfer peaksof the Peierls couplings.To actually answer the questions about real world systems, one needsdetailed information about the system. We have already shown that thesensitivity of the transfer rate to the Peierls phonon couplings even in thesmall interaction regime. 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For example , for the∏n1,n2et22[Hn1 ,Hn2 ] term, we couldexpand it in powers of 1/N and take expectation valuestrb( ∏n1>n2et22[H+n1 ,H+n2]∏n1,n2et22[H−n1 ,H−n2])= trb(e−J2t2N∑n1>n2(γ†kn1−γkn1 )(γ†kn2−γkn2 ))≈ 1− J4t4N2N(N − 1).(A.3)As we can see in the main text this already happens in the J2 order. There-fore in the weakly interacting region, our calculation can be a good approx-imation to the exact result.156A.2. Adiabatic DecouplingA.2 Adiabatic DecouplingThe goal is to solve the asymptotic solution to the following equationa(t) = ∓√NJ(t) b˙(t) (A.4)¨b(t)− (iEn + J˙(t)J(t))b˙(t) + J2(t)N b(t) = 0. (A.5)We first study the evolution of the state (0, 1). Actually, the state (1, 0) canbe treated in the same way just with an opposite sign of J(t) and En. Thenwe have the initial conditionb(t0) = 1 (A.6)b˙(t0) = 0. (A.7)Define the following quantity for simplification, keeping in mind thatJ20N isvery small =J20N. (A.8)The general solution of the equation (2.95) isb(t) = C1e−iEnt/2−iEn√1+4/E2n/2(1 + ekt)i√/kf1(−ekt)+ C2e−iEnt/2+iEn√1+4/E2n/2(1 + ekt)i√/kf2(−ekt).(A.9)Heref1(x) = 2F1(1 +iEn2k +i√k − iEn2k√1 + 4E2n,− iEn2k + i√k − iEn2k√1 + 4E2n; 1− iEn2k√1 + 4E2n;x) (A.10)f2(x) = 2F1(1 +iEn2k +i√k +iEn2k√1 + 4E2n,− iEn2k + i√k +iEn2k√1 + 4E2n; 1 + iEn2k√1 + 4E2n;x). (A.11)We set our initial time t0 → −∞, then ekt → 0 andf1,2(−ekt)→ 1. (A.12)Thereforeb(t0) = 1= C1e−iEnt0/2−iEnt0√1+4/E2n/2 + C2e−iEnt0/2+iEnt0√1+4/E2n/2;(A.13)157A.2. Adiabatic Decouplingb˙(t0) =1=C1(−iEn/2− iEn√1 + 4/E2/2)e−iEnt0/2−iEnt0√1+4/E2n/2+ C2(−iEn/2 + iEn√1 + 4/E2n/2)e−iEnt0/2+iEnt0√1+4/E2n/2.(A.14)Then we can get the constant C1, C2 asC1 =12eiEnt0/2+iEnt0√1+4/E2n/2(−1 + (1 + 4E2n)−12 ) (A.15)C2 =12eiEnt0/2−iEnt0√1+4/E2n/2(1 + (1 +4E2n)−12 ). (A.16)After we get the solution , we could study its behavior when t → ∞. Weuse the fact that when t→∞f1(−ekt) → AeiEnt/2−i√t+ itEn2√1+4/E2n/2 (A.17)f2(−ekt) → BeiEnt/2−i√t− itEn2√1+4/E2n/2 (A.18)withA =Γ(1− iEnk√1 + 4/E2n)Γ(1 +iEnk )Γ(1 + iEn2k − i√k − iEnk√1 + 4/E2n)× 1Γ(1 + iEn2k +i√k − iEnk√1 + 4/E2n)(A.19)B =Γ(1 + iEnk√1 + 4/E2n)Γ(1 +iEnk )Γ(1 + iEn2k − i√k +iEnk√1 + 4/E2n)× 1Γ(1 + iEn2k +i√k +iEnk√1 + 4/E2n). (A.20)Recalling that  =J20N is a very small number, we can expand the solutionaccording to the order of .Zeroth order : if  = 0, then C1 → 0, C2 → 1, andB → γ(1 +iEnk )γ(1 +iEnk )γ(1 + iEnk )γ(1 +iEnk )→ 1. (A.21)⇒ b(t)→ 1, which is exactly what we expected.158A.2. Adiabatic DecouplingFirst order Since√1 + 4E2n≈ 1 + 2E2, we haveC1 → eiEnt0(1 + itEn)(− E2) = − E2neiEnt0 (A.22)C2 → e−it0/E2n(1− En) = 1− E2n− itE2n. (A.23)As a resultA → Γ(1− iEnk)Γ(1 +iEnk)×[1− ikEn(Γ′(1− iEn/k)Γ(1− iEn/k) + iΓ′(1))+E2n(Γ′(1)− Γ′′2(1))](A.24)B → 1− Γ′2(1 + iEnk )Γ(1 + iEnk )2k2+Γ′′(1 + iEk )Γ(1 + iEnk )k2; (A.25)⇒ b(t)→ 1− k2eiEnt0Γ(1− iEnk)Γ(1 +iEnk)− E2n− itE2n− k2Γ′2(1 + iEnk )Γ(1 + iEnk )2+k2Γ′′(1 + iEk )Γ(1 + iEnk )+O(32 ).(A.26)Then the trace of tr(eiH+teiH−t) is |b(t)|2−|a(t)|2. Since |b(t)|2−|a(t)|2 = 1,tr(eiH+teiH−t) = 2|b(t)|2 − 1. For adiabatic decoupling we mentioned in themain text, we take k → 0, then ddtJ(t)→ 0. The system is slowly decoupledfrom the bath. We use the fact that when |z| → ∞(ln Γ(z))′ = ln z +z − 1/2z+O(z−2) (A.27)(ln Γ(z))′′ =Γ′′(z)Γ(z)−(Γ′(z)Γ(z))2(A.28)(ln Γ(z))′′ =1z+12z2+O(z−3) (A.29)(A.30)We can get the factor b(t) as well as the decoherence factor κn for thissubspace as|b(t)|2 → 2(1− E2n) (A.31)κn(t) = tr(eiH+n teiH−n t)→ 1− 2J20NE2n(A.32)159Appendix BPure Phase DecoherenceB.1 Free ParticleWe consider first the free particle for the N -site symmetric ring, with Hamil-tonianHo =∑<ij>[∆oc†icj eiA0ij +H.c.]. (B.1)and band dispersion kn = 2∆o cos(kn − Φ/N).For this free particle the dynamics is entirely described in terms of thebare 1-particle Green functionGojj′(t) ≡ 〈j|Go(t)|j′〉 ≡ 〈j|e−iHot|j′〉=1N∑ne−i2∆0t cos(kn−Φ/N)eikn(j′−j). (B.2)which gives the amplitude for the particle to propagate from site j′ at timezero to site j at time t. This can be written as a sum over winding numbersm, viz.,Gjj′(t) =1N∑kne−i2∆0t cos(kn−Φ/N)eikn(j−j′)=∞∑`=0∑`m=0(−i∆ot)`m!(`−m)!eiΦ/N(`−2m)× 1NN−1∑n=0e−i2pin(`−2m−j+j′)N . (B.3)This sum may be evaluated in various forms, the most useful being in terms160B.1. Free Particleof Bessel functions:Gojj′(t)=1NN−1∑n=0+∞∑m=−∞Jm(2∆ot)(−i)meim(kn−Φ/N)+ikn(j−j′)=+∞∑m=−∞Jm(2∆ot)(−i)me−imΦ/NδNp,m+j−j′=+∞∑p=−∞JNp+j′−j(2∆ot)e−i(Np+j′−j)(Φ/N+pi/2)(B.4)which can also be written asGojj′(t) =∑peipΦ+iΦN(j−j′)INp+j−j′(−2i∆ot) , (B.5)where we use the hyperbolic Bessel function Iα(x), defined as Iα(x) =(i)−αJα(ix), and we drop the modulus signs for the hyperbolic Bessel func-tions Iα, since I−α = iαJ−α = iα(−1)αJα = i−αJα = Iα as long as α ∈ Z.Consider now the free particle density matrix. As discussed in the maintext, we have in general some initial density matrix ρol,l′ = 〈l|ρ(t = 0)|l′〉 attime t = 0 (where l and l′ are site indices), so at a later time t we haveρojj′(t) =∑l,l′ρl,l′Gojl(t)Goj′l′(t)†. (B.6)Now the most obvious way of evaluating this is by using the result forthe Green function, to produce a double sum over winding numbers:ρojj′(t) =∑l,l′ρl,l′∑pp′ei(p−p′)ΦeiΦ(j−j′+l−l′)/N (−i)|Np+j−l|(i)|Np′+j′−l′|J|Np+j−l|(2∆ot)J|Np′+j′−l′|(2∆ot)=∑l,l′ρl,l′∑pp′ei(p−p′)ΦeiΦ(j−j′+l−l′)/NINp+j−l(−2i∆ot)INp′+j′−l′(2i∆ot) , (B.7)However this expression is somewhat unwieldy, particularly for numericalevaluation, because of the sum over pairs of Bessel functions. It is thenuseful to notice that we can also derive the answer as a single sum over161B.1. Free Particlewinding numbers, as follows:〈j|ρ(t)|j′〉=1N2∑l,l′N−1∑n,n′=0ρol,l′e−i(kn(j−l)−kn′ (j′−l′))+4i∆ot sin[φ−(kn+kn′ )/2] sin[(kn−kn′ )/2]=1N2N−1∑n,m=0N∑l,l′∞∑p=−∞ρol,l′Jp[4∆ot sin(km/2)]× eip(φ−kn+km/2)−ikn(j−l)+i(kn−km)(j′−l′)=N∑l,l′=1ρol,l′N∞∑p′=−∞N−1∑m=0(JNp′+j′−j+l−l′ [4∆ot sin(km/2)]eikm(l=l′−j−j′+Np′)/2)· eiφ(Np′+j′−j+l−l′) . (B.8)In the second step we replaced n′ = m − n. In the third step we also usedthe identity∑N−1n′=0 eikn′` ≡∑∞p′=−∞Nδ`,Np′ . If we start with ρ(0) = |0〉〈0|,the expression is shortened toρjj′ =1NN−1∑m=0∞∑p′=−∞JNp′+j′−j [4∆ot sin(km/2)]× eiφ(Np′+j′−j)−ikm(j+j′−Np′)/2.(B.9)It is useful and important to show that these two expressions ((B.7)) and((B.8))are equivalent to each other. To do this we use Graf’s summationtheorem for Bessel functions[166]Jν(2x sinθ2)(−e−iθ) ν2 =+∞∑µ=−∞Jν+µ(x)Jµ(x)eiµθ. (B.10)We set θ = 0, 2piN , ...2pimN , ...2pi(N−1)N , which is the km in (B.8) and multiplye−ijθ on each side. We then haveJν(2x sinkm2)e−i(km+pi)ν2 e−ijkm=+∞∑µ=−∞Jν+µ(x)Jµ(x)ei(µ−j)km . (B.11)Noticing then thatN−1∑m=0eikmn =∑pδNp,n, (B.12)162B.2. Including Phase Decoherencewe do the sum over m; only µ− j = Np survives, and thus1NN−1∑m=0Jν(2x sinkm2)e−i(km+pi)ν2 e−ijkm=1N∑pJNp+j+ν(x)JNp+j(x). (B.13)Setting ν = Np′+j′−Np−j, x = 2∆ot, we then substitute back into (B.7),to getρojj′(t) =∑pp′ei(Φ/N+pi/2)(Np′−Np+j′−j)JNp+j(2∆ot)JNp′+j′(2∆ot)=1N∑pe+i(Np+j′−j)( ΦN+pi2)N−1∑m=0JNp+j′−j(4∆ot sinkm2)· e−i(km+pi)Np+j′−j2 e−ijkm=1N∑pN−1∑m=0JNp+j′−j(4∆ot sinkm2)ei(Np+j′−j) ΦN−ikm(j+j′+Np)/2.(B.14)The density matrix ρ is Hermitian, ie., ρjj′ = ρ∗j′j ; setting p′ = −p, we thenhaveρojj′(t)=1N∑p′N−1∑m=0J−Np′+j−j′(4∆ot sinkm2)e−i(−Np′+j−j′) ΦN+ikm(j+j′−Np′)/2=1NN−1∑m=0∞∑p′=−∞JNp′+j′−j [4∆ot sin(km/2)]eiφ(Np′+j′−j)−ikm(j+j′−Np′)/2.(B.15)where in the last line, we use set km → −km, and use the fact that for integerorder n, Jn(−x) = J−n(x). Thus we have demonstrated the equivalence ofthe single and double sum forms for the density matrix.B.2 Including Phase DecoherenceTo calculate the reduced density matrix for the particle in the presence ofthe spin bath, we need to average over the spin bath degrees of freedom.163B.2. Including Phase DecoherenceWe will do this in a path integral technique, adapting the usual Feynman-Vernon [14] theory for oscillator baths to a spin bath; the following is ageneralization of the method discussed previously [1]. We can parametrize apath for the angular coordinate Θ(t) which includes m transitions betweensites in the formΘ(m)(t) = Θ(t = 0) +m∑i=1∑qi=±qiθ(t− ti) , (B.16)where θ(x) is the step-function; we have transitions either clockwise (withqj = +1) or anticlockwise (with qj = −1) at times t1, t2, . . . , tm. The prop-agator K(1, 2) for the particle reduced density matrix between times τ1 andτ2 is thenK(1, 2) =∫ Θ2Θ1dΘ∫ Θ′2Θ′1dΘ′ e−i~ (So[Θ]−So[Θ′])F [Θ,Θ′] , (B.17)where So[Θ] is the free particle action, and F [Θ,Θ′] is the “influence func-tional” [14], defined byF [Θ,Θ′] =∏k〈Uˆk(Θ, t)Uˆ †k(Θ′, t)〉 . (B.18)Here the unitary operator Uˆk(Θ, t) describes the evolution of the k-th envi-ronmental mode, given that the central system follows the path Θ(t) on its“outward” voyage, and Θ′(t) on its “return” voyage. Thus F [Θ,Θ′] acts asa weighting function, over different possible paths (Θ(t),Θ′(t)). The aver-age 〈...〉 is performed over environmental modes - its form depends on whatconstraints we apply to the initial full density matrix. In what follows wewill assume an initial product state for the full particle/environment densitymatrix.For the general Hamiltonian in eqtns. (3.1)-(3.12), the environmentalaverage is a generalisation of the form that appears when we average overa spin bath for a central 2-level system, or “qubit” (see ref. ([1]), and alsoref. ([97])). The essential result is that we can calculate the reduced densitymatrix for a central system by performing a set of averages over the baredensity matrix. For a spin bath these can be reduced to phase averagesand energy averages; and for the present case it reduces to a simple phaseaverage.Let us write the “bare” free particle density matrix in the form of adouble sum over winding numbersρojj′(t) =∑pp′ρojj′(p, p′; t). (B.19)164B.2. Including Phase DecoherenceThen the key result is that in the presence of phase coupling to the spinbath, the reduced density matrix takes the formρjj′(t) =∑pp′∑ll′ρoj−l,j′−l′(p, p′; t)F l,l′j,j′(p, p′)ρll′ , (B.20)where the influence functional, initially over the entire pair of paths forthe reduced density matrix, has now reduced to a much simpler weightingfunction Fj,j′(p, p′) over winding numbers. We can do this because theeffect of the pure phase coupling to the spin bath is to accumulate an simpleadditional phase in the path integral each time the particle hops. Just asfor the free particle, we can then classify the paths by winding number; fora path with winding number p which starts at site l (the initial state) andends at site j, the additional phase factor can then be written asexp{−ip∑k 〈N0〉∑〈mn〉=〈01〉−i〈j−1,j〉∑〈mn〉=〈l,l+1〉 (αmnk · σk)} (B.21)and for fixed initial and final sites, this additional phase only depends onthe winding number. The weighting function F l,l′j,j′(p, p′) is just the influencefunctional, but now it is an ordinary function, which we will henceforthcall the “influence function”. In the same way as the original influencefunctional, it depends in general on the initial state ρl,l′ of the density matrixat time t = 0. Performing the sums over the two paths as before, but nowincluding the phase factors (B.21), we get:F l,l′j,j′(p, p′) =〈e−i(p−p′)∑k∑〈N−1,N〉〈mn〉=〈0,1〉αmnk ·σk · e−i(p−p′)∑k∑〈l−1,l〉〈mn〉=〈l′,l′+1〉αmnk ·σk· e−i∑k∑〈j−1,j〉〈mn〉=〈j′,j′+1〉αmnk ·σk〉 (B.22)for the case of general phase couplings αijk to the bath.In the purely symmetric case where αijk → αk for every link, the influencefunction reduces to the much simpler resultF l,l′j,j′(p, p′) = 〈e−i[N(p−p′)+(j−j′+l−l′)]∑kαk·σk〉 (B.23)which for a particle being launched from the origin gives the result (B.23)quoted in the main text.165B.2. Including Phase DecoherenceNow consider the current Ij,j+1(t), which is given in general by:Ij,j+1 = −i〈∆˜j,j+1ρj+1,j − ∆˜j+1,jρj,j+1〉, (B.24)where we define∆˜j,j+1 = ∆oeiΦ/Nei∑kαj,j+1k ·σk . (B.25)Using the results derived above for the density matrix, we can derive ex-pressions for Ij,j+1(t) in both single and double winding number forms. Thedouble Bessel function form isIj,j+1 = −2∆o∑pp′JNp+j−l(2∆ot)JNp′+j+1−l′(2∆ot)× Re〈ρl,l′iN(p−p′)ei[(p−p′)+1N]Φ e−i(p−p′)∑k∑〈N0〉〈mn〉=〈01〉αmnk ·σke2i∑k∑〈j−1,j〉〈mn〉=〈j′,j′+1〉αj,j+1k ·σk〉. (B.26)Again, we make the assumption of a completely ring-symmetric bath, sothat αijk → αk. Then we getIj+1,j = 2∆o∑pp′∑l,l′JNp+j−l(2∆ot)JNp′+j+1−l′(2∆ot)Fl,l′(p′, p)× Re[ρll′eiΦ[p′−p+(l−l′)/N)]]. (B.27)From this we can derive the single Bessel Function summation form as fol-lows. Using the equation∑pJNp+n−l(x)JNp+n−l+ν(x) =1NN−1∑m=0Jk(2x sinkm2)e−i(n−l)km−i(km−pi)ν/2(B.28)which is another form of Graf’s identity[166], we set ν = N(p′−p)+1+l−l′,x = 2∆ot; thenIj+1,j =2∆oNN−1∑m=0∑p∑l,l′JNp+1+l−l′(4∆ot sinkm2)e−ikm[Np+12+n−(l+l′)/2]· iNp+1+l−l′Fll′(p)Re[ρl,l′eiΦ[(p′−p+l−l′)/N)]](B.29)where we define Fll′(p, 0) ≡ Fll′(p).166B.2. Including Phase DecoherenceIf we make the assumption that the particle starts at the origin, theseresults simplify considerably; one getsIj+1,j=2∆o∑pp′JNp+j(2∆ot)JNp′+j+1(2∆ot)F0(p′, p) cos[(pi2N + Φ)(p′ − p)]=2∆oNN−1∑m=0∑pJNp+1(4∆ot sinkm2)e−ikm(Np+12+j)iNp+1F0(p) cos[(pi2N + Φ)p](B.30)for the double and single sums over winding numbers, respectively; andF0(p) ≡ Fj,j(p, 0). The latter expression is used in the text for practicalanalysis.167Appendix COrthogonal BlockingApproximationC.1 A Useful IntegralIn this section we are going to evaluate the integralI(t) =∫ ∞−∞e−iξ2pidξL∏j=0(ei(ξ−Mjht) − 1i(ξ −Mjht)). (C.1)When L ≥ 2, this integral does converge. So we are free to choose anyprincipal value and get the same result. We replacing iξ with z and shiftthe denominator by an infinitesimal −η < 0:I(t) =∫ i∞−i∞e−z2piidzL∏j=0(ez−iMjht − 1z − iMjht− η). (C.2)This function now has poles at z = iMjht+ η. If on expands the numeratorone gets a polynomial of ez with only one negative power term e−z(−1)L+1.We use the residue theorem to evaluate the integral. The contour we choosegoes along the imaginary axis. But whether it encloses the left or right halfof the complex plane depends on the power of ez. Noticing that the polesare all in the right half plane, only the negative power term requires thecontour being closed in the right half plane. We only need to consider thee−z term in this choice. If Mj 6= Mk, ∀k 6= j, the integral equalsI(t) = (−1)LL∑k=0L∏j 6=ke−iMkhti(Mk −Mj)ht, (C.3)which is inversely proportional to (Mk−Mj). If all the Mj are equal to eachother, Mj = M, ∀j, we haveI(t) =e−iMhtL!. (C.4)168C.2. OBA Expansion of the SystemThe residue calculation when not all the Mj are the same is tedious but canbe complicated. In the main part of the thesis we use the result when Mjcan only take two values: either M or M ′. In this case, the integral turnsout to be :∫ ∞−∞e−iξ2pidξn+m−1∏j=0(ei(ξ−Mht) − 1i(ξ −Mht))n(ei(ξ−M ′ht) − 1i(ξ −M ′ht))m=n−1∑a=0(−1)m(m+ a− 1)!(n− 1− a)!(m− 1)!a!e−iMht(i(M −M ′)ht)m+a+m−1∑a=0(−1)n(n+ a− 1)!(m− 1− a)!(n− 1)!a!e−iM ′ht(i(M ′ −M)ht)n+a ,(C.5)here n,m > 0 and n+m = L+ 1.C.2 OBA Expansion of the SystemTo evaluate this quantity, we define the operatorTL ≡ δ(∑kσzk−M)ei∑k αiLiL−1k σxk ...δ(∑kσzk−M)ei∑k αi1i0k σxk δ(∑kσzk−M) .(C.6)We then substitute (3.66) into (C.6). We can decompose TL intoTL = (∏kT kL)e−iM∑Ln=0 ξn (C.7)T kL = eiξLσzkeiαiLiL−1k σxk ......eiξ1σzkeiαi1i0k σxkeiξ0σzk . (C.8)Afterwards, we replace αinin−1k by αink , since this does not cause confusionwithin a single path. Thereafter, we do a Taylor expansion to second order169C.2. OBA Expansion of the Systemin αk:T kL| ↑〉 =ei∑Ln=0 ξn{| ↑〉+ iL∑n=1αink e−2i∑Lm=n ξm | ↓〉− (12L∑n=1(αink )2 +∑m<nαink αimk e−2i∑n−1j=m ξj )| ↑〉}T kL| ↓〉 =e−i∑Ln=0 ξn{| ↓〉+ iL∑n=1αink e2i∑Lm=n ξm | ↑〉− (12L∑n=1(αink )2 +∑m<nαink αimk e2i∑n−1j=m ξj )| ↓〉}.(C.9)Here we still assume equally distributed initial states, i.e., the initial densitymatrix of the spin bath itself is an identity matrix. Given a certain totalpolarization M, all configurations with K+M2 spin-ups andK−M2 spin-downscompose the∑k σzk = M subspace. The weight of these configurations arep(M) =K!(K+M2 )!(K−M2 )!2K. (C.10)The average amplitude within this subspace is defined in the following way:A±k,M (L,L′) ≡ 〈↑ (↓)|T kLT †kL′ | ↑ (↓)〉. (C.11)170C.2. OBA Expansion of the SystemTherefore, Ak can be written asA±k (L,L′) = e±i(∑Ln=0 ξn−∑L′n′=0 ξ′n′ )[1− 12L∑n=1(αink )2 − 12L′∑n=1(αi′nk )2−L∑m<nαink αimk e∓2i∑n−1j=m ξj − L′∑m<nαi′nk αi′mk e±2i∑n−1j=m ξ′j+L∑n=1L′∑n′=1αink αi′n′k e∓2i(∑n−1m=0 ξm−∑n′−1m′=0 ξ′m′ )]≈ e±i(∑Ln=1 ξn−∑Ln′=1 ξ′n′ ) exp[−12L∑n=1(αink )2 − 12L′∑n=1(αi′nk )2−L∑m<nαink αimk e∓2i∑n−1j=m ξj − L′∑m<nαi′nk αi′mk e±2i∑n−1j=m ξ′j+L∑n=1L′∑n′=1αink αi′n′k e∓2i(∑n−1m=0 ξm−∑n′−1m′=0 ξ′m′ )].(C.12)Here A+k is for spin-up ↑ at kth spin; A−k is for spin-down ↓ at kth spin.Then the product over all environment spins isAM (L,L′) =∏kA±k e−iM(∑Ln=0 ξn−∑L′n′=0 ξ′n′ ). (C.13)This product is for a certain configuration of environment spins with po-larization M. The configuration determines the plus minus sign of A±k . Wecan see that the overall phase factor e−iM(∑Ln=0 ξn−∑L′n′=0 ξ′n′ ) is canceled bythe collective contribution from terms e±i(∑Ln=0 ξn−∑Ln′=0 ξ′n′ ) acting on thisconfiguration.The next step is to take the average of this product in the∑k σzk = Msubspace. This product is for a particular arrangement of spins. Notice thatsmall M configurations have a way larger phase space, which allows morepossible configurations. We could expand (C.10), according to the standardmethod in the large K case aslogp(M)p(0)= log(K2 )!(K2 )!(K+M2 )!(K−M2 )!≈ −M22K(C.14)The leading order is of order −M2/K2. We will then expand the resultinto a power series in MK .Within theMK ≈ 0 region, we can find a complex171C.2. OBA Expansion of the Systemconjugate of almost every Ak. Actually, we can findK+M2 plus sign termsand K−M2 minus sign terms. To zeroth order inMK we getA0(L,L′) = exp{ −L∑n=1∑k[12(αink )2 +L∑m<nαink αimk cos(2n−1∑j=mξj)]−L′∑n=1∑k[12(αi′nk )2 +L′∑m<nαi′nk αi′mk cos(2n−1∑j=mξ′j)]+L∑n=1L′∑n′=1∑kαink αi′n′k cos(2n−1∑j=0ξj − 2n′−1∑j′=0ξ′j′).}(C.15)The first order term isA1(L,L′) = exp{+ iMK∑k[L∑n=1L∑m<nαink αimk sin(2n−1∑j=mξj)−L′∑n=1L′∑m<nαi′nk αi′mk sin(2n−1∑j=mξ′j)+L∑n=1L′∑n′=1αink αi′n′k sin(2n−1∑j=0ξj − 2n′∑j′=0ξ′j′)].}(C.16)The contribution of all polarization M configurations to first order isA(L,L′) = p(0)A0(L,L′) ·A1(L,L′)e−M22K (C.17)Summing over all possible M, we have the resultWi0...iLi′L′ ....i′0=eiΦ(∆t)L+L′iL−L′L!L′!(∏n∫ 2pi0dξn2pi)(∏n′∫ 2pi0dξ′n′2pi)A0(L,L′)·K∑M=−Ke−M22Kθ(K)AM (L,L′).(C.18)Here θ(K) =∑KM=−K e−M22K is the normalization factor for this summation.To evaluate an exact quantity, we make the following variable transforma-172C.2. OBA Expansion of the Systemtions :χn = 2n−1∑p=0ξp ;χ′n = 2n′−1∑p=0ξ′p ;snk = αink (cosχn, sinχn) ;s′nk = αi′nk (cosχ′n, sinχ′n) .(C.19)The transformation of integrations is∏n∫ 2pi0dξn2pi=∏n∫ χn+1+2piχn+1dχn2pi=∏n∫ 2pi0dχn2pi(C.20)Here χn does not need to have a particular order. Another thing we shouldnotice here is that this transformation is only valid when αink 6= 0. If αk = 0for some k , then Ak is simply equal to 1. All the following calculations arebased on the fact that αink 6= 0, ∀k, in.DefiningS =∑n,ksnk −∑n,ks′nk , (C.21)then we can rewrite A’s in the formA0(L,L′) =∫dSδ2(S−∑n,ksnk −∑n,ks′nk)e− 12|S|2 (C.22)A1(L,L′) =1 +iMK∑k[L∑n=1L∑m<n(snk × smk )−L′∑n=1L′∑m<n(s′nk × s′mk )−L∑n=1L′∑n′=1(snk × s′n′k )]−M22K2{∑k[L∑n=1L∑m<n(snk × smk )−L′∑n=1L′∑m<n(s′nk × s′mk )−L∑n=1L′∑n′=1(snk × s′n′k )]}2(C.23)Here we include a constraintδ2(S−∑n,ksnk +∑n′,ks′n′k ) =∫d2z4pi2eiz·(S−∑n,k snk+∑n′,k s′n′k ) (C.24)173C.3. Zero PolarizationC.3 Zero PolarizationIn this section we will only deal with the lowest order in MK , i.e. the A0(L,L′)term. We will see that actually the higher order correction from AM (L,L′)is proportional to α4, which is higher than the order of Taylor expansion wedid in (C.9). Therefore we can simply neglect AM (L,L′) and evaluate thewhole density matrix explicitly.(∏n∫ 2pi0dξn2pi)(∏n′∫ 2pi0dξ′n′2pi)A0(L,L′) = (∏n∫ 2pi0dχn2pi)(∏n′∫ 2pi0dχ′n′2pi)·∫d2S∫d2z4pi2eiz·(S−∑n,k snk+∑n′,k s′n′k )e−12|S|2=∫d2S∫d2z4pi2(∏nJ0(∑kaink z))(∏n′J0(∑kαi′n′k z))eiS·z− 12|S|2=∫ ∞0z dz(∏nJ0(∑kαink z))(∏n′J0(∑kαi′n′k z))e− 12z2 .(C.25)Here we use the integral presentation of Bessel functionsJn(x) =12pi∫ pi−pie−i(nτ−x sin τ) dτ (C.26)Then substituting back to (C.18), we have the result of each path to zerothorder in MK ,W 0i0...iLi′L′ ....i′0=(i∆ot)L+L′iL−L′L!L′!∫ ∞0z dz2pi(∏nJ0(∑kαink z))(∏n′J0(∑kαi′n′k z))e− 12z2 .(C.27)If we assume the perfectly symmetric case αinin+1k = αk for every site,the zeroth order result becomesW 0i0...iLi′L′ ....i′0=(∆ot)L+L′iL−L′L!L′!∫ ∞0z dz2piJ0(∑kαkz))L+L′e−12z2=eiΦiL−L′L!L′!∫ ∞0z dz(J0(∑kαkz)∆ot)L+L′e−12z2 .(C.28)174C.4. M = 1 PolarizationC.4 M = 1 PolarizationIn this section we would like to deal with the A1(L,L′) terms in (3.79). Wewill verify our statement in 3.3.2 that the contribution from the M = 1polarization group is of O(α4) order and therefore can be safely neglected.Higher polarization groups can be dealt in the same way but are more com-plicated.At first we use the expansion of A1(L,L′) derived in the previous sectionto second order in MK , i.e.A1(L,L′) =1 +iMK∑k[L∑n=1L∑m<n(snk × smk )−L′∑n=1L′∑m<n(s′nk × s′mk )−L∑n=1L′∑n′=1(snk × s′n′k )]−M22K2{∑k[L∑n=1L∑m<n(snk × smk )−L′∑n=1L′∑m<n(s′nk × s′mk )−L∑n=1L′∑n′=1(snk × s′n′k )]}2.(C.29)The cross product here simply means the value of its z-component, since thesnk are all 2-D vectors. To evaluate the correction of AM (L,L′) , which is(∏n∫ 2pi0dξn2pi)(∏n′∫ 2pi0dξ′n′2pi)e−iz·(∑n,k snk−∑n′,k s′n′k )AM (L,L′), (C.30)we need the following relations〈sn〉 ≡ (∫ 2pi0dχ2pie−iz·ssn)/(∫ 2pi0dχ2pie−iz·s) = isJ ′0(sz)znzJ0(sz)(C.31)〈sms′n〉 = −sJ ′0(sz)zmzJ0(sz)s′J ′0(s′z)znzJ0(s′z)= −ss′zmznJ ′0(sz)J ′0(s′z)z2J0(sz)J0(s′z)(C.32)〈smsn〉 = iJ0(sz)∂2∂zn∂zmJ0(sz)=sJ ′0(sz)z3J0(sz)(znzm − δm,nz2)− s2J ′′0 (sz)J0(sz)zmznz2. (C.33)Here z = (z1, z2),s = (s1, s2) = s(cosχ, sinχ), m,n = 1, 2. Then we caneasily see that 〈s × s′〉 = 〈s1s′2 − s2s′1〉 = 0. Therefore the first order termdoes not contribute. To calculate the second term, we need to evaluate175C.4. M = 1 Polarizationterms like 〈(sn × sm)(sn′ × sm′)〉, with n > m,n′ > m′.If n = n′,m = m′,〈(sn × sm)(sn′ × sm′)〉 = 〈(sxn)2(sym)2 + (syn)2(sxm)2 − 2sxnsynsxmsym〉=snsmzJ0(snz)J0(smz)(smJ′′0 (smz)J′0(snz) + snJ′′0 (snz)J′0(smz)).(C.34)If n = n′,m 6= m′,〈(sn × sm)(sn′ × sm′)〉 = (sxn)2symsym′ + (syn)2sxmsxm′ − sxnsyn(sxmsym′ + symsxm′)=snsmsm′J′0(snz)J′0(smz)J′0(sm′z)zJ0(snz)J0(smz)J0(sm′z).(C.35)If none of the n, n′,m,m′ is identical to another, then〈(sn × sm)(sn′ × sm′)〉 = 0 (C.36)Then we can deal with the second order terms. Before we go to the calcula-tion details, we introduce a new sequence wnk , n = 1, 2, ....L, L+ 1....L+ L′based on snk , s′n′k :wnk = snk , , 1 ≤ n ≤ L; (C.37)wnk = s′L+L′+1−nk , , L+ 1 ≤ n ≤ L+ L′; (C.38)wn =∑kwnk (C.39)wn =∑kαink , , 1 ≤ n ≤ L; (C.40)wn =∑kαi′L+L′+1−nk , , L+ 1 ≤ n ≤ L+ L′; (C.41)In this notation,L+L′∑n=1L∑m<n(wnk ×wmk )=L∑n=1L∑m<n(snk × smk )−L′∑n=1L′∑m<n(s′nk × s′mk )−L∑n=1L′∑n′=1(snk × s′n′k ).(C.42)176C.4. M = 1 PolarizationThen it is easy to calculate the correction ∼ M2K2, for the n = n′,m = m′, k =k′ case, the contributions are equal to∑kL+L′∑n=1∑m<n1z[(wmk wnk )2wnJ ′′0 (wmz)J0(wmz)J ′0(wnz)J0(wnz)+(wmk wnk )2wmJ ′′0 (wnz)J0(wnz)J ′0(wmz)J0(wmz)]=∑k1z[L′+L∑m6=n(wmk wnk )2wnJ ′′0 (wmz)J0(wmz)J ′0(wnz)J0(wnz)](C.43)For the n = n′,m 6= m′, k = k′ case; the contributions are∑kL+L′∑l<m<n2z[(wlk)2wmk wnkwlJ ′0(wlz)J ′0(wmz)J ′0(wnz)J0(wlz)J0(wmz)J0(wnz)]. (C.44)Therefore Wi0...iLi′L′ ....i′0to the second order isWi0...iLi′L′ ....i′0= W 0i0...iLi′L′ ....i′0×K∑M=−Ke−M22Kθ(K){1− M2zK2∑k[L′+L∑m6=n(wmk wnk )2wnJ ′′0 (wmz)J0(wmz)J ′0(wnz)J0(wnz)+ 2L+L′∑l<m<n(wlk)2wmk wnkwlJ ′0(wlz)J ′0(wmz)J ′0(wnz)J0(wlz)J0(wmz)J0(wnz)].}.(C.45)As a result, the correction due to the M 6= 0 part isWi0...iLi′L′ ....i′0≈W 0i0...iLi′L′ ....i′0√11 + a(z, L+ L′). (C.46)If we take the prefect symmetric case |wnk | = α/K, ∀k, n, noticing that ourexpression is an even function of ωnk , we geta(z, L+ L′) =2α3J ′0(αz)zK3J30 (αz)(L+ L′)(L+ L′ − 1)[J ′′0 (αz)J0(αz)+13J ′0(αz)2(L+ L′ − 2)].(C.47)We can shorten the expression into the following form up to first order:ρij(t) =∫ ∞0z dz e−z22 (ρ0ij(tJ0(∑kαkz))− κt2 ∂2∂t2ρ0ij(tJ0(∑kαkz))).(C.48)177C.5. General CouplingHere we have another parameter for this term as κ =∑k α4k which is ofO(α4) order. This justifies our statement at the beginning of this section.C.5 General CouplingBefore we start calculating the problem with couplings in arbitrary direc-tions, we change our definition of notation a little bit:αink =√(αinkx)2 + (αinky)2 (C.49)αink+ = αinkx + iαinky (C.50)αink− = αinkx − iαinky. (C.51)Then , we can write the T kL operators asT kL| ↑〉 =ei∑Ln=0 ξn{| ↑〉+ iL∑n=1αink+e−2i∑Lm=n ξm | ↓〉 − (12L∑n=1((αink )2+ (αinkz)2 − 2iαinkz) +∑m<n(αink αimk e−2i∑n−1j=m ξj + αimkzαinkz))| ↑〉}T kL| ↓〉 =e−i∑Ln=0 ξn{| ↓〉+ iL∑n=1αink−e2i∑Lm=n ξm | ↑〉 − (12L∑n=1((αink )2+ (αinkz)2 + 2iαinkz) +∑m<n(αink αimk e2i∑n−1j=m ξj + αimkzαinkz))| ↓〉}.(C.52)Here we ignore the second order correction to the flipping terms, since itwill only enter third order corrections of our final expression. Following the178C.5. General Couplingsame procedure, we getA±k (L,L′) ≈e±i(∑Ln=1 ξn−∑Ln′=1 ξ′n′ ) exp[−12L∑n=1αink+αink− −12L′∑n=1αi′nk+αi′nk−−L∑m<nαink+αimk−e∓2i∑n−1j=m ξj − L′∑m<nαi′nk−αi′mk+e±2i∑n−1j=m ξ′j+L∑n=1L′∑n′=1αink+αi′n′k−e∓2i(∑n−1m=0 ξm−∑n′−1m′=0 ξ′m′ )− 12(L∑n=1αinkz −L′∑n=1αi′nkz)(L∑n=1αinkz −L′∑n=1αi′nkz ∓ 2i)].(C.53)We use the same tricks by definingθn,k = arg(αink+);θ′n,k = arg(αi′nk+);χn = 2n−1∑p=0ξp ;χ′n = 2n′−1∑p=0ξ′p + θ′n,k ;snk = αink (cos(χn + θn,k), sin(χn + θn,k)) ;s′nk = αi′nk (cos(χ′n + θ′n,k), sin(χ′n + θ′n,k)) ., (C.54)We can write our zeroth order correction asA0(L,L′) = e−12∑k(∑n αinkz−∑n αi′nkz)2∫dSδ2(S−∑n,ksnk −∑n,ks′nk)e− 12|S|2 ,(C.55)W 0i0...iLi′L′ ....i′0=eiΦ(i∆ot)L+L′iL−L′L!L′!∫ ∞0z dz2pi(∏nJ0(finin+1z))× (∏n′J0(fi′ni′n+1z))e− 12z2e− 12∑k(∑n αinkz−∑n αi′nkz)2 .(C.56)179C.6. General Initial State in the High Field LimitHere fij is a function associated with the z-direction coupling on each hop-ping term. In terms of αink , this function isfinin+1 =√(∑kαink cos θn,k)2 + (∑kαink sin θn,k)2=√(∑kαinkx)2 + (∑kαinky)2.(C.57)The value of this function depends on the inhomogenity of αink and θn,k.The maximum value is∑k αink and the minimum value is determined bythe average value of |αink − αin′k |. We can also define the inhomogenitydecoherence quantity asκ = 〈finin+1〉. (C.58)This is basically a generalization of the previous κ in (3.84).C.6 General Initial State in the High Field LimitFor general initial state ρo = ⊗∏k ρk(nˆ), in which |ρk(nˆ) ≡ 12I + 12 nˆ · σ.Then as before, we can write down the expression for A¯0(L,L′) asA¯0(L,L′) =∏ktr(T †kL′ (12I +12nˆk · σk)T kL). (C.59)DefineT kL =(1− ak ibkib∗k 1− a∗k)(C.60)withak =12∑n(αnk)2 +∑m<nαink αimk e−2i∑n−1j=m ξj (C.61)bk =∑nαnke−2i∑Lm=n ξm . (C.62)180C.6. General Initial State in the High Field LimitThen up to second order, we havetr(T †kL′ ITkL) = tr((1− a′∗k −ib′k−ib′∗k 1− a′k)I(1− ak ibkib∗k 1− a∗k))≈ 2− a′∗k − ak − a′k − a∗k + b′kb∗k + b′∗k bk (C.63)tr(T †kL′ σxTkL) = tr((1− a′∗k −ib′k−ib′∗k 1− a′k)σx(1− ak ibkib∗k 1− a∗k))≈ i(b′∗k + b∗k − b′k − bk) (C.64)tr(T †kL′ σyTkL) = tr((1− a′∗k −ib′k−ib′∗k 1− a′k)σy(1− ak ibkib∗k 1− a∗k))≈ b′∗k − b∗k + b′k − bk (C.65)tr(T †kL′ σzTkL) = tr((1− a′∗k −ib′k−ib′∗k 1− a′k)σz(1− ak ibkib∗k 1− a∗k))≈ −a′∗k − ak + a′k + a∗k − b′kb∗k + b′∗k bk. (C.66)This expression is really hard to solve, but we could separate each term tosee its effect individually.For the pure ny case, by doing the same trick as in previous sections, wecan deduce that(∏n∫ 2pi0dξn2pi)(∏n′∫ 2pi0dξ′n′2pi)A0(L,L′)= (∏n∫ 2pi0dχn2pi)(∏n′∫ 2pi0dχ′n′2pi)∫dS∫dz4pi2× eiz·(S−∑n,k snk+∑n′,k s′n′k )e−12|S−~ny |2+ 12n2y=∫dS∫dz4pi2(∏nJ0(∑kaink z))(∏n′J0(∑kai′n′k z))eiS·z− 12|S~ny |2+ 12n2y=∫ ∞0z dzJ0(nyz)(∏nJ0(∑kaink z))(∏n′J0(∑kai′n′k z))e− 12z2+ 12n2y .(C.67)In this expression the zeroth order correction will be the (3.105) in the maintext:ρij(t) =∫ ∞0z dz J0(nyz)e− z2−n2y2 ρ0ij(tJ0(∑kαkz)) (C.68)181C.6. General Initial State in the High Field LimitHere ny acts like an extra decoherence coefficient which quenches the valueof this integral for longer path, since J0(nyz) is a larger quenching term thanthe boost e12n2y .182Appendix DOscillator Bath model: Roleof Peierls Coupling inEnergy TransferD.1 Green FunctionIn this section, we show the derivation of Green functions for the wholeHamiltonian (4.1) in both the frequency and time domain. At first werewrite the Hamiltonian as H = Ho + HI + HB. Here HB is the Bethelattice Hamiltonian in (4.1), and HI = t1(c†1d0 + d†0c1). The rest of theterms in the Hamiltnoian go to Ho. In the following derivation, we usep, q, r to label different phonon states. At first we use Dyson’s equation forthe Green functionG = G0 +GHIG0 (D.1)with G(z) = 1z−H , and Go(z) = 1z−Ho−HB . In the one exciton subspace, wehave〈p|ciG(z)c†j |q〉= 〈p|ciG0(z)c†j |q〉+∑rt1〈p|ciG(z)d†0|, r〉〈, r|c†1G0(z)c†j |aj , q〉= 〈p|ciG0(z)c†j |q〉+∑rt21〈p|ciG(z)c†1|r〉〈r|c1G0(z)c†j |q〉G0B(z, r).(D.2)Here G0B(z, r) = G0B(z − E(r)). E(r) is the total phonon energy of thephonon state r and G0B(z) =2z+√z2−D2 is the Hubbard Green function. Inthe last line of (D.2), we use the fact that〈p|ciG(z)d†0|, r〉 = t1〈p|ciG(z)c†1|r〉〈r|d0G0B(z)d†0|r〉= t1〈p|ciG(z)c†1|r〉G0B(z − E(r)).(D.3)183D.1. Green FunctionBy setting j = 1, we get〈p|ciG(z)c†1|, q〉 =〈p|ciG0(z)c†1|q〉+∑rt1〈p|ciG(z)c†1|r〉〈r|c1G0(z)c†1|q〉. (D.4)By defining matrices[G(i, j; z)]pq = 〈p|ciG(z)c†j |aj , q〉 (D.5)[G0(i, j; z)]pq= 〈p|ciG0(z)c†j |q〉 (D.6)[A(z)]pq = δpqG0B(z, p), (D.7)we can write the solution of (D.4) asG(i, 1; z) = G0(i, 1; z)11− t21A(z)G0(1, 1; z). (D.8)Substituting (D.8) back to (D.2) gives us the general expression of Greenfunction matrices, viz.,G(i, j; z) = G0(i, j; z)+G0(i, 1; z)t211− t21A(z)G0(1, 1; z)A(z)G0(1, j; z).(D.9)If we know the Green’s function of a bare system Ho with phonon coupling,we can calculate the Green function for the whole system. Since there arenot many phonon modes in this system, we diagonalize the Ho Hamiltonianexactly up to a maximum of 10 phonons in the system. Various tests showthat results already converge around 5 phonons due to the large phononenergy. The time domain evolution can be acquired by the Fourier transfor-mation:G(τ) =i2pi∫ ∞−∞e−iωτdω G(ω + iη)=1pi∫ ∞−∞e−iωτdω ImG(ω + iη).(D.10)The spectral function is S(ω) = −ImG(ω+ iη) . The actual results for G(τ)and S(ω) are acquired by tracing over all the phonon modes in G(τ) andS(ω).184D.2. Bethe Lattice Reaction CenterD.2 Bethe Lattice Reaction CenterIn this section, we describe in detail how the reaction center model worksin the main paper. To avoid unnecessary complexities, we consider thesimplest case where there is only one site coupled to the reaction center.The Hamiltonian is nowH0 = ε1c†1c1 + t1(c†1d0 + d†0c1) +HB (D.11)with HB =∑〈ij〉 V (d†idj + h.c.) being the Hamiltonian of a Bethe latticewith branching number Z → ∞. Without phonon, the matrix defined in(D.5) is a number,i.e.G(1, 1; z) = G0(1, 1; z)11− t21A(z)G0(1, 1; z)(D.12)with G0 = 1z+iη−ε1 and A(z) =2z±√z2−D2 . The ± sign is determined by thereal part of z. D = 2V√Z − 1 is half the bandwidth of the RC. The timedomain propagator is calculated through (D.10):G(1, 1; t) =i2pi∫ ∞−∞e−iωτdω1ω − ε1 − 2t21ω±√ω2−D2. (D.13)The contour of the right-handed integral is chosen to be the half circle inthe lower half of the complex plane. The denominator has a branch cutfrom (−D,+D) and up to two possible poles on the real axis. The integralaround the branch cut is− 12pii∫ +D−Ddωe−iωτ (1ω − ε1 − 2t21ω+i√D2−ω2− 1ω − ε1 − 2t21ω−i√D2−ω2)=2t21pi∫ +D−Ddωe−iωτ√D2 − ω24ω(ω − ε1)t21 + 4t41 + (w − ε1)2D2=12pi∫ +1−1dxcos(xDτ)√1− x2(x− ε1D )(x+ D24t21(x− ε1D )) +t21D2,(D.14)when τ →∞, the integral → 0 according to the Riemann-Lebesgue lemma.Therefore, this part is a decaying part which eventually goes to zero.185D.2. Bethe Lattice Reaction CenterThe remaining part of the integral is determined by the poles at thelower half of the complex plane. i.e., the solutions of equationz − ε1 − 2t21z +√z2 −D2 = 0 (D.15)with non-positive imaginary parts. We can prove that this equation doesnot have any real roots which coincide with the previous branch-cut. Whenε21 + 4t21 < D2, there are no roots for this equation. The transfer rate issolely determined by the branch cut integral (D.14). When ε21 + 4t21 > D2,there are real roots of the euqation (D.15). If ε1 ≥ D − 2t21D , the rootz1 =ε1(t21 − D22 ) +√2(t21 − D24 )(D.16)is a real root of (D.15). Here  = t41ε21 + 4t61 −D2t41. If ε1 ≤ 2t21D −D, thenthe rootz2 =ε1(t21 − D22 )−√2(t21 − D24 )(D.17)is a real root of (D.15).0.5 1.0 1.5 2.0 2.5Ε1D0. D.1: Illustration of phase diagram of single site system. The greyzone indicates there is no real root of (D.15). In this zone the exciton wouldbe transferred into the sink completely. In the green zone there is one realroot of (D.15) . There are two real roots of (D.15) in the rest of the area.Fig. D.1 is an illustration of the phase diagram of our single site systemcoupled to an RC. In the grey zone, with small coupling t1 and small devi-ation ε1, the site c1 is on resonance with the RC and there is no localized186D.3. Logarithm Fitting0.8 0.9 1.0 1.1 1.2Ω/∆γgP=0.05gH=0.50.8 0.9 1.0 1.1Ω/∆0.9980.9991.000Correlation CoefficientgP=0.05gH=0.5(a) (b)Figure D.2: The plots of fitting (1 − ρRC) as a exponential function of twhen Ω ≈ ∆. The parameters of the bare model is same as the valuesin previous figures. The range of t is taken from 0 to 200. (a) red star:gH = 0, gP = 0.05, the decay rate γ as a function of Ω/∆; blue circle star:gP = 0, gH = 0.5, the decay rate γ as a function of Ω/∆. (b) the linearcorrelation coefficient between log(1− ρRC) and t.state formed around that site c1. Any exciton would be transferred into theRC eventually. In the green zone, the deviation becomes stronger and thereis a local state formed around site c1. The probability to stay at site c1 goesto a finite value when t → ∞. In the rest area, t1 becomes so strong thattwo local states are formed around the site c1 and site d0. In this region,the probability to find the exciton as that site will oscillate at a frequencyf ≈√ε21+4t21−D21−D24t21.The transfer efficiency of LHC, i.e. the ratio between the amount ofphotons transported to RC and the amount of photons absorbed by theantennae, is fantastically high, almost 100% efficient[4]. We also requirethat our RC model has a very high transfer rate. Therefore, in this paper,we always work in the grey region.D.3 Logarithm FittingIn this section, the transfer speed γ is a quantity acquired by fitting log(1−ρRC) as a linear function of time t, i.e. we assume the exponential transferfrom the molecule to the RC:(1− ρRC) ∝ e−γt. (D.18)Fig.D.2 is the result when the phonon frequency is near the resonant pointΩ ≈ ∆. The quantity γ represents how fast the exciton transfers into the187D.3. Logarithm FittingRC. We can see the huge increase of the transfer rate near the resonantpoint. The linear correlation coefficient has a dip near the resonant pointtoo, but still better than 0.998. The results of the pure Holstein case and0.0 0.5 1.0 1.5g0.γgHgP0.0 0.1 0.2t00.γgHgP0.0 0.5 1.0 1.5g0.940.960.981.00Correlation CoefficientgHgP0 50 100 150 200t-10.0-5.00.0Log(1-ρRC )gP=0.50gP=0.53gP=0.56gP=0.72gP=0.77gP=0.80(a) (b)(c)(d)Figure D.3: The plots of fitting (1 − ρRC) as a exponential function of tfor the pure Holstein case and the pure Peierls case. The parameters ofthe bare model is same as the values in previous figures. The range of t istaken from 0 to 200. (a) the decay rate F as a function of gH(blue circle)and gP (red star); (b) the decay rate F as a function of t˜H0 (blue circle) andt˜P0 (red star);(c) the linear correlation coefficient between log(1 − ρRC) andt; (d) the log(1− ρRC) v.s. t plot in two transition regions: the lines withinthe dashed circle represent the region where the exponential model start tofail; the line within the solid circle represent where the exponential behavioris restored.the pure Peierls case are illustrated in Fig.D.3. We can see from the linearcorrelation coefficient Fig.D.3(c) that (4.7) is a good approximation in theweak interaction for both type of couplings. It is also a good approximationin the pure Holstein case in the intermediate to large coupling region. In theintermediate coupling region of the Peierls coupling case (0.7 > gP > 0.4),the linearity drops down. We can see from Fig.D.3(d) how the behavior of(1−ρRC) deviates from the exponential decay at the beginning and the endof this parameter region. For the initial drop, I think the reason is that188D.3. Logarithm Fittingthe branch cut integral in (D.13)( 12pi∫ +1−1 dxcos(xDτ)√1−x2(x− ε0D)(x+D24t21(x− ε0D))+t21D2) decaysat a rate between τ−1 to τ−2 in certain parameter region. Therefore, after asufficient long time this part would become dominant. However, the resonantenergy transfer process happens fast in the light harvesting molecules (nomore than a few pico seconds, which is equivalent to the order of 102 in ourmodel. ) Therefore we believe (4.7) is still a good approximation in theshort time region.When the coupling goes beyond 0.7, the transfer rate decreases and thelinearity is restored. The reason for this is probably due to the fact that inthe large coupling case, the spectrum of the whole interacting system is sodifferent from the bare system that the resonance condition no longer holds.In both cases, the correlation coefficient is still larger than 0.998, whichmakes the approximation in (4.7) still a good measure of the transfer speedof the model.189


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