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Fluctuation driven phenomena in spinor Bose gas Song, Junliang 2010

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Fluctuation Driven Phenomena in Ultracold Spinor Bose Gas by Junliang Song B. Sc., Peking University, 2004 M. Phil., The Chinese University of Hong Kong, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2010 c
 Junliang Song 2010 Abstract In this thesis, we have investigated several 
uctuation-driven phenomena in ultracold spinor Bose gases. In Bose-Einstein condensates of hyperne spin- two (F=2) atoms, it is shown that zero-point quantum 
uctuations com- pletely lift the accidental continuous degeneracy in quantum spin nematic phases predicted by mean eld analysis, and these 
uctuations select out two distinct spin nematic states with higher symmetries. It is further shown that 
uctuations can drive a novel type of coherent spin dynamics which is very sensitive to the variation of quantum 
uctuations controlled by magnetic elds or potential depths in optical lattices. These results have indicated fundamental limitations of precision measurements based on mean eld the- ories. In addition, 
uctuation-driven coherent spin dynamics studied here is a promising tool to probe correlated 
uctuations in many body systems. In another system { a two-dimension super
uid of spin-one (F=1) 23Na atoms { we have investigated spin correlations associated with half quan- tum vortices. It is shown that when cold atoms become super
uid below a critical temperature a unique nonlocal topological order emerges simultane- ously due to 
uctuations in low dimensional systems. Our simulation have indicated that there exists a nonlocal softened -spin disclination structure associated with a half-quantum vortex although spin correlations are short ranged. We have also estimated 
uctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps. These results indicate that the strongly 
uctuating ultracold spinor system is a promising candidate for studying topological orders that are the focus of many other elds. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Fluctuations in a Harmonic Oscillator and Beyond . . . . . 3 1.2 Fluctuation-Driven Phenomena . . . . . . . . . . . . . . . . 6 1.3 Bose-Einstein Condensates of Ultracold Atoms . . . . . . . . 9 1.4 Ultracold Spinor Condensates . . . . . . . . . . . . . . . . . 13 1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . 20 2 Interatomic Interactions and Low-Energy Scattering . . . 21 2.1 Interatomic Interactions . . . . . . . . . . . . . . . . . . . . . 21 2.2 Low Energy Scattering . . . . . . . . . . . . . . . . . . . . . 24 2.3 Interactions with Hyperne Spin Degrees of Freedom . . . . 25 2.4 Vector and Tensor Algebra and Representations . . . . . . . 27 2.4.1 F=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 F=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Linear and Quadratic Zeeman Shift . . . . . . . . . . . . . . 32 3 Fluctuation-Induced Uniaxial and Biaxial Nematic States in F = 2 Spinor Condensates . . . . . . . . . . . . . . . . . . . . 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Mean Field Theory of F=2 Condensates . . . . . . . . . . . 36 3.2.1 Ferromagnetic States . . . . . . . . . . . . . . . . . . 38 iii Table of Contents 3.2.2 Cyclic States . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.3 Nematic States . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Zero-Point Quantum Fluctuations . . . . . . . . . . . . . . . 42 3.4 Uniaxial and Biaxial Spin Nematic States . . . . . . . . . . . 45 3.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . 46 4 Quantum Fluctuation Controlled Coherent Spin Dynamics in F=2 87Rb Condensates . . . . . . . . . . . . . . . . . . . . . 48 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Microscopic Hamiltonian for F = 2 87Rb in Optical Lattices 50 4.3 Range of Interaction Parameters for the Lattice Hamiltonian 51 4.4 Mean Field Nematic States and Coherent Dynamics . . . . 54 4.5 Eective Hamiltonian for QFCSD . . . . . . . . . . . . . . . 55 4.5.1 Nematic Manifold . . . . . . . . . . . . . . . . . . . . 55 4.5.2 Collective Coordinates and Conjugate Momentum . . 57 4.5.3 Spectra of Collective Modes . . . . . . . . . . . . . . 58 4.5.4 Eective Hamiltonian . . . . . . . . . . . . . . . . . . 59 4.6 Potentials Induced by Quantum Fluctuations . . . . . . . . 61 4.7 Oscillations Induced by Quantum Fluctuations . . . . . . . 64 4.8 Eects of Thermal Fluctuations . . . . . . . . . . . . . . . . 70 4.9 Eects of Spin Exchange Losses . . . . . . . . . . . . . . . . 72 4.10 Estimates of Condensate Fraction . . . . . . . . . . . . . . . 77 4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Half Quantum Vortex (HQV) in F = 1 Super
uids of Ultra- cold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 HQV in Spinor Bose-Einstein Condensates . . . . . . . . . . 82 5.2.1 Magnetic Properties and Energetics of HQV . . . . . 83 5.2.2 HQV Lattice in Rotating Traps . . . . . . . . . . . . 89 5.3 HQV in 2D Super
uid of Ultracold F = 1 Atoms . . . . . . . 93 5.3.1 Eective Hamiltonian . . . . . . . . . . . . . . . . . . 93 5.3.2 HQV and Topological Order - A Gauge Field Descrip- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.3 HQV and Topological Order - Monte Carlo Simula- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3.4 Critical Frequency for HQV Nucleation . . . . . . . . 104 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 iv Table of Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Appendices A Band Structure in Optical Lattices . . . . . . . . . . . . . . . 124 B The -function of the Nonlinear Sigma Model . . . . . . . 127 v List of Tables 2.1 Scattering lengths (in atomic units) of various alkali atoms. . 26 3.1 Magnetic orderings in mean eld ground states of F = 2 spinor condensates. . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Scattering lengths (in atomic units) of various alkali isotopes with hyperne spin F = 2. . . . . . . . . . . . . . . . . . . . . 38 3.3 Collective modes of nematic states specied by . . . . . . . . 43 vi List of Figures 2.1 Interaction energy of two 85Rb atoms as a function of inter- nuclear distance r. . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Hyperne energy levels of a 87Rb atom in its electronic ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Wavefunctions of spin nematics and phase diagram of F = 2 spinor condensates . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 The zero-point energy of spin nematics per lattice site as a function of  . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1 Coupling constants in optical lattices. . . . . . . . . . . . . . 52 4.2 An artistic view of the ve-dimension manifold of spin nematics. 56 4.3 Barrier height Bqf (in units of pK) as a function of optical potential depth V . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Eective potential as a function of  for various experimental parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Fluctuation-driven coherent spin dynamics when the quadratic Zeeman coupling is absent. . . . . . . . . . . . . . . . . . . . 64 4.6 Fluctuation-driven coherent spin dynamics around various spin nematic states at the presence of the quadratic Zeeman coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.7 Fluctuation-driven coherent spin dynamics around biaxial spin nematic states at the presence of the quadratic Zeeman cou- pling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.8 Stability of 
uctuation-driven coherent spin dynamics . . . . 69 4.9 Induced free energy barrier height as a function of tempera- ture for dierent lattice potentials. . . . . . . . . . . . . . . . 73 4.10 Estimated time evolution of rubidium population at j2; 0i hy- perne spin state when the quadratic Zeeman coupling or Zeeman eld varies. . . . . . . . . . . . . . . . . . . . . . . . 74 4.11 Estimated time evolution of population at j2; 0i state at dif- ferent temperatures. . . . . . . . . . . . . . . . . . . . . . . . 76 vii List of Figures 4.12 Condensation fraction versus optical lattice potential depth V (in units of ER). . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 Two quantum vortices separated at a distance of d . . . . . . 84 5.2 Magnetic structure of an individual half quantum vortex and interaction energy between two half quantum vortices . . . . 86 5.3 Creation of a half quantum vortex in rotating traps . . . . . . 90 5.4 Dynamical creation of half quantum vortex lattice in rotating traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Numerical solution to the self-consistent equation Eq.5.34 . . 98 5.6 A half quantum vortex in the Z2 lattice gauge eld model. . . 100 5.7 Correlations and loop integrals in 2D nematic super
uids . . 102 5.8 Free energy and creation of half quantum vortex in rotating traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 viii Acknowledgements First and foremost, I would like to thank my supervisor Fei Zhou for his constant guidance and patience over the past four years. He has led me into this fascinating research eld of cold atomic gases. I have been fortunate to work with someone with ability and integrity as Fei. He has always been my source of inspiration and encouragement for research and life. I would like to thank the members of my committee: Kirk Madison, David Jones, Moshe Rozali and George Sawatzky for their helpful sug- gestions of improving the thesis and my oral presentation. I owe partic- ular thanks to Kirk Madison for his careful proofreading of my thesis and thoughtful comments on experimental issues. I would like to thank my collaborators: Gordon Semeno, Wu-Min Liu, An-Chun Ji for their important contributions to this thesis. I would also like to thank Tin-Lun Ho, Xiaoling Cui, Michiel Snoek, Hui Zhai for their valuable suggestions and stimulating discussions. I am grateful to the people in the Physics and Astronomy department at University of British Columbia, for providing a pleasant and supportive atmosphere to work in. I would like to thank my loving family and my wonderful friends. As always, my deepest thanks are owed to my parents, who have supported me throughout my years of education. During the workshop on condensed matter physics of cold atoms at Bei- jing in 2009, I was supported by the Kavli Institute for Theoretical Physics China under the visiting graduate student fellowship. ix Dedication To my parents x Chapter 1 Introduction Fluctuation, or to shift back and forth irregularly, is one the most common phenomena in the physical world. For example, the sea level rises and falls every day, the temperature of our earth varies wildly in the last decade. It seems that nothing stands still and everything 
uctuates. According to the laws of modern physics, this is indeed the case since 
uctuation is be- lieved to be an intrinsic nature of all physical systems. For example, the density at any point inside a cup of hot water can not remain constant, although the amount of variation might be small. The density 
uctuation is caused by the thermal movement of the constituent molecules and can only be suppressed by decreasing the water's temperature. One would expect that 
uctuations can be eliminated when the temperature goes down to the absolute zero where everything is supposed to be still. However, even in this limit the molecules, as well as any other particles, are never at rest: they are in a seemingly random motion called \zero-point motion", as a direct con- sequence of the Heisenberg uncertainty principle of quantum mechanics[1]. Not only particles 
uctuate as their positions and velocities are not denite, the vacuum which assumed to be empty of matter and energy also 
uctu- ates. Physicists picture the vacuum as a sea with 
uctuating foam of virtual particles which are repeatedly created and annihilated spontaneously. And according to modern cosmology, even our whole universe is 
uctuating. What causes the 
uctuation, especially the mysterious quantum 
uctu- ation associated with the \zero-point motion"? This is a deep and con- troversial question in fundamental physics. Some physicists insist that the 
uctuation is a result of an incompleteness or a drawback of the quantum theory(for example [2]). They believe that a more fundamental theory be- yond the quantum theory should be a deterministic one in which predictions of physical quantities should be denite, as Einstein once put it \God does not play dice"[3]. Others, perhaps more and more, believe that this 
uctua- tion is not due to any correctable 
aws in measurement, but rather re
ects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of various quantum elds. Regardless of the controversies about the origin of 
uctuations, the cur- 1 Chapter 1. Introduction rent theory turns out be rather successful. Fluctuation-driven phenomena have attracted a lot of research interests in a variety of areas. One famous example is the Brownian motion[4] { the seemingly random movement of particles suspended in a 
uid. This random movement of the suspended particle is driven by thermal 
uctuations of the constituent molecules in the 
uid and can be explained by the kinetic theory of thermal equilibrium. The importance of the Brownian motion is many-fold: it indirectly conrms the atomic nature of matter that atoms or molecules do exist; it reveals a deep and general relation between thermal 
uctuations and dissipation, as the Brownian motion is related to the viscosity of the 
uid; practically it can be used for measurement of the pulling force on a single DNA molecule shown recently. Another example is the Casimir force[5], a physical force arising from the quantum 
uctuations of electromagnetic elds between two re
ecting bodies placed in vacuum. The Casimir force is the physical man- ifestation of the zero-point energy and can be measured quite accurately between metallic objects separated at a distance of a few micrometers or nanometers. It has been suggested that the Casimir force have applica- tions in micro-technology and nano-technology. Besides, the Casimir force serves as an important conceptual tool for in modern theoretical physics. It is believed that the origin of the dark matter and the cosmology constant is related to the zero-point 
uctuations of gravitons. The concept of the Casimir eect can be also applied to emergent degrees of freedom in con- densed matter physics, resulting in the \order by disorder" phenomena in quantum magnetism and more exotic phenomena in the strongly coupling regime. In this thesis, we are interested in the 
uctuation-driven phenomena in ultracold atomic gases, in particular the Bose-Einstein condensates (BEC) of ultracold atoms with spin degrees of freedom. BEC is a novel form of matter which has attracted a lot of research interests since the early days of quantum mechanics. Many fascinating quantum phenomena can be ob- served in a BEC, such as super
uidity and macroscopic coherent dynamics, because the majority of the constituent atoms in a BEC occupy the same mi- croscopic state. Atoms that are not in that particular microscopic state give rise to 
uctuations in density correlations, which is related to energy and mo- mentum dissipation, low dimensional phases, and super
uid-insulator phase transitions, among which many have been observed and studied in ultracold atoms. However, spin 
uctuations are less studied and not well understood in Bose-Einstein condensates, These 
uctuations can be studied in spinor BEC systems { Bose-Einstein condensates of atoms with spin degrees of freedom. 2 1.1. Fluctuations in a Harmonic Oscillator and Beyond The coherence and interplay between dierent spin components within the spinor BEC have promised a wealth of interesting and rich phenomena, such as spin correlated states, macroscopic coherent spin dynamics, spin domains and spin textures, and a variety of topological defects. However, for a long time the accessibility of spinor BEC has been restricted to a small num- ber of physical systems such as super
uid helium-3. The spinor Bose gas of ultracold atoms created in recent experiments provides us new opportu- nities to study spin correlated matter and especially the spin 
uctuations. Spin 
uctuations have several interesting and unique eects which can not be studied in the single-component BEC. These eects are the main top- ics of this thesis, such as 
uctuation-driven spin correlations, macroscopic coherent dynamics and fractionalization of topological defects. They are also connected to a number of 
uctuation driven phenomena found in other physical systems. The studies of 
uctuations in spinor ultracold atoms here can improve our general understanding of 
uctuations and may shed light on open problems related to 
uctuations. In the following sections of this chapter, I give a more specic introduc- tion to the topics mentioned above. In Sec.1.1, I go over the 
uctuations in a harmonic oscillator, which is an important and useful conceptual tool to illustrate the zero-point motion; then I examine the 
uctuations of lattice vibrations based on the picture of harmonic oscillators. In Sec.1.2, I list a number of 
uctuation-driven phenomena that people were and are interested in. In Sec.1.3, I go over Bose gas of ultracold atoms with an emphasis on many-body physics. In Sec.1.4, I present a brief survey of the spinor Bose gas. In the last section of this chapter, I provide an outline of the thesis's main contents. 1.1 Fluctuations in a Harmonic Oscillator and Beyond The harmonic oscillator is one of the simplest models, yet turns out be one of the most useful models in theoretical physics. With very few exceptions almost any mass or eective mass that subjects to a force in a stable equi- librium acts as a harmonic oscillator in small vibrations. The energy E of a harmonic oscillator is given by E = p2 2m + m!2x2 2 : (1.1) Here p is the momentum of the oscillator, x is the position of the oscillator, m is the mass or the eective mass and ! is the angular oscillation frequency. 3 1.1. Fluctuations in a Harmonic Oscillator and Beyond According to the classical theory, in the ground state the oscillator sits still at the center with zero momentum and zero energy. The oscillator has a denite position and velocity without any 
uctuations. However, the quantum theory suggests a dierent picture. According the Heinsenberg's uncertainty principle, the oscillator can not have denite values for both position and momentum at the same time. If one tries to pin the oscillator to the center exactly, its momentum will 
uctuate wildly; if one tries to suppress the kinetic energy by keeping p = 0, then the position will 
uctuate wildly leading to a huge potential energy cost. The state with the lowest energy in the quantum mechanics is called the zero-point motion. The energy associated with the zero-point motion is called zero- point energy which is rst proposed by Einstein and Stern in 1913[6]. The zero-point energy of a quantum harmonic oscillator is E = 1 2 ~!: (1.2) And the 
uctuations in the oscillator's position and momentum are given by x = r ~ 2m! (1.3) p = r ~m! 2 : (1.4) The simple picture and even the above equations can be extended to a large number of systems, where the whole system can be viewed as one eective harmonic oscillator or a set of uncoupled eective harmonic oscilla- tors. For example let's look at the zero-point motion of phonons, which are quantized oscillations of the crystal lattices. In the simplest picture, a solid can be described by the displacement UR of each constituent atom around the lattice site R from their equilibrium positions. rR = R+ UR: (1.5) The total energy includes the kinetic energy and the elastic potential energy: E = X R P 2R 2m + m!2 2 X hR;R0i jUR  UR0 j2: (1.6) Here we assume a quadratic form of the elastic potential which is a reason- able approximation for small vibrations. And without loss of generality we 4 1.1. Fluctuations in a Harmonic Oscillator and Beyond also restrict the elastic potential energy to exist only between nearest neigh- bor sites. In this picture, the solid is not dierent from a set of coupled harmonic oscillators. One can nd out the normal modes of the oscillating system by introducing collective coordinates UQ and PQ, in which Q is the crystal momentum of the lattice. UR = 1p NT X Q UQe iQR (1.7) PR = 1p NT X Q PQe iQR: (1.8) Here NT is the total number of the lattice sites. One can rewrite the energy in the following form E = X Q " P 2Q 2m + 1 2 m 2QU 2 Q # : (1.9) Here Q is a function of !, Q. In this picture, we have a set of uncoupled harmonic oscillators with dierent frequencies. The quantized version of each oscillator is called a phonon with momentum Q and energy Q. The zero-point of motion of phonons has an important eect: the transla- tional symmetry cannot be spontaneously broken in a one-dimension innite chain described by Eq.1.6. If the translational symmetry was broken and the crystalline order existed, Q would vanish linearly when jQj ! 0. The 
uctuation of UR would become divergence due to the 
uctuations at long wavelengths: hU2Ri = 1 NT X Q hjUQj2i = 1 NT X Q ~ 2m Q (1.10) = 1 2a Z dQ ~ 2m Q  ~ 4am Z  0 dQ Q : (1.11) In the second line of the above equation, the summation is replaced by an integral in the continuum limit whereNT !1. Here a is the lattice spacing. The integral includes 
uctuations of wave vectors from zero to the Debye cuto . The infrared divergence of the average displacement indicates that the crystalline order can not exist in this case. In fact, this is an example of the more general Mermin-Wagner theorem, which states that continuous symmetry can not be spontaneously broken in dimension d  2 at the zero temperature. 5 1.2. Fluctuation-Driven Phenomena 1.2 Fluctuation-Driven Phenomena Fluctuation is one the most profound and rudimentary concepts in mod- ern physics. The origin of 
uctuation is deeply rooted in the fundamental laws of modern physics. It can be found in quantum mechanics, in particu- lar the Heisenberg uncertainty principle; it can also be found in statistical physics that sets the limit of chaotic and thermal motion or usually known as thermal 
uctuations. In the strict sense, everything is more or less re- lated to 
uctuations according the rst principles in quantum mechanics. Still, there are some phenomena which are explicitly related to 
uctuations, or even driven by 
uctuations. Some of the 
uctuation-driven phenomena have played an important role in the development of the quantum theory and statistical mechanics. Some of them still attract interests as they are related to unresolved problems. In the following I will provide a number of well known examples.  Lamb shift in atomic physics Lamb shift is a small energy dierence between the two energy levels 2S1=2 and 2P1=2 in the hydrogen atom. These two levels are supposed to have the same energy according to the Dirac theory. The experi- mental discovery of the Lamb shift by Lamb and Retherford in 1947[7] provided a stimulus for the renormalization theory of quantum elec- trodynamics(QED) in which zero-point 
uctuations of electromagnetic elds must be taken into account. In the picture of QED, the physical vacuum is not empty anymore. It is lled with photons, the quan- tized objects of electromagnetic waves, in the zero-point motion. The Lamb shift can be interpreted as the in
uence of virtual photons that have been emitted and absorbed by the atom (see, for example, the textbook by Peskin and Schroeder[8]).  Casimir eect The Casimir force[5] is one of the best known examples of 
uctuation- driven phenomena. Consider two uncharged metallic plates in a vac- uum separated by a few micrometers. According to the classical theory of elds there should be no forces between two plates since there are simply no sources of electric elds. When zero-point motion or quan- tum 
uctuation of vacuum is taken into account, the above classical picture is changed and there is either attractive or repulsive net forces between two plates, depending on the the geometry and placement of the two plates. This mechanism can be explained again by using virtual photons in quantum electrodynamics. An alternative simple 6 1.2. Fluctuation-Driven Phenomena explanation is that the energy of quantum 
uctuations or zero-point motion depends on the specic arrangement of the two plates and thus produces an eective potential for the two plates. The Casimir eect is very general and nds applications in various elds of physics. Ac- curate measurements were made recently by using nano-scale metal- lic and semiconductor devices of dierent geometries[9]. In particu- lar, ultracold atoms turned out to be natural candidates for measur- ing Casimir-Polder eects at considerably large distances, i.e. a few macrometers. For example, in an experiment by Obrecht et al.[10], the frequency of the dipole oscillations is shifted by the Casimir-Polder force between the rubidium atoms and a dielectric substrate separated a micrometers away.  Coleman{Weinberg mechanism of spontaneous symmetry breaking In 1972, Coleman and Weinberg[11] considered the electrodynamics of massless mesons. They pointed out that quantum 
uctuations of electromagnetic elds, or more specically radiative corrections in this case, may produce spontaneous symmetry breaking and may lead to a mass generation of mesons. In the terminology of quantum eld theory, it is a form of spontaneous symmetry breaking which does not show up at the tree level. Equivalently one may say that the Coleman-Weinberg model possesses a rst-order phase transition. The Coleman-Weinberg mechanism is very general and nds applications in high energy physics as well as condensed matter physics.  Order by disorder in quantum magnetism Fluctuation eects in frustrated magnetism have attracted a lot of interest in recent years. The frustration, either of geometrical ori- gin or as a result of competition between dierent couplings, usually lead to a family of degenerate classical ground states. Fluctuations will lift the accidental degeneracy and select out a particular type of classical ground states, usually with higher symmetries. In the condensed matter community, this mechanism is commonly known as \order by disorder" [12, 13, 14]. For example, let's consider spins in an anti-ferromagnetic body-cubic-centered lattice with strong next- nearest neighbor couplings and weaker nearest neighbor couplings. In this case, one has two simple cubic anti-ferromagnetic sublattices. In the classical version or the mean eld approximation the system's en- ergy is independent of relative orientations of the two sublattices. In the quantum version, one must consider the 
uctuations (spin waves 7 1.2. Fluctuation-Driven Phenomena in this case) around dierent classical ground states. It is shown that quantum 
uctuations favour a collinear alignment of the sub- lattices. Examples of ground state selection via quantum 
uctuation have been demonstrated in a number of frustrated magnetic systems such as frustrated Heisenberg model, body-centered tetragonal model, and Kagome lattices (see [15] for a review).  The Mermin{Wagner{Hohenberg theorem and 
uctuations in low di- mensional systems The Mermi{Wagner{Hohenberg theorem [16, 17] (also known as Mermin{ Wagner theorem) states that continuous symmetries cannot be spon- taneously broken at nite temperature in systems with suciently short-range interactions in dimensions d  2. Strong 
uctuations in low dimensional systems will destroy those ordered states that could exist in the higher dimensional counterparts. For example, gapless 
uctuations of scalar elds lead to a logarithmic divergent correlation function. Excitations of gapless long-range 
uctuations are always fa- vored because they can increase the entropy at very low energy cost. The entropy of a gapless mode with momentum k scales as kd1 log k and it always dominates the energy cost (scaling as jkj) for any nite temperatures in d  2 systems. This theorem points out the impor- tance of 
uctuations in low dimensional systems and nds applications in many systems. In the context of spinor ultracold atoms, there is no true long-range two-dimension condensate at nite temperatures; and the 
uctuations can destroy the long-range coherence of spin correla- tions in 2d, leading to short-range spin correlations.  The Fluctuation-dissipation theorem The 
uctuation-dissipation theorem is a simple, yet quite deep, con- cept originated from statistical physics. It reveals a remarkably simple relation between the dynamical system's response to small external perturbations and its inherent thermal noise. The underlying prin- ciple of the 
uctuation-dissipation theorem is that a non-equilibrium state may have been reached either as a result of a random 
uctuation or of an external force, and that the evolution toward equilibrium is the same in both cases. One example is the Brownian motion of tiny particles suspended in 
uids. It was realized that random forces or 
uctuations that resulted in thermal motions of molecules or atoms composing the 
uid are responsible for both driving the particle into the Brownian motion and also dragging the particle if pushed through 8 1.3. Bose-Einstein Condensates of Ultracold Atoms the 
uid. Einstein pointed out the underlying link with his famous Einstein-relation[18] that the diusion constant D is related to tem- perature T and particle's mobility p as D = pkBT . The 
uctuation- dissipation theorem enables transport coecients to be calculated in terms of response to external elds. 1.3 Bose-Einstein Condensates of Ultracold Atoms The successful experimental realization of the long-predicted Bose-Einstein condensates (BEC) in dilute alkali atomic gases[19, 20, 21] in 1995 has opened up a new era in which many-body physics has become one of the pri- mary concerns in ultracold atomic gases. Breakthrough technologies in cool- ing, trapping and manipulating methods have enabled us to explore quan- tum many-body phenomena in which statistics, interactions, correlations and collective behaviors of a large number of atoms rather than properties of single atoms or photons are of main interest. Remarkable progress has been made in the past decade in this fast-evolving eld as a large number of ultracold atomic systems have been created and a wide range of many-body phenomena have been realized and studied in experiments. Attaining a suciently cold temperature is a crucial prerequisite and a great technical challenge for the study of quantum many-body eects in di- lute atomic gases. This is because the quantum degeneracy temperature, the threshold below which quantum statistics become important, is extremely low in dilute atomic gases as compared to other condensed matter systems. The degeneracy temperature of atomic gases of typical densities  1013cm3 is of order 100nK, which is seven orders of magnitude smaller than that of the super
uid helium, and ten orders of magnitude smaller than the electronic Fermi energy in metals. Such a low temperature scale is needed to real- ize BEC, super
uidity, degenerate Fermi gas. For more subtle many-body phenomena such as superconductivity or quantum magnetism, the relevant temperature scale crucially depends on the interaction strength and may be even far below the quantum degeneracy temperature. Eorts of cooling atomic gases in pursuit of atomic BEC began in the late 1970s, when people rst realized that lasers could be used to trap and cool atoms[22]. However, the lowest temperature in laser cooling is set by the recoil energy (about 1K), which is the kinetic energy an atom acquires through the sponta- neous emission of a single photon. It turns out that to go beyond the laser cooling limits and reach the ultracold regime, one needs new technologies 9 1.3. Bose-Einstein Condensates of Ultracold Atoms such as evaporative cooling with magnetic or magneto-optical trappings. These breakthrough technologies have nally amounted to the realization of BEC in alkali atomic gases, with a temperature of 50nK that is one third of the condensate transition temperature[19]. Up to now, a large number of atomic systems have been cooled to close to or even below the degeneracy temperatures. Atomic BEC was created in a number of isotopes of alkali elements as well as other elements of 1H[23], 4He[24] in the lowest triplet state, 7Li[20], 23Na[21], 41K[25], 52Cr[26], 84Sr[27], 85Rb[28, 29], 87Rb[19], and 133Cs[30]. Degenerate fermionic gas as well as Bose-Fermi mixtures were created in 40K[31], 40K-87Rb[32], 6Li-7Li[33, 34], 6Li-23Na[35]. In particular, all-optical trapping allows us to create quantum degenerate gas with internal degrees of freedom arising from hyperne spins: hyperne spin F = 1[36] and F = 2[37] 23Na, F = 1[38] and F = 2[39] 23Rb, F = 3 52Cr[26], and a pseudo-spin one-half (two hyperne levels are involved) systems such as jF = 1;m = 2i and jF = 1;m = 1i in 87Rb BEC[40], and jF = 12 ;m = 12i in 6Li Fermi gas[41]. Cold molecules were also created in experiments using Feshbach resonance (see review articles [42] and [43]). Besides being ultracold, a number of other features have made dilute atomic gases a perfect playground for studying many-body quantum physics. From theorists' point of view, the dilute atomic gas is \pure" and \clean" in a sense that the interatomic interactions, at least the relevant low-energy properties, are clearly understood and accurately modeled. Besides, dilute gases in experiments are well isolated from environment; thus they are free from uncontrolled disorder, defects or noise that are ubiquitous in traditional materials studied in condensed matter physics. Furthermore, highly devel- oped and sophisticated technologies of atomic, molecular and optical physics have enabled us to prepare, manipulate and measure ultracold atomic gases with unprecedented control and precision. A large number of controlling parameters, including those that are hard to change in traditional experi- ments, can be easily and independently tuned. Examples include geometries of trapping potentials, dimensionality of quantum gases, interaction strength between atoms, tunneling rates between arrays of localized atoms, popula- tion and relative phase of atoms in dierent species, total momentum as in moving condensates, angular momentum as in rotating gas and disorders. Early experimental studies were focused on quantum Bose gases and were aimed at investigating the important consequences of Bose-Einstein conden- sation. Major achievements of these studies have been, among others, the hydrodynamic nature of the collective oscillations [44, 45, 46], Josephson- like eects [47, 48], the realization of quantized vortices [49, 50, 51], the observation of interference of matter waves [52], the study of coherence phe- 10 1.3. Bose-Einstein Condensates of Ultracold Atoms nomena in atom laser congurations [53, 54, 55, 56], the observation of four- wave mixing [57] the realization of spinor condensates [38] and coherent spin dynamics[58, 39]; the propagation of solitons [59, 60, 61, 62]. On the theoretical side, these phenomena can be well understood in the framework of mean eld theory in which one assumes the existence of a macroscopic wavefunction 	(r) ( also known as order parameter) and a nonlinear Schrodinger-like equation that governs the macroscopic wavefunc- tion. This mean eld theory is an analogue of the Ginzburg-Landau theory of superconductors(see the book by Tinkham [63] for a reference), in which the macroscopic wavefunction describes the bosonic 
uid made up of Cooper pairs of electrons. Here the macroscopic wavefunction 	(r) is directly re- lated to the super
uid of bosonic atoms. The super
uid density s and the super-current's velocity vs are related to 	(r) = j	(r)jei(r) as s = j	(r)j2 (1.12) vs = ~ m r (1.13) In weakly-interacting Bose gases at zero temperature, the order parame- ter coincides with one of the eigen-states of the one-body density matrix. The eigenvalue associated with this eigen-state is exactly the fraction of the total atoms in the Bose-Einstein condensate. Spatial structures and temporal evolutions of the macroscopic wavefunction are governed by the Gross-Pitaevskii equation[64, 65]: i @ @t =  ~ 2 2m r2	+ gj	j2	+ V (r)	 (1.14) This nonlinear Schrodinger-like equation includes the eects of interaction in its nonlinear term, as well as the consequences of trapping potentials V (r). The mean eld theory has provided a complete description of low energy degrees of freedom in weakly-interacting Bose gases. It has proven capable of accounting for most of the relevant experimentally measured quantities in Bose-Einstein condensed gases such as density proles, collective oscillations, structure of vortices, etc. (see review papers and books[66, 67, 68, 69]). The realization of super
uid-insulator phase transition in Bose gas con- ned in optical lattices[70, 71] opens up opportunities to go beyond the mean eld picture and to reach the strongly coupling limit in quantum gases. Ex- perimental developments such as Feshbach resonance and optical lattices have substantially extended the range of many-body physics that can be explored in ultracold gases. 11 1.3. Bose-Einstein Condensates of Ultracold Atoms Optical lattices are made up of several pairs of counter-propagating laser beams. The standing wave pattern provides an periodic potential which mimics the crystalline lattice in solids. In deep optical lattices, the local wells (local minimum in the periodical potential) have an eective conn- ing frequency of order 10kHz which is 10 100 times larger than the typical frequency in atomic traps. The strong connement not only enhances the in- teraction energy via increasing the overlap of two atoms occupying the same well, but also reduces the eective hopping of atoms between dierent wells. These two eects can both increase the 
uctuations in Bose-Einstein conden- sates, leading a quantum phase transition. In a seminal paper Jaksch et al. [70] proposed to realize a quantum phase transition from super
uid phase to Mott-insulator phase by loading cold bosonic atoms in optical lattices and increasing the optical potential depth. The experimental observation of super
uid-insulator transition rst reported by Bloch's group [71] marked the beginning of exploring strongly correlated quantum gases in optical lat- tices. Several other groups have observed super
uid-insulator transitions in clean bosonic systems, disordered bosonic systems, boson-fermion mixtures. In particular, optical lattices can be used to create low dimensional atomic gases. For example, a two-dimension conguration (in the xy plane) can be obtained by applying a strong connement in the z direction. In this conguration the spatial degree freedom in the z direction is actu- ally \frozen" in the low temperature regime, while atoms can still move around in the other two directions. Using similar methods one can create one-dimension gas. The one-dimension hardcore Bose gas was created and observed in ultracold atoms loaded in one-dimension tubes made by optical lattices by Paredes et al.[72] and Kinoshitaet al.[73]. These experiments are the rst experimental realization of Tonks-Girardeau gas after its prediction [72, 73] 40 years ago. The Berezinskii-Kosterlitz-Thouless phase transition [74, 75]in two dimensional super
uid was also observed in experiments, in particular the vortex-anti vortex connement-proliferation transition are ex- amined [75]. Another, or perhaps more direct way to reach the strongly interacting regime is tuning two-body scattering properties via Feshbach resonance. Feshbach resonances refer to an energy-dependent enhancement of interpar- ticle collision cross sections due to the existence of a metastable bound state. Such phenomena have been thoroughly studied in nuclear physics[76, 77, 78]. In ultracold atomic gases, the associated resonance energy depend on the strength of magnetic elds via Zeeman energy of dierent hyperne levels. Near these magnetic eld tuned resonances, the values of two-body scatter- ing lengths become comparable to or even larger than average interatomic 12 1.4. Ultracold Spinor Condensates distances, leading to strongly-correlated atomic gases. In Bose gases, Fesh- bach resonance was rst observed in Bose-Einstein condensates in 1997 by Cornish et al.[79]. On one side of resonance where scattering lengths are neg- ative, collapse growth cycles due to thermal clouds and collapsing-exploding dynamics due to mechanical instabilities have been observed by Bradley et al.[20], Donley et al.[80] and Roberts et al.[81]. Despite the reduced lifetime of Bose gases near Feshbach resonance due to enhanced three body recombi- nation, quite remarkable progress[82, 83] has been made recently to explore the physics of Bose gases at large scattering lengths where quantum 
uctu- ations can no longer be accounted for by perturbations around mean eld theories. On the few-body level, evidence of fascinating three-body Emov states and universal four-body states near resonance are observed in recent compelling experiments[84, 85, 86]. In these strongly-coupling quantum systems, mean eld theories don't suce, sometimes even break down, due to the strong 
uctuations arising from strong optical connement, resonate interactions and other methods. 1.4 Ultracold Spinor Condensates In early experiments, Bose-Einstein condensates were usually conned in magnetic traps. The conning potential in magnetic traps is produced by Zeeman energy of hyperne spins. One limitation of magnetic trapping is that only \low eld seekers" - atoms with particular spins that favor low magnetic eld - can be trapped. During inelastic scattering processes, \low eld seekers" can 
ip into \high eld seekers" resulting in a decrease in the number of atoms conned in the trap (trap loss). When atoms with dier- ent hyperne spins are simultaneously conned, trap loss may dramatically increase through spin relaxation collisions, restricting experimental studies of spinor condensates. Alternatively, one can say that the spin degrees of freedom are \frozen" in magnetic traps. This limitation is overcome by using optical trapping methods in which the spin independent conning potential is produced by couplings between laser lights and atoms' electrical dipoles. The spinor Bose-Einstein conden- sates were rst created in hyperne F = 1 23Na atoms [38, 36] in optical traps. In these experiments, spinor condensates of F = 1 23Na atoms were prepared in several steps. Laser cooling and evaporative cooling was used to produce sodium condensates in the mF = 1 state in a cloverleaf magnetic trap. The condensates were then transferred into an optical dipole trap consisting of a single focused infrared laser beam. In the nal step of prepa- 13 1.4. Ultracold Spinor Condensates ration, desired hyperne levels were then populated by transferring atoms in mF = 1 via Rabi oscillations. Linear superposition of hyperne levels such as 1p 2 (jmF = 1i+ jmF = 1i) was prepared by tuning frequencies and durations of the driving elds in Rabi oscillations. The evolution and life- time of the spinor condensate were measured by the standard time-of-
ight method, in which hyper levels were separated by applying a Stern-Gerlach magnetic gradient eld. It turns out the spinor condensate in this case had a relatively long lifetime of 110 seconds, which made itself a perfect exper- imental candidate for studying rich physics predicted in spinor condensates. Spinor condensates were later created in other systems, including hyperne F = 2 23Na [37], hyperne F = 1 and F = 2 87Rb[58, 87, 39, 88]. Although bosons with half integer spins are prohibited by quantum statistics, spinor condensates with pseudo-spin 1=2 were created in binary mixtures of F = 1 and F = 2 Rb atoms with the help of Rabi oscillations driven by external microwave elds [40, 89, 90]. The successful creation of stable spinor condensates have led to fruitful experimental studies covering dierent topics: magnetic orderings and as- sociated quantum phase transitions [91, 92, 93, 94], coherent spin dynamics [49, 95, 39, 88, 58, 87, 96, 97, 98, 99, 100, 101, 102, 103], exotic topological defects [104]. The central theme of these phenomena in the spinor condensates is the interplay between coherence and spin dependent interaction. Coherence is the hallmark of Bose-Einstein condensates in which most of the particles share the same wavefunction. In spinor condensates, coherence is further extended to internal degrees of freedom such that atoms in dierent hyper- ne levels have stationary phase dierences. Coherence of internal degrees of freedom is also the distinction between spinor condensates and fragmented condensates[105, 106]. For example, let's consider a nematic spin conden- sate of N atoms:j	si  ( y+1 +  y1)N j0i and a fragmented condensate as j	f i  ( y+1) N 2 ( y1) N 2 j0i. Although these two states have the same aver- age atom number in each hyperne level, their physical properties are very dierent. 	s depends on the relative phase between two components such that by shifting  +1 to e i +1 one would have another condensate which is related to the original one by a rotation; for fragmented condensates, such a phase shift merely leads to a global gauge phase shift, leaving the state unchanged. Interatomic interactions between atoms with internal degrees of freedom are more complicated, leading to a variety of structures and dynamics in spinor condensates. The structure and symmetry of the magnetic ordered 14 1.4. Ultracold Spinor Condensates ground states depend on details of interatomic interactions. Rich phase dia- grams of a variety of magnetic ordered states are mapped out for hyperne spin F = 1, F = 2[107, 108], and arbitrary F spinor condensates[109] in principle. When magnetic elds are present, a competition between Zeeman energy and inter-atomic interaction will lead to more complicated and rich phase diagrams and quantum phase transitions which are experimentally accessible by tuning external magnetic elds [38, 99, 93, 100, 94]. On the two-body level, spin-dependent interaction is responsible for spin- changing scattering processes. Two atoms can change their hyperne spins during elastic scattering processes. For example, two jmF = 0i atoms can become jmF = +1i and jmF = 1i and vice versa. Rabi-like oscillations of spins due to the elastic spin-changing scattering are observed and studied in the few-body limit[96, 97]. These oscillations are driven by the interatomic interactions rather than external driving forces. The inter-convertibility of dierent internal degrees of freedom is the distinction between spinor sys- tems and interspecies mixtures(although spinor condensates can be treated as a binary mixture in some cases). In spinor condensates, coherent spin-changing scattering can drive co- herent spin dynamics, which is a class of macroscopic quantum phenom- ena similar to Josephson oscillations between coupled superconductors[110]. When the spinor condensate are prepared in a state other than its ground state, the condensate will change its spin as a whole, resulting in a time- reversible macroscopic spin dynamics that could be observed in experiments. Coherent spin dynamics are the focus of a number of experimental studies in spinor condensates of 23Na atoms with hyperne spin F = 1 [100, 58] and F = 2 [58], and of 87Rb atoms with hyperne spin F = 1 [87, 98] and F = 2 [39, 99, 95, 102]. Spin coherent dynamics is studied in many theoretical investigations[111, 112, 113, 114, 94, 115, 116, 117, 118, 119, 120, 121, 122]. Simply speaking, coherent spin dynamics can be roughly described by a classical pendulum living in the internal spin space[58]. In this analogy, the spatial position of the pendulum represents the spin population and relative phases between dierent spin levels. The oscillation of the pendulum is driven by an ex- ternal force, and spin dynamics is driven by interaction energy dierences of spinor condensates. The initial potential energy can convert into the ki- netic energy of a pendulum, corresponding to the kinetic spin energy with an eective mass determined by interactions. In Chap.4, I will present the equation of motion of spin coherent dynamics which makes this analogy more transparent. Note that although the condensate and the formation of the order parameter depends on quantum mechanics, the dynamics of the 15 1.4. Ultracold Spinor Condensates order parameter is classical in many spinor condensates. As shown later in some cases, such a classical treatment of order parameter is not sucient as 
uctuations set in. The interplay of coherence and spin-dependent interaction can also lead to exotic topological defects such as fractionalized quantum vortices in spinor condensates. In conventional quantum super
uids, the circulation of super- current along any closed loop must be integer times of a basic quanta, i.e.H vs  dl = C 2~m ; C = 0;1;2;    (m is the mass of the atom). This quantization rule is related to the phase coherence throughout the conden- sate or single-valuedness of the wavefunction, requiring that the phase wind- ing along any closed loop must be multiples of 2. Fractionalized quantum vortices with circulation quanta jCj < 1 are energetically forbidden, as the energy cost for a phase discontinuity is proportional to the size of the whole condensate. In spinor condensates, fractionalized quantum vortices can be stabilized by additional structures brought in by spin degrees of freedom. For example, a half quantum vortex[123] composed of a  spin dislocation and C = 1=2 phase winding is an energetically stable topological defect in F = 1 nematic spinor condensates. The phase discontinuity of  brought by phase winding C = 1=2 is annihilated by superimposing a spin rotation which generates a Berry phase . C = 1=2 quantum vortices are also predicted in biaxial nematic F = 2 spinor condensates[124], and C = 1=3 quantum vortices are predicted in cyclic phase in F = 2 spinor condensates[125]. High dimensional topological defects such as skyrmions and monopoles are pos- sible stable topological defects in spinor condensates. A systematic way to classify topological defects is to analyze the topology and symmetry of the ground states' manifold (the geometrical space in which the order parame- ter lives)[126]. For example, the ground state manifold of single-component BEC is U(1) or S1 or a unit circle, representing the phase. The ground state manifold of nematic condensate is more complicated: (S1S2)Z2 , where S1 is a unit circle for the phase, S2 is a unit sphere for the spin nematic vector, the gauge group Z2 is for the gauge symmetry [123] in the S1  S2 representa- tion and is closely related to the half quantum vortices. Similarly, a discrete gauge symmetry group of the C = 1=3 quantum vortex was predicted in cyclic phase[125]. From the theoretical point of view, the coherence with spin degrees of freedom can be well understood in the mean eld theory. Ground state prop- erties, low-energy excitations as well as spin dynamics can be described by the Gross-Pitaevskii equation (see Sec.1.3 in this chapter) in which a multi- component order parameter is introduced to account for internal degrees of 16 1.4. Ultracold Spinor Condensates freedom. The validity of mean eld theory relies on the assumption that 
uctuations around mean eld solutions are small and have little eects on physical properties. This is indeed the case in many experiments of dilute spinor Bose gas, in which 
uctuations only provide a small correction to the mean eld result. However, there are certain cases when 
uctuations are important and the mean eld picture will not suce. 1. Broken symmetry restoring In early theoretical investigations of spinor condensates, it was noticed that the ground state predicted by the quantum rotor model method [118] is dierent from the mean eld ground states. In F = 1 spinor gas with anti-ferromagnetic interactions, the mean eld ground state [107, 108] is a nematic state. They are rotational symmetry broken states and are hugely degenerate. The broken sym- metry is specied by a nematic vector n such that all the atoms are in the hyperne mF = 0 levels if n is the projection axis. On the other hand, the quantum rotor model[113] predicted a spin- singlet ground state which is rotationally invariant and non-degenerate. The state vector in the Zeeman basis is given by  1 2 ; 1p 2 ; 12  . Further- more, the quantum rotor model predicts spin dynamics that would mix nematic states with dierent n. This dynamics is absent in the mean eld theory. It turns out that results obtained by the quantum motor model can be reproduced by including quantum 
uctuations in the mean eld nematic states. The broken rotational symmetry is restored by quan- tum 
uctuations at wavelengths of the condensate size through the nematic-mixing dynamics. In fact, the spin singlet state can be viewed as a state with 
uctuating nematic vectors: 	s = Z dn jni : (1.15) More importantly, the eects of 
uctuations at wavelengths of the condensate size will eventually vanish in the thermodynamic limit N ! 1, thus making mean eld description exact. For example (pointed out in [123]), the nematic-mixing dynamics takes place at a time scale proportional to N , and the energy dierence between spin singlet state and nematic state is proportional to 1N of the total energy. 17 1.4. Ultracold Spinor Condensates The degeneracy in ground states predicted in the mean eld theory is also exact in the thermodynamic limit , in a sense that there are numer- ous low-lying states with excitation energies of  1N in a condensate of N atoms. In the thermodynamic limit, mean eld theory provides an accurate and convenient theoretical description for ground states and low energy degrees of freedom. On the other hand the quantum rotor model is a more natural choice to describe few-body systems, as the quantum 
uctuations in nematic states are strong in these sys- tems. The size-dependent nematic-mixing dynamics is a good example to reveal the dierence and connection between few-body physics and many-body physics. It also clearly demonstrates that spontaneous symmetry breaking is a many-body eect that only happens in the thermodynamic limit. On the theoretical side, full numerical simulations [113, 118, 116, 120, 114] showed that 
uctuations lead to slow dynamics in mean eld ground states as well as damping or de-phasing in mean eld spin dynamics. These results are also obtained in [119, 115] by introducing an eective quantum Hamiltonian in which the spin population and condensate phase forms a canonical conjugate pair. Peculiar behaviors in long-time spin dynamics due to 
uctuations are studied by Diener et. al. [117] who pointed out the \quantum carpet" spin-time structure with self-similar properties in spin dynamics. On the experimental side, the observation of 
uctuation-driven dynam- ics is a challenging task, because these dynamics such as damped os- cillations take place at timescales > 10s under usual conditions, which is beyond the lifetime of the condensate. Besides, damped oscillations observed in experiments may be attributed to other factors such as the breakdown of single mode approximation [119, 115], and dissipa- tive dynamics[94]. It is suggested to make observations in relatively small condensates loaded in high-nesse optical cavities, which can detect spin changes with precision[127]. In a recent experiment on spin coherent dynamics in F = 1 sodium condensate, Liu et al.[94] di- rectly measured the atom number 
uctuation or \shot-noise" in spin projections, and revealed the dissipative nature of the spin dynamics. 2. Spinor gas in optical lattices Quantum 
uctuations in single-component Bose-Einstein condensates can be greatly enhanced in optical lattices, leading to a super
uid- insulator quantum phase transition [71] in which the phase coherence 18 1.4. Ultracold Spinor Condensates is completely destroyed by quantum 
uctuations. For a spinor gas in the optical lattice, similar quantum phase transitions would take place when the optical potential is increased, except here interactions between spin degrees freedom give rise to richer and more subtle phase diagrams and transitions. On the Mott insulator side, additional pos- sible quantum phase transitions between spin-ordered Mott insulating and spin-disordered Mott insulating states are predicted and stud- ied in F = 1 cold atoms [128, 129, 130, 131, 132, 133], and F = 2 atoms[134, 135]. On the super
uid side, coherent spin dynamics pre- dicted in the mean eld theory is greatly modied by enhanced quan- tum 
uctuations, especially approaching the phase transition[136]. In Chap. 4, I am going to examine in detail another type of coherent spin dynamics driven solely by 
uctuations in the optical lattice. Optical lattices can also be used to study low dimensional ultracold gases where 
uctuations are strong and dier from the high dimen- sional cases qualitatively. For example, Mott insulating phases and exotic spin correlations are studied in F = 1 antiferromagnetic spinor gas conned in low dimensional geometries[128, 129, 131, 130]. In Chap.5 and [137], half quantum vortices in two-dimension topologi- cally ordered super
uids are examined in details. On the experimen- tal side, disk and wave-guide connement can be realized via optical lattices. Strongly repulsive one-dimension Bose gas (Tonks-Girardeau gas)[72, 73] and BKT[74, 75] phase transition in two-dimension Bose gas have been observed and are the focus of many recent experimental studies. 3. Order by disorder phenomena In F = 2 nematic spinor condensates, the accidental degeneracy of a family of nematic states predicted in the mean eld theory is lifted by quantum 
uctuations and thermal 
uctuations, giving rise to nematic states with higher symmetries such as the uniaxial nematic state which is rotationally invariant along its easy-axis or the bi-axial nematic state with dihedra-4 symmetry. This mechanism is an analogue of \order by disorder" in quantum magnetism or the Lamb shift in electronic spectrum of hydrogen atoms where 
uctuations around the mean eld \vacuum" lead to observable eects. Fluctuation can also drive a coherent spin dynamics which is otherwise absent in the mean eld theory. This novel type of dynamics can be controlled in optical lattice and is highly sensitive to 
uctuations. I will examine this topic in detail 19 1.5. Organization of the Thesis in Chap.3. 1.5 Organization of the Thesis The rest of the thesis is organized as follows. In Chap.2, I brie
y discuss some of the atomic properties of alkaline atoms with an emphasis on spin- dependent low-energy scattering. In particular, vector or tensor basis are introduced. The following chapters are divided into two parts, for two kinds of 
uctuation-driven phenomena in spinor systems. The rst part consists of Chap. 3 and Chap.4, which focus on F = 2 nematic spinor condensates. In Chap.3, I present one of the major results in this thesis | 
uctuation-driven spin nematic states in spinor condensates. After examining the mean eld phase diagram of spin correlated phases in the rst section, the collective excitations of spin nematic states are examined in detail. Eective poten- tial of dierent nematic states at zero temperature is obtained, leading to the main conclusion of this chapter. In the following chapter, I study the coherent spin dynamics driven and controlled by 
uctuations in the same system. An eective Hamiltonian including the leading order of 
uctuations is obtained for spin dynamics in the rst several sections of this chapter. Using this Hamiltonian, coherent spin dynamics under various experimental conditions are examined in details in the following section. Eects of nite temperatures and spin loss are also discussed at the end of the chapter. In Chap.5, I turn to the second part of the thesis | fractionalized quantum vortices in F = 1 nematic spinor super
uid. In the rst several sections of this chapter I present the spatial and magnetic structures of half quan- tum vortices and dynamical creation of vortex lattice in spinor condensates where the order parameter is well dened. In the following sections, I turn to the half quantum vortex in the strongly 
uctuating limit | 2D spinor super
uids. A heuristic argument using gauge eld formulation is carried out to illustrate the basic ideas which are later conrmed in Monte Carlo simulations. Nucleation of half quantum vortices in rotating traps is dis- cussed later as one of the experimental signatures of the topological order. In the conclusion, I summarize the main results and list some related topics in the future. In the appendix, I include some calculations that are not presented in the main text for reference. 20 Chapter 2 Interatomic Interactions and Low-Energy Scattering Atomic properties of alkaline atoms play an key role in both theory and experiment of atomic gases. In this chapter I will brie
y discuss some of the atomic properties of alkaline atoms with an emphasis on spin-dependent low- energy scattering processes. In Sec.2.1, I describe the interatomic interac- tions between alkaline atoms. In Sec.2.2, I go over the low-energy scattering theory in which the s-wave scattering length is introduced. In Sec.2.3, I de- scribe the eects of hyperne spin degrees of freedom on scattering. Atoms may change their hyperne spins during the elastic scattering processes due to the exchange eects of electronic spins. The elastic spin-changing scat- tering is exactly the microscopic origin of spin 
uctuations coherent spin dynamics in spinor condensates. In the low-energy scattering theory, spin- changing collisions can be modeled by introducing spin-dependent scattering lengths. In Sec.2.3, I describe the vector and the tensor representations of the spins. Compared to the standard Zeeman basis, the vector and tensor representations are more convenient for theoretical analysis when rotational symmetries are concerned. In the last section of this chapter I describe the quadratic Zeeman shift, which is the leading order nontrivial eect of magnetic elds on spinor gases. 2.1 Interatomic Interactions Fig.2.1 is the interatomic interaction energy of two 85Rb atoms, which is a typical example of interatomic potential of alkali atoms. The long-range part of interatomic interaction between two alkali atoms is of van der Waals type (' C6r6), due to an induced dipole-dipole attractive interaction between two unpolarized atoms. At short distances, a strong repulsion arises due to the large overlapping of the electrons' wavefunction. The crossover between the repulsion at small distances and attraction at long distances is marked by an equilibrium position rmin, with a typical value of 1  10aB in alkali 21 2.1. Interatomic Interactions atoms. The long-range van der Waals interaction denes another length scale lvdW , at which the strength of interaction energy C6 r6 is equal to the kinetic energy ~ 2 2mr2 . The van der Waals characteristic length scale ldvW is of order  100aB in alkali atoms, which roughly sets the scale for scattering lengths. The interatomic interaction depends on the spin states of the electrons. For two alkali atoms, two electrons outside the closed shells can form either a spin singlet S = 0 or a spin triplet S = 1. The spin triplet state usually has a higher energy as a result of its anti-symmetric spatial wavefunction. Therefore, two electrons in the triplet state are not likely to reduce their total energy by occupying the same orbital as in the singlet case. For atoms with denite hyperne spins, the electronic spin state is a linear combination of spin singlet and spin triplet, leading to a hyperne spin dependent interaction. For example in Fig.2.1, when two atoms are far apart, there is an energy dierence between atoms with dierent F . Such a spin dependent interaction is the source of spin-dependent low-energy collisions and spin-changing scattering processes that are going to be examined later. Besides spin-exchange interactions, another type of spin-dependent inter- action in ultracold gases is magnetic dipole-dipole interaction. The nuclear dipole moment is three orders of magnitude smaller than electronic dipole moment and is neglected. The dipole-dipole interaction between two atoms is Ud = 1 2 4~2ad m Z dr1dr2 S1  S2  3(S1  r̂12)(S2  r̂12) r312 ; (2.1) where the dipole length ad is introduced to represent the strength of dipole- dipole interaction [138] as ad = 0g 2 F 2 Bm 12~2 : (2.2) The scattering length which characterizes the strength of the interac- tion for Rb87 is around 5nm, which is about two orders of magnitude larger than the dipole length ad = 0:01nm [139]. In 87Rb, the eective scatter- ing length for the hyperne spin dependent exchange interaction is about 0:03nm for F = 1 and 0:08nm for F = 2[97], which marginally dominates the dipole-dipole interaction in the latter case. For spinor condensates with ferromagnetic correlations, the dipole-dipole interaction can lead to observ- able eects such as spin textures as recently reported by [139]. For spinor condensates with antiferromagnetic interactions as considered in this thesis, the eects due to the dipole-dipole interaction are suppressed 22 2.1. Interatomic Interactions 0 10 20 30 r [aBohr] -5000 -4000 -3000 -2000 -1000 0 1000 2000 30 60 90 -0.2 0 0.2 1Σg + 3Σ u + 2S1/2 + 2S1/2 f=3 + f=3 f=2 + f=3 f=2 + f=2 l vdW V/(hc) [cm-1] r u Figure 2.1: (Color online) Interaction energy of two 85Rb atoms as a function of internuclear distance r [42]. The interaction energy in the electronic spin singlet state 1+g (in red) and triplet state 3+u (in blue). The inset shows the hyperne structures at large distances. The hyperne splitting between F = 2 and F = 3 is 3GHz for the 85Rb isotope. The 87Rb isotope has a similar curve, but the hyperne spin structure is dierent. The values of hyperne spin are F = 1 and F = 2, with a splitting 6.8 GHz. An energy associated with wave number unit 1cm1 corresponds to 30 GHz or 1.44K. 23 2.2. Low Energy Scattering because the local spin moment S vanishes in nematic spinor BECs. To induce a nonzero local spin moments, one need considerably large magnetic elds. For the cases considered in Chap.4 one needs around 400mG to match the energy dierence between the ferromagnetic states and nematic spin states in optical lattices. Hence, for nematic spinor condensates under small magnetic elds, one can neglect the dipole-dipole interactions. 2.2 Low Energy Scattering In this section, we brie
y review the scattering theory, which provides a theoretical framework to describing low energy collisions. Spin dependent interaction lead to spin-changing processes that can be described in multi- channel scattering theory. Let's rst consider two spinless bosons scattering in 3D space through a short-range isotropic interaction V (jrj). Potentials falling faster than r3 at long distance can also be treated as short-range potential[1]. Thus van der Waals potential decaying at r6 can be treated as short-range potential with an eective range  lvdW . The asymptotic behavior of the total wavefunction in the center of mo- tion frame is given by  (r) = eikz + f() eikr r : (2.3) Here the relative velocity in the incoming wave is along z direction. f() is the scattering amplitude as a function of the angle between the incoming wave and the scattering wave. For isotropic potentials, scattering waves can be decomposed into un- coupled partial waves according to their angular momentum. f() = 1X l=0 flPl(cos()): (2.4) It turns out that for low energy scattering in Bose gas, s-wave (l = 0) scat- tering dominates partial waves with higher angular momentum. Therefore we only consider s-wave scattering throughout the following chapters. The general s-wave solution (up to a phase) to the Schrodinger equation in 3D is given by  (r) = sin(kr + k) r ; (2.5) 24 2.3. Interactions with Hyperne Spin Degrees of Freedom Here k is the s-wave phase shift which is determined by the details of the potential and the energy of scattering atoms. In the low energy limit, the phase shift k can be expanded in a series of k 2[1], k cot k = 1=as + 1 2 rsk 2 +    (2.6) Here as is the scattering length of the potential and rs is the eective range of the interaction potential. By comparing Eq.2.5 and Eq.2.3, one can express the scattering ampli- tude in terms of the scattering length: fk = 1 k cot k  ik (2.7) '  1 a1s + ik (2.8) ! as; when k ! 0: (2.9) Thus the scattering amplitude is a constant as in low energy collisions, when the relevant energy in the question is much less than ~ 2 ma2s . This con- dition is usually met in dilute gases except that as become divergent near resonances. In theory it is often useful to replace the real and complicated interaction potential (as in Fig.2.1) by a simpler and analytical trackable potential, as long as these two potentials have the same scattering length. The new simple potential is often called eective potential, which can reproduce all the low energy physics as the original one. One simple choice of eective potential is the -function: U(r) = 4~a m (r): (2.10) In rst order Born approximation, the scattering matrix element is a con- stant a, which coincides with the physical interaction with scattering length a. 2.3 Interactions with Hyperne Spin Degrees of Freedom Let's consider scattering problems with spin degrees of freedom. The hyper- ne spin F = I+ S is the sum of the nuclear spin I and the electronic spin S(electronic orbital angular momentum L is zero in our consideration). In 25 2.3. Interactions with Hyperne Spin Degrees of Freedom Table 2.1: Scattering lengths (in atomic units) of various alkali atoms. (Data sources: [140, 141, 142, 143, 68, 137]) isotopes a0 a2 a4 F = 1, 23Na 50:0 1:6 55:0 1:7 N.A. F = 1, 87Rb 101:8 0:2 100:4 0:1 N.A. F = 2, 23Na 34:9 1:0 45:8 1:1 64:5 1:3 F = 2, 87Rb 89:4 3:0 94:5 3:0 106:0 4:0 F = 2, 85Rb 445:0+100300 440:0+150225 420:0+100140 F = 2, 83Rb 83:0 3:0 82:0 3:0 81:0 3:0 alkali atoms the electronic spin is always 1=2, therefore the hyperne spin F can be only in two sectors: F = I  1=2. The coupling between nuclear and electronic magnetic moments will split these two sectors by a energy sepa- ration of several gigahertz or several millikelvins in many alkali atoms. This hyperne splitting energy is three or four orders of magnitude larger than the typical temperature in ultracold experiments. Therefore the individual atom's hyperne spin can be consider to be conserved in elastic scattering. Note that in inelastic scattering processes atoms in the higher hyperne spin sectors can be gradually brought to the lower hyperne sectors, resulting in a relatively short lifetime of spinor gas in its higher hyperne sector. In this case, the conservation of individual atom's hyperne spin is still meaningful before severe spin loss takes place. The total hyperne spin of the two atoms is also conserved due to the rotational invariance of the central interaction. The selection rule arising from the rotational invariance is that two atoms's total spin F and its zeeman componentsm F is conserved. For example, two F = 1 bosons can have total spin F = 0; 2 ( F = 1 is forbidden by the symmetry of Bose's statistics). Two atoms with total spin F = 2;m F = 0 can not be scattered into F = 0 or F = 2;m F = 1 and vice versa. As shown above, scattering channels labeled by F;m F are independent in low energy collisions, and we can introduce scattering amplitudes -a F ;m F for each channel. These scattering amplitudes should be independent of m F since all the channels with the same F can be transformed to each other by rotations. Spin-mixing scattering can be understood in this framework, for exam- ple, let's consider two F = 1 bosonic atoms initially in mF = +1 and mF = 1 state. It is a linear superposition of two channels as j F = 2;m F = 0i + j F = 0;m F = 0i. Since these two channels usually scatter 26 2.4. Vector and Tensor Algebra and Representations with dierent scattering lengths, the states after scattering will be 0j F = 2;m F = 0i+ 0j F = 0;m F = 0i with 0 0 6=  , including dierent spin states such as jmF = 0i  jmF = 0i. Physically, the hyperne spin-mixing scattering is due to the electronic spin exchange eects, in particular at short distances. From Fig.2.1 one can see that at short distances the energy dierence between electronic spin sin- glet and electronic spin triplet is of order of 103K which is large compared to hyperne splitting of order 103K. When two atoms are close enough, the electronic spin exchange energy dominates the hyperne splitting inter- action, resulting in a coupling of dierent hyperne levels. These spin-dependent scattering lengths contain the complete informa- tion of low energy collisions. Actually one can construct an eective inter- action that serves as the starting point of many theoretical analysis. The two-body eective interaction can be expressed in the following form, U(r) = 4 m (r) X F a FP F : (2.11) Here P F is the operator that projects out the total spin F . It is also more useful to write spin projection operator explicitly by using second quantiza- tion forms, especially when F is specied. 2.4 Vector and Tensor Algebra and Representations For a given F , one can introduce annihilation and creation operators  ̂m and  ̂ym associated with each zeeman level m = F;F + 1;    ; F  1; F . They satisfy the following commutation rules: [ ̂m(r);  ̂m0(r 0)] = [ ̂ym(r);  ̂ y m0(r 0)] = 0 (2.12)h  ̂m(r);  ̂ y m0(r 0) i = m;m0(r  r0) (2.13) The key to write down the eective interaction with hyperne spins is the rotational symmetry. A general form of two-body contact interaction is A
 ̂ y  ̂ y  ̂
 ̂: (2.14) Here ; ; 
;  are hyperne sub-states. The rotational symmetry has put a strong constraint on the form of A
 such that after a rotation R in 27 2.4. Vector and Tensor Algebra and Representations which  ̂ ! R0 ̂0 , the eective interaction should be in the same form as Eq.2.14. It requires that A
 = R0R0R

0R0A00
00 (2.15) The number of invariant interactions under the above constraint is lim- ited,compared to situations without such a constraint. Actually the number of possible interaction terms is F + 1, which is exactly the number of inde- pendent scattering channels that two hyperne spin F atoms can have. For example it can be shown there are only two terms for F = 1 atoms and only three terms for F = 2 atoms. In the following, I will present the second quantization form of the eec- tive interaction for hyperne spin F = 1 and F = 2, which are frequently used in later chapters. For F = 3 (53Cr)and even higher hyperne spins, similar formulas can be obtained(see [144, 145]). 2.4.1 F=1 The eective interaction can be written as: Û(r) = 1 2 h c0(̂ 2  ̂) + c2(F̂ 2  2̂) i : (2.16) Here ̂ and F̂ 2 is the total density and total spin operator: ̂ =  y1 1 + y 0 0 + y 1 1 (2.17) F̂ 2 = 2̂+ 2 ̂y0 ̂0( ̂ y 1 ̂1 +  ̂ y 1 ̂1) + ( ̂ y 1 ̂1   ̂y1 ̂1)2 (2.18) +2( ̂y0 ̂ y 0 ̂1 ̂1 + h:c:): (2.19) The second line in the above expansion of F 2 clearly shows the spin-mixing coupling as j0; 0i  j+ 1;1i. Sometimes it is more convenient to introduce a vector representation via which one can write a compact form of interaction:0B@  ̂ y x  ̂yy  ̂yz 1CA = 0B@ 1p 2 1p 2 0 ip 2 ip 2 0 0 0 1 1CA 0B@  ̂ y 1  ̂y1  ̂y0 1CA ; (2.20) and 0B@  ̂ y 1  ̂y1  ̂y0 1CA = 0B@ 1p 2 ip 2 0 1p 2 ip 2 0 0 0 1 1CA 0B@  ̂ y x  ̂yy  ̂yz 1CA : (2.21) 28 2.4. Vector and Tensor Algebra and Representations The total density ̂ and spin operator F̂ along  = x; y; z can be written as: ̂ =  ̂y ̂ (2.22) F̂ = i
 ̂y ̂
 (2.23) F̂ 2 = F̂F̂ = ̂ 2   ̂y ̂y ̂ ̂ + 2̂: (2.24) Here 
 is the antisymmetric unit tensor. The coupling constant appearing in the interaction c0 and c2 can be expressed in terms of spin-dependent scattering lengths: c0 = 4~2 m a0 + 2a2 3 (2.25) c2 = 4~2 m a2  a0 3 : (2.26) 2.4.2 F=2 The eective interaction can be written as Û = a 2 (̂2  ̂) + b 2 (F̂ 2  6̂) + 5cD̂yD̂ (2.27) Here ̂ is the total density, F̂ is the total spin, and D̂y is the spin dimer creation operator which creates a spin singlet dimer(with total spin zero). One can write down explicitly in terms ̂ = +2X m=2  ̂ym ̂m (2.28) D̂ = 1p 10  2 ̂+2 ̂2  2 ̂+1 ̂1 +  ̂0 ̂0  (2.29) F̂ 2 = (2 ̂y+2 ̂+1 + p 6 ̂y+1 ̂0 + p 6 ̂y0 ̂1 + 2 ̂ y 1 ̂2) (2.30) (2 ̂y+1 ̂+2 + p 6 ̂y0 ̂+1 + p 6 ̂y1 ̂0 + 2 ̂ y 2 ̂1) (2.31) + (2 ̂y+2 ̂+2 +  ̂ y +1 ̂+1   ̂y1 ̂1  2 ̂y2 ̂2)2 (2.32)  (2 ̂y+2 ̂+2 +  ̂y+1 ̂+1   ̂y1 ̂1  2 ̂y2 ̂2) (2.33) As one can see, there are spin-mixing matrix elements corresponding to j0; 0i  j+1;1i  j+2;2i and also j+1;+1i  j+2; 0i. These matrix elements can be directly studied and measured in few-body spin-mixing dynamics in experiments[96, 97]. 29 2.4. Vector and Tensor Algebra and Representations It is convenient to introduce a tensor representation[146] As one can see later, this representation is useful when discussing magnetic structures in rotational symmetry breaking states.  ̂yxx = 1p 2 ( ̂y2 +  ̂ y 2) 1p 3  ̂y0; (2.34)  ̂yyy =  1p 2 ( ̂y2 +  ̂ y 2) 1p 3  ̂y0; (2.35)  ̂yzz = 2p 3  ̂y0; (2.36)  ̂yxy = ip 2 ( ̂y2   ̂y2) (2.37)  ̂yxz = 1p 2 ( ̂y1   ̂y1) (2.38)  ̂yyz =  ip 2 ( ̂y1 +  ̂ y 1); (2.39) The corresponding annihilation operators can be introduced in the same way. These operators have the following properties [134]: [ ̂ ;  ̂ y 00 ] = 00 + 00  2 3 00 (2.40)X   ̂ = 0: (2.41) This last property puts a constraint on the constructions of linear operators. For operators Tr[ ̂] = X ;  ̂; (2.42) the tensor  can be always reduced to a traceless one, i.e., Tr[] = X   = 0: (2.43) This constraint is needed, because when introducing this new basis we have enlarged the Hilbert space by constructing six operators out of ve. This constraint brings the size of the physical Hilbert space back to the original one. The density operator ̂ in terms of the new operators can be derived as: ̂ = X m  ̂ym ̂m = 1 2 Tr[ ̂y ̂] = 1 2 X ;  ̂y ̂: (2.44) 30 2.4. Vector and Tensor Algebra and Representations The factor 12 appears here because the trace involves a double sum over the operator  ̂ . This same factor will appear later when deriving the hopping term in the Hamiltonian. The spin operator is straightforwardly derived as: F̂ = i
 ̂y ̂
 : (2.45) It has the following properties: [F̂; F̂ ] = i
F̂
 ; (2.46) [F̂; ̂] = 0 (2.47) [F̂;Tr[( ̂ y)n]] = 0: (2.48) The total spin operator is then given by: F̂ 2 = F̂F̂ (2.49) =  ̂y ̂
 ̂ y 
 ̂   ̂y ̂
 ̂y ̂
 (2.50) The dimer creation operator in the new representation is: D̂y = 1p 40 Tr[( ̂y)2]; (2.51) which has the following properties: [D̂; D̂y] = 1 + 2 5 ̂: (2.52) This operator creates two particles which together form a spin singlet. In the same way one can construct an operator which creates three particles which together form a singlet. This is called the trimer operator and dened as T̂ y = 1p 140 Tr[( ̂y)3]: (2.53) The coupling constants a b and c in Eq.2.27 can be related to scattering lengths aF in total spin F = 0; 2; 4 channels as: a = 4~2 m 4a2 + 3a4 7 (2.54) b = 4~2 m a4  a2 7 (2.55) c = 4~2 m  a0  a4 5  2a2  a4 7  : (2.56) 31 2.5. Linear and Quadratic Zeeman Shift These equations are obtained by matching aF to the scattering amplitudes calculated from from Eq. 2.27. For example for the spin-single state 	0 = D̂yj0i, one can have 4~2a0 m = h	0jÛ j	0i = a 6b+ 5c (2.57) Similarly, for the F = 4 state such as 	4 = 1p 2  ̂y2 ̂ y 2j0i, and the F = 2 state such as 	2 = 1 10 p 6 ̂y2 ̂ y 0   ̂y1 ̂y1  j0i one can have 4~2a4 m = h	4jÛ j	4i = a+ 4b (2.58) 4~2a2 m = h	2jÛ j	2i = a 3b: (2.59) By solving the three equations above Eq.2.57-2.59, one can immediately obtain the relations shown above. 2.5 Linear and Quadratic Zeeman Shift Let's consider an atom with nuclear spin I and electronic spin S in the presence of a magnetic eld along the z direction. Here we assume that the atom is in its electronic ground state so the orbital angular momentum is zero. Hs = AI  S + CSz +DIz: (2.60) For 87Rb, jIj = 32 and jSj = 12 . A is the hyperne coupling constant and is equal to one half of the hyperne splitting hf between F = 1 and F = 2 states. Constants C and D are given by C = 2BB;D =  I B; (2.61) where  is the nuclear magnetic moment and is equal to 2:7N for 87Rb. We have C ' 2MHz and D ' 1kHz when B = 1G. Since jCD j  mpme ' 2000, for most applications D can be neglected. The eigenstates of the above spin Hamiltonian can be labeled by their total spin F and Fz. E(F; Fz) as a function of A, C are given by: E(2;+2) = 3 4 A+ 1 2 C; 32 2.5. Linear and Quadratic Zeeman Shift Figure 2.2: (a)-(c) Shift of hyperne spin energy levels due to the quadratic Zeeman eects in hyperne F = 1 and F = 2 87Rb atoms. (d) Correspond- ing Zeeman energy for selected combinations of two-particle states versus magnetic elds[97]. E(2;2) = 3 4 A 1 2 C; E(2;+1) = 1 4 A+ r 3 4 A2 + 1 4 (A+ C)2; E(1;+1) = 1 4 A r 3 4 A2 + 1 4 (A+ C)2; E(2;1) = 1 4 A+ r 3 4 A2 + 1 4 (A C)2; E(1;1) = 1 4 A r 3 4 A2 + 1 4 (A C)2; E(2; 0) = 1 4 A+ r A2 + 1 4 C2; E(1; 0) = 1 4 A r A2 + 1 4 C2: (2.62) When the zeeman coupling C is much less than the hyperne coupling A (which is true for many experiments), one can make an expansion in terms 33 2.5. Linear and Quadratic Zeeman Shift of CA and keeping terms to the second order: E(2;2) = 3 4 A 1 2 C; E(2;1) = 3 4 A 1 4 C + 3C2 32A ; E(2; 0) = 3 4 A+ C2 8A ; E(1;1) = 5 4 A 1 4 C  3C 2 32A ; E(1; 0) = 5 4 A C 2 8A : (2.63) We can rewrite the energy levels in a more compact form as E(2;mF ) = 3 4 A+ C2 8A + C 4 mF  C 2 32A m2F E(1;mF ) = 5 4 A C 2 8A  C 4 mF + C2 32A m2F : (2.64) The linear Zeeman eects are not important in spin-changing collisions. Although the mF for each spin is changed during the collision, the total spin moment is conserved. As a result, the linear Zeeman energy does not change in these collisions. In this case the leads eect of magnetic elds is thus coming from the second order Zeeman eects or the quadratic Zeeman eects. In the second quantization form introduced in the previous section, sec- ond order Zeeman eects can be cast HQB = B +FX m=F m2 ym m: (2.65) B =  C 2 32A = (BB) 2 4hf ; F = 1 or 2 (2.66) We can also write the HB in the vector representation. For F = 1 HQB = B    yz z  ; (2.67) and for F = 2 HQB = B  2 2 yzz zz + X =x;y  yz z ! : (2.68) 34 Chapter 3 Fluctuation-Induced Uniaxial and Biaxial Nematic States in F = 2 Spinor Condensates 3.1 Introduction The spin nematic condensate is a state that breaks the rotational symme- try but not the translational symmetry, in which most of the spins align themselves along a particular axis. Spin nematic states are fascinating be- cause of the rich topology of their order parameter space (also known as \manifold"), which gives rise to exotic topological defects such as quan- tum vortices with fractional circulation integrals. Although spin nematics haven't been successfully realized in conventional solid state systems, it is widely believed that at least some of them are likely to be created and observed in cold atomic gases. For example, nematic states are predicted to be the ground state in the spin-one condensates with anti-ferromagnetic interactions[107, 108], and they have also been studied in experiments[38]. For spin-two condensates a family of degenerate nematic states with d-wave symmetries[147] are predicted by the mean eld theory[107, 108]. In this chapter I will examine the eects of zero-point 
uctuations on nematic correlations. Zero-point 
uctuations are known to lead a wide range of physical phenomena, including the Lamb shift[7], the Coleman- Weinberg mechanism of spontaneously symmetry breaking[11], 
uctuation- induced rst order transitions in liquid crystals and superconductors[148], and \order-by-disorder" in magnetic systems[12, 13, 14]. It turns out that the zero-point spin 
uctuations in F = 2 nematic condensates can lift the accidental degeneracy and select out two states with distinct symmetries: a uniaxial nematic state which is rotational invariant along its symmetry axis, and a biaxial nematic state with Dih4 symmetry. In Sec.3.2, I present the results obtained previously in mean eld theories. In Sec.3.3, I examine the quantum 
uctuations in the nematic states. The 35 3.2. Mean Field Theory of F=2 Condensates quantum 
uctuations studied here can be viewed as zero-point motion of mean eld collective modes. Bogoliubov quasi-particles, quantized version of the mean eld modes, are also analyzed. In Sec.3.4, I examine the topology of the uniaxial and biaxial nematic states. In Sec.3.5, I conclude the chapter by a brief discussion of the amplitude of 
uctuations, which will be studied in more detail in the next chapter. 3.2 Mean Field Theory of F=2 Condensates In this section, we will examine spontaneous magnetic ordering of mean eld ground states, without the presence of Zeeman elds. These results are obtained previously in [141, 112]. It is shown that there are three types of possible phases for a spin-two spinor Bose condensate. These phases are spontaneous rotation symmetry broken states characterized by their mag- netizations, spin quadrupole moments and singlet pair amplitudes (summa- rized in Tab.3.1). We will also look at the topology of ground state manifolds which are possible to host fractionalized quantum vortices. We consider atoms loaded in optical lattices described by the following Hamiltonian: H = X k  aL 2  ̂2k  ̂k  + bL 2  F̂2k  6̂k  + 5cLDykDk   tL X <kl>   yk; l; + h:c:    X k ̂k: (3.1) Here k is the lattice site index and < kl > are the nearest neighbor sites,  is the chemical potential and tL is the one-particle hopping amplitude; aL, bL and cL are three interaction constants that are dependent both on optical lattices and atomic species. In the mean eld approximation, one can replace the operator by their expectation values or the associated condensates amplitudes. The amplitude of a condensate is a traceless symmetric tensor, ~ = h ̂k;i: (3.2) In this condensate, atoms occupy a one-particle spin state j~i that is dened as a linear superposition of ve F = 2 hyperne states j2;mF i or  mF j~i = X ;mF ~C;mF mF ; C;mF = r 15 4 Z d   nn  1 3   Y 2;mF (; ): (3.3) 36 3.2. Mean Field Theory of F=2 Condensates Table 3.1: Magnetic orderings in mean eld ground states of F = 2 spinor condensates. Dk, F̂k and Ôk are spin single annihilation operator, spin operator and spin quadrupole moment respectively. Their denition can be found in Sec.2.4.2 of Chap.2 and in Eq.3.15. Nematic Cyclic Ferromagnetic h mF i   sin ; 0; p 2 cos ; 0; sin    1; 0;3p2i; 0; 1 (1; 0; 0; 0; 0) hDki 6= 0 0 0 hF̂ki 0 0 6= 0 hÔki 6= 0 0 6=0 Here n,  = x; y; z, are components of a unit vector n(; ); nx = sin  cos, ny = sin  sin and nz = cos . Y2;mF , mF = 0;1;2 are ve spherical harmonics with l = 2. The mean eld energy per site as a function of ~ can be obtained as EMF = aL 8 Tr(~ ~)Tr(~ ~) + cL 8 Tr(~ ~)Tr(~~) + bL 4 Tr[~; ~]2  ztLTr(~ ~)  2 Tr(~ ~); (3.4) where z is the coordination number. Minimization of EMF with respect to the tensor ~ yields nematic, cyclic and ferromagnetic phases. In the following sections we will examine these states and their magnetic properties in detail. Here, we plot the wavefunctions of various spin nematic states in Fig.3.1(a) and also the mean eld phase diagram including all three types of phases in Fig.3.1(b). Spin wavefunctions 	S are represented by the usual spher- ical harmonics. For example, from a spinor condensate with condensate amplitude  m one can construct the following spin wavefunction 	S(; ) = lX m=l  mYl;m(; ): (3.5) One can also use the tensor form of condensate amplitude  to get a compact expression 	S = r 15 4 X  nn : (3.6) Here n,  = x; y; z are components of a unit vector n(; ). 37 3.2. Mean Field Theory of F=2 Condensates Table 3.2: Scattering lengths (in atomic units) of various alkali isotopes with hyperne spin F = 2. (Data sources: [141, 68, 137]) isotopes a0 a2 a4 23Na 34:9 1:0 45:8 1:1 64:5 1:3 87Rb 89:4 3:0 94:5 3:0 106:0 4:0 85Rb 445:0+100300 440:0+150225 420:0+100140 83Rb 83:0 3 82:0 3 81:0 3 Note that the coordinates  and  represent the spin degrees of freedom, not the spatial orientations. However, these spin wavefunctions transform the same way as spatial orbital wavefunction under an SO(3) rotation, pro- viding an intuitive picture of symmetry properties of nematic states. 3.2.1 Ferromagnetic States In the ferromagnetic phase(when bL < 0; cL > 4bL), the spinor condensate breaks the rotation symmetry spontaneously through its non-zero magneti- zation. hF̂ki 6= 0 hDki = 0 (3.7) The condensate amplitude is specied by the following matrix up to an SO(3) rotation and phase. ~ = r M 2 0@ 1 i 0i 1 0 0 0 0 1A Here M is the average number of atoms per site or M = h ̂ki. In the representation of Zeeman spin levels, we have h ̂mF i = p M (1; 0; 0; 0; 0) (3.8) One can see only integer quantum vortices are allowed in ferromagnetic states. 3.2.2 Cyclic States The cyclic state(when bL > 0; cL > 0) also breaks the rotation symme- try through a nonzero spin quadrupole moment hÔk;i = hF̂k;F̂k;i  38 3.2. Mean Field Theory of F=2 Condensates Figure 3.1: (Color online) (a) Wavefunctions of spin nematics at various .  = 0; 3 ; 2 3 ;    are wavefunctions of uniaxial spin nematics(labeled as `U'), and  = 6 ; 5 6 ; 9 6    are biaxial spin nematics(labeled as `B') with Dih4 symmetries. These spin wavefunctions 	S are represented by spherical har- monic functions (see text). Colors indicate the phase of wavefunctions. (b) Ferromagnetic(labeled as `F'), cyclic(labeled as `C') and nematic states in the mean eld theory are separated by blue boundaries; uniaxial and biaxial nematics when taking into account of quantum 
uctuations are separated by red lines. The locations of various isotopes on this diagram are located as:  =23Na, 4 =85Rb,  =87Rb and 
 =83Rb (see Tab.3.2). 39 3.2. Mean Field Theory of F=2 Condensates 1 3hF̂2i. hF̂ki = 0 hDki = 0 hÔki 6= 0 (3.9) The condensate amplitude up to a SO(3) rotation and a phase is: ~ = r 2M 3 0B@ 1 0 00 e 2i3 0 0 0 e 4i 3 1CA or  mF = r M 20 e  6 i  1; 0;3 p 2i; 0; 1  (3.10) It is pointed out that 13 quantum vortex is possible in the cyclic states due to the special topology of the submanifold[125]. A detailed analysis shows that the manifold of the cyclic states have the structure: SO(3)U(1)T . Here U(1) is for the phase, SO(3) is the rotation, and T is the Tetrahedral group composed of all twelve rotations that keep a tetrahedral invariant. Distinct quantum vortices then can be identied with their free homotopy classes, which coincide with conjugacy classes of the based homotopy group [126, 149]. In a more physically transparent way, the fractionalized quantum vortex here is possible due to the Berry phase associated with rotations. Note that the cyclic state specied by Eq.3.10 is invariant up to a Berry phase under an SO(3) rotation R around x = y = z of 23 , ~!R~RT = e23 i ~; R = 0@ 0 0 11 0 0 0 1 0 1A Therefore, a 1=3 quantum vortex is constructed by arranging the spinor condensates along a closed loop such that the total phase winding is 23 and total spin rotation is R given above. 3.2.3 Nematic States Particularly when cL < 0; cL < 4bL, minimization of energy in Eq.(3.4) requires that ~ be a real symmetric tensor (up to a phase). An arbitrary 40 3.2. Mean Field Theory of F=2 Condensates solution ~ can be obtained by applying an SO(3) rotation and a U(1) gauge transformation to a real diagonal matrix , ~ = p 4MeiRR1; (3.11) R is an SO(3) rotation matrix, and  is a normalized real diagonal traceless matrix[134],  = 0@ xx 0 00 yy 0 0 0 zz 1A ; Tr() = 1 2 : (3.12) The matrix elements can be parameterized by  2 [0; 2]: xx = 1p 3 sin     6  ; yy = 1p 3 sin    5 6  ; zz = 1p 3 sin    9 6  : (3.13) In the mean eld approximation, all of the quantum spin nematics specied by dierent diagonal matrices Eq.(3.12) or dierent  have the same energy, i.e. are exactly degenerate. For instance, when  = 0 or 2xx = 2yy = zz = 1= p 3, up to an overall SO(3) rotation, all atoms are condensed in the hyperne state j2; 0i. This choice of  represents a uniaxial spin nematic. When xx = yy = 1=2; zz = 0 or  = 2 , the atoms are condensed in the state 1p 2 (j2; 2i+ j2;2i). In the Zeeman basis, the nematic states are:  mF = r M 2  sin ; 0; p 2 cos ; 0; sin   (3.14) It is worth pointing out that spin nematics here are time-invariant; the expectation value of the hyperne spin operator Fk in these states is zero. However, all nematics have the following nonzero quadrupole spin order (up to an SO(3) rotation), O =< FkFk > 1 3  < F2k >; O = 2M sin(2 + ) : (3.15) Two kinds of nematic states have higher symmetries than other nematic states, namely the uniaxial nematic states and biaxial nematic states. These states will be selected out by quantum 
uctuations and will be discussed in the next two sections. 41 3.3. Zero-Point Quantum Fluctuations 3.3 Zero-Point Quantum Fluctuations Quantum 
uctuations of mean eld nematic states can be viewed as the zero- point motion of mean eld collective modes. These modes can be treated as a set of decoupled oscillations with denite frequencies and wavefunctions, which are the eigen-modes of the linearized Gross-Pitaevskii equation. They have been thoroughly studied in single-component condensates in experi- ments. For example, various shape oscillations including breathing modes and quadrupole modes were created by modulating the trapping potentials and were measured by monitoring time evolution of density proles[45, 44]. The propagation of sound waves created by laser pulses were also studied in a similar way[150]. The Bogoliubov quasi-particles have also been studied using Bragg spectroscopy[46], in which one can read out wavefunctions of Bogoliubov excitations from a condensate's response to optically imprinted phonons. However, these mean eld modes in spinor condensates haven't been fully explored in experiments yet. In principle they can be detected and identied by the time evolution of spin populations. For example, nor- mal modes in dierent branches usually have distinctive spin wavefunctions, as summarized in Tab.3.3 for spin-two nematic states. To study zero-point quantum 
uctuations, we rst examine the energy spectra of collective modes. For this, we expand  y about the mean eld ~,  ̂yk; = p 4M() + X  L()̂ y k; (3.16) where the superscript  = x; y; z; t; p labels zero-point motions of  along ve orthogonal directions: three SO(3) spin modes (x-,y-,z-mode) for rotations about the x, y and z axes, respectively, a spin mode (t-mode) for the motion along the unit circle of , and a phase mode (p-mode) describing 
uctuations of the condensate's overall phase. The corresponding matrices are Lx = 0@ 0 0 00 0 1 0 1 0 1A ; Ly = 0@ 0 0 10 0 0 1 0 0 1A ; Lz = 0@ 0 1 01 0 0 0 0 0 1A ; 42 3.3. Zero-Point Quantum Fluctuations Table 3.3: Collective modes of nematic states specied by . Distinctive spin waves functions of collective modes are listed in the Zeeman basis. v is the velocity of the -th mode. The uniaxial nematics are specied by  = 0; =3;    and the biaxial nematics are specied by  = =6; =2;   . mode() spin wavefunctions v2=(4MtLd 2 L) phase(p) (sin ; 0; p 2 cos ; 0; sin ) aL + cL deformation(t) (cos ; 0; p2 sin ; 0; cos ) cL spin(x) (0; + 1; 0; + 1; 0) 4bL sin 2(  2=3) cL spin(y) (0; + 1; 0;  1; 0) 4bL sin2( + 2=3) cL spin(z) (+1; 0; 0; 0;  1) 4bL sin2   cL Lt = 2p 3 0@ sin( + 3 ) 0 00 sin(  3 ) 0 0 0 sin(  23 ) 1A ; Lp = 2p 3 0@ sin(  6 ) 0 00 sin(  56 ) 0 0 0 sin(  96 ) 1A : (3.17) These matrices are mutually orthogonal, Tr(LL) = 2 . The ve modes are decoupled and the corresponding operators obey the bosonic commuta- tion relations, h k;;  y l; i = kl . We expand the Hamiltonian in Eq. (3.1) using Eq. (3.16) and keep the lowest order non-vanishing terms. The result is a Hamiltonian for the 
uctu- ations which is bilinear in yk; , k; . It can be diagonalized by a Bogoliubov transformation. The result can be be expressed in terms of Bogoliubov operators, ~yq; and ~q; , H = X q; q q(2mBNv2 + q)  ~yq; ~q; + 1 2  : (3.18) Here q = 4tL P (1 cos qdL) is the kinetic energy of an atom with crys- tal quasi-momentum q = (qx; qy; qz); dL is the lattice constant. mBN = 1=4tLd 2 L is the eective band mass. v ( = x; y; z; t; p) is the sound veloc- ity of the -mode in the small-jqj limit; v2=x;y;z = 4MtLd2L (bLG  cL), v2t = 4MtLd 2 L (cL), v2p = 4MtLd2L (aL + cL). The velocities of three spin modes depend on a ~-dependent 3  3 symmetric matrix G, G = (1=2)Tr [L; Lp] L; Lp. More explicitly, G is a diagonal matrix with elements: Gxx = 4 sin2(  23 ), Gyy = 4 sin2( + 23 ), Gzz = 4 sin2 . No- tice that the velocity of the phase mode vp can be written as p L=mBN , 43 3.3. Zero-Point Quantum Fluctuations 0.00 0.33 0.67 1.00 1.33 1.67 2.00  ξ/pi -0.002 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 E( ξ)− Ε(0 ) bL = -2.8 cL bL = 0.24 cL bL= 0 -10 -8 -6 -4 -2 0 0.2 bL/cL -0.005 0 5 10 15 20 g(b L/ c L ,  y) Figure 3.2: (Color online) The zero-point energy of spin nematics per lattice site as a function of  (in units of jMcLj5=2=t3=2L , cL < 0). Inset is g(x; y), the amplitude of E( = 6 )E(0) as a function of x(= bL=cL). y(= jcLj=tL) has been set to be 8 103. A quantum rst order phase transition occurs at bL = 0 as a result of zero-point quantum 
uctuations. For a positive bL, the energy minima correspond to uniaxial spin nematics while the maxima to biaxial spin nematics with Dih4 symmetries. only depending on the band mass mBN and the chemical potential L; it is independent of . Unlike the mean eld energy EMF that is independent of , the zero- point energy of spin nematics per lattice site E is -dependent (  summed over x; y; z), E() = 1 2NT X q; q q(2mBNv2() + q): (3.19) Here NT is the number of lattice sites. The main contribution to the - dependence of energy is from 
uctuations of wave vector jqj  mBNv(); for condensates, this characteristic momentum is much smaller than ~=dL. 44 3.4. Uniaxial and Biaxial Spin Nematic States The -dependent energy is proportional to P  v 5 =(d 5 Lt 4 L); the amplitude of it can be expressed as jMcLj5=2=t3=2L g(bL=cL; jcLj=tL), where g(x; y) is a dimensionless function that is studied numerically. In Fig. 3.2, we plot the energy as a function of . We nd that there are two distinct phases when the parameters occur in the region cL < 0 and bL > cL=4. They are separated by bL = 0 line. When bL is positive,  = 0; =3; 2=3; ::: are ground states. These are uniaxial nematic states which are rotationally invariant along an easy axis (see Fig. (3.1 a)). When bL is negative, the points of  = =6; =2; 5=6; ::: are the stable ground states while the uniaxial nematics are unstable. The energy as a function of  has following symmetries E() = E(); (3.20) E() = E(  3 + ): (3.21) These are a result of the rotation and gauge invariance of the energy function. They are found by examining the mean eld solutions in Eq.(3.11),(3.13). A rotation R of 120 around the x = y = z line eectively transforms a solution at  to + =3, or RT()R= (+ 3 ). And because the energy is invariant under SO(3) rotations and is an even function of , one nds that E() = E(+=3). In addition, a rotation of 180 around the x+y = 0 line in the xy-plane transforms () to (), i.e. RT()R = (); so similarly one nds that E() = E(). These symmetries only depend on rotational and gauge invariance and are exact. So the energy is an even and periodic function of , with the period equal to =3. For an analytic function with these symmetries,  = 0, =6 and =3 are always extrema. Our calculations of the zero-point energy are consistent with this. The above observation also indicates that up to an SO(3) rotation and a phase factor, states at  and at  + =3 are equivalent. 3.4 Uniaxial and Biaxial Spin Nematic States We now examine the topology of the manifold of 
uctuation-induced spin nematics. Without loss of generality, we rst consider the uniaxial nematic state at  = 0, i.e., xx = yy = 1=2 p 3 and zz = 1= p 3; in this case, the nematic easy axis specied by a unit vector e is pointing along the z- direction (See Fig. 3.1 (a)). Such a uniaxial state is invariant under an arbitrary rotation around the nematic axis e; it is further invariant under an inversion of the nematic axis e! e which is evident in the plot for the 45 3.5. Discussion and Conclusion wavefunction. The vacuum manifold for the uniaxial nematic is therefore simply S 2 Z2  S1; here S2=Z2 is where the nematic director lives and S1 is the unit circle of condensate phase variable. Unlike in uniaxial spin nematics of F = 1 atoms [123], here the spin orientation and condensate phase are not entangled. For the biaxial nematic state at  = =2, xx = yy = 1=2 and zz = 0. The state is invariant under the eight element dihedral-four group Dih4. The seven rotations that leave the state invariant (up to a phase shift) are 900, 1800 and 2700 rotations about the z-axis, 1800 rotations about the x- and y-axes and about the xy = 0 lines in the xy-plane. Four of these rotations, the 900, 2700 rotations around z-axis, and 1800 rotation around x  y = 0 lines must be accompanied by a shift of the phase of the condensate by . The manifold therefore is [SO(3) S1]=Dih4, where the Dihedral elements in the denominator contain the rotations mentioned plus the -phase shifts. These nematics contain half-vortices. On the other hand, a generic biaxial spin nematic (with no accidental symmetries) is only invariant under a rotation of 1800 around x-, or y- or z-axis and the invariant subgroup is the Klein-four or dihedral-two group; it has lower symmetries than either of the spin nematics selected by zero-point quantum 
uctuations. 3.5 Discussion and Conclusion The perturbative calculation carried out here is valid when 
uctuations of the order parameter  are small. Here we estimate the 
uctuations along each of the ve directions  = x; y; z; t; p. Using Eq. 3.16, we eval- uate the amplitude of local 
uctuations in an orthogonal mode, A1() = h0jyk;k; j0i. Taking into account the expression for v , we have the follow- ing estimate for the relative amplitude of 
uctuations, A1() M M1=2(~a ~ap aL + cL tL )3=2: (3.22) ~a is the eective scattering length of each mode which is a linear combina- tion of scattering lengths in F = 0; 2; 4 channels: ~a = 2G 7 (a2  a4) 1 5 (a0  a4) ; (3.23) ~at = 2 7 (a2  a4) 1 5 (a0  a4) ; (3.24) ~ap = 1 7 (2a2 + 5a4) + 1 5 (a0  a4) : (3.25) 46 3.5. Discussion and Conclusion For 87Rb atoms in optical lattices, ~a=~ap  102, aL  50nk and when tL  300nk, the relative amplitude of 
uctuations in x-, y-, z-, or t-mode is typically less than one percent(see next chapter (Chap.4) for numerical estimations of in spin-two 87Rb). One of experimental signatures of quantum 
uctuations is number 
uc- tuations in spin populations. The microscopic origin of spin-population 
uctuation is the two-body spin-changing scattering processes. For example atoms in the uniaxial nematic condensate(j2; 0i) are possible to be scat- tered into other spins via these channels j2; 0i + j2; 0i  j2;+2i + j2;2i and j2; 0i + j2; 0i  j2;+1i + j2;1i. The relative amplitude of the spin- population 
uctuations is directly related to the expression in Eq.3.22. In this chapter, we have found that zero-point quantum 
uctuations lift a continuous degeneracy in spin nematics. Only uniaxial spin nematics or biaxial nematics with dihedral-four symmetries are selected as the true ground states. For Rb87 atoms in the hyperne spin-two manifold, bL is be- tween 3cL and 10cL. According to the above analysis, the ground state should be a uniaxial spin nematic. The 
uctuation-induced energy land- scape can be experimentally mapped out by investigating the macroscopic quantum dynamics of condensates prepared in certain initial states. A con- densate of rubidium atoms initially prepared at state 1p 2 (j2; 2i + j2;2i) (corresponding to  = =2 point), because of the 
uctuation-induced poten- tial shown in Fig.3.2, could evolve towards a condensate with atoms at state 1 2 j2; 0i + p 3 2 p 2 (j2; 2i + j2;2i) (corresponding to  = =3 point). This leads to a temporal oscillation of the population of atoms at state j2; 0i, which is the main topic of the following chapter. 47 Chapter 4 Quantum Fluctuation Controlled Coherent Spin Dynamics in F=2 87Rb Condensates 4.1 Introduction Temporal evolution of a macroscopic condensate of ultra-cold atoms is usu- ally driven by mean eld potentials, either due to scattering between atoms or due to coupling to external elds; and coherent quantum dynamics of this type have been observed in various cold atom experiments. In this chapter, we report results of studies of a class of quantum spin dynamics which are purely driven by zero-point quantum 
uctuations of spin collec- tive coordinates. Unlike the usual mean-eld coherent dynamics, quantum 
uctuation-controlled spin dynamics or QFCSD studied here are very sen- sitive to variation of quantum 
uctuations and the corresponding driving potentials induced by zero-point motions can be tuned by four to ve orders of magnitude using optical lattices. These dynamics have unique depen- dence on optical lattice potential depths and quadratic Zeeman elds. We also nd that thermal 
uctuations generally can further enhance the in- duced potentials although the enhancement in deep optical lattices is much less substantial than in traps or shallow lattices. QFCSD can be potentially used to calibrate quantum 
uctuations and investigate correlated 
uctua- tions and various universal scaling properties near quantum critical points. Spin correlated macroscopic quantum dynamics have also been a focus of many cold atom experiments carried out recently. Spin ordering and spin- relaxation collisions were rst investigated in condensates of sodium atoms by Inouye et al. and Miesner et al. from Ketterle's group at MIT[38, 151]. Coherent spin dynamics driven by various mean eld interactions or external elds were later demonstrated in condensates of hyperne spin-two rubid- 48 4.1. Introduction ium atoms [39], and hyperne spin-one rubidium atoms[87, 92]. Ordering in spinor gases is usually induced by hyperne spin dependent two-body scattering[107, 108]. Coherent spin dynamics observed in experiments are related to the coherent quantum dynamics explored in solid state supercon- ductors, and earlier experiments on ultra cold gases of atoms[152, 54]. They are explicit manifestations of fascinating macroscopic quantum states and can be potentially applied towards constructing high precision interferome- ters. Remarkably, coherent dynamics also provide a unique direct measure of interaction energies or scattering lengths as emphasized before [152, 39, 87]. Quantum 
uctuation-controlled spin dynamics or QFCSD we are go- ing to study in this chapter on the other hand are a direct measure of quantum 
uctuations; they can be potentially used to calibrate quantum 
uctuations and investigate correlated 
uctuations near quantum critical points or universal scaling properties. Furthermore, QFCSD of cold atoms can be designed to simulate many other quantum-
uctuation-induced phe- nomena such as Coleman-Weinberg mechanism of spontaneous symmetry breaking[11], and order due to disorder in antiferromagnets[13, 14]. The system we are examining to understand QFCSD is a condensate of rubidium atoms (87Rb) in hyperne spin-two (F = 2) states. The two- body scattering lengths between rubidium atoms have been estimated using both photoassociation data[153, 28, 154, 143], and elastic scattering data near Feshbach resonances[29, 142]. Most recently, Rabi oscillations between dierent two-spin states have also been used to measure spin dependent interactions[96]. However it is quite challenging to probe QFCSD in traps without op- tical lattices, if not impossible. For dilute gases in the absence of optical lattices, the eective driving potential induced by quantum 
uctuations is about 105pK per particle (see section III for more discussions) because the relative amplitude of 
uctuations is very small. The corresponding dy- namics driven by such a small potential are only visible at a time scale of a few thousand seconds, too slow to be observed in current cold atom ex- periments. In addition, a tiny external magnetic eld of strength 1mG can result in a quadratic Zeeman coupling of order of 6  103pk for rubidium atoms which is three orders of magnitude larger than the induced potential in dilute gases. In most experiments because of noises in lasers, the eective quadratic Zeeman coupling can be controlled only up to an uncertainty that is equivalent to a magnetic eld of an order of 1mG. This further compli- cates future experimental studies of QFCSD . To resolve these diculties, we propose to enhance the eect of QFCSD using optical lattices. To vary the amplitude of quantum 
uctuations and optimize the eect of 
uctua- 49 4.2. Microscopic Hamiltonian for F = 2 87Rb in Optical Lattices tions, we study QFCSD in optical lattices where the optical potential depth V is a convenient tunable parameter[70, 71, 155, 156, 157]. The rest of the chapter is organized as follows. In Sec.4.2 and Sec.4.3, we introduce a lattice Hamiltonian to study dynamics of 87Rb atoms in optical lattices and discuss the range of parameters we have used to investigate this phenomenon. In Sec.4.4, we brie
y discuss the structure of spin nematic states. In Sec.4.5, we derive an eective Hamiltonian for QFCSD by in- troducing low energy collective coordinates and their conjugate momentum. In Sec.4.6 and Sec.4.7, we present our main numerical results on quantum- 
uctuation induced potentials, frequencies of coherent dynamics, and how potentials and frequencies depend on optical lattice potential depth. We also study the dynamical stabilities of coherent oscillations when a quadratic Zeeman coupling is present. In Sec.4.8, we discuss eects of thermal 
uctu- ations and analyze the potential-depth dependence of thermal enhancement of induced driving potential. In Sec.4.9, we further investigate eects of spin exchange losses and propose how to observe quantum-
uctuation-controlled spin dynamics within a relatively short life time of F = 2 rubidium atoms. In Sec.4.10, we estimate the condensation fraction or depletion fraction to verify the validity of our perturbative calculations. In the last section, we conclude our studies of QFCSD. 4.2 Microscopic Hamiltonian for F = 2 87Rb in Optical Lattices We are considering the following Hamiltonian for F = 2 atoms in optical lattices: H = X k  aL 2  ̂2k  ̂k  + bL 2  F̂2k  6̂k  + 5cLDykDk   tL X <kl>   yk; l; + h:c:    X k ̂k + qB X k Qk;zz: (4.1) Here k is the lattice site index and < kl > are the nearest neighbor sites,  is the chemical potential and tL is the one-particle hopping amplitude, qB is the quadratic Zeeman coupling constant. aL, bL and cL are three interaction constant which we will discuss later. We have employed the traceless symmetric matrix operator  y that was introduced previously in Sec.2.4.2 of Chap.2. Components  y, ;  = x; y; z are linear superpositions of ve spin-two creation operators,  ymF , 50 4.3. Range of Interaction Parameters for the Lattice Hamiltonian mF = 0;1;2. It is advantageous to use this tensor representation if one is interested in rotational symmetries of condensate wavefunctions, or con- struction of rotationally invariant operators. We use it to analyze collective spin modes that correspond to small rotations around various axes. The number operator ̂k, the dimer or singlet pair creation operator Dyk, the to- tal spin operator F̂k; are dened as ̂k = 1 2Tr   yk k  , Dyk = 1p40Tr   yk y k  , F̂k; = i
 yk; k;
 (See Chap.2 for details). The quadratic Zeeman operator Qk;zz is dened as Qk;zz = Tr   ykQ k  Q = 0@ 0 0 00 0 0 0 0 1 1A : (4.2) 4.3 Range of Interaction Parameters for the Lattice Hamiltonian Spin correlations between hyperne spin-two rubidium atoms are determined by three two-body s-wave scattering lengths aF , F = 0; 2; 4. In optical lat- tices, local spin-dependent interactions contain two contributions as shown in Eq.(4.1); one is, bLF2k=2, the energy of having total hyperne spin Fk at site k, and the other is the energy of creating spin singlet pairs (dimers), 5cLDykDk where Dk is the dimer creation operator. The usual contact in- teraction at site k is of the form aL( 2 k  k)=2, where k is the number of atoms. Three eective coupling constants aL; bL; cL which characterize vari- ous interactions are functions of two-body scattering lengths aF , F = 0; 2; 4 and on-site orbitals  0(r), aL(bL; cL) = a(b; c) 4~2 m Z dr ( 0(r) 0(r)) 2 : (4.3) Here a = (4a2 + 3a4)=7; b = (a4  a2)=7; c = (7a0  10a2 + 3a4)=35 (4.4) are three eective scattering lengths;  0 is the localized Wannier function obtained by solving the Schrodinger equation for an atom in periodical po- tentials. The band structures and Wannier functions are calculated through 51 4.3. Range of Interaction Parameters for the Lattice Hamiltonian 2 4 6 8 100 16 t L [n K] 2 4 6 8 100 80 a L [n K] 2 4 6 8 10 V[ER] 0.0 1.0 b L [n K] 2 4 6 8 10 V[ER] -0.4 0 c L [n K] Figure 4.1: (Color online) Coupling parameters aL, bL, cL and hoping in- tegral tL (all in units of nK) as a function of optical potential depth V (in units of recoil energy ER) in 3D (black circles) and 2D (red squares) optical lattices. ER = 3:18kHz for  = 850nm lasers. standard methods outlined in the Appendix. aL; bL; cL can then be calcu- lated using the estimates of scattering lengths obtained in Ref.[142]. The range of these parameters for rubidium atoms in optical lattices is plotted in Fig.4.1. The range of lattice potential depth is chosen to be from zero up to ten recoil energy or 10ER. (ER = h 2=2m2, where  is the wavelength of lasers.) Below 2ER, quantum 
uctuations turn out to be too weak to induce substantial potential and dynamics. And also the above Hamiltonian is not valid for shallow optical lattices as a large overlapping of Wannier functions will lead complications such as interaction term between adjacent sites. For these reason, we only show numerical results for V larger than 2ER but less than 10ER. Within this range, we nd that quantum depletion is usually small, less than twenty percent and our lowest order calculation should suce. Although quantum 
uctuations can be further enhanced above 10ER and a super
uid-Mott phase transition should take place at 13ER[70, 71], close to a critical region we however expect a perturbative calculation like the one carried out in this chapter becomes invalid. The range of atom number density here, or the number of atoms per lattice 52 4.3. Range of Interaction Parameters for the Lattice Hamiltonian site M is from zero to three. Most of data are shown for typical values M = 1:0; 2:0; 3:0. The single band Hamiltonian used here is valid when all interaction constants above are much smaller than the energy spacing between centers of two lowest bands. Note that a gap between the rst band and higher bands is not necessarily required for the single band Bose model as for the fermionic models. This is because in the condensate phase when depletion is small, most of the bosons are staying at the bottom of the rst band that very few atoms are occupying higher levels and can see the gap. On the contrary, properties of fermions in the optical lattice are sensitive to lling factors due to pauli exclusion principles that the existence or absence of a band gap can make a crucial dierence. For F = 2 rubidium atoms, the quadratic Zeeman coupling qB is related to a uniform external magnetic eld B via qB = 3(BB=2) 2=,  = 6:8GHz (see Appendix). Here B is the Bohr magneton and  is the hyperne splitting. For the purpose of studying QFCSD, we set the range of B to be 70mG > B > 1mG where eects of quantum 
uctuations are most visible; beyond 70mG the dynamics are mainly driven by the mean eld quadratic coupling qB (see section VI for more discussions). The corresponding range for the quadratic Zeeman coupling is then from 102pK to 100pK. Here we assume that the optical lattice potential does not dependent on hyperne spin levels. Practically, the optical dipole potential has a week hyperne spin dependence for alkali atoms in many cases because dipole transitions depend on which hyperne Zeeman levels are involved. For 87Rb, the hyperne spin dependent dipole interaction up to the leading order of mF is given by [158, 159, 160] V (r) = V0 241 + gFmF 1=2 3=2 1=2 + 23=2 X q=1;0;1 q Iq(r) I(r) 35 (4.5) Here mF is the Zeeman state (with gyromagnetic ratio gF ) of the atom; 3=2 (1=2) is the detuning relative to the S ! P3=2 (S ! P1=2) transition; q refers to the three possible polarizations of light. Iq is the intensity of laser beams with polarization q and I is the total intensity. The small parame- ter controlling the strength of hyperne spin dependent term is 1=23=2 1=2+23=2 which is about 0.07 for 87Rb loaded in optical lattices operating at a wave- length of 850nm. By choosing a suitable polarization of the laser beams[160], one can make the rst order term vanish, resulting in an even weaker hy- perne dependence appearing in the quadratic term of mF . In this case, 53 4.4. Mean Field Nematic States and Coherent Dynamics the leading contribution is from the residual quadratic term of mF , which is estimated to be 101 at maximum of the spin-dependent two-body in- teraction. More importantly, this residual quadratic term can be absorbed into the quadratic Zeeman term in Eq.4.1 by introducing a small shift in the denition of qB given in the previous paragraph. For these two reasons, we will neglect the eects of spin dependent optical potential. 4.4 Mean Field Nematic States and Coherent Dynamics The coherent spin dynamics of a uniform condensate can be described by the evolution of a condensate wavefunction ~, i.e. the expectation value of matrix operator  y. The corresponding equation of ~ is i 2 @ ~ @t = @Hsc @ ~ + @HQ @ ~ + @Hqf @ ~ ; @Hsc @ ~ = aL 4 ~Tr(~ ~) + cL 4 ~Tr(~~) + bL 2 [~; [ ~; ~]] (ztL + =2)~; @HQ @ ~ = qBQ~: (4.6) Here Hsc is the mean eld (semiclassical) Hamiltonian (of matrix ~, ~) obtained previously[134, 125, 124], HQ is the quadratic Zeeman coupling term with qB being the coupling strength. Hsc and HQ(per lattice site) are given by: Hsc = aL 8 Tr(~ ~)Tr(~ ~) + cL 8 Tr(~ ~)Tr(~~) + bL 4 Tr[~; ~]2  (ztL + =2)Tr(~ ~); (4.7) HQ = qBTr(~ Q~): (4.8) z is the coodination number of optical lattices. For a eld along the z- direction, matrix Q is dened as Q = zz. And Hqf is the quantum- 
uctuation-induced Hamiltonian discussed below. For rubidium atoms, scattering lengths estimated in Ref.[28, 154, 143] lead to bL = 10cL while results in Ref.[29, 142] yield bL = 2:8cL. Both calculations show that interaction parameters satisfy cL < 0 and 4bL > cL. As pointed out in the previous chapter, in this parameter region without 54 4.5. Eective Hamiltonian for QFCSD quadratic Zeeman coupling, ground states are spin nematics characterized by real and symmetric tensor wavefunctions ~ (up to an overall phase). Any condensate that is initially prepared in this submanifold has no mean eld dynamics because the potential gradient @Hsc=@ ~  vanishes. 4.5 Eective Hamiltonian for QFCSD To investigate the kinetic energy and quantum 
uctuation-induced potential energy for dynamics of a condensate initially prepared in this submanifold, we expand tensor  y about a reference condensate wavefunction () in terms of ve collective coordinates X introduced above and their conju- gate operators P̂ . We furthermore separate the macroscopic dynamics of condensates (q = 0-mode) from the microscopic zero-point quantum 
uc- tuations (q 6= 0 mode). And we restrict ourselves to the dynamics of a condensate in a linear regime. 4.5.1 Nematic Manifold To highlight the structure of degenerate ground states, we consider an ar- bitrary condensate amplitude ~ in the spin nematic submanifold. It can be parameterized using an SO(3) rotation, R and a U(1) phase shift , and a real diagonal traceless matrix ; i.e., ~(Xx; Xy; Xz; ; ) = p 4MeiRT ()R; R(Xx; Xy; Xz) = exp(T xXx + T yXy + T zXz) (4.9) where M is the number density or average number of atoms per lattice site. R is an SO(3) rotation matrix dened by three spin angles X,  = x; y; z, and antisymmetric generators T
 = 
 . (),  2 [0; 2], are normalized real diagonal traceless matrices that form a family of solutions specied by a single parameter [124];  = sin(  )p 3 ; (4.10) and x = =6, y = 5=6 and z = 3=2. Nematics with dierent  exhibit dierent spin congurations. Following the denition of tensor operator in Eq.(2.34), one can easily show that these solutions represent conden- sates of spin-two atoms specied by ve-component wavefunctions  T = ( 2;  1;  0;  1;  2) and  T = p M( sin p 2 ; 0; cos ; 0; sin p 2 ): (4.11) 55 4.5. Eective Hamiltonian for QFCSD Figure 4.2: (Color online) An artistic view of the ve-dimension manifold of spin nematics. Three SO(3) rotation degrees and U(1)-phase degree are represented by a cross section at a given spin deformation angle . We have also plotted the nematic wavefunctions  S(; ) = P  nn as a function ;  in a spherical coordinate. n, is the th component of a unit vector n(; ) and colors indicate phases of wavefunctions. So in a more conventional representation, these states labeled by value  correspond to condensates where all atoms occupy a particular spin-two state, j >= cos j2; 0i+ sin p 2 (j2; 2i+ j2;2i): (4.12) Therefore, the nematic submanifold is eectively a ve-dimension space that is characterized by ve collective coordinates X ,  = x; y; z; ; p: three spin rotational angles X ( = x; y; z), one spin deformation -angle X (or ) specifying spin congurations and one phase angle Xp(= ). Further- more, it is easy to verify that up to an SO(3) rotation and a phase factor, states at  and at +=3 are equivalent and the fundamental period of this characterization is =3. When the system is rotationally invariant (i.e. no external elds), we also nd that E() = E( + =3) andE() = E()(See previous discussion in Chap.3). E() is the energy of a state dened by . 56 4.5. Eective Hamiltonian for QFCSD 4.5.2 Collective Coordinates and Conjugate Momentum To study QFCSD, we expand  y about a reference condensate ,  ̂yk; = p 4M() + 1p NT X  L(̂ y (0) + X q6=0 eiqrk ̂y(q)): (4.13) Here NT is the number of lattice sites andM is the number of atoms per site. The superscripts (or subscripts)  = x; y; z; ; p specify condensate motion along the ve orthogonal directions of the submanifold. ̂(q) (̂ y (q)) are annihilation (creation) operators of the collective excitations of mode  with lattice crystal momentum q. One can further include bilinear terms in the expansion. The motion along these ve orthogonal directions is studied by introducing ve mutually orthogonal matrices L , i.e. Tr(LL) = 2 . Explicit forms of these orthogonal matrices were introduced in the previous chapter. We are restricting ourselves to the dynamics of a condensate in a linear regime. By expanding the original Hamiltonian in Eq.(4.1) using the decom- position introduced in Eq.(4.13), we obtain an eective Hamiltonian that is bilinear in terms of y and . It is convenient to introduce ve collective Hermitian operators ~X dened in terms of  y (0), (0) for q = 0 modes, X̂p = yp  p 2i p N ; X̂t = yt + t 2 p N ; X̂x = yx + x 4 p N(yy  zz) ; X̂y = yy + y 4 p N(zz  xx) ; X̂z = yz + z 4 p N(xx  yy) (4.14) where N = NTM is the total number of atoms in a lattice with NT sites. By comparing Eq.(4.13) and the expression for ~ in terms of ve spin coor- dinates X (see discussions before Eq.(4.10)), one identies the semiclassical collective coordinates X as the expectation value of harmonic oscillator op- erators X̂ . In the following, we do not distinguish between X̂ and X . 57 4.5. Eective Hamiltonian for QFCSD The conjugate momentum operators P̂ can be introduced accordingly so that the usual commutating relations are obeyed, i.e [X̂ ; P̂] = i. P̂p = p N  ̂yp + ̂p  ; P̂t = i p N  ̂t  ̂yt  ; P̂x = 2i p N (yy  zz)  ̂x  ̂yx  ; P̂y = 2i p N (zz  xx)  ̂y  ̂yy  ; P̂z = 2i p N (xx  yy)  ̂z  ̂yz  : (4.15) 4.5.3 Spectra of Collective Modes Five modes of zero-point motions (microscopic) around () can also be labeled by the same set of indices. For collective excitations of mode- with lattice momentum q(6= 0), creation operators are y(q). These operators obey the bosonic commutation relations, h k;;  y l; i = kl . Note that only the properties of three rotation modes and deformation mode (-mode) depend on the value of . The collective mode dispersion around  = 0 or  = =2 has a very simple form, E;q(;X; qB) = q ;q[2mBNv2( +X) + ;q]: (4.16) And in the expression for E;q, ;q = 4tL X  (1 cos qdL) + qB (4.17) is the energy of an atom with crystal quasi-momentum q = (qx; qy; qz); dL is the lattice constant. mBN = 1=(4tLd 2 L) is the eective band mass.  ,  = x; y; z;  are coecients introduced in Eq.(4.19),(4.20). v is the sound velocity of the -mode in the small-jqj limit, v2() = M (4bLG() cL) mBN ;  = x; y; z; t (4.18) G() and  are a function of 8>><>>: Gxx = sin 2    23  Gyy = sin 2   + 23  Gzz = sin 2 () Gtt = 0 58 4.5. Eective Hamiltonian for QFCSD 8>><>>: x =  2 xx  2zz y =  2 yy  2zz z = 0 t = 4( _ 2 zz  2zz) Here _zz = dzz=d. When the quadratic Zeeman coupling is zero, the energy dispersion is also derived in the previous chapter. The dispersion of the t-mode in this case is independent of  or X. And among four gapless spin modes, only the x-,y- and z- spin rotational modes contribute to the -dependence of the quantum 
uctuation-induced potential Vqf . Substituting Eq.(4.16) into Eq.(4.19), one obtains the quantum-
uctuation-induced potential Vqf . When the quadratic Zeeman coupling is present, we nd that for  = 0, x;y;z;t are not positive dened and collective spin modes can be unstable. To calculate induced-potential Vqf , we only include stable modes (q 6= 0) which contribute to the renormalization of adiabatic condensate dynamics. At  = =2, all collective modes are stable. 4.5.4 Eective Hamiltonian In the previous section, using the decomposition introduced in Eq.(4.13) we expand the Hamiltonian in Eq.(4.1) in terms of collective coordinates X , P̂ and obtain an eective Hamiltonian for submanifold dynamics. Up to the quadratic order, the Hamiltonian contains two sectors, one involving operators of q = 0-mode and the other one only involving operators of q 6= 0. The Hamiltonian for the q = 0 sector generates the kinetic energy needed for the dynamics along ve orthogonal directions. The corresponding eective masses can be expressed in terms of scattering lengths a0;2;4, and quadratic Zeeman coupling qB. In addition, the expansion of the quadratic Zeeman term HQ for q = 0 mode also generates a mean eld potential VQ. This potential VQ as illustrated below always favors a biaxial nematic with  = =2. The biaxial nematic has dihedral-four (Dih4) symmetries with easy axes in the xy-plane[124]. On the other hand, the sector of Hamiltonian for q 6= 0 collective modes contains zero-point energies of those modes. These energies in general depend on spin congurations or the values of . So quantum 
uctuations of collective coordinates eectively induce a - dependent potential Vqf in the submanifold; this potential alone selects out a unique ground state as recently pointed out by Song et al [124] and Turner et al [161]. For rubidium atoms with a positive bL, previous calculations show that the ground state is a uniaxial nematic in the absence of quadratic 59 4.5. Eective Hamiltonian for QFCSD Zeeman coupling. Turner et al also pointed out that thermal 
uctuations further enhance the amplitude of induced potentials and this order-from- disorder phenomenon is robust against nite temperatures[161]. We now study the dynamical consequences of both quadratic Zeeman coupling HQ and zero-point quantum 
uctuations Hqf . For simplicity, we are mainly focused on dynamics around a) a uniaxial nematic at  = 0 or a condensate with rubidium atoms occupying hyperne spin state j2; 0i; b) a biaxial nematic at  = =2 or a condensate with rubidium atoms occupying hyperne spin state (j2; 2i+ j2;2i)=p2. The resultant Hamiltonian for oscillations around a state  ( = 0 or =2) can be cast in the following form, Hqf = X  P̂ 2 2NTm +NT (Vqf (;X) + VQ(;X)); Vqf (;X) = 1 2NT X q6=0 X  E;q(;X; qB); VQ(;X) = 4MqB 3 cos2( +X) + 4MqB X  X 2 : (4.19) In Eq.(4.19), NT is the number of lattice sites and M is the average number of atoms per site. The masses for ve directions are calculated to be mp = 1 aL + cL ; mt = 2M qB  2McL ; m = 8MG 8MbLG  2McL + qBG1 : (4.20) t is a function of  and is 4=3 when  = 0, and 4=3 when  = =2. E;q(;X; qB) is the energy of a mode- ( = x; y; z; t; p) collective excita- tion with crystal momentum q, and is a function of parameters ;X, and quadratic Zeeman coupling qB. Only spin modes with  = x; y; z;  con- tribute to the -dependence of potential Vqf . Fluctuations of phase modes are independent of parameter  or Zeeman coupling qB and are irrelevant for discussions of spin dynamics as a result of spin-phase separation. Note that Vqf is a function of , qB and X; and VQ is a function of , qB and X ,  = x; y; z; . 60 4.6. Potentials Induced by Quantum Fluctuations 4.6 Potentials Induced by Quantum Fluctuations We rst consider a situation where the quadratic Zeeman coupling is ab- sent and the potential VQ vanishes. As argued before, generally speaking Vqf (; 0) = Vqf ( + =3; 0) and Vqf (; 0) = Vqf (; 0). The explicit form of Vqf calculated here indeed is consistent with this general requirement. We also nd that the main contribution to Vqf is from 
uctuations of wave- length 1=(mBNv) that is much longer than the lattice distance dL. More specically, the -dependent induced potential can be written as a sum of polynomials of spin-wave velocities, i.e. P  v d+2  for d-dimension optical lattices. Spin-wave velocities are functions of cL; bL and tL. We numerically integrate over all wavelengths and obtain the -dependence of Vqf as shown in the inset of Fig.4.3. Consequently, we nd that in d-dimension lattices (d = 2; 3), the barrier height Bqf which is dened as Vqf (  6 ; 0)  Vqf (0; 0), the energy dierence between  = =6 and  = 0 satises the following simple scaling function, Bqf = jMcLj d+22 t d 2 L gd  bL cL ; M jcLj tL  : (4.21) Here gd(x; y) is a dimensionless function that can be studied numerically. This scaling function is either insensitive to the variation of y as for d = 3 [124] or independent of y as for d = 2. In 3d optical lattices, we nd that the -dependent energy potential is proportional to P  v 5 =(d 5 Lt 4 L), dL is the lat- tice constant and tL is the hopping integral. In 2d lattices, the -dependent energy potential Vqf () is proportional to P  v 4 =(d 4 Lt 3 L) ln tLdL=v. Eq.4.21 suggests that one can increase the quantum 
uctuation-induced potential by increasing cL. This is consistent with our intuition that stronger two-body interactions lead to higher quantum 
uctuations. As pointed out in Sec.3.3, the microscopic origin of the zero-point spin 
uctuations in con- densates is the two-body spin-changing scattering, of which the energy scale is the spin-dependent coupling jcLj in the optical lattices. Alternatively, one can enhance the quantum 
uctuations by increasing the ratio cLtL , which is a dimensionless quantity measuring the interaction strength. As a comparison, the interaction strength in single-component Bose gases is measured by the gas parameter p a3s, in which  is the density and as is the scattering length. The \gas parameter" for spin dependent interaction strength in optical lattices is q M jcj3L=t3L. One striking consequence of Eq.4.21 is that one can enhance the quan- tum 
uctuation induced potential very eectively by increasing the optical 61 4.6. Potentials Induced by Quantum Fluctuations 2.0 4.0 6.0 8.0 10.0 V[ER] 0.001 0.01 0.1 1 10 B qf [p K] M=1.0 M=2.0 M=3.0 0 pi/3 2pi/3 piξ0 0.4 0.8 V qf [p K] Figure 4.3: (Color online) Barrier height Bqf (in units of pK) as a function of optical potential depth V (in units of recoil energy ER). Inset is for the -dependence of Vqf , a potential induced by quantum 
uctuations when V = 6ER and M = 2:0. potential depth V . As shown in Fig.4.1 jcLj increases and tL decreases as V increases, both of the changes are enhancing 
uctuations. When V increases from 2ER to 10ER, jcLj increases by 4 times, and tL becomes one-fth, so Bqf is estimated to be enhanced by 1000 times according to Eq.4.21. We study the barrier height Bqf as a function of V , the potential depth of optical lattices. We nd that in the absence of lattice potentials or in traps, for a density that is equivalent to one particle (M = 1:0) per lattice site the barrier height is of order of 105 to 104pK and is negligible in experiments. When the optical lattice potential depth V is varied, the barrier height typically increases by four or ve orders of magnitude. Particularly, as V increases from 2ER to 10ER, mean eld interaction energies aL; bL; cL vary by less than a factor of three; however, for the same range of V , the barrier height Bqf varies from 10 3pK to a few pK. The energy shift between the uniaxial state at  = 0 and biaxial state at  = =6 is analogous to the Lamb shift observed in atoms[7]. In Fig.4.3, we show the barrier height Bqf of induced potential versus lattice potential depth V . In Fig. 4.4, we further plot the eective potential Veff = VQ + Vqf as a function of optical potential depth V . When V is less than Vc, the 62 4.6. Potentials Induced by Quantum Fluctuations 0 0.2pi 0.4pi 0.6pi 0.8pi pi  ξ -0.02 -0.01 0 0.01 V ef f[n K] V=2ER V=5ER V=8ER V=9ER V=11ER (a) 2 4 6 8 10 V[ER] 0.3pi 0.5pi 0.7pi ξ (b) Figure 4.4: (Color online) (a) Eective potential Veff = VQ + Vqf (in units of nK) as a function of  for various optical potential depth V (in units of recoil energy ER) of 3D lattices. The quadratic Zeeman coupling strength is set to be 10pk(31mG) and number of atoms per lattice site M is equal to one. (b) Values of  at which global potential minima in (a) are found are plotted as a function of optical potential depth V . 63 4.7. Oscillations Induced by Quantum Fluctuations 2 4 6 8 10 V[ER] 0 8 16 24 f 0 [H z] M=1.0 M=2.0 M=3.0 (a) 2 4 6 8 10 V[ER] 0 8 16 24 f 0 [H z] M=1.0 M=2.0 M=3.0 (b) Figure 4.5: (Color online) (a),(b), frequencies f0 = 0=(2) for population oscillations around j2; 0i state are shown as a function of potential depth V in (a) 2D and (b) 3D optical lattices for dierent atom number density M ; here the quadratic Zeeman coupling is absent. biaxial nematic with  = =2 is stable and as V > Vc, it becomes locally unstable. For 87Rb atoms with one atom per lattice site (M = 1:0), Vc is about 5:5ER (ER is the recoil energy of optical lattices) when the quadratic Zeeman coupling is 10pK. Almost opposite behaviors are found for uniaxial nematics at  = 0. Values of  at which global potential minima are found are shown as a function of Veff in Fig. 4.4b. 4.7 Oscillations Induced by Quantum Fluctuations To understand the dynamical consequences of Vqf , we consider coherent dynamics around the uniaxial nematic state at  = 0 that is selected out by the 
uctuation-induced potential Vqf or the biaxial nematic state at  = =2 that is favored by the quadratic Zeeman potential VQ. Particularly we are interested in oscillations along the Xt-direction around state  = 0 or =2. These motions correspond to the following time evolution of condensates, jti  (cos   sin X0F (t)) j2; 0i+ 64 4.7. Oscillations Induced by Quantum Fluctuations 1p 2 (sin  + cos X0F (t))(j2; 2i+ j2;2i); F (t) = cos   2 t+ i m  4M sin   2 t (4.22) where X0 is the initial deviation from state  = 0 or =2 and m is the eec- tive mass given in Eq.(4.20). By solving the equation of motion in Eq.(4.19) including the quadratic Zeeman eects, we derive a general expression for oscillation frequencies (= 2f). The oscillation frequency (qB; V ) in general is a function of the quadratic Zeeman coupling qB, optical lattice potential depth V , and the number density M . For population oscillations around  = 0 or =2, we obtain the following frequencies   = 2 r ( 8MqB 3 + V 00 qf )( 2qB 3M + jcLj); (4.23) here  = 1 for  = 0 and =2 respectively. V 00qf is the curvature of potentials at  = 0 or =2. Note that both the real and imaginary part of F (t) oscillate as a function of time; thus unless m  4M is exactly equal to one which only occurs when atoms are noninteracting or the quadratic Zeeman coupling qB is innite, the magnitude of F (t) oscillates as the time varies. Practically, in most cases we examine below, m  turns out to be much less than unity because both the curvature V 00 qf and the Zeeman coupling qB are smaller than the inverse of eective mass m, or the spin interaction energy cL; thus oscillations of the imaginary part of F (t) can be neglected. Oscillations along the direction of X therefore always lead to temporal oscillations in population of rubidium atoms in either j2; 0i or j2;2i states that can be observed in experiments. We rst examine the cases when the quadratic Zeeman coupling is ab- sent i.e. qB = 0. The frequency in this limit is a direct measure of quantum 
uctuations and 0 = (qB = 0; V ) = 2 q V 00 qf jcLj. In the absence of lattice potentials, or in traps we nd that oscillation frequencies are about 103Hz. At nite temperatures, thermal 
uctuations do enhance oscillation frequen- cies by a factor of two to four; however, this is far from sucient for the experimental study of QFCSD within the life time of these isotopes[39]. In optical lattices, we nd that the enhancement of spin-dependent in- teractions bL; cL, and especially the rapid increasing of band mass mBN can result in oscillation frequencies of order of a few Hz, which are about three to four orders of magnitude higher than those in traps. The scaling behavior 65 4.7. Oscillations Induced by Quantum Fluctuations of 0 is closely related to that of Bqf . Taking into account the expression for eective mass m, we nd  0  M d+2 4 jcLj d+44 t d 4 L g 1 2 d ( bL cL ; M jcLj tL ): (4.24) In Fig. 4.5, we show the plot of f0 = 0=2, the oscillation frequency versus V in the absence of Zeeman coupling qB. Experimental studies of QFCSD can also be carried out by investigat- ing dynamics of rubidium atoms in the presence of nite quadratic Zeeman coupling qB. The submanifold dynamics now are driven by both quadratic Zeeman coupling and quantum 
uctuations. There are three main modi- cations to the eective Hamiltonian Heff . Firstly, according to Eq.(4.20), eective masses now depend on the quadratic Zeeman coupling qB. Sec- ondly, following discussions in the appendix, among four spin collective modes ( = x; y; z; t) and one phase mode ( = p), two of spin modes, x- and y-mode, can be gapped and one, z-mode, always remains gapless. Details of spectra of -mode depend on the value of ; excitations are unsta- ble when  is negative and are gapped when  are positive. The energy gap of stable -mode excitations depends on  and becomes maximal at  = =2. Finally, a mean eld quadratic Zeeman potential that depends on X and X,  = x; y; z, is also present and modies the spin dynamics. Oscillation frequencies  as a function of quadratic Zeeman coupling qB for various potential depth V can be calculated explicitly using Eq.(4.23). Plots of these results are shown in Fig.4.6,4.7. In the limit of weak quadratic Zeeman coupling, QFCSD can be most conveniently studied around the uniaxial nematic state at  = 0, or j2; 0i state. For rubidium atoms, this is the ground state when the quadratic Zeeman coupling is absent, and remains to be locally stable along the di- rection of Xt up to a nite qc1. When the quadratic Zeeman coupling qB is much smaller than qc1, the eective potential Veff is mainly due to the quantum-
uctuation induced one, Vqf . Above qc1, a dynamical instability occurs and a perturbation along theX-direction around j2; 0i starts to grow exponentially. We nd that in traps where V = 0, qc1 is about 10 2pK; in optical lattices when V varies between 2ER to 10ER, the value of qc1 in- creases from 0:1pK up to about 0:1nK. And as qB approaches zero, the oscillation frequency (qB; V ) saturates at the value 0 shown in Fig.4.5. The qB-dependence of frequencies for oscillations around  = 0 is shown in Fig.4.6; near qB = qc1, the frequencies scale as p qc1  qB. Alternatively, one can also study oscillations around states at which global potential minima 66 4.7. Oscillations Induced by Quantum Fluctuations 0.00 0.05 0.10 0.15 qB[nK] 0 10 20 30 f[H z] M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER (a) 2 4 6 8 10 V[ER] 0.00 0.15 q C 1[n K] M=1.0 M=2.0 M=3.0 (b) 0.00 0.10 0.20 qB[nK] 0 10 20 30 f[H z] M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER (c) Figure 4.6: (Color online) Frequencies f = =(2) for population oscillations as a function of quadratic Zeeman coupling qB(in units of nK) for various potential depth V in 2D optical lattices withM atoms per lattice site. (a) for oscillations along the Xt-direction around state j2; 0i; dynamics are unstable above a threshold qc1(V ). The V-dependence of qc1 (in units of nK) is shown in (b). (c) is for oscillations around states with  = m(< =2) at which global potential minima shown in Fig.4.4(a) are found. (see also Eq.(4.22)) 67 4.7. Oscillations Induced by Quantum Fluctuations 0.0 0.1 0.2 qB[nK] 0 10 20 30 40 f[H z] M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER (a) 0.0 1.5qB[nK] 0 100 f[H z] (b) 2 4 6 8 10 V[ER] 0.0 0.2 q C 2[n K] M=1.0 M=2.0 M=3.0 (c) Figure 4.7: (Color online) (a), (b) Frequencies f = =(2) for oscillations along the Xt-direction around biaxial nematic state j2; 2i + j2;2i versus quadratic Zeeman coupling qB (in units of nK) for various optical potential depth V in 2D optical lattices with M atoms per lattice site. (a) is for population oscillations close to the threshold qc2 and (b) is for away from qc2. Oscillations are unstable below a threshold qc2; in (c), qc2 (in units of nK) as a function of V is shown. 68 4.7. Oscillations Induced by Quantum Fluctuations 0.00 0.05 0.10 qB[nK] 2 4 6 8 10 V[ER] qC1qC2 Uniaxial Biaxial Figure 4.8: (Color online) Regions of dynamical stability in the V qB plane for three particles per lattice site (M = 3:0). Oscillations around uniaxial nematic or state j2; 0i (biaxial nematic or state j2; 2i + j2;2i) are locally stable along the X-direction in the yellow shaded (purple patterned) area; dynamical instabilities occur when qB > qc1(V ) (qB < qc2(V )). qB is in units of nK. of Veff are found; similar qB-dependence is shown in Fig.4.6c. At relatively high frequencies, convenient oscillations to investigate are the ones along the Xt direction around the biaxial nematic state at  = =2 point or (j2; 2i + j2;2i)=p2 state. This state becomes stable when qB is larger than qc2 and oscillations are well dened only in this limit. When qB is smaller than this critical value, the dynamics are mainly driven by quantum 
uctuations and oscillations are unstable. In the vicinity of qc2, the oscillation frequency again scales as p qB  qc2. When qB is much bigger than qc2 but smaller than cL, the frequency is proportional to p qB; in this limit, the potential Veff is already dominated by the quadratic Zeeman term, or VQ, but the eective mass is mainly induced by spin dependent interactions. When the quadratic Zeeman coupling further increases well above the value of cL, relatively fast dynamics are now mainly driven by the exter- nal coupling and the frequency  approaches 8qB=3, which is equal to the quadratic Zeeman splitting between state j2; 0i and j2;2i. In this limit, scattering between atoms during a short period of 2=  becomes 69 4.8. Eects of Thermal Fluctuations negligible; the population oscillations due to scattering or interactions are therefore signicantly suppressed as here hyperne spin two atoms are eec- tively noninteracting. Indeed, following the expression for F (t) in Eq.(4.22) and Eq.(4.20),(4.23), one nds that the amplitude of population oscillations scales as 1=qB at large qB limit and F (t) becomes exp(i4qBt=3) when qB approaches innite, i.e., a pure phase factor with a constant modulus. In Fig.4.7, we show that the qB-dependence of oscillation frequencies both close to the threshold qc2 and away from qc2. As we have mentioned before, QFCSD is rather sensitive to the variation in V . qc1 and qc2 as functions of V are shown in Fig. 4.6,4.7. So far we have studied coherent dynamics driven by potentials induced by quantum 
uctuations at zero temperature. In the following section, we analyze the eect of temperatures, or thermal 
uctuations. 4.8 Eects of Thermal Fluctuations We now turn to the eect of temperatures and focus on three-dimension op- tical lattices. At a nite temperature, collective modes including spin waves are thermally excited. The occupation number of th spin-wave excitations with momentum ~q and energy E;q is n(; q) = 1 e E;q kT  1 : (4.25) The free energy density (or per lattice site) of a condensate characterized by  is F (; T; V ) = 1 NT X ;q  E;q 2 + kT ln(1 eE;qkT )  : (4.26) Just as in the zero temperature case because of the -dependence of spin wave velocities, the free energy density also depends on the values of  and the main contribution to the -dependence of free energy density is again from 
uctuations of wavelength 1=(mBNv), or of a characteristic energy T that is of an order of jcLj (or bL). Thermal 
uctuations become more important than quantum ones when temperatures are much higher than T. The other temperature eect is from the temperature dependence of spin wave velocities v. Because of the thermal depletion, spin wave velocities v( = x; y; z) decrease as temperatures increase and become zero when the BEC temperature TBEC is approached. Therefore the temperature 70 4.8. Eects of Thermal Fluctuations dependence of free energy are determined by two dimensionless quantities, T=T and T=TBEC . In all cases we are going to study below, T (of order of cL) is less or much less than TBEC . We study the asymptotics of free energy in low temperature (T  T) and high temperature (T  T) limits. When T  T, only modes with energy much smaller than T are thermally occupied. Therefore contribu- tions from these thermal excitations can be calculated by substituting quasi- particle spectra E;q approximately with phonon-like spectra vq. One then obtains the following expression for the free energy per lattice site (up to a constant) F (; T; V ) = Vqf (1 ( T TBEC ) 3 2 ) 5 2   2kT 90 X   dLkT v 3 : (4.27) The barrier height, Bth(T; V ) = F ( =  6 ; T; V )F ( = 0; T; V ), can be calculated accordingly and the asymptotics at low temperatures is Bth(T; V )Bth(0; V )  jcLj 5=2 t 3=2 L ( kT jcLj) 4: (4.28) Here we have neglected the term depending on T=TBEC because we are interested in temperatures much less than TBEC .(Such a term is only im- portant at very low temperatures that are of an order of jcLj(jcLj=TBEC)3=5 ( T)). Therefore, the low temperature enhancement of Bth is mainly from the "black-body" radiation of spin wave excitations. When T  T, spin modes with energy much bigger than T are ther- mally occupied. For modes of energy T, we can approximate n(; q) with a classical result n(; q) = kT=E;q. Our scaling analysis of the free energy per lattice site shows that in this limit, F (; T; V ) = 2kT 3 X  ( v 4dLtL )3(1 ( T TBEC ) 3 2 ) 3 2 : (4.29) The barrier height, Bth(T; V ) = F ( =  6 ; T; V )F ( = 0; T; V ), can be calculated accordingly and asymptotics at high temperatures are Bth(T; V )  c 5=2 L t 3=2 L kT cL (1 ( T TBEC ) 3 2 ) 3 2 : (4.30) 71 4.9. Eects of Spin Exchange Losses The above result shows that at relatively high temperatures thermal spin wave 
uctuations enhance induced potentials and this enhancement is char- acterized by a linear function of kT=jcLj or kT=T. However, because of the thermal depletion of condensates at nite temperatures, there is an overall suppression given by a factor 1  (T=TBEC)3=2. The competition between these two eects results in a maxima in the free energy density. That is at temperatures larger than T but much smaller than TBEC , the free energy increases linearly as a function of temperature until the ther- mal depletion becomes signicant. The maximal enhancement is therefore about TBEC=T. At further higher temperatures, the free energy density decreases and near TBEC , its temperature dependence is mainly determined by a factor (1 (T=TBEC)3=2)3=2 as shown in Eq.(4.30). Evidently, the magnitude of maximal enhancement is very much depen- dent on the ratio between TBEC and T. In optical lattices, the magnitude of BEC transition temperatures for, on average, one or two particles per lattice is mainly set by the band width tL while T is determined by jcLj. This ratio and therefore the enhancement is quite sensitive to the optical lattice potential depth V . We nd that the enhancement in the absence of lattice potentials is substantial and thermal 
uctuations increase the in- duced potential by about a factor of fty. This is consistent with a previous calculation[161]. However, when the lattice potential depth increases, cL increases but tL decreases, the thermal enhancement becomes much less sig- nicant. At V = 10ER, we nd that the thermal eect only increases the potential by a few tens of percent and the potential can be predominately due to quantum 
uctuations. 4.9 Eects of Spin Exchange Losses For F = 2 rubidium atoms, the life time is mainly due to spin exchange losses that take place at a relatively short time scale of about 200ms[39, 87]. Note that three-body recombination usually occurs at a much longer time scale and has little eect on the time evolution studied here. Spin exchange losses put a very serve constraint on possible observations of full coherent oscillations driven by quantum 
uctuations at frequencies well below 5Hz. To overcome this diculty, we suggest to apply a quadratic Zeeman coupling close to qc1 (qc2) and study time evolution of state j2; 0i or 1p2(j2; 2i + j2;2i). Regions of dynamical stability of these states can be obtained by studying small oscillations around them as shown in section III. Results are summarized in Fig.4.8. 72 4.9. Eects of Spin Exchange Losses 0 10 20 30 40 50 60 T[nK] 0.01 0.1 1 10 B qf [p K] V=2ER V=4ER V=6ER V=8ER V=10ER (a) 0 5 10 15 20 25 T[nK] 0 1 2 3 4 B qf [p K] V=6ER High T asymptotics (b) Figure 4.9: (Color online) Induced free energy barrier height as a function of temperature for dierent lattice potentials. The dashed line in (b) is a plot for the high temperature asymptotics derived in Eq.(4.29). Note that thermal enhancement of barrier height is much less signicant in deep optical lattices. 73 4.9. Eects of Spin Exchange Losses 0 100 200 300 400 5000.0 0.2 0.4 0.6 0.8 1.0 re la tiv e po p.  in  m F= 0 1mG 17mG 35mG 43mG 64mG (a) 0 100 200 300 400 5000.0 0.2 0.4 0.6 0.8 1.0 re la tiv e po p.  in  m F= 0 1mG 17mG 35mG 43mG 64mG (b) 0 100 200 300 400 500 time[ms] 0.5 0.6 0.7 0.8 0.9 1.0 re la tiv e po p.  in  m F= 0 1mG 5.8mG 9.8mG 12mG 19mG (c) Figure 4.10: (Color online) Estimated time evolution of rubidium population at j2; 0i hyperne spin state when the quadratic Zeeman coupling or Zeeman eld varies. (a) and (b) are for optical lattices with potential depth V = 10ER and (c) is for V = 6ER. In (a), we assume the density is uniform and there are no spin losses while (b) and (c) are for traps with inhomogeneous density and a nite loss rate 1= = (200ms)1. In (a),(b),(c), magnetic elds which yield dynamical critical quadratic Zeeman coupling qc1( qc2) are 35mG, 35mG and 10mG respectively. 74 4.9. Eects of Spin Exchange Losses Dynamical stabilities of j2; 0i state below the threshold qc1, or instabil- ities of (j2; 2i + j2;2i)=p2 state below threshold qc2 are directly induced by quantum 
uctuations. Coherent quantum dynamics around state j2; 0i ((j2; 2i+ j2;2i)=p2) below and above dynamical critical coupling qc1(qc2) are qualitatively dierent and it is therefore plausible to probe these dynam- ics that are driven mainly by quantum 
uctuations at relative low frequen- cies. To explore this possibility, we have studied strongly damped popula- tion oscillations around a condensate j2; 0i or a uniaxial state, at dierent quadratic Zeeman couplings when the spin exchange loss time is set to be 200ms. For this part of calculations we also take into account the density in- homogeneity of rubidium atoms in traps. Main results are shown in Fig.4.10 for dierent magnetic elds; at a given eld, the corresponding quadratic Zeeman coupling is given as qB = 3(BB=2) 2=(6:8GHz). Our major ndings are three-folded and outlined below. Firstly, for a condensate with small deviations from j2; 0i state, because of fast spin exchange losses complete coherent oscillations around j2; 0i are hard to ob- serve. Secondly, dynamics can be dramatically modied by a nite quadratic Zeeman coupling. When the coupling is much less than qc1 (dened for the density at the center of a trap), dynamics become slower when the quadratic Zeeman coupling increases. Beyond qc1, dynamics become faster while the quadratic Zeeman coupling increases. These behaviors are consistent with results for the case of uniform density shown in the previous section; there, the oscillation frequency becomes zero at a dynamical critical coupling. Thirdly (and perhaps most importantly), the population of atoms at state j2; 0i always grows initially at short tome scales when the quadratic Zeeman coupling is zero or smaller than qc1, as a direct consequence of dy- namical stabilities of a j2; 0i-condensate. By contrast, beyond qc1 the popu- lation of atoms at state j2; 0i always decreases initially at short time scales as a result of dynamical instabilities in this limit. These two distinct short- time behaviors of population at state j2; 0i are signatures of a transition from quantum-
uctuation driven dynamics to a mainly quadratic Zeeman coupling driven dynamics. In the presence of strong spin losses, observing these distinct short-time asymptotics is an eective way to probe many- body quantum 
uctuations in condensates we have considered. At very large quadratic Zeeman coupling, we have found expected rapid oscillations corresponding to mean eld coherent dynamics and we don't show those results here. Finally, we also estimate the time-dependence of population oscillations at nite temperatures using thermal-
uctuation induced poten- tials discussed in a previous section. Thermal 
uctuations usually speed up the coherent dynamics around state j2; 0i when the quadratic Zeeman cou- 75 4.9. Eects of Spin Exchange Losses 0 100 200 300 400 500 time[ms] 0.75 0.80 0.85 0.90 0.95 1.00 re la tiv e po p.  in  m F= 0 T=0 T=0.2TBEC T=0.4TBEC T=0.6TBEC T=0.8TBEC (a) 0 100 200 300 400 500 time[ms] 0.00 0.20 0.40 0.60 0.80 1.00 re la tiv e po p.  in  m F= 0 T=0 T=0.2TBEC T=0.4TBEC T=0.6TBEC T=0.8TBEC (b) Figure 4.11: (Color online) Estimated time evolution of population at j2; 0i state at dierent temperatures. (a) is for a magnetic eld of 13mG that is below the dynamical critical eld corresponding to qc1 and (b) is for 38mG eld which is above qc1. The optical lattice potential depth is V = 8ER, TBEC = 10:9nK and the spin exchange life time is again set to be 200ms. Density inhomogeneity in traps has also been taken into account to obtain these plots. 76 4.10. Estimates of Condensate Fraction pling is suciently weak (smaller than qc1); on the other hand, they can also slow down coherent dynamics around state j2; 2i+ j2;2i when a quadratic Zeeman coupling larger than qc2 is present. These results are shown in Fig.4.11. In obtaining these results, we have extrapolated the linear dynam- ical analysis into a regime where oscillation amplitude is substantial. So strictly speaking, results for 35mG, 44mG, 65mG magnetic elds shown in Fig.4.10(b) are qualitative. However, short-time asymptotics that are the focuses of our discussions here are accurate. 4.10 Estimates of Condensate Fraction In this section, we estimate the condensate in optical lattices both at zero temperature and nite temperatures. An important reason for such an es- timation is to verify the validity of perturbative calculation that we have carried out in the previous section. The perturbative calculation or Bogoli- ubov approximation only works well when depletion fraction is much smaller than one. The major results are summarized in Fig.4.12. In the range when 2 < VER < 10 the quantum depletion is smaller than 1 5 , and our perturba- tive calculation should suce. Another reason to estimate the condensation fraction is to understand how the 
uctuation-induced barrier depends on temperature, which is studied in the previous section. To study the condensate depletion in the optical lattice, we start from the eective Hamiltonian. Five collective modes on top of a nematic state are described by the following: H = X ;q E;q  ~y;q ~;q + 1 2  : (4.31) Here ~y;q and ~;q are bosonic creation and annihilation operators for the -th mode. These operators can be obtained by Bogoliubov transformations: ~;q = u;q;q + v;q y ;q; (4.32) ~y;q = u  ;q y ;q + v  ;q;q: (4.33) The coecients u;q and v;q are given: u;q = 1p 2  s 1 + mBNv2;q E;q + 1 !1=2 ; (4.34) v;q = 1p 2  s 1 + mBNv2;q E;q  1 !1=2 : (4.35) 77 4.10. Estimates of Condensate Fraction 2.0 4.0 6.0 8.0 10.0 V[ER] 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Co nd en sa te  F ra ct io n M=1.0 M=2.0 M=3.0 0 5 10 15 20 25 T[nK] 0.0 0.2 0.4 0.6 0.8 1.0 C. F. Figure 4.12: (Color online) Condensation fraction versus optical lattice po- tential depth V (in units of ER). The inset shows the condensate fraction at dierent temperatures when V is equal to 6ER. Condensate deletions from the -th mode with crystal momentum q are hy;q;qi =  u2;q + v 2 ;q  h~y;q ~;qi+ v2;q: (4.36) Here h~y;q ~;qi is the occupation number of the th mode and it is given by the Bose-Einstein statistics: h~y;q ~;qi = 1 exp (E;q=kT ) 1 : (4.37) Putting all these together, we reach the nal equation to determine the condensate density: M M0(T ) = 1 NT X ;q 6=0  v2;q + u2;q e E;q kT  1 ! : (4.38) M is the atom number density or number of atoms per lattice site and M0 is the condensed number density. The rst term in the summation gives quantum depletion number due to the two body scattering and the second term gives the thermal depletion number. We solve the above equation numerically and the results are shown in Fig.4.1. 78 4.11. Conclusion At zero temperature, quantum depletion is given by the term v2;q which captures the eect of interactions or two body scattering processes. Our estimate shows that 1 M0 M  X =p;;x;y;z 1 M  mBNv tLdL 3  1 M  MaL tL 3=2 : (4.39) We have noticed that major contributions are from the phase 
uctuations since aL  jbLj; jcLj. In optical lattices when the potential depth V in- creases, the bandwith tL decreases exponentially as a function of V and aL increases; as a result, quantum depletion grows as (aLtL ) 3=2. At nite temperatures, M0 decreases as temperature T increases and becomes zero at the transition temperature TBEC . In the weakly interacting limit (tL >> aL; bL; jcLj), the condensate fraction is approximately equal to M0(T ) M0(T = 0) = 1  T TBEC  3 2 ; (4.40) and TBEC  5M2=30 tL in optical lattices. 4.11 Conclusion In conclusion, we have demonstrated that unlike the usual mean-eld-interaction driven coherent dynamics, QFCSD is a novel class of coherent dynamics fully driven by quantum 
uctuations. These dynamics are conveniently tunable in optical lattices where oscillation frequencies can be varied by three to four order of magnitude. Frequencies of oscillations driven by quantum 
uctuations have both distinct lattice-potential dependence and quadratic Zeeman-coupling dependence that can be studied experimentally. For ru- bidium atoms, QFCSD can be directly probed either at frequencies of a few Hz or when the quadratic Zeeman coupling is about 101nK. One of poten- tial applications of QFCSD perhaps is the possibility of performing precise measurements of strongly correlated quantum 
uctuations and critical expo- nents near quantum critical points. Current studies of critical correlations are based on analyzing statistics of interference fringes[162, 74]. Given the great control of coherent dynamics recently demonstrated for cold atoms[10], we would like to believe that quantum-
uctuation controlled dynamics might 79 4.11. Conclusion be an alternative and promising path towards probing critical correlations. Finally, we have also investigated the eects of nite temperatures, spin ex- change losses and studied time evolution of condensates mainly driven by quantum 
uctuations. 80 Chapter 5 Half Quantum Vortex (HQV) in F = 1 Super
uids of Ultracold Atoms 5.1 Introduction Topological excitations such as quantized vortices have been fascinating for quite a few decades and recently have also been well studied in Bose-Einstein condensates (BECs) of ultracold atoms [49, 50, 51, 163, 164, 165, 166, 167, 168]. In spinor condensates the rich magnetic structure can give rise to many exotic topological defects such as quantum vortices with fractionalized circulations, monopoles and skyrmions. The existence of stable topological defects is related to the topology of magnetic order parameters. In a conven- tional super
uid, the complex-valued scalar order parameter has a topology equivalent to a unit circle S1 which supports quantum vortices; in spinor condensates, the vector order parameter can have complicated topologies such as SO(3)U(1)T [125] in the cyclic phase of F = 2 spinor condensates mentioned in Chap.3 or S2S1Z2 [123] in nematic phase of F = 1 spinor sys- tems which is the focus of this chapter. Topological defects other than the integer quantized vortices were previously studied in other physical systems such as super
uid helium-3[169, 170] and liquid crystals[171]. In this chapter we are going to examine half quantum vortices (HQVs) in F = 1 spinor systems, especially the emergent topological order and frac- tionalization phenomena due to strong spin 
uctuations in two-dimension super
uids. Quantum number fractionalization has been one of the most fundamental and exciting concepts studied in modern many-body physics and topological eld theories[172, 173, 174, 175]. In more recent theoret- ical studies of strongly correlated two-dimension electrons [176, 177, 178, 179, 180, 181], spin-charge separated fractionalized excitations in spin liq- uids and the underlying topological order have also been focuses of many investigations although their experimental realization is still under exten- 81 5.2. HQV in Spinor Bose-Einstein Condensates sive debate[182]. Very recently, low dimensional fractionalized quantum states have further been proposed to be promising candidates for carry- ing out fault tolerant quantum computation[183]. Given the availability of low dimensional cold gases[74], in this chapter we propose to explore fractionalization phenomena in super
uids of ultracold sodium atoms which can potentially be studied in experiments[38]. Particularly, we investigate 
uctuation-driven fractionalization of circulation quantum and topological order in two-dimension quantum gases where spin correlations are short ranged. Fractionalized phases in spinor gases characterized by the topological order are previously studied in [128] and [129, 131], where a novel spin singlet condensate with distinct symmetries and excitations has been found. It breaks the U(1) gauge symmetry but not the SO(3) rotational symmetry; spin-charge separation have been also found in the spin single condensate in some one-dimension Mott states[129, 131]. In the rst part of this chapter I describe the structure and interactions of HQVs in the spinor condensates using mean eld theory. In the following part I study the HQVs with strong 
uctuations in 2D spinor super
uids. Finally I study the nucleation of the HQVs in 2D spinor super
uids in rotating traps. 5.2 HQV in Spinor Bose-Einstein Condensates Topological excitations such as quantized vortices have been fascinating for quite a few decades and recently have also been thoroughly studied in Bose- Einstein condensates (BECs) of ultracold atoms [49, 50, 51, 163, 184, 164, 165, 166, 167, 168]. For vortices in single-component BECs, the circulation of supercurrent velocity vs along a closed curve  around a vortex line C = Z  dl  vs (5.1) is quantized in units of 2~m (m is the atomic mass), with C = 1;2; ::: as a consequence of analyticity of single-valued wavefunctions of coherent quantum states. Furthermore, only a vortex with circulation C = 1 or an elementary vortex is energetically stable. A secondary vortex with circula- tion C = 2;3; ::: spontaneously splits into a few elementary ones which interact via long-range repulsive potentials. A conguration with its circulation smaller than the elementary value (C = 1) has to be described by a singular wavefunction and it always turns 82 5.2. HQV in Spinor Bose-Einstein Condensates out to be energetically catastrophic. A most obvious example is a two- dimension conguration, where the condensate phase angle (r; ) rotates slowly and uniformly by 180 in the r   plane around a vortex center but jumps from  to 2 when the polar angle  is equal to 2. The - phase jump here eectively induces a singular cut in the wavefunction. The corresponding circulating velocity eld is simply vs(r; ) = ~ 2mr e (5.2) , leading to C = 1=2 that is one-half of an elementary value. The energy of a cut per unit length along  = 2 line where the phase jumps is nite and therefore the overall energy of a cut in an individual fractionalized vortex scales as L, L is the size of system, while the energy for an integer vortex only scales as a logarithmic function of L. Consequently, a cut that connects two singular fractionalized vortices mediates a linear long rang attractive potential that connes all fractionalized excitations(see Fig.5.1(a)). So in a single-component condensate, vortices with C = 1 are fundamental ones which do not further split into smaller constituent elements as a result of connement of fractionalized vortices. Hyperne-spin degrees of freedom can drastically change the above ar- guments about elementary vortices. In condensates of sodium (23Na) or rubidium (87Rb) atoms in optical traps [38], hyperne spins of cold atoms are correlated because of condensation. A pure spin defect (or a spin discli- nation in the case of 23Na) where spins of cold atoms slowly rotate but no supercurrents 
ow, can carry a cut, i.e. a line along which a -phase jump occurs as a result of Berry's phases induced by spin rotations [149]. Such a spin defect can then terminate a cut emitted from a singular half-quantum vortex (HQV) conguration, which consequently leads to a linear conning potential between the spin defect and HQV(see Fig.5.1(c) and (d)). For instance, a HQV with C = 1=2 conned to a spin defect does exist as a fundamental excitation in spin nematic condensates [123, 185]. In this section we will explore the magnetic structure of a single HQV and interactions between HQVs. I will also present the simulation of dynamical creation of fractionalized HQVs in a rotating BEC of sodium atoms, and brie
y formulate an experimental procedure for the realization of such exotic topological excitations. 5.2.1 Magnetic Properties and Energetics of HQV Unlike conventional integer vortices, HQVs have a rich magnetic structure. We will demonstrate that far away from its core a HQV has vanishing lo- 83 5.2. HQV in Spinor Bose-Einstein Condensates Figure 5.1: Two quantum vortices separated at a distance. (a) A HQV in a single-component BEC. (b) Two IQVs in a single-component BEC. (c) Two HQVs (both are positive types) in a spinor BEC. (d) Two HQVs (one positive and one negative) in a spinor BEC. Red arrows represent the phase vectors, and green arrows represent the spin vectors. 84 5.2. HQV in Spinor Bose-Einstein Condensates cal spin densities but is accompanied by slowly rotating spin quadrupole moments; within the core, spin densities are nonzero. The Hamiltonian for interacting sodium atoms is H = Z dr y(r) ~2r2 2m  (r) + c2 2 Z drŜ2(r) + c0 2 Z dr̂2(r): (5.3) Here  y( ),  = x; y; z are creation (annihilation) operators for sodium atoms in hyperne states ji; they are dened as linear superpositions of creation operators for three spin-one states, j1;mF i, mF = 0;1.  yx = ( y1   y1)= p 2,  yy = ( y1 + y 1)=i p 2 and  yz =  y0. c0;2 are interaction parameters that depend on two-body s-wave scattering lengths a0;2 for to- tal spin 0,2: c0 = 4~2(a0 + 2a2)=3m and c2 = 4~2(a2  a0)=3m. For condensates of 23Na atoms, a0 ' 50 aB and a2 ' 55 aB (aB is the Bohr ra- dius). Ŝ = i
		
 and ̂ =  y  are local spin-density and density operators(see Chap.2). Let's rst nd out the mean eld ground state of the system. Generally, a condensate wavefunction 	 = h yi (5.4) for spin-one atoms is a complex vector. For sodium atoms, interactions favor states with a zero spin density. This leads to a spin nematic ground state which does not break the time reversal symmetry and 	 = exp (i) p  n: (5.5) Here n is a unit vector with three components n, and  is the number density of sodium atoms. This spin nematic state is invariant under an inversion of n and a -phase shift [123] n ! n (5.6)  ! +  (5.7) The spinor condensate specied in Eq.5.5 is a spontaneously symmetry breaking state which breaks both the gauge symmetry and rotational sym- metry. Although local spin densities hŜ(r)i in a nematic state vanish, a nematic condensate carries a spin quadrupole moment dened as Q(r) = hŜŜi  1 3 hŜ2i; (5.8) 85 5.2. HQV in Spinor Bose-Einstein Condensates 0 10 20 30 40 d [ξ] 0 4 8 12 16 in te ra ct io n en er gy  [µ ] Vpm Vpp (b) -40 -20 0 20 40 r [ξ] -1.0 0.0 1.0 ρ + 1− ρ − 1 -20 -10 0 10 20 r [ξ] 0.0 0.2 0.4 0.6 0.8 1.0 de ns ity  ρ −1ρ +1ρ +1−ρ−1 (a) Figure 5.2: (color online). (a) Density (1 = j 1j2) and spin-density (11) proles of an individual plus half-quantum vortex (HQV) centered at r = 0. (b) Interaction potentials (in units of chemical potential ) between two HQVs as a function of separation distance d. Vpm is the potential between a plus HQV and a minus HQV; Vpp is the strong repulsive potential between two plus HQVs. Inset is for spin densities in a pair of plus-minus HQVs separated at d = 30,  is the healing length. which can be calculated and is specied by the nematic unit director n introduced above: Q =   nn  1 3   : (5.9) From the experimental point of view, the nematic spin vector has a special meaning that all the atoms in the condensate are in the m0F = 0 level where m0F is the spin projection along the n axis. After we examining the ground states, let's turn to the HQV. Without loss of generality, we assume that the HQV is centered at the origin and oriented along the z-direction, and nematic directors lying in a perpendicular r   plane rotate by 180 forming a spin-disclination. The corresponding condensate wavefunction far way from the vortex core is 	(; r !1) = exp  i 2 p   cos  2 ; sin  2 ; 0  : (5.10) 86 5.2. HQV in Spinor Bose-Einstein Condensates The 180 rotation of nematic director n() =  cos 2 ; sin  2 ; 0  around the vortex illustrates that a -spin disclination where spin quadrupole moments Q() slowly rotate is indeed conned to this HQV. In the Zeeman basis of j1;mF i states, mF = 0;1, the above vortex state is equivalent to  +1 = exp(i)f(r) r  2 ; (5.11)  1 = g(r) r  2 ; (5.12)  0 = 0: (5.13) When it is far away from the vortex core, f(r !1) = g(r !1) = 1. The core structure can be studied by numerically solving the multiple- component Gross-Pitaevskii (GP) equation  ~ 2 2m r2 + (c0 + c2)1 + (c0  c2)1   1 = 0; (5.14) here mF = j mF j2. The corresponding boundary conditions when r ! 1 are set by the asymptotic behaviors of a HQV far away from the core as discussed before Eq.(5.14). Notice that the last term in Eq.(5.14) indicates that mutual interactions between j1;1i atoms induced by scattering are repulsive since c0  c2 is positive. The numerical solution of the above GP equation is presented in Fig.5.2(a). The results show that within the core, j1; 1i atoms state are completely de- pleted while j1;1i atoms are not. This is because supercurrents are only present in j1; 1i component; in fact the density of j1;1i atoms has an ad- ditional small bump at the center of core, to further take advantage of the depletion of j1; 1i atoms in the same region to minimize the overall repulsive interactions between j1;1i atoms. So the core has a nonzero local spin density hŜz(r)i(= j 1j2  j 1j2) with excess atoms at j1;1i state and we dene it as a minus HQV. Similarly, one can construct a plus HQV with an identical vorticity (i.e. r vs) but with excess j1; 1i atoms in its core (See Fig.5.2(a)). Generally, HQVs have distinct spatial magnetic structures: a HQV core carries excess spins while far away from the core spin quadrupole moments slowly rotate around the vortex. Interactions between two HQVs very much depend on species of HQVs involved. When both are plus ones (or minus) as in Fig.5.1(c), the corre- sponding interaction potential Vpp(mm) is a logarithmic long-range one due to interference between coherent supercurrents. However, the interaction be- tween a plus and a minus HQV Vpm (as in Fig.5.1(d)) is repulsive and short 87 5.2. HQV in Spinor Bose-Einstein Condensates ranged only extending over a scale of vortex cores. In this case, supercur- rents 
ow in dierent components and they don't interfere; the short-range interaction is entirely due to inter-component interactions between j1; 1i and j1;1i atoms. Indeed, the amplitude of potential Vpm is proportional to, when it is small, c0  c2 which characterizes mutual interactions between j1;1i atoms. An integer vortex or a pair of plus-minus HQVs centered at a same point therefore is unstable and further fractionalizes into elementary HQVs. In Fig.5.2, we summarize results of an individual HQV and two HQVs. It is also useful to describe the interaction between two HQVs in the language of nematic spin vectors n and phase  instead of Zeeman basis. The leading term of the interaction energy is mainly from the kinetic part Ekin which can be easily split into two parts when far away from the cores, Ekin ' Z dr ~2 2m jr	j2 (5.15) ' Z dr ~2 2m jrj2 + jrnj2 : (5.16) The rst term is the supercurrent's kinetic energy which also exists in the conventional quantum vortices. The second term is from spin gradient which is unique in spinor condensates. This formulation suggests that we can separate the spin part and the phase part when evaluating energies. The interaction energy from the phase part or the supercurrent has the same form of conventional vortices (see [68]): C1C2~ 2Lz 2m log R d : (5.17) Here two vortices with circulation C1 and C2 are parallel to each other and Lz is the length of the system along the vortex line. And we also assume the separation d is much larger than the healing length  and much smaller than the system's radical size R. The interaction energy from the spin gradient term is much more complicated. However in the simple example shown above in which the spin vector n is always pinned down in a plane, we can use the same formula by introducing the circulation number ~C1 and ~C2 for spin vectors. Therefore the interaction between two HQVs is Vint(d) ' ~ 2Lz 2m log R d  C1C2 + ~C1 ~C2  : (5.18) For example, in Fig.5.1(c), we have C1 = C2 = ~C1 = ~C2 = 12 , leading to logarithmic repulsion between two HQVs. In Fig.5.1(d), we have C1 = C2 = 88 5.2. HQV in Spinor Bose-Einstein Condensates ~C1 =  ~C2 = 12 , leading to an cancelation of the logarithmic part. In this case the interaction between two HQVs are mainly coming from the contact of the core structures, therefore are short-ranged. 5.2.2 HQV Lattice in Rotating Traps To dynamically create HQVs, we numerically solve the time-dependent cou- pled GP equations of spin-one BEC (i 
)~@ 1 @t = [ ~ 2 2m r2 + Vtr    Lz + c0 + c2(1 + 0  1)+W] 1+c2 20  1; (i 
)~@ 0 @t = [ ~ 2 2m r2 + Vtr   Lz + c0 + c2(1 + 1)] 0 + 2c2 1 1  0; (5.19) where = P mF mF is the total condensate density, Vtr(r) is a spin-independent conning potential of an optical trap, and W(r) are pulsed magnetic trap- ping potentials which we further apply in order to create HQVs.  and  are the Lagrange multipliers used to preserve the total number and magne- tization of atoms respectively; 
 is a phenomenological damping parameter which is necessary for studies of quasi-stationary states [166]. We restrict ourselves to a cigar-shaped potential with the aspect ratio  = !?=!z  14 which was also used in early experiments [50]. We consider a two-dimension cylindrical trap which is characterized by two dimension- less parameters C0 = 8N(a0+2a2) 3Lz and C2 = 8N(a2a0) 3Lz , with Lz the size of system along the z-axis and N = 3106 the total number of sodium atoms. When combined with a nonaxisymmetric dipole potential that can be cre- ated using stirring laser beams [50, 51], the optical trapping potential in a rotating frame is given by Vtr(r) = m! 2 ?f(1+)x2+(1)y2g=2. Here !? = 2250 Hz, and anisotropic parameter is set to be  = 0:025. We also include an additional magnetic trapping potential: W(r) = m!2?(x2 + y2)=2, which could be realized in an Ioe-Pritchard trap via a Zeeman splitting mF gFBB with the Lande factor gF=1 for sodium atoms. We start our numerical simulations with an initial state where j1;1i are equally populated. Experimentally, it was demonstrated that when the bias eld is small and the gradient eld along the z-axis of the trap is almost canceled, such a state can be prepared and the j1;1i components are completely miscible (however, with the immiscibility between j1;1i and j1; 0i components) [38]. We then study the time evolution of this initial 89 5.2. HQV in Spinor Bose-Einstein Condensates Figure 5.3: Creation of a half-quantum vortex. Density proles of j +1j2, j 1j2, j +1j2+j 1j2, and spin-density prole j +1j2{j 1j2(bold line) are shown. The rotating frequency is suddenly decreased from an initial value  =0:65!? to =0:3!? at t=800 ms. The bottom panel shows that a single half-quantum vortex is formed at t=1600 ms after the magnetic trapping potential has been adiabatically switched o. state using the Crank-Nicolson implicit scheme [166]. The unit of length is ah= p ~=2m!?= 0:48 m and the period of the trap is !1? = 4 ms; the interaction parameters are C0=500 and C2= 450 and the damping rate is 
=0:03. Further, we include symmetry breaking eects in our simulations by allowing the trap center to randomly jump within a region [; ][; ] (=0:001h, where h is the grid size), which is crucial for vortices to enter the condensate one by one [168] rather than in opposing pairs [166, 167]. However, without additional pulsed magnetic potentials, our results in- dicate that dynamic instabilities for creation of integer vortices in rotating BECs occur almost at same frequencies as for HQVs and a triangular integer- vortex lattice is formed (see Fig.5.4(a)). This integer-vortex lattice is locally stable with respect to the non-magnetic perturbations, by applying an ad- ditional optical trapping potential with an oscillating trapping frequency to eectively shake them, indicating their metastability. For this reason, a time-dependent magnetic trapping potential with harmonic form is applied; and we nd that when a pulsed magnetic eld with  > 0:005 is applied, HQV lattices could be formed. Here we set  = 0:1, which is suitable for both generating a single HQV and demonstrating the dynamical evolution 90 5.2. HQV in Spinor Bose-Einstein Condensates of HQV lattices formation. After the magnetic trapping potentials are on, j1; 1i component further spreads to the edge, while j1;1i component re- mains at the center of the trap and surfaces of equally populated j1;1i components become mismatched with two dierent Thomas-Fermi radii. First, we dynamically create a single half-quantum vortex in condensates, which can be used for the study of dynamics of a HQV. We switch on abruptly a rotating drive with  = 0:65!? and the trap anisotropy  is increased rapidly from zero to its nal value 0.025 in 20 ms. At t = 800 ms, only one vortex in j1;1i component appears. Afterwards, We decrease  to 0:3!? suddenly, and switch o the magnetic trapping potential adiabatically within 200 ms. We then nd a stable single HQV formed at t = 1600 ms as shown in Fig.5.3. The 0:3!? frequency used after t = 1600 ms is within the stable region estimated earlier [186] and our simulations of dynamics are consistent with the energetic analysis. Fractionalized-vortex lattices can be created in a similar setup. The main experimental procedure and results of our simulations for creation of HQV lattices are presented in Fig.5.4. After a rotation with frequency  = 0:7!? starts abruptly and anisotropy  is set to its nal value 0.025, we can see that the cloud is initially elongated and at the same time rotates with the trap. At about 150 ms, surface ripples due to quadrupole excitations occur in j1; 1i component of the condensate, while no surface oscillations appear on the surface of j1;1i component. At t = 240 ms, we nd that the density prole of j1; 1i component is along a short-axis while the j1;1i component is along a long-axis due to repulsive interactions between two components, and the surface of j1;1i component is not always buried in the inner region of j1; 1i. The surfaces of two components oscillate independently and are de- coupled dynamically. At t = 430 ms, we nd that two plus HQVs with excess j1; 1i atoms inside cores have nucleated at the center. Correspondingly, we observe two small regions near the center where j1;1i atoms are completely depleted and the density of j1; 1i atoms remains high. At t = 800 ms, two components are phase separated, but the structure of HQV cores remains almost unchanged. Finally, we switch o the additional magnetic potential adiabatically within 200 ms and a HQV lattice with interlaced square con- guration becomes visible. In this structure, to minimize strong repulsive interactions Vpp(mm) between plus or minus HQVs, the vorticity is evenly distributed among plus and minus HQVs, or between j1;1i components. This also indicates that spatially each plus HQV prefers to be adjacent to minus HQVs and vice versa to avoid stronger interactions Vpp; Vmm and to take advantage of relative weaker interactions Vpm (See Fig.5.2). To further minimize repulsive interactions Vpm between nearest neighboring vortices, 91 5.2. HQV in Spinor Bose-Einstein Condensates Figure 5.4: (a) Creation of a triangular integer-vortex lattice in a rotating optical trap at t = 1600 ms. (b) Creation of half-quantum vortex lattices when an additional pulsed magnetic trapping potential is applied. Here time evolution of various condensate densities is shown. The optical trap rotates at  = 0:7!? with a magnetic trapping potential on until t = 800 ms; afterwards, the magnetic trap is adiabatically switched o within 200 ms. The bottom panel shows the half-quantum vortex lattice formation at t = 1600 ms; a square lattice in the spin-density prole is clearly visible. 92 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms a plus HQV is displaced away from adjacent minus HQVs by a maximal distance. Generally because of the asymmetry between Vpp and Vpm, a bi- partite vortex lattice should be favored over frustrated geometries such as triangular lattices where a plus HQV could be adjacent to another plus HQV resulting in much stronger repulsion. In our simulations of 23Na atoms in rotating traps with c0 ' 30c2, square vortex lattices have always been found. Equilibrium energetics of rectangular or square lattices were also considered in the quantum Hall regime [187, 188], in two-component BECs coupled by an external driving eld where vortex molecules are formed [189], and also observed in condensates of pseudo-spin-1=2 rubidium atoms [190]. Here we have mainly focused on dynamical creation of HQV lattices conned to a spin-density-wave structure at relatively low frequencies; this structure can be conveniently probed by taking absorption images of ballistically expand- ing cold atoms in a Stern-Gerlach eld [38]. 5.3 HQV in 2D Super
uid of Ultracold F = 1 Atoms In this section we will examine the 
uctuation-driven vortex fractionaliza- tion and topological order in 2D nematic super
uids of cold sodium atoms. The Monte Carlo simulations suggest that a softened -spin disclination be conned to a half-quantum vortex when spin correlations are short ranged. Both this result and direct calculations of winding-number operators in- dicate that a non-local topological spin order emerges simultaneously as cold atoms become a super
uid below a critical temperature. We have also obtained 
uctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps and discussed probing these excitations in experiments. In the following we will rst introduce the system and Hamiltonian, and get an insight from a gauge eld formulation, then we will present the results from numerical simulations and nally the nucleation of HQVs in rotations traps. 5.3.1 Eective Hamiltonian We employ the Hamiltonian introduced previously for F=1 sodium atoms in optical lattices[129, 128], H = X k bL 2 ̂2k + cL 2 Ŝ2k 93 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms  tL X hkli ( yk; l; + h:c:) X k ̂k: (5.20) Here k is the lattice site index and hkli are the nearest neighbor sites.  is the chemical potential and tL is the one-particle hopping amplitude. Two coupling constants are bL(cL) = b(c) 4~2 m R dr(w(r)w(r))2; b, c are eective s-wave scattering lengths, w is the Wannier function for atoms in a period- ical potential. Operators  y,  = x; y; z create hyperne spin-one atoms in 1p 2 (j1i j1i), i 1p 2 (j1i+ j  1i) and j0i states respectively. The spin and number operators are dened as Ŝ = i
 y 
 , and ̂ =  y . Spin correlations are mainly induced by interaction cLŜ2. Minimization of this spin-dependent interaction requires that the order parameter 	(= h yk;i) be a real vector up to a global phase, i.e. 	 = p Nn exp(i) where n is a unit director on a two-sphere, exp(i) represents a phase director and N is the number of atoms per site. All low energy degrees of freedom are characterized by congurations where n and  vary slowly in space and time[128]. Low lying collective modes include spin-wave excitations with energy dispersion !(q) = vsq, vs = p cLtLa and phase-wave excitations with !(q) = vpq, vp = p bLtLa (here a is the lattice constant). In one-dimensions, low energy quantum 
uctuations destroy spin order leading to quantum spin disordered super
uids[123]. In two-dimensions, the amplitude of quantum spin 
uctuations is of order of cL=tL and is negligible in shallow lattices as tL is order-of-magnitude bigger than cL. At nite temperatures, spin correla- tions are mainly driven by long wave length thermal 
uctuations, analogous to quantum 1D cases. This aspect was also paid attention to previously and normal-super
uid transitions were investigated[191]. We therefore study the following Hamiltonian which eectively captures long wave length thermal 
uctuations H = J X hkli (nk  nl) (k l) ; (5.21) here states at each site are specied by two unit directors: a nematic di- rector, n = (cos0 sin 0; sin0 sin 0; cos 0) and a phase director,  = (cos; sin). J = 2NtL is the eective coupling between two neighbor- ing sites and depends on N , the number of atoms per site. The model is invariant under the following local Ising gauge transformation: ni ! sini, ! sii, and si = 1. In the following, we present results of our simula- tions on 2D super
uids, especially spin structures, energetics of HQVs and nucleation of HQVs in rotating optical traps using the eective Hamiltonian in Eq.5.21. 94 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms 5.3.2 HQV and Topological Order - A Gauge Field Description In this section we are aiming at providing an intuitive and qualitative picture of the emergent degrees of freedom. And this can be done by introducing a Z2 lattice gauge eld model in which the softened spin disclination is repre- sented by an excitation in the Z2 gauge elds. The Z2 lattice gauge model has also been introduced for spin singlet condensates in low dimensional optical lattices [128, 129], where the Z2 gauge eld is to represent that an even or odd number of spin singlets are formed between two neighbor sites. The quantitative analysis is carried out in the next section via a full monte carlo simulation of Eq.5.21 which supports the intuitive picture obtained in this section. We start from the following classical Hamiltonian introduced in the last section: H = J X hiji (i j)(ni  nj) (5.22) Here i and j are phase vectors and their inner product is given by cos (i  j). ni is the nematic spin vector at site i. The summation runs over all the nearest neighbor sites hiji. J is a positive coupling constant which measures the strength of the phase and nematic vector twisting en- ergy. Let's rst look at the continuum limit of the system where only 
uctua- tions of long wavelengths are concerned. In this case we can treat n and as a smoothed function of space coordinates: i j ' 1 a 2 2 (i j) a2 ' 1 a 2 2  @ @x    @ @x  (5.23) ni  nj ' 1 a 2 2 (ni  nj) a2 ' 1 a 2 2  @n @x    @n @x  (5.24) The eective Hamiltonian in the continuum limit is (up to a constant) H ' Z d2x  J 2 jrnj2 + J 2 jrj2  : (5.25) The above equation shows at large lengths scales, the spin vector and phase vectors are decoupled. It also shows that the spin vectors can be described by the O(3) non-linear sigma model and the phase vectors are described by the O(2) non-linear sigma model or XY model. According to renormalization 95 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms group method, in two-dimension space there is a quasi-long order in O(2) model and no long-range order in the O(3) model(see appendix). In other words, spin wave excitations open up an energy gap and they will not appear in our low energy eective theory. Although the nonlinear sigma model captures the long-wavelength physics of spin and phase 
uctuations, HQVs are missing in the continuum model in which spin and phase are decoupled. To obtain a low energy eective Hamiltonian which allows for HQV structures, we can introduce a gauge eld by using the Hubbard-Stratonovich transformation[192]. We introduce a real-valued auxiliary gauge eld ij dened on bonds hiji in the lattice. The eective Hamiltonian with the gauge eld can be dened as [128]: H() = J 2 X hiji 2ij + (i j)2 + (ni  nj)2 + J X hiji ij (i j) + ij (ni  nj) (5.26) It can be shown that the gauge eld Hamiltonian listed above is equivalent to the original one. In fact, an integration of the  eld in the new Hamiltonian will recover the original Hamiltonian as eH  Z D[]eH(): (5.27) The integration on the right hand side is performed straightforwardly by noticing that it can be decoupled to independent Gaussian integrals. We can dene a local gauge transformation associated with the Z2 sym- metry in the nematic representation. The Hamiltonian is invariant under the following transformation: ni ! rini (5.28) i ! rii; (5.29) ij ! rirjij (5.30) Here ri = 1. An eective Hamiltonian ~H includes phase vectors and gauge elds can be obtained by integrating the spin degrees of freedom. Since an exact analytical expression is impossible in this case, let's estimate this integral using the molecule eld approximation. Namely the correlation between ni and nj can be replaced by a vector coupled to a eective eld which is the average of all the other nematic vectors in the system. We assume that 96 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms hnii = xez, where 1  x  1. The correlation term can be approximated as: ni  nj = hnii  nj + hnji  ni  hnii  hnji: (5.31) For simplicity let's rst choose the gauge eld to be ij = . The Hamilto- nian of the nematic spin elds is Hn = X i 2Jx2n2i;z  4Jxni;z  Jx4  2Jx2 (5.32) = X i ni;z  4Jx 2Jx2ni;z  : : : (5.33) Here (4Jx 2Jx2ni;z) plays a role of an eective eld coupling to nematic spin vector. The value of x can be determined from the self consistent equation x = hni;ziT = R dni;zni;z exp  J  4x 2x2ni;z  ni;z R dni;z exp [J (4x 2x2ni;z)ni;z] (5.34) Numerical solutions of the above self consistent equation are shown in Fig.5.5. There are two dierent phases in this system, and a phase tran- sition occurs when we tune the parameter to across the boundary line. A ferromagnetic phase with nz 6= 0 appears at low temperatures or for large  values where the thermal 
uctuations are too small to destroy the spin orders and the eective molecule eld is large enough to hold the spin order; a paramagnetic phase appears in the opposite limit: high temperatures and low  eld. The phase transition line separated two phases is exactly given by the following: Jc = 3 4 : (5.35) After the integration of spin degrees of freedom the free energy of the system now is F = X <ij> J 2 (i j  )2 + Fn(J; ) (5.36) Fn = X i T ln Z dni;ze Hn  (5.37) 97 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 J 0 0.12 0.24 0.36 0.48 0.60 0.72 0.84 0.96 0 1 2 3 4 5 6 σ -9 -8 -7 -6 -5 -4 -3 -2 fre e en er gy βJ=0.1 βJ=0.2 βJ=0.3 βJ=0.4 βJ=0.5 βJ=0.6 Figure 5.5: (a) Solution of the self consistent equation at dierent temper- atures J and for gauge eld of dierent strength . (b) Free energy as a function of  at dierent temperatures. 98 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms Here Fn is the free energy due to the 
uctuations of the spin vectors. Given J and , one rst solve Eq.5.34 to get values of x, then Hn and Fn can be evaluated by the above equations, thus Fn is function of J and  only. The overall free energy F as a function of J and  can be obtained by minimizing with F with respect to i j rst with a given . The resultant free energy as a function of  at dierent temperatures is plotted in Fig. 5.5. The free energy's dependence on  is very dierent at low temperatures and high temperatures. The average value of the sigma eld hi determined from the local minimum of free energy can be nonzero at low temperatures. This result suggests that there is a nonzero coupling between the gauge eld and phase vectors at low temperatures. In another words, it suggest a nonzero ij(i  j) coupling term other the usual (i  j)2 phase correlation term in the low energy eective Hamiltonian ~H. In the above heuristic argument we consider the uniform gauge eld ij = hi which we believe to describe be the low energy states. One may expect that a gauge eld with ij =  is close to uniform gauge eld in their free energy, since a gauge transformation can change the sign of ij . In this case, the sign of the gauge eld sij = ij hi appear to be relevant low energy degrees of freedom. One can obtain the eective Hamiltonian in terms of sij and i as [128] ~H =  X hiji h ~J2 (i j)2 + ~Jsij (i j) i + ~K X  Y ij2 sij : (5.38) Here ~J , ~J2 and ~K are renormalized coupling constants. The last term is the energy of pure gauge elds dened as the product of sij on four edges of a plaquette. In the language of the gauge eld model, a HQV can be viewed as a combination of a gauge defect and a  phase winding. In Fig.5.6 we plot the phase vectors and the gauge elds around a single HQV. A gauge defect located at the center is put in the center by setting ij = 1 along the line from the vortex core to the system's boundary. The phase vectors on one side of the cut are anti-parallel to the ones on the other side, such a conguration won't raise too much energy. In fact, the energy cost of adding a HQV in this system is approximately  ~J2d 2 log R 2 +  4 ~Jd2 log R  + ~K: (5.39) Here R is the system's size, d is the lattice constant, and  is the core size of the vortex. The ~K term is coming from the gauge defect located at the 99 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms Figure 5.6: A HQV in the Z2 lattice gauge model. Phase vectors i are plotted as red arrows. Gauge elds with ij = 1 are marked in green lines and ij = 1 are not marked. center and is originating from the spin 
uctuations in the original picture. Note that in the condensate case, energy cost of a HQV due to the spin vectors is also logarithmic function of the system size (Eq.5.16). The nite contribution of spin 
uctuations in the 2D super
uids is consistent with short-ranged spin correlations. The presence of gauge defects in a system can be characterized by the Wilson-loop variable WC : WC = Y hiji2C sij : (5.40) Here C is a closed loop add hiji are edges on the loop. For a given gauge eld, WC = 1 if there is even numbers of gauge defects within the closed loop and WC = 1 otherwise. The expectation value of the Wilson-loop variable is used to characterize the connement or deconnement of the gauge elds, namely according to weather the variable increases with with the circumference of the loop (known as \perimeter law") , or with the area enclosed within the closed loop (known as \area law") . In the gauge eld connement phase, an individual gauge defect can not move freely in the system. The gauge defect is bounded to either another gauge defect or a phase defect. In the latter case, such a combination leads to a HQV. Previous studies on the eective gauge model Eq.5.38 [193] shows that there is indeed a gauge connement phase when the temperature is not too high. The above heuristic arguments illustrated the central idea in this sec- 100 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms tion: a HQV can be stabilized by a gauge defect in two-dimension spinor super
uid. Considering the gauge defect, a quantity dened globally (such as quantum circulation), is actually coming from short-ranged spin correla- tions, it implies there exists a nontrivial global order or topological order in spin correlations. In the next section, we will carry out a monte carlo sim- ulation to quantitatively study HQV and the associated topological order. 5.3.3 HQV and Topological Order - Monte Carlo Simulations An early analysis suggests that around a HQV, both phase and nematic directors rotate slowly by 1800[123]; in nematic coordinates (; ), a HQV in condensates is represented by 	(; ) = p N() exp(i=2)n(), with n = (cos(=2); sin(=2); 0). The question here is whether, when nematic directors are not ordered, a spin disclination is still present in a HQV. To fully take into account 2D thermal 
uctuations, we carry out Monte Carlo simulations on a square lattice of 128 128 sites and study spatial correlations between a HQV and a -spin disclination, and topological order. We rst identify critical temperatures of the normal-super
uid phase transition by calculating correlations and the phase rigidity. The gauge- invariant quadrupole-quadrupole correlation functions we have studied are f s;p(r1; r2) = hQs;p(r1)Qs;p(r2)i: (5.41) Here Qs(r1) = n1;n1;  (1=3) ,  = x; y; z; Qp(r1) = 1;1;  (1=2),  = x; y. In simulations, we have studied these correlation func- tions and found that the phase correlation length for fp(r1; r2) becomes divergent at a temperature 0:35J which is identied as a critical temper- ature Tc. We also calculate the phase rigidity or the renormalized phase coupling Jp Jp = @2F @2 ; (5.42) here  is a small phase dierence applied across the opposite boundaries of the lattice and F is the corresponding free energy. We indeed nd that it approaches zero at Tc while at T = 0 Jp takes a bare value J . Meanwhile, the spin correlation function f s(r1; r2) remains to be short ranged across Tc. By extrapolating our data to lower temperatures, we nd that the spin correlation length diverges only at T = 0 (see Fig.5.7a). Our simulations for 101 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms 0.35 0.40 0.45 T/J 100 101 102 103 (a) 0 64 128 192 256 loop size 0.01 0.1 1 (b) 0 64 128 192 256 loop size -1 -0.5 0 0.5 1 (c) 0 64 128 192 256 loop size -1 -0.5 0 0.5 1 (d) 0 64 128 192 256 loop size -1 0 1 (e) 0 64 128 192 256 loop size -1 -0.5 0 0.5 1 (f) 0 0.2 0.4 0.6 T/J 0 0.5 1 Figure 5.7: a)Spin(circle) and phase(square) correlation lengths versus tem- peratures; inset is for the renormalized phase coupling constant Jp(in units of J). Dashed lines are the ts to exp(A= p T  Tc) for phase correlations with Tc  0:35J and exp(B=T ) for spin correlations. b) Thermal average hWsi(circle) and hWpi(square) versus the loop perimeter at T = 0:33J(solid line) and T = 0:45J(dashed line). c)-f): hWs;pihv and hCihv(diamond) aver- aged over congurations with half-vortex boundary conditions. c)-d),f) are for T=J = 0:45,0:33,0:20 respectively; in e), we also show hWs;pihv normal- ized in terms of back ground values hWs;pibg at T = 0:33J . The background values hWs;pibg(dashed lines) are obtained by averaging over congurations with a uniform phase distribution at the boundary. 102 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms correlation lengths are consistent with the continuum limit of the model in Eq.5.22 which is equivalent to an XY model and an O(3) nonlinear-sigma model. In order to keep track of the winding of nematic directors in a wildly 
uc- tuating back ground, we introduce the following gauge invariant -rotation checking operator, which is essentially a product of sign-checking operators Ws = Y hkli2C sign(nk  nl); Wp = Y hkli2C sign(k l): (5.43) Here the product is carried out along a closed square-shape path C centered at the origin of a 2D lattice. Ws;p can be either +1 or 1; and Ws(p) is minus one when C encloses a -spin disclination (HQV). The gauge invariant circulation of supercurrent velocity (in units of ~=m) is dened as C = 1  X 2C sign(nk  nl) sin(k  l): (5.44) This quantity is equal to one in a HQV. In our simulations, we investigate the winding number hWs;pihv averaged over congurations where phase directors rotate by 1800 around the bound- ary of the lattice and the center plaquette. At temperatures above the normal-super
uid transition temperature Tc, both winding numbers Ws;p and circulation C are averaged to zero within our numerical accuracy(see Fig.5.7). And our choice of boundary conditions does not lead to a vortex or disclination conguration in the absence of phase rigidity. Below Tc, the circulation C is averaged to one indicating that the boundary conditions eectively project out HQV congurations. Meanwhile, we observe loop- perimeter dependent hWs;pihv which can be attributed to the background 
uctuations of HQV or disclination pairs. The loop-perimeter dependence of hWs;pihv here is almost identical to that for uniform boundary condi- tions, i.e. the back ground value. After normalizing hWs;pihv in terms of background winding numbers hWs;pibg, we nd both hWsihvhWsibg and hWpihv hWpibg ap- proach 1 (see Fig.5.7). We thus demonstrate that a softened disclination is spatially correlated with a HQV. At the temperatures we carry out these simulations the spin correlation length is suciently short compared to the size of the lattice. At further lower temperatures, the spin correlation length becomes longer than the lattice size and 
uctuations of pairs of disclination- 103 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms anti-disclination are strongly suppressed; hWs;pihv are equal to 1 for almost all loops, which corresponds to a mean eld result. Results in Fig.5.7 indicate that a HQV is conned to a softened spin disclination and vice versa to form a fundamental excitation. Thus, - disclinations like HQVs have logarithmically divergent energies and are fully suppressed in ground states. Our results also illustrate that although the average local spin quadrupole moments Qs vanish because of strong 
uc- tuations, an overall -rotation of nematic directors in disclinations is still conserved because of a coupling to the super
uid component. This coupling between a HQV and disclination can also be attributed[137] to a coupling between Higgs matter and discrete gauge elds[194]. Furthermore, the ab- sence of unbound -disclinations in super
uids indicates a topological or- der, similar to the one introduced previously for an isotropic phase of liquid crystal[193]. Consequently, once a conventional phase order appears below a critical temperature, a topological spin order simultaneously emerges while spin correlations remain short ranged. The emergent topological order can be further veried by examining the average of product-operator Ws;p over all congurations (with open bound- aries). Above the normal-super
uid transition temperature Tc, we again nd that Ws;p both are averaged to zero within our numerical accuracy im- plying proliferation of unbound HQVs or disclinations. Below Tc, we study the loop-perimeter dependence of average winding numbers hWs;pi and nd that both lnhWpi and lnhWsi are linear functions of loop-perimeter anal- ogous to the Wilson-loop-product of deconning gauge elds[195]; if there were unbound disclinations, one should expect that lnhWpi is proportional to, instead of the loop-perimeter, the loop-area which represents the number of unbound disclinations enclosed by the loop. 5.3.4 Critical Frequency for HQV Nucleation Let us now turn to the nucleation of those excitations in rotating traps[50]. To understand the critical frequency for nucleation, we study the free energy of a vortex, in a rotating frame, as a function of the distance r from the axis of a cylindrical optical trap (the axis is along the z-direction), Fh:v:(r) = F 0 h:v:(r) Lz(r): (5.45) Here F 0h:v:(r) is the free energy of a HQV located at distance r from the trap axis in the absence of rotation, Lz(r) is the angular momentum of the vortex state and  is the rotating frequency. In a 2D lattice without a trapping 104 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms 35 40 45 -100 0 100 -64 -32 0 32 64 r -2000 -1000 0 1000 2000 F h .v .[n K] 0 0.08ωtr 0.23ωtr 0.32ωtr (a) 0.1 0.2 0.3 0.4 T[J] 0 20 40 60 80 Ω cD  [H z] HQV IQV (b) 0.2 0.3 0.4 T[J] 0.2 0.3 0.4 0.5 R at io Figure 5.8: a) The free energy of a HQV at a distance r from the trap center at rotation frequencies 0, 0:08, 0:23 and 0:32 (in units of trap frequency !tr = 120Hz) at T = 0:33J . Details near the edge are shown in the inset. At the center, the exchange coupling is J = 154nk. b) The critical frequency  Dc (solid line) varies from 0:32!tr (or 38Hz) to 0:17!tr (or 20Hz) as the temperature T increases. As a reference, we also show Dc versus T for integer quantum vortices (dashed line). Inset is the temperature-dependence of the ratio between the HQV energy F 0h:v: and the IQV energy F 0 v . potential, the energy of a HQV is F 0h:v: = (Jp=4+Js=4) ln(L=a), with con- tributions from phase winding and spin twisting; here Jp;s are renormalized phase and spin coupling respectively and L is the size of system. For an integer-quantum vortex (IQV), the energy is F 0v = Jp ln(L=a). The ratio between F 0h:v: and F 0 v depends on the ratio Js=Jp or spin 
uctuations; in the limit L approaches innity, the ratio F 0h:v:=F 0 v changes discontinuously from 1=2 at T = 0 where Jp  Js = J to 1=4 at nite low temperatures in 2D where Js vanishes. In simulations of a nite trap (see below), because of a nite size eect we nd that this ratio varies from 1=2 to 1=4 smoothly as temperatures increase from 0 to Tc. To study nucleation of half-quantum vortices in an optical trap, we as- 105 5.3. HQV in 2D Super
uid of Ultracold F = 1 Atoms sume a nearly harmonic trapping potential V (r) = 1=2m!2trr 2, with !tr being the trap frequency. The average number of particles per site N(r) has a Thomas-Fermi prole; N(r) = N0(1  r2=R2TF ), here N0 is the number density at the center and RTF is the Thomas-Fermi radius. Furthermore, the optical lattice potential along the axial direction is suciently deep so that atoms are conned in a two-dimension xy plane; the in-plane lattice po- tential depth is set to be 5ER(ER is the photon recoil energy) and tL = 77nk, bL = 187nk and cL = 10nk. For N0 = 1:2 and trap frequency !tr = 120Hz, we nd that RTF = 43a and Lh:o: = 4a where Lh:o:(= 1= p 2m!tr) is the har- monic oscillator length. The coupling Jkl depends on the distance from the center of trap and at the center, the coupling is about 154nk. In non-rotating or slowly rotating traps, the free energy maximum is located at the center and there should be no vortices in the trap. As frequencies are increased, a local energy minimum appears at the center and becomes degenerate with the no-vortex state at a thermodynamic critical frequency (which is about 0:08!tr at T = 0:33J); however because of a large energy barrier separating the two degenerate states as shown in Fig.5.8, vortices are still prohibited from entering the trap. Further speeding up rotations results in an energetically lower and spa- tially narrower barrier. Within the range of temperatures studied, thermal activation turns out to be insignicant within an experimental time scale ( 100ms) because of low attempt frequencies. So only when the spa- tial width of barrier becomes comparable to a hydrodynamic breakdown length[196], the barrier can no longer be felted and vortices start to pen- etrate into the trap. The hydrodynamic breakdown length LB is about (L4h:o:=2RTF ) 1=3, which in our case turns out to be about 2a (a is the lattice constant). We use this criterion to numerically determine the dynamical critical frequency for vortex nucleation Dc ; for IQVs, the estimated  D c is a 
at function of T (see Fig.5.8b) which is qualitatively consistent with earlier estimates[197, 198]. For HQVs, however, as F 0h:v: depends on the amplitude of spin 
uctuations, Dc varies from about 0:32!tr at T = 0 where Js  Jp due to a nite size eect, to about 0:17!tr at temperatures close to Tc where Js  0. Note when Js approaches zero as in the thermodynamic limit,  Dc ( 0:17!tr) for HQVs is about one-half of the critical frequency for IQVs (about 0:37!tr for the trap studied here). The interaction between two HQVs with the same vorticity at a sep- aration distance d contains two parts. One, Vcc(> 0) is from interactions between two supercurrent velocity elds which is logarithmic as a function of d; and the other, Vss is from interaction between two spin twisting elds accompanying HQVs. For a disclination-anti disclination pair, in the dilute 106 5.4. Conclusion limit one nds that Vcc  Vss resulting in a cancelation of long-range inter- actions. The resultant short-range repulsions lead to square vortex lattices found in numerical simulations[199]. For 
uctuation-driven fractionalized vortices, Vss is almost zero and the overall interactions are always logarith- mically repulsive. HQVs nucleated in a rotating trap should therefore form a usual triangular vortex lattice. Individual vortex lines can be probed either by studying a precession of eigenaxes of surface quadrupole mode in rotating super
uids[200, 201, 202]. In the later approach, one studies the angular momentum carried per particle in a HQV state. When a HQV is nucleated in the trap, super
uids are no longer irrotational and the angular momentum per particle is h=2 rather than h per particle for an integer vortex state. When a surface quadrupole oscillation across a rotating super
uid is excited, larger axes of quadrupole oscillation start to precess just as in the case of integer vortices. However, the precession rate is only one half of the value for an integer vortex state which can be studied in experiments. 5.4 Conclusion In conclusion, 2D super
uids of sodium atoms can have a non-local topo- logical spin order associated with half quantum vortices. In rotating traps, 
uctuation-driven fractionalized vortices can nucleate at a critical frequency which is about half of that for integer vortices. In the future, it is interest- ing to explore 2D-3D dimension transition that can be realized in optical lattices. When a super
uid of cold atoms in a 3D lattice is gradually sep- arated into independent layers of 2D super
uids, possible phase transitions accompanied by the emergence of the topological order may occur. It is also expected that across this phase transition there is interesting vortex lat- tice transition connecting square lattices in 3D condensates and triangular lattices in 2D super
uids. 107 Chapter 6 Conclusion In conclusion, we have studied 
uctuation-driven phenomena in spinor su- per
uids of ultracold systems. We have shown that 
uctuations in spinor systems of ultracold atoms studied in the thesis have completely changed the picture suggested by mean eld analysis and have given rise to several novel 
uctuation-driven phenomena. In hyperne F = 2 nematic condensates, quantum 
uctuations can change the mean eld prediction in magnetic correlations and spin dynamics dramatically. It is shown that zero-point quantum 
uctuations completely lift the accidental continuous degeneracy that is found in mean eld analy- sis of quantum spin nematic phases of hyperne spin-two cold atoms. Two distinct spin nematic states with higher symmetries are selected by quan- tum 
uctuations. In ultracold experiments, it is possible to probe quantum 
uctuations directly via studying coherent spin dynamics of ultracold atoms in optical lattices. We have carried out a detailed study of 
uctuation- driven spin dynamics. Unlike the mean eld coherent dynamics, quantum 
uctuation-controlled spin dynamics are very sensitive to the variation of quantum 
uctuations. They have peculiar dependence of Zeeman elds and potential depths in optical lattices. In particular this novel type of spin coherent dynamics can be potentially explored under accessible experimen- tal conditions. Our results not only point out the fundamental limitations of previous measurement based on mean eld coherent dynamics, but also provide a new type of tool to investigate 
uctuation-driven phenomena. In the future, it would be interesting to use 
uctuation-driven dynamics to investigate correlated 
uctuations and various universal scaling properties near quantum critical points. In another spinor system, 2D nematic super
uids of cold sodium atoms, spin structures of 
uctuation-driven fractionalized vortices and topological spin order are studied. Monte Carlo simulations suggest a softened -spin disclination structure in a half quantum vortex when spin correlations are short ranged; in addition, calculations indicate that a unique non-local topo- logical spin order emerges simultaneously as cold atoms become a super
uid below a critical temperature. We have also proposed an experimental study 108 Chapter 6. Conclusion of half quantum vortices in rotating optical traps where the critical frequen- cies for vortex nucleation are 
uctuation dependent. In the future, it would be interesting to explore 2D-3D dimension transition that can be realized in optical lattices. Interesting phase transitions should occur when a super
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For a cubic lattice in 3D, the following periodical potential is used in the calculation of the band structure. V (r) = V0  sin2  2  x  + sin2  2  y  + sin2  2  z  (A.1) Three counter-propagating laser beams are slightly mismatched in their fre- quencies to avoid interference between laser elds of dierent directions.  is the wavelength of the laser, and V0 is the depth of optical lattice which is proportional to the intensity of laser elds. V0 is often measured in units of the recoil energy associated with the laser beams as ER = ~2 2m2 . Here for simplicity we assume V0 is the same for all three directions. In experiments they can be varied independently to generate other geometries such as quasi 2D to support the system studied in Chap.5. A single atom loaded in an optical lattice can be described by the Schrodinger equation: ~ 2r2 2m + V (r)   (r) = E (r): (A.2) The solution to the above problem has the form of Bloch waves and can be labeled by a crystal momentum k and a band index n.  (r) = e ikrun;k(r) (A.3) E = En(k) (A.4) Here un;k(r) is a periodic function with a period =2 and is normalized asZ u:c: dr ju2n;k(r)j = 1: (A.5) 124 Appendix A. Band Structure in Optical Lattices Here the integration is performed within the unit cell(u.c.). The Wannier function forming a convenient basis in deep optical lattices can be constructed from Bloch waves as wn;R(r) = Z k2BZ dk eik(rR)un;k(r): (A.6) It can be easily shown that Wannier functions labeled by the localization center R and the band index n forms an orthogonal basis,Z u:c: dr wn;R(r)wn0;R0(r) = R;R0n;n0 : (A.7) In the second quantization form one can also introduce creation (anni- hilation) operators  yn;R( n;R) for Wannier functions  y(r) = X nR wn;R(r) y n;R  yn;R = Z drwn;R(r) y(r) [ n;R; y n0;R0 ] = n;n0R;R0 (A.8) One can rewrite the following many-body Hamiltonian of interacting bosons by using the expansion given in Eq.A.8: H = Z dr   y(r)  ~ 2r2 2m  + V (r)   (r) + 2~2as m  y(r) y(r) (r) (r)  ' tL X hRR0i   yR0 R + h:c:   L X R  yR R + aL 2 X R  yR y R R R: (A.9) When only the lowest band is concerned we can obtain the above single band Hubbard model in which only the leading terms are kept. Other terms such as next-nearest neighbor hopping and interaction from atoms occupying dierent sites turns out to be much smaller than the leading terms in deep optical lattices. Such a single band model serves as starting point of many calculations in this thesis. Two important parameters of this model tL and aL are given 125 Appendix A. Band Structure in Optical Lattices by tL = Z u:c: dr wn=0;R(r)wn=0;R0(r) = Z dk En=0(k)e ik(RR0) aL = 4as m Z u:c: dr jw4n=0;R(r)j: (A.10) Using the methods outlined above, one can determine tL and aL as a function of potential depth V in optical lattice, as shown in Fig.4.1. Notice that in Eq.A.6 there is undetermined k-dependent phase factor in the Bloch waves un;k(r). The Wannier function actually depends on this choice. In our calculation, we choose the so-called maximum localized Wan- nier function which can reduce overlapping terms such as R drjw2n;Rw2n;R0 j that may appear in the lattice model. 126 Appendix B The -function of the Nonlinear Sigma Model We derive the  function of O(3) nonlinear sigma model in this section. Let's consider a unit vector j~sj = 1 which has three components (s1; s2; s3). H = J 2 Z ddx (r~s(x))2 (B.1) It is convenient to rescale the Hamiltonian by introducing a coupling con- stant g = 1J , H = 1 2g Z ddx (r~s(x))2 (B.2) The coupling constant has a dimension of (length)d2, and one can also introduce a dimensionless constant u as g = uad20 ; (B.3) here a is the smallest length scale in the system and it is inverse proportional to the momentum cuto . Let's consider small 
uctuations around the mean eld ground state s = (1; 0; 0) in terms of two small parameters s3 and : s1 = q 1 s23 cos() (B.4) s2 = q 1 s23 sin() (B.5) (B.6) The energy density H can be written in terms of  and s3 as: H(x) = 1 2g  (rs3)2 + (1 s23)(r)2 + (s3rs3)2 1 s23  (B.7) 127 Appendix B. The -function of the Nonlinear Sigma Model Notice the 
uctuations of the s3 eld has a magnitude of  pg. Hence we can rescale the s3 eld as p g = s3; (B.8) and expand the Hamiltonian according to the orders of g H(x) ' 1 2 (r )2 + 1 2g (r)2  1 2  2(r)2 (B.9) + g 2 ( r )2 + g 2 2  2( r )2 +O(g3) (B.10) Let's now perform the integration over the \fast" elds or elds with momentum within the shell b < jpj < . To obtain the leading order contributions in terms of g, we can assume that the  eld is smooth and we only need to integrate over the  eld within the shell:Z b<jpj< D eH[ ;] = (B.11)Z b<jpj< D (p) e 1 2 R ddx h (r )2+ 1 g (r)2 2(r)2+O(g) i (B.12) After the Fourier transformation of  eld, we can perform a Gaussian integral and then rewrite the result in the exponential form.Z b<jpj< D (p) e 1 2 R ddp (2)d [p2(r)2]j (p)j2 (B.13) ' Y b<jpj<  2 p2  (r)2  1 2 (B.14) ' exp " 1 2 Z b<jpj< ddp (2)d log 2 p2  (r)2 # (B.15) ' exp " 1 2 Z b<jpj< ddp (2)d  log 2 p2 + (r)2 p2 !# (B.16) The coarse grained Hamiltonian H0 up to the rst order of g is H0(x) = 1 2 Z b<jpj< ddp (2)d log 2 p2 + 1 2 (r )2  1 2  2(r)2 + 1 2 " 1 g  Z b<jpj< ddp (2)d 1 p2 # (r)2 +O(g) (B.17) 128 Appendix B. The -function of the Nonlinear Sigma Model When we rescale the Hamiltonian, the new cuto 0 is reduced and the new length scale a0 is increased(b < 1), indicating the system is dilated. 0 = b; a00 = a0 b (B.18) The coupling term g0 in the coarse grained HamiltonianH 0 can be written as: 1 u0a0d20 = 1 uad20  Z b<jpj< ddp (2)d 1 p2 (B.19) By dierentiating the above equation with the length scale a0, one can obtain the renormalization group function or -function for the coupling constant u. (u) = a0 du da0 = (2 d)u+ u 2 2 +O(u3) (B.20) The  function of u is very dierent for d  2 and d > 2. In the former cases, the coupling constant goes to innite as one increases the length scale indicating the spins are always short-ranged (T > 0). On the other hand, in d > 2 cases, the system has two dierent phases separated by the xed point uc((uc) = 0). At d = 2, one can also estimate the spin correlation length  by solving the equation of the beta function:  = a0 exp  2 u  = a0 exp  2J T  ; (B.21) here a0 and J are the lattice constant and the coupling constant of the lattice model. 129

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