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Quantum control of dynamics of quasiparticles in periodic and disordered lattice potentials Xiang, Ping 2014

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Quantum control of dynamics of quasiparticles in periodicand disordered lattice potentialsbyPing XiangBachelor of Science, Nankai University, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemistry)The University Of British Columbia(Vancouver)August 2014c© Ping Xiang, 2014AbstractThis thesis describes research on controlling the dynamics of quasiparticles in pe-riodic and disordered lattice potentials. Working with model systems of arrays ofatoms and molecules trapped in optical lattices, I focus on, but not limited to, therotational excitons of polar molecules and propose to use external fields to controlthe binding and propagation of quasiparticles.First, we study the binding of rotational excitons in a periodic potential. Weshow that non-linear interactions of such excitons can be controlled by an elec-tric field. The exciton-exciton interactions can be tuned to induce exciton pairing,leading to the formation of biexcitons and three-body bound states of excitons. Inaddition, we propose a non-optical way to create biexcitons by splitting a high-energy exciton into two low-energy excitons.Second, we present schemes to control the propagation of a collective excitedstate in ordered and disordered aggregates of coupled particles. We demonstratethat the dynamics of these excitations can be controlled by applying a transientexternal potential which modifies the phase of the quantum states of the individualparticles. The method is based on an interplay of adiabatic and sudden time scalesin the quantum evolution of the many-body states. We show that specific phasetransformations can be used to accelerate or decelerate quantum energy transferand spatially focus delocalized excitations onto different parts of arrays of quan-tum particles. For the model systems of atoms and molecules trapped in an opticallattice, we consider possible experimental implementations of the proposed tech-nique and study the effect of disorder, due to the presence of impurities, on itsfidelity. We further show that the proposed technique can allow control of energytransfer in completely disordered systems.iiFinally, in an effort to refine the theoretical tools to study dynamics of quasipar-ticles, I extend calculations of lattice Green’s functions to disordered systems. Wedevelop a generic algorithm that can be easily adapted to systems with long-rangeinteractions and high dimensionalities. As an application of the method, we pro-pose to use the Green’s function to study the tunneling of biexciton states throughimpurities.iiiPrefacePart of the material in Chapter 3 was published in the article: P. Xiang, M. Litin-skaya and R. V. Krems, Tunable exciton interactions in optical lattices with polarmolecules, Phys. Rev. A 85, 061401(R) (2012). The project was identified anddesigned by Roman Krems, Marina Litinskaya and the author. The author per-formed all the numerical calculations and derived all the analytical expressions,except the expression for the biexciton wavefunction, which is the work of MarinaLitinskaya. Marina Litinskaya also contributed to the numerical calculations in anindirect way by providing analytical results that were used to check the results ofnumerical computation.Part of the material in Chapter 4 was published in the paper: P. Xiang, M.Litinskaya, E. A. Shapiro, and R. V. Krems, Non-adiabatic control of quantumenergy transfer in ordered and disordered arrays, New J. Phys. 15, 063015 (2013).The project was designed by E. A. Shapiro, M. Litinskaya, R. V. Krems and theauthor. The idea of phase kicking was initially proposed by E. A. Shapiro andthe idea of controlling the group velocity of exciton wavepackets by external fieldscame from Marina Litinskaya. The author performed all the numerical calculationsand analytical derivations, except the derivations in Section 4.3.1 and Section 4.3.3,which are the work of Marina Litinskaya, and the estimates in Section 4.4.1, whichwas done by Evgeny Shapiro.Chapter 5 is unpublished work by the author. The project was designed by theauthor under the guidance of Roman Krems. We are preparing a research publica-tion based on these results.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Implications of ultracold temperatures . . . . . . . . . . . . . . . 11.2 External field control of ultracold molecules . . . . . . . . . . . . 41.3 Ultracold polar molecules on optical lattices . . . . . . . . . . . . 51.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Background material . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Review of angular momentum theory . . . . . . . . . . . . . . . . 112.1.1 Symmetry groups . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Rotation operators and spherical harmonics . . . . . . . . 122.1.3 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . 172.1.4 Properties of rotation matrices . . . . . . . . . . . . . . . 192.1.5 Coupling of angular momenta . . . . . . . . . . . . . . . 212.1.6 Clebsch-Gordan series . . . . . . . . . . . . . . . . . . . 242.1.7 Spherical tensor operators . . . . . . . . . . . . . . . . . 26v2.1.8 Coupling of spherical tensors . . . . . . . . . . . . . . . . 302.1.9 Matrix elements of tensor operators . . . . . . . . . . . . 322.2 Application of angular momentum theory . . . . . . . . . . . . . 352.2.1 Diatomic molecule in external field . . . . . . . . . . . . 352.2.2 Dipole-dipole interaction . . . . . . . . . . . . . . . . . . 372.3 Introduction to excitons in molecular crystals . . . . . . . . . . . 432.3.1 Commutation relation . . . . . . . . . . . . . . . . . . . 442.3.2 Exciton Hamiltonian in second quantization . . . . . . . . 482.3.3 Eigenstates of the exciton Hamiltonian in the Heitler-Londonapproximation . . . . . . . . . . . . . . . . . . . . . . . 513 Tunable exciton interactions in optical lattices with polar molecules 563.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Exciton–exciton interactions in an optical lattice . . . . . . . . . . 583.3 Biexcitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.1 Method to calculate biexciton energies . . . . . . . . . . . 603.3.2 Analytical derivation of the biexciton wavefunction . . . . 683.3.3 Properties of biexciton states . . . . . . . . . . . . . . . . 763.4 Non-optical creation of biexcitons . . . . . . . . . . . . . . . . . 793.5 Extension to exciton trimers . . . . . . . . . . . . . . . . . . . . 843.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 Quantum energy transfer in ordered and disordered arrays . . . . . 904.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Sudden phase transformation . . . . . . . . . . . . . . . . . . . . 934.2.1 Group velocity of wave packet . . . . . . . . . . . . . . . 944.2.2 Phase kicking in quasimomentum space . . . . . . . . . . 964.3 Focusing of a delocalized excitation . . . . . . . . . . . . . . . . 1014.3.1 Focusing to a single site . . . . . . . . . . . . . . . . . . 1024.3.2 Focusing a broad wavepacket in coordinate space . . . . . 1044.3.3 Focusing a plane wave in coordinate space . . . . . . . . 1084.3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . 1104.4 Experimental feasibility of phase transformation . . . . . . . . . 113vi4.4.1 Suppressing spontaneous emission in system of ultracoldatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.4.2 Phase kicking of collective excitation in arrays of ultracoldmolecules . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.3 Focusing in system of ultracold molecules . . . . . . . . . 1204.5 Control of energy transfer in dipolar systems . . . . . . . . . . . . 1234.5.1 The effect of long-range interaction . . . . . . . . . . . . 1244.5.2 Anisotropy of dipolar interaction . . . . . . . . . . . . . . 1254.5.3 Computation details . . . . . . . . . . . . . . . . . . . . 1274.6 Energy transfer in the presence of vacancies . . . . . . . . . . . . 1374.7 Focusing in the presence of strong disorder . . . . . . . . . . . . 1394.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455 Green’s function for two particles on a lattice . . . . . . . . . . . . . 1485.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.2 Equation of motion for Green’s function . . . . . . . . . . . . . . 1495.3 Recursive calculation of Green’s function . . . . . . . . . . . . . 1525.3.1 With the nearest neighbor approximation . . . . . . . . . 1525.3.2 Extension to long-range interactions . . . . . . . . . . . . 1575.3.3 Extension to high-dimensional systems . . . . . . . . . . 1615.3.4 Comparison with other methods . . . . . . . . . . . . . . 1625.4 Application to the problem of biexciton scattering . . . . . . . . . 1665.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . 1736.2 Limitations and possible extension . . . . . . . . . . . . . . . . . 1756.3 Future research directions . . . . . . . . . . . . . . . . . . . . . . 1776.3.1 Influence of exciton-exciton interaction on polariton lasing 1776.3.2 Parallel computation of Green’s functions . . . . . . . . . 179Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183viiList of FiguresFigure 3.1 (a) The parameters D and J (in units of Vdd = d2/a3) as func-tions of the electric field strength. (b) The ratio D/J as a func-tion of the electric field strength. The field is perpendicularto the intermolecular axis. For LiCs molecules possessing thedipole moment d=5.529 Debye, the value E f d/Be = 1 corre-sponds to E f = 2.12 kV/cm. (c) Schematic depiction of theangle θ between the field (represented by blue arrows) and themolecular array (represented by red dots). (d) D and J for a 1Darray of LiCs molecules separated by 400 nm as functions ofθ for E f = 6 kV/cm. . . . . . . . . . . . . . . . . . . . . . . 77Figure 3.2 (a) and (b): Two-excitation spectra of a 1D array of LiCs moleculeson an optical lattice: NNA (dashed lines) and exact solutions(solid lines). The shaded regions encapsulate the bands of thecontinuum two-exciton states. (c)θ -dependence of the biex-citon binding energy ∆. The electric field magnitude is 6.88kV/cm, θ0 = arccos√2/3, θ ∗ = arccos√1/3. (d) Biexcitonwave function vs the lattice site separation |r|= |n−m| of thetwo excitations for K = 0. Inset: Mean width of the biexci-ton wave function 〈r〉 calculated as the width of ψ2K(r) at halfmaximum. Numbers on each curve indicate the value of D/2J. 78Figure 3.3 The rotational energies of a closed-shell polar molecule as afunction of the strength of a DC field. The dashed lines repre-sent other rotational states with MN 6= 0. . . . . . . . . . . . 81viiiFigure 3.4 Population dynamics for the transition from |g〉 → | f 〉 exci-ton (middle panel) and from an f state localized on a sin-gle molecule (lower panel) to coherent |g〉 → |e〉 excitons andbiexcitons. The green dashed curves denote the population ac-cumulated in the pairs of non-bound |g〉 → |e〉 exciton states,the red solid curves the biexciton state and the blue dot-dashedcurves the f state. The shaded region in the upper panel en-capsulates the band of the continuum two-exciton states. Thecalculation is for a 1D ensemble of Nmol = 501 LiCs moleculeson an optical lattice with lattice separation a = 400 nm. Theelectric field of magnitude 6.88 kV/cm is perpendicular to themolecular array. . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.5 Three-excitation spectra of a 1D array of molecules on an op-tical lattice. The calculation is done for a system of 20 latticesites with the hopping interaction J = 10 kHz and the dynamicinteraction D = 30 kHz. The black dots represent energies ofall three-exciton states, the red curves denote the boundaries ofenergy continuum of three free excitons, and the blue curvesrepresent the boundaries of energy continuum for a biexcitonplus a free exciton. . . . . . . . . . . . . . . . . . . . . . . . 88Figure 4.1 The dispersion curve of an exciton. The interaction betweensite m and site n is proportional to 1/|n−m|3. . . . . . . . . 96ixFigure 4.2 Example of controlled energy transfer in a one-dimensionalarray of quantum monomers subjected to a linear phase trans-formation. The graph illustrates the evolution of the excitonwave packet centered at k = 0 and initially positioned at thecenter of the array. The phase of the wave function is shownby color. The brightness of color corresponds to the amplitudeof the excitation with white color corresponding to zero am-plitude. The calculation is for a one-dimensional array of 201monomers with α = 22.83 kHz and ∆Ee−g = 12.14 GHz, andthe linear phase transformation Φn ' Φ0− 1.29n. The corre-sponding experimental setup is illustrated in Fig. 4.6 . . . . . 99Figure 4.3 “Bloch oscillation” of the exciton wave packet in the momen-tum and coordinate spaces. The phase of the wave functionis shown by color as in Fig. 4.2. A low laser intensity of106 W/cm2 is used and all other parameters are the same asin Fig. 4.2. Part (a) shows that the wave packet moves in kspace in response to the linear laser field. However, since thewave vector is limited in the first Brillouin zone, when thewave packet reaches −pi or pi , it disappears at the boundaryand reappears at the other boundary. Part (b) presents the mo-tion of the wave packet in coordinate space in accordance withthe phase kicking in k space. Note that the wave packet spreadsin both the momentum and coordinate spaces because the laserintensity profile along the molecular array is not perfect andthis is amplified over an extended time period. . . . . . . . . 100xFigure 4.4 Focusing of a completely delocalized collective excitation (pan-els a and b) and a broad Gaussian wave packet of Frenkel ex-citons (panels c and d) in a one-dimensional array using thequadratic phase transformations at t = 0 as described in Sec-tion 4.3.2 and Section 4.3.3. In panels (b) and (d), the excita-tion probability distribution is displayed by color. The dashedlines show the initial distribution magnified by 20 and 5 re-spectively in (a) and (c). The solid curves in panels (a) and(c) correspond to two different phase transformations focusingthe same wave packet onto different parts of the array. Thecalculations are performed with the same parameters α , a, and∆Ee−g as in Fig. 4.2. The results are computed with all cou-plings accounted for. . . . . . . . . . . . . . . . . . . . . . . 111Figure 4.5 Focusing of a delocalized excitation in a 2D array shown att = 0 in panel (a) onto different parts of the lattice (panels b–d).For better visualization, the probability distribution in panel (a)is magnified by a factor of 60. The calculations are performedwith the same parameters α , a, and ∆Ee−g as in Fig. 4.2 andthe quadratic phase transformation at t = 0. . . . . . . . . . . 112Figure 4.6 The experimental setup for the calculation corresponding toFig. 4.2. A 1D array of LiCs molecules trapped on an opticallattice with lattice constant a = 400 nm is subjected to a ho-mogeneous DC field of 1 kV/cm directed perpendicular to theintermolecular axis. The kicking potential leading to the phasetransformation presented by Eq. (4.54) can be provided by aλ = 1064 nm Gaussian laser beam that is linearly polarized inthe direction of the DC field, with the propagation directionalong the array axis, focused to a radius of 5 µm, with the in-tensity at the focus equal to 107 W/cm2. The laser pulse is onbetween 0 and 3µs. . . . . . . . . . . . . . . . . . . . . . . 118xiFigure 4.7 An illustration to show the orientations of 1D and 2D molec-ular arrays inside the Gaussian beam. (a) the 1D lattice liesalong the x-axis of the z = 0 plane and the DC field is at someangle with the x-axis such that the coupling α between moleculesis negative. (b) the 2D square lattice is at the center of the z = 0plane and the DC field is at 45◦ degree with the x-axis. In bothcases, the laser is linearly polarized along the direction of theDC field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 4.8 Control of excitation transfer in a 1D many-body system withdipolar interactions by varying the direction of an external elec-tric field. (a) Exciton dispersion curves for a 1D ensembleof diatomic molecules on an optical lattice for different an-gles θ between the direction of the external DC electric fieldand the axis of the molecular array. In 1D, the coupling α ∝(1/3− cos2 θ). (b) Propagation of a wave packet centered atak = −pi/3 controlled by tuning the electric field direction.The thin dotted line depicts the corresponding angle variationswith time. The brightness of the color corresponds to the prob-ability of the excitation. . . . . . . . . . . . . . . . . . . . . 126Figure 4.9 Control of excitation transfer in a 2D many-body system withdipolar interactions by varying the direction of an external elec-tric field. Panels (a) and (b) show the trajectories of the centerof an exciton wave packet in a 2D lattice during the time from0 to 3 ms; Panels (c) and (d) represent the changing of thedressing DC field orientation (θ ,φ) associated with (a) and (b)respectively. The initial wave packet is a 2D Gaussian distribu-tion centered around akx = aky = pi/2 and has a width of ∼60lattice sites in coordinate space. The magnitude of the DC fieldis fixed to 6 kV/cm while its direction is changing. The calcu-lations are done for a 2D array of LiCs molecules in a latticewith a = 400 nm. . . . . . . . . . . . . . . . . . . . . . . . 128xiiFigure 4.10 The orientations of the DC field and molecular arrays in thecoordinate systems. (a) The DC field is along the z-axis andthe 1D molecular array is in the direction represented by θRand φR. (b) The 2D molecular array is on the XY plane and theorientation of the DC field is represented by (θ ,φ). For clarity,we have only drawn the molecules (as blue dots) along a par-ticular axis which is at angle γ with respect to the X-axis. It isto be understood that there are also other intermolecular axesat different angles with the X-axis. Note part (a) and part (b)have different coordinate systems and the meaning of (θR,φR)is different from that of (θ ,φ). In fact, the angle θR betweenthe DC field and the molecular array in (b) is related to θ andφ by cosθR = cosθ cos(φ − γ). . . . . . . . . . . . . . . . . 130Figure 4.11 Enhancement factors η (red symbols) and χ (blue symbols)as functions of vacancy percentage in a 2D lattices. See textfor the definitions of η and χ . The error bars are for 95% ofconfidence interval. . . . . . . . . . . . . . . . . . . . . . . 138Figure 4.12 Time snapshots of a collective excitation in a 2D array witha vacancy concentration of 10 % (a) The distribution of thevacant sites; (b) The initial probability distribution of the ex-cited state; (c) The probability distribution of the excitation atthe focusing time when the focusing scheme is applied. Thefocusing time is found numerically as the time when the prob-ability at the target molecule (71, 71) reaches maximum fora given phase transformation. (d) The probability distributionof the wave function at the focusing time when the focusingscheme is not applied. The calculations are performed withthe same parameters as in Fig. 4.5. The probabilities in (b) and(d) are magnified by 16 and 6, respectively. . . . . . . . . . . 140xiiiFigure 4.13 Focusing of a collective excitation in a strongly disordered sys-tem with 60% of lattice sites unoccupied. Panel (a) shows dif-ferent phases applied to different blocks of the lattice beforethe time evolution. (b) The initial probability distribution ofthe excited state. (c) The probability distribution of the excitedstate at the focusing time T = 3 ms with the phase transfor-mation depicted in panel (a) before the time evolution. (d) Theprobability distribution of the excited state at the focusing timeT = 3 ms with no phase transformation applied. The calcula-tions are performed with the same parameters as in Fig. 4.5.The probabilities in (b) and (d) are magnified by 60 and 5, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure 4.14 Efficiency of focusing collective excitations in strongly dis-ordered 2D lattices. The molecular array is divided into 400blocks as shown in Fig.4.13(a). The focusing time t∗ is arbi-trarily set to 4 ms. For each realization of disorder, we useEq.(4.99) with T = t∗ to find the phase mask applied to differ-ent blocks. Shown are the enhancement factors η (red sym-bols) defined in Eq.(4.90), as a function of the vacancy per-centage. The error bars are for 95% confidence interval. . . . 144Figure 5.1 Green’s function G(10,71,50,51,E + iη) with η = 0.1 as afunction of energy E. The calculation was done for a finite or-dered 1D crystal of 101 lattice sites with the nearest-neighborapproximation. The energies of the particles en were set tozero, and the dynamic interaction and the hopping interac-tion were set to 5. The upper panel shows the real part ofthe Green’s function and the lower panel shows the imagi-nary part. The results calculated by the brute-force methodare marked with empty circles while the results obtained byour recursive method are marked by “X”. This figure clearlydemonstrates that the recursive method produces the same re-sults as the brute-force method. . . . . . . . . . . . . . . . . 164xivFigure 5.2 Green’s function G(33,90,50,51,E + iη) with η = 0.1 as afunction of energy E. The calculation was done for a finite dis-ordered 1D crystal of 101 lattice sites with the nearest-neighborapproximation. The dynamic interaction and the hopping in-teraction were set to 5, and allow the energies of the particles ento fluctuate in range [−5,5] randomly. The upper panel showsthe real part of the Green’s function and the lower panel showsthe imaginary part. The results calculated by the brute-forcemethod are marked with empty circles while the results ob-tained by our recursive method are marked by “X”. This figureclearly demonstrates that the recursive method produces thesame results as the brute-force method. . . . . . . . . . . . . 165xvAcknowledgmentsFirst of all, I would like to thank my supervisor, Prof. Roman Krems for acceptingme into his group and giving me helpful guidance throughout my PhD study. AlsoI want to thank Prof. Yan Alexander Wang, Prof. Mark Thachuk and Prof. GrenPatey for their help during my time in Vancouver.I truly appreciate fruitful discussions and collaborations with Marina Litin-skaya, Evgeny Shapiro and Felipe Herrera. In addition, I want to thank Yu Zhang,Yakun Chen and Zhiying Li for their useful advice on life and career.Especially, I am grateful to my wife for her love, support and encouragement.She brings a lot of happiness to my life and I am very lucky to have such a lovelyand smart life partner. And last but not least, I thank my father for inspiring myinterest in science since my childhood, and my mother for being supportive allthese years. I owe too much to my parents for their everlasting love and support.xviTo my parents and my wife.xviiChapter 1IntroductionThis thesis is a theoretical study on controlling the quantum dynamics of quasipar-ticles in periodic and disordered lattice potentials. Although some work presentedin the thesis is quite general and can be applied to any system with a periodic ordisordered lattice potential, the main systems considered here are ultracold atomsand molecules, as their experimental manipulation is relatively easy. The controlschemes proposed in the thesis are also related closely to recent developments inthe laser cooling and trapping of atoms and molecules. Therefore, to give thereader a broad context in which to understand the thesis, this chapter presents anintroductory overview of the field of ultracold atoms and molecules. In Section 1.1,I briefly introduce ultracold atoms and molecules and discuss the unique featuresof ultracold ensembles. After that, in Section 1.2, I discuss the controllability ofultracold molecules. In Section 1.3, I introduce a particular ultracold system withpolar molecules trapped on optical lattices. Finally, I outline the research presentedin the thesis in Section Implications of ultracold temperaturesCooling gaseous ensembles of atoms and molecules to extremely low temperatureshas revolutionized the field of atomic, molecular and optical physics. The develop-ment of experimental techniques for cooling atoms to ultracold temperatures hasled to many ground-beaking applications[1, 2], including the fundamental studies1of Bose−Einstein condensation of weakly interacting particles[3], nonlinear andquantum atom optics[4], collisions at ultracold temperatures[5], the developmentof all-optical atomic clock[6], and quantum information processing with atomsand photons[7]. Inspired by this success with ultracold atoms, researchers are nowaiming to extend the experimental methods to cooling molecules. The experimen-tal work with ultracold molecules is likewise expected to have a great impact ondifferent areas of science and technology, which is covered by numerous reviewarticles[8–17]. By convention, a distinction is made between two ranges of tem-perature T : cold 1 mK < T < 1 K, and ultracold T < 1 mK. At ultracold tempera-tures, the translational motion of particles has a velocity that can be as low as a fewcm/s, in sharp contrast with the speed of a few hundred m/s at room temperatures.The small kinetic energies associated with the ultra-slow translational motion havesome important consequences, summarized below.First, ultracold temperatures allow for novel macroscopic quantum states ofmatter. A particle in the gaseous phase behaves as a quantum-mechanical wavepacketwith the extension on the order of its de Broglie wavelength, given byΛ=hmv. (1.1)For the ultra-slow velocity v at ultracold temperatures, the corresponding de Brogliewavelength is larger than at room temperatures by several orders of magnitude,and is comparable to or larger than the mean distance between particles in the gasphase. Under these conditions, the particles in a gaseous ensemble become indis-tinguishable and their wavefunctions overlap heavily, leading to the formation ofa macroscopic coherent matter wave, that is, a Bose-Einstein condensate[3, 18]for bosonic particles in which all particles occupy the lowest quantum state, anda quantum degenerate Fermi gas[19, 20] for fermionic particles of same kind inwhich every accessible energy level is occupied by exactly one particle due to thePauli exclusion principle.Second, ultracold temperatures lead to new collision phenomena. Because ofthe large de Broglie wavelengths of ultracold particles, their low-energy collisionis qualitatively different from the collision at thermal temperatures. At ultracoldtemperatures, particles do not have well-defined trajectories in phase space, the2spherical harmonics Ylm(θ ,φ), also called partial waves with l = 0,1,2, · · · thatrepresent the angular momenta of the rotational motion of the collision complex,are commonly used to describe the collisions. In the absence of external fields, asT → 0, the quantum threshold regime is reached and only the lowest allowed partialwave contributes to the collision[21, 22]. For collisions between reactive particlesat ultracold temperatures, the quantum phenomena are even more dramatic. As thekinetic energy is so small, the tunneling under reaction barriers becomes the dom-inant mechanism of chemical reactions, giving temperature-independent reactionrate constants that can be very large at zero Kelvin[23–26]. Moreover, quantumstatistics governs reactive collisions at ultracold temperatures. For example, de-pending on whether the colliding molecules are in the same internal quantum state,the collisions can be dominated by s-wave collisions or s-wave and p-wave colli-sions, giving very different reaction rates[27].Third, ultracold temperatures enable numerous control possibilities. It has beena long-sought goal to control atomic and molecular dynamics by utilizing theirinteractions with external fields. However, at normal temperatures, the thermalmotion of atoms and molecules randomizes their encounters and diminishes theeffect of external fields. As the temperature approaches a few µK, the kineticenergy of ultracold molecules becomes so small that it is even less than the en-ergy splittings of hyperfine structures of molecules. In this case, an external fieldof moderate strength can produce a perturbation in energy that is larger than thetranslational energy of molecules, creating dramatic changes in the collisions prop-erties of molecules[22]. Normally, ultracold particles are obtained by first coolingthem to cold temperatures and then loading them into a trap for further evaporativecooling[28]. This inspired the development of a variety of traps, such as electro-static traps[29], AC electric traps[30], optical and microwave traps[31, 32], andmagnetic and magneto-optical traps[33–35]. Within the traps, the thermal motionof particles is restricted, so their interaction with external fields can be more pre-cisely controlled. This leads to unprecedented precision of spectroscopic measure-ment because of the much longer interrogation time[36–38]. Because the shapesof the traps can be changed, the trapped ultracold atoms and molecules can be con-fined in restricted geometries. This may be used for quantum simulations of funda-mental many-body systems[17] and various setups for quantum computing[39–45].31.2 External field control of ultracold moleculesPerhaps the most important feature of ultracold atoms and molecules is their con-trollability via interactions with external fields. Since the corresponding researchfield is so vast, as reviewed by Ref.[46], we limit the discussion to ultracold moleculesas they are more relevant for our research. Compared with atoms, molecules haveadditional vibrational and rotational degrees of freedom, which leads to rich con-trol possibilities. In the following, I briefly outline the possibilities from the aspectof single molecule and from the aspect of intermolecular interaction separately.First of all, the internal states of ultracold molecules can be selected by theirinteraction with external fields. This is very important because it enables the pro-duction of a near quantum degenerate gas of molecules in their rovibronic groundstate[47, 48], which provides a basis for other research that requires a supply ofultracold molecules. Recently, it has been demonstrated that a rovibronic ground-state molecular quantum gas can be prepared in a single hyperfine state or in anarbitrary coherent superposition of hyperfine states[27] by using a two-photon mi-crowave transition in the presence of a magnetic field. This control over hyperfinestructures, together with the manipulation of electronic, vibrational, and rotationalstates, provides the full control over all internal degrees of freedom of molecules.As quantum information can be encoded into the internal states of molecules, theability to manipulate the internal states constitutes the basis of quantum computa-tion. For instance, as presented in Ref.[49], universal quantum gates can be im-plemented using the hyperfine levels of ultracold heteronuclear polar moleculesin their electronic, vibrational, and rotational ground state. These high-fidelitylogic gates, driven by microwave pulses between hyperfine states, offer versatileencoding possibilities as a consequence of the richness of the energy structuresand the state mixing of molecules in external fields. The precise control of inter-nal states also facilitates quantum simulation of many-body systems using ultra-cold molecules. For example, it has been shown that hyperfine structures of polarmolecules in an one-dimensional lattice can be controllably accessed and manipu-lated as a resource for generating complex quantum dynamics [50].Moreover, the permanent electric dipole moment of polar molecules enablesthe tuning of long-range intermolecular interactions. The electric dipole moment4describes the separation of positive and negative electrical charges in a moleculeand is given byµ =∑nqnrn , (1.2)where rn gives the positions of charges qn and the sum is over all charges inthe molecule including the electrons and nuclei. For molecules in a particularelectronic and vibrational state, it is convenient to define the dipole operator asd = 〈ψ|µ |ψ〉 by integrating over the electronic and vibrational wavefunction ψ .The dipole moment operator only couples states with different parity. Take a sim-ple 1Σ molecule for example, its rotational eigenstate |J,M〉 has a well-definedparity determined by the angular momentum quantum number J in the followingway:P|J,M〉= (−1)J|J,M〉 , (1.3)where P is the parity operator. It follows that molecules in a rotational eigen-state have no net dipole moment as 〈J,M|d|J,M〉 = 0 in the laboratory frame. Ifmolecules are prepared in their electronic, vibrational and rotational ground state,they possess no net dipole moment and interact at large intermolecular distancesvia a van der Waals attraction which falls off as 1r6 with respect to the interparticleseparation r. With the application of electric fields that mix rotational states ofdifferent parity, static or oscillating dipoles are induced in molecules, leading tononzero interaction – dipole-dipole interaction, which falls off as 1r3 and dominatesover the van der Waals interaction at long range. The dipole-dipole interactioncan be modified by external fields to change the strength and shape of the interac-tion potential between polar molecules[44, 51]. This tunability, combined with theanisotropy characteristic of the dipole-dipole interaction, has inspired a large bodyof research on the condensed matter theory of dipolar quantum gases[17, 52–54].1.3 Ultracold polar molecules on optical latticesIn the past few years, our group has been interested in ensembles of ultracold po-lar molecules trapped on optical lattices. Optical lattices are periodic potentialsformed by the standing wave patterns of laser beams[55, 56]. The simplest op-tical lattice is an one-dimensional lattice created by superimposing two counter-5propagating laser beams with the same frequency, intensity and polarization. If thetwo laser beams propagate in the +z and −z directions respectively, the resultingtotal electric field is given byE(z, t) = E0 cos(kz−ωt)eˆ+E0 cos(−kz−ωt)eˆ= 2E0 cos(ωt)cos(kz)eˆ , (1.4)where eˆ is the polarization vector. This oscillating electric field induces an oscillat-ing dipole moment in a particle and at the same time interacts with the dipolemoment, creating a time-averaged periodic potential V (z) ∝ E20 cos2(kz) whosetroughs can be used as microtraps in the z direction. The depth of those micro-traps is proportional to the light intensity and their effective sizes can be as smallas several tens of nm. If the two laser beams are Gaussian beams, their Gaussianintensity profiles in the xy plane will provide an additional weak radial confinementpotential. This can be combined with the periodic standing wave pattern to form anone-dimensional array of disk-like trapping potentials. Optical lattices with higherdimensions can be created by using additional pairs of laser beams. For example,adding a pair of counter-propagating laser beams along the x direction to the cur-rent one-dimensional lattice can produce a two-dimensional array of traps, eachof which looks like a tube in the y direction. These tube-like potentials can bebroken into small segments along the y direction by adding a third pair of laserbeams in that direction. This leads to a three-dimensional periodic potential. Inaddition to dimensionality, the geometry of optical lattices can also be changed byadjusting the angles between different pair of laser beams. For instance, if threepairs of laser beams are arranged in the xy plane with an angle 2pi/3 with respectto each other and they are all polarized along the z direction, the resulting latticepotential V (x,y)∝ 3+4cos(3kx/2)cos(√3ky/2)+2cos(√3ky) is a triangular lat-tice in two dimensions. Apart from dimensionality and geometry, the trap depthsand the lattice separations of optical lattices can also be controlled by changing thelaser intensities and wavelengths. This kind of flexibility doesn’t exist in naturalsolid state crystals. Moreover, compared with natural solid state crystals with thelattice constants on the order of a few angstroms, optical lattices have much largerlattice site separations on the order of an optical wavelength (micron), which open6the possibility to address individual sites in optical lattices[57, 58]. Due to allthe above features, optical lattices with polar molecules trapped inside, provide avery versatile platform for studying controllable many-body phenomena[17] andfor implementing new schemes of quantum computing[59–61].The low-energy physics of particles trapped in an optical lattice can be de-scribed by the Hubbard HamiltonianH =− ∑i, j,σJσi j b†i,σb j,σ + ∑i, j,σ ,σ ′Uσσ′i j2b†i,σbi,σb†j,σ ′b j,σ ′ , (1.5)where b†i,σ (bi,σ ) are the creation (annihilation) operators for a particle at site i inthe internal state σ , Jσi j describes the hopping of a particle from site j to site i,and Uσσ′i j is the onsite (i = j) or offsite (i 6= j) interaction between the particle atsite i in state σ and the particle at site j in state σ ′. By tuning the trap depth,the ratio between J and U can be controlled. When the particle number is com-mensurate with the number of lattice sites and the trap depth is deep enough suchthat JU , a phase transition from a superfluid phase or a Fermi liquid to a Mottinsulator can be observed[62–66]. In the Mott insulator phase, the tunneling ofparticles between sites is suppressed and the number of particles at a site is aninteger value. For the system that our group is interested – ensembles of polarmolecules trapped on optical lattices, a lot of effort has been devoted to explor-ing exotic quantum phases (see Ref.[17] and the references therein) because ofthe long-range and anisotropic character of the dipole-dipole interaction betweentrapped polar molecules. In those studies, many novel phase diagrams, associatedwith different rates of particle tunneling and various types of interparticle interac-tions, have been discovered.We instead focus on the Mott insulator phase and explore the tunability of thesystem of ultracold molecules in the context of collective excitations. The formof the relevant Hamiltonian is still the same as Eq. (1.5), but the creation and an-nihilation operators are defined with respect to the quasiparticles – the molecularexcitations rather than the real particles. So J now describes the propagation ofthe excitation energy in the crystal and U represents the interaction between twoor more molecular excitations. This is a new line of research which may be ex-7ploited to study collective excitations in new interaction regimes. For example,previous work by our group members has demonstrated that the artificial crystalsof polar molecules in the Mott insulator phase can be used to investigate control-lable exciton-impurity interactions[67], to control collective spin excitations[68],to engineer open quantum systems with tunable coupling to the bath[69], and tostudy new polaron models[70].1.4 Thesis outlineThe current thesis extends the work on collective excitations in several new direc-tions. Here is the overview of the thesis.Chapter 2 introduces the background material that can help the reader to under-stand the thesis. I first give a review of angular momentum theory, and then applythe theory to obtain the rotational levels of a diatomic molecule in external fieldsand to calculate the dipole-dipole interaction. After that, I present a brief introduc-tion to the exciton theory. More specifically, I explain the commutation relationsfor excitons, derive the exciton Hamiltonian in second quantization, and introducethe Heitler-London approximation.Chapter 3 investigates the interactions between multiple collective excitations,and in particular the binding mechanism of these excitations. We study a particularkind of quasiparticles – rotational Frenkel excitons in a periodic lattice potential.These quasiparticles are the collective excitonic modes of polar molecules trappedon an optical lattice in the Mott insulator phase, and are induced by the dipole-dipole interaction which couples the rotational states of different molecules. Weshow that the application of a moderate electric field, through mixing rotationalstates of different parity, can give rise to a non-linear dynamic interaction D be-tween the rotational excitons. For the system considered here, the dynamic inter-action is always attractive and its strength can be tuned by the external DC field.This leads to controllable formation of biexciton states with tunable binding energy,as demonstrated numerically for a 1D array of LiCs molecules on an optical lat-tice. We also obtain the two-excitation spectra of the rotational excitons and deriveanalytical expressions for the wavefunction of biexciton states using the nearest-neighbor approximation. In an effort to extend the theoretical model of exciton8binding, we calculate three-excitation spectra of rotational excitons and observethe three-body bound states of the excitons. To make our theoretical study of biex-citon states more relevant to experiments, we propose an nonoptical way to createthe rotational biexciton states, avoiding the difficulty associated with direct exci-tation. This method makes use of the resonance between the high-energy (N = 2)excitonic states and the biexciton states of low-energy (N = 1) excitons and canproduce biexcitons with high efficiency.Chapter 4 focuses on the propagation properties of excitations. This chapterproposes a new scheme to control the energy transfer in ordered and disorderedcrystals. We show that the energy transfer through an array of coupled quantummonomers can be controlled by applying a transient external potential which mod-ifies the phase of the quantum states of the individual monomers. The success ofthe method relies on two very different time scales in the quantum evolution of themany-body states, namely the fast time scale corresponding to the excitation of asingle monomer and the slow one associated with the excitation hopping betweenmonomers. Because of these two different time scales, it is possible to find a suit-able local perturbation to a single monomer that is adiabatic with respect to themonomer excitation but is sudden with respect to the excitation hopping. In an or-dered crystal, if such perturbations are applied to give different phases to differentmonomers, the momentum of the collective excitation is modified and its propaga-tion behavior is influenced as well. Our work shows that different phase transfor-mations can accelerate or decelerate quantum energy transfer and spatially focusdelocalized excitations onto different parts of those ordered arrays. On the otherhand, for a completely disordered array, random scattering at numerous lattice sitesdisturbs the above phase transformations. In this case, inspired by the “transfer ma-trix” methods for focusing a collimated light beam in an opaque medium[71–80],we develop another scheme of phase transformation that can achieve effective fo-cusing of a delocalized excitation in the presence of strong disorder. To make con-nection with the current study of ultracold molecules, we also consider possibleexperimental implementations of the proposed technique in an array of ultracoldatoms or molecules trapped on an optical lattice, and demonstrate the feasibility ofthe proposed phase transformations.9Chapter 5 extends the recursive method[81–83] for computing the Green’sfunctions for particles in periodic potentials to calculate two-particle Green’s func-tions in disordered crystals. This provides a powerful tool to study the quantumdynamics of quasiparticles in a disordered potential. This chapter shows that bygrouping Green’s functions into different set of vectors, the equation of motion ofGreen’s function can be rewritten as a recursion relation that links three consecu-tive vectors. Provided that certain boundary conditions are assumed, the recursionrelation enables the recursive calculations of Green’s functions. I present the re-cursive method in the form of a generic algorithm which can be easily adapted tosystems with long-range interactions and high dimensionalities, and describe itsadvantage over the conventional methods to calculate Green’s functions. As an ap-plication of the method, I propose to use the Green’s function to study the tunnelingof biexciton states through impurities.Chapter 6 gives a summary of the thesis, points out the limitations of the currentresearch, and comments on future research directions.10Chapter 2Background materialThis chapter briefly outlines the background material that is directly related to theresearch in later chapters.2.1 Review of angular momentum theoryIn this section, we briefly review the theory of angular momentum. Instead ofgiving a comprehensive picture as various books[84–92] have already done, wetake a pragmatic approach that is more concerned with intuitive understanding,discussing only the concepts and ideas that are relevant to research in this thesis.2.1.1 Symmetry groupsThe theory of angular momentum is essentially the study of symmetry under rota-tions in quantum mechanics. As group theory is an important tool that is used todetermine symmetry, we introduce the concepts of group theory in this subsection.A set of operations {A,B,C, · · ·} is called a group if it satisfies the followingproperties:• there is an identity operation I in the set such that AI = A• each operation A has an inverse operation A−1 in the same set such thatAA−1 = A−1A = I• the multiplication of two operations is also an operation in the same set11• operations are associative such that (AB)C = A(BC)2.1.2 Rotation operators and spherical harmonicsAs the theory of angular momentum concerns symmetry under rotations, we startby defining the rotation operator in quantum mechanics. Given a rotation R, weassociate a rotation operator D(R) that transforms a state from the original systemto the rotated system|ψ〉R = D(R)|ψ〉 . (2.1)To construct the rotation operator, we denote the rotation operator D(nˆ,dφ) forinfinitesimal rotation by angle dφ about an axis nˆ. Because D(nˆ,dφ) must be linearin dφ for small rotation angle and becomes the identity operator when dφ = 0, itis useful to define the angular momentum operator Jnˆ for rotations about the axis nˆasJnˆ ≡ ih¯ limφ→0D(nˆ,φ)−1φ , (2.2)such thatD(nˆ,dφ) = 1− i(Jnˆh¯)dφ . (2.3)From the above equation, we can regard Jnˆ as the generator of rotation in the sensethat Jnˆ produces the increment of a general function f due to a rotation R by dφabout the axis nˆ, that is,− idφ(Jnˆh¯)f = D(nˆ,dφ) f − f . (2.4)By compounding successively infinitesimal rotations about the same axis, we canobtain the finite rotation operatorD(nˆ,α) = limm→∞[1− i(αm)(Jnˆh¯)]m= exp(−iαJnˆh¯)= 1−iαJnˆh¯−12!α2J2nˆh¯2+ · · · . (2.5)12where in the second step the following equation,limx→∞[1+(1x)]x= e , (2.6)was used. It is clear that the set of all rotation operations satisfy the four conditionsof a group (see Section 2.1.1). Correspondingly, the infinite set of rotation opera-tors also satisfy the same conditions and thus form a group which is called the fullrotation group.Rotations have the following important property: rotations about the same axiscommute whereas rotations about different axes do not. This property leads tothe commutation relation of the angular momentum operator J and its Cartesiancomponents. To illustrate this point, let’s consider rotations Rx, Ry, and Rz about thex, y and z axes respectively. In coordinate space, these rotations can be representedby 3× 3 matrices. For instance, the rotation by angle φ about the z axis can beexpressed asRz(φ) =cosφ −sinφ 0sinφ cosφ 00 0 1 . (2.7)In the case of an infinitesimal angle φ → ε , the above equation can be written asRz(ε) =1− ε22 −ε 0ε 1− ε22 00 0 1 . (2.8)By comparing the effect of a y-axis rotation followed by an x-axis rotation withthat of an x-axis rotation followed by a y-axis rotation, we can show thatRx(ε)Ry(ε)−Ry(ε)Rx(ε) = Rz(ε2)− I . (2.9)Assuming the corresponding rotation operators satisfy a similar equation as Eq. (2.9)and making use of Eq. (2.3), we obtain[Jx,Jy] = ih¯Jz . (2.10)13Repeating the above analysis for other axes, we obtain the commutation relationsfor the components of angular momentum[Ji,J j] = ih¯εi jkJk , (2.11)where i, j, k can be any one of x, y and z, and εi jk is defined asεi jk =1 if (i, j,k) is an even permutation of (x,y,z)−1 if (i, j,k) is an odd permutation of (x,y,z)0 if two or more indices are equal. (2.12)Many important properties follow from the angular-momentum commutationrelation represented by Eq. (2.11). For example, by making use of Eq. (2.11), wecan show that the operator J2 defined byJ2 ≡ J2x + J2y + J2z , (2.13)commutes with any one of Jx, Jy and Jz, namely,[J2,Jk]= 0, (k = x,y,z) . (2.14)Since J2 and Jz commute, the eigenstates of J2 can be chosen to be also the eigen-states of Jz. We denote these states by | j,m〉 such thatJ2| j,m〉 = j( j+1)h¯2| j,m〉 ,Jz| j,m〉 = mh¯| j,m〉 . (2.15)To determine the allowed values for j and m, it is convenient to work with thenon-Hermitian operatorsJ± = Jx± iJy , (2.16)where J+ is called the raising operator and J− is called the lowering operator. J±14satisfy the following commutation relations[J+,J−] = 2h¯Jz ,[Jz,J±] = ±h¯J± ,[J2,J±]= 0 , (2.17)all of which can easily be derived from Eq. (2.11).As a special case of the angular momentum operator, the orbital angular mo-mentum operator L = r× p has components that satisfy the same commutationrelations[Li,L j] = ih¯εi jkLk . (2.18)Using the fact that momentum is the generator of translationT (dx)|r〉=[1− i(ph¯)·dx]|r〉= |r+dx xˆ〉 , (2.19)we have[1− i(dφh¯)Lz]|x,y,z〉=[1− i(dφh¯)(xpy− ypx)]|x,y,z〉=[1− i( pyh¯)xdφ + i( pxh¯)ydφ]|x,y,z〉= |x− ydφ ,y+ xdφ ,z〉 . (2.20)The above equation shows exactly the effect of an infinitesimal rotation about thez axis, as one would expect. For an arbitrary state |ψ〉, an infinitesimal rotationabout z axis changes the wavefunction 〈x,y,z |ψ〉 to〈x,y,z|[1− i(dφh¯)Lz]|ψ〉= 〈x+ ydφ ,y− xdφ ,z |ψ〉 . (2.21)By changing to spherical coordinates (r,θ ,φ), the above equation can be converted15to〈r,θ ,φ |[1− i(dφh¯)Lz]|ψ〉= 〈r,θ ,φ −dφ |ψ〉= 〈r,θ ,φ |ψ〉−dφ ∂∂φ 〈r,θ ,φ |ψ〉 . (2.22)Therefore, we obtainLz =−ih¯∂∂φ . (2.23)In the way, we can also derive the expressions for the other orbital angular momen-tum operators, givingLx = −ih¯(−sinφ ∂∂θ − cotθ cosφ∂∂φ),Ly = −ih¯(cosφ ∂∂θ − cotθ sinφ∂∂φ). (2.24)The eigenfunctions of L2 and Lz are known as the spherical harmonics, whichare given by〈nˆ|l,m〉= Ylm(θ ,φ) , (2.25)where θ and φ specify the orientation of nˆ. The dependence of spherical harmonicson angles can be separated so thatYlm(θ ,φ) =Θlm(θ)Φm(φ) , (2.26)whereΦm(φ) =√12pi exp(imφ) , (2.27)andΘlm(θ) = (−1)m[2l +12(l−m)!(l +m)!]1/2Pml (cosθ) , (2.28)for m≥ 0 andΘlm(θ)= (−1)mΘl−m(θ) for m< 0, and Pml (cosθ) are the associatedLegendre polynomials. Sometimes, it is more convenient to work with modified16spherical harmonics Clm defined byClm(θ ,φ) =√4pi2l +1Ylm(θ ,φ) . (2.29)2.1.3 Rotation matricesSince an arbitrary rotation can be decomposed into rotations about three coordinateaxes (x, y, and z axis), it is usually convenient to express the orientations of a bodyin terms of rotations about some fixed axes. For this to work, we have to choosean initial orientation as a reference on which the rotations are operating. For avector described by the two spherical polar angles θ and φ , we use a referenceorientation parallel to the space-fixed Z axis. It is obvious to see that a rotation ofthis reference orientation through an angle θ about the space-fixed Y axis and arotation by an angle φ about the space-fixed Z axis can reproduce the orientationof the vector. For a general body, three angles φ , θ and χ are needed to describean arbitrary rotation. We attach a second coordinate system (x,y,z) to the bodyand refer to it as the body-fixed axis system. This axis system is called body-fixed as by construction it moves and rotates along with the body as a whole. Incontrast, the original coordinate system (X ,Y,Z) is fixed in space, so it is called thespace-fixed axis system. To describe the orientation of a body, we imagine that thebody is initially at a position with its body-fixed axis system coincident with thespace-fixed axis system, and then we carry out the following three rotations:1. rotate by χ about the space-fixed Z axis,2. rotate by θ about the space-fixed Y axis,3. rotate by φ about the space-fixed Z axis.These three rotations can produce arbitrary orientation of the body. For an intuitiveunderstanding of the Euler angles, we link the orientation of the body directly withφ , θ and χ . It can be shown that θ and φ can be used to define the orientationof the body-fixed z axis in the space-fixed axis system, in the same way as θ andφ are used to define the orientation of a vector. The angle χ measures a rotation17about the body-fixed z axis. It is an azimuthal angle about the z axis just as φ is anazimuthal angle about the Z axis.Based on Eq. (2.5), the rotation operator that corresponds to the three Eulerangles can be written asD(φ ,θ ,χ) = exp(−iφJZh¯)exp(−iθJYh¯)exp(−iχJZh¯). (2.30)Because of the closure property of the full rotation group, the multiplication ofthree rotation operators in Eq. (2.30) is equivalent to another rotation operatorD(R) = exp(−iαJnˆh¯)= D(φ ,θ ,χ) , (2.31)which corresponds to a rotation R through an angle α about an axis nˆ.Now we study the matrix elements of the rotation operator D(R). As a conse-quence of that fact that J2 commutes with any component of J, the rotation operatorD(R), a function of Jnˆ, also commutes with J2. Thus the eigenfunctions | j,m〉 ofJ2 are also eigenfunctions of the rotation operator and rotations don’t change thetotal angular momentum. As a result, to see the effect of a rotation on a state witha definite angular momentum, we only need to calculate the matrix elements of therotation operator between two states with the same j value, that isD jm′m = 〈 j,m′|exp(−iαJnˆh¯)| j,m〉 . (2.32)The (2 j+1)× (2 j+1) matrix formed by D jm′m in the above equation is called therotation matrix. Making use of Eq. (2.30), we can easily show thatD jm′,m(φ ,θ ,χ) = exp(−iφm′− iχm)〈 j,m′|exp(−iθJY/h¯)| j,m〉≡ exp(−iφm′− iχm)d jm′,m(θ) , (2.33)where d jm′,m(θ) is the element of a reduced rotation matrix. As the above equationshows, the reduced rotation matrix can be calculated from the matrix representation18of JY and its expression is[87]:d jm′,m(θ) = ∑t(−1)t[( j+m′)!( j−m′)!( j+m)!( j−m)!]1/2( j+m′− t)!( j−m− t)!t!(t +m−m′)!×(cosθ2)2 j+m′−m−2t(sinθ2)2t−m′+m, (2.34)where the sum over t is for all integers for which the factorial arguments are non-negative. It can be shown that d jm′,m(θ) satisfies the following symmetry relations[87]:d jm′,m(θ) = (−1)m′−md jm,m′(θ) = (−1)m′−md j−m′,−m(θ) . (2.35)To understand the physical meaning of the rotation matrix, we start from a state| j,m〉 and rotate it. Even though the rotation R doesn’t change the j value, we willgenerally expect the m value to be different after the rotation. Consequently, onewould be interested in knowing the probability of the system being found in a state| j,m′〉. By using the identity relation, I = ∑ j′,m′ | j′,m′〉〈 j′,m′|, the final state canbe written asD(R)| j,m〉 = ∑j′∑m′| j′,m′〉〈 j′,m′|D(R)| j,m〉= ∑m′| j,m′〉〈 j′,m′|D(R)| j,m〉δ j, j′= ∑m′| j,m′〉〈 j,m′|D(R)| j,m〉= ∑m′| j,m′〉D jm′,m(R) . (2.36)From the above equation, one can see that the matrix element D jm′,m(R) is sim-ply the probability amplitude for the rotated state to be found in | j,m′〉 when theoriginal state is | j,m〉.2.1.4 Properties of rotation matricesLet’s consider some important properties of the rotation matrices. From the lastsubsection, we know the basis ket | j,m〉 are orthonormal and remain so on rotation,this means that the matrices that represent rotations are unitary, that is, D−1 = D†.19More specifically, we haveD jm′,m(R−1) =(D†) jm′,m (R) =[D jm′,m(R)]∗, (2.37)where R−1 denotes the inverse of the rotation R.From Eq. (2.34), we can see the matrix elements of reduced rotation matricesare always real. Taking this fact into account, and using Eq. (2.33) and Eq. (2.35),we arrive atD j∗m′,m(R) = exp[−iφ(−m′)− iχ(−m)]d jm′,m(θ)= exp[−iφ(−m′)− iχ(−m)](−1)m−m′d j−m′,−m(θ)= (−1)m−m′D j−m′,−m(R) . (2.38)As will be seen later, this equation is very useful because it helps us to deal withthe complex conjugates of matrix elements of rotation matrices.Another important property is that the rotation matrix elements Dlm′,m reducesto spherical harmonics when l is an integer and m′ or m is zero[87], that isDlm,0(φ ,θ ,χ) = C∗lm(θ ,φ) =√2l +14pi Y∗lm(θ ,φ) ,Dl0,m(φ ,θ ,χ) = Cl−m(θ ,φ) =√2l +14pi Yl−m(θ ,φ) . (2.39)Given a general state |nˆ〉 with the unit vector nˆ with the orientation specified by(θ ,φ), we know from previous discussion that it can be constructed from the state|zˆ〉 by a rotation about y axis by angle θ and a rotation about z axis by angle φ .Thus we obtain|nˆ〉 = D(φ ,θ ,χ)|zˆ〉= ∑l′∑m′D(φ ,θ ,χ)|l′,m′〉〈l′,m′|zˆ〉 (2.40)where χ is undetermined. Multiplying both sides of the above equation by 〈l,m|,we get〈l,m|nˆ〉=∑m′Dlm,m′(φ ,θ ,χ)〈l,m′|zˆ〉 . (2.41)20Based on the definition of the spherical harmonics, 〈l,m′|zˆ〉 is just Y ∗lm′(θ ,φ) withθ = 0 and φ undetermined. For θ = 0 the associated Legendre polynomials satisfyPml (1) = δm,0, so that the spherical harmonics of Eq. (2.26) gives rise to〈l,m′|zˆ〉 = Y ∗lm(θ = 0,φ)δm,0=√2l +14pi δm,0 . (2.42)Substituting this equation into Eq. (2.41) yields Eq. (2.39).Finally, we consider the integral of rotation matrices. From Eq. (2.33) andEq. (2.34), it can shown that the integral of a rotation matrix over dω = sinθdθdφdχis a product of delta functions[92], that is∫D jm′,m(φ ,θ ,χ)dω = 8pi2δ j,0δm′,0δm,0 . (2.43)Similarly, the integral of two rotation matrices is given by[87]∫D j1∗m′1,m1(φ ,θ ,χ) D j2m′2,m2(φ ,θ ,χ)dω =8pi22 j1 +1δ j1, j2δm′1,m′2δm1,m2 . , (2.44)which describes the normalization condition for rotation matrices.2.1.5 Coupling of angular momentaSuppose a system can be divided into two parts with different angular momentumoperators J1 and J2, respectively. When the two parts of the system interact throughsome physical mechanism, J1 and J2 become coupled and we define the additionof the two angular momentum operator asJ = J1 +J2 . (2.45)It is easy to verify that J also satisfies the commutation rules of angular momentum(Eq. (2.11)) and thus the sum of two angular momentum operator is also an angularmomentum operator.In quantum mechanics, the state of a system is described by the simultaneouseigenfunctions of a complete set of commuting operators. For the current system21of angular momenta, there are two complete sets of angular momentum operators.One set is J21, J22, J1Z , and J2Z , and therefore we can use their simultaneous eigen-states | j1,m1; j2,m2〉 ≡ | j1,m1〉| j2,m2〉 to describe the system. This representationis called the uncoupled representation. The other complete set of commuting an-gular momentum operators is J21, J22, J2, and JZ . Their simultaneous eigenstates|( j1 j2) j,m〉 which we sometimes write as | j,m〉 are used to describe the system.This representation is called the coupled representation as the quantum numbers jand m of the coupled angular momentum are used. Note that the bracket “()” in|( j1 j2) j,m〉 indicates the coupling of the two angular momenta J1 and J2 and theresult of the coupling is represented by the number j immediately after the bracket.The two representations describe the same set of states of the system so theycan be related by an unitary transformation connecting two bases, that is|( j1 j2) j,m〉= ∑m1,m2| j1,m1; j2,m2〉〈 j1,m1; j2,m2|( j1 j2) j,m〉 , (2.46)where 〈 j1,m1; j2,m2|( j1 j2) j,m〉 are called the Clebsch-Gordan coefficients. Dueto symmetry considerations, it is often convenient to express the coefficients bythe 3- j symbols. The relation between the Clebsch-Gordan coefficients and 3- jsymbols is[87](j1 j2 j3m1 m2 m3)≡ (−1) j1− j2−m3(2 j3 +1)− 12 〈 j1m1, j2m2| j3 −m3〉 , (2.47)〈 j1m1, j2m2| j3 m3〉 ≡ (−1)j1− j2+m3(2 j3 +1)12(j1 j2 j3m1 m2 −m3). (2.48)The 3- j symbols are more symmetric than the Clebsch-Gordan coefficients. Wedo not discuss the symmetry properties here as they are not important for under-standing the thesis. The interested readers should refer to Zare’s book[87]. The 3- jsymbols are subject to the selection rules of angular momentum, that is(j1 j2 j3m1 m2 m3)= 0 unless m1 +m2 +m3 = 0 and | j1− j2| ≤ j3 ≤ j1 + j2,22which physically means that only certain angular momenta are coupled. Many cal-culations in molecular spectroscopy boil down to the evaluation of 3- j symbols.For details on the evaluation, readers should refer to Zare’s book[87], which men-tions some efficient algorithms for calculating 3- j symbols and gives the algebraicexpressions for the commonly encountered 3- j symbols in Table 2.5.Different from the coupling of two angular momenta, the coupling of three an-gular momenta has more than one possible coupling scheme, and all these couplingschemes are related by unitary transformations. We might couple J1, J2 and J3 insuch a way that J1 + J2 = J12, J12 + J3 = J. The eigenfunctions in this couplingscheme are given by | [( j1 j2) j12 j3] j,m〉, where the brackets “[]” and “()”, indicatethe coupling of two angular momenta j12 and j3 to produce j. We can also cou-ple j2 and j3 to produce a new angular momentum j23, and then couple j23 withj1 to produce the total angular momentum j. The corresponding eigenstates aregiven by | [ j1( j2 j3) j23] j,m〉. The transformation between the two different cou-pling schemes is|( j1 j23) j,m〉=∑j12〈( j12 j3) j,m|( j1 j23) j′,m′〉|( j12 j3) j′,m′〉δ j, j′δm,m′ , (2.49)where |( j1 j23) j,m〉 is a short-hand way to write | [ j1( j2 j3) j23] j,m〉. In the aboveequation, since 〈( j12 j3) j,m|( j1 j23) j′,m′〉 is a scalar product, it doesn’t depend onthe orientation of the coordinate system and is independent of the projection quan-tum numbers. Thus we can drop m and m′ and write it as 〈( j12 j3) j|( j1 j23) j′〉.These recoupling coefficients can be replaced with the so-called 6- j symbols de-fined as[87]{j1 j2 j12j3 j j23}= (−1) j1+ j2+ j3+ j [(2 j12 +1)(2 j23 +1)]− 12 〈( j12 j3) j|( j1 j23) j′〉 .(2.50)Similarly, the coupling of four angular momenta also has more than one pos-sible coupling scheme. For example, | [( j1 j2) j12( j3 j4) j34] j,m〉 is associated with23the coupling schemeJ1 +J2 = J12 , J3 +J4 = J34 , J12 +J34 = J ,and | [( j1 j4) j14( j2 j3) j23] j,m〉 is associated with another coupling scheme:J1 +J4 = J14 , J2 +J3 = J23 , J14 +J23 = J .The relation between the states corresponding to those two coupling schemes is| [( j1 j4) j14( j2 j3) j23] j,m〉 = ∑j12∑j34〈( j1 j2) j12( j3 j4) j34 j|( j1 j4) j14( j2 j3) j23 j〉×| [( j1 j4) j14( j2 j3) j23] j,m〉 , (2.51)and the correspondingly the 9- j symbol is defined by[87]〈( j1 j2) j12( j3 j4) j34 j|( j1 j4) j14( j2 j3) j23 j〉=√(2 j12 +1)(2 j34 +1)(2 j14 +1)(2 j23 +1)j1 j2 j12j3 j4 j34j14 j23 j.(2.52)The last column of the 9- j symbol is related to the first coupling scheme and thelast row is associated with the second coupling scheme.2.1.6 Clebsch-Gordan seriesAfter discussing the coupling of angular momenta, we now return to the rotationmatrix and further develop its properties in the context of angular momentum cou-pling. We consider the connection between the uncoupled | j1,m1〉| j2,m2〉 and thecoupled | j,m〉 representations under a rotational transformation. Applying the ro-tation transformation on | j1,m1〉, | j1,m1〉 and | j,m〉 in Eq. (2.46) individually and24using Eq. (2.36) we obtain∑m′1∑m′2D j1m′1,m1(R)D j2m′2,m2(R)| j1,m′1〉| j2,m′2〉=∑j∑m′〈 j1,m1; j2,m2| j,m〉Djm′,m(R)| j,m′〉 . (2.53)Multiplying both sides by 〈 j1,m′1|〈 j2,m′2| and making use of Eq. (2.38), we obtainthe so-called Clebsch-Gordan series:D j1m′1,m1(R) D j2m′2,m2(R) = ∑j〈 j1,m1; j2,m2| j,m〉〈 j1,m′1; j2,m′2| j,m′〉D jm′,m(R)= ∑j(2 j+1)(j1 j2 jm1 m2 m)(j1 j2 jm′1 m′2 m′)D j∗m′,m(R)(2.54)Similarly, the inverse Clebsch-Gordan series are given byD j∗m′,m(R) = ∑m1∑m′1∑m2∑m′2(2 j+1)(j1 j2 jm1 m2 m)(j1 j2 jm′1 m′2 m′)×D j1m′1,m1(R) D j2m′2,m2(R) . (2.55)The Clebsch-Gordan series can help us to evaluate the integral over a product ofthree rotation matrix elements. Multiplying both sides of Eq. (2.54) by D j3m′3,m3(R)we haveD j1m′1,m1(R) D j2m′2,m2(R) D j3m′3,m3(R) = ∑j(2 j+1)(j1 j2 jm1 m2 m)(j1 j2 jm′1 m′2 m′)× D jm′,m(R)∗ D j3m′3,m3(R) . (2.56)Integrating over dω = sinθdθdφdχ and utilizing the normalization condition of25Eq. (2.44) then gives∫D j1m′1,m1(φ ,θ ,χ) D j2m′2,m2(φ ,θ ,χ) Dj3m′3,m3(φ ,θ ,χ)dω= 8pi2(j1 j2 j3m′1 m′2 m′3)(j1 j2 j3m1 m2 m3). (2.57)The above equation is very important because alternative ways to evaluate the in-tegral are very laborious. We will use this equation in the derivation of the Wigner-Eckart theorem. Since the rotation matrix is proportional to the spherical harmon-ics under some special conditions (see Eq. (2.39)), we have∫Dl1m1,0(φ ,θ ,χ) Dl2m2,0(φ ,θ ,χ) Dl3m3,0(φ ,θ ,χ)dω= 8pi2(l1 l2 l3m1 m2 m3)(l1 l2 l30 0 0)=√4pi ·4pi ·4pi(2l1 +1)(2l2 +1)(2l3 +1)×2pi∫Y ∗l1m1(φ ,θ) Y∗l2m2(φ ,θ) Y∗l3m3(φ ,θ)sinθdφdθ , (2.58)which gives rise to∫Yl1m1(φ ,θ) Yl2m2(φ ,θ) Yl3m3(φ ,θ)sinθdθdφ=√(2l1 +1)(2l2 +1)(2l3 +1)4pi(l1 l2 l3m1 m2 m3)(l1 l2 l30 0 0).(2.59)2.1.7 Spherical tensor operatorsWe have seen how the angular momentum wavefunction | j,m〉 transforms under arotation. Specifically, based on Eq. (2.36), a general state |α〉 is changed under arotation R according to|α〉 → D(R)|α〉 . (2.60)26Now we study how an operator transforms under a rotation. Let’s consider a so-called vector operator, for example J, which is composed of operators JX , JY andJZ . We know a vector in classical physics with three components transforms likeVj → ∑ j Ri, jVj under a rotation R. It is reasonable to expect that the expectationvalue of a vector operator V transforms like a classical vector, that is〈α|Vi|α〉 → 〈α|D†(R)ViD(R)|α〉=∑jRi, j〈α|Vj|α〉 , (2.61)where Eq. (2.60) has been used. From the above equation, it follows that the trans-formed operator in the original basis is given byD†(R)ViD(R) =∑jRi, jVj . (2.62)By generalizing the definition of a vector Vj→ ∑ j Ri, jVj, we define a tensor asa quantity which transforms likeTi jk···→∑i′∑j′∑k′∑···Ri,i′R j, j′Rk,k′ · · ·Ti′ j′k′··· . (2.63)The number of indices is called the rank of a tensor. Such a tensor is called a Carte-sian tensor because its components Ti jk··· are defined with respect to the Cartesianaxes.The problem with a Cartesian tensor is that it is reducible meaning it containsparts that transform differently under rotations. Take for example a Cartesian tensorT formed from the product of two vectors U and VTi j ≡UiVj . (2.64)It can be shown that the tensor can be decomposed into the following parts:UiVj =U ·V3δi, j +UiVj−U jVi2+(UiVj +U jVi2−U ·V3δi, j). (2.65)The first term on the right hand side is a scalar and has 1 independent component.The second term looks like a cross product of two vectors and has 3 independent27components. The third term is more complicated and we can rewrite it as(1−δi, j)UiVj−U jVi2+δi, j(UiVi−U ·V3),where the first term represents the off-diagonal part of the 3× 3 tensor and thesecond term represent the diagonal part. Now, it can be easily seen that this termis symmetric and its trace is zero. So it contains 5 independent components. Thenumber of independent components associated with the three terms in Eq. (2.65) is1, 3, and 5 respectively, which are precisely the multiplicities of angular momentawith l = 0, l = 1, and l = 2 respectively. This suggests that the tensor T can bedecomposed into tensors that can transform like spherical harmonics with l = 0, 1,and 2.From the above example, it seems like the spherical harmonics can be used asirreducible tensors to represent any reducible tensor. This motivates us to defineirreducible spherical tensors based on the spherical harmonics. Before presentingthe definition of a spherical tensor, let’s first investigate how the spherical harmon-ics transform under rotations. For a direction eigenket |nˆ〉, a rotation transforms itto another direction eigenket |nˆ′〉, that is|nˆ′〉= D(R)|nˆ〉 . (2.66)By taking the hermitian adjoint of the above equation and then multiplying by |lm〉on the right, we obtainYlm(nˆ′) =∑m′Ylm′(nˆ)(D†)lm′m (R) . (2.67)The spherical harmonics can be used as both functions and operators just like thecoordinates x, y, z can also be used as position operators. Treating the sphericalharmonics as operators, Ylm(nˆ′) on the left hand side of Eq. (2.67) are the operatorsafter the rotation transformation, which can be written in terms of the originaloperator D†(R)Ylm(nˆ)D(R) based on Eq. (2.62). So that Eq. (2.67) becomesD†(R)Ylm(nˆ)D(R) =∑m′Ylm′(nˆ)Dl ∗mm′(R) . (2.68)28Similarly, we define an irreducible spherical tensor operator of rank k with (2k+1)components asD†(R)T (k)q (nˆ)D(R) =k∑q′=−kDk∗qq′(R)T(k)q′ (nˆ) . (2.69)Replacing the rotation R by its inverse R−1 and using Eq. (2.37), the above defini-tion can be recast asD(R)T (k)q (nˆ)D†(R) =k∑q′=−kT (k)q′ (nˆ)Dkq′q(R) . (2.70)Note that T (k)q and T(k)q′ here are defined with respect to the same axis system.Because the spherical harmonics and the irreducible spherical tensors transform inthe same way under rotations, they are proportional to each other, that is[87]T (k)q (nˆ) =(4pik!(2k+1)!!)1/2Ykq(nˆ) , (2.71)wheren!! =n · (n−2) · · ·5 ·3 ·1 if n is odd and positiven · (n−2) · · ·6 ·4 ·2 if n is even and positive1 if n is −1 or 0(2.72)From Eq. (2.71), it follows that the first-rank irreducible spherical tensor is just themodified spherical harmonics, namelyT (1)q =C1q . (2.73)In quantum mechanics, operators are usually written in terms of Cartesian vec-tors, like the position vector r and the electric dipole moment operator d, and wewant to know their corresponding spherical tensor operators. In such cases, thedefinition of the spherical tensors is not very useful and we need to find alternativeways. It turns out that the irreducible spherical tensors can be written in terms ofCartesian coordinates. For example the first-rank irreducible spherical tensor is29given byT (1)1 =−1√2(Xˆ + iYˆ ) , (2.74)T (1)0 = Zˆ , (2.75)T (1)−1 =1√2(Xˆ− iYˆ ) , (2.76)where Xˆ , Yˆ , and Zˆ represents the units lengths for the Cartesian coordinate system.For every Cartesian vector V, there is one corresponding first-rank spherical tensorT (1)(V). Similar to the irreducible spherical tensor, T (1)(V) can also be written interms of Cartesian vectors V, that isT (1)1 (V) =−1√2(VX Xˆ + iVY Yˆ ) , (2.77)T (1)0 (V) =VZZˆ , (2.78)T (1)−1 (V) =1√2(VX Xˆ− iVY Yˆ ) . (2.79)2.1.8 Coupling of spherical tensorsAs we have seen, the spherical tensors behave like spherical harmonics. As a result,the spherical tensors couple in the same way as angular momenta. For example,two spherical tensors Rk1 and Sk2 can be combined to form a tensor of rank K,T (K)P (Rk1 ,Sk2) = ∑p1,p2〈k1, p1;k2, p2|K,P〉T(k1)p1 (R)T(k2)p2 (S) . (2.80)In practice, we can regard the above equation as the coupling of two angular mo-menta j1 and j2 to form j. Comparing Eq. (2.46) with Eq. (2.80), we can easilysee that j1 corresponds to k1, p1 to m1, j2 to k2, p2 to m2, j to K, and m to P.Similarly, K can only takes the values from |k1− k2| to k1 + k2. We call T(K)P thetensor product of Rk1 and Sk2 and sometimes denote it asT (K)P (Rk1 ,Sk2) =[R(k1)⊗S(k2)](K)P.30For later reference, the two most important cases of Eq. (2.80) are[87][A(k)⊗B(k)](0)0= (2k+1)−12 ∑q(−1)k−qT (k)q (A)T(k)−q (B) , (2.81)[A(1)⊗B(1)](2)q= ∑qA,qB〈1,qA;1,qB|2,q〉T(1)qA (A)T(1)qB (B)= ∑qA,qB(−1)q√5(1 1 2qA qB −q)T (1)qA (A)T(1)qB (B) .(2.82)It is helpful to examine further the meaning of[A(1)⊗B(1)](0)0 . Based onEq. (2.81)) we obtain[A(1)⊗B(1)](0)0=1√3[T (1)1 (A)T(1)−1 (B)−T(1)0 (A)T(1)0 (B)+T(1)−1 (A)T(1)1 (B)].(2.83)Substituting Eqs. (2.77), (2.78) and (2.79) into the above equation gives[A(1)⊗B(1)](0)0=−1√3(AX BX +AY BY +AZBZ) =−1√3A ·B . (2.84)Therefore, the rank-zero tensor product of two rank-one tensors is equivalent to thescalar product of the corresponding vectors except for a factor. Similarly, it can beshown that the tensor product of rank k = 1 is related to the cross product by[A(1)⊗B(1)](k)q= (A×B) · eˆq , (2.85)whereeˆ+1 = −1√2(Xˆ+ iYˆ) ,eˆ0 = Zˆ ,eˆ−1 =1√2(Xˆ− iYˆ) . (2.86)31It is worthwhile mentioning that all the results so far assume that the spher-ical tensor operators are defined with respect to one specific fixed axis system.However, in practice, this assumption is usually not valid. Some tensor opera-tors are naturally described in body-fixed coordinates, for example the moleculardipole moment operator, and other operators are more conveniently expressed inspace-fixed coordinates, for instance the external field operator. This leads to thequestion, “How the spherical harmonics defined in the space-fixed axis system berelated to those in body-fixed axis system?”. Suppose the molecule-fixed axes canbe obtained by a rotation R of the space-fixed axes. Equation (2.70) will still bevalid. Now, the tensor operators are all defined in the space-fixed axis system andthe body-fixed axes can be obtained by rotation R of the space-fixed axes. We re-call that a transformation S turns a wavefunction ψ into Sψ and turns an operator Tinto ST S−1 in the transformed basis set. In the case of the rotation transformationD, the new operator in the rotated basis set is given by DT D−1 = DT D†. There-fore, the left hand side of Eq. (2.70) can be interpreted as new tensor operator inthe rotated axis system (or body-fixed axis system), and Eq. (2.70) can be writtenasT (k)b (T) =k∑s=−kDksb(R)T(k)s (T) , (2.87)where T (k)b is the tensor operator defined in body-fixed axis system and T(k)s is thetensor operator defined in space-fixed axis system. Note that s labels space-fixedcomponents and q labels body-fixed components. Replacing R with its inverseR−1, using Eq. (2.37) and exchanging b and s, we obtain the inverse of Eq. (2.87),namelyT (k)s (T) =k∑s=−kDksb(R)∗ T (k)b (T) . (2.88)2.1.9 Matrix elements of tensor operatorsWe now consider the evaluation of matrix elements of tensor operators with respectto angular-momentum eigenstates. Given a matrix element 〈η , j,m|T (k)q (A)|η ′, j′,m′〉where η and η ′ denotes quantum numbers other than rotational quantum numbers,we can rotate the bra, operator and ket using Eq. (2.36) and Eq. (2.87), the result32must be the same. Therefore we obtain〈η , j,m|T (k)q (A)|η ′, j′,m′〉= ∑n,n′,pD jnm(R)∗ Dkpq(R) Dj′n′m′(R)〈η , j,n|T(k)p (A)|η ′, j′,n′〉 . (2.89)Integrating over dω = sinθdθdφdχ , we obtain on the left hand side 8pi2× “origi-nal term” as the matrix element is a scalar and independent of any angles, and wehave on the right hand side the integral over three rotation matrices, that is∫dωD j∗nm(R) Dkpq(R) Dj′n′m′(R) . (2.90)Rewriting D j∗nm(R) in terms of Dj−n−m(R) using Eq. (2.38) and evaluating the inte-gral using Eq. (2.57) gives the value as(−1)n−m8pi2(j k j′−m q m′)(j k j′−n p n′), (2.91)which then used in Eq. (2.89) gives〈η , j,m|T (k)q (A)|η ′, j′,m′〉= (−1) j−m(j k j′−m q m′){∑n,n′,p(−1) j−n(j k j′−n p n′)×〈η , j,n|T (k)p (A)|η ′, j′,n′〉}. (2.92)The term in the braces is independent of the projection quantum numbers and wewrite it as 〈η , j||T (k)p (A)||η ′, j′〉. Thus, we have[92]〈η , j,m|T (k)q (A)|η ′, j′,m′〉= (−1) j−m(j k j′−m q m′)〈η , j||T (k)(A)||η ′, j′〉 .(2.93)This is the Wigner-Eckart theorem and states that the matrix element of a ten-sor operator in angular-momentum eigenstates can be separated into a part whichdescribes all the angular dependence and another part which is independent of pro-33jection quantum numbers and hence of orientation.To make the Wigner-Eckart theorem useful, we need to know how to evaluatethe reduced matrix element 〈η , j||T (k)p (A)||η ′, j′〉. The usual approach is to calcu-late the matrix element for some special cases and then derive the reduced matrixelement from the result. For example, if we want to calculate the matrix elementsassociated with the tensor operator of the modified spherical harmonics Ckq(θ ,φ),〈l1,m1|Ckq(θ ,φ)|l2,m2〉, we can consider the special case when q = 0, m1 = 0 andm2 = 0. The corresponding matrix element can be converted to an integral over aproduct of three spherical harmonics, that is〈l1,0|Ck0(θ ,φ)|l2,0〉=√4pi2k+1〈Yl10|Yk0|Yl20〉 ,which gives rise to[(2l1 +1)(2l2 +1)]1/2(l1 k l20 0 0)2, (2.94)based on Eq. (2.59). Alternatively, from the Wigner-Eckart theorem we have〈l1,0|Ck0(θ ,φ)|l2,0〉= (−1)l1(l1 k l20 0 0)〈l1||C(k)||l2〉 . (2.95)Comparing Eq. (2.94) with Eq. (2.95) gives〈l1||C(k)||l2〉= (−1)l1 [(2l1 +1)(2l2 +1)]1/2(l1 k l20 0 0). (2.96)From this reduced matrix element, any matrix elements can be calculated using theWigner-Eckart theorem, so that Eq. (2.93) gives〈l1,m1|Ckq(θ ,φ)|l2,m2〉 = (−1)m1 [(2l1 +1)(2l2 +1)]1/2(l1 k l20 0 0)×(l1 k l2−m1 q m2). (2.97)342.2 Application of angular momentum theoryWe have reviewed the theory of angular momentum in Section 2.1. To lay thefoundation for later chapters, we apply the theory of angular momentum to solvesome problems closely related to the research in this thesis.2.2.1 Diatomic molecule in external fieldThe first problem involves calculating the dressed rotational states of molecules inan external field. For simplicity, we only consider 1Σ molecules with a permanentdipole momentum and no spins. Since only rotational states are relevant here, thetotal Hamiltonian can be written asH = Hrot +HI , (2.98)where Hrot describes the rotational states of the system without any external fieldand HI describes the interaction with the external field. For 1Σ molecules,Hrot = BNˆ2 , (2.99)where Nˆ is the rotational angular momentum operator and B is the rotational con-stant obtained by averaging over all quantum states other than rotational states. Toobtain the eigenstate of the total Hamiltonian, we can use the bare rotational states|N,M〉 as a basis set. We want to know the matrix representation of H and calcu-late its eigenvalues and eigenvectors. The rotational part of the Hamiltonian onlycontributes to the diagonal part of the matrix and is given by〈N,M|Hrot|N′,M′〉= δN,N′δM,M′BN(N +1) . (2.100)The evaluation of the contribution from the interaction part is more involved andwe discuss it in the following two cases.First, consider a static electric (DC) field for whichHI =−d ·E = dEDC cosθ , (2.101)where d is the permanent dipole, E is the external field, and θ is the angle between35d and E. To evaluate the corresponding matrix elements, we need to calculate〈N,M|cosθ |N′,M′〉. Because cosθ can be expressed in terms of a spherical har-moniccosθ =√4pi3Y10(θ ,φ) =C10 , (2.102)Eq. (2.97) gives〈N,M|HI|N′,M′〉= Ed〈N,M|C10|N′,M′〉= (−1)MdEDC√(2N +1)(2N′+1)(N 1 N′0 0 0)(N 1 N′−M 0 M′).(2.103)Second, we consider a laser (AC) field which is off-resonant with any vibra-tional or electronic states of the molecule. The strength of the AC field is givenbyEAC(t) = EAC,0 cos(ωt) , (2.104)where ω is the frequency of the laser field. In this case, the interaction Hamiltonianis given by [93]HI(t) =−dEAC(t)cosθAC−12E 2AC(t)(α‖ cos2 θAC +α⊥ sin2 θAC), (2.105)where θAC is the angle between the dipole moment of the molecule and the ACfield, and α‖ and α⊥ are the parallel and perpendicular polarizabilities of themolecule, respectively. The interaction Hamiltonian is a function of time, andchanges very fast as EAC(t) oscillates at optical frequency. During one period ofrotational motion of the molecule, the interaction Hamiltonian has oscillated manytimes. Therefore, we can average HI(t) over one oscillation period of the light andobtain the effective interaction HamiltonianHI =−14E 2AC,0(∆α cos2 θAC +α⊥), (2.106)where ∆α = α‖−α⊥. As we did in the case of the DC field, to calculate the matrixelements of the interaction Hamiltonian, we rewrite cos2 θAC in terms of spheri-36cal harmonics and express the matrix elements in terms of those of the modifiedspherical harmonics. Doing this and using Eq. (2.97) gives〈N,M|HI|N′,M′〉=−14E 2AC,0〈N,M|(∆α cos2 θAC +α⊥)|N′,M′〉=−14E 2AC,0〈N,M|[∆α(23C20 +13)+α⊥]|N′,M′〉=−E 2AC,012[(−1)M2∆α√(2N +1)(2N′+1)(N 2 N′0 0 0)(N 2 N′−M 0 M′)+(∆α+3α⊥)δN,N′δM,M′]. (2.107)2.2.2 Dipole-dipole interactionThe second problem we consider calculates the matrix elements of the dipole-dipole interaction between two molecules by using the theory of angular momen-tum. Consider two molecules A and B, in rotational states |NAMA〉 and |NBMB〉,respectively. In free space, one molecule can rotate around the other and the stateof the two molecules can be expressed as|NAMA〉|NBMB〉|lm〉 ,where l is the orbital angular momentum of A around B. However, in the solidstate or optical lattices, the position of molecules are fixed to a good approxima-tion, and they cannot rotate around each other. In this case, their states are givenby |NAMA〉|NBMB〉 if there is no interaction between them. If we consider the inter-action between the two molecules, we can always use |NAMA〉|NBMB〉 as basis set.By expanding the Hamiltonian in this basis set and diagonalizing it, we can get thenew eigenstates.As the first step, we have to know the interaction between the two molecules.If A has a permanent dipole dA and B has a permanent dipole dB and the vectorconnecting their centers of mass is R, then the dipole-dipole interaction between A37and B is given byVˆdd =(1R3)[dA ·dB−3(dA · Rˆ)(dB · Rˆ)]. (2.108)Here we only consider the dipole moment. There also exists quadrupole, octopole,and higher order moments, but their magnitudes are so small that we can ignorethem at large distances, say, 400 nm. Our final goal is to evaluate the matrixelement 〈NAMA|〈NBMB|Vˆdd|N′AM′A〉|N′BM′B〉 by rewriting Vˆdd in terms of sphericalharmonics, and then in terms of 3- j symbols which can be calculated numerically.The first term in the square bracket of Eq. (2.108), can be expressed in termsof a tensor product using Eq. (2.84), that isdA ·dB =−√3[d(1)A ⊗d(1)B](0)0. (2.109)The second term in the square bracket of Eq. (2.108) is the product of two scalars(dA · Rˆ)(dB · Rˆ), which we consider as a special case of the dot product of twovectors in one-dimensional space. So that according to Eq. (2.109), we can write(dA · Rˆ)(dB · Rˆ) = 3[[d(1)A ⊗ Rˆ(1)](0)⊗[d(1)B ⊗ Rˆ(1)](0)](0)0. (2.110)There is a simple understanding of the above equation: an angular momentumdA with j1 = 1 couples another angular momentum Rˆ with j2 = 1 to give riseto a new angular momentum, and an angular momentum dB with j3 = 1 coupleswith another angular momentum Rˆ with j4 = 1 to give rise to another new angularmomentum, and then these two new angular momenta couple with each other.The form of Eq. (2.110) is not convenient for calculations. We know onlydA and dB operate on the rotational states |NA,MA〉|NB,MB〉, but they are cou-pled with Rˆ, which makes the calculation of the matrix element associated withEq. (2.110) cumbersome. Therefore, it is necessary to regroup these operatorsso that the dipole operators are separated from the position operators. Let us ex-pand[[d(1)A ⊗ Rˆ(1)](0)⊗[d(1)B ⊗ Rˆ(1)](0)](0)0in terms of another coupling scheme,where dA couples with dB and Rˆ couples with Rˆ. This involves the definition of389- j symbols. Based on Eq. (2.51) and Eq. (2.52) we have(dA ·Rˆ)(dB ·Rˆ)= 3∑k(2k+1)1 1 k1 1 k0 0 0[[d(1)A ⊗d(1)B](k)⊗[Rˆ(1)⊗ Rˆ(1)](k)](0)0.(2.111)The 9- j symbols in the above equation can be expressed in terms of 6- j symbols(related to the coupling of 3 angular momenta). In the special case where the finalangular momentum j9 = 0, we have[87]j1 j2 j3j4 j5 j6j7 j8 j9= (−1) j2+ j3+ j4+ j7 [(2 j3 +1)(2 j7 +1)]− 12×{j1 j2 j3j5 j4 j7}δ j3 j6δ j7 j8 , (2.112)so that1 1 k1 1 k0 0 0= (−1)k+2(2k+1)−12{1 1 k1 1 0}= (−1)k+2(2k+1)−12{13(−1)−k 0≤ k ≤ 20 otherwise.(2.113)Because k results from the coupling of j1 = 1 and j2 = 1, it is in the range of| j1− j2|, · · · , | j1 + j2|, and the above equation can be simplified as1 1 k1 1 k0 0 0=(2k+1)−123. (2.114)39Substituting Eq. (2.114) into Eq. (2.111) and using Eq. (2.81) gives(dA · Rˆ)(dB · Rˆ) = ∑k=0,1,2∑q(−1)k−q[d(1)A ⊗d(1)B](k)q[Rˆ(1)⊗ Rˆ(1)](k)−q. (2.115)Since the tensor product of rank 1 is related to the cross product (see Eq. (2.85)),[Rˆ(1)⊗ Rˆ(1)](k=1)−q is associated with Rˆ× Rˆ = 0 and is zero. So that Eq. (2.115)simplifies to(dA · Rˆ)(dB · Rˆ) =[d(1)A ⊗d(1)B](0)0[Rˆ(1)⊗ Rˆ(1)](0)0+ ∑|q|=0,1,2(−1)q[d(1)A ⊗d(1)B](2)q[Rˆ(1)⊗ Rˆ(1)](2)−q= (−1√3dA ·dB)(−1√3Rˆ · Rˆ)+∑q(−1)q[d(1)A ⊗d(1)B](2)q[Rˆ(1)⊗ Rˆ(1)](2)−q=dA ·dB3+∑q(−1)q[d(1)A ⊗d(1)B](2)q[Rˆ(1)⊗ Rˆ(1)](2)−q.(2.116)Substituting Eq. (2.116) into Eq. (2.108) yieldsVˆdd(R) =−3R32∑q=−2(−1)q[d(1)A ⊗d(1)B](2)q[Rˆ(1)⊗ Rˆ(1)](2)−q. (2.117)In the above equation, Rˆ can be viewed as a first-order irreducible spherical ten-sor, therefore the contraction of two Rˆ gives rise to the second-order irreduciblespherical tensor[Rˆ(1)⊗ Rˆ(1)](2)−q= T (2)−q (Rˆ) . (2.118)Based on Eq. (2.71) we obtain[Rˆ(1)⊗ Rˆ(1)](2)−q=√8pi15Y2−q(θR,φR) , (2.119)where θR and φR describe the orientation of Rˆ in the space-fixed axis system, and40then the dipole-dipole operator becomes:Vˆdd(R) =−2√6pi51R32∑q=−2(−1)qY2−q(θR,φR)[d(1)A ⊗d(1)B](2)q. (2.120)To calculate the matrix elements of the dipolar operator, we now only need toconsider the tensor product[d(1)A ⊗d(1)B](2)q. Using Eq. (2.82), we can express thetensor product in terms of the tensor components in some axis system[d(1)A ⊗d(1)B](2)q= ∑qA,qB(−1)q√5(1 1 2qA qB −q)T (1)qA (dA)T(1)qB (dB) . (2.121)The rotational states |NA,MA〉|NB,MB〉 are defined in the space-fixed axis system,so only tensor operators defined in the same axis system can operate on them di-rectly. Therefore, for the convenience of calculation, the tensor components inEq. (2.121) should also be in the space-fixed axis system. However, the dipole mo-ment of a molecule is most conveniently expressed in the body-fixed axis system.In the case of a diatomic polar molecule, we can choose the body-fixed z-axis to bein the same direction as the dipole moment. So that the only nonzero body-fixed bcomponent is given byT (1)b=0(d) = dzˆ , (2.122)where d is the magnitude of the dipole moment. Now the problem is obtainingthe space-fixed dipole moment component from the body-fixed component. FromEq. (2.88), we haveT (1)s (d) =1∑b=−1D1∗sb T(1)b (d)= D1∗s0 d= d0C1s , (2.123)where T (1)s is the tensor component in space-fixed axis system and the last equality41comes from Eq. (2.39). Substituting Eq. (2.123) into Eq. (2.121) then gives[d(1)A ⊗d(1)B](2)s= ∑sA,sB(−1)s√5dAdB(1 1 2sA sB −s)C1sAC1sB . (2.124)Using Eq. (2.97), the corresponding matrix elements are given by〈NA,MA|〈NB,MB|[d(1)A ⊗d(1)B](2)s|N′A,M′A〉|N′B,M′B〉= (−1)s+MA+MBdAdB√5√(2NA +1)(2NB +1)(2N′A +1)(2N′B +1)×(NA 1 N′A0 0 0)(NB 1 N′B0 0 0)× ∑sA,sB(NA 1 N′A−MA sA M′A)(NB 1 N′B−MB sB M′B)(1 1 2sA sB −s).(2.125)Finally, after considering every possible value of q in Eq. (2.120) and makinguse of Eq. (2.120), we arrive at〈NA,MA|〈NB,MB|Vˆdd(R)|N′A,M′A〉|N′B,M′B〉=−F2R3{3sin2 θRe−2iφRDA−DB−−6sinθR cosθRe−iφR[DA0 DB−+DA−DB0]+3sin2 θRe2iφRDA+DB++6sinθR cosθReiφR[DA0 DB++DA+DB0]+√6(3cos2 θR−1)[DA+DB−+DA−DB++DA0 DB0]}, (2.126)whereF = (−1)MA+MBdAdB√5√(2NA +1)(2NB +1)(2N′A +1)(2N′B +1)×(NA 1 N′A0 0 0)(NB 1 N′B0 0 0), (2.127)42and DA±,0DB±,0 are defined usingfAB(sA,sB) =(NA 1 N′A−MA sA M′A)(NB 1 N′B−MB sB M′B)(1 1 2sA sB −s),(2.128)in the following wayDA+DB+ = fAB(1,1) ,DA+DB0 = fAB(1,0) ,DA+DB− = fAB(1,−1) ,DA0 DB− = fAB(1,−1) ,DA0 DB+ = fAB(0,1) ,DA0 DB0 = fAB(0,0) ,DA0 DB− = fAB(0,−1) ,DA−DB+ = fAB(−1,1) ,DA−DB0 = fAB(−1,0) ,DA−DB− = fAB(−1,−1) . (2.129)The factor F describes the selection rule of rotational quantum number N for thedipole-dipole interaction, which says that only two consecutive rotational levelsare coupled. DA±,0DB±,0 describes the change of the projection quantum number Mby the dipole-dipole interaction. It can be easily seen from Eq. (2.128) that DAX DBYis nonzero only if the projection quantum numbers change in a certain way. Forexample, DA+DB+ is nonzero if only MA−M′A = 1 and MB−M′B = 1.2.3 Introduction to excitons in molecular crystalsIn this section, we aim to give a very brief introduction to excitons. For simplic-ity, we consider a 1D molecular crystal with every lattice site occupied by one43molecule. The Hamiltonian is given byH =∑nHn +12∑n ∑m6=nVnm , (2.130)where n is the molecule index, Hn is the Hamiltonian for an isolated molecule n,and Vnm describes the interaction between the two molecules labeled n and m. Theextension to 2D and 3D molecular arrays is straightforward: one only needs toreplace the number index n by a vector index n in the 2D or 3D dimensional space.2.3.1 Commutation relationIn a molecular crystal, there are a large number of identical interacting molecules.To study the excited states of the system, it is suitable to use the second-quantizationformalism which is convenient for describing many-particle systems.The second-quantization representation uses a basis that describes the numberof particles occupying each state in a complete orthonormal set of single-particlestates. Assuming the distance between any two molecules is large such that theirwavefunctions hardly overlap, we can choose the single-particle states to be theeigenfunctions ψ fn of the isolated molecule Hamiltonian Hn. The basis set for thesecond-quantization formalism is then given by the wavefunction that describes acrystal formed by non-interacting molecules, that is|N1 f ′N2 f ′′ ...Nn f ...〉,where Nn f represents the occupation number of state f in molecule n and it is either0 or 1. Accordingly, the occupation number operator is defined asNˆn f |...Nn f ...〉= Nn f |...Nn f ...〉 . (2.131)For the crystal formed by interacting molecules, its eigenstates can be expressed interms of linear combinations of the above basis set.The molecule at site n must be in some state, therefore the occupation operators44of all states sum up to the identity operator:∑fNˆn f = 1 , (2.132)where f denotes any (ground or excited) states of molecule n. It is convenient tointroduce two operators b†n f and bn f associated with molecule n and state f suchthatb†n f |...Nn f ...〉 = (1−Nn f )|...(Nn f +1)...〉 ,bn f |...Nn f ...〉 = Nn f |...(Nn f −1)...〉 , (2.133)and express the occupation number operator as the product of these two operators,Nˆn f = b†n f bn f . (2.134)The physical meaning of the two operators is clear: b†n f creates the state f inmolecule n and bn f destroys the state f in molecule n. It follows from Eq. (2.133)that the commutation relation of the b operators for the same energy level of thesame molecule is given bybn f b†n f + b†n f bn f = 1 ,bn f bn f = b†n f b†n f = 0 . (2.135)Because an operator for a specific molecule n and state f does not operate onother molecules and other states, any two operators that correspond to differentmolecules n and m or different states f and f ′ commute.The b operators defined above are not very convenient as they are associatedwith both the ground and excited states. Since we want to focus on the excitedstates, we define the so-called excitation creation and annihilation operators asP†ne = b†nebng , Pne = b†ngbne , (2.136)where g denotes the ground state and e represents any excited state. Based on thephysical meaning of b†n f and bn f , it is clear that P†ne creates an excited state from the45vacuum state |0〉 and Pne does the reverse. Making use of Eq. (2.135), we obtainfrom the above definitions thatP†nePne = b†nebngb†ngbne= b†ne(1−b†ngbng)bne= Nˆne , (2.137)PneP†ne = b†ngbneb†nebng= b†ng(1−b†nebne)bng= Nˆng . (2.138)Note that the above derivations are valid because bngbne and bnebng produce zerowhen operating on any states. Based on Eq. (2.132), the subtraction and additionof Eq. (2.137) and Eq. (2.138) yield:PneP†ne−P†nePne = 1−∑f 6=gNˆn f − Nˆne , (2.139)PneP†ne +P†nePne = 1−∑f 6=gNˆn f + Nˆne . (2.140)Because Pn f and P†n f don’t operate on a different molecule n′, any exciton operatorscorresponding to different molecules or different states commute with each other,so that Eq. (2.139) can be extended asPneP†n′e′−P†n′e′Pne = δnn′δee′(1−∑f 6=gNˆn f − Nˆne). (2.141)Eq. (2.140) and Eq. (2.141) describe the exact statistics of excitons. Unfortunately,it is cumbersome to take into account the exact statistics in most cases. Usually,only a small number of molecules in a crystal is excited, so the following inequality〈Nne〉  1 (2.142)holds for any excited state e. This means we can ignore the operators Nˆne in prac-46tice. However, we cannot ignore all Nˆne in both Eq. (2.139) and Eq. (2.140) as itwill lead to contradictory results, that isPneP†ne−P†nePne = 1 , (2.143)which describes two bosons at the same site, andPneP†ne +P†nePne = 1 , (2.144)which describes two fermions at the same site. The above two equations cannot besatisfied at the same time, so we have to choose either one of them. Consideringthe fact that two excitations cannot reside at the same site as a molecule cannotbe excited twice, it seems more reasonable to choose Eq. (2.144) to be valid. Inaddition, in the Heitler-London approximation where only the ground and firstexcited states are considered, Eq. (2.144) becomes exact as Nˆng + Nˆne = 1. Asthe Heitler-London approximation is usually a good approximation, it also makessense to choose Eq. (2.144). Therefore, we usually approximate the exact statisticsof Eq. (2.140) and Eq. (2.141) by the following commutation relations:PneP†ne +P†nePne = 1 , (2.145)Pn′e′P†ne−P†n′e′Pne = 0 for n 6= n′ or e 6= e′ . (2.146)This is called the Pauli commutation relation as Eq. (2.145) looks like the commu-tation relation for fermions and Eq. (2.146) looks like the commutation relation forbosons. For the convenience of calculations, the commutation relation is rewrittenas{Pne,P†me}= δm,n +(1−δm,n)2P†mePne ,[Pne,Pme] = [P†ne,P†me] = 0 ,[Pne,Pne′ ] = [P†ne,P†ne′ ] = 0 ,PnePne = P†neP†ne = 0 , (2.147)where n and m, e and e′ are assumed to be different, {A,B}= AB+BA, and [A,B] =47AB−BA.2.3.2 Exciton Hamiltonian in second quantizationThe Hamiltonian in Eq. (2.130) can be rewritten in terms of the exciton creationand annihilation operators. Here, we first derive the Hamiltonian in the two-levelapproximation and then give the Hamiltonian for the most general case.In the first step, we express the Hamiltonian in terms of the b operators. Theresult is[87]H =∑iεib†i f bi f +12 ∑i, j 6=i∑f ′, f ,l′,lb†i f ′b†jl′bi f b jl〈 f′l′|Vi j| f l〉 , (2.148)where i and j are site indices, f , f ′, l and l′ represent any molecular state, and〈 f ′l′|Vi j| f l〉 is a shorthand for 〈 f ′|i〈l′| jVi j| f 〉i|l〉 j.Now we consider the two-level approximation where each molecule has onlytwo energy levels: the ground state g and one excited state e such that f , f ′, l and l′in Eq. (2.148) can only take g or e. Making use of the definitions of exciton creationand annihilation operators(Eq. (2.136)), we can easily show that Eq. (2.148) can bewritten asH =∑i(εgb†igbig + εeb†iebie)+ ∑i, j 6=i[12b†igbigb†jgb jg〈gg|Vi j|gg〉+(P†ie +Pie)b†jgb jg〈eg|Vi j|gg〉+P†ieP†je〈ee|Vi j|gg〉+b†iebieb†jgb jg〈eg|Vi j|eg〉+P†iePje〈eg|Vi j|ge〉+b†iebie(P†je +Pje)〈eg|Vi j|ee〉+12b†iebieb†jeb je〈ee|Vi j|ee〉]. (2.149)There are terms like b†igbig and b†iebie in the above equation. We want to get rid ofthose terms. Noticing thatb†iebie = P†iePie = Nˆie , (2.150)48andb†igbig = Nˆig = 1− Nˆie = 1−P†iePie , (2.151)we can rewrite the Hamiltonian asH = H0 +H1 +H2 +H3 +H4 +H5 +H6 , (2.152)whereH0 = εgNmol +12 ∑i, j 6=i〈gg|Vi j|gg〉 , (2.153)H1 =∑i{(εe− εg)+∑j 6=i[〈eg|Vi j|eg〉−〈gg|Vi j|gg〉]}P†iePie , (2.154)H2 = ∑i, j 6=i〈eg|Vi j|ge〉P†iePje , (2.155)H3 =12 ∑i, j 6=i〈eg|Vi j|ge〉(P†ieP†je +PiePje), (2.156)H4 =12 ∑i, j 6=i[〈ee|Vi j|ee〉+ 〈gg|Vi j|gg〉−2〈eg|Vi j|eg〉]P†iePieP†jePje , (2.157)H5 = ∑i, j 6=i〈eg|Vi j|gg〉(P†ie +Pie), (2.158)H6 = ∑i, j 6=i[〈eg|Vi j|ee〉−〈eg|Vi j|gg〉](P†ie +Pie)P†jePje . (2.159)For the general case with multiple energy levels, the derivation of the Hamilto-nian in terms of exciton operators is very similar but tedious. So I choose to use the49Mathematica software[94] to write a symbolic program to handle the derivations.For reference, the result is given below. The Hamiltonian can be divided asH = H(0)+H(1)+H(2)+H(3)+H(4) , (2.160)where the superscripts “(n)” mean that the corresponding term contains n excitonoperators. In the following, we use capital characters L, L′, M and M′ to representany excited state of molecule, and G to denote the ground state. The first two termcan be expressed asH(0) = εGNmol +12 ∑i, j 6=i〈GG|Vi j|GG〉 , (2.161)H(1) = ∑i, j 6=i∑M[〈GG|Vi j|GM〉PjM + 〈GM|Vi j|GG〉P†jM], (2.162)neither of which doesn’t conserve particle numbers. H(2) can be further dividedinto a part that conserves particle numbers and a part that doesn’t, that isH(2) = H(2)conserving +H(2)non−conserving , (2.163)whereH(2)conserving =∑i∑M{(εM− εG)+∑j 6=i[〈MG|Vi j|MG〉−〈GG|Vi j|GG〉]}P†iMPiM+ ∑i, j 6=i∑L,M〈GL|Vi j|MG〉PiMP†jL+ ∑i, j 6=i∑L,M(1−δM,L)〈LG|Vi j|MG〉P†iLPiM , (2.164)H(2)non−conserving =12 ∑i, j 6=i∑L,M[〈GG|Vi j|ML〉PiMPjL + 〈ML|Vi j|GG〉P†iMP†jL].(2.165)50H(3) doesn’t conserve particle numbers and is given byH(3) = ∑i, j 6=i∑M,M′,L{[〈MG|Vi j|M′L〉−δM,M′〈GG|Vi j|GL〉]P†iMPiM′PjL+[〈ML|Vi j|M′G〉−δM,M′〈GL|Vi j|GG〉]P†iMPiM′PjL}. (2.166)H(4) conserves particle numbers and is given byH(4) =12 ∑i, j 6=i∑M,M′,L,L′[δM,M′δL,L′〈GG|Vi j|GG〉+ 〈ML|Vi j|M′L′〉−2δL,L′〈MG|Vi j|M′G〉]P†iMPiM′P†jLPjL′ . (2.167)2.3.3 Eigenstates of the exciton Hamiltonian in the Heitler-LondonapproximationThe Heitler-London approximation was first used by Frenkel in his study [95, 96]of electronic excitations in molecular crystals. Considering only two levels of amolecule, the ground state g and excited state e, the approximation assumes thefollowing:• the ground-state wavefunction of the crystal is the product of the ground-state wavefunction of individual molecules that constitute the crystal• an excited state of the crystal is a superposition of products · · · |g〉i|e〉 j|g〉k · · ·in which only one molecule j is excited and all other molecules are in theground stateFrom the above two statements, we see that the Heitler-London approximation isexpected to be good when the intermolecular interaction is weak and the moleculesin the crystal preserve a large part of their individuality.We now consider the lowest excited state of a crystal, which corresponds tothe basis set { · · · |g〉i|e〉 j|g〉k · · · } with only one molecule excited. In this case, thenumber of excitations is conserved, so that only the parts of the Hamiltonian in51Eq. (2.152) that conserve particle numbers need to be included, and we can rewritethe Hamiltonian asH = H0 +H1 +H2 +H4 . (2.168)To understand the problem better, we analyze this Hamiltonian. H0 is a constantand thus we can set it as zero energy and ignore it. H1 can be written asH1 =∑i(∆εe +De)P†iePie , (2.169)where∆εe = εe− εg (2.170)is the excitation energy of an isolated molecule, andDe =∑j 6=i[〈eg|Vi j|eg〉−〈gg|Vi j|gg〉](2.171)is the difference of the two interactions: one for the excited molecule with allremaining ground-state molecules in the crystal, and the other for the same nonex-cited molecule with all remaining ground-state molecules. De is also called thegas-condensed matter shift and it exists when the molecules in the crystal are in-teracting with each other. H2, as given byH2 = ∑i, j 6=i〈eg|Vi j|ge〉P†iePje , (2.172)destroys an excitation in molecule j and then creates an excitation in molecule i.Therefore, it describes the propagation of an excitation in the crystal and we callit the hopping term. The strength of the hopping interaction is given by the ma-trix elements 〈eg|Vi j|ge〉. Usually, the interaction between two nearest moleculesis strongest and it decays fast with respect to the separation of two molecules,so sometimes it is justified to consider only the interaction between two nearestneighbors. We call this the nearest-neighbor approximation. H4 is the so-called thedynamic interaction, which describes the interaction between two excitations. It52can be written asH4 =12 ∑i, j 6=iΦi jP†iePieP†jePje , (2.173)and its strength is determined by the matrix elementsΦi j = 〈ee|Vi j|ee〉+ 〈gg|Vi j|gg〉−2〈eg|Vi j|eg〉 .In molecular crystals, the dipole-dipole interaction is an important kind of interac-tion Vi j between molecules. For molecules with a center of inversion, their eigen-states (for example |e〉 and |g〉) have well-defined parity, and the above matrixelements are zero. Under these circumstance, the dynamic interaction is zero andH4 can be ignored. In the case studied in this thesis, when we are concerned withthe one-excitation subspace, we can safely ignore the dynamic interaction becauseit requires two excitations in the crystal. From another perspective, we can ignorethe dynamic interaction because its matrix elements in the one-excitation subspacevanish. For illustration purposes, we give the derivation here. Based on Eq. (2.147),it can be shown that the matrix element of H4 is given by〈en,gm|12 ∑i, j 6=iΦi jP†iePieP†jePje|gn,em〉=12 ∑i, j 6=iΦi j〈0|PneP†iePieP†jePjeP†me|0〉=12 ∑i, j 6=iΦi j〈0|[δi,n +(1−2δi,n)P†iePne]P†jePie[δ j,m +(1−2δ j,m)P†jePme]|0〉 .(2.174)Without proceeding further, we can easily see that the above equation gives 0 as itconsists of terms in which either P† operates on 〈0| or P operates on |0〉.The above analysis shows that Eq. (2.160) can be simplified toH =∑n(∆εe +De)P†nePne + ∑n,m 6=n〈eg|Vnm|ge〉P†nePme , (2.175)53if one is only interested in the lowest-energy excited state of the crystal. Eq. (2.175)gives the exciton Hamiltonian in the Heitler-London approximation. The corre-sponding eigenstates are called Frenkel excitons, which are many-body excitedstates of the whole crystal.The Hamiltonian in the Heitler-London approximation can be diagonalized an-alytically. Before showing this, we first introduce the concepts of site represen-tation and quasimomentum representation. Previously, the excitation creation andannihilation operators Pn f and P†n f were defined with respect to the lattice site n.These operators form the site representation. Due to the periodicity of the crystal,we can use the quasimomentum representation, to represent excitations by takingthe Fourier transforms of Pn f and P†n f , that isPf (k) =1√N∑ne−iknPn f ,P†f (k) =1√N∑neiknP†n f , (2.176)where N is the number of molecules in the crystal. Accordingly, the inverse Fouriertransforms arePn f =1√N∑keiknPf (k) ,P†n f =1√N∑ke−iknP†f (k) . (2.177)The conversion between the site representation and the quasimomentum represen-tation usually involves the equalities1N ∑keik(n−m) = δn,m ,1N ∑nein(k−k′) = δk,k′ . (2.178)These two equalities are very important and we will use them frequently in laterderivations.54Substituting Eq. (2.177) into Eq. (2.175) givesH =∑n(∆εe +De)1N ∑ke−iknP†f (k)∑k′eik′nPf (k′)+∑n∑m6=n〈eg|Vnm|ge〉1N ∑ke−ikmP†f (k)∑k′eik′nPf (k′)=∑k,k′(1N ∑nein(k′−k))P†f (k)Pf (k′)+∑n∑l=n−m〈eg|Vn,n−l|ge〉eikl(1Nein(k′−k))P†f (k)Pf (k′) .(2.179)Because of the periodicity of the crystal, the interaction Vn,m only depends on theseparation of the two molecules at site n and m. Thus 〈eg|Vn,n−l|ge〉 in the aboveequation is independent of the index n, so we rewrite it as 〈eg|V (l)|ge〉. Afterthat, making use of Eq. (2.178), we can easily show that the diagonal form of theHamiltonian isH =∑kE f (k)P†f (k)Pf (k) , (2.180)where E f (k) is the eigenenergy of the system and is given byE f (k) = ∆εe +De +∑l〈eg|V (l)|ge〉eikl , (2.181)and thus the eigenstates of the Hamiltonian are|k〉= P†f (k)|0〉=1√N∑neikn|n〉 , (2.182)where |0〉 is the vacuum state and |n〉 represents the single-excitation state in whichmolecule n is in the excited state and all other molecules are in the ground state.The state |k〉 is a Frenkel exciton with quasimomentum (or wavevector) k.55Chapter 3Tunable exciton interactions inoptical lattices with polarmoleculesThe rotational excitation of polar molecules trapped in an optical lattice gives riseto rotational excitons. Here we show that non-linear interactions among such ex-citons can be controlled by an electric field. The exciton–exciton interactions canbe tuned to induce exciton pairing, leading to the formation of biexcitons. Tunablenon-linear interactions between excitons can be used for many applications rangingfrom the controlled preparation of entangled quasiparticles to the study of polaroninteractions and the effects of non-linear interactions on quantum energy transportin molecular aggregates.3.1 IntroductionThe absorption of photons by a solid-state crystal gives rise to quasiparticles calledexcitons. There are two limiting models of excitons: Wannier-Mott excitons andFrenkel excitons. Wannier-Mott excitons occur in crystals with band structure lead-ing to collective excitations with an effective radius much greater than the latticeconstant, while Frenkel excitons are typical for molecular crystals, where collectiveexcitations are superpositions of elementary excitations localized on different lat-56tice sites. These properties lead to important differences in the non-linear excitoninteractions for the two models. The interactions between Wannier-Mott excitonsare determined by Coulomb interactions and phase space filling [97, 98], whilethe interactions of Frenkel excitons are determined by shorter range dynamicalcouplings [99]. Multiple experiments have demonstrated that Wannier-Mott exci-tons can form two-exciton bound states called biexcitons [100–106]. By contrast,despite many theoretical studies [107–110], Frenkel biexcitons have eluded exper-imental observation, with one notable exception [111, 112]. In the present work,we show that the rotational excitation of ultracold molecules trapped on an opticallattice gives rise to Frenkel excitons with controllable non-linear interactions. Wedemonstrate that the exciton–exciton interactions can be tuned to induce the forma-tion of Frenkel biexcitons and that the biexciton binding energy can be controlledby an external electric field.Several experiments have recently demonstrated that ultracold molecules canbe trapped in the periodic potential of an optical lattice [113–115]. Such systemscan be used for the study of quantum energy transfer [67, 116], non-linear photon–photon interactions [117], novel quantum memory devices [118, 119] and, mostnotably, for quantum simulation of lattice models [43, 50, 120–127]. Although de-scribing different phenomena, the Hamiltonians presented in these references, andin the present work, can be cast in the same form. For example, the exciton Hamil-tonian discussed here can be mapped onto the t-V model, which is a special case ofthe Heisenberg-like models studied in the context of ultracold molecules in Refs.[43, 121, 122, 126, 127]. The key difference between the present work and thatin Refs. [43, 121, 122, 126, 127] is that we explore phenomena associated withthe excitation spectrum of the many-body system in the limit of a small numberof excitations. Because we consider a simpler Hamiltonian, our scheme is con-ceptually simpler, requiring fewer molecular states and external field parameters.We use the rotational states of molecules as a probe of the collective interactions,i.e. the experiments proposed here can be carried out by measuring site-selectivepopulations of the rotational states. This can be achieved by applying a gradientof an electric field and detecting resonant transitions from Stark-shifted levels, asdescribed in Ref. [59].573.2 Exciton–exciton interactions in an optical latticeWe consider an ensemble of polar diatomic molecules in the 1Σ electronic statetrapped on an optical lattice in the ro-vibrational ground state. The rotational statesof the molecules |NMN〉 are described by the rotational angular momentum Nˆ andits projection on the quantization axis MN . We assume that the molecules are inthe Mott-insulator phase [113–115] and that each lattice site contains only onemolecule. We consider the rotational excitation |N = 0,MN = 0〉 → |N = 1,MN =0〉 of molecules in the lattice [128]. For simplicity, we denote the ground state ofthe molecule in site n by |gn〉 and the excited state by |en〉. Because the moleculesare coupled by the dipole-dipole interaction, the rotational excitation gives rise toa rotational Frenkel exciton [67], which is an eigenstate of the HamiltonianHˆexc = E0Nmol∑n=1Pˆ†n Pˆn +Nmol∑n,m 6=nJ(n−m)Pˆ†n Pˆm, (3.1)where J(n−m)= 〈en,gm|Vˆdd(n−m)|gn,em〉with Vˆdd(n−m) representing the dipole-dipole interaction between molecules in sites n and m, E0 is the energy differencebetween the states |g〉 and |e〉, and the operators Pˆ†n and Pˆn are defined by the rela-tions Pˆ†n |gm〉 = δnm|en〉 and Pˆn|em〉 = δnm|gn〉. The Pˆn and Pˆm operators satisfy thePauli commutation relation, as shown by Eq. (2.145) and Eq. (2.146). More specif-ically, by omitting the subscript “e” in these two equations, we have the fermioniccommutationPˆnPˆ†m + Pˆ†mPˆn = 1 , (3.2)if n = m, and the bosonic commutationPˆnPˆ†m− Pˆ†mPˆn = δn,m = 0 , (3.3)if n 6= m [99]. It is useful to combine the two commutation relations asPˆnPˆ†m + Pˆ†mPˆn = δm,n +(1−δm,n)2Pˆ†mPˆn , (3.4)which gives rise toPˆnPˆ†m = δm,n +(1−2δm,n)Pˆ†mPˆn . (3.5)58We will make frequent usage of Eq. (3.5) when evaluating the matrix elements ofHamiltonians.Multiple excitations lead to dynamical exciton–exciton interactions describedby [99]Hˆdyn =12Nmol∑n,m6=nD(n−m)Pˆ†n Pˆ†mPˆnPˆm (3.6)whereD(n−m) = 〈en,em|Vˆdd(n−m)|en,em〉+ 〈gn,gm|Vˆdd(n−m)|gn,gm〉−2〈en,gm|Vˆdd(n−m)|en,gm〉, (3.7)and the factor 1/2 is there to cancel the effect of double counting of the pairs (n,m).In the above summations, we have the restriction that m cannot equal to n becausetwo excitations cannot sit at the same lattice site (or in other words a molecule can’tbe excited twice to the same excited state). The dipole - dipole interaction operatorVˆdd(n−m) can only couple states of different parity [92]. If |g〉 and |e〉 are statesof well-defined parity, such as the rotational states |N,MN〉, the matrix elementsD(n−m) must be zero.The inversion symmetry (parity) of molecules on an optical lattice can be bro-ken by applying an external dc electric field. In an electric field, |gn〉 and |en〉are eigenstates of the Hamiltonian Hˆmoln = BeNˆ2n −dn ·E f , where dn is the dipolemoment of molecule in site n and E f is the electric field vector. They can be ex-pressed as |g〉=∑N aN |N,MN = 0〉 and |e〉=∑N bN |N,MN = 0〉, where aN and bNare determined by the electric field strength. Note that we choose MN = 0 statesbecause they adiabatically connect to nondegenerate rotational bare states as theexternal field goes to zero, leading to a single isolated exciton band. For otherMN 6= 0 states, for example MN = 1 states, there will be two crossing exciton bandscorresponding to the excitation from the dressed state |0,1〉 to |1,1〉 (See figure 2in Ref. [67]).As shown in Section 2.3.3, the Hamiltonian of Eq. (3.1) can be diagonal-ized by the Fourier transforms: Pˆ†(k) = 1√Nmol∑neiknPˆ†n and Pˆ(k) =1√Nmol∑ne−iknPˆn,where k is the wave vector of the exciton. This transformation leads to Hˆexc =59∑kE(k)Pˆ†(k)Pˆ(k), where E(k) = E0 + J(k) with J(k) = ∑nJ(n)e−ikn, and Pˆ†(k) andPˆ(k) create and annihilate Frenkel excitons with energies E(k). The interaction ofEq. (3.6) in the momentum representation isHˆdyn =1Nmol∑k1,k2,qD˜(q)Pˆ†(k1 +q)Pˆ†(k2−q)Pˆ(k1)Pˆ(k2), (3.8)whereD˜(q) =∑nD(n)e−iqn . (3.9)3.3 BiexcitonsThe exciton–exciton interactions generally have little effect on the energy spectrumof two-particle continuum states E(k1)+E(k2). However, under certain conditionsdiscussed below, non-linear interactions may result in the formation of a boundtwo-exciton complex, a biexciton. The biexciton state is split from the two-particlecontinuum. The splitting is the biexciton binding energy.3.3.1 Method to calculate biexciton energiesIn the following, we discuss how to calculate the energy of a biexciton. We startfrom the HamiltonianHˆ = Hˆexc + Hˆdyn= E0Nmol∑n=1Pˆ†n Pˆn︸ ︷︷ ︸1+Nmol∑n,mJ(n−m)Pˆ†n Pˆm︸ ︷︷ ︸2+12Nmol∑n,mD(n−m)Pˆ†n Pˆ†mPˆnPˆm︸ ︷︷ ︸3, (3.10)where the constraint that m 6= n is removed by assuming J(0) = 0 and D(0) = 0.The task is to find the eigenvalues and eigenfunctions of the above Hamiltonian inthe two-excitation basis sets|ψ〉=∑n,mCn,mPˆ†n Pˆ†m|0〉 . (3.11)60Normally, we would exclude the terms in which n = m from the above expansion.But in the current case, we assume Cn,n = 0 and keep those terms so that the Fouriertransformation for the coefficients Cn,m to a quasimomentum spaceCn,m =1Nmol∑k1,k2Ck1,k2ei(k1n+k2m) (3.12)is well-defined. More importantly, by doing so we can derive an equation that willcorrespond to the case of two noninteracting bosons. This will become evidentlater. Based on the physical meaning of the coefficients Cn,m, we conclude thatCn,m satisfy the symmetry relationCn,m =Cm,n , (3.13)and are to be normalized by∑n,m|Cn,m|2 = 1 . (3.14)Assuming the wavefunction ψ satisfies the Schro¨dinger equationHˆ|ψ〉= E|ψ〉 , (3.15)we can obtain the equations for the coefficients Cn,m. Since the terms Cn,n areadded into Eq. (3.12) for artificial purposes, their values are not determined by theSchro¨dinger equation. To derive the equations for Cn,m, we let the Hamiltonian Hˆoperate on the wavefunction term by term. The first term in Eq. (3.10) gives1 |ψ〉 = E0 ∑n′,n,mCn,mPˆ†n′Pˆn′Pˆ†n Pˆ†m|0〉= E0 ∑n′,n,mCn,mPˆ†n′[δn′,n +(1−2δn′,n)Pˆ†n Pˆn′]Pˆ†m|0〉= E0 ∑n′,n,mCn,m[δn′,nPˆ†n′Pˆ†m +(1−2δn′,n)Pˆ†n′Pˆ†n Pˆn′Pˆ†m]|0〉= E0 ∑n′,n,mCn,m[δn′,nPˆ†n′Pˆ†m +(1−2δn′,n)Pˆ†n′Pˆ†n δn′,m]|0〉= E0∑n,mCn,m[Pˆ†n Pˆ†m +(1−2δm,n)Pˆ†mPˆ†n]|0〉 , (3.16)61where |0〉 is the vacuum state where every particle is in the ground state. In theabove derivation, Eq. (3.5) and Pˆn|0〉= 0 has been used. Similarly, the second termand the third term in Eq. (3.10) give rise to2 |ψ〉=∑n,m[∑n′J(n′−n)Cn′,mPˆ†n Pˆ†m +∑n′J(n′−m)Cn,n′(1−2δn′,n)Pˆ†mPˆ†n]|0〉(3.17)and3 |ψ〉= 12 ∑n,m[D(n−m)Cn,mPˆ†mPˆ†n +D(n−m)Cn,m(1−2δm,n)Pˆ†n Pˆ†m]|0〉 (3.18)respectively. The right hand side of Eq. (3.15) isE|ψ〉= E∑n,mCn,mPˆ†n Pˆ†m|0〉 . (3.19)Given the fact that Pˆ†n Pˆ†m and Pˆ†mPˆ†n are equivalent, a comparison of both sides ofthe Schro¨dinger equation yields the following equation for Cn,m when n 6= m(E−2E0)Cn,m−∑n′J(n′−n)Cn′,m−∑n′J(n′−m)Cn,n′ = D(n−m)Cn,m . (3.20)When n = m, the above equation becomes(E−2E0)Cn,m−∑n′J(n′−n)Cn′,m−∑n′J(n′−m)Cn,n′= (E−2E0)Cn,n−2∑n′J(n−n′)Cn,n′ . (3.21)Note the two sides of Eq. (3.21) are equivalent, so we cannot obtain the value ofCn,n from it. This is consistent with the previous assumption that Cn,n = 0 becauseEq. (3.21) allows us to take an arbitrary value for Cn,n. By combining Eq. (3.20)62for n 6= m and Eq. (3.21) for n = m, we obtain a new equation(E−2E0)Cn,m︸ ︷︷ ︸4−∑n′J(n′−n)Cn′,m︸ ︷︷ ︸5−∑n′J(n′−m)Cn,n′︸ ︷︷ ︸6= δn,m(E−2E0)Cn,m︸ ︷︷ ︸7−2δn,m∑n′J(n−n′)Cn,n′︸ ︷︷ ︸8+D(n−m)Cn,m︸ ︷︷ ︸9, (3.22)which is valid for all coefficients Cn,m. The left hand side of Eq. (3.22) now cor-responds to the case of two noninteracing bosons since the summations includeterms like Cm,m and Cn,n. On the right hand side, the first two terms describe thekinematic interaction of excitons due to Eq. (3.4) and Eq. (3.5), and the last termrepresents the dynamical interaction of excitons.The dimension of the system of coupled equations (3.22) is about Nmol×Nmol,which is large even for a lattice with moderate size. To reduce the number ofequations that we need to solve, we make use of the translational symmetry ofthe lattices and convert Eq. (3.22) into the quasimomentum representation. Sincethe conversion involves some nontrivial derivations, we give the details for 8 in63Eq. (3.22) which are characteristic for the whole calculations, namely8 = δn,m[∑n′J(n′−n)Cn′,m +∑n′J(n′−m)Cn,n′]=δn,mNmol[∑n′J(n′−n) ∑k1,k2Ck1,k2ei(k1n′+k2m)+∑n′J(n′−m) ∑k1,k2Ck1,k2ei(k1n+k2n′)]=δn,mNmol[∑k1,k2Ck1,k2ei(k2m+k1n)∑n′J(n′−n)eik1(n′−n)+ ∑k1,k2Ck1,k2ei(k1n+k2m)∑n′J(n′−m)eik2(n′−m)]=δn,mNmol[∑k1,k2Ck1,k2ei(k2m+k1n) ∑n′−nJ(n′−n)eik1(n′−n)+ ∑k1,k2Ck1,k2ei(k1n+k2m) ∑n′−mJ(n′−m)eik2(n′−m)]=δn,mNmol∑k1,k2ei(k1n+k2m)Ck1,k2[J˜(k1)+ J˜(k2)]=1N2mol∑k3eik3(n−m) ∑k1,k2ei(k1n+k2m)Ck1,k2[J˜(k1)+ J˜(k2)]=1N2mol∑k1,k2,k3ei(k1+k3)nei(k2−k3)mCk1,k2[J˜(k1)+ J˜(k2)]=1N2mol∑k′1=k1+k3,k′2=k2−k3ei(k′1n+k′2m)Ck′1−k3,k′2+k3∑k3[J˜(k′1− k3)+ J˜(k′2 + k3)]=1N2mol∑k1,k2ei(k1n+k2m)Ck1−k3,k2+k3∑k3[J˜(k1− k3)+ J˜(k2 + k3)]. (3.23)In the above derivation, we have made use of Eq. (3.12), the definition of J˜(k):J˜(k)≡∑neiknJ(n) (3.24)and the normalization relation of plane wave:1N ∑keik(n−m) = δn,m . (3.25)64In the last step of Eq. (3.23), since k′1 and k′2 have the same range as k1 and k2, wehave dropped their prime superscripts. Similar derivations like the above yield4 =(E−2E0)Nmol∑k1,k2Ck1,k2ei(k1n+k2m) , (3.26)5 =1Nmol∑k1,k2Ck1,k2ei(k1n+k2m)J˜(k1) , (3.27)6 =1Nmol∑k1,k2Ck1,k2ei(k1n+k2m)J˜(k2) , (3.28)7 =(E−2E0)N2mol∑k1,k2ei(k1n+k2m)∑k3Ck1−k3,k2+k3 , (3.29)and9 =1N2mol∑k1,k2ei(k1n+k2m)∑k3D˜(k3)Ck1−k3,k2+k3 . (3.30)With all these results, Eq. (3.22) reduces to an equation for Ck1,k2 , namely[E− ε(k1)− ε(k2)]Ck1,k2 = ∑k3E− ε(k1− k3)− ε(k2 + k3)NmolCk1−k3,k2+k3+1Nmol∑k3D˜(k3)Ck1−k3,k2+k3 , (3.31)where ε(k) is the energy of an exciton and is given byε(k) = E0 +∑nJ(n)eikn . (3.32)There is an additional constraint for Ck1,k2 which is due to the assumption that65Cn,n = 0, that is∑k1+k2=KCk1,k2 = ∑k1,k21N ∑n,mCn,me−i(k1n+k2m)= ∑k11N ∑n,mCn,me−i[k1n+(K−k1)m]= ∑n,mCn,meiKm 1N ∑k1e−ik1(n−m)= ∑n,mCn,meiKmδn,m= ∑nCn,neiKn = 0 . (3.33)Because of Eq. (3.33), we can eliminate E terms in the first summation on the righthand side of Eq. (3.31) and obtain[E− ε(k1)− ε(k2)]Ck1,k2 =1Nmol∑k3[D˜(k3)− ε(k1− k3)− ε(k2 + k3)]Ck1−k3,k2+k3 .(3.34)Note in the above equation, the range of the wavectors k1, k2 and k3 is [−pi,pi) sothat the range of k1−k3 and k2+k3 is (−2pi,2pi) which is outside the first Brillouinzone. However, due to the symmetry property of the lattice, we can always bringk1−k3 or k2+k3 back to the first Brillouin zone by adding or subtracting 2pi . Thus,we define two new wavevectorsk′1 = k1− k3±2pi , (3.35)andk′2 = k2 + k3±2pi , (3.36)in which + or − is used to make sure the values of k′1 and k′2 are within the firstBrillouin zone. With this change, Eq. (3.34) becomes[E− ε(k1)− ε(k2)]Ck1,k2 =1Nmol∑k′1+k′2=k1+k2[D˜(k1− k′1)− ε(k′1)− ε(k′2)]Ck′1,k′2.(3.37)To avoid double counting exciton pairs (k1,k2), we restrict the summation to k1 ≥66k2 and k′1 ≥ k′2, reducing the dimension of Eq. (3.37) by half, leading to[E− ε(k1)− ε(k2)]Ck1,k2 =1Nmol∑k′1≥k′2k′1+k′2=k1+k2{D˜(k1− k′1)+(1−δk′1,k′2)D˜(k1− k′2)− (2−δk′1,k′2)[ε(k′1)+ ε(k′2]}Ck′1,k′2. (3.38)As can be seen from the above equation, each coefficient Ck1,k2 is only coupledto other coefficients Ck′1,k′2when k1 + k2 = k′1 + k′2. Therefore we can separateEq. (3.38) into different sets and solve them independently. This will reduce thenumber of equations by a factor of Nmol, which is the reason that we want to con-vert Eq. (3.22) from the site representation to the quasimomentum representation.For each set of equations, the summation of wavevectors k1 + k2 = K is fixed forall pairs, so Eq. (3.38) can be written in the form of an eigenvalue equationA(K)C = E(K)C(K) , (3.39)where the elements of the matrix A(K) are given byA(k1,k2),(k′1,k′2)=1Nmol{D˜(k1− k′1)+(1−δk′1,k′2)D˜(k1− k′2)− (2−δk′1,k′2)[ε(k′1)+ ε(k′2)]}+ δk1,k′1δk2,k′2 [ε(k1)+ ε(k2)] , (3.40)and C is a vector composed of all the relevant coefficients Ck1,k2C =...Ck1,k2...Ck′1,k′2.... (3.41)By solving Eq. (3.39) under the constraint presented in Eq. (3.33), we can ob-tain all the eigenvalues and eigenvectors of the two-exciton system. Under some67conditions, a set of eigen-energies is split from the energy continuum of two freeexcitons and correspond to the biexciton states.3.3.2 Analytical derivation of the biexciton wavefunctionTo gain a deeper understanding of the biexciton state, we aim to derive the biexcitonwavefunction in site representation using the nearest neighbor approximation. Inthis section, I will sketch the entire derivation and give some details of the difficultpart.To begin, let’s examine the coefficients Cn,m in the most general two-excitonwavefunction of Eq. (3.11). These Cn,m’s are related to Ck1,k2 by Eq. (3.12). Fromthe analysis in Section 3.3.1, we know the sum of wavevectors of two excitons isa good quantum number for a biexciton state. Thus, it makes sense to define twonew wavevectors in terms of k1 and k2, namelyK = k1 + k2 , (3.42)k =k1− k22. (3.43)Substituting the above two equations into Eq. (3.12), and using the Fourier trans-formCKk =1√Nmol∑lCK(l)e−ikl , (3.44)andCK(l) =1√Nmol∑kCKk eikl , (3.45)we arrive atCn,m =1Nmol∑k1,k2Ck1,k2ei(K/2+k)nei(K/2−k)m=1Nmol∑K,kCKk ei(K/2+k)nei(K/2−k)m=1√Nmol∑KCK(n−m)eiK(n+m)/2 . (3.46)68The two-exciton wavefunction (Eq. (3.11)) can then be written as|ψ〉= 1√Nmol∑n,m∑KCK(n−m)eiK(n+m)/2Pˆ†n Pˆ†m|0〉 . (3.47)The above equation inspires us to introduce a wavefunction that corresponds to aparticular K|ψKb 〉=1√Nmol∑n,mCK(n−m)eiK(n+m)/2Pˆ†n Pˆ†m|0〉 , (3.48)and we are expecting it to be the wavefunction for a biexciton state. To verify ourguess, we check the orthonormality of the wavefunction. The calculation of theoverlap between two K-wavefunctions gives rise to〈ψQb |ψKb 〉= 2δQ,K∑l∣∣CK(l)∣∣2 . (3.49)This indicates that the wavefunctions for different K’s are orthogonal and the nor-malization of the wavefunction can be satisfied as long as2∑l∣∣CK(l)∣∣2 = 1 . (3.50)Now the problem of finding the biexciton wavefunction becomes the problemof finding the values of CK(l) that satisfy Eq. (3.50). As usual, wavefunctions canbe obtained by solving the Schro¨dinger equation, so we work with Eq. (3.37) whichis equivalent to the Schro¨dinger equation for the biexciton states. The main ideais to substitute Eq. (3.44) into Eq. (3.37) and get rid of the individual wavevectork1, k2, k′2 and k′2. In this way, we will finally obtain an equation for CK(l) thatdepends on the summation of the wavevectors K = k1 + k2 rather than the indi-vidual wavevectors. However before doing that, we need to introduce the Green’sfunctionGKk1,k2 =1E− ε(k1)− ε(k2), (3.51)69and its fourier transformGK(n) =1Nmol∑qeiqnE− ε(K/2+q)− ε(K/2−q)=1Nmol∑k1+k2=Kei(k1−k2)n/2E− ε(k1)− ε(k2). (3.52)In the nearest neighbor approximation where only the interaction between the near-est sites is considered, the energy ε(k1) of an exciton with wavevector k1 is givenbyε(k1) = E0 +2J cos(k1) , (3.53)and Eq. (3.52) reduces toGK(n) =1Nmol∑qeiqn(E−2E0)−2J cos(K/2+q)−2J cos(K/2−q)=1Nmol∑qeiqn(E−2E0)−2JK cos(q), (3.54)where JK can be understood as the half bandwidth of the two free excitons withwavevector K/2+q and K/2−q, and is given byJK = 2J cos(K/2) . (3.55)When the number of sites Nmol becomes very large, Eq. (3.54) can be approximatedby the integralGK(n) =12pi∫ pi−pidqeiqn(E−2E0)−2JK cos(q), (3.56)which can be transformed into another integral over the complex variable w = eiqalong the unit circleGK(n) =12pii|JK |∮dww|n|w2 +2xw+1, (3.57)where x = E/2JK . Equation (3.56) can be calculated using the Residue theorem.70In the case of a biexciton state, its energy EKb is outside the energy continuum oftwo free excitons ε(K/2+ q)+ ε(K/2− q). There are two scenarios: first, whenE−2E0 > 2JK > 0, GK(n) is given byGK(n) =1√(E−2E0)2− (2JK)2−E−2E02JK+√(E−2E02JK)2−1|n|; (3.58)second, when E−2E0 < 2JK < 0, GK(n) is given byGK(n) =1√(E−2E0)2− (2JK)2−E−2E02JK−√(E−2E02JK)2−1|n|. (3.59)For the derivations of the above two equations, please refer to page 88 of Economou’sbook on Green’s function [129]. Since the analysis for the two scenarios are verysimilar, we will only concern ourselves with the first one in the following discus-sions.The reason we want to calculate GK(n) before working with Eq. (3.37) is thatEq. (3.37) contains GKk1,k2 and GKk1,k2, the Fourier transform of GK(n), that isGKk1,k2 =∑mGK(m)e−i(k1−k2)m/2 . (3.60)To utilize the known analytical expression of GK(n), we divide both sides of Eq. (3.37)by [E− ε(k1)− ε(k2)] and rewrite D˜(k) in terms of D(n) using Eq. (3.9). The resultis then given by71Ck1,k2︸ ︷︷ ︸10=1NmolGKk1,k2 ∑k′1+k′2=k1+k2∑nD(n)ei(k1−k′1)nCk′1,k′2︸ ︷︷ ︸11−1NmolGKk1,k2 ∑k′1+k′2=k1+k2ε(k′1)Ck′1,k′2︸ ︷︷ ︸12−1NmolGKk1,k2 ∑k′1+k′2=k1+k2ε(k′2)Ck′1,k′2︸ ︷︷ ︸13. (3.61)This is the equation we work with to derive an equation for CK(n). Since thederivation is not straightforward, we illustrate some difficult points by dealing withthe equation term by term.Substituting Eq. (3.44) and Eq. (3.60) into Eq. (3.61) gives for each term:10 =1√Nmol∑lCK(l)e−i(k1−k2)l/2 , (3.62)7211 =1Nmol∑mGK(m)e−i(k1−k2)m/2 ∑k′1+k′2=k1+k2∑nD(n)ei(k1−k′1)n×1√Nmol∑lCK(l)e−i(k′1−k′2)l/2=1Nmol√Nmol∑m,n,l∑k′1+k′2=k1+k2GK(m)e−i(k1−k2)m/2D(−n)e−i(k1−k′1)n×CK(l)e−i[k′1−(k1+k2−k′1)]l/2=1(Nmol)3/2 ∑m,n,lGK(m)e−i(k1−k2)m/2D(n)e−ik1nCK(l)ei(k1+k2)l/2 ∑k′1+k′2=k1+k2eik′1(n−l)=1√Nmol∑m,n,lGK(m)e−i(k1−k2)m/2D(n)e−ik1nCK(l)ei(k1+k2)l/2δn,l=1√Nmol∑m,lGK(m)e−i(k1−k2)m/2D(l)CK(l)e−i(k1−k2)l/2 , (3.63)12 =1Nmol∑mGK(m)e−i(k1−k2)m/2 ∑k′1+k′2=k1+k2[E0 +∑nJ(n)eik′1)n]×1√Nmol∑lCK(l)e−i(k′1−k′2)l/2=1√Nmol∑m,n,lGK(m)e−i(k1−k2)m/2ei(k1+k2)l/2×E0CK(l)1Nmol∑k′1+k′2=k1+k2e−ik′1l + J(l)CK(l)1Nmol∑k′1+k′2=k1+k2eik′1(n−l)=1√Nmol∑m,n,lGK(m)e−i(k1−k2)m/2ei(k1+k2)l/2[E0CK(l)δl,0 + J(l)CK(l)δl,n]=1√Nmol∑m,lGK(m)e−i(k1−k2)m/2J(l)CK(l)eiKl/2 (3.64)73and13 =1√Nmol∑m,lGK(m)e−i(k1−k2)m/2J(l)CK(l)e−iKl/2 . (3.65)A comparison between the left hand side and the right hand side of Eq. (3.61)indicates that we can eliminate the exponent e−i(k1−k2)m/2 from 12 and 13 byexchanging the index l with m. However, term 11 will cause a problem becauseafter exchanging l with m it becomes11 =1√Nmol∑m,lGK(l)e−i(k1−k2)l/2D(m)CK(m)e−i(k1−k2)m/2 . (3.66)After dividing both sides of Eq. (3.61) by the common factor e−i(k1−k2)l/2, the ex-ponent e−i(k1−k2)m/2 will still remain in term 13 . As mentioned before, we wantto obtain an equation for CK(n) that has only dependence on the summation ofwavevectors K = k1 + k2, so we don’t want e−i(k1−k2)m/2 to appear. It turns out thisproblem can be resolved by making use of the periodicity of the lattices. Due tothe translational symmetry of the lattices, l−m takes the same range (−∞,∞) asl for a crystal with infinite size, so we can replace the summation over l with thesummation over l−m, that is∑lGK(l)e−i(k1−k2)l/2 = ∑l−mGK(l−m)e−i(k1−k2)(l−m)/2= ∑lGK(l−m)e−i(k1−k2)(l−m)/2 . (3.67)Although this equation is only strictly valid for a lattice with infinite number ofsites, we can still use it for the case of sufficiently large lattices with a large Nmol.Substituting Eq. (3.67) into Eq. (3.66), we obtain11 =1√Nmol∑m,lGK(l−m)D(m)CK(m)e−i(k1−k2)l/2 , (3.68)so that Eq. (3.61) reduces toCK(l) =∑mCK(m)[GK(l−m)D(m)−2GK(l)J(m)cos(Km/2)]. (3.69)74In the nearest neighbor approximation, the above equation for CK(m) can be sim-plified asCK(l) =CK(1){D(1)[GK(l−1)+GK(l +1)]−2GK(l)JK}, (3.70)where GK(l) can be calculated from Eq. (3.58) and JK is defined by Eq. (3.55).Substituting Eq. (3.58) into Eq. (3.70) and taking l to be 1, we obtain an equa-tion for the biexciton energy EKb which can be solved to giveEKb = 2E0 +D+J2KD. (3.71)Substituting this solution into Eq. (3.58), we can obtain an expression for GK(n) interms of D and JGK(n) =DD2− J2K(JKD)|n|. (3.72)Up to this point, everything except CK(l) in Eq. (3.70) has been known, thereforeevery other CK(l) can be written in terms of CK(1), so that the normalization con-dition for CK(l), i.e., Eq. (3.50), becomes an equation for CK(1) only. We caneasily solve Eq. (3.50) by assuming Nmol is sufficiently large and D > JK to obtainCK(n) =√D2− J2K2D(JKD)|n|−1. (3.73)So the biexciton wave function in the site representation in the nearest neighborapproximation because|Ψb(K)〉= ∑n,m6=neiK(n+m)/2ψK(|n−m|)|Pˆ†n Pˆ†m〉,ψK(r) =√D2−4J2 cos2(K/2)2D√Nmol(2J cos(K/2)D)|r|−1, (3.74)where r = n−m is the distance between two excitations. Since JK = 2J cos(K/2)<D, we can see from the biexciton wavefunction that the amplitude ψK(r) for twoexcitations to appear at two sites decays exponentially with the distance betweenthe two sites, which indicates that biexciton states are indeed bound states.753.3.3 Properties of biexciton statesReference [107] shows that biexcitons can generally form in 1D and 2D crystals if|D| ≥ 2|J| , (3.75)where D and J are the coupling constants D(n−m) and J(n−m) for n−m = 1. In3D crystals, biexcitons can form if |D|> 6|J| [110]. For 1D crystals, the biexcitonenergy in the nearest neighbor approximation (NNA) isEb(K) = 2E0 +D+4J2 cos2(aK/2)D, (3.76)where K is the total wave vector for two interacting excitons and a is the latticeconstant, and the biexciton binding energy is ∆ = (D− 2J)2/D. The maximumnumber of exciton–exciton bound states is equal to the dimensionality of the crys-tal, i.e. one biexciton state for 1D, two for 2D and three for 3D [107, 130].For molecules in an optical lattice, the magnitudes of J and D depend on thestrength of the applied electric field and the angle between the field and the molec-ular array (θ ), as seen in Fig. 3.1 (c). We calculate these parameters for a 1D arrayof 1Σ polar molecules (such as alkali metal dimers produced in ultracold moleculeexperiments) trapped on an optical lattice with the lattice separation a = 400 nm.Fig. 3.1 (b) shows that for a fixed angle θ the ratio D/J increases as the electricfield magnitude increases. For LiCs molecules, the condition of Eq. (3.75) is sat-isfied for electric fields > 3.6 kV/cm. Note that the ratio D/J is independent ofθ .Frenkel excitons are quasiparticles characterized by an effective mass, meff.The sign of J determines the sign of the effective mass [67]: negative J correspondsto positive meff and vice versa, as seen in Fig. 3.1 (d). Due to the linearity of theSchro¨dinger equation, a positive potential is attractive for particles with negativemass, just like a negative potential is attractive for particles with positive mass.Because the signs of J and D are the same (and consequently the signs of D andmeff are opposite), the dynamical interaction of Eq. (3.6) between excitons in thissystem is attractive.To demonstrate the formation of the biexciton and calculate the biexciton en-76-60-40-2002040Energy (kHz)-80 10 30 50 70 90θ  (degree)DJDJ0 2 4 6 8 10ε  d / Bm     > 0effm     < 0eff(d) e f0 2 4 6 8 10123456D/J00. (units of V  )0 ddθ*90oθ = (a)(b)(c)ε  d / B e fθε fFigure 3.1: (a) The parameters D and J (in units of Vdd = d2/a3) as functionsof the electric field strength. (b) The ratio D/J as a function of the elec-tric field strength. The field is perpendicular to the intermolecular axis.For LiCs molecules possessing the dipole moment d=5.529 Debye, thevalue E f d/Be = 1 corresponds to E f = 2.12 kV/cm. (c) Schematic de-piction of the angle θ between the field (represented by blue arrows)and the molecular array (represented by red dots). (d) D and J for a1D array of LiCs molecules separated by 400 nm as functions of θ forE f = 6 kV/cm.77Energy (kHz)Energy (kHz)Binding energy (kHz)-40-200204060-60-40-2002040-30-20-1001020(a)(c)(b)(d)D > 0D < 090oθ = θ0θ = 2- 2 0Ka2- 2 0KaΔ(θ)θ*θ010 30 50 70 90θ  (degree)50| r | D / 2J1.051.252.5 51015201 2 3rψ (r)K0.250.50 1Nmol1001Figure 3.2: (a) and (b): Two-excitation spectra of a 1D array of LiCsmolecules on an optical lattice: NNA (dashed lines) and exact solutions(solid lines). The shaded regions encapsulate the bands of the contin-uum two-exciton states. (c)θ -dependence of the biexciton binding en-ergy ∆. The electric field magnitude is 6.88 kV/cm, θ0 = arccos√2/3,θ ∗ = arccos√1/3. (d) Biexciton wave function vs the lattice site sepa-ration |r|= |n−m| of the two excitations for K = 0. Inset: Mean widthof the biexciton wave function 〈r〉 calculated as the width of ψ2K(r) athalf maximum. Numbers on each curve indicate the value of D/2J.78ergy, we diagonalize the Hamiltonian Hˆexc + Hˆdyn for a one-dimensional array ofNmol = 501 LiCs molecules. The matrix of the Hamiltonian is evaluated by ex-panding the biexciton states as|Ψb(K)〉 = ∑k1+k2=Kk1≥k2Ck1,k2 |k1,k2〉= ∑k≥0CKk |Pˆ†(K/2+ k)Pˆ†(K/2− k)〉, (3.77)where K = k1+k2 and k = (k1−k2)/2, and k1 and k2 denote the wavevectors of theinteracting excitons. The Hamiltonian matrix is diagonalized numerically for fixedvalues of K, which is conserved. Figure 3.2 (c) shows that for θ = 90o > θ ∗ =arccos(1/√3) the biexciton energy is above the two-exciton continuum (bindingfor particles with negative mass), and for θ = arccos√2/3 < θ ∗ below it (bindingfor particles with positive mass). The binding energy of the biexciton changes signat θ = θ ∗. The biexciton wave function ψK(r) is plotted in Fig. 3.2 (d). Figure 3.1and Fig. 3.2 thus illustrate that the biexciton binding energy and size can be tunedby varying the strength and orientation of the electric field.3.4 Non-optical creation of biexcitonsThe possibility of forming Frenkel biexcitons has been proposed quite a long timeago[110]. However, in contrast to the well-known Wannier-Mott biexcitons insemiconductors, Frenkel biexcitons in solid-state molecular crystals are very diffi-cult to observe. We now explain the reasons. First, many molecular crystals, suchas anthracene or naphthalene, possess inversion symmetry. In these crystals, theconstant D as defined in the line after Eq. (3.6) must vanish and Eq. (3.75) is notsatisfied. Second, it is difficult to excite biexciton states optically: it was shownin Ref.[108] that the oscillator strength for the photon-induced transitions to thebiexciton state must decrease with the increasing binding energy of the biexcitons.Therefore, two-photon excitation can only produce unstable weakly bound biexci-tons. Third, excitons in molecular crystals decay via bimolecular annihilation pro-cesses into higher-energy states and subsequent relaxation accompanied by emis-sion of phonons. This process is prohibited by conservation of energy in an optical79lattice with diatomic molecules. Figure 3.1 demonstrates that the first obstacle canbe removed by tuning the electric field. To overcome the second obstacle, we pro-pose a non-optical method of populating deeply bound biexciton states based on theunique structure of 1Σ polar molecules. At zero electric field, the rotational states|g〉 and |e〉 are separated by the energy ∆εe−g = 2Be, while the energy separation be-tween state |e〉 and the next rotationally excited state | f 〉 ≡ |N = 2,MN = 0〉 is equalto ∆εe− f = 4Be. As the electric field increases, ∆εe−g increases faster than ∆εe− f ,as shown in Fig. 3.3. When E f d/Be ' 3.24 (corresponding to E f ' 6.88 kV/cmfor LiCs), ∆ε f−g = 2∆εe−g. At electric fields near this magnitude, two |g〉 → |e〉excitons can undergo the transition to the | f 〉 state, and, inversely, the |g〉 → | f 〉excitation can produce a pair of |g〉 → |e〉 excitons or a biexciton state depicted inFig. 3.2. The coupling between states |e〉 and | f 〉 is Hˆ12 = ∑n6=mM(n−m)RˆnPˆ†n Pˆ†m,where M(n−m) = 〈en,em|Vdd(n−m)| fn,gm〉, and the operator Rˆn annihilates the| f 〉 excitation in lattice site n. The total Hamiltonian describing this three-levelsystem isHˆg−e− f = Hˆexc + Hˆdyn + Hˆ2 + Hˆ12 , (3.78)whereHˆ2 = E f ∑nRˆ†nRˆn + ∑n,m6=nJg− f (n−m)Rˆ†nRˆm (3.79)andJg− f (n−m) = 〈gn, fm|Vdd(n−m)| fn,gm〉 . (3.80)In order to calculate the probability of the population transfer from state f tothe biexciton state, we solve the time-dependent Schro¨dinger equation with theHamiltonian Hˆg−e− f evaluated in the basis of products of the eigenstates of Hˆexc +Hˆdyn and the eigenstates of Hˆ2. This leads to coupled differential equationsih¯C˙ = HC , (3.81)where C is a vector that represents a wavefunction in the basis set and C˙ is itsderivative with respect to time. Since the total Hamiltonian H is time-independent,80024681012ε f d/B e-4-3-2-1012345Energy (units of 2Be)(N=0, MN=0)(N=1, MN =0)(N=2, MN =0)gefFigure 3.3: The rotational energies of a closed-shell polar molecule as a func-tion of the strength of a DC field. The dashed lines represent other rota-tional states with MN 6= 0.81we can make a basis-set transformation to diagonalize HUTHU = D , (3.82)so that C in the new basis set can then be solved by direct integration and thenC in the original basis set can be found by the inverse basis-set transformation.Multiplying both sides of Eq. (3.81) by UT, we obtainih¯UTC˙ = UTHUUTC , (3.83)where UUT = I has been used. Let’s defineA≡ UTC , (3.84)thenA˙ = UTC˙ (3.85)because H is time-independent and UT must be time-independent too. So Eq. (3.81)can be rewritten asih¯A˙ = DA , (3.86)which can solved formally by direct integration to giveA(t) = e−ih¯ D tA(0) . (3.87)Substituting Eq. (3.84) into Eq. (3.87), we getC(t) = Ue−ih¯ D tUTC(0) , (3.88)where D is a diagonal matrix whose non-zero elements are the eigenvalues of H,and each column of U is an eigenvector of H. Therefore, given the eigenvaluesand eigenvectors of the total Hamiltonian and the initial wavefunction C(0), thewavefunction C(t) at any time t can be calculated from Eq. (3.88). In our numericalcalculations, we use subroutines from the LAPACK library[131] to calculate theeigenvalues and eigenvectors of the matrix.The magnitude of Jg− f is about ten times smaller than J. In the absence of de-82-80-60-40-2000 50 100 150 200Time ( μs) 2-KabΨlocal60%f20Energy (kHz)plane wave87%0 50 100 150 20000. ( μs)Figure 3.4: Population dynamics for the transition from |g〉 → | f 〉 exciton(middle panel) and from an f state localized on a single molecule (lowerpanel) to coherent |g〉 → |e〉 excitons and biexcitons. The green dashedcurves denote the population accumulated in the pairs of non-bound|g〉 → |e〉 exciton states, the red solid curves the biexciton state andthe blue dot-dashed curves the f state. The shaded region in the upperpanel encapsulates the band of the continuum two-exciton states. Thecalculation is for a 1D ensemble of Nmol = 501 LiCs molecules on anoptical lattice with lattice separation a = 400 nm. The electric field ofmagnitude 6.88 kV/cm is perpendicular to the molecular array.83coherence, the |g〉 → | f 〉 excitation gives rise to the Frenkel exciton and the transi-tion from the f states to the biexciton state is a coherent exciton–exciton transition.In the presence of decoherence, the exciton states become localized. If the deco-herence rate is larger than Jg− f /h, but smaller than J/h, the |g〉 → | f 〉 excitationis localized, while the biexciton states remain coherent. Figure 3.4 presents thecalculations of the population transfer probabilities for both scenarios. The resultsshow that the biexciton states can be populated with high efficiency. The equilib-rium populations (in the limit of large t) depend on the relative energies of the fstate, the biexciton bound state and exciton–exciton continuum states, which canbe tuned by varying the electric field magnitude. The efficiency of the populationtransfer can be maximized if the electric field is detuned far away from resonancewhen the biexciton population oscillations reach the first maximum. Detuning theelectric field to low magnitudes effectively decouples the f state from the states inthe {g,e} subspace and interrupts the population dynamics. This corresponds toswitching off the channel for bimolecular annihilation of excitons, which is one ofthe reasons of the biexciton population depletion in solids. We have confirmed thatthe calculations with electric fields < 5.0 kV/cm yield no noticeable populationtransfer.3.5 Extension to exciton trimersIn Section 3.3, it was shown that Frenkel rotational biexcitons exist under someconditions in optical lattices. The question arises naturally, “If two excitons canbind together, what about three excitons?”. In this section, we extend the methodto handle the three-exciton case and answer the question about exciton trimers.Similar to the case for biexcitons, we define a three-exciton wavefunction inthe site representation as|Ψ〉= ∑n1,n2,n3Cn1,n2,n3Pˆ†n1Pˆ†n2Pˆ†n3 |0〉 , (3.89)and start from the Scho¨dinger equation with the same Hamiltonian Hˆ = Hˆexc +HˆdynHˆ|Ψ〉= Etrimer|Ψ〉 . (3.90)84Since two excitations can’t exist at the same molecule, we have the constraint onCn1,n2,n3 thatCn1,n2,n3 = 0 if n1 = n2 or n2 = n3 or n1 = n3 . (3.91)After derivations similar to those used for the biexciton case, Eq. (3.90) and Eq. (3.91)lead to[3E0D(n1−n2)+D(n1−n3)+D(n2−n3)−Etrimer]Cn1,n2,n3+∑n[J(n−n1)Cn,n2,n3 + J(n−n2)Cn1,n,n3 + J(n−n3)Cn1,n2,n]= 2δn1,n2∑nJ(n1−n)Cn,n2,n3 +2δn2,n3∑nJ(n2−n)Cn1,n,n3+2δn1,n3∑nJ(n3−n)Cn1,n2,n .(3.92)Note that this equation can’t be used to determine the values of Cn1,n2,n3 when anytwo of n1, n2, and n3 are equal. Using the Fourier transformCn1,n2,n3 =1(√Nmol)3 ∑k1,k2,k3C(k1,k2,k3)ei(k1n1+k2n2+k3n3) , (3.93)we can transform Eq. (3.92) to1Nmol∑qD˜(q) [C(k1−q,k2 +q,k3)+C(k1−q,k2,k3 +q)+C(k1,k2−q,k3 +q)]+[ε(k1)+ ε(k2)+ ε(k3)−Etrimer]C(k1,k2,k3)=2Nmol∑q[J˜(k1−q)C(k1−q,k2 +q,k3)+ J˜(k2−q)C(k1,k2−q,k3 +q)+J˜(k3−q)C(k1 +q,k2 +q,k3−q)], (3.94)which can be written as an eigenvalue problem∑k′1+k′2+k′3=KAk1k2k3;k′1k′2k′3C(k′1,k′2,k′3) = EtrimerC(k1,k2,k3) , (3.95)85whereAk1k2k3;k′1k′2k′3= δk1,k′1δk2,k′2δk3,k′3 [ε(k1)+ ε(k2)+ ε(k3)]+1Nmol{δk1,k′1[D˜(k2− k′2)−2J˜(k′2)]+δk2,k′2[D˜(k3− k′3)−2J˜(k′3)]+δk3,k′3[D˜(k1− k′1)−2J˜(k′1)]}. (3.96)The above equations clearly show that only those coefficients C(k1,k2,k3) whosewavevectors add up to a fixed K are coupled to each other. Therefore, the eigen-values for each value of K can be calculated independently. However, since weassumed that Cn1,n2,n3 = 0 when any two of n1,n2,n3 are equal, we need to takecare of this constraint when solving Eq. (3.95).Let’s examine the effect of the constraint on Cn1,n2,n3 . Assuming k3 is fixed, wehave∑k1+k2=K−k3C(k1,k2,k3) = ∑k1∑n1,n2,n31(√Nmol)3 ei(k1n1+k2n2+k3n3)Cn1,n2,n3= ∑n1,n2,n31√Nmol(1Nmol∑k1eik1(n1−n2))ei(K−k3)n2eik3n3Cn1,n2,n3= ∑n1,n2,n31√Nmolδn1,n2ei(K−k3)n2eik3n3Cn1,n2,n3= ∑n2,n31√Nmolei(K−k3)n2eik3n3Cn2,n2,n3= 0 . (3.97)Since the fixed wavector k3 can be chosen arbitrarily, a condition similar toEq. (3.97) will hold for a fixed k1 and a fixed k2 as well. This means for a par-ticular K, summing all the C(k′1,k′2,k′3) whose first (second or third) wavevector86k′1(k′2 or k′3) is equal to a fixed k, will give zero. Thus, we can add the zero1Nmol∑k′1,k′2,k′3[−2J˜(k2)δk′1,k1C(k′1,k′2,k′3)−2J˜(k3)δk′2,k2C(k′1,k′2,k′3)−2J˜(k1)δk′3,k3C(k′1,k′2,k′3)]= 0 , (3.98)to the left hand side of Eq. (3.95) to produce a new equation∑k′1+k′2+k′3=KBk1k2k3;k′1k′2k′3C(k′1,k′2,k′3) = EtrimerC(k1,k2,k3) , (3.99)whereBk1k2k3;k′1k′2k′3= δk1,k′1δk2,k′2δk3,k′3 [ε(k1)+ ε(k2)+ ε(k3)]+1Nmol{δk1,k′1[D˜(k2− k′2)−2J˜(k2)−2J˜(k′2)]+δk2,k′2[D˜(k3− k′3)−2J˜(k3)−2J˜(k′3)]+δk3,k′3[D˜(k1− k′1)−2J˜(k1)−2J˜(k′1)]}.(3.100)The advantage of Eq. (3.99) over Eq. (3.95) is that the matrix B is symmetric,which leads to easier diagonalization.The energies of three-exciton states can be obtained by diagonalizing the ma-trix B in Eq. (3.99) and filtering out the eigenvalues whose corresponding eigen-vectors don’t satisfiy Eq. (3.97). As a proof-of-principle example, we have solvedEq. (3.99) for a small lattices under the nearest-neighbor approximation and presentthe results in Fig. 3.5. As can be seen from the figure, three excitons may form athree-body bound state or a biexciton plus a free exciton. Since the energies ofthree-body bound states are located outside the energy continuum of a biexcitonplus a free exciton, an interesting question arise, “Do three-body bound states ofexcitons exist when two-body bound states don’t form, that is, does the Efimov ef-fect occur in the current case?”. Unfortunately, numerical investigation shows thatthe three-body and two-body bound states always occur at the same time when D/Jreaches 2, indicating no Efimov effect. However, this is not a conclusive result as87only the interaction between nearest neighbors are included in the calculation andthe effect of long-range interactions needs to be examined.-3-2-10123 K-60-40-20020406080Energy (kHz)Figure 3.5: Three-excitation spectra of a 1D array of molecules on an opti-cal lattice. The calculation is done for a system of 20 lattice sites withthe hopping interaction J = 10 kHz and the dynamic interaction D = 30kHz. The black dots represent energies of all three-exciton states, thered curves denote the boundaries of energy continuum of three free exci-tons, and the blue curves represent the boundaries of energy continuumfor a biexciton plus a free exciton.3.6 DiscussionWe have shown that rotational excitation of molecules trapped on an optical lat-tice gives rise to rotational excitons whose interactions can be controlled by anexternal electric field. The exciton–exciton interactions can be tuned to producetwo-exciton bound states. A biexciton is an entangled state of two Frenkel exci-88tons. The creation of biexcitons and tuning of the electric field to the regime of zerobinding energy can thus be used for the controlled preparation of entangled pairsof non-interacting excitons. In order to observe the biexcitons, one could measurecorrelations between the populations of the rotationally excited states of moleculeson different lattice sites by applying a gradient of an electric field and detectingresonant transitions from Stark-shifted levels, as proposed in Ref. [59].The present work suggests several interesting questions. For example, it wasrecently shown that Frenkel excitons in shallow optical lattices can be coupled tolattice phonons, leading to polarons [69]. Coupling a Frenkel biexciton to phononswould produce strongly interacting polarons. It would be interesting to explorewhether these interactions lead to the formation of bipolarons.We have repeated the calculations presented here for a system of three excitonsand similarly observed the formation of three-exciton bound states. It would be in-teresting to explore the effect of tunable exciton–exciton interactions on excitationcorrelations, both as a function of D/J and the density of excitations, to understandfundamental limits of exciton clustering [132].The creation of biexcitons with tunable binding energy and measuring quan-tum energy transport for different ratios D/J can be used to study the effects ofexciton–exciton entanglement on energy transfer in molecular aggregates [133–135]. The ability to tune exciton–exciton interactions can be used to explore therole of multiple excitation correlations on energy transfer in disordered systems(the confining lattice potential can be tilted or the molecules can be perturbed by adisorder potential produced by an inhomogeneous electric field).89Chapter 4Quantum energy transfer inordered and disordered arraysAn elementary excitation in an aggregate of coupled particles generates a collectiveexcited state. We show that the dynamics of these excitations can be controlled byapplying a transient external potential which modifies the phase of the quantumstates of the individual particles. The method is based on an interplay of adiabaticand sudden time scales in the quantum evolution of the many-body states. We showthat specific phase transformations can be used to accelerate or decelerate quantumenergy transfer and spatially focus delocalized excitations onto different parts ofarrays of quantum particles. We consider possible experimental implementationsof the proposed technique and study the effect of disorder due to the presence ofimpurities on its fidelity. We further show that the proposed technique can allowcontrol of energy transfer in completely disordered systems.4.1 IntroductionThe experiments with ultracold atoms and molecules trapped in optical latticeshave opened a new frontier of condensed-matter physics research[8–17]. Theunique properties of these systems – in particular, large (> 400 nm) separationof lattice sites, the possibility of tuning the tunneling amplitude of particles be-tween lattice sites by varying the trapping field and the possibility of controlling90interparticle interactions with external electric or magnetic fields – offer many ex-citing applications ranging from quantum simulation of complex lattice models[43, 50, 66, 120–127, 136–138] to the study of novel quasi-particles [139] that can-not be realized in solid-state crystals. In the limit of strong trapping field, each siteof an optical lattice is populated by a fixed number of ultracold atoms or molecules.Such states can be produced with either bosonic or fermionic particles [62, 138].Here, we consider an optical lattice fully or partially filled with one particle per lat-tice site, and assume that tunneling between lattice sites is completely suppressed.Such an array can be thought of as a prototype of a system, in which a single latticesite (or a small number of lattice sites) can be individually addressed by an externalfield of a focused laser beam. This can be exploited for engineering the propertiesof quantum many-body systems by changing the energy of particles in individuallattice sites [140].In the present chapter, we consider the generic problem of energy transfer –i.e. the time evolution of an elementary quantum excitation – in such a system.In particular, we explore the possibility of controlling energy transfer through anarray of coupled quantum monomers by applying monomer-specific external per-turbations. This is necessary for several applications. First, collective excitations inmolecular arrays in optical lattices have been proposed as high-fidelity candidatesfor quantum memory [118, 119]. The ability to manipulate collective excitations isnecessary for building scalable quantum computing networks [141]. Second, ultra-cold atoms and molecules in optical lattices can be perturbed by a disorder potentialwith tunable strength [142]. Engineering localized and delocalized excitations insuch systems can be used to investigate the role of disorder-induced perturbationson quantum energy transfer, a question of central importance for building efficientlight-harvesting devices [143]. Third, the possibility of controlling energy transferin an optical lattice with ultracold atoms or molecules can be used to realize inelas-tic scattering processes with both spatial and temporal control. Finally, control overenergy transfer in quantum systems can be used for studying condensed-matter ex-citations and energy transport without statistical averaging.An excitation of a coupled many-body system generates a wave packet rep-resenting a coherent superposition of single-particle excitations. The method pro-posed here is based on shaping such many-body wave packets by a series of sudden91perturbations, in analogy with the techniques developed for strong-field alignmentand orientation of molecules in the gas phase [144]. Alignment is used in molecularimaging experiments and molecular optics [144–147], and is predicted to providecontrol over mechanical properties of molecular scattering [148, 149]. Here, weconsider the use of similar techniques for controlling quantum energy transfer inan interacting many-body system. When applied to a completely ordered system,the proposed method is reminiscent of the techniques used to move atoms in op-tical lattices, where a uniform force is applied for a short period of time [150].The conceptual difference comes from the fact that in the present case the momen-tum is acquired by a quasi-particle – a collective excitation distributed over manymonomers. During the subsequent evolution, the particles do not move – rather,the excitation is transferred from one monomer to another. In order to control suchexcitations, we exploit an interplay of the adiabatic and sudden time scales, whichcorrespond to single-monomer and multi-monomer evolution. We also exploit thewave-like nature of the excitation wave function to draw on the analogy with waveoptics. This analogy, too, is not complete due to the discrete nature of the lattice.In order to emphasize the generality of the proposed method, we formulatethe problem in terms of the general Hamiltonian parameters. We then describe indetail how the required external perturbations can be realized in experiments withultracold atoms and molecules. The possibility of using rotational excitations inmolecular arrays is particularly interesting due to the long lifetime of rotationallyexcited states. Electronic excitations of atoms in an optical lattice may also giverise to collective excitations [151]. However, the lifetime of these excited statesis limited by fast spontaneous emission [152, 153]. We propose a mechanism forsuppressing spontaneous decay by tailoring the properties of the excitation wavepackets.The chapter has the following structure. Section 4.2 and Section 4.3 presentthe results in terms of the general Hamiltonian parameters. Section 4.4 addressesthe particular case of ultracold atoms and molecules. Section 4.5 discusses con-trolled energy transfer in systems with, specifically, dipole - dipole interactions.Section 4.6 considers the effects of lattice vacancies on the possibility of control-ling energy transfer and Section 4.7 extends the proposed technique to control ofexcitation dynamics in strongly disordered arrays with a large concentration of im-92purities. Section 4.8 presents the conclusions.4.2 Sudden phase transformationConsider, first, an ensemble of N coupled identical monomers possessing two in-ternal states arranged in a one-dimensional array with translational symmetry. TheHamiltonian for such a system is given byHexc = ∆Ee−g∑n|en〉〈en|+∑n,mα(n−m)|en,gm〉〈gn,em| , (4.1)where |gn〉 and |en〉 denote the ground and excited states in site n, ∆Ee−g is themonomer excitation energy and α(n−m) represents the coupling between twomonomers at sites n and m. Any singly excited state of the system is given by|ψexc〉=N∑n=1Cn|en〉∏i6=n|gi〉. (4.2)In general, the expansion coefficients Cn are complicated functions of n determinedby the properties of the system, in particular, the translational invariance or lackthereof as well as the strength of the disorder potential. If an ideal, periodic sys-tem with lattice constant a is excited by a single-photon transition, the expansioncoefficients are Cn = eiakn/√N and |ψexc〉 ⇒ |k〉 represents a quasi-particle calledFrenkel exciton, characterized by the wave vector k [99]. The magnitude of thewave vector k is determined by the conservation of the total (exciton plus photon)momentum. The energy of the exciton is given byE(k) = ∆Ee−g +α(k) , (4.3)with α(k) = ∑nα(n)e−iakn. In the nearest neighbor approximationE(k) = ∆Ee−g +2α cosak, (4.4)where α = α(1).With atoms or molecules on an optical lattice, it is also possible to generate alocalized excitation placed on a single site (or a small number of sites) by apply-93ing a gradient of an external electric or magnetic field and inducing transitions inselected atoms by a pulse of resonant electromagnetic field [59]. The presence ofa disorder potential, whether coming from jitter in external fields or from incom-plete population of lattice sites, also results in spatial localization. Similar to howEq. (4.2) defines the collective excited states in the basis of lattice sites, the sameexcitation state |ψexc〉 can be generally written as a coherent superposition of theexciton states |k〉 with different k:|ψ〉=∑kGk|k〉, (4.5)where Gk’s are the Fourier transforms of Cn’s in Eq. (4.2).Control over energy transfer in an ordered array can be achieved by (i) shift-ing the exciton wave packets in the momentum representation (which modifies thegroup velocity and the shape evolution of the wave packets) and (ii) focusing thewave packets in the coordinate representation to produce localized excitations inan arbitrary part of the lattice. To achieve this, we propose to apply a series of site-dependent perturbations that modify the phases of the quantum states of spatiallyseparated monomers. These phase transformations change the dynamics of thetime evolution of the collective excitations. Here we consider the transformationsleading to acceleration or deceleration of collective excitations, while the focusingphase transformations are described in Section Group velocity of wave packetBefore we discuss how to accelerate or decelerate the motion of collective excita-tions, we first clarify what determines their propagation behavior. The propagationdynamics of an exciton wave packet is determined by the exciton dispersion curve.More specifically, the first derivative of the dispersion curve gives approximatelythe propagation speed of a wave packet in real space. This can be easily shown bysimple derivations.In a perfect crystal, the eigenstates of the system are characterized by a wavevec-94tor k and their time evolution is determined by E(k) through|k(t)〉 = e−iE(k)h¯ t |k(t = 0)〉=1√N∑nei[kan− E(k)h¯ t]|n(t = 0)〉 , (4.6)where N is the number of sites in the crystal and |n(t = 0)〉 represents the state ofsystem in which only site n is in the excited state. It can be seen from Eq. (4.6) thatthe probability of excitation at each site is always 1/N for state |k〉, so the planewave |k〉 doesn’t move in real space, as expected. Different from a plane wave state,a wave packet composed of different |k(t)〉 states is not stationary in real space.This is because different |k(t)〉 components correspond to different energy E(k), sothey will have different time evolutions, leading to a time-dependent interferencepattern in real space. To illustrate this point, we consider a Gaussian wave packetin k space|ψ(t)〉 = A∑ke−(k−k0)22σ2k |k(t)〉=A√N∑ke−(k−k0)22σ2k ∑nei[kan− E(k)h¯ t]|n(t = 0)〉 , (4.7)where A is the normalization constant. Expanding E(k) around the point k = k0and ignoring terms with second and higher-order powers of (k−k0), and replacingthe summation over k by integration, an equation describing the time-evolution ofthe wave packet in real space is obtained, that is|ψ(t)〉= A′∑ne−(n−vgt)22(1/σk)2 |n(t = 0)〉 , (4.8)where A′is some factor that has nothing to do with the shape of the wavepacket inreal space. The group velocityvg =1h¯dE(k)dk∣∣∣∣k=k0(4.9)determines how fast the center of the wave packet moves in real space. There-95fore, once the exciton dispersion curve is known, the propagation speed of a wavepacket centered around k0 in momentum space can be calculated from the slope ofdispersion curve at k0.4.2.2 Phase kicking in quasimomentum spaceAs an example, Fig. 4.1 presents an exciton dispersion curve. It can seen thatdifferent regions of the dispersion curve have different slopes, corresponding todifferent group velocities. Direct optical excitation can create an exciton wavepacket which centers around k ≈ 0 in the k space. However, such a wave packethardly travels in real space as a whole since its group velocity is zero. Based onFig. 4.1 , if we want to accelerate a wave packet centered around k = 0, a sensibleway is to move the wave packet to the region of the Brillouin zone near k = pi/2where the slope of the dispersion curve is steep. Similarly, to decelerate a wavepacket we should move it to a place where the slope is more shallow. Now theproblem of accelerating/decelerating an exciton wave packet becomes the problemof moving the wave packet as a whole in k space.Figure 4.1: The dispersion curve of an exciton. The interaction between sitem and site n is proportional to 1/|n−m|3.96But how do we move an exciton wave packet in k space without changing itsshape? We can get a hint from the expression of the plane wave state|k(t)〉=1√N∑neikan|n(t)〉 . (4.10)This equation suggests that we can change |k〉 to |k+δ 〉 by adding a factor eiδan toeach term in the expansion. Since the increment δ is independent of the |k〉 state,each |k〉 component in a wave packet (Eq. (4.2)) can be transformed into |k + δ 〉in this way. This transformation shifts the wave packets by δ in k-space whilepreserving their shape. As a result, one can engineer wave packets to probe any partof the dispersion E(k) leading to different group velocities and shape evolution.Knowing that adding the proper phases is a key step, we now study how toadd those phases. There are two different time scales involved: one is related tothe evolution of the free monomer states, Tm = h¯/∆Ee−g, and the other is relatedto the excitation population transfer between monomers, Te = h¯/α . Usually, Tm issmaller than Te by several orders of magnitude. This huge difference in magnitudeenables us to work with the adiabatic and sudden time scales at the same time asdescribed below. Consider the n-th monomer subjected to an external field En(t)which varies from 0 to some value and then back to 0 in time T . If the variation isadiabatic with respect to the evolution of the free monomer states, T  Tm, eacheigenstate | f 〉 of the monomer acquires a state-dependent phase shift [154]| fn(T )〉= eiφ fn | fn(0)〉, (4.11)whereφ fn =−1h¯∫ T0E fn (t)dt , (4.12)and E fn (t) is the instantaneous eigenenergy and f can be e or g. Now considerthe action of such a phase change on the collective excitation state of Eq. (4.2).If T  Te, the change is sudden with respect to the excitation transfer betweenmonomers, so during the time T the excitation probability doesn’t have enoughtime to propagate between different monomers and the Cn’s in Eq. (4.2) remainalmost the same. Since the Gk’s are just Fourier transforms of Cn’s, Gk will remain97the same as well, conserving the shape of the wave packet |ψexc〉 in k space. Thewave packet |ψexc〉 then acquires a site-dependent phase Φn = φ en − φgn that willinfluence each |k〉 component in the same way. If Φn = Φ0 + naδ , then the mo-mentum δ is imparted onto the excitonic wavefunction without changing its shape.By analogy with the “pulsed alignment of molecules” [144], we call this transfor-mation a “phase kick” or “momentum kick”. Its action is also similar to that of athin prism on a wavefront of a monochromatic laser beam.In order to illustrate the shifting of exciton wave packets in the momentumspace, we solve numerically the time-dependent Schro¨dinger equation with theunperturbed Hamiltonian of Eq. (4.1), subjected to a transient site-dependent ex-ternal perturbation that temporarily modulates ∆Ee−g. We choose the parameters∆Ee−g = 12.14 GHz, α = 22.83 kHz and the lattice constant a = 400 nm that cor-respond to an array of polar molecules trapped in an optical lattice, as described indetail in Section 4.4. The time-dependent perturbation has the form of a short pulsewith the duration T = 3 µs. The phase acquired by the particles during this timeis given by Φn 'Φ0−1.29n, which can be achieved with a focused laser beam, asdescribed in Section 4.4.The excitation at t = 0 is described by a Gaussian wave packet of the excitonstates |ψexc(k)〉, with the central wavevector k = 0. Figure 4.2 shows that the en-tire wave packet acquires momentum during the external perturbation pulse (leftpanels). This is manifested as a phase variation in the coordinate representation,and as a shift of the central momentum in the k-representation. After the externalperturbation is gone, the wave packet does not evolve in the k-representation butmoves with the acquired uniform velocity in the coordinate representation.As a side note, if the laser intensity remains the same after reaching the maxi-mum, the wave packet oscillates in momentum and coordinate spaces (see Fig. 4.3).This phenomenon is analogous to conventional Bloch oscillations although herethe force is acting on a quasi-particle rather than on real particles[155]. This sug-gests the possibility of studying condensed-matter excitations and energy transportwithout statistical averaging.The results presented in Fig. 4.2 and all subsequent results of this work arefor a single collective excitation in an interacting many-particle system. A generalexperimental implementation may result in multiple excitations, leading to non-98   pipi2 pi2pi0--ka   0 1.5 3201150100501Site indexTime (µs)3 100 200Figure 4.2: Example of controlled energy transfer in a one-dimensional ar-ray of quantum monomers subjected to a linear phase transformation.The graph illustrates the evolution of the exciton wave packet centeredat k = 0 and initially positioned at the center of the array. The phaseof the wave function is shown by color. The brightness of color corre-sponds to the amplitude of the excitation with white color correspondingto zero amplitude. The calculation is for a one-dimensional array of 201monomers with α = 22.83 kHz and ∆Ee−g = 12.14 GHz, and the linearphase transformation Φn 'Φ0−1.29n. The corresponding experimen-tal setup is illustrated in Fig. 4.699pi0-pika(a)0 400 800 1200 1600 2000Time (µs)600500400Site index (b)Figure 4.3: “Bloch oscillation” of the exciton wave packet in the momentumand coordinate spaces. The phase of the wave function is shown bycolor as in Fig. 4.2. A low laser intensity of 106 W/cm2 is used andall other parameters are the same as in Fig. 4.2. Part (a) shows thatthe wave packet moves in k space in response to the linear laser field.However, since the wave vector is limited in the first Brillouin zone,when the wave packet reaches −pi or pi , it disappears at the boundaryand reappears at the other boundary. Part (b) presents the motion of thewave packet in coordinate space in accordance with the phase kicking ink space. Note that the wave packet spreads in both the momentum andcoordinate spaces because the laser intensity profile along the moleculararray is not perfect and this is amplified over an extended time period.100linear exciton interactions. There are two mechanisms for exciton - exciton inter-actions: kinematic interactions arising from the statistical properties of excitonsand dynamical interactions determined by the matrix elements of the inter-particleinteractions in the Hilbert sub-space of binary excitations [99, 156]. The effectsof the kinematic interactions on energy transfer in molecular crystals have neverbeen observed so these interactions are considered to be weak, especially in thelimit of a small number of excitations easily achievable in experiments [157]. Formolecules on an optical lattice, the dynamical interactions are important in thepresence of strong external electric fields where molecular states of different par-ity are strongly mixed [67, 139]. At weak parity-mixing fields considered here,the exciton-exciton interactions insignificantly mix different k states of the individ-ual excitons, contributing weakly to localization. These effects are expected to bemuch smaller than the disorder-induced perturbations, discussed in Section 4.6 andSection Focusing of a delocalized excitationIn order to achieve full control over excitation transfer, it is desirable to find a par-ticular phase transformation that focuses a delocalized many-body excitation ontoa small part of the lattice, ideally a single lattice site. In optics, a thin lens focusesa collimated light beam by shifting the phase of the wavefront, thus converting aplane wave to a converging spherical wave. Similarly, a phase kick can serve asa time domain “lens” for collective excitations: an excitation initially delocalizedover a large number of monomers can be focused onto a small region of the arrayafter some time. By analogy with optics, a concave or convex symmetric site-dependent phase Φ(n) applied simultaneously to all monomers may turn a broadinitial distribution Cn(t = 0) into a narrow one.The dynamics of the excitation state in the lattice is determined by the timedependence of the coefficients Cn(t) in Eq. (4.2). In order to find the expression forCn(t), we expand the amplitudes at t = 0 in a Fourier seriesCn(t = 0) =∑qeiqan√NC(q; t = 0) , (4.13)101and apply the propagator e−iE(q)t/h¯ to each q-component with E(q) representingthe exciton energy given by Eq. (4.4). Transforming the amplitudes C(q) back tothe site representation then yieldsCm(t) =1N ∑n,kCn(t = 0)ei[Φ(n)+ka(m−n)−E(k)t/h¯], (4.14)where Φ(n) is a site-dependent phase applied at t = 0, as described in the previoussection. Note that the phase Φ(n) does not have to be applied instantaneously. Thephase Φ(n, t) can be applied continuously over an extended time interval as longas the accumulated phase gives the desired outcome∫ T0 Φ(n, t)dt =Φ(n).4.3.1 Focusing to a single siteAs Eq. (4.14) shows, the focusing efficiency is determined by the phase transfor-mation and the shape of the dispersion curve E(k). Given the cosine dispersionof excitons in Eq. (4.4), is it possible to focus a delocalized excitation onto a sin-gle lattice site? To answer this question, we assume an initial condition where theexcitation is localized to the site n0, that isCn(t = 0) = δn,n0 , (4.15)and run a backward-in-time evolution to calculate the coefficients Cm(t) at t =−τ:using the expansion of an exponent in Bessel functions (Jacobi-Anger expansion,see Ref. [158])eiacosx =∑ne−i(x−pi/2)nJn(a) , (4.16)102we obtainCm(−τ) =1N ∑n,kCn(t = 0)ei[ka(m−n)−E(k)(−τ)/h¯]=1N ∑n,kCn(t = 0)eika(m−n)∑n′e−i(ka−pi/2)n′Jn′(2ατ/h¯)=1N ∑n,kδn,n0eika(m−n)∑n′e−i(ka−pi/2)n′Jn′(2ατ/h¯)=1N ∑keika(m−n0)∑n′e−i(ka−pi/2)n′Jn′(2ατ/h¯)= ∑n′eipin′Jn′(2ατ/h¯)(1N ∑keika(m−n0−n′))= ∑n′eipin′Jn′(2ατ/h¯)δn′,m−n0= eipi(m−n0)/2Jm−n0 (2ατ/h¯) , (4.17)where Jn are Bessel functions of the first kind. This calculation shows that awavepacket in the site representation, Cm(0) = eipi(m−n0)/2Jm−n0 (2αt/h¯), will fo-cus onto the lattice site n0 after evolving for time t. We can easily verify the resultby running a forward-in-time evolution starting with the initial wave packet andmaking use of the orthonormality of the Bessel functions∑nJn(x)Jn−m(x) = δm,0 . (4.18)Equation 4.17 shows that the focusing of a wave packet onto a single site wouldrequire not only adding the phase Φ(n) = Arg[Cm(−τ)], but also the amplitudemodulations of the coefficients equal to the absolute values of Cm(−τ). This sec-ond task is beyond the manipulation tools considered here. Creating such a wavepacket may require multiple and complicated phase transformations, which may bedifficult to realize in experiments.1034.3.2 Focusing a broad wavepacket in coordinate spaceA simpler procedure can be implemented if the phase transformations are restrictedto a particular part of the exciton dispersion. From wave optics, waves with quadraticdispersion can be focused, while those with linear dispersion propagate withoutchanging the wave packet shape [159, 160]. It is this interplay of the quadratic(at low k) and linear (at k ≈ ±pi/2a) parts of the cosine-like exciton dispersion(Eq. (4.4)) that precludes perfect focusing of a general collective excitation. Inorder to avoid the undesirable amplitude modulations, it may be possible to focusdelocalized excitations by a phase transformation that constrains the wave packetof Eq. (4.5) to the quadratic part of the dispersion E(k). For such wave packets,adding a quadratic phase Φ(n) = Φ0(n− n0)2 must lead to focusing around siten0. Below we illustrate the effect of the quadratic phase transformation for a broadwave packet in coordinate space. In the next subsection (Section 4.3.3), we con-tinue to explore the case of a plane wave.Consider a Gaussian wave packet with a narrow width σk,0 centered aroundk = 0 in k space withCk = Ae− a2k22σ2k,0 , (4.19)where A is some normalization factor. Since the normalization factor doesn’t mat-ter for the discussion here, we will ignore it in the following derivations. Thewave packet represented by Eq. (4.19) is also a Gaussian wave packet in coordi-nate space. This can be verified by the transformation from k space to coordinate104space , that isCn =1√N∑kCk(t = 0)eikan≈A√N∫ ∞−∞dk e− a2k22σ2k,0+ikan=A√Ne− n22σ2k,0∫ ∞−∞dk e− 12σ2k,0(ka−inσ2)2=A√Nσk,0√2piae−n2σ2k,02∝ e− n22(1/σk,0)2 , (4.20)where in the second step the integration approximates the summation over k in thefirst Brillouin zone with its range extended from (−pi/a,pi/a] to (−∞,∞) since thewidth σk is very small. Eq. (4.20) also shows that the width of the wave packet incoordinate space is the inverse of its width in k space, that is, σx,0 = 1/σk,0.To focus the wave packet, we apply an inhomogeneous phase Φ(n) = Φ0(n−n0)2 at t = 0 so that the wave packet becomesCn(t = 0) ∝ e− n22(1/σk,0)2 eiΦ0(n−n0)2, (4.21)which upon a transformation from coordinate space to k space givesCk(t = 0) =1√N∑nCn(t = 0)e−ikan≈1√N∫ ∞∞dn Cn(t = 0)e−ikan∝ e−a2k2+4akn0Φ0+2in20σ2k,0Φ02(σ2k,0−2iΦ0) . (4.22)The exponent in Eq. (4.22) can be separated into a real part and an imaginary part.Since the imaginary part contributes only a trivial phase to the wavefunction and105doesn’t change the shape of the wave packet, we can ignore it and obtainCk(t = 0) ∝ e−σ2k,0(ka−2n0Φ0)22(σ4k,0+4Φ20) . (4.23)So the width of the wave packet after adding the phase isσk(t = 0) =√σ2k,0 +4Φ20σ2k,0, (4.24)which is obviously larger than the initial width σk,0. This indicates that the phaseapplied for focusing cannot be too large. Otherwise the wave packet will bebroadened beyond the quadratic region of the dispersion curve where the focus-ing scheme doesn’t work. Note that the phase does not affect the width of thewave packet in coordinate space since it only adds some phase to the coefficientsCn(t = 0) in Eq. (4.20).To see how the wave packet evolves after the phases are added, we can applythe propagator e−iE(k)t/h¯ to Eq. (4.23) and convert the wavefunction into coordinatespace. Assuming the width σk(t = 0) is very small so that a large proportion of thewave packet is within the quadratic region of the dispersion curve, then the eigenenergy of the |k〉 state can be approximated asE(k)≈ E0 +βa2k2 . (4.25)Using this equation, we obtain the wavefunction of the wave packet after time tCn(t) =1√N∑keikane−iE(k)t/h¯Ck(0)∝ ∑keikane−iE(k)t/h¯e−σ2k,0(ka−2n0Φ0)22(σ4k,0+4Φ20)∝ e−(n+4tβΦ0n0)22σx(t)2 , (4.26)106where σx(t) is a time-dependent width given byσx(t) = σx,0√(1+4tβΦ0)2 +4t2β 2σ4x,0. (4.27)Taking the derivative of σx(t) with respect to time and setting it to zero, we canfind the time at which the wave packet is most focused, namelyTF =−Φ0σ4x,0β(1+4Φ20σ4x,0) . (4.28)At time TF, the minimum width isσx,F =σx,0√1+4Φ20σ4x,0, (4.29)and the center of the wave packet is atnc = 4TFβΦ0n0 =4Φ20σ4k,0 +4Φ20n0 ≈ n0 . (4.30)Equation 4.29 shows that the wave packet will become more focused at site n0 atTF since σx,F < σx,0 and that a larger phase Φ0 will leads to a better focusing effect.However, as we discussed before, large values of Φ0 may take the wave packetoutside the quadratic part of the dispersion making Eq. (4.25) invalid. Therefore,the best focusing effect can only be achieved by balancing the two factors.In the case of excitons with the cosine dispersion curve represented by Eq. (4.4),β = −αa2. To find the optimal phase Φ∗0 that keeps the wave packet within thequadratic dispersion while focusing it, we use the condition σk,0 . 1, which yieldsΦ∗0 =±1/2σx,0 for the optimal focusing. At timet∗ ≈ 1/4αΦ∗0 , (4.31)107the wave packet is most focused and has a widthaσx,F(Φ∗0) =aσx,0√1+4Φ∗20 σ4x,0∼ a. (4.32)For the time t∗ in Eq. (4.31) to be positive, α and Φ∗0 must have the same sign.Therefore, a convex quadratic phase profile Φ(n) with Φ0 > 0 must focus collec-tive excitations in a system with repulsive couplings between particles in differentlattice sites (α > 0), and a concave quadratic phase profile Φ(n) with Φ0 < 0 mustfocus excitations in a system with attractive couplings (α < 0).4.3.3 Focusing a plane wave in coordinate spaceConsider a completely delocalized excitation of Eq. (4.2) with Cn(k; t = 0)= eiakn/√Ndescribing an eigenstate of an ideal system of N coupled monomers. Suppose E(k)in Eq. (4.14) can be approximated as E(k) = ∆Ee−g−αa2k2. As we saw in thederivation of last section, the effect of E(k) on the wave packet is to add a phasedue to time evolution, i.e. e−iE(k)t . Since ∆Ee−g in E(k) contributes the same phaseto all k components and thus adds a trivial phase to the whole wavefunction, wecan safely ignore it if only the shape of the wavefunction is concerned. Therefore,we use E(k) = −αa2k2 instead in the following derivation. To focus the planewave, the quadratic phase Φ(n) =Φ0n2 is applied at t = 0. This changes the initialwavefunction toCn(k; t = 0) =1√NeikaneiΦ0n2, (4.33)which renders the wavefunction a wave packet in k space with the coefficientsCq(t = 0) =1√N∑ne−iqanCn(k; t = 0) . (4.34)Each q component in this equation evolves according toCq(t) = e−iE(q)tCq(0) = eiαa2q2Cq(0) , (4.35)108and the wavefunction in coordinate space after time t is given byCm(t) =1√N∑qeiqamCq(t)=1N√N∑qeiqameiαa2q2∑ne−iqaneikaneiΦ0n2=e−iαa2k2N√ipiNΦ0∑qei[a2(k−q)2(αt−1/4Φ0)+qa(m+2αak)]×Θ(−NΦ0a< k−q <NΦ0a), (4.36)where Θ(z) = 1 if z is true and zero otherwise. In order to derive Eq. (4.36), weused the approximate equalityM∫−Mdx e−i(ax2+bx) ≈√piiaeib2/4a Θ(−2Ma < b < 2Ma), (4.37)obtained by approximating the error function of a complex argument Erf(√ix) bythe sign function, which is accurate for large argument x.At time t∗ = 1/4αΦ0, the terms quadratic in q in Eq. (4.36) are canceled, andthe sum over q reduces to a delta-function, if the summation limits are from −pi/ato pi/a. Therefore, the choice Φ0 = pi/N yieldsCm(t) =√ie−iNa2k2/4piδm,−νk , (4.38)where νk is the index of the initial wave vector k = 2piνk/Na, quantized due tothe discreteness of the lattice. According to the step function Θ(z) in Eq. (4.36),the dimensionless width of the wave packet in the wave vector space is ∆k(Φ0) ≡aσk(Φ0)≈ 2NΦ0. When Φ0 = pi/N, the wave packet spreads over the entire Bril-louin zone, including the linear parts of the exciton dispersion. Using Eq. (4.37)we find that for an arbitrary value of ∆k(Φ0), the site amplitudes at the time of109focusing t∗ = 1/4αΦ0 areCn(k; t = t∗)≈ei∆(Φ0)n2/2Nn√2ipi∆k(Φ0)sin(n∆k(Φ0)/2). (4.39)In order to keep the linear part of the dispersion spectrum unpopulated, we choosethe optimal focusing phase Φ∗0 ∼ 1/2N, so that ∆k(Φ∗0)∼ Numerical resultsIn Section 4.3.2 and Section 4.3.3, Eq. (4.32) and Eq. (4.39) are derived basedon the cosine dispersion curve which is valid for a many-body system with near-est neighbor interactions only. In most physical systems, the energy dispersion ismodified by long-range couplings. In order to confirm that the above predictionsare also valid for systems with long-range interactions and illustrate the focusingof delocalized excitations, I compute the time evolution of the wave packets bysolving the wave equation numerically for a system with long-range dipole-dipoleinteractions. In the calculations, I expanded the Hamiltonian of Eq. (4.1) in the sitebasis, i.e. {|en〉∏i 6=n |gi〉}, and applied the corresponding phase to the state |en〉 ofeach site at t = 0 and then calculated the time evolution to obtain the wavefunctionat each time point. To find the time for the optimal focusing, I searched for thelargest probability at the target focusing site while scanning the wavefunctions atevery time point. Figure 4.4 illustrates the focusing dynamics of a completely delo-calized excitation (panels a and b) and a broad Gaussian wave packet (panels c andd) in a system with all (first neighbor, second neighbor, etc.) couplings explicitlyincluded in the calculation. The results show that the collective excitations can befocused to a few lattice sites. The role of the long-range coupling will be explicitlydiscussed in Section 4.6.The focusing scheme demonstrated above can be generalized to systems ofhigher dimensionality. To illustrate this, we repeated the calculations presented inFig. 4.4 (c) and Fig. 4.4 (d) for a delocalized excitation placed in a square 2D latticewith an external potential that modulates the phase as a function of both x and y.Figure 4.5 shows the focusing of an initially broad wave packet onto different partsof a 2D lattice induced by the quadratic phase transformation Φ(x,y) = Φ0[(nx−1101 200 400 600 800 1001Site index00.0360.0720.1080.144Probability(a)× 201 200 400 600 800 1001Site index00.0090.0180.0270.036Probability(c)× 50 1.25 2.50 3.75 5.00Time (ms)12505007501001Site index(b)0.00.0070.0140.1380 0.45 0.90 1.35 1.80Time (ms)200350500650800Site index(d)0.00.0050.0110.035Figure 4.4: Focusing of a completely delocalized collective excitation (pan-els a and b) and a broad Gaussian wave packet of Frenkel excitons (pan-els c and d) in a one-dimensional array using the quadratic phase trans-formations at t = 0 as described in Section 4.3.2 and Section 4.3.3. Inpanels (b) and (d), the excitation probability distribution is displayed bycolor. The dashed lines show the initial distribution magnified by 20 and5 respectively in (a) and (c). The solid curves in panels (a) and (c) cor-respond to two different phase transformations focusing the same wavepacket onto different parts of the array. The calculations are performedwith the same parameters α , a, and ∆Ee−g as in Fig. 4.2. The results arecomputed with all couplings accounted for.111nx0)2 +(ny− ny0)2], where nx and ny are the lattice site indices along the x and ydirections. The calculations include all long-range couplings as in Fig. 4.4. Thecomparison of Fig. 4.4 and Fig. 4.5 illustrates that the focusing efficiency in 2Dis greater. The results also demonstrate that the delocalized excitations can beeffectively focused on different parts of the lattice simply by varying the referencesite (nx0 ,ny0) in the phase transformation.(a) ×6003.7×10−37.4×10−31.1×10−21.5×10−2            Probability(b)1 25 50 75 101Site index (x-axis)1255075101Site index (y-axis) (c) (d)Figure 4.5: Focusing of a delocalized excitation in a 2D array shown at t = 0in panel (a) onto different parts of the lattice (panels b–d). For bettervisualization, the probability distribution in panel (a) is magnified by afactor of 60. The calculations are performed with the same parametersα , a, and ∆Ee−g as in Fig. 4.2 and the quadratic phase transformation att = 0.1124.4 Experimental feasibility of phase transformationIn Section 4.2 and Section 4.3, we discussed the idea of phase kicking and itsapplication in focusing delocalized excitations, but were mainly concerned withthe theoretical perspective. In this section, we focus on experimental feasibility andshow that the techniques proposed in Section 4.2 and Section 4.3 can be realizedwith ultracold atoms or molecules trapped in an optical lattice with one particle perlattice site [57, 58, 161, 162]. Three general requirements must be satisfied:• (i) The time required for a phase transformation must be shorter than thespontaneous decay time of the excited state.• (ii) The overall coherence of the system must be preserved on the time scaleof the excitonic evolution in the entire array, set by Kh¯/α , where K is thenumber of monomers participating in the dynamics of the collective excita-tion.• (iii) The lattice constant must be large enough to allow considerable variationof the external perturbation from site to site.Optical lattices offer long coherence times (> 1 sec) and large lattice constants(> 400 nm) [31]. The lifetime of the collective excitations depends on the internalstates of the particles used in the experiment and the momentum distribution of theexcitonic states in the wave packet. In the following two subsections, the case ofultracold atoms and molecules are discussed, separately.4.4.1 Suppressing spontaneous emission in system of ultracold atomsFor ultracold alkali metal atoms in an optical lattice, an optical excitation may gen-erate the collective states of Eq. (4.2), as discussed in Ref. [152, 153]. The lifetimeof these excited states is limited by the spontaneous emission of the electronicallyexcited atoms and is in the range of 10 - 30 ns. However, the collective excitedstates can be protected from spontaneous emission if the wave vector range pop-ulated by excitons in the wave packet of Eq. (4.5) is outside of the so-called lightcone, so that k > ∆Ee−g/h¯c. To understand why this is the case, we investigate the113interaction of the collective excitation state of atoms with light. A photon can ex-cite an exciton state of ultracold atoms and the exciton can also decay as a photon.During these optical processes the energy is conserved, which leads toEph = h¯ck = h¯c√k2‖+ k2⊥ = Eex(kex) , (4.40)where k and kex represent the magnitudes of the wave vectors of the photon andthe exciton, respectively. As written in Eq. (4.40), the wave vector of the photoncan be decomposed into two parts: the part (k‖) that is parallel to kex; and the otherpart (k⊥) that is perpendicular to kex. Since only the parallel component of k isconserved in the optical excitation, we havek‖ = kex < k . (4.41)The exciton energy Eex(kex) in Eq. (4.40) can be approximated by the correspond-ing excitation energy ∆Ee−g since the interaction α between atoms is much smallerthan the energy gap between the ground state and the excited state of a single atom(see Eq. (4.3)). With this approximation, the wave vector of the photon can becalculated from Eq. (4.40) ask∗ =∆Ee−gh¯c. (4.42)Since kex is just the parallel component of k∗, only the excitonic states with kex≤ k∗can satisfy both the energy conservation (Eq. (4.40)) and the momentum conserva-tion (Eq. (4.41)) at the same time, and thus interact with light. By comparing k∗and the largest value of kex in the first Brillouin zone (−pi/a,pi/a], we can obtainthe number of excitonic k-states in the bright and dark regions. For Frenkel exci-tons originating from electronic transitions in solid-state crystals, the wavelengthof excitation light λ0 (∼ several hundred nm) is much bigger than the lattice con-stant a (∼ a few A˚), so the bright region is narrow: kex < k∗ = 2pi/λ0 ≈ 0. Foratoms trapped in optical lattices, the typical values of λ0 are of the same magnitudeas the lattice constant a (several hundred nm), and a large portion of the dispersioncurve lies in the bright region and the dark region may be narrow. In an ideal infi-nite system those states in the dark region have infinitely long radiative lifetimes,as there is no free-space photon they can emit, assuming the conservation of both114energy and momentum [99, 152, 153, 163]. In finite and/or disordered systems,the emission of photons may occur at the array boundaries or due to perturbationsbreaking the translational symmetry. In this case, the time scale for spontaneousdecay must depend on the size of the system (i.e. the probability of the excitationto reach the array boundary) and the disorder potential breaking the translationalsymmetry.Once collective excited states are created, the phase-kicking technique intro-duced in Section 4.2, if implemented on a time scale faster than the radiative life-time of a single atom, can be used to shift the excited states in the wave vectorspace away from the bright region (cf. Fig. 4.2) and thus protect the excited statesfrom fast spontaneous decay. This phase transformation can be induced by a pulseof an off-resonant laser field EAC, detuned from the e↔ g resonance by the valueδω , leading to the AC Stark shift (see e.g. Ref.[159])∆EAC = E2ACV 2eg4δω , (4.43)where Veg is the matrix element of the dipole-induced transition. By choosingVeg = 1 a.u., δω = 3Veg, and the laser intensity I = 5× 1010 W/cm2, we obtainthat the shift φ = ∆EAC × Tpulse = pi can be achieved in less than 1 ns. Anotherphase transformation can bring the excited state back to the bright region, whereit can be observed via fast spontaneous emission. The experiments with ultracoldatoms have demonstrated a lattice filling factor reaching 99 % [57, 58, 161]. Thephase transformations proposed here can be used to stabilize excitonic states inultracold atomic ensembles against spontaneous emission for multiple interestingapplications [151, 164].4.4.2 Phase kicking of collective excitation in arrays of ultracoldmoleculesThe spontaneous decay problem can be completely avoided by using rotationalexcitations in an ensemble of ultracold polar molecules trapped in an optical lattice.The rotational states are labeled by the quantum number of the rotational angularmomentum J and the projection MJ of J on the space-fixed quantization axis Z.We choose the rotational ground state |J = 0,MJ = 0〉 as |g〉 and the rotational115excited state |J = 1,MJ = 0〉 as |e〉. The state |J = 1,MJ = 0〉 is degenerate with thestates |J = 1,MJ =±1〉. This degeneracy can be lifted by applying a homogeneousDC electric field, making the |g〉 and |e〉 states an isolated two-level system. Themolecules on different lattice sites are coupled by the dipole-dipole interactionVdd(n−m). The magnitude of the coupling constant α(n−m) = 〈en,gm|Vdd(n−m)|gn,em〉 between molecules with a dipole moment of 1 Debye separated by 500nm is on the order of 1 kHz [67]. Due to the low value of ∆Ee−g, the spontaneousemission time of rotationally excited molecules exceeds 1 second.For molecules on an optical lattice, one can implement the phase kicks bymodifying the molecular energy levels with pulsed AC or DC electric fields. Therotational energy levels for 1Σ molecules in a combination of weak AC and DCelectric fields are given by [93]EJ,MJ ≈ BJ(J +1)+µ2E 2DC2BG(J,MJ)−α⊥E 2AC4+(α||−α⊥)E 2AC4F(J,MJ) , (4.44)where B is the rotational constant, G(J,MJ) is given byG(J,MJ) =J2−|MJ|2J(2J +1)(2J−1)−(J +1)2−|MJ|2(J +1)(2J +1)(2J +3), (4.45)F(J,MJ) is given byF(J,MJ) =−12[1−(2|MJ|−1)(2|MJ|+1)(2J−1)(2J +3)], (4.46)EAC is the envelope of the quickly oscillating AC field, α‖ and α⊥ are the paralleland perpendicular polarizabilities, and µ is the permanent dipole moment of themolecule.The momentum shift of the exciton wave packets can be achieved by applyinga time-varying DC electric field E (t) = E∗+E (n)sin2(pit/T ), where E (n) is linearwith respect to n. Assuming E (n) = (n−n0)A and E (n) E∗, and using Eq. (4.12)and Eq. (4.44), we obtain the phase accumulated by the excited state |1,0〉n at site116n with respect to the ground state:φ(n) = −1h¯∫ T0[Een(t)−Egn (t)]dt= −1h¯∫ T0µ2E 2DC2B[G(1,0)−G(0,0)]dt= −4µ215h¯B∫ T0[E∗+Asin2(pit/T )(n−n0)2]dt≈ −4µ215h¯B∫ T0[E 2∗ +2E∗Asin2(pit/T )(n−n0)]dt= −4µ215h¯B∫ T0[E 2∗ T +TE∗A(n−n0)]. (4.47)In the above equation, the term that is linear with lattice index n will give the phasekickδ =−4AE∗µ2T/15h¯Ba . (4.48)I have confirmed this result by a numerical computation showing that for LiCsmolecules in an electric field of E∗ = 1 kV/cm, an electric field pulse with A =7.434× 10−4 kV/cm and T = 1 µs results in a kick of δ = −pi/2a, bringing anexcitonic wave packet from the k = 0 region to the middle of the dispersion zone.An alternative strategy is to use a pulse of an off-resonant laser field, as foratoms. The phase transformations can be induced by a Gaussian laser beam withthe intensity profileI(r,z) =I01+ z2z2Rexp−2r2w20(1+ z2z2R) , (4.49)where I0 is the light intensity at the beam center, r is the radial distance from thecenter axis of the beam, z is the axial distance from the beam center, zR = piw20/λis Rayleigh range, w0 is the beam waist and λ is the wavelength. In order for thedressed rotational states |J,MJ〉 to have a definite space quantization axis, the laseris linearly polarized in the direction of the DC field. As shown in Fig. 4.6, weput the 1D molecular array along the z-axis of the beam so that r = 0 for all the117TEMPLATE DESIGN © 2008 Non-adiabatic control of quantum energy transfer in ordered and disordered arrays Ping Xiang,   Marina Litinskaya,   Evgeny A. Shapiro,   Roman V. Krems Department of Chemistry, University of British Columbia, Vancouver Main results  We propose a method for controlling collective excitations in an aggregate of coupled particles by changing the phases of the quantum states of the individual monomers.  We show that specific phase transformations can be used to  • Accelerate and decelerate quantum energy transfer  • Focus a delocalized excitation  • Realize controlled energy transfer in disordered systems  Reference: P. Xiang et al, New J. Phys. 15 063015 (2013) Theory  Quadratic phase transformation to focus excitation Effect of linear phase transformation in ordered arrays Focusing in completely disordered arrays eigenstate of an ordered array 1 1| ( ) | | | ( )ia nNiakienn in nk e e g kN         • Converts the wave front of a spreading wave packet     to that of a converging wave packet  • An example with LiCs molecules in an optical lattice   exc,| | ( ) | , , |e g n n n m n mn n mH E e e n m e g g e      coupling between particle n and  particle m energy difference between  e and g states • System  An ordered or disordered array of coupled identical monomers possessing two internal states with the Hamiltonian: m57 20  W/c10 3pulseI T s 5 m lattice constant: 400 nmA  excited state is    exc1| | | .Nn n in i nC e g     controlling the phase of the individual states = controlling collective excitations • Phase transformation   1.  Apply a pulse of external potential with the proper       duration: pulse/ /e gE T   slow enough so that it is adiabatic with respect to  the evolution of free monomer states fast enough so that there is no population transfer between different monomers 3. Different phase transformations  produce different results  a) Linear phases                to accelerate or decelerate       collective excitations  a) Quadratic phases                               to focus excitations  c) Block-dependent phases optimizing energy transfer in      disordered systems n n 20( )n n n  2. Monomer states acquire the phase:  pulsepulse 0exp[ ]|| ( ) (0) exp ( )niT fi i nif T f E t dt      depends on the external field at monomer n • Can be used to accelerate and decelerate energy      transfer Figure 1. Upper panel: an off-resonance laser pulse of duration 3 µ s causes the rotational exciton wavepacket to shift in the k-space. Lower panel: the movement of the wavepacket in the coordinate space Figure 2.  Focusing of a completely delocalized collective excitation in  a1D array. Panel (a) shows that the initial plane wave (dashed line) can be focused to very narrow area and enhance the probability by a factor much larger than 20. Panel (b) demonstrates how a plane wave evolves after the addition of  a quadratic phase.  Figure 3.  Focusing of a delocalized collective excitation in  a 2D array. The initial broad probability distribution (a) can be focused to different parts of the array in (b), (c) and (d). The probability in (a) has been magnified by a factor of 60 for better visualization.  • Similar to the “T-matrix” method in optics  • Apply the optimal phase mask for blocks of monomers   such that the contributions from various  parts of the     lattice can add up constructively at a target molecule Figure 4.  After applying the phase mask in (a), the initial random probability (b) in a lattice with 60% of vacancies focuses to a small area in (c) and the probability at the target molecule has increased by a factor much larger than 60. If no phases were applied, the vacancies will localize the excitation at random.  Figure 5. The t-matrix focusing scheme greatly enhances the probability at the target molecule even for very high vacancy percentages.   small group velocity large group velocity Figure 4.6: The experimental setup for the calculation corresponding toFig. 4.2. A 1D array of LiCs molecules trapped on an optical lattice withlattice constant a = 400 nm is subjected to a homogeneous DC field of1 kV/cm directed perpendicular to the intermolecular axis. The kickingpotential leading to the phase transformation presented by Eq. (4.54)can be provided by a λ = 1064 nm Gaussian laser beam that is linearlypolarized in the direction of the DC field, with the propagation directionalong the array axis, focused to a radius of 5 µm, with the intensity atthe focus equal to 107 W/cm2. The laser pulse is on between 0 and 3µs.molecules. Based on Eq. (4.49), the intensity for each molecule of the array isI(n) =I01+ (z0+na)2z2R, (4.50)where n is the molecule index and z0 is the distance between the center of the wavepacket and the beam center. Suppose the width of the wave packet in coordinatespace is about Sx (positive integer) times of the lattice constant a and we wantto use the laser gaussian beam to give the wave packet a momentum kick. Forconvenience, the molecule at the center of the wave packet is indexed as 0, themolecules to the left are indexed by negative integers and the molecules to the rightby positive integers. In practice, we can assume that the wave packet is confinedwithin the range [−Sxa,Sxa] for a time period much shorter than 1/α . To makethe intensity I(n) vary linearly with n, the natural way is to put the molecular arrayfar away from the beam center such that na << z0. In this case, a Taylor series118expansion of the intensity function givesI(n) = I0{z2Rz20 + z2R−2anz0z2R(z20 + z2R)2 +a2n2z2R(3z20− z2R)(z20 + z2R)3 +O(n3)}. (4.51)When z0 is large enough that omission of terms higher than linear is justified, theslope of the approximately linear intensity function I(n) is small, meaning a morepowerful laser field is needed to give the momentum kick. This is undesirable forexperiments. Instead, we find that one does not need to put the molecular array veryfar away from the beam center to obtain a linear intensity profile. The trick is tolet z0 = zR/√3 so that some higher terms (like the second-order term) in Eq. (4.51)vanish. Rather than working with the Taylor series, it is more convenient to workwith the original intensity function directly. Substituting z0 = zR/√3 and assuming1−√32anzR6= 0 givesI(n) =I01+ (z0+na)2z2R=I0(1−√32anzR)[1+(1√3+ nazR)2](1−√32anzR)=1−√32anzR43[1− 3√38(anzR)3] . (4.52)For I(n) to be linear within the range [−Sx,Sx], the second term in the square brack-ets in Eq. (4.52) should be much less than 1. This leads to a restriction on the widthof the wave packet, namelySxa . 0.5zR . (4.53)Therefore if the two conditions z0 = zR/√3 and Sxa . 0.5zR are satisfied, we canimplement the momentum kick for the wave packet. Using Eq. (4.11) , Eq. (4.44)and Eq. (4.52), we carry out a calculation similar to that in Eq. (4.47) and estimatethe momentum kick by such a pulse asδ =−√3T I0(α‖−α⊥)/80zR . (4.54)To demonstrate the feasibility of momentum kick by a laser pulse, a simulation119with z0 = 45 µm and zR = 73.8 µm was performed. The momentum kick of thewavepacket in k space, as shown in Fig. 4.2, deviates only by about 7% from theanalytical estimates.4.4.3 Focusing in system of ultracold moleculesThe Gaussian intensity profile of Eq. (4.49) can also be used to implement thequadratic phase transformations needed for focusing collective excitations. Toachieve this, a molecular array (1D or 2D) must be arranged in the z = 0 planeof the Gaussian beam. As mentioned in Section 4.4.2, an extra DC field is alsoneeded here to lift the degeneracy of rotational states and to avoid mixing of differ-ent exciton bands. In addition, the laser is linearly polarized along the direction ofthe DC field in order for the system to have a well-defined space quantization axis.Figure 4.7: An illustration to show the orientations of 1D and 2D moleculararrays inside the Gaussian beam. (a) the 1D lattice lies along the x-axisof the z = 0 plane and the DC field is at some angle with the x-axis suchthat the coupling α between molecules is negative. (b) the 2D squarelattice is at the center of the z = 0 plane and the DC field is at 45◦ degreewith the x-axis. In both cases, the laser is linearly polarized along thedirection of the DC field.First, we consider the case of a 1D molecular array. The wave packet we wantto focus is a Gaussian wave packet of width σx. We put the beam center at the120center of the wave packet and define the x-axis of the z = 0 plane to be alongthe intermolecular axis (see Fig. 4.7 (a)). Then the molecules in the lattice canbe indexed by theirs x coordinates as nxa. If the dimension of the wave packet issmaller than one third of the beam waist, the Gaussian intensity profile in Eq. (4.49)can be approximated asI(x,y = 0,z = 0)≈ I0(1−2n2xa2w20), (4.55)with an accuracy of 90%. Equation 4.55 presents a concave quadratic intensityprofile which can be used to focus a wave packet with attractive coupling (α < 0)(see Section 4.3.2). This indicates that we need to make the coupling α be-tween molecules negative. A calculation of the dipole-dipole interaction betweenmolecules in DC field shows that α is proportional to (1/3−cos2 θ) where θ is theangle between the DC field and the intermolecular axis (or x-axis). Therefore, wecan orientate the DC field to a proper direction to achieve the attractive couplingbetween molecules. As shown in Fig. 4.8 (a), θ must be in the range [0,57.3◦) toensure the attractive coupling.From previous discussions in Section 4.3.2, good focusing occurs when thequadratic phase profile is Φ(n) = Φ0n2 with Φ0 = −a/2σx if α < 0. So we willuse this particular value of Φ0 to estimate the strength of the laser field neededfor the focusing. Specifically, we are proposing to use a laser pulse with a shortramp-up and ramp-down time Ts and a long steady time Tl:I0(t) =Im sin2(piτ2Ts)(0≤ τ < Ts)Im (Ts ≤ τ ≤ Ts +Tl)Im sin2[pi(τ−Ts−Tl)2Ts+ pi2](Ts +Tl < τ ≤ 2Ts +Tl)(4.56)Based on Eq. (4.12) and Eq. (4.44), the phase acquired by a molecule at site nx121with respect to the molecule at the beam center is given byφ(nx) = −2(α‖−α⊥)I0n2xa215w20(Ts2+Tl)≈ −2(α‖−α⊥)I0n2xa2Tl15w20, (4.57)where Tl  Ts is assumed in the last step. Equating φ(nx) with the optimal phaseprofile Φ(nx) = Φ0n2x = −a/2σxn2x , and assuming the laser power is P and theinitial width of the wave packet is σx = Sa, we can estimate the duration of thelaser pulse to beTl ≈15piw404Sa2(α‖−α⊥)P. (4.58)For LiCs molecules trapped on an optical lattice with the lattice constant a =400nm, if the initial width of the wave packet is about 100 lattice sites and thebeam waist is 3 times larger than the width, the pulse duration calculated from Eq.(4.58) is about 335 µs for a laser Gaussian beam with the power of 10 W. To en-sure the focusing works as expected, the wave packet should be exposed to the laserfield for long enough time to accumulate the required phases, so the focusing timeTf for the wave packet to become most focused must be longer than the duration ofthe laser pulse. This is not a problem considering that Tf is inversely proportionalto the coupling strength αTf ≈14αΦ0, (4.59)and α can be tuned to be very small values by changing the magnitude of the DCfield [67] by only a few kV/cm or by increasing the angle θ . For instance, whenthe DC field is about 1 kV/cm and orientated along the intermolecular axis, thefocusing timeTf ≈14αΦ0=14(−20 kHz)(− 1200)= 2.5 ms (4.60)is much longer than the duration of the laser pulse, leaving enough time for theaccumulation of the phases.Second, we consider the case of a 2D molecular array. The situation is similar122to the 1D case except the anisotropy of the dipole-dipole interaction comes intoplay. Instead of only considering the coupling between molecules in the x-axis,we also have to consider the dipole-dipole interactions along the y-axis and alongother directions between the x-axis and y-axis. This is because the DC field willbe at different angles with different chains of molecules, giving rise to differentdipole-dipole interactions in different directions. For simplicity, we use the nearestneighbor approximation here and only consider the dipole-dipole interactions alongthe x-axis and y-axis. Assuming the dimension of the wave packet is smaller thanone third of the beam waist, we can approximate the Gaussian intensity profile inEq. (4.49) byI(x,y = 0,z = 0)≈ I0[1−2(n2x +n2y)a2w20]. (4.61)Similar to Eq. (4.55) in the 1D case, Eq. (4.61) is also a concave quadratic intensityprofile which can be used to focus a wave packet with attractive coupling. But thedifference here is that we need to make sure the couplings along both the x-axis andy-axis are negative. Since the interactions along x-axis and y-axis are independentof each other, the dependence of α on θ in the 1D case can be applied for the 2Dcase. Therefore, in order for the focusing scheme to work, the angle θx betweenthe DC field and the x-axis and the angle θy between the DC field and the y-axisshould be in the range [0,57.3◦). As an example, Fig. 4.7 shows one configuration(θx = θy = 45◦) which will ensure attractive coupling along both axes.4.5 Control of energy transfer in dipolar systemsDipolar interactions play a central role in the study of long-range interaction effectsusing ultracold systems [8]. While, in general, the coupling constant α in Eq. (4.1)can be determined by a variety of interactions, the dominant contribution to α foratoms and molecules on an optical lattice is determined by the matrix elements ofthe dipole - dipole interaction. It is therefore particularly relevant to discuss thespecifics of energy transfer in systems with dipolar interactions.1234.5.1 The effect of long-range interactionDipolar interactions are long-range and anisotropic. This long-range charactermanifests itself in the modification of the exciton dispersion. While Eq. (4.4) isvalid for a system with nearest neighbor couplings only, higher-order couplingsin the case of α(n−m) ∝ 1/|n−m|3 modify the exciton dispersion leading to acosine-like, but non-analytic dispersion relation, both in 1D and 2D.To see why the dispersion curve is non-analytic, we start from the expressionfor E(k)E(k) = ∆Eeg +∑nα(n)e−ikan= ∆Eeg +∑nα|n|3e−ikan , (4.62)where n is the difference between the two molecular indexes and the summationgoes from −∞ to −1 and from 1 to ∞ for a crystal of infinite size. Restricting n tobe a positive integer, Eq. (4.62) can be written asE(k) = ∆Eeg +∞∑n=12α cos(kan)n3, (4.63)giving the first derivative of the dispersion curve asdE(k)d(ka)=−∞∑n=12α sin(kan)n2, (4.64)and the second derivative asd2E(k)d(ka)2=−∞∑n=12α cos(kan)n. (4.65)The series in Eq. (4.63) and Eq. (4.64) converge with respect to n, but the series inEq. (4.65) doesn’t. For example, when ka= 0 or pi , the series in Eq. (4.65) becomesthe harmonic series and diverges with respect to n. Therefore, the dispersion curve124cannot be Taylor series expanded asE(k) = E(k0)+dE(k)dk∣∣∣∣k0(k− k0)+d2E(k)dk2∣∣∣∣k0(k− k0)2 + · · · . (4.66)Conventionally, we only consider the interactions between nearest neighborsand take n = 1 in Eq. (4.63). This might be justified if only a rough estimate of thedispersion curve is wanted because the series in Eq. (4.63) converges very fast withrespect to the distance n. Given the consideration on the convergence, higher-orderneighbors should be included in the summation to calculate the properties depen-dent on the first derivative of the dispersion curve, and all neighbors are probablyneeded to calculate properties determined by the second derivative.To investigate the effect of this nonanalyticity in dispersion curve, a series ofcalculations were performed with the long-range couplings neglected after a certainlattice site separation n−m for the 1D system. The results become converged (towithin 0.2 %) when each molecule is directly coupled with 20 nearest molecules.While the calculations with only the nearest neighbor couplings are in good agree-ment with the analytical predictions given by Eq. (4.31) and Eq. (4.32), the fullcalculations reveal that long-range couplings somewhat decrease the focusing ef-ficiency. The long-range couplings also decrease the focusing time, by up to afactor of 2. The dynamics of collective excitations leads to interference oscillationpatterns clearly visible in panels a and c of Fig. 4.4. These oscillations are muchless pronounced when all but nearest neighbor couplings are omitted. Given thatthe analytical derivation in Section 4.3 are based on a possible invalid Taylor seriesexpansion of the dispersion curve (Eq. (4.66)), the numerical results of Fig. 4.4 andFig. 4.5 are particularly important because they demonstrate that the phase trans-formations introduced in the present work are effective for systems with dipolarinteractions.4.5.2 Anisotropy of dipolar interactionThe anisotropy of the dipolar interactions can be exploited for controlling energytransfer in dipolar systems by varying the orientation of a dressing external DCelectric field. For example, for polar molecules on an optical lattice, the matrix125elements α(n−m) = 〈en,gm|Vdd(n−m)|gn,em〉 depend not only on the choice ofthe states |g〉 and |e〉, but also on the magnitude and orientation of an external dcelectric field [67, 139]. Since the value of α determines the exciton dispersion,the exciton properties can be controlled by varying the angle θ between the inter-molecular axis and the applied DC field. This is illustrated in Fig. 4.8.ak-π -π/2 ππ/2Time (μs)Site indexE(k) (kHz)150-3030-9090-150700340460220580100600 18001200 24000 3000Figure 4.8: Control of excitation transfer in a 1D many-body system withdipolar interactions by varying the direction of an external electric field.(a) Exciton dispersion curves for a 1D ensemble of diatomic moleculeson an optical lattice for different angles θ between the direction of theexternal DC electric field and the axis of the molecular array. In 1D, thecoupling α ∝ (1/3−cos2 θ). (b) Propagation of a wave packet centeredat ak =−pi/3 controlled by tuning the electric field direction. The thindotted line depicts the corresponding angle variations with time. Thebrightness of the color corresponds to the probability of the excitation.The results presented in Fig. 4.8 are for a 1D array of LiCs molecules in a lattice126with a = 400 nm. As before, |g〉 is the absolute ground state of the molecule and |e〉is the rotationally excited state that adiabatically correlates with the rotational state|J = 1,MJ = 0〉 in the limit of vanishing electric field. The upper panel of Fig. 4.8shows that the angle θ between the electric field vector and the molecular array axisdetermines the sign and magnitude of α , and therefore the shape of the dispersioncurve. This enables control over the sign and magnitude of the group velocity of anexcitonic wave packet containing contributions with k 6= 0. Dynamically tuning θ ,one can propagate a localized excitation to different parts of the lattice, as shownin Fig. 4.8 (b).In a 2D lattice, the intermolecular interactions depend on an additional az-imuthal angle φ that describes the rotation of the electric field axis around the axisperpendicular to the lattice. The numerical calculations presented in Fig. 4.9 showthat the energy flow in two dimensions can be controlled by varying both θ and φ .In addition to the phase transformation discussed earlier, this allows for a dynam-ical energy transfer in quantum many-body systems with anisotropic interparticleinteractions.4.5.3 Computation detailsThis subsection describes details of the calculations presented in Section 4.5.2. Wefirst evaluate the dipole-dipole interaction in the 1D and 2D molecular arrays , thendiscuss how to form the Hamiltonian matrix in an efficient way, and finally presentsome approximations that can reduce the computation cost significantly.As discussed in Section 2.2.2, the dipole-dipole operator for molecule A andmolecule B connected by a vector R in optical lattices is given byVˆdd(R) =−2√6pi5(1R)3∑q(−1)qY2,−q(θR,φR)[d(1)A ⊗d(1)B](2)q, (4.67)where the angles (θR,φR) describe the orientation of the vector R in the coordinatesystem whose z-axis coincides with the direction of DC field and Y2,−q are the127250 125 0 125 250Site index (x-axis)2501250125250Site index (y-axis)t=0t=3 ms(a)250 125 0 125 250Site index (x-axis)2501250125250Site index (y-axis)t=0t=3 ms(b)0 0.5 1.0 1.5 2.0 2.5 3.0Time (ms)0153045607590Angle (degree)θφ(c)0 0.5 1.0 1.5 2.0 2.5 3.0Time (ms)0153045607590Angle (degree)φθ(d)Figure 4.9: Control of excitation transfer in a 2D many-body system withdipolar interactions by varying the direction of an external electric field.Panels (a) and (b) show the trajectories of the center of an exciton wavepacket in a 2D lattice during the time from 0 to 3 ms; Panels (c) and(d) represent the changing of the dressing DC field orientation (θ ,φ)associated with (a) and (b) respectively. The initial wave packet is a 2DGaussian distribution centered around akx = aky = pi/2 and has a widthof ∼60 lattice sites in coordinate space. The magnitude of the DC fieldis fixed to 6 kV/cm while its direction is changing. The calculations aredone for a 2D array of LiCs molecules in a lattice with a = 400 nm.128spherical harmonicsY2,−2 =14√152pi sin2 θRe−2iφR ,Y2,−1 =12√152pi sinθR cosθRe−iφR ,Y2,0 =14√5pi (3cos2 θR−1) ,Y2,1 = −12√152pi sinθR cosθReiφR ,Y2,2 =14√152pi sin2 θRe2iφR . (4.68)To calculate the dipole-dipole interaction, we evaluate the matrix element of thedipole-dipole operator in the basis of bare rotational states:〈NA,MA|〈NB,MB|Vˆdd(R)|N′A,M′A〉|N′B,M′B〉=−F2R3{3sin2 θRe−2iφRDA−DB−−6sinθR cosθRe−iφR[DA0 DB−+DA−DB0]+3sin2 θRe2iφRDA+DB++6sinθR cosθReiφR[DA0 DB++DA+DB0]+√6(3cos2 θR−1)[DA+DB−+DA−DB++DA0 DB0]}, (4.69)where F and DA−/0/+DB−/0/+ are given in Section 2.2.2. In Eq. (4.69), the super-script of DA−/0/+DB−/0/+ denotes molecule X(= A or B), and the subscript S(= + or− or 0) indicates that it is nonzero only if the projection of the rotational quantumnumber of molecule X changes in a certain way: “+” means MX = M′X + 1; “−”means MX = M′X −1; 0 means MX = M′X . As depicted in Fig. 4.10 (a), rotating thecoordinate system around its z-axis does not change the physical properties of themolecular array as the relative orientation of the the DC field with respect to theintermolecular axis remains the same. Therefore we have the freedom to chooseφR = 0 in Eq. (4.69) for the 1D molecular array. However, a 2D molecular arrayhas many intermolecular axes that are at different angles (θR,φR) with the externalfield, which means one cannot make all φR zero by rotating the coordinate systemwhile keeping the relative orientation of the external field with respect to the molec-129ular array. So in general we have to consider both θR and φR when calculating thedipole-dipole interaction in a 2D molecular array.D C  f i e l dY( a ) ( b ) D C  f i e l dXZFigure 4.10: The orientations of the DC field and molecular arrays in the co-ordinate systems. (a) The DC field is along the z-axis and the 1Dmolecular array is in the direction represented by θR and φR. (b) The2D molecular array is on the XY plane and the orientation of the DCfield is represented by (θ ,φ). For clarity, we have only drawn themolecules (as blue dots) along a particular axis which is at angle γwith respect to the X-axis. It is to be understood that there are alsoother intermolecular axes at different angles with the X-axis. Notepart (a) and part (b) have different coordinate systems and the mean-ing of (θR,φR) is different from that of (θ ,φ). In fact, the angle θRbetween the DC field and the molecular array in (b) is related to θ andφ by cosθR = cosθ cos(φ − γ).Keeping track of both angles θR and φR is complicated. Fortunately, we canget rid of the dependence on φR in Eq. (4.69) by choosing to deal with only certainrotational states. In a DC field or a linearly polarized laser field, the dressed rota-tional states |N˜, |M|〉 of the system are a linear combinations of the bare rotationalstates with the same projection along the direction of the DC field or along thedirection of laser polarization, that is|N˜, |M|〉=∑NCN,M|N,M〉 (4.70)130Note that the coefficients CN,M only depend on the magnitude of the external fieldand have nothing to do with the angles θR and φR. Because those dressed stateswith different projections |M| have different energies, we can choose to work withthe states with a particular value of |M| by tuning the excitation energy to matchthe corresponding energy levels. In the current study, |M| is chosen to be zero,therefore only the bare rotational states with MX = 0 are involved, so only the termassociated with DA0 DB0 in Eq. (4.69) is nonzero, giving〈NA,0|〈NB,0|Vˆdd(R)|N′A,0〉|N′B,0〉=−√6F2R3(3cos2 θR−1)DA0 DB0 . (4.71)The dipole-dipole interaction between two dressed states with |M|= 0 can then beexpressed as linear combinations of Eq. (4.71), that is〈N˜A,0|〈N˜B,0|Vˆdd(R)|N˜′A,0〉|N˜′B,0〉=∑NA∑NB∑N′A∑N′BC∗NA,0C∗NB,0CN′A,0CN′B,0〈NA,0|〈NB,0|Vˆdd(R)|N′A,0〉|N′B,0〉 .(4.72)Equation (4.71) and Eq. (4.72) show that we only need to consider the angle θRbetween the intermolecular axis and the external field if only the dressed rotationalstates with |M|= 0 are involved. This is valid for both the 1D and 2D cases.Since a 1D molecular array can be treated as a limiting case of a 2D moleculararray with only a single intermolecular axis, we focus on the discussion of the2D case. As illustrated by Fig. 4.10, we suppose the 2D molecular array is inthe XY plane, and the orientation of the external field is given by (θ ,φ). For anytwo molecules whose intermolecular axis is at angle γ with respect to the X-axis,cos2 θR can be calculated ascos2 θR =[cos(pi2−θ)cos(φ − γ)]2= sin2 θ (cosφ cosγ+ sinφ sinγ)2 . (4.73)Therefore the dipole-dipole interaction between two molecules can be calculatedfrom Eq. (4.71) and Eq. (4.72) provided the angle between the intermolecular axis131and the X-axis is known.Once the dipole-dipole interaction is known, we can form the Hamiltonian ma-trix by expanding the Hamiltonian in the basis of individual sites. Suppose the 2Dmolecular array is a square lattice with N×N sites and the coordinate system isoriented such that its X-axis and Y -axis coincide with the bottom and left edge ofthe 2D lattice respectively. The position of the molecule in the array can then berepresented by its coordinates (x,y). Without loss of generality, we consider onlythe case where the external field is oriented such that 0≤ θ ≤ 90◦ and 0≤ φ ≤ 90◦,and give the indexes 0,1,2, · · · ,N2−1 to every molecule in the 2D array, startingfrom the bottom to the top of the array and from left to right in each row. Based onEq. (4.71) and Eq. (4.72), the matrix element Hi, j corresponding to two sites i andj is given byHi, j = 〈0˜,0|i〈1˜,0| jH0 +Vˆdd|1˜,0〉i|0˜,0〉 j= ∑NA∑NB∑N′A∑N′B−√6F2R3i, j(3cos2 θR(i, j)−1)DA0 DB0C∗NA,0C∗NB,0CN′A,0CN′B,0,(4.74)where Ri, j is the distance between molecules in sites i and j, and θR(i, j) is theangle between the external field and the intermolecular axis. Assuming the unitlength of the coordinate system is the lattice constant of the 2D array, the distancebetween the two sites i and j can be calculated from their coordinatesRi, j = a√(xi− x j)2 +(yi− y j)2 , (4.75)wherexi = i%Nyi = i/N , (4.76)and % is the mod operator. Since the angle γ between the external field and the132intermolecular axis connecting molecules in sites i and j is given byγ(i, j) = arccos[x j− yi√(xi− x j)2 +(yi− y j)2], (4.77)the value of cos2 θR(i, j) in Eq. (4.74) can be calculated from Eq. (4.73).With the Hamiltonian matrix H, we can now run the simulations presented inSection 4.5.2. The essential problem is to solve the time-dependent Schro¨dingerequationΨ˙(t) =1ih¯H(t)Ψ(t) . (4.78)Solving the above differential equations involves the computation of the Hamil-tonian matrix at many different time points. To make the simulation as fast aspossible, I now explore how to evaluate H(t) efficiently. In the calculations, thedirection of the DC field is changed adiabatically with respect to the time scale ofthe excitation hopping while the magnitude of the field is kept constant, thereforethe coefficients CNA,0, CNB,0, CN′A,0, and CN′B,0in Eq. (4.74) remain the same and thetime dependence of the Hamiltonian matrix is solely due to the time dependenceof θR(t). Based on this observation, the Hamiltonian matrix can be separated intotwo parts:H(t) = H1 [θ(t),φ(t)]∗H2 , (4.79)where “*” means element-wise multiplication, H1 is the time-dependent part thatneeds to be updated whenever θ(t) and φ(t) change, and H2 is the time-independentpart that can be computed once and saved into memory for later retrieval. It is easyto derive from Eq. (4.74) the expression for H2, that isH2 =∑NA∑NB∑N′A∑N′B−√6F2R3i, jDA0 DB0C∗NA,0C∗NB,0CN′A,0CN′B,0. (4.80)Because the angle γ in Eq. (4.73) is only dependent on the positions of the lat-tice sites, we can further separate the evaluation of H1 into different parts for the133purpose of computing efficiency. The expression for H1 is given byH1(θ(t),φ(t)) = 3cos2[θ(t)]∗{cos[φ(t)]∗Kcos + sin[φ(t)]∗Ksin}∗∗2−1 ,(4.81)where “**2” means element-wise square, and Kcos is a matrix whose elements aregiven byKcosi, j = cos(γ(i, j)) , (4.82)and Ksin is a matrix whose elements are given byKsini, j = sin(γ(i, j)) . (4.83)Since Kcos and Ksin are independent of time, we compute them once and savethem into memory for later usage. In summary, the efficient computation of H(t)is carried out in three steps: 1. calculate H2, Kcos, and Ksin and save them intomemory; 2. compute θ(t) and φ(t) and then use the values of Kcos, and Ksin inmemory to calculate H1; 3. use the value of H2 in memory to compute H.For the simulations, I used a complex ordinary differential equation solvercalled “ZVODE”[165, 166] to solve Eq. (4.78). Given an initial state Ψ(t = 0),the derivative Ψ˙ of the wavefunction with respect to time, and the desired accuracyfor the calculation, the solver gives the wavefunction Ψ(t) at a later time t. In myexperience, the ZVODE solver is not efficient for solving a rapidly changing sys-tem of differential equations where the potential energy change is very steep. Ifthis is the case, it is better to convert the time-dependent Schro¨dinger equation intoa real ordinary differential equation and use the corresponding ODE solver called“DVODE”[165, 166] that can handle rapidly changing systems much better. As areference for future students, I show below the conversion from the time-dependentSchro¨dinger equation to a real ordinary differential equation. The starting point isthe Schro¨dinger equationih¯C˙ = HC , (4.84)where C is a vector representing the state of the system. After rewriting C asC = Creal + i Cimag , (4.85)134and H asH = Hreal + i Himag , (4.86)and collecting the real part and imaginary part separately in Eq. (4.84), we obtaintwo real equationh¯C˙real = HimagCreal +HrealCimag ,h¯C˙imag = −HrealCreal +HimagCimag . (4.87)The above two equations can be written in block matrix form ash¯(C˙realC˙imag)=(Himag Hreal−Hreal Himag)(CrealCimag), (4.88)which leads to a real differential equationh¯C˙new = HnewCnew , (4.89)equivalent to the original Schro¨dinger equation.Finally, I discuss a few tricks and approximations that enable us to do the 2Dsimulations efficiently. For a 2D molecular array with N×N sites, since the cou-plings between every two sites are included, the dimension of the Hamiltonianmatrix is O(N2×N2). For a medium-sized 2D array with N = 100, the numberof coupled differential equations is about 104, which is close to the limit that ourcode can handle. As N becomes larger, the number of differential equations willincrease dramatically. This poses a dilemma. On one side, we want to limit thesize of the crystal so that the computation is fast. On the other side, as we arestudying the motion of an exciton wave packet in a DC field whose orientation ischanging adiabatically with respect to the excitation hopping, the simulation timeperiod can be long. This requires a large crystal with enough space in which thewave packet can propagate for a relatively long time without hitting the boundaries.In addition, we also want the initial size of the wave packet to be large enough sothat it is narrow in k-space and doesn’t spread very fast in coordinate space. Afterexperimenting with different arrays, we found a wave packet with a width of ∼ 60135lattice sites in coordinate space is good for illustrative purposes. If such a wavepacket is used for simulations, the crystal size must be much larger than 100×100,which is beyond the limit size that our code can handle. There are two ways toresolve the small-or-large-crystal dilemma. The most obvious choice is to use thenearest neighbor approximation so that each molecule will only interact with thefour molecules that are closest to it. The dimension of the Hamiltonian matrix willthen scale like O(N×N), and the number of differential equations drops signifi-cantly. However, this approach fails to account for the long-range interaction andmisses some characteristics of the anisotropy of the dipole-dipole interaction as itonly considers the interactions only along the X-axis and the Y -axis. In the simula-tion, I take another approach by treating different regions of the crystal separately.As the dipole-dipole interaction decays fast with distance, the lattice sites that areclose to the position of the wave packet are most important and should be alwaystaken into account. So I define a computation zone as the square box that enclosesthe wavepacket, and define the rest of the crystal as the supplement zone. At thebeginning of the simulation, I choose a large enough computation zone such thatthe probability to find excitation at the edges of the zone is almost zero, and com-pute all the interactions between any two molecules inside the zone to form theHamiltonian matrix. All the other lattice sites beyond the computation zone areignored. In this way, the size of the Hamiltonian matrix will be determined by thesmaller computation zone rather than the whole lattice and the most important partof the long-range interaction is also kept. After each time step, I find the new po-sition of the wave packet center and shift the computation zone such that its centercoincides with the wave packet center. In the process of the shifting, I lose a smallportion of the wave packet because the sites near the boundaries of the computationzone will be removed out of the computation zone at the next time point and theexcitation population at those sites will also be lost. However, if the computationzone is large enough, this should not be a problem. Therefore maintaining a smallcomputation zone and dynamically moving it based on the movement of the wavepacket allows for the calculation of the motion of a wave packet in a much biggercrystal. In Fig. 4.9, we have used a computation zone with the size 101×101 andthe size of the whole crystal is 1001×1001 (not shown in the figure). During thesimulation, I noticed that the wave packet spreads in coordinate space as time goes136on, but 99% of the wavepacket is still inside the computation zone. Because thewave packet spreads over time, our simulation will eventually break up at somepoint as more and more part of the wave packet gets lost at the boundaries of thecomputation zone.4.6 Energy transfer in the presence of vacanciesWhile experiments with ultracold atoms have produced states with one atom perlattice site with 99% fidelity [57, 58, 161], the latest experiments with moleculesyield lattice-site populations about 10% [162]. Multiple experiments are currentlyunderway to trap polar molecules on an optical lattice with close to the full pop-ulation of the lattice. However, lattice vacancies may be unavoidable in the bestexperiments. In this section, we examine the effect of vacancies on the possibil-ity of focusing collective excitations to a desired region of the lattice by the phasetransformations discussed in Section 4.3. For concreteness, we perform calcula-tions for the system described in Section 4.4, namely a 2D array of LiCs moleculeson a square optical lattice with a = 400 nm.To explore the effect of vacancy-induced interactions, we performed simula-tions for different vacancy numbers using the same parameters for molecule-fieldand inter-molecular interactions as in the calculations presented in Fig. 4.5 (b). Foreach vacancy concentration, we carried out 48 calculations with random distribu-tions of empty lattice sites. The quadratic phase transformations are applied, asdescribed in Section 4.3, in order to focus the collective excitation at time t∗ to themolecule in the middle of the 2D array.Vacancies disturb the translational symmetry of the system and produce an ef-fective disorder potential that tends to localize collective excitations [68]. Becausethe natural time evolution of the wave packet in a disorder potential may lead toenhancement of the probability in certain regions of the lattice, it is necessary todistinguish the effect of the vacancy-induced localization and the effect of the fo-cusing phase transformation. To quantify these two effects, we define two factors:the enhancement of the probability at the target molecule with respect to the initialvalue, that isη = p′(t = t∗)p(t = 0), (4.90)1370 10 20 30 40 50 60 70 80 90Vacancy percentage (%)11020304050607080Enhancement factorFigure 4.11: Enhancement factors η (red symbols) and χ (blue symbols) asfunctions of vacancy percentage in a 2D lattices. See text for the defi-nitions of η and χ . The error bars are for 95% of confidence interval.and the ratio of the probability to find the excitation on the target molecule with(p′) and without (p) the phase transformation, that isχ = p′(t = t∗)p(t = t∗). (4.91)The time t∗ is the focusing time found numerically for the corresponding vacancy-free system. The quantity η illustrates the actual enhancement of the probability tofocus a collective excitation, while the quantity χ illustrates the effect of the focus-ing phase transformation. Figure 4.11 presents the values of η and χ as functionsof the vacancy concentration. It illustrates two important observations. First, thedisorder potential with vacancy concentrations > 20 % renders the phase transfor-mation ineffective. In the presence of strong disorder, the dynamics of the system138is entirely determined by the disorder potential and the energy transfer becomeshighly inefficient (however, see Section 4.7). On the other hand, vacancy concen-trations of less than 10 % appear to have little effect on the efficacy of the focusingphase transformation.Our calculations indicate that the focusing time may be somewhat modifiedby the disorder potential, even if the concentration of vacancies is less than 10%. Figure 4.12 depicts the excitation wave functions at the time of the maximalenhancement on the target molecule, chosen as molecule (71,71). Figure 4.12shows that despite the presence of multiple vacancies, the focusing transformationenhances the probability to find the excitation on the target molecule by 16 times.4.7 Focusing in the presence of strong disorderAlthough the focusing method demonstrated in Section 4.3 and Section 4.6 appearsto be robust in the presence of a disorder potential induced by a small concentrationof vacancies, it is important for practical applications to also consider controlledenergy transfer in quantum arrays under a strong disorder potential. To considerfocusing in a strongly disordered system, we employ an analogy with the “transfermatrix” methods for focusing of a collimated light beam in opaque medium [71–80].In optics, a collimated laser beam passing through an opaque medium resultsin a random pattern of speckles arising from random scattering of light inside themedium [167]. Likewise, the random distribution of empty sites in an optical lat-tice with molecules scatters the exciton wavepackets, resulting in a completelyrandom excited state. However, in optics, the randomness of the scattering cen-ters inside the opaque medium can be compensated for by shaping the incidentwavefront with a spatial light modulator such that the contributions from variousparts of the medium can add constructively upon exit from the medium, producinga focus. We suggest that the same can be achieved with a many-body system ona lattice by separating the entire lattice into multiple blocks and applying properphase transformations to those individual blocks.The initial state for an ensemble of molecules on a lattice with multiple vacan-139(a) (b) × 1601.2×10−32.3×10−33.5×10−34.7×10−3            Probability1 25 50 75 101Site index (x-axis)1255075101Site index (y-axis) (c) (d) × 6Figure 4.12: Time snapshots of a collective excitation in a 2D array with a va-cancy concentration of 10 % (a) The distribution of the vacant sites; (b)The initial probability distribution of the excited state; (c) The prob-ability distribution of the excitation at the focusing time when the fo-cusing scheme is applied. The focusing time is found numerically asthe time when the probability at the target molecule (71, 71) reachesmaximum for a given phase transformation. (d) The probability dis-tribution of the wave function at the focusing time when the focusingscheme is not applied. The calculations are performed with the sameparameters as in Fig. 4.5. The probabilities in (b) and (d) are magnifiedby 16 and 6, respectively.140cies can be written as|ψ(t = 0)〉=∑ici(t = 0)|i〉 , (4.92)where|i〉= |ei〉∏j 6=i|g j〉 , (4.93)and the indexes i and j run over all occupied sites. After a long evolution time T ,the probability amplitude for the excitation to reside on a particular target moleculeis given byco(T ) =∑iUo,i(T )ci(t = 0)≡∑icoi(T ), (4.94)where Uo,i(t) = 〈o|exp[−iHexct]|i〉 is a matrix element of the time evolution oper-ator. In a disordered system, the transfer coefficients Uo,i are not a-priori knownand depend on the disorder potential. The phasors coi(T ) have quasi-random am-plitudes and phases. While the amplitude of each phasor cannot be controlledexperimentally, their phases are controllable via the phases of the coefficients ciat t = 0, which can be tuned using the phase-kicking transformations introducedabove. To achieve the highest probability at the target molecule, it is necessary toensure that the contribution coi = Uo,i(T )ci(t = 0) from every site i has the samephase so that they add up constructively.In a practical implementation, it may be difficult to control the phase of eachmolecule in each individual site. It may be more desirable to work with blocks ofseveral lattice sites. Assuming that the entire array of molecules can be divided intoM blocks, each containing many molecules, and that the blocks can be perturbedindividually, the excitation probability amplitude at the target molecule at time Tisco(T ) =M∑γ=1cγ(T ) , (4.95)where the contribution from block γ is given bycγ(T )≡ |cγ |eiφγ =∑i∈γUo,i(T )ci(t = 0) . (4.96)141In Eq. (4.96), the time evolution operator U depends on the randomness of vacancysites and thus is out of our control, but we can manipulate the initial state ci(t = 0).It turns out that the contributions from different blocks can be made to interfereconstructively by adding a phase exp(−iφγ) to each occupied site in block γ . ForM blocks in the array and quasi-random evolution matrix, simply setting all thephases equal may lead to an ∼M-fold increase of the excitation probability at thetarget molecule, as compared to the sum of M quasi-random phasors in Eq. (4.95)[71].Similarly to optics, the phases −φγ which must be added in each block, canbe found experimentally provided that the same (or similar) realization of disorderpersists in a series of trials. A straightforward optimization would scan throughthe strengths of phase kicks applied to different blocks. In each experiment onewould measure the excitation probability at the target molecule |co(T )|2, e.g. viaresonance fluorescence from the target molecule at the end of the experiment. Moresophisticated optimization techniques, aimed at fast focusing multi-frequency lightin optical systems, are currently under rapid development [72–80].For a proof-of-principle calculation, we consider a 2D lattice of size 101×101with 60% of the sites vacant and each non-vacant site occupied by a single LiCsmolecule. First, we determine the phase exp(−iφγ) applied to block γ . Due to timereversibility of the time evolution operator U(T ) and Eq. (4.96), we obtain|cγ |exp(−iφγ) =[n∑j=1U γo, j(T )cγj(0)]∗=[n∑j=1U γj,o(−T )cγj(0)]∗. (4.97)The matrix element U γj,o(−T ) can be calculated by performing a backward timepropagation starting from a local excitation at site “o” and calculating the coef-ficient c j(t) at time −T . Alternatively, one can propagate the evolution equationsforward in time, finding c j(T ): Since the Hamiltonian of Eq. (4.1) is real, its eigen-functions are real, and the evolution matrix U is symmetric, Uo, j = U j,o. Thus wefindc j(T ) =∑iU γj,i(T )ci(0) =Uγj,o(T ) , (4.98)since co(0) = 1 and all other coefficients are zero. For a completely delocalized1421 25 50 75 1011255075101(a) Phase0pi2pi3pi22pi (b)× 6008.2×10−31.6×10−22.5×10−23.3×10−2            Probability1 25 50 75 101Site index (x-axis)1255075101Site index (y-axis) (c) (d)× 5Figure 4.13: Focusing of a collective excitation in a strongly disordered sys-tem with 60% of lattice sites unoccupied. Panel (a) shows differentphases applied to different blocks of the lattice before the time evo-lution. (b) The initial probability distribution of the excited state. (c)The probability distribution of the excited state at the focusing timeT = 3 ms with the phase transformation depicted in panel (a) beforethe time evolution. (d) The probability distribution of the excited stateat the focusing time T = 3 ms with no phase transformation applied.The calculations are performed with the same parameters as in Fig. 4.5.The probabilities in (b) and (d) are magnified by 60 and 5, respectively.143initial state, we assume that all coefficients in Eq. (4.92) are equal, so that thephases φγ required for block γ are|cγ |exp(−iφγ) =[∑jc j(T )]∗, (4.99)where the index j runs over all occupied sites in block γ . Figure 4.13 shows thatthis choice of phases leads to effective focusing of the collective excitation in astrongly disordered system.0 10 20 30 40 50 60 70 80 90 100Vacancy percentage (%)150100150200250300350Enhancement factorFigure 4.14: Efficiency of focusing collective excitations in strongly disor-dered 2D lattices. The molecular array is divided into 400 blocks asshown in Fig.4.13(a). The focusing time t∗ is arbitrarily set to 4 ms.For each realization of disorder, we use Eq.(4.99) with T = t∗ to findthe phase mask applied to different blocks. Shown are the enhance-ment factors η (red symbols) defined in Eq.(4.90), as a function of thevacancy percentage. The error bars are for 95% confidence interval.144To illustrate the efficiency of the focusing method described above, we havecarried out a series of calculations with different vacancy concentrations. For eachvacancy concentration, we performed 48 calculations with random distributions ofempty lattice sites. The phase transformations are calculated individually for eachrandom distribution of vacancy sites as described above. The results are shown inFig. 4.14. As can be seen, the transformations proposed above are effective forvacancy concentration < 70%. At higher concentrations of vacancies, the excitedstates become strongly localized and immobile. The focusing efficiency at vacancyconcentrations 10% and 20% appears to be higher than that in the absence of va-cancies, which we attribute to the effect of the boundaries.4.8 ConclusionWe have proposed a general method for controlling the time evolution of quantumenergy transfer in ordered 1D and 2D arrays of coupled monomers. Any elemen-tary excitation in an aggregate of coupled monomers can be represented as a coher-ent superposition of Frenkel exciton states. We propose shaping the exciton wavepackets using nonadiabatic perturbations that temporarily modulate the energy lev-els of the monomers leading to monomer-dependent linear phase transformationand a displacement of the wave packets in the wave vector representation. This,combined with the possibility of focusing a collective excitation on a particularpart of the lattice by a quadratic phase transformation and with the directed propa-gation of collective excitations, allows for control of energy transfer in the lattice.An experimental observation of the excitations described here can be achieved bymeasuring site-selective populations of the molecular or atomic states by applyinga gradient of an electric field and detecting resonant transitions from Stark-shiftedlevels [59].We have presented numerical calculations for an ensemble of polar moleculestrapped on an optical lattice that demonstrate the feasibility of both momentum-shifting and focusing of collective excitations by applying external laser fields, withparameters that can be easily achieved in the laboratory. We have also investigatedthe effect of the disorder potential arising from incomplete population of the lattice.Our results show that the phase transformations leading to focusing of collective145excitations on different regions of a 2D lattice remain effective in the presenceof vacancies with concentrations not exceeding 10 %. For systems with largerconcentrations of vacancies and affected by strong disorder potentials, we proposean alternative procedure based on engineering constructive interference of the wavefunction contributions arising from difference parts of the lattice.The momentum-shifting technique proposed here can be used to protect collec-tive excitations of ultracold atoms from spontaneous emission. The spontaneousdecay processes, which in the case of an ordered many-body system must satisfyboth the energy and wave vector conservation rules, can be restricted by shiftingthe exciton wave packets to a region of the dispersion curve, where the wave vectorconservation cannot be satisfied. If performed faster than the spontaneous emissiontime, such phase transformations should create collective excitations with muchlonger lifetimes, which opens a variety of new applications for ultracold atoms onan optical lattice.As was proposed by multiple authors [168–173], molecular wavepackets canbe used to encode quantum information. Similarly, collective excitations can beused for quantum memory [118, 119]. Control over excitation transfer is neededfor creating networks of quantum processors where information is transmitted overlarge distances with photons and stored in arrays of quantum monomers via one ofthe quantum memory protocols [174]. Momentum kicking can be used for wavepacket transport within a single array. Focusing excitonic wave packets would en-able local storage of information, while directed propagation combined with con-trolled interactions of multiple excitons [139] or excitons with lattice impurities[175] may be used to implement logic gates. Controlled energy transfer in molec-ular arrays may also be used for the study of controlled chemical interactions for aclass of reactions stimulated by energy excitation of the reactants. Directing energyto a particular lattice site containing two or more reagents can be used to induce achemical interaction [176], an inelastic collision or predissociation [177] with thecomplete temporal and spatial control over the reaction process.Finally, the present work may prove to be important for simulations of openquantum systems. We have recently shown [67, 70] that the rotational excitationsof ultracold molecules in an optical lattice can, by a suitable choice of the trappinglaser fields, be effectively coupled to lattice phonons. The exciton - phonon cou-146plings can be tuned from zero to the regime of strong interactions [67, 70]. Thepossibility of shaping (accelerating, decelerating and focusing) collective excita-tions as described in the present work combined with the possibility of couplingthese excitations to the phonon bath opens an exciting prospect of detailed studyof controlled energy transfer in the presence of a controllable environment. Ofparticular interest would be to study the effect of the transition from a weakly cou-pled Markovian bath to a strongly coupled non-Markovian environment on energytransfer with specific initial parameters.We note that the effect of site-dependent phase transformations on quantumtransport was independently considered in Ref. [178] from the point of view oftime-reversal symmetry breaking. The authors of Ref. [178] propose an experi-mental realization based on ions in a linear Paul trap. Their method relies on thepossibility of tuning time-dependent phases, leading to new effects. The presentwork and Ref. [178] should be considered complementary.147Chapter 5Green’s function for two particleson a lattice5.1 IntroductionCalculating Green’s function for particles on lattices is important in the study ofcondensed matter systems. Many physical quantities can be obtained or expressedin terms of lattice Green’s functions, so they are used in a wide context such as,for example, the transport and diffusion properties of solids [129], band structure[179], statistical models of ferromagnetism [180–182], random walk theory [183],and analysis of infinite electric networks [184–187]. However, the numerical eval-uation of the Green’s function is usually a cumbersome task[81]. Berciu has re-cently developed a recursive method [81–83] to calculate the Green’s function forparticles on a lattice. This method is numerically more efficient than other existingmethods and could potentially widen the application of Green’s functions in morechallenging problems.In this chapter, I first extend Berciu’s method to a disordered system, and thenemploy the Green’s function to study the tunneling of biexciton states throughimpurities. The scattering of biexciton states is particularly interesting becausethey are compound particles and their transport behavior in a disordered lattice isnot well-understood[188].1485.2 Equation of motion for Green’s functionTo derive the equation of motion for Green’s function, we start from the Hamilto-nian for the system. We are considering a 1D periodic lattice with the HamiltonianHˆ =∑lelPˆ†l Pˆl +∑l∑r 6=0tl,l+rPˆ†l Pˆl+r +∑l∑r 6=0dl,l+rPˆ†l Pˆ†l+rPˆlPˆl+r , (5.1)where el is the energy of the particle at site l, Pˆ†l creates a particle at site l, tl,l+ris the amplitude for hopping between site l and site l + r, and dl,l+r representsthe dynamic interaction between particles in site l and site l + r. Note that thisHamiltonian Hˆ is very general as there is no restriction on the values of el , tl,l+r,and dl,l+r, and thus it can describe both an ordered system and a disordered system.The Green’s operator is defined in terms of the Hamiltonian, that isGˆ(z)≡ (z− Hˆ)−1 , (5.2)where z = E + iη is a complex energy with a small imaginary part η . From thedefinition of the Green’s operator, it is clear that(z− Hˆ)Gˆ(z) = 1 . (5.3)The above equation describes the system compactly.Depending on the system under consideration, the creation and annihilationoperators will satisfy different commutation rules. For instance, if the particles arefermions, Pˆ†n and Pˆm satisfy the anticommutation relations{Pˆ†n , Pˆm}= Pˆ†n Pˆm + PˆmPˆ†n = δn,m ,{Pˆ†n , Pˆ†m}= {Pˆn, Pˆm}= 0 , (5.4)and if the particles are bosons, Pˆ†n and Pˆm satisfy the commutation relations[Pˆ†n , Pˆm] = Pˆ†n Pˆm− PˆmPˆ†n = δn,m ,[Pˆ†n , Pˆ†m] = [Pˆn, Pˆm] = 0 . (5.5)149In the current study, we are considering collective excitations of atom or moleculesin a lattice. They have the characteristics of both fermions and bosons. When twoexcitations reside on different sites they behave like bosons, that isPˆ†n Pˆm− PˆmPˆ†n = 0 , (5.6)if n 6= m; when excitations are on the same site, they behave as fermions, that is,Pˆ†n Pˆm + PˆmPˆ†n = 0 , (5.7)if n = m. For convenience, we can combine Eq. (5.6) and Eq. (5.7) into one equa-tion:PˆmPˆ†n = δm,n +(1−2δm,n)Pˆ†n Pˆm . (5.8)Using the two-particle basis set |n,m〉 (and n 6= m), Eq. (5.3) can be rewrittenas〈n,m|(z− Hˆ)Gˆ(z)|n′,m′〉= 〈n,m|I|n′,m′〉 . (5.9)To simplify the above equation, we make use of Eq. (5.8) and evaluate the effect ofthe Hamiltonian operating on the basis states, givingHˆ|n,m〉 = (en + em)|n,m〉+∑r 6=0(1−δn−r,m)tn−r,n|n− r,m〉+∑r 6=0(1−δn,m−r)tm−r,m|n,m− r〉+∑r 6=0dn,mδn−m,±r|n,m〉 ,(5.10)where the factors like (1−δn−r,m) appear because only the two-particle states |n,m〉with n 6= m are included in the basis set. The left-hand side of Eq. (5.9) becomes〈n,m|(z− Hˆ)Gˆ(z)|n′,m′〉= (z− en− em)G(n,m,n′,m′;z)−∑r 6=0(1−δn−r,m)tn−r,nG(n− r,m,n′,m′;z)−∑r 6=0(1−δn,m−r)tm−r,mG(n,m− r,n′,m′;z)−∑r 6=0dn,mδn−m,±rG(n,m,n′,m′;z) ,(5.11)150where G(n,m,n′,m′;z) represents the matrix element of the Green’s operator in thetwo-particle basis setG(n,m,n′,m′;z) = 〈n,m|Gˆ(z)|n′,m′〉 . (5.12)Similarly, substituting Eq. (5.8) into the right hand side of Eq. (5.9) gives rise to〈n,m|I|n′,m′〉 = 〈0|PˆnPˆmPˆ†n′Pˆ†m′ |0〉= 〈0|Pˆn[δm,n′+(1−2δm,n′)Pˆ†n′Pˆm]Pˆ†m′ |0〉= δm,n′δn,m′+δn,n′δm,m′ . (5.13)Therefore, the equation of the motion for the Green’s function is(z− en− em−∑r 6=0dn,mδn−m,±r)G(n,m,n′,m′;z)−∑r 6=0(1−δn−r,m)tn−r,nG(n− r,m,n′,m′;z)−∑r 6=0(1−δn,m−r)tm−r,mG(n,m− r,n′,m′;z)= δm,n′δn,m′+δn,n′δm,m′ . (5.14)Because two excitations cannot reside at the same lattice site, we know the Green’sfunction G(n,m,n′,m′;z) is meaningless whenever n = m or n′ = m′. It is clearthat Eq. (5.14) can’t guarantee all the Green’s functions will be physical even if werestrict the basis set to be |i〉| j〉 where i 6= j. That’s why factors like (1− δn,m−r)appear.Since all the Green’s functions in Eq. (5.14) have the values of n′ and m′, wedefineG˜(n,m;z) = G(n,m,n′,m′;z) (5.15)151to simplify the notation. So that Eq. (5.14) becomes(z− en− em−∑r 6=0dn,mδn−m,±r)G˜(n,m;z)−∑r 6=0(1−δn−r,m)tn−r,nG˜(n− r,m;z)−∑r 6=0(1−δn,m−r)tm−r,mG˜(n,m− r;z)= δm,n′δn,m′+δn,n′δm,m′ . (5.16)5.3 Recursive calculation of Green’s function5.3.1 With the nearest neighbor approximationThe equation of motion for the Green’s function, Eq. (5.16), is our starting point tocalculate the Green’s function. In the nearest neighbor approximation, the distancebetween two interacting sites r can only be 1 or −1 This simplifies Eq. (5.16) to(z− en− em−dn,mδn−m,±1) G˜(n,m;z)−(1−δn−1,m)tn−1,nG˜(n−1,m;z)− (1−δn+1,m)tn+1,nG˜(n+1,m;z)−(1−δn,m−1)tm−1,mG˜(n,m−1;z)− (1−δn,m+1)tm+1,mG˜(n,m+1;z)= δm,n′δn,m′+δn,n′δm,m′ . (5.17)As mentioned before, factors like (1− δn−1,m) in Eq. (5.17) are used to eliminatethe unphysical Green’s functions like G˜(i, i;z). For instance, when n = m− 1,G˜(n+1,m;z) and G˜(n,m−1;z) will disappear in Eq. (5.17) because of the factorsin front of them. Equation 5.17 shows that G˜(n,m;z) only relates directly withat most four other Green’s functions in the nearest neighbor approximation. This152relationship can be shown schematically as...G˜(n−1,m+1;z) →{G˜(n−2,m+1;z), G˜(n−1,m;z), G˜(n,m+1;z), G˜(n−1,m+2;z)}G˜(n,m;z) →{G˜(n−1,m;z), G˜(n,m−1;z), G˜(n+1,m;z), G˜(n,m+1;z)}G˜(n+1,m−1;z) →{G˜(n,m−1;z), G˜(n+1,m−2;z), G˜(n+2,m−1;z), G˜(n+1,m;z)}...(5.18)For the Green’s function G˜(i, j;z) on the left, the summation of parameters i+ j =n+m is fixed. For the Green’s functions G˜(i′, j′;z) on the right, the summation ofparameters i′+ j′ can only be either n+m− 1 or n+m+ 1. This inspires us togroup some Green’s functions into a vector VK according to their values of n andm, that is,VK =...G˜(i−1, j+1;z)G˜(i, j;z)G˜(i+1, j−1;z)..., (5.19)with K = i+ j. So Eq. (5.17) can be written asVK = αK(z)VK−1 +β K(z)VK+1 , (5.20)if n′+m′ 6= K, and asVK = αK(z)VK−1 +β K(z)VK+1 +C , (5.21)if n′+m′ = K. In the above two equations, αK(z) and β K(z) are matrices and C isa constant vector, and their values can be determined from Eq. (5.17). Because thetwo excitations are indistinguishable from each other, G˜(n,m;z) is equivalent to153G˜(m,n;z). In the following discussion, we require n < m in all G˜(n,m;z) to reducethe dimension of VK by half.Deriving expressions for αK(z) and β K(z) by hand is tedious. Instead wechoose to do that in a programmable way using the pseudocode1: Form all the vectors VK for K = 1 to 2N−12: Generate the index I(n,m) of every G˜(n,m;z) such that the I(n,m)th elementof Vn+m is G˜(n,m;z), that is Vn+m[I(n,m)] = G˜(n,m;z)3: for every G˜(i, j;z) in VK do4: nth← I(i, j)5: Set every element of αK to 06: if i−1≥ 0 then7: mth← I(i−1, j)8: αK(nth,mth)← ti−1,iz−ei−e j−di, jδ j−i,19: end if10: if j−1 > i then11: mth← I(i, j−1)12: αK(nth,mth)←t j−1, jz−ei−e j−di, jδ j−i,113: end if14: Set every element of β K to 015: if i+1 < j then16: mth← I(i+1, j)17: β K(nth,mth)←ti,i+1z−ei−e j−di, jδ j−i,118: end if19: if j+1≤ N then20: mth← I(i, j+1)21: β K(nth,mth)←t j, j+1z−ei−e j−di, jδ j−i,122: end if23: end forAs we will show in the next subsection, the above algorithm is even more usefulwhen the coupling equations like Eq. (5.17) become more complicated in the casesof longer range interaction and high-dimensional systems.It is clear from Eq. (5.20) and Eq. (5.21) that VK relates only with VK−1 and154VK+1. This gives the opportunity to solve for individual VK recursively providedthat a starting point is given. For example, given the values for two particular VKand VK−1, we can obtain VK+1, and then calculate VK+2 from VK and VK+1, andetc. But how do we start? The hint comes from the following physical arguments:G(n+δn,m+δm,n,m;z) is expected to approach zero as |δn| →∞ and |δm| →∞because its Fourier transform G(n+ δn,m+ δm,n,m; t) represents the amplitudefor the two particles to move a distance δn and δm respectively in time t [81, 129].To figure out which of the Green’s functions can be approximated by zero, wefix the values of n′ and m′. Suppose Kc = n′+m′, then we can assume that VKapproaches zero when |K−Kc| → ∞. For the purpose of numerical computations,a cutoff distance M is chosen such that VKc−M and VKc+M are very small. In thiscase, the infinite crystal is replaced by a finite crystal whose central region containssite n′ and site m′. To ensure the convergence of the results, the finite crystal mustbe large enough such that its boundaries are far away from both site n′ and sitem′. Assuming the 1D finite crystal has N +1 lattice sites indexed by 0,1, · · · ,N−1,N, the starting point for the recursive calculation of the Green’s function are thefollowing two approximationsV2N−1 ≈ 0 ,V1 ≈ 0 . (5.22)Given these two approximations, we are ready to calculate the Green’s func-tions recursively. Instead of working with the vectors VK directly, we define twoquantities AK and A˜K that relate two consecutive vectors VK and VK−1. Forn≥ Kc +1, we haveVn+1 = An+1Vn , (5.23)and for n≤ Kc−1, we haveVn = A˜nVn+1 . (5.24)The validity of the above two equations can be verified by substituting Eq. (5.22)into Eq. (5.20) and solving VK recursively. Because VKc satisfies Eq. (5.21) rather155than Eq. (5.20), Eq. (5.23) and Eq. (5.24) are not valid for n = Kc. At the leftboundary of the crystal for which n = 1, we haveV1 = β 1(z)V2 , (5.25)because of Eq. (5.20), and comparing it with Eq. (5.24), we concludeA˜1 = β 1(z) . (5.26)There is a recursive relation between different A˜n. Substituting Vn−1 = A˜n−1Vninto Eq. (5.20) givesVn =[1−α n(z)A˜n−1]−1β n(z)Vn+1 , (5.27)which upon comparison with Eq. (5.24) givesA˜n =[1−α n(z)A˜n−1]−1β n(z) . (5.28)Since this equation involves matrix inversion, which is difficult to compute numer-ically, we instead calculate A˜n by solving the linear equation[1−α n(z)A˜n−1]A˜n = β n(z) . (5.29)Similarly at the right boundary of the crystal for which n = 2N−1, we obtain thestarting value of AKA2N−1 = α 2N−1(z) , (5.30)and the recursive relationAn = [1−β n(z)An+1]−1α n(z) . (5.31)To calculate the Green’s functions, we start from the left boundary of the crystaland calculate A˜1, A˜2, · · · , from left to center until A˜Kc−1 is reached. We have tostop there because Eq. (5.24) is not valid for VKc . We then proceed from the rightboundary to the center and calculate A2N−1, A2N−2, · · · and stop when we reach156AKc+1 beyond which Eq. (5.23) is not valid. Knowing the value of AKc+1 andA˜Kc−1, we rewrite VKc+1 and VKc−1 in terms of VKc usingVKc+1 = AKc+1VKc ,VKc−1 = A˜Kc−1VKc , (5.32)and substitute them into Eq. (5.21) to obtain the solution for VKc , that isVKc =[1−αKc(z)A˜Kc−1−β Kc(z)AKc+1]−1C . (5.33)Once VKc is known, all Vn can be calculated from Eq. (5.23) and Eq. (5.24) giventhe values for An+1 and A˜n.In summary, the described calculation of Green’s function is carried out in thefollowing steps:1. fix the value of n′ and m′ in the Green’s function G(n,m,n′,m′;z), assumethe two approximations in Eq. (5.22), and calculate A˜1 from Eq. (5.26) andA2N−1 from Eq. (5.30)2. start from A˜1 and calculate A˜2, A˜3, · · · , A˜Kc−1 from Eq. (5.28)3. start from A2N−1 and calculate A2N−2, A2N−3, · · · , AKc+1 from Eq. (5.31)4. use the values of A˜Kc−1 and AKc+1 to calculate VKc from Eq. (5.33)5. start from VKc and calculate VKc−1, VKc−2, · · · , V1 from Eq. (5.24)6. start from VKc and calculate VKc+1, VKc+2, · · · , V2N−1 from Eq. (5.23)5.3.2 Extension to long-range interactionsAs mentioned in Section 4.5.1, the dipole-dipole interaction is long-range and thenearest neighbor approximation may not represent the physical picture accurately.So it is desirable to extend the calculation method in Section 5.3.1 to the case oflong-range interactions.First, we consider a 1D lattice with the same Hamiltonian of Eq. (5.1) withboth first nearest-neighbor and second nearest-neighbor interactions, for which the157equation of motion for the Green’s function becomes(z− en− em−dn,mδm−n,±1−dn,mδm−n,±2) G˜(n,m;z)−(1−δn−1,m)tn−1,nG˜(n−1,m;z)− (1−δn+1,m)tn+1,nG˜(n+1,m;z)−(1−δn,m−1)tm−1,mG˜(n,m−1;z)− (1−δn,m+1)tm+1,mG˜(n,m+1;z)−(1−δn−2,m)tn−2,nG˜(n−1,m;z)− (1−δn+2,m)tn+2,nG˜(n+2,m;z)−(1−δn,m−2)tm−2,mG˜(n,m−2;z)− (1−δn,m+2)tm+2,mG˜(n,m+2;z)= δm,n′δn,m′+δn,n′δm,m′ . (5.34)The above equation shows that the Green’s functions G˜(n′,m′;z) are only coupledwith the Green’s function G˜(i, j;z) whose parameters i and j sum up to n+m−1or n+m+1 or n+m−2 or n+m+2. As for the nearest neighbor approximation,we can group the Green’s functions according to the summation of their parametersand obtainZKVK = MK,K+1VK+1 +MK,K−1VK−1 +MK,K+2VK+2 +MK,K−2VK−2 , (5.35)where VK has the same definition as in Eq. (5.48). Different from the NNA casewhere the recurrence relation links three consecutive terms VK , VK−1 and VK+1,Eq. (5.35) has two extra terms VK−2 and VK+2. At first glance, adding the secondnearest-neighbor interactions invalidates the method presented in Section 5.3.1 asthe calculation of Green’s function relies on a recurrence relation linking threeconsecutive terms. But if we work with combinations of VK instead of individualVK , we can obtain the proper recurrence relation. Replacing K with K + 1 inEq. (5.35) givesZK+1VK+1 = MK+1,K+2VK+2 +MK+1,KVK +MK+1,K+3VK+3 +MK+1,K−1VK−1 .(5.36)Equations 5.35 and 5.36 can be written asW2K+1(VKVK+1)= α 2K−3(VK−2VK−1)+β 2K+5(VK+2VK+3), (5.37)158whereW2K+1 =(ZK −MK,K+1−MK+1,K ZK+1), (5.38)α 2K−3 =(MK,K−2 MK,K−10 MK+1,K−1), (5.39)β 2K+5 =(MK,K+2 0MK+1,K+2 MK+1,K+3). (5.40)Comparing Eq. (5.37) with Eq. (5.20) shows that the method introduced in Sec-tion 5.3.1 will also work for the current case ifV˜2K+1 ≡(VKVK+1)(5.41)and use series of vectors V˜1, V˜5, V˜9, · · · is used rather than V1, V2, V3, · · · .To further illustrate the point, we consider an even longer range interaction withthe first nearest-neighbor, second nearest-neighbor, and the third nearest-neighborcouplings. Similarly, we work with a set of equations that relate different VK , thatisZKVK = MK,K+1VK+1 +MK,K−1VK−1 +MK,K+2VK+2 +MK,K−2VK−2+MK,K+3VK+3 +MK,K−3VK−3 , (5.42)ZK+1VK+1 = MK+1,K+2VK+2 +MK+1,KVK +MK+1,K+3VK+3 +MK+1,K−1VK−1+MK+1,K+4VK+4 +MK+1,K−2VK−2 , (5.43)ZK+2VK+2 = MK+2,K+3VK+3 +MK+2,K+1VK+1 +MK+2,K+4VK+4 +MK+2,KVK+MK+2,K+5VK+5 +MK+2,K−1VK−1 . (5.44)159These equations give rise to a recurrence relation that links three vectors, namelyZK −MK,K+1 −MK,K+2−MK+1,K ZK+1 −MK+1,K+2−MK+2,K −MK+2,K+1 ZK+2VKVK+1VK+2=MK,K−3 MK,K−2 MK,K−10 MK+1,K−2 MK+1,K−10 0 MK+2,K−1VK−3VK−2VK−1+MK,K+3 0 0MK+1,K+3 MK+1,K+4 0MK+2,K+3 MK+2,K+4 MK+2,K+5VK+3VK+4VK+5 (5.45)The above recurrence relation is in the form of Eq. (5.37), and a new vector can bedefined asV¯3K+3 ≡VKVK+1VK+2 . (5.46)We can then use the series of vectors V¯3, V¯12, V¯21, · · · in the recursive calculations.It is clear from the above discussion that the recursive method to calculate theGreen’s function can be applied to any finite range interactions and the dimensionsof the matrices increase linearly with respect to the number of neighbors included.As the coupling equations get more and more complicated, it becomes more diffi-cult to derive the expressions for Z’s and M’s by hand. Fortunately the algorithmpresented in Section 5.3.1 can be easily adapted to do this kind of computation. Forexample, the following algorithm can be used to calculate MK,K−2 and MK,K+2:1: for every G˜(i, j;z) in VK do2: nth ← I(i, j)3: Set every element of MK,K−2 to 04: if i−2≥ 0 then5: mth← I(i−2, j)6: MK,K−2(nth,mth)← ti−2,i7: end if1608: if j−2 > i then9: mth← I( j−2, i)10: MK,K−2(nth,mth)← t j−2, j11: end if12: if j−2 < i then13: mth← I(i, j−2)14: MK,K−2(nth,mth)← t j−2, j15: end if16: Set every element of MK,K+2 to 017: if i+2 < j then18: mth← I(i+2, j)19: MK,K+2(nth,mth)← ti+2,i20: end if21: if i+2 > j then22: mth← I( j, i+2)23: MK,K+2(nth,mth)← ti+2,i24: end if25: if j+2≤ N then26: mth← I(i, j+2)27: MK,K+2(nth,mth)← t j, j+228: end if29: end for5.3.3 Extension to high-dimensional systemsIn the last two subsections, we only discuss the 1D lattice. It turns out that extend-ing the method to a high-dimensional system is straightforward. For instance, inthe case of a 2D lattice, we want to calculate the Green’s functionG˜(nx,ny,mx,my;z)≡ 〈nx,ny,mx,my|Gˆ(z)|n′x,n′y,m′x,m′y〉 , (5.47)where |nx,ny〉 represents the state of the particle at site (nx,ny) and n′x, n′y, m′x,and m′y are fixed. Due to the structure of the equation of motion for the Green’sfunction, one Green’s function is only coupled with a certain set of other Green’s161functions. In the nearest neighbor approximation, the Green’s functions can begrouped into different vectors by definingVix+iy+ jx+ jy =· · ·G˜(ix−1, iy +1, jx, jy;z)G˜(ix, iy, jx−1, jy +1;z)G˜(ix, iy, jx, jy;z)G˜(ix, iy, jx +1, jy−1;z)G˜(ix +1, iy−1, jx, jy;z)..., (5.48)and the same procedure outlined in Section 5.3.1 then follows. Note that the algo-rithm in Section 5.3.1 can also be applied here with only a small change.5.3.4 Comparison with other methodsConventionally, Green’s function can be calculated by a brute-force approach. Thefirst step is to diagonalize the Hamiltonian H and calculate the spectrumH|φn(R)〉= λn|φn(R)〉 , (5.49)where n indicates the nth eigenvector of the system and R represents the positionof the particles. From the definition of the Green’s operator Gˆ = (z−H)−1, theGreen’s functionG(R,R′;z) =∑n′φn(R)φ ∗n (R′)z−λn, (5.50)can be obtained where ∑n′ represents summation over the discrete spectrum andintegration over the continuous spectrum.This brute-force method is simple but the computational cost is large. Considera two-particle state |i〉| j〉 in a 1D lattice with N sites. The number of two-particlebasis sets used to represent the system is O(N2) and the dimension of the matrixis O(N2×N2). As most of the algorithms for eigenvalue computations scale likeO(n3) for a n× n matrix, the computational cost of Eq. (5.49) scales like O(N6).To calculate one Green’s function from Eq. (5.50) requires computing the inner162product of two vectors N2 times, which costs O(N6) operations in total. So thetotal cost of the brute-force approach is O(N6). In contrast, the recursive methodis much more efficient. Of all the vectors VK , VKc has the largest size which isabout N. Thus, the most time-consuming step is solving Eq. (5.33), which requiresa run time that scales like O(N3). Since there are about N similar equations tosolve, the total computational cost is O(N4) which is much less than the cost of thebrute-force approach. In addition, the recursive method yields N Green’s functionsat the same time.To verify our recursive method, it results were compared with brute-force cal-culations for an ordered array and a disordered array for a variety of complex en-ergies z. As shown in Fig. 5.1 and Fig. 5.2, both methods give identical results,which provides strong evidence to the correctness of the recursive method.The brute-force approach is not often used in computations. Instead, the re-cursion method developed by Haydock [189, 190] is most widely used. For com-parison purpose, we give a very brief description of the method and compare itwith our method. The main idea of Haydock’s method is to generate a series oforthonormal vectors |u0〉, |u1〉, |u2〉, · · · from the iterationsH|un〉= b∗n|un−1〉+an|un〉+bn+1|un+1〉 , (5.51)and calculate the values of an and bn until convergence. Because the Hamiltonianmatrix is tridiagonal in this basis, the Green’s functions can be expressed in termof an and bn as〈u0|Gˆ(z)|u0〉=1z−a0−|b1|2z−a1−···. (5.52)For a tight-binding model with only nearest-neighbor interactions, the vectors |un〉are a linear combinations of local states |R〉 with particles residing at R. TheHamiltonian has the following useful feature:H|R〉= tR,R−1|R−1〉+hR,R|R〉+ tR,R+1|R+1〉 , (5.53)where hR,R is the on-site energy, t are the matrices related to the hopping terms,and |R−1〉 and |R+1〉 denote the states whose particle positions are 1 site dif-16320 10 0 10 200.0150.0100.0050.0000.0050.0100.0150.020Re[G(10,71,50,51,E+0.1∗i)]20 10 0 10 20E (arbitrary unit)0.0150.0100.0050.0000.0050.0100.0150.020Im[G(10,71,50,51,E+0.1∗i)]Figure 5.1: Green’s function G(10,71,50,51,E+ iη) with η = 0.1 as a func-tion of energy E. The calculation was done for a finite ordered 1D crys-tal of 101 lattice sites with the nearest-neighbor approximation. Theenergies of the particles en were set to zero, and the dynamic interactionand the hopping interaction were set to 5. The upper panel shows thereal part of the Green’s function and the lower panel shows the imagi-nary part. The results calculated by the brute-force method are markedwith empty circles while the results obtained by our recursive methodare marked by “X”. This figure clearly demonstrates that the recursivemethod produces the same results as the brute-force method.16420 10 0 10 200.0080.0060.0040.0020.0000.0020.0040.0060.0080.010Re[G(33,90,50,51,E+0.1∗i)]20 10 0 10 20E (arbitrary unit)0.0100.0050.0000.0050.0100.015Im[G(33,90,50,51,E+0.1∗i)]Figure 5.2: Green’s function G(33,90,50,51,E+ iη) with η = 0.1 as a func-tion of energy E. The calculation was done for a finite disordered 1Dcrystal of 101 lattice sites with the nearest-neighbor approximation. Thedynamic interaction and the hopping interaction were set to 5, and al-low the energies of the particles en to fluctuate in range [−5,5] ran-domly. The upper panel shows the real part of the Green’s function andthe lower panel shows the imaginary part. The results calculated by thebrute-force method are marked with empty circles while the results ob-tained by our recursive method are marked by “X”. This figure clearlydemonstrates that the recursive method produces the same results as thebrute-force method.165ferent from that of |R〉. As a specific example, Eq. (5.10) clearly demonstratesthis property of the local state |R〉= |n〉|m〉. As a result of Eq. (5.53), to calculateG(R,R;z), we can initiate the recursion by setting |u0〉= |R〉 and b0 = 0. Assum-ing orthonormality of local states, a0 can be obtained by taking the product of 〈u0|with Eq. (5.51),a0 = 〈u0|H|u0〉= hR,R . (5.54)Comparing Eq. (5.51) with Eq. (5.53), we can easily see that |u1〉 is a superpositionof |R−1〉 and |R+1〉 and b1 can be computed by requiring |u1〉 to be normalized.To continue the recursion, we take n = 1 in Eq. (5.51) to obtainH|u1〉= b∗1|u0〉+a1|u1〉+b2|u2〉 , (5.55)and let H operate on |u1〉, a superposition of |R−1〉 and |R+1〉, according toEq. (5.53), which gives rise to a linear combination of |R−2〉, |R−1〉, |R〉, |R+1〉and |R−2〉 on the left hand side of Eq. (5.55). Now it is clear that |u2〉 will alsobe a superposition of |R−2〉, |R−1〉, |R〉, |R+1〉 and |R−2〉, and b2 can becalculated from the normalization condition of |u2〉. Follow the above analysis, weconclude that |un〉 is a superposition of these local states: |R−n〉, |R−n+1〉, · · · ,|R+n−1〉 and |R+n〉, which are within n-sites distance from R.Haydock’s method and our method both rely on recursive calculations, but theydiffer a lot in numerical efficiency. Consider the calculation of the two-particleGreen’s function as an example. If Haydock’s method terminates the recursion atstep N, it has to deal with vectors of size up to ∼ N2. To employ the same numberof local states as in Haydock’s method, our method sets A2N+1 = 0 and A˜1 = 0,and the largest size of the vectors is about N, which is much smaller. Therefore ourmethod involves much smaller matrices and can handle much larger crystal sizes.5.4 Application to the problem of biexciton scatteringAs a powerful analytical tool, Green’s functions are often used to develop analyti-cal formulations for all sorts of physical problems. However, due to difficulties intheir numerical computation, these formulations often become something of theo-retical merit only, and are rarely used for computations. For example, as will be166shown below, the Green’s function formulation of scattering theory is quite generaland elegant. It can handle all types of scattering potentials and treat single-particlescattering and multiple-particle scattering in the same way. But still the Green’sfunctions are not used very often in scattering problems because of the numericaldifficulties involved in their calculation[81]. Using the new recursive method inSection 5.3, which makes the calculation easier, the Green’s function formulationof scattering will be used to develop a numerical method to calculate the tunnel-ing of a biexciton state through disorder potentials. This method is numericallyefficient and can handle multiple impurities.We consider a periodic lattice with hopping interactions and dynamic interac-tions. Although the lattice is regular, disorders can still appear due to differenton-site energies and different coupling strengths at different sites. We write theHamiltonian in two parts,Hˆ = Hˆ0 + Hˆ1 , (5.56)where Hˆ0 represents an ordered array and is given byHˆ0 = E0∑nPˆ†n Pˆn + ∑n,m6=nJn,mPˆ†n Pˆm +12 ∑n,m6=nDn,mPˆ†n Pˆ†mPˆnPˆm , (5.57)and Hˆ1 describes the disorders and is given byHˆ1 = ∑i∈S0(Ei−E0)Pˆ†i Pˆi + ∑i, j 6=i(i, j)∈SJ(J′i, j− Ji, j)Pˆ†i Pˆj+12 ∑i, j 6=i(i, j)∈SD(D′i, j−Di, j)Pˆ†i Pˆ†j PˆiPˆj , (5.58)where S0 is the set of sites whose energies are different from E0, SJ is the set of pairsof sites whose hopping terms J′ are different from J, and SD is the set of pairs ofsites whose dynamic interactions D′ are different from D. As shown in Chapter (3),if the dynamical interaction is 2 times larger than the hopping interaction, H0 willhave a set of states with energies that are located outside the spectrum of two freeexciton pairs. These states are correlated states of two excitations and are called167biexcitons. The presence of H1 in the Hamiltonian causes the eigenstates to bedifferent from the free-biexciton state.We are interested in the scattering of a biexciton state through disorder. Forconvenience, the disorder is assumed to appear at the center region of the lattice.The starting state is a biexciton state |φ(K)〉 which is an eigenstate of Hˆ0,Hˆ0|φ(K)〉= E(K)|φ(K)〉 . (5.59)If the scattering process is elastic, we want to obtain a solution to the full-HamiltonianSchro¨dinger equation with the same energy as the initial biexciton state(Hˆ0 + Hˆ1)|ψ〉= E(K)|ψ〉 . (5.60)It can be shown that the desired solution might be|ψ〉= 1E(K)− Hˆ0Hˆ1|ψ〉+ |φ(K)〉 . (5.61)There is a problem with the above solution: the operator 1E(K)−Hˆ0will generate sin-gular results because E(K) is an eigenenergy of Hˆ0. To resolve this complications,we can make the energy complex and the solution is then given by|ψ±〉= 1E(K)± iη− Hˆ0Hˆ1|ψ±〉+ |φ(K)〉 . , (5.62)where η is an infinitesimal positive number. The above equation is the Lippmann-Schwinger equation[91]. |ψ+〉 and |ψ−〉 are outgoing and incoming solutions,respectively. In most cases, only |ψ+〉 is interesting because it corresponds to themeasurement at a position far away from the scatterers. Based on the definition ofGreen’s function, Eq. (5.62) can also be written as|ψ+〉= |φ(K)〉+ Gˆ0(E(K)+ iη)Hˆ1|ψ±〉 . (5.63)Substituting Eq. (5.63) into itself and iterating gives|ψ+〉= |φ〉+ Gˆ+0 Hˆ1|φ〉+ Gˆ+0 Hˆ1Gˆ+0 Hˆ1|φ〉+ · · · , (5.64)168whereGˆ+0 (E) = Gˆ0(E + iη) . (5.65)Based on Eq. (5.56), we can express the Green’s operator Gˆ for the total Hamilto-nian in terms of Gˆ0 and H1. The derivation is presented as follows:Gˆ(z) = (z− Hˆ0− Hˆ1)−1={(z− Hˆ0)[1− (z− Hˆ0)−1Hˆ1]}−1=[1− (z− Hˆ0)−1Hˆ1]−1(z− Hˆ0)−1=[1− Gˆ0(z)Hˆ1]−1Gˆ0(z)=[1+ Gˆ0(z)Hˆ1 + Gˆ0(z)Hˆ1Gˆ0(z)Hˆ1 + · · ·]Gˆ0(z)= Gˆ0(z)+ Gˆ0(z)Hˆ1Gˆ0(z)+ Gˆ0(z)Hˆ1Gˆ0(z)Hˆ1Gˆ0(z)+ · · · . (5.66)Substituting Eq. (5.66) into Eq. (5.64) gives|ψ+〉= |φ〉+ Gˆ(E(K)+ iη)Hˆ1|φ〉 . (5.67)This is the equation that can be used in numerical calculations given the initial state|φ〉 and the Green’s function G(z).Based on the wavefunction of a biexciton state, Eq. (3.74), derived in Sec-tion 3.3.2, we may write the biexciton state in the two-particle basis set as|φ(K)〉 = ∑n,m 6=neiK(n+m)/2 fK(|n−m|)Pˆ†n Pˆ†m|0〉= ∑n,m 6=neiK(n+m)/2 fK(|n−m|)|n,m〉 , (5.68)where fK(|n−m|) is a function that decreases fast with increasing |n−m|. In prin-ciple, all possible two-particle states |n,m〉 with n 6= m should be used to representthe biexciton state, but for practical purposes a cutoff distance L for |n−m| is usedto greatly reduce the number of basis sets. This will hardly affect the final resultas long as fK(|n−m|) decays fast enough such that fK(L) ≈ 0. In this case the169biexciton state can be rewritten as|φ(K)〉 ≈∑nn±L∑m=n±1eiK(n+m)/2 fK(|n−m|)|n,m〉 . (5.69)Because the two-particle states in Eq. (5.69) can effectively represent a biexcitonstate, they are called the biexciton basis and denoted by Sbiexciton.To calculate the scattering state from Eq. (5.67), let’s analyze Hˆ1|φ〉 first. Ex-panding the disorder term in a two-particle basis set givesHˆ1 = ∑i∈S0∑j 6=i(Ei−E0)|i, j〉〈i, j|+ ∑i, j 6=i(i, j)∈SJ∑l 6=i, j(J′i, j− Ji, j)| j, l〉〈i, l|+12 ∑i, j 6=i(i, j)∈SD(D′i, j−Di, j)|i, j〉〈i, j| . (5.70)The above equation can be derived by inserting the identity relation ∑i |i〉〈i| = Iinto the first and second terms of Eq. (5.58). Suppose the hopping disorder andthe dynamic disorder only appear between two sites that are within the cutoff dis-tance L, we can easily see that in Eq. (5.70) only the two-particle states |n,m〉 with|n−m| ≤ L will play a role, by letting Hˆ1 operate on Eq. (5.69). Since only thesetwo-particle states are needed to represent the effective disorder the biexciton ex-periences, we call them the disorder basis Sdisorder. In the case of a small numberof disorder sites, Sdisorder is a small subset of Sbiexciton by construction. Thus Hˆ1|φ〉gives rise to a new wavefunction|φHˆ1〉= Hˆ1|φ〉= ∑(i, j)∈Sbiexcitonc(i, j)|i, j〉 , (5.71)where c(i, j) can be nonzero only if (i, j) ∈ Sdisorder. Given |φHˆ1〉, we can continueto calculate the scattering state according to Eq. (5.67). Since only a small numberof coefficients c(i, j) is nonzero in Eq. (5.71), only a small portion of the matrixthat represents the Green’s operator Gˆ(E(K)+ iη) is needed, that is, the matrixelements 〈i′, j′|Gˆ(E(K)+ iη)|i, j〉 with |i, j〉 being states in the disorder basis. Thistask can be efficiently handled by the recursive method in Section 5.3 by setting170the initial particle positions (n′,m′) to be every site pairs (i, j) in Sdisorder.To prevent the processes of a biexciton breaking into two free excitons, we as-sume the dynamic interaction is more than 2 times larger than the hopping interac-tion such that the energy of a biexciton is well separated from the energy spectrumof two free excitons. During the scattering process, the energy must be conserved,and thus the scattering wavefunction has the following asymptotic behavior|ψ〉 →{|φ(K)〉+R|φ(−K)〉 for n,m→−∞T|φ(K)〉 for n,m→ ∞, (5.72)if the initial biexciton state |φ(K)〉 incidents from the left with its eigenenergyE(K). This asymptotic behavior can be used to calculate the transmission coeffi-cient of a biexciton through disorder. In principle, scattering calculations usuallyassume a infinitely large space, but in practice, a finite crystal with sufficientlylarge size is used. We start the calculation from an initial biexciton state φ(K) witheigenenergy E(K), and then calculate the scattering state |ψ〉 using Eq. (5.67). Thetransmission coefficient can then be deduced from Eq. (5.72) by choosing two sitesn and m to the far right of the disorder region, that ist =〈n,m|ψ〉〈n,m|φ(K)〉 . (5.73)Note that the scattering wavefunction |ψ〉must be properly normalized with respectto the finite crystal.5.5 SummaryIn this chapter, a recursive method to calculate lattice Green’s functions is de-scribed. The essential idea is to group Green’s functions into different sets of vec-tors and rewrite the equation of motion of Green’s function as a recursion relationlinking three consecutive vectors. Based on physical reasoning, certain vectors areassumed to be zero and then substituted into the corresponding recursion relationsto express one set of Green’s function in terms of another set of Green’s function.By iterating on the recursion relation, a chain of relationships between each twoconsecutive vectors can be obtained. Finally, the vector at the end of this relation-171ship chain can be calculated by solving a linear equation, and all the vectors can becalculated one by one through the relationship chain. The method can handle a sys-tem with arbitrary disorder, and can be easily extended to systems with interactionsof longer (but finite) range, and to systems with high dimensionality. To show thenumerical efficiency of the method, it was compared with the brute-force methodand Haydock’s method. Our method involves vectors of much smaller sizes. Asan application, we described using the recursive method to calculate the Green’sfunctions to study the scattering of a biexciton state by impurities.172Chapter 6ConclusionTrapping ultracold atoms and molecules in optical lattices has opened an excitingfrontier of condensed matter research[17]. In the limit of strong trapping fieldsthat completely suppress particle tunneling in optical lattices, an artificial crystalof atoms and molecules, called the Mott insulator phase, appears. These artificialcrystals possess unique properties – in particular, they offer the ability to addresssingle lattice sites[57, 58] and the possibility for controlling interparticle interac-tions with external electric or magnetic fields[16, 17], making them a very goodplatform to study collective phenomena[8, 16, 17]. The thesis, by utilizing thecontrollability of the artificial crystals of ultracold particles, explores theoreticallythe mechanisms for controlling the quantum dynamics of quasiparticles.6.1 Summary of the thesisThe control schemes presented in this thesis can be categorized into two majortypes: 1) mixing internal states by external fields; 2) changing the phase of internalstates by temporal perturbations.The first type of control schemes is commonly used to control the interparticleinteractions in the field of ultracold atoms and molecules. For example, it has beenshown that both the shape and strength of the dipole-dipole interaction betweenmolecules can be controlled by external fields[51], which inspires numerous studyon using molecules trapped on optical lattices for quantum simulation[43, 50, 120–173127]. Instead, the current thesis studied the possibility of using the external fieldsto control inter-quasiparticle interactions rather than interparticle interactions. Inthe case of the rotational excitation of polar molecules trapped in an optical lattice,the dipole-dipole operator leads to different kinds of inter-quasiparticle interac-tions, namely the hopping interaction that is responsible for excitation propaga-tion, the attractive dynamic interaction that can induce exciton binding, and theconversion interaction that can split a high-energy exciton into two low-energy ex-citons. These interactions are associated with different dressed rotational states ofmolecules, which correspond to different mixing of bare rotational states in ex-ternal fields. So the different inter-quasiparticle interactions respond differently tothe change of external fields. This thesis showed that the ratio between the hoppingand the dynamic interaction can be tuned by an electric field to induce the forma-tion of biexciton and triexciton states with tunable binding energies, and found thatthe conversion interaction leads to an non-optical way to create biexciton statesfrom the high-energy (N = 2) excitonic states under the resonance condition whena high-energy excitonic state have the same energy as two low-energy (N = 1)excitonic states. These results demonstrated that the physics of dipole-dipole inter-actions in the context of quasiparticles is also rich and opened up the possibility tocontrol the binding and creation of collective excitation modes of polar moleculestrapped on optical lattices.The second type of control schemes is reminiscent of the techniques used forstrong-field alignment and orientation of molecules in the gas phase[144]. The es-sential idea is to use an external field to perturb the internal states of moleculesfor a short time, adding phases to the wavefunctions of the internal states. Thisperturbation does not change the interparticle interactions as the internal states re-main the same except that their phases are changed. Though the phase changes areseemingly unimportant, they play an important role in determining the dynamicsof quasiparticles. Since a collective excitation in an array of coupled monomersis a superposition of internal states of monomers, any phase change of the inter-nal states will translate to a change of quasimomentum of the collective excitation,and influence the excitation energy propagation in the array. The thesis showedthat specific phase transformations can be used to accelerate or decelerate excita-tion energy transfer and spatially focus delocalized excitations in arrays of quantum174particles. The proposed control scheme is very general as the excitations can beof any type and the array can be ordered and disordered. This thesis also demon-strated the feasibility of the control scheme in the system of ultracold atoms andmolecules trapped on optical lattices. All the above results showed that the phasesof individual monomer state in an array of coupled monomers can be used to ma-nipulate the quantum energy transfer in the array, which pointed out a new way tocontrol the dynamics of quasiparticles.In addition, the thesis extended calculations of lattice Green’s functions to dis-ordered systems. The proposed method is numerically more efficient than conven-tional methods and may be used to study the dynamics of quasiparticles in stronglydisordered potentials.6.2 Limitations and possible extensionIn this section, I discuss the limitations of the thesis and explain how the currentwork can be extended by going beyond those limitations.Conventionally, the 1D and 2D crystals of ultracold atoms and molecules arecreated in 3D optical lattices. In a 3D optical lattice, since the intensity of everylaser pair can be tuned independently, the trap depths along the x, y and z directionscan be adjusted separately. By making the trap depths along the x and y directionsso deep that the particle tunneling along these two directions are completely sup-pressed, an effective 1D crystal can be constructed in which the trapped particlesare only allowed to move in the periodic potentials along the z direction. Simi-larly, an effective 2D crystal can be created by suppressing the motion of trappedparticles along just one direction. In this thesis, I study the excitations, rather thanreal particles, trapped in one and two dimensions. To confine excitations in re-duced dimensionalities, it is not enough to use the effective 1D and 2D systemsdescribed previously since excitations can still leak into other dimensions even ifthe tunneling of real particles along that direction is completely suppressed. Theo-retically, one needs a truely 1D or 2D crystal in a 3D optical lattice, where only thesites along a single line or on a single plane are occupied by atoms or molecules.In practice, one may use a 3D optical lattice with one polar molecule per site inthe Mott insulator phase. To approximate the 1D and 2D crystals required in the175thesis, the wavelengths of laser beams in some directions can be increased, pro-ducing larger lattice separations along those directions and reducing the strengthof coupling between molecules in the corresponding dimensions. This leads to sup-pression of the excitation propagation along those directions, creating the quasi-1Dand quasi-2D arrays. However, limited by the available wavelengths of the laserbeams used to form optical lattices, the ratio between the large and small latticeseparations along different directions is typically about 2∼3. This means that theinteraction between nearest neighbors along the direction with the large lattice con-stant will be of the same magnitude as the interaction between next nearest neigh-bors or next next nearest neighbors along the direction with small lattice constant.As shown in Chapter 4, the long-range interaction between molecules can play arole in determining the dynamics of quasiparticles, therefore it may be necessaryto include the weak interactions along the suppressed dimension(s). In addition,because the dipole-dipole interaction is anisotropic and the relative orientations ofthe dipole moments with respect to the intermolecular axis are different, the weakinteractions along the suppressed dimension(s) generally have different signs fromthe strong interaction within the 1D and 2D arrays. So it would be interesting to in-vestigate whether the results obtained for purely 1D and 2D arrays are valid for thequasi-1D and quasi-2D crystals and especially the effect of the anisotropic weakinteractions.In the thesis, I have assumed that the optical lattices are deep and ignoredthe translational motion of molecules within the trap. Going beyond this assump-tion may lead to interesting results. Recently, Herrera and Krems showed thatthe rotational excitons can interact with the translational motion of molecules toform polarons[69]. The Hamiltonian for these polarons includes a combination ofbreathing-mode and Su Schrieffer-Heeger couplings, leading to interesting sharptransitions in the polaron phase diagram[70]. It might be possible to extend Her-rera’s work to include the dynamic interaction between excitons.Chapter 3 has shown the existence of the two-body and three-body bound statesof excitons under certain conditions. However, this does not mean that biexcitonand triexciton states will appear at any exciton concentrations. In the presence ofmultiple excitons, the combination of a biexciton and an exciton might be unstableto triexciton formation and triexciton + exciton might be unstable to bigger bound176complexes. So it is necessary to extend our theoretical model to handle n-bodybound states and to investigate the fundamental limits of exciton clustering. Refer-ences [108, 193, 194] have already studied the problem of exciton clustering underthe nearest-neighbor approximation and implied the existence of excitonic n-stringwhere n can be any positive integer. It might be interesting to extend their analysisto the case of long-range interactions.The work on energy transfer presented in the thesis has so far ignored the inter-actions between excitons. This is justified for the case of the rotational excitationsin small external fields as the exciton–exciton interaction is small compared withthe hopping interaction. However as Chapter 3 shows, the dynamic interaction be-tween excitons can be comparable and even a few times larger than the hoppinginteraction at certain electric field strengths. In this situation, it is important toinvestigate if the control scheme works in the presence of exciton-exciton interac-tions. Particularly, it would be very interesting to see whether we can control thepropagation of the biexciton states.6.3 Future research directions6.3.1 Influence of exciton-exciton interaction on polariton lasingThe coupling between cavity photons and excitons gives rise to the half-light andhalf-matter quasiparticles called cavity polaritons[195]. Unlike the weakly inter-acting cavity photons, polaritons can strongly interact with each other due to the in-herited strong-interacting characteristics from the matter (or exciton) components.This polariton-polariton interaction produces effective strong photon-photon inter-action, and leads to the emerging field of quantum fluids of light[196]. One branchof the field focuses on using the Bose-Einstein condensate of cavity polaritons tocreate polariton laser, a new coherent source of light that may lead to new opto-electronic devices[197]. Polaritons decay in a cavity because their photonic com-ponents leak out through cavity mirrors. When polaritons form a condensate, thedecay of polaritons in the macroscopic coherent state produces photons in the samecoherent state. This process is similar to the conventional photon lasing except thepopulation inversion, therefore polariton lasing has a much lower threshold energy177for coherent emission[198].In experiments with polariton lasers, both inorganic and organic semiconduc-tors can be used as a basis for microcavities[197]. However, they have very dif-ferent excitonic modes. In inorganic semiconductors, the excitons are loosely-bound electron-hole pairs called Wannier-Mott excitons, while in organic mate-rials, the excitons are tightly-bound electron-hole pairs called Frenkel excitons.Compared with Wannier-Mott excitons, Frenkel excitons are highly stable at roomtemperature[199]. As a result, there is a trend towards using organic materials asthe microcavities[197], with the notable achievement of room-temperature polari-ton lasing[200, 201].Polariton lasers requires the Bose-Einstein condensate of polaritons. To form apolariton condensate, efficient relaxation mechanisms are required to transfer pop-ulations from other polariton states to the ground polariton state. The polariton-polariton scattering[202, 203], usually resulting from both the Coulomb interac-tion between excitons and the exciton saturation, is a very important relaxationmechanism. In organic materials, due to the strong screening effect, the Coulombinteraction is very weak. So the exciton saturation or the kinematic interaction (seeChapter 3), originating from the fact that two excitons cannot sit on the same latticesite, plays the dominant role and it is well investigated[204–206]. To the contrary,another type of exciton-exciton interaction, namely the dynamic interaction (seeChapter 3), is not discussed with regard to polariton lasers in organic semicon-ductors. In an organic crystal, if the molecules possess static dipole moments, thedynamic interaction can be as large as the width of the exciton bandwidth[99]. Thisstrong dynamic exciton interaction can have a significant effect on the polariton-polariton interaction. It would be very interesting to investigate whether the dy-namic interaction can enhance the relaxation towards the ground polariton stateand further reduce the threshold energy for polariton lasing.To simulate the polaritons in organic semiconductors, one could use a dipolarcrystal of ultracold polar molecules trapped on the surface of a high-Q supercon-ducting stripline cavity, like the setup described in Ref. [118] and Ref. [119]. Therotational excitons of the molecular array resemble the Frenkel electronic excitonsin the organic crystal. The cavity photons can be provided by a microwave striplinecavity. The rotational excitons may couple strongly with the microwave photons to178form polaritons. (Note that the conventional microwave cavities like those in Ref.[207] cannot be used here because their large size results in very weak interactionbetween excitons and photons.) In the current system, an external dc field is re-quired to induce electric dipole moments of polar molecules to maintain the dipolarcrystal. This external field can be tuned to change the strength of exciton-excitoninteraction. Because of this tunability, it would be very interesting to study thecondition of polariton lasing in the current system and to investigate the influenceof exciton-exciton interaction on polariton relaxation. Recently, Cristofolini et al.have demonstrated polaritons with static dipole moments[208]. It is expected thatthe static dipole moments will reinforce polariton-polariton interactions, just likethe dynamic interaction discussed before, and may lead to new effects as discussedin Ref.[209] and Ref.[210]. The current system may also be used as a platform tostudy new polariton physics of the long-range dipolar interactions.6.3.2 Parallel computation of Green’s functionsWith the advance of technology, multi-core processors are becoming more power-ful and less expensive, and they are expected to become the mainstream for largescientific computations. Calculating the Green’s functions for multiple particles isalways a computation intensive task due to the large number of degrees of freedominvolved, so it is desirable to develop parallel numerical methods to speed up thecomputation. However, the recursive method to calculate the Green’s functions, aspresented in Chapter 5, can only be implemented in a sequential way. This can beeasily seen from Eq. (5.28) and Eq. (5.31): the calculation of A˜n(An) can only bedone after A˜n(An+1) is calculated. In the following, I sketch a parallel way to cal-culate the Green’s function as a future research direction. This method is inspiredby the recursive doubling algorithm for solving the tridiagonal systems[211].For simplicity, I only discuss the case with the nearest-neighbor coupling. Itis straightforward to extend the discussion to cases with long-range interactions.Based on the analysis in Chapter 5, the equation of motion for the Green’s functioncan be rewritten as a set of recursive relations:WKVK = αKVK−1 +β KVK+1 + γK (6.1)179where γK can be a zero vector or nonzero vector. For a finite crystal, the value ofK is from 1 to Kmax. Rewriting the above recursion relation asVK+1 = β−1K WKVK−β−1K αKVK−1−β−1K γK , (6.2)one can easily derive the following equation:VK+1VK1=β−1K WK −β−1K αK −β−1K γK1 0 00 0 1VKVK−11 . (6.3)If we defineXK ≡VKVK−11 (6.4)andBK ≡β−1K WK −β−1K αK −β−1K γK1 0 00 0 1 , (6.5)we haveXK+1 = BKXK for 1≤ K ≤ Kmax . (6.6)So once X1 is known, all other XK can be calculated by repeated application ofEq. (6.6):X2 = B1X1X3 = B2X2 = B2B1X1· · ·XKmax+1 = BKmaxBKmax−1 · · ·B2B1X1 . (6.7)To simplify the notation, we can define the products of the B matrices asCK = BKBK−1 · · ·B2B1X1 for 1≤ K ≤ Kmax , (6.8)180which allows us to writeXKmax+1 = CKmaxX1 . (6.9)It can be easily shown that all the CK matrices have the following pattern:CK =gK11 gK12 gK13gK21 gK22 gK230 0 1 (6.10)where the g blocks can take zero or nonzero values. Therefore Eq. (6.9) becomesVKmax+1VKmax1= CKmaxV1V01=g11 g12 g13g21 g22 g230 0 1V1V01 . (6.11)By definition, VKmax+1 = 0 and V0 = 0, then from Eq. (6.11), we obtain0 = g11×V1 +g13×1 , (6.12)from which V1 can be obtained. After that, all other VK can be calculated fromEq. (6.7).The most time-consuming part of the above method is to calculate the productsof the B matrices: C1, C2, · · · , CKmax . If one can make this computation parallel,the above algorithm can be accelerated. It turns out that the computation of theseries of CK matrices is very similar to the all-prefix-sums operation in parallelalgorithms[212]. Given an array of objects[a1,a2, · · · ,an]and a binary associative operator ⊕, the all-prefix-sums operation returns the or-dered set[a1,(a1⊕a2) , · · · ,(a1⊕a2⊕·· ·⊕an)] .As one of the simplest and most useful building blocks in parallel algorithms[212],the all-prefix-sums operation is well studied and has been implemented in variousways[213]. However, many existing algorithms implicitly assume the computation181cost of the binary operation ⊕ between any two consecutive objects ai and ai+1to be almost the same. This is not the case here because the dimension of the BKcan vary a lot depending on the value of K. For example, BK has the largest sizewhen K ≈ Kmax/2. 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