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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 fluctuation-driven phenomena in ultracold spinor Bose gases. In Bose-Einstein condensates of hyperfine spintwo (F=2) atoms, it is shown that zero-point quantum fluctuations completely lift the accidental continuous degeneracy in quantum spin nematic phases predicted by mean field analysis, and these fluctuations select out two distinct spin nematic states with higher symmetries. It is further shown that fluctuations can drive a novel type of coherent spin dynamics which is very sensitive to the variation of quantum fluctuations controlled by magnetic fields or potential depths in optical lattices. These results have indicated fundamental limitations of precision measurements based on mean field theories. In addition, fluctuation-driven coherent spin dynamics studied here is a promising tool to probe correlated fluctuations in many body systems. In another system – a two-dimension superfluid of spin-one (F=1) 23 Na atoms – we have investigated spin correlations associated with half quantum vortices. It is shown that when cold atoms become superfluid below a critical temperature a unique nonlocal topological order emerges simultaneously due to fluctuations in low dimensional systems. Our simulation have indicated that there exists a nonlocal softened π-spin disclination structure associated with a half-quantum vortex although spin correlations are short ranged. We have also estimated fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps. These results indicate that the strongly fluctuating ultracold spinor system is a promising candidate for studying topological orders that are the focus of many other fields.  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 Fluctuations in a Harmonic Oscillator and Beyond 1.2 Fluctuation-Driven Phenomena . . . . . . . . . . 1.3 Bose-Einstein Condensates of Ultracold Atoms . . 1.4 Ultracold Spinor Condensates . . . . . . . . . . . 1.5 Organization of the Thesis . . . . . . . . . . . . .  . . . . . . . . . . .  2 Interatomic Interactions and Low-Energy Scattering 2.1 Interatomic Interactions . . . . . . . . . . . . . . . . . 2.2 Low Energy Scattering . . . . . . . . . . . . . . . . . 2.3 Interactions with Hyperfine Spin Degrees of Freedom 2.4 Vector and Tensor Algebra and Representations . . . 2.4.1 F=1 . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 F=2 . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Linear and Quadratic Zeeman Shift . . . . . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  1 3 6 9 13 20  . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  21 21 24 25 27 28 29 32  3 Fluctuation-Induced Uniaxial and Biaxial Nematic States in F = 2 Spinor Condensates . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mean Field Theory of F =2 Condensates . . . . . . . . . . . 3.2.1 Ferromagnetic States . . . . . . . . . . . . . . . . . .  35 35 36 38 iii  Table of Contents  3.3 3.4 3.5  3.2.2 Cyclic States . . . . . . . . . . . . 3.2.3 Nematic States . . . . . . . . . . . Zero-Point Quantum Fluctuations . . . . Uniaxial and Biaxial Spin Nematic States Discussion and Conclusion . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  4 Quantum Fluctuation Controlled Coherent Spin Dynamics in F=2 87 Rb Condensates . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Microscopic Hamiltonian for F = 2 87 Rb in Optical Lattices 4.3 Range of Interaction Parameters for the Lattice Hamiltonian 4.4 Mean Field Nematic States and Coherent Dynamics . . . . 4.5 Effective Hamiltonian for QFCSD . . . . . . . . . . . . . . . 4.5.1 Nematic Manifold . . . . . . . . . . . . . . . . . . . . 4.5.2 Collective Coordinates and Conjugate Momentum . . 4.5.3 Spectra of Collective Modes . . . . . . . . . . . . . . 4.5.4 Effective Hamiltonian . . . . . . . . . . . . . . . . . . 4.6 Potentials Induced by Quantum Fluctuations . . . . . . . . 4.7 Oscillations Induced by Quantum Fluctuations . . . . . . . 4.8 Effects of Thermal Fluctuations . . . . . . . . . . . . . . . . 4.9 Effects of Spin Exchange Losses . . . . . . . . . . . . . . . . 4.10 Estimates of Condensate Fraction . . . . . . . . . . . . . . . 4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .  38 40 42 45 46  48 48 50 51 54 55 55 57 58 59 61 64 70 72 77 79  5 Half Quantum Vortex (HQV) in F = 1 Superfluids of Ultracold 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 Superfluid of Ultracold F = 1 Atoms . . . . . . . 93 5.3.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . 93 5.3.2 HQV and Topological Order - A Gauge Field Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.3 HQV and Topological Order - Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 β-function of the Nonlinear Sigma Model  . . . . . . . 127  v  List of Tables 2.1  Scattering lengths (in atomic units) of various alkali atoms. .  3.1  Magnetic orderings in mean field ground states of F = 2 spinor condensates. . . . . . . . . . . . . . . . . . . . . . . . . Scattering lengths (in atomic units) of various alkali isotopes with hyperfine spin F = 2. . . . . . . . . . . . . . . . . . . . . Collective modes of nematic states specified by ξ. . . . . . . .  3.2 3.3  26  37 38 43  vi  List of Figures 2.1 2.2  3.1 3.2  4.1 4.2 4.3  Interaction energy of two 85 Rb atoms as a function of internuclear distance r. . . . . . . . . . . . . . . . . . . . . . . . . Hyperfine energy levels of a 87 Rb atom in its electronic ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavefunctions of spin nematics and phase diagram spinor condensates . . . . . . . . . . . . . . . . . . The zero-point energy of spin nematics per lattice function of ξ . . . . . . . . . . . . . . . . . . . . .  of F = 2 . . . . . . site as a . . . . . .  Coupling constants in optical lattices. . . . . . . . . . . . . . An artistic view of the five-dimension manifold of spin nematics. Barrier height Bqf (in units of pK) as a function of optical potential depth V . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effective potential as a function of ξ for various experimental parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fluctuation-driven coherent spin dynamics when the quadratic Zeeman coupling is absent. . . . . . . . . . . . . . . . . . . . 4.6 Fluctuation-driven coherent spin dynamics around various spin nematic states at the presence of the quadratic Zeeman coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Fluctuation-driven coherent spin dynamics around biaxial spin nematic states at the presence of the quadratic Zeeman coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Stability of fluctuation-driven coherent spin dynamics . . . . 4.9 Induced free energy barrier height as a function of temperature for different lattice potentials. . . . . . . . . . . . . . . . 4.10 Estimated time evolution of rubidium population at |2, 0⟩ hyperfine spin state when the quadratic Zeeman coupling or Zeeman field varies. . . . . . . . . . . . . . . . . . . . . . . . 4.11 Estimated time evolution of population at |2, 0⟩ state at different temperatures. . . . . . . . . . . . . . . . . . . . . . . .  23 33  39 44 52 56 62 63 64  67  68 69 73  74 76 vii  List of Figures 4.12 Condensation fraction versus optical lattice potential depth V (in units of ER ). . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8  78  Two quantum vortices separated at a distance of d . . . . . . 84 Magnetic structure of an individual half quantum vortex and interaction energy between two half quantum vortices . . . . 86 Creation of a half quantum vortex in rotating traps . . . . . . 90 Dynamical creation of half quantum vortex lattice in rotating traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Numerical solution to the self-consistent equation Eq.5.34 . . 98 A half quantum vortex in the Z2 lattice gauge field model. . . 100 Correlations and loop integrals in 2D nematic superfluids . . 102 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 field 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 suggestions of improving the thesis and my oral presentation. I owe particular 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 Semenoff, 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 Beijing 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 fluctuates. According to the laws of modern physics, this is indeed the case since fluctuation is believed 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 fluctuation 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 fluctuations 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 consequence of the Heisenberg uncertainty principle of quantum mechanics[1]. Not only particles fluctuate as their positions and velocities are not definite, the vacuum which assumed to be empty of matter and energy also fluctuates. Physicists picture the vacuum as a sea with fluctuating foam of virtual particles which are repeatedly created and annihilated spontaneously. And according to modern cosmology, even our whole universe is fluctuating. What causes the fluctuation, especially the mysterious quantum fluctuation associated with the “zero-point motion”? This is a deep and controversial question in fundamental physics. Some physicists insist that the fluctuation is a result of an incompleteness or a drawback of the quantum theory(for example [2]). They believe that a more fundamental theory beyond the quantum theory should be a deterministic one in which predictions of physical quantities should be definite, as Einstein once put it “God does not play dice”[3]. Others, perhaps more and more, believe that this fluctuation is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of various quantum fields. Regardless of the controversies about the origin of fluctuations, the cur1  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 fluid. This random movement of the suspended particle is driven by thermal fluctuations of the constituent molecules in the fluid and can be explained by the kinetic theory of thermal equilibrium. The importance of the Brownian motion is many-fold: it indirectly confirms the atomic nature of matter that atoms or molecules do exist; it reveals a deep and general relation between thermal fluctuations and dissipation, as the Brownian motion is related to the viscosity of the fluid; 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 fluctuations of electromagnetic fields between two reflecting bodies placed in vacuum. The Casimir force is the physical manifestation 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 applications 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 fluctuations of gravitons. The concept of the Casimir effect can be also applied to emergent degrees of freedom in condensed 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 fluctuation-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 observed in a BEC, such as superfluidity and macroscopic coherent dynamics, because the majority of the constituent atoms in a BEC occupy the same microscopic state. Atoms that are not in that particular microscopic state give rise to fluctuations in density correlations, which is related to energy and momentum dissipation, low dimensional phases, and superfluid-insulator phase transitions, among which many have been observed and studied in ultracold atoms. However, spin fluctuations are less studied and not well understood in Bose-Einstein condensates, These fluctuations 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 different 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 number of physical systems such as superfluid helium-3. The spinor Bose gas of ultracold atoms created in recent experiments provides us new opportunities to study spin correlated matter and especially the spin fluctuations. Spin fluctuations have several interesting and unique effects which can not be studied in the single-component BEC. These effects are the main topics of this thesis, such as fluctuation-driven spin correlations, macroscopic coherent dynamics and fractionalization of topological defects. They are also connected to a number of fluctuation driven phenomena found in other physical systems. The studies of fluctuations in spinor ultracold atoms here can improve our general understanding of fluctuations and may shed light on open problems related to fluctuations. In the following sections of this chapter, I give a more specific introduction to the topics mentioned above. In Sec.1.1, I go over the fluctuations in a harmonic oscillator, which is an important and useful conceptual tool to illustrate the zero-point motion; then I examine the fluctuations of lattice vibrations based on the picture of harmonic oscillators. In Sec.1.2, I list a number of fluctuation-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 effective mass that subjects to a force in a stable equilibrium acts as a harmonic oscillator in small vibrations. The energy E of a harmonic oscillator is given by p2 mω 2 x2 + . (1.1) 2m 2 Here p is the momentum of the oscillator, x is the position of the oscillator, m is the mass or the effective mass and ω is the angular oscillation frequency. E=  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 definite position and velocity without any fluctuations. However, the quantum theory suggests a different picture. According the Heinsenberg’s uncertainty principle, the oscillator can not have definite values for both position and momentum at the same time. If one tries to pin the oscillator to the center exactly, its momentum will fluctuate wildly; if one tries to suppress the kinetic energy by keeping p = 0, then the position will fluctuate 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 zeropoint energy which is first proposed by Einstein and Stern in 1913[6]. The zero-point energy of a quantum harmonic oscillator is E=  1 ω. 2  (1.2)  And the fluctuations in the oscillator’s position and momentum are given by √ δx = √ δp =  2mω mω . 2  (1.3) (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 effective harmonic oscillator or a set of uncoupled effective harmonic oscillators. 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=  ∑ P2 mω 2 ∑ R + |UR − UR′ |2 . 2m 2 ′ R  (1.6)  ⟨R,R ⟩  Here we assume a quadratic form of the elastic potential which is a reasonable 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 neighbor sites. In this picture, the solid is not different from a set of coupled harmonic oscillators. One can find 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 = PR =  1 ∑ UQ e−iQ·R NT Q 1 ∑ √ PQ eiQ·R . NT Q √  (1.7) (1.8)  Here NT is the total number of the lattice sites. One can rewrite the energy in the following form [ ] ∑ PQ2 1 2 2 + mΩQ UQ . E= (1.9) 2m 2 Q  Here ΩQ is a function of ω, Q. In this picture, we have a set of uncoupled harmonic oscillators with different frequencies. The quantized version of each oscillator is called a phonon with momentum Q and energy ΩQ . The zero-point of motion of phonons has an important effect: the translational symmetry cannot be spontaneously broken in a one-dimension infinite chain described by Eq.1.6. If the translational symmetry was broken and the crystalline order existed, ΩQ would vanish linearly when |Q| → 0. The fluctuation of UR would become divergence due to the fluctuations at long wavelengths: ⟨UR2 ⟩ = =  1 ∑ 1 ∑ ⟨|UQ |2 ⟩ = NT NT 2mΩQ Q Q ∫ ∫ Λ 1 dQ ∼ . dQ 2πa 2mΩQ 4πam 0 Q  (1.10)  (1.11)  In the second line of the above equation, the summation is replaced by an integral in the continuum limit where NT → ∞. Here a is the lattice spacing. The integral includes fluctuations of wave vectors from zero to the Debye cutoff Λ. 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 modern physics. The origin of fluctuation is deeply rooted in the fundamental laws of modern physics. It can be found in quantum mechanics, in particular 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 fluctuations. In the strict sense, everything is more or less related to fluctuations according the first principles in quantum mechanics. Still, there are some phenomena which are explicitly related to fluctuations, or even driven by fluctuations. Some of the fluctuation-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 difference between the two energy levels 2S 2 1/2 and P1/2 in the hydrogen atom. These two levels are supposed to have the same energy according to the Dirac theory. The experimental discovery of the Lamb shift by Lamb and Retherford in 1947[7] provided a stimulus for the renormalization theory of quantum electrodynamics(QED) in which zero-point fluctuations of electromagnetic fields must be taken into account. In the picture of QED, the physical vacuum is not empty anymore. It is filled with photons, the quantized objects of electromagnetic waves, in the zero-point motion. The Lamb shift can be interpreted as the influence of virtual photons that have been emitted and absorbed by the atom (see, for example, the textbook by Peskin and Schroeder[8]). • Casimir effect The Casimir force[5] is one of the best known examples of fluctuationdriven phenomena. Consider two uncharged metallic plates in a vacuum separated by a few micrometers. According to the classical theory of fields there should be no forces between two plates since there are simply no sources of electric fields. When zero-point motion or quantum fluctuation 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 fluctuations or zero-point motion depends on the specific arrangement of the two plates and thus produces an effective potential for the two plates. The Casimir effect is very general and finds applications in various fields of physics. Accurate measurements were made recently by using nano-scale metallic and semiconductor devices of different geometries[9]. In particular, ultracold atoms turned out to be natural candidates for measuring Casimir-Polder effects 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 fluctuations of electromagnetic fields, or more specifically radiative corrections in this case, may produce spontaneous symmetry breaking and may lead to a mass generation of mesons. In the terminology of quantum field 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 first-order phase transition. The Coleman-Weinberg mechanism is very general and finds applications in high energy physics as well as condensed matter physics. • Order by disorder in quantum magnetism Fluctuation effects in frustrated magnetism have attracted a lot of interest in recent years. The frustration, either of geometrical origin or as a result of competition between different 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 nextnearest 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 field approximation the system’s energy is independent of relative orientations of the two sublattices. In the quantum version, one must consider the fluctuations (spin waves 7  1.2. Fluctuation-Driven Phenomena in this case) around different classical ground states. It is shown that quantum fluctuations favour a collinear alignment of the sublattices. Examples of ground state selection via quantum fluctuation have been demonstrated in a number of frustrated magnetic systems such as frustrated Heisenberg model, body-centered tetragonal model, and Kagom´e lattices (see [15] for a review). • The Mermin–Wagner–Hohenberg theorem and fluctuations in low dimensional systems The Mermi–Wagner–Hohenberg theorem [16, 17] (also known as Mermin– Wagner theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Strong fluctuations in low dimensional systems will destroy those ordered states that could exist in the higher dimensional counterparts. For example, gapless fluctuations of scalar fields lead to a logarithmic divergent correlation function. Excitations of gapless long-range fluctuations are always favored because they can increase the entropy at very low energy cost. The entropy of a gapless mode with momentum k scales as k d−1 log k and it always dominates the energy cost (scaling as |k|) for any finite temperatures in d ≤ 2 systems. This theorem points out the importance of fluctuations in low dimensional systems and finds applications in many systems. In the context of spinor ultracold atoms, there is no true long-range two-dimension condensate at finite temperatures; and the fluctuations can destroy the long-range coherence of spin correlations in 2d, leading to short-range spin correlations. • The Fluctuation-dissipation theorem The fluctuation-dissipation theorem is a simple, yet quite deep, concept 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 principle of the fluctuation-dissipation theorem is that a non-equilibrium state may have been reached either as a result of a random fluctuation 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 fluids. It was realized that random forces or fluctuations that resulted in thermal motions of molecules or atoms composing the fluid 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 fluid. Einstein pointed out the underlying link with his famous Einstein-relation[18] that the diffusion constant D is related to temperature T and particle’s mobility µp as D = µp kB T . The fluctuationdissipation theorem enables transport coefficients to be calculated in terms of response to external fields.  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 primary concerns in ultracold atomic gases. Breakthrough technologies in cooling, trapping and manipulating methods have enabled us to explore quantum 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 field 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 sufficiently cold temperature is a crucial prerequisite and a great technical challenge for the study of quantum many-body effects in dilute 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 ∼ 1013 cm3 is of order 100nK, which is seven orders of magnitude smaller than that of the superfluid helium, and ten orders of magnitude smaller than the electronic Fermi energy in metals. Such a low temperature scale is needed to realize BEC, superfluidity, 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. Efforts of cooling atomic gases in pursuit of atomic BEC began in the late 1970s, when people first 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 1µK), which is the kinetic energy an atom acquires through the spontaneous 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 finally 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 1 H[23], 4 He[24] in the lowest triplet state, 7 Li[20], 23 Na[21], 41 K[25], 52 Cr[26], 84 Sr[27], 85 Rb[28, 29], 87 Rb[19], and 133 Cs[30]. Degenerate fermionic gas as well as Bose-Fermi mixtures were created in 40 K[31], 40 K-87 Rb[32], 6 Li-7 Li[33, 34], 6 Li-23 Na[35]. In particular, all-optical trapping allows us to create quantum degenerate gas with internal degrees of freedom arising from hyperfine spins: hyperfine spin F = 1[36] and F = 2[37] 23 Na, F = 1[38] and F = 2[39] 23 Rb, F = 3 52 Cr[26], and a pseudo-spin one-half (two hyperfine levels are involved) systems such as |F = 1, m = 2⟩ and |F = 1, m = −1⟩ in 87 Rb BEC[40], and |F = 21 , m = ± 12 ⟩ in 6 Li 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 developed 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 experiments, 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, population and relative phase of atoms in different 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 condensation. Major achievements of these studies have been, among others, the hydrodynamic nature of the collective oscillations [44, 45, 46], Josephsonlike effects [47, 48], the realization of quantized vortices [49, 50, 51], the observation of interference of matter waves [52], the study of coherence phe10  1.3. Bose-Einstein Condensates of Ultracold Atoms nomena in atom laser configurations [53, 54, 55, 56], the observation of fourwave 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 field theory in which one assumes the existence of a macroscopic wavefunction Ψ(r) ( also known as order parameter) and a nonlinear Schrodinger-like equation that governs the macroscopic wavefunction. This mean field 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 fluid made up of Cooper pairs of electrons. Here the macroscopic wavefunction Ψ(r) is directly related to the superfluid of bosonic atoms. The superfluid density ρs and the super-current’s velocity vs are related to Ψ(r) = |Ψ(r)|eiΘ(r) as ρs = |Ψ(r)|2  (1.12)  ∇Θ  (1.13)  vs =  m  In weakly-interacting Bose gases at zero temperature, the order parameter 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  2 ∂Ψ =− ∇2 Ψ + g|Ψ|2 Ψ + V (r)Ψ ∂t 2m  (1.14)  This nonlinear Schrodinger-like equation includes the effects of interaction in its nonlinear term, as well as the consequences of trapping potentials V (r). The mean field 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 profiles, collective oscillations, structure of vortices, etc. (see review papers and books[66, 67, 68, 69]). The realization of superfluid-insulator phase transition in Bose gas confined in optical lattices[70, 71] opens up opportunities to go beyond the mean field picture and to reach the strongly coupling limit in quantum gases. Experimental 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 effective confining frequency of order 10kHz which is 10 − 100 times larger than the typical frequency in atomic traps. The strong confinement not only enhances the interaction energy via increasing the overlap of two atoms occupying the same well, but also reduces the effective hopping of atoms between different wells. These two effects can both increase the fluctuations in Bose-Einstein condensates, leading a quantum phase transition. In a seminal paper Jaksch et al. [70] proposed to realize a quantum phase transition from superfluid phase to Mott-insulator phase by loading cold bosonic atoms in optical lattices and increasing the optical potential depth. The experimental observation of superfluid-insulator transition first reported by Bloch’s group [71] marked the beginning of exploring strongly correlated quantum gases in optical lattices. Several other groups have observed superfluid-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 configuration (in the xy plane) can be obtained by applying a strong confinement in the z direction. In this configuration the spatial degree freedom in the z direction is actually “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 first 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 superfluid was also observed in experiments, in particular the vortex-anti vortex confinement-proliferation transition are examined [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 interparticle 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 fields via Zeeman energy of different hyperfine levels. Near these magnetic field tuned resonances, the values of two-body scattering 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, Feshbach resonance was first observed in Bose-Einstein condensates in 1997 by Cornish et al.[79]. On one side of resonance where scattering lengths are negative, 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 recombination, quite remarkable progress[82, 83] has been made recently to explore the physics of Bose gases at large scattering lengths where quantum fluctuations can no longer be accounted for by perturbations around mean field theories. On the few-body level, evidence of fascinating three-body Efimov states and universal four-body states near resonance are observed in recent compelling experiments[84, 85, 86]. In these strongly-coupling quantum systems, mean field theories don’t suffice, sometimes even break down, due to the strong fluctuations arising from strong optical confinement, resonate interactions and other methods.  1.4  Ultracold Spinor Condensates  In early experiments, Bose-Einstein condensates were usually confined in magnetic traps. The confining potential in magnetic traps is produced by Zeeman energy of hyperfine spins. One limitation of magnetic trapping is that only “low field seekers” - atoms with particular spins that favor low magnetic field - can be trapped. During inelastic scattering processes, “low field seekers” can flip into “high field seekers” resulting in a decrease in the number of atoms confined in the trap (trap loss). When atoms with different hyperfine spins are simultaneously confined, 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 confining potential is produced by couplings between laser lights and atoms’ electrical dipoles. The spinor Bose-Einstein condensates were first created in hyperfine F = 1 23 Na atoms [38, 36] in optical traps. In these experiments, spinor condensates of F = 1 23 Na 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 final step of prepa13  1.4. Ultracold Spinor Condensates ration, desired hyperfine levels were then populated by transferring atoms in mF = −1 via Rabi oscillations. Linear superposition of hyperfine levels such as √12 (|mF = 1⟩ + |mF = −1⟩) was prepared by tuning frequencies and durations of the driving fields in Rabi oscillations. The evolution and lifetime of the spinor condensate were measured by the standard time-of-flight method, in which hyper levels were separated by applying a Stern-Gerlach magnetic gradient field. It turns out the spinor condensate in this case had a relatively long lifetime of 1−10 seconds, which made itself a perfect experimental candidate for studying rich physics predicted in spinor condensates. Spinor condensates were later created in other systems, including hyperfine F = 2 23 Na [37], hyperfine F = 1 and F = 2 87 Rb[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 fields [40, 89, 90]. The successful creation of stable spinor condensates have led to fruitful experimental studies covering different topics: magnetic orderings and associated 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 different hyperfine levels have stationary phase differences. 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† † N sate of N atoms:|Ψs ⟩ ∼ (ψ+1 + ψ−1 ) |0⟩ and a fragmented condensate as  † † ) 2 |0⟩. Although these two states have the same aver|Ψf ⟩ ∼ (ψ+1 ) 2 (ψ−1 age atom number in each hyperfine level, their physical properties are very different. Ψs depends on the relative phase between two components such that by shifting ψ+1 to eiθ ψ+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 N  N  14  1.4. Ultracold Spinor Condensates ground states depend on details of interatomic interactions. Rich phase diagrams of a variety of magnetic ordered states are mapped out for hyperfine spin F = 1, F = 2[107, 108], and arbitrary F spinor condensates[109] in principle. When magnetic fields 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 fields [38, 99, 93, 100, 94]. On the two-body level, spin-dependent interaction is responsible for spinchanging scattering processes. Two atoms can change their hyperfine spins during elastic scattering processes. For example, two |mF = 0⟩ atoms can become |mF = +1⟩ and |mF = −1⟩ 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 different internal degrees of freedom is the distinction between spinor systems 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 coherent spin dynamics, which is a class of macroscopic quantum phenomena 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 timereversible 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 23 Na atoms with hyperfine spin F = 1 [100, 58] and F = 2 [58], and of 87 Rb atoms with hyperfine 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 different spin levels. The oscillation of the pendulum is driven by an external force, and spin dynamics is driven by interaction energy differences of spinor condensates. The initial potential energy can convert into the kinetic energy of a pendulum, corresponding to the kinetic spin energy with an effective 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 sufficient as fluctuations 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 superfluids, the circulation of supercurrent along any closed loop must be integer times of a basic quanta, i.e. vs · dl = C 2π m , C = 0, ±1, ±2, · · · (m is the mass of the atom). This quantization rule is related to the phase coherence throughout the condensate or single-valuedness of the wavefunction, requiring that the phase winding along any closed loop must be multiples of 2π. Fractionalized quantum vortices with circulation quanta |C| < 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 π spin dislocation and C = 1/2 phase winding is an energetically stable topological defect in F = 1 nematic spinor condensates. The phase discontinuity of π brought by phase winding C = 1/2 is annihilated by superimposing a spin rotation which generates a Berry phase π. 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 possible 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 parameter 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 2) , where S1 is a manifold of nematic condensate is more complicated: (S1Z×S 2 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 representation 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 field theory. Ground state properties, 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 multicomponent order parameter is introduced to account for internal degrees of  16  1.4. Ultracold Spinor Condensates freedom. The validity of mean field theory relies on the assumption that fluctuations around mean field solutions are small and have little effects on physical properties. This is indeed the case in many experiments of dilute spinor Bose gas, in which fluctuations only provide a small correction to the mean field result. However, there are certain cases when fluctuations are important and the mean field picture will not suffice. 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 different from the mean field ground states. In F = 1 spinor gas with anti-ferromagnetic interactions, the mean field ground state [107, 108] is a nematic state. They are rotational symmetry broken states and are hugely degenerate. The broken symmetry is specified by a nematic vector n such that all the atoms are in the hyperfine mF = 0 levels if n is the projection axis. On the other hand, the quantum rotor model[113] predicted a spinsinglet ground state which is rotationally invariant(and non-degenerate. ) The state vector in the Zeeman basis is given by 12 , √12 , 12 . Furthermore, the quantum rotor model predicts spin dynamics that would mix nematic states with different n. This dynamics is absent in the mean field theory. It turns out that results obtained by the quantum motor model can be reproduced by including quantum fluctuations in the mean field nematic states. The broken rotational symmetry is restored by quantum fluctuations at wavelengths of the condensate size through the nematic-mixing dynamics. In fact, the spin singlet state can be viewed as a state with fluctuating nematic vectors: ∫ Ψs = dn |n⟩ . (1.15) More importantly, the effects of fluctuations at wavelengths of the condensate size will eventually vanish in the thermodynamic limit N → ∞, thus making mean field description exact. For example (pointed out in [123]), the nematic-mixing dynamics takes place at a time scale proportional to N , and the energy difference between spin singlet state and nematic state is proportional to N1 of the total energy. 17  1.4. Ultracold Spinor Condensates The degeneracy in ground states predicted in the mean field theory is also exact in the thermodynamic limit , in a sense that there are numerous low-lying states with excitation energies of ∼ N1 in a condensate of N atoms. In the thermodynamic limit, mean field 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 fluctuations in nematic states are strong in these systems. The size-dependent nematic-mixing dynamics is a good example to reveal the difference and connection between few-body physics and many-body physics. It also clearly demonstrates that spontaneous symmetry breaking is a many-body effect that only happens in the thermodynamic limit. On the theoretical side, full numerical simulations [113, 118, 116, 120, 114] showed that fluctuations lead to slow dynamics in mean field ground states as well as damping or de-phasing in mean field spin dynamics. These results are also obtained in [119, 115] by introducing an effective quantum Hamiltonian in which the spin population and condensate phase forms a canonical conjugate pair. Peculiar behaviors in long-time spin dynamics due to fluctuations 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 fluctuation-driven dynamics is a challenging task, because these dynamics such as damped oscillations 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 dissipative dynamics[94]. It is suggested to make observations in relatively small condensates loaded in high-finesse 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] directly measured the atom number fluctuation or “shot-noise” in spin projections, and revealed the dissipative nature of the spin dynamics. 2. Spinor gas in optical lattices Quantum fluctuations in single-component Bose-Einstein condensates can be greatly enhanced in optical lattices, leading to a superfluidinsulator quantum phase transition [71] in which the phase coherence 18  1.4. Ultracold Spinor Condensates is completely destroyed by quantum fluctuations. 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 possible quantum phase transitions between spin-ordered Mott insulating and spin-disordered Mott insulating states are predicted and studied in F = 1 cold atoms [128, 129, 130, 131, 132, 133], and F = 2 atoms[134, 135]. On the superfluid side, coherent spin dynamics predicted in the mean field theory is greatly modified by enhanced quantum fluctuations, 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 fluctuations in the optical lattice. Optical lattices can also be used to study low dimensional ultracold gases where fluctuations are strong and differ from the high dimensional cases qualitatively. For example, Mott insulating phases and exotic spin correlations are studied in F = 1 antiferromagnetic spinor gas confined in low dimensional geometries[128, 129, 131, 130]. In Chap.5 and [137], half quantum vortices in two-dimension topologically ordered superfluids are examined in details. On the experimental side, disk and wave-guide confinement 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 field theory is lifted by quantum fluctuations and thermal fluctuations, 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 fluctuations around the mean field “vacuum” lead to observable effects. Fluctuation can also drive a coherent spin dynamics which is otherwise absent in the mean field theory. This novel type of dynamics can be controlled in optical lattice and is highly sensitive to fluctuations. 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 briefly discuss some of the atomic properties of alkaline atoms with an emphasis on spindependent low-energy scattering. In particular, vector or tensor basis are introduced. The following chapters are divided into two parts, for two kinds of fluctuation-driven phenomena in spinor systems. The first 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 — fluctuation-driven spin nematic states in spinor condensates. After examining the mean field phase diagram of spin correlated phases in the first section, the collective excitations of spin nematic states are examined in detail. Effective potential of different 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 fluctuations in the same system. An effective Hamiltonian including the leading order of fluctuations is obtained for spin dynamics in the first several sections of this chapter. Using this Hamiltonian, coherent spin dynamics under various experimental conditions are examined in details in the following section. Effects of finite 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 superfluid. In the first several sections of this chapter I present the spatial and magnetic structures of half quantum vortices and dynamical creation of vortex lattice in spinor condensates where the order parameter is well defined. In the following sections, I turn to the half quantum vortex in the strongly fluctuating limit — 2D spinor superfluids. A heuristic argument using gauge field formulation is carried out to illustrate the basic ideas which are later confirmed in Monte Carlo simulations. Nucleation of half quantum vortices in rotating traps is discussed 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 briefly discuss some of the atomic properties of alkaline atoms with an emphasis on spin-dependent lowenergy scattering processes. In Sec.2.1, I describe the interatomic interactions 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 describe the effects of hyperfine spin degrees of freedom on scattering. Atoms may change their hyperfine spins during the elastic scattering processes due to the exchange effects of electronic spins. The elastic spin-changing scattering is exactly the microscopic origin of spin fluctuations coherent spin dynamics in spinor condensates. In the low-energy scattering theory, spinchanging 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 effect of magnetic fields on spinor gases.  2.1  Interatomic Interactions  Fig.2.1 is the interatomic interaction energy of two 85 Rb 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 (≃ −C6 r−6 ), 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 defines another length scale lvdW , at which the strength of interaction energy Cr66 is equal to the kinetic 2 energy 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 definite hyperfine spins, the electronic spin state is a linear combination of spin singlet and spin triplet, leading to a hyperfine spin dependent interaction. For example in Fig.2.1, when two atoms are far apart, there is an energy difference between atoms with different 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 interaction 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 ∫ 1 4π 2 ad S1 · S2 − 3(S1 · ˆr12 )(S2 · ˆr12 ) Ud = , (2.1) dr1 dr2 3 2 m r12 where the dipole length ad is introduced to represent the strength of dipoledipole interaction [138] as ad =  µ0 gF2 µ2B m . 12π 2  (2.2)  The scattering length which characterizes the strength of the interaction for Rb8 7 is around 5nm, which is about two orders of magnitude larger than the dipole length ad = 0.01nm [139]. In 87 Rb, the effective scattering length for the hyperfine 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 observable effects such as spin textures as recently reported by [139]. For spinor condensates with antiferromagnetic interactions as considered in this thesis, the effects due to the dipole-dipole interaction are suppressed 22  2.1. Interatomic Interactions  -1  V/(hc) [cm ] 2000 1000  3 + Σu  2  0  ru  -1000 -2000  f=3 + f=3 f=2 + f=3 f=2 + f=2  0.2  1 + Σg  -3000  0  -4000  -0.2  -5000 0  2  S1/2 + S1/2  lvdW 30  60 20  10  90 30  r [aBohr] Figure 2.1: (Color online) Interaction energy of two 85 Rb atoms as a function of internuclear distance r [42]. The interaction energy in the electronic spin 3 + singlet state 1 Σ+ g (in red) and triplet state Σu (in blue). The inset shows the hyperfine structures at large distances. The hyperfine splitting between F = 2 and F = 3 is 3GHz for the 85 Rb isotope. The 87 Rb isotope has a similar curve, but the hyperfine spin structure is different. The values of hyperfine spin are F = 1 and F = 2, with a splitting 6.8 GHz. An energy associated with wave number unit 1cm−1 corresponds to 30 GHz or 1.44K.  23  2.2. Low Energy Scattering because the local spin moment Sα vanishes in nematic spinor BECs. To induce a nonzero local spin moments, one need considerably large magnetic fields. For the cases considered in Chap.4 one needs around 400mG to match the energy difference between the ferromagnetic states and nematic spin states in optical lattices. Hence, for nematic spinor condensates under small magnetic fields, one can neglect the dipole-dipole interactions.  2.2  Low Energy Scattering  In this section, we briefly 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 multichannel scattering theory. Let’s first consider two spinless bosons scattering in 3D space through a short-range isotropic interaction V (|r|). Potentials falling faster than r−3 at long distance can also be treated as short-range potential[1]. Thus van der Waals potential decaying at r−6 can be treated as short-range potential with an effective range ∼ lvdW . The asymptotic behavior of the total wavefunction in the center of motion frame is given by ψ(r) = eikz + f (θ)  eikr . r  (2.3)  Here the relative velocity in the incoming wave is along z direction. f (θ) 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 uncoupled partial waves according to their angular momentum. f (θ) =  ∞ ∑  fl Pl (cos(θ)).  (2.4)  l=0  It turns out that for low energy scattering in Bose gas, s-wave (l = 0) scattering 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 ψ(r) =  sin(kr + δk ) , r  (2.5) 24  2.3. Interactions with Hyperfine Spin Degrees of Freedom Here δk 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 δk can be expanded in a series of k 2 [1], 1 k cot δk = −1/as + rs k 2 + · · · 2  (2.6)  Here as is the scattering length of the potential and rs is the effective range of the interaction potential. By comparing Eq.2.5 and Eq.2.3, one can express the scattering amplitude in terms of the scattering length: fk  1 k cot δk − ik 1 ≃ − −1 as + ik → −as , when k → 0. =  (2.7) (2.8) (2.9)  Thus the scattering amplitude is a constant −as in low energy collisions, 2 when the relevant energy in the question is much less than ma2 . This cons 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 effective potential, which can reproduce all the low energy physics as the original one. One simple choice of effective potential is the δ-function: U (r) =  4π a δ(r). m  (2.10)  In first order Born approximation, the scattering matrix element is a constant −a, which coincides with the physical interaction with scattering length a.  2.3  Interactions with Hyperfine Spin Degrees of Freedom  Let’s consider scattering problems with spin degrees of freedom. The hyperfine 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 Hyperfine 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, 23 Na 50.0 ± 1.6 55.0 ± 1.7 N.A. F = 1, 87 Rb 101.8 ± 0.2 100.4 ± 0.1 N.A. F = 2, 23 Na 34.9 ± 1.0 45.8 ± 1.1 64.5 ± 1.3 F = 2, 87 Rb 89.4 ± 3.0 94.5 ± 3.0 106.0 ± 4.0 +100 85 F = 2, Rb −445.0−300 −440.0+150 −420.0+100 −225 −140 83 F = 2, Rb 83.0 ± 3.0 82.0 ± 3.0 81.0 ± 3.0 alkali atoms the electronic spin is always 1/2, therefore the hyperfine 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 separation of several gigahertz or several millikelvins in many alkali atoms. This hyperfine splitting energy is three or four orders of magnitude larger than the typical temperature in ultracold experiments. Therefore the individual atom’s hyperfine spin can be consider to be conserved in elastic scattering. Note that in inelastic scattering processes atoms in the higher hyperfine spin sectors can be gradually brought to the lower hyperfine sectors, resulting in a relatively short lifetime of spinor gas in its higher hyperfine sector. In this case, the conservation of individual atom’s hyperfine spin is still meaningful before severe spin loss takes place. The total hyperfine 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 components mF¯ 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, mF¯ = 0 can not be scattered into F¯ = 0 or F¯ = 2, mF¯ = 1 and vice versa. As shown above, scattering channels labeled by F¯ , mF¯ are independent in low energy collisions, and we can introduce scattering amplitudes -aF¯ ,mF¯ for each channel. These scattering amplitudes should be independent of mF¯ 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 example, 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 α|F¯ = 2, mF¯ = 0⟩ + β|F¯ = 0, mF¯ = 0⟩. Since these two channels usually scatter 26  2.4. Vector and Tensor Algebra and Representations with different scattering lengths, the states after scattering will be α′ |F¯ = ′ 2, mF¯ = 0⟩ + β ′ |F¯ = 0, mF¯ = 0⟩ with αβ ′ ̸= αβ , including different spin states such as |mF = 0⟩ ⊗ |mF = 0⟩. Physically, the hyperfine spin-mixing scattering is due to the electronic spin exchange effects, in particular at short distances. From Fig.2.1 one can see that at short distances the energy difference between electronic spin singlet and electronic spin triplet is of order of 103 K which is large compared to hyperfine splitting of order 10−3 K. When two atoms are close enough, the electronic spin exchange energy dominates the hyperfine splitting interaction, resulting in a coupling of different hyperfine levels. These spin-dependent scattering lengths contain the complete information of low energy collisions. Actually one can construct an effective interaction that serves as the starting point of many theoretical analysis. The two-body effective interaction can be expressed in the following form, U (r) =  ∑ 4π δ(r) aF¯ PF¯ . m ¯  (2.11)  F  Here PF¯ is the operator that projects out the total spin F¯ . It is also more useful to write spin projection operator explicitly by using second quantization forms, especially when F is specified.  2.4  Vector and Tensor Algebra and Representations  For a given F , one can introduce annihilation and creation operators ψˆm † and ψˆm associated with each zeeman level m = −F, −F + 1, · · · , F − 1, F . They satisfy the following commutation rules: † † ′ [ψˆm (r), ψˆm′ (r′ )] = [ψˆm (r), ψˆm ′ (r )] = 0 [ ] † ′ ψˆm (r), ψˆm = δm,m′ δ(r − r′ ) ′ (r )  (2.12) (2.13)  The key to write down the effective interaction with hyperfine spins is the rotational symmetry. A general form of two-body contact interaction is Aαβγδ ψˆα† ψˆβ† ψˆγ ψˆδ .  (2.14)  Here α, β, γ, δ are hyperfine sub-states. The rotational symmetry has put a strong constraint on the form of Aαβγδ such that after a rotation R in 27  2.4. Vector and Tensor Algebra and Representations which ψˆα → Rαα′ ψˆα′ , the effective interaction should be in the same form as Eq.2.14. It requires that Aαβγδ = Rαα′ Rββ ′ Rγγ ′ Rδδ′ Aα′ β ′ γ ′ δ′  (2.15)  The number of invariant interactions under the above constraint is limited,compared to situations without such a constraint. Actually the number of possible interaction terms is F + 1, which is exactly the number of independent scattering channels that two hyperfine 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 effective interaction for hyperfine spin F = 1 and F = 2, which are frequently used in later chapters. For F = 3 (53 Cr)and even higher hyperfine spins, similar formulas can be obtained(see [144, 145]).  2.4.1  F=1  The effective interaction can be written as: ] 1[ 2 2 ˆ ˆ U (r) = c0 (ˆ ρ − ρˆ) + c2 (F − 2ˆ ρ) . 2  (2.16)  Here ρˆ and Fˆ 2 is the total density and total spin operator: † ρˆ = ψ−1 ψ−1 + ψ0† ψ0 + ψ1† ψ1 † ˆ † ˆ Fˆ 2 = 2ˆ ρ + 2ψˆ0† ψˆ0 (ψˆ1† ψˆ1 + ψˆ−1 ψ−1 ) + (ψˆ1† ψˆ1 − ψˆ−1 ψ−1 )2 +2(ψˆ† ψˆ† ψˆ ψˆ + h.c.). 0 0 1 −1  (2.17) (2.18) (2.19)  The second line in the above expansion of F 2 clearly shows the spin-mixing coupling as |0, 0⟩ | + 1, −1⟩. Sometimes it is more convenient to introduce a vector representation via which one can write a compact form of interaction:     1  √ √1 ˆ† 0 ψ ψˆx† 2 2   1†   ˆ†   √ (2.20)  ψy  =  −i2 √i2 0   ψˆ−1  , † † ψˆ ψˆz 0 0 1 0  and      1 √ ψˆ1†  ˆ†   √12  ψ−1  =  2 ψˆ0† 0  √i 2 −i √ 2  0    0 ψˆx†   0   ψˆy†  . ψˆz† 1  (2.21)  28  2.4. Vector and Tensor Algebra and Representations The total density ρˆ and spin operator Fˆα along α = x, y, z can be written as: ρˆ = ψˆα† ψˆα Fˆα = −iϵαβγ ψˆβ† ψˆγ  (2.22)  Fˆ 2 = Fˆα Fˆα = ρˆ2 − ψˆα† ψˆα† ψˆβ ψˆβ + 2ˆ ρ.  (2.24)  (2.23)  Here ϵαβγ 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 = c2 =  2.4.2  4π 2 a0 + 2a2 m 3 4π 2 a2 − a0 . m 3  (2.25) (2.26)  F=2  The effective interaction can be written as ˆ U  =  a 2 b ˆ †D ˆ (ˆ ρ − ρˆ) + (Fˆ 2 − 6ˆ ρ) + 5cD 2 2  (2.27)  ˆ † is the spin dimer Here ρˆ is the total density, Fˆ is the total spin, and D creation operator which creates a spin singlet dimer(with total spin zero). One can write down explicitly in terms ρˆ =  +2 ∑  † ˆ ψˆm ψm  (2.28)  m=−2  ) 1 ( ˆ ˆ √ 2ψ+2 ψ−2 − 2ψˆ+1 ψˆ−1 + ψˆ0 ψˆ0 10 √ √ † † ˆ † ˆ = (2ψˆ+2 ψ+1 + 6ψˆ+1 ψˆ0 + 6ψˆ0† ψˆ−1 + 2ψˆ−1 ψ−2 ) √ √ † † † † ·(2ψˆ ψˆ + 6ψˆ ψˆ + 6ψˆ ψˆ + 2ψˆ ψˆ )  ˆ = D Fˆ 2  + −  +1 +2 † (2ψˆ+2 ψˆ+2 + † ˆ (2ψˆ+2 ψ+2 +  0 +1 † ψˆ+1 ψˆ+1 − † ˆ ψˆ+1 ψ+1 −  −1 0 −2 −1 † † ψˆ−1 ψˆ−1 − 2ψˆ−2 ψˆ−2 )2 † ˆ † ˆ ψˆ−1 ψ−1 − 2ψˆ−2 ψ−2 )  (2.29) (2.30) (2.31) (2.32) (2.33)  As one can see, there are spin-mixing matrix elements corresponding to |0, 0⟩ | + 1, −1⟩ | + 2, −2⟩ and also | + 1, +1⟩ | + 2, 0⟩. 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. † ψˆxx = † ψˆyy = † ψˆzz = † ψˆxy = † ψˆxz = † ψˆyz =  1 1 † √ (ψˆ−2 + ψˆ2† ) − √ ψˆ0† , 2 3 1 ˆ† 1 − √ (ψ−2 + ψˆ2† ) − √ ψˆ0† , 2 3 2 ˆ† √ ψ0 , 3 i ˆ† √ (ψ−2 − ψˆ2† ) 2 1 ˆ† √ (ψ−1 − ψˆ1† ) 2 i † − √ (ψˆ−1 + ψˆ1† ), 2  (2.34) (2.35) (2.36) (2.37) (2.38) (2.39)  The corresponding annihilation operators can be introduced in the same way. These operators have the following properties [134]: 2 [ψˆαβ , ψˆα† ′ β ′ ] = δαα′ δββ ′ + δαβ ′ δβα′ − δαβ δα′ β ′ 3 ∑ ˆ ψαα = 0.  (2.40) (2.41)  α  This last property puts a constraint on the constructions of linear operators. For operators ∑ ˆ = ∆αβ ψˆβα , Tr[∆ψ] (2.42) α,β  the tensor ∆ can be always reduced to a traceless one, i.e., ∑ Tr[∆] = ∆αα = 0.  (2.43)  α  This constraint is needed, because when introducing this new basis we have enlarged the Hilbert space by constructing six operators out of five. This constraint brings the size of the physical Hilbert space back to the original one. The density operator ρˆ in terms of the new operators can be derived as: ρˆ =  ∑ m  ∑ † 1 † ˆ ˆ =1 ψˆm ψm = Tr[ψˆ† ψ] ψˆαβ ψˆβα . 2 2  (2.44)  α,β  30  2.4. Vector and Tensor Algebra and Representations The factor 12 appears here because the trace involves a double sum over the operator ψˆαβ . This same factor will appear later when deriving the hopping term in the Hamiltonian. The spin operator is straightforwardly derived as: † ˆ Fˆα = −iϵαβγ ψˆβη ψηγ .  (2.45)  It has the following properties: [Fˆα , Fˆβ ] = iϵαβγ Fˆγ , [Fˆα , ρˆ] = 0 [Fˆα , Tr[(ψˆ† )n ]] = 0.  (2.46) (2.47) (2.48)  The total spin operator is then given by: Fˆ 2 = Fˆα Fˆα † ˆ ˆ† ˆ † ˆ ˆ† ˆ = ψˆβη ψηγ ψγξ ψξβ − ψˆβη ψηγ ψβξ ψξγ  (2.49) (2.50)  The dimer creation operator in the new representation is: 1 Dˆ† = √ Tr[(ψˆ† )2 ], 40  (2.51)  which has the following properties: ˆ D ˆ † ] = 1 + 2 ρˆ. [D, 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 defined as 1 Tˆ † = √ Tr[(ψˆ† )3 ]. (2.53) 140 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 = b = c =  4π 2 4a2 + 3a4 m 7 4π 2 a4 − a2 m ( 7 ) 4π 2 a0 − a4 a2 − a4 −2 . m 5 7  (2.54) (2.55) (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 Ψ0 = ∧† |0⟩, one can have D 4π  2a 0  m  ˆ |Ψ0 ⟩ = a − 6b + 5c = ⟨Ψ0 |U  (2.57)  Similarly, for the F = 4 state such as Ψ4 = √12 ψˆ2† ψˆ2† |0⟩, and the F = 2 state (√ ) such as Ψ2 = 1 6ψˆ† ψˆ† − ψˆ† ψˆ† |0⟩ one can have 10  2 0  4π  2a 4  m 4π  2a 2  m  1 1  ˆ |Ψ4 ⟩ = a + 4b = ⟨Ψ4 |U  (2.58)  ˆ |Ψ2 ⟩ = a − 3b. = ⟨Ψ2 |U  (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 field 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 87 Rb, |I| = 23 and |S| = 12 . A is the hyperfine coupling constant and is equal to one half of the hyperfine splitting ∆hf between F = 1 and F = 2 states. Constants C and D are given by µ C = 2µB B, D = − B, I  (2.61)  where µ is the nuclear magnetic moment and is equal to 2.7µN for 87 Rb. We m C have C ≃ 2M Hz and D ≃ 1kHz when B = 1G. Since | D | ∼ mpe ≃ 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 1 A + C, 4 2 32  2.5. Linear and Quadratic Zeeman Shift  Figure 2.2: (a)-(c) Shift of hyperfine spin energy levels due to the quadratic Zeeman effects in hyperfine F = 1 and F = 2 87 Rb atoms. (d) Corresponding Zeeman energy for selected combinations of two-particle states versus magnetic fields[97]. E(2, −2) = E(2, +1) = E(1, +1) = E(2, −1) = E(1, −1) = E(2, 0) = E(1, 0) =  1 3 A − C, 4 2√ 1 3 2 1 − A+ A + (A + C)2 , 4 4 4 √ 1 3 2 1 − A− A + (A + C)2 , 4 4 4 √ 1 3 2 1 − A+ A + (A − C)2 , 4 4 4 √ 1 3 2 1 − A− A + (A − C)2 , 4 4 4 √ 1 1 − A + A2 + C 2 , 4 4 √ 1 1 − A − A2 + C 2 . 4 4  (2.62)  When the zeeman coupling C is much less than the hyperfine coupling A (which is true for many experiments), one can make an expansion in terms  33  2.5. Linear and Quadratic Zeeman Shift of  C A  and keeping terms to the second order:  1 3 A ± C, 4 2 3 1 3C 2 E(2, ±1) = A± C + , 4 4 32A C2 3 A+ , E(2, 0) = 4 8A 5 1 3C 2 E(1, ±1) = − A ∓ C − , 4 4 32A 5 C2 E(1, 0) = − A − . 4 8A We can rewrite the energy levels in a more compact form as E(2, ±2) =  (2.63)  3 C2 C C2 2 A+ + mF − m 4 8A 4 32A F C2 C C2 2 5 − mF + m . (2.64) E(1, mF ) = − A − 4 8A 4 32A F The linear Zeeman effects 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 effect of magnetic fields is thus coming from the second order Zeeman effects or the quadratic Zeeman effects. In the second quantization form introduced in the previous section, second order Zeeman effects can be cast E(2, mF ) =  +F ∑  † m2 ψ m ψm .  (2.65)  (µB B)2 C2 = ± =± , F = 1 or 2 32A 4∆hf  (2.66)  HQB = δB  m=−F  δB  We can also write the HB in the vector representation. For F = 1 ( ) HQB = δB ρ − ψz† ψz ; and for F = 2  ( HQB = δB  † 2ρ − 2ψzz ψzz +  ∑  (2.67)  ) † ψzη ψηz  .  (2.68)  η=x,y  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 symmetry but not the translational symmetry, in which most of the spins align themselves along a particular axis. Spin nematic states are fascinating because of the rich topology of their order parameter space (also known as “manifold”), which gives rise to exotic topological defects such as quantum 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 field theory[107, 108]. In this chapter I will examine the effects of zero-point fluctuations on nematic correlations. Zero-point fluctuations are known to lead a wide range of physical phenomena, including the Lamb shift[7], the ColemanWeinberg mechanism of spontaneously symmetry breaking[11], fluctuationinduced first 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 fluctuations 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 field theories. In Sec.3.3, I examine the quantum fluctuations in the nematic states. The  35  3.2. Mean Field Theory of F =2 Condensates quantum fluctuations studied here can be viewed as zero-point motion of mean field collective modes. Bogoliubov quasi-particles, quantized version of the mean field 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 fluctuations, 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 field ground states, without the presence of Zeeman fields. 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 magnetizations, spin quadrupole moments and singlet pair amplitudes (summarized 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: ) ) ∑ ( aL ( ) bL ( 2 † 2 ˆ ρˆk − ρˆk + H = Fk − 6ˆ ρk + 5cL Dk Dk 2 2 k ) ∑( † ∑ − tL ψk,αβ ψl,βα + h.c. − µ ρˆk . (3.1) <kl>  k  Here k is the lattice site index and < kl > are the nearest neighbor sites, µ 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 field 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, χ ˜αβ = ⟨ψˆk,αβ ⟩.  (3.2)  In this condensate, atoms occupy a one-particle spin state |χ⟩ ˜ that is defined as a linear superposition of five F = 2 hyperfine states |2, mF ⟩ or ψmF ∑ |χ⟩ ˜ = χ ˜αβ Cαβ,mF ψmF , αβ,mF  √  Cαβ,mF =  15 4π  ∫  (  ) 1 ∗ dΩ nα nβ − δαβ Y2,m (θ, ϕ). F 3  (3.3) 36  3.2. Mean Field Theory of F =2 Condensates  Table 3.1: Magnetic orderings in mean field ground states of F = 2 spinor ˆ k are spin single annihilation operator, spin condensates. Dk , Fˆk and O operator and spin quadrupole moment respectively. Their definition can be found in Sec.2.4.2 of Chap.2 and in Eq.3.15. Nematic Cyclic Ferromagnetic √ √ ) ( ) ( ⟨ψmF ⟩ ∼ (1, 0, 0, 0, 0) sin ξ, 0, 2 cos ξ, 0, sin ξ 1, 0, −3 2i, 0, 1 ⟨Dk ⟩ ̸= 0 0 0 ⟨Fˆk ⟩ 0 0 ̸= 0 ˆk ⟩ ⟨O ̸= 0 0 ̸=0 Here nα , α = x, y, z, are components of a unit vector n(θ, ϕ); nx = sin θ cos ϕ, ny = sin θ sin ϕ and nz = cos θ. Y2,mF , mF = 0, ±1, ±2 are five spherical harmonics with l = 2. The mean field energy per site as a function of χ ˜ can be obtained as cL aL Tr(χ ˜∗ χ)Tr( ˜ χ ˜∗ χ) ˜ + Tr(χ ˜∗ χ ˜∗ )Tr(χ ˜χ) ˜ 8 8 bL µ + Tr[χ ˜∗ , χ] ˜ 2 − ztL Tr(χ ˜∗ χ) ˜ − Tr(χ ˜∗ χ), ˜ 4 2 EM F =  (3.4)  where z is the coordination number. Minimization of EM F with respect to the tensor χ ˜ 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 field phase diagram including all three types of phases in Fig.3.1(b). Spin wavefunctions ΨS are represented by the usual spherical harmonics. For example, from a spinor condensate with condensate amplitude ψm one can construct the following spin wavefunction ΨS (θ, ϕ) =  l ∑  ψm Yl,m (θ, ϕ).  (3.5)  m=−l  One can also use the tensor form of condensate amplitude ξ to get a compact expression √ 15 ∑ ΨS = χαβ nα nβ . (3.6) 4π αβ  Here nα , α = x, y, z are components of a unit vector n(θ, ϕ). 37  3.2. Mean Field Theory of F =2 Condensates  Table 3.2: Scattering lengths (in atomic units) of various alkali isotopes with hyperfine spin F = 2. (Data sources: [141, 68, 137]) isotopes a0 a2 a4 23 Na 34.9 ± 1.0 45.8 ± 1.1 64.5 ± 1.3 87 Rb 89.4 ± 3.0 94.5 ± 3.0 106.0 ± 4.0 85 Rb −445.0+100 −440.0+150 −420.0+100 −300 −225 −140 83 Rb 83.0 ± 3 82.0 ± 3 81.0 ± 3 Note that the coordinates θ and ϕ 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, providing 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 magnetization. ⟨Fˆk ⟩ ̸= 0 ⟨Dk ⟩ = 0  (3.7)  The condensate amplitude is specified by SO(3) rotation and phase. √  1 −i M −i −1 χ ˜= 2 0 0  the following matrix up to an  0 0  0  Here M is the average number of atoms per site or M = ⟨ ρˆk ⟩. In the representation of Zeeman spin levels, we have √ (3.8) ⟨ψˆmF ⟩ = M (1, 0, 0, 0, 0) 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 symmeˆ k,αβ ⟩ = ⟨Fˆk,α Fˆk,β ⟩ − try through a nonzero spin quadrupole moment ⟨O 38  3.2. Mean Field Theory of F =2 Condensates  Figure 3.1: (Color online) (a) Wavefunctions of spin nematics at various ξ. ξ = 0, π3 , 2π 3 , · · · are wavefunctions of uniaxial spin nematics(labeled as ‘U’), 9π and ξ = π6 , 5π 6 , 6 · · · are biaxial spin nematics(labeled as ‘B’) with Dih4 symmetries. These spin wavefunctions ΨS are represented by spherical harmonic functions (see text). Colors indicate the phase of wavefunctions. (b) Ferromagnetic(labeled as ‘F’), cyclic(labeled as ‘C’) and nematic states in the mean field theory are separated by blue boundaries; uniaxial and biaxial nematics when taking into account of quantum fluctuations are separated by red lines. The locations of various isotopes on this diagram are located as: ♢ =23 Na, △ =85 Rb, =87 Rb and ⃝ =83 Rb (see Tab.3.2).  39  3.2. Mean Field Theory of F =2 Condensates 1 ˆ2 3 δαβ ⟨F ⟩.  ⟨Fˆk ⟩ = 0 ⟨Dk ⟩ = 0 ˆ k ⟩ ̸= 0 ⟨O  (3.9)  The condensate amplitude up to a SO(3) rotation and a phase is:   √ 1 0 0 2πi 2M   0  χ ˜=  0 e 3 3 4πi 0 0 e 3 or  √ ψmF =  ) √ M −πi ( e 6 1, 0, −3 2i, 0, 1 20  (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 identified 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 specified by Eq.3.10 is invariant up to a Berry phase under an SO(3) rotation R around x = y = z of 2π 3 ,   0 0 1 −2π χ ˜ → RχR ˜ T = e 3 i χ, ˜ R= 1 0 0  0 1 0 Therefore, a 1/3 quantum vortex is constructed by arranging the spinor condensates along a closed loop such that the total phase winding is 2π 3 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 χ ˜ be a real symmetric tensor (up to a phase). An arbitrary 40  3.2. Mean Field Theory of F =2 Condensates solution χ ˜ can be obtained by applying an SO(3) rotation and a U (1) gauge transformation to a real diagonal matrix χ, √ χ ˜ = 4M eiϕ RχR−1 ; (3.11) R is an SO(3) rotation matrix, and χ is a normalized real diagonal traceless matrix[134],   χxx 0 0 1 (3.12) χ =  0 χyy 0  , Tr(χχ) = . 2 0 0 χzz The matrix elements can be parameterized by ξ ∈ [0, 2π]: ( 1 π) χxx = √ sin ξ − , 6 3 ( ) 1 5π √ χyy = sin ξ − , 6 3 ( ) 1 9π χzz = √ sin ξ − . 6 3  (3.13)  In the mean field approximation, all of the quantum spin nematics specified by different diagonal matrices Eq.(3.12) or different ξ have the same energy, i.e. are exactly degenerate. For instance, when ξ = 0 or 2χxx = 2χyy = √ −χzz = 1/ 3, up to an overall SO(3) rotation, all atoms are condensed in the hyperfine state |2, 0⟩. This choice of χ represents a uniaxial spin nematic. When χxx = −χyy = 1/2, χzz = 0 or ξ = π2 , the atoms are condensed in the state √12 (|2, 2⟩ + |2, −2⟩). In the Zeeman basis, the nematic states are: √ ( ) √ M ψmF = (3.14) sin ξ, 0, 2 cos ξ, 0, sin ξ 2 It is worth pointing out that spin nematics here are time-invariant; the expectation value of the hyperfine spin operator Fk in these states is zero. However, all nematics have the following nonzero quadrupole spin order (up to an SO(3) rotation), 1 Oαβ =< Fkα Fkβ > − δαβ < Fk2 >, 3 Oαβ = 2M sin(2ξ + ξα )δαβ .  (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 fluctuations and will be discussed in the next two sections. 41  3.3. Zero-Point Quantum Fluctuations  3.3  Zero-Point Quantum Fluctuations  Quantum fluctuations of mean field nematic states can be viewed as the zeropoint motion of mean field collective modes. These modes can be treated as a set of decoupled oscillations with definite frequencies and wavefunctions, which are the eigen-modes of the linearized Gross-Pitaevskii equation. They have been thoroughly studied in single-component condensates in experiments. 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 profiles[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 field modes in spinor condensates haven’t been fully explored in experiments yet. In principle they can be detected and identified by the time evolution of spin populations. For example, normal modes in different branches usually have distinctive spin wavefunctions, as summarized in Tab.3.3 for spin-two nematic states. To study zero-point quantum fluctuations, we first examine the energy spectra of collective modes. For this, we expand ψ † about the mean field χ, ˜ ∑ √ † † ψˆk,αβ = 4M χαβ (ξ) + Lναβ (ξ)θˆk,ν (3.16) ν  where the superscript ν = x, y, z, t, p labels zero-point motions of χ along five 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 ξ, and a phase mode (p-mode) describing fluctuations of the condensate’s overall phase. The corresponding matrices are   0 0 0 Lx =  0 0 1  , 0 1 0   0 0 1 Ly =  0 0 0  , 1 0 0   0 1 0 Lz =  1 0 0  , 0 0 0 42  3.3. Zero-Point Quantum Fluctuations  Table 3.3: Collective modes of nematic states specified by ξ. Distinctive spin waves functions of collective modes are listed in the Zeeman basis. vν is the velocity of the ν-th mode. The uniaxial nematics are specified by ξ = 0, π/3, · · · and the biaxial nematics are specified by ξ = π/6, π/2, · · ·. mode(ν) spin wavefunctions vν2 /(4M tL d2L ) √ phase(p) (sin ξ, 0, √ 2 cos ξ, 0, sin ξ) aL + cL deformation(t) (cos ξ, 0, − 2 sin ξ, 0, cos ξ) −cL 2 spin(x) (0, + 1, 0, + 1, 0) 4bL sin (ξ − 2π/3) − cL spin(y) (0, + 1, 0, − 1, 0) 4bL sin2 (ξ + 2π/3) − cL spin(z) (+1, 0, 0, 0, − 1) 4bL sin2 ξ − cL  sin(ξ + π3 ) 0 0 2 , √  0 sin(ξ − π3 ) 0 3 2π 0 0 sin(ξ − 3 )   π sin(ξ − 6 ) 0 0 2 . √  0 sin(ξ − 5π 0 6 ) 3 9π 0 0 sin(ξ − 6 )   Lt =  Lp =  (3.17)  These matrices are mutually orthogonal, Tr(Lµ Lν ) = 2δµν . The five modes are decoupled and operators obey the bosonic commuta[ the corresponding ] † tion relations, θk,µ , θl,ν = δkl δµν . 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 fluctu† ations which is bilinear in θk,ν , θk,ν . It can be diagonalized by a Bogoliubov transformation. The result can be be expressed in terms of Bogoliubov † operators, θ˜q,ν and θ˜q,ν , ( ) ∑√ 1 † ˜ 2 ˜ H = ϵq (2mBN vν + ϵq ) θq,ν θq,ν + . (3.18) 2 q,ν ∑ Here ϵq = 4tL α (1 − cos qα dL ) is the kinetic energy of an atom with crystal quasi-momentum q = (qx , qy , qz ); dL is the lattice constant. mBN = 1/4tL d2L is the effective band mass. vν (ν = x, y, z, t, p) is the sound veloc2 = 4M tL d2L (bL Gαα − cL ), ity of the ν-mode in the small-|q| limit; vα=x,y,z 2 2 2 2 vt = 4M tL dL (−cL ), vp = 4M tL dL (aL + cL ). The velocities of three spin modes depend ˜ ]) 3 × 3 symmetric matrix Gαβ , Gαβ = ( α on [a βχ-dependent p p −(1/2)Tr [L , L ] L , L . More explicitly, Gαβ is a diagonal matrix with yy = 4 sin2 (ξ + 2π ), Gzz = 4 sin2 ξ. Noelements: Gxx = 4 sin2 (ξ − 2π 3 ), G 3 √ tice that the velocity of the phase mode vp can be written as µL /mBN , 43  3.3. Zero-Point Quantum Fluctuations  1.4 1.2 E(ξ)−Ε(0)  1  bL = -2.8 cL bL = 0.24 cL bL= 0  20 15 10 5 0 -0.005 g(bL/cL, y)  1.6  -10 -8 -6 -4 -2 bL/cL  0.8  0 0.2  0.6 0.4 0.2 0  -0.002 0.00  0.33  0.67  1.00 ξ/π  1.33  1.67  2.00  Figure 3.2: (Color online) The zero-point energy of spin nematics per lattice 3/2 site as a function of ξ (in units of |M cL |5/2 /tL , cL < 0). Inset is g(x, y), the amplitude of E(ξ = π6 ) − E(0) as a function of x(= bL /cL ). y(= |cL |/tL ) has been set to be 8 × 10−3 . A quantum first order phase transition occurs at bL = 0 as a result of zero-point quantum fluctuations. 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 µL ; it is independent of ξ. Unlike the mean field energy EM F that is independent of ξ, the zeropoint energy of spin nematics per lattice site E is ξ-dependent ( α summed over x, y, z), √ 1 ∑ E(ξ) = ϵq (2mBN vα2 (ξ) + ϵq ). (3.19) 2NT q,α Here NT is the number of lattice sites. The main contribution to the ξdependence of energy is from fluctuations of wave vector |q| ∼ mBN vα (ξ); for condensates, this characteristic momentum is much smaller than /dL . 44  3.4. Uniaxial and Biaxial Spin Nematic States ∑ 5 5 4 The ξ-dependent energy is proportional to α vα /(dL tL ); the amplitude 3/2 of it can be expressed as |M cL |5/2 /tL g(bL /cL , |cL |/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 ξ. We find 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, ξ = 0, π/3, 2π/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 ξ = π/6, π/2, 5π/6, ... are the stable ground states while the uniaxial nematics are unstable. The energy as a function of ξ has following symmetries E(ξ) = E(−ξ), π E(ξ) = E( + ξ). 3  (3.20) (3.21)  These are a result of the rotation and gauge invariance of the energy function. They are found by examining the mean field solutions in Eq.(3.11),(3.13). A rotation R of 120◦ around the x = y = z line effectively transforms a solution at ξ to ξ + π/3, or RT χ(ξ)R= −χ(ξ + π3 ). And because the energy is invariant under SO(3) rotations and is an even function of χ, one finds that E(ξ) = E(ξ +π/3). In addition, a rotation of 180◦ around the x+y = 0 line in the xy-plane transforms χ(ξ) to χ(−ξ), i.e. RT χ(ξ)R = χ(−ξ); so similarly one finds that E(ξ) = E(−ξ). These symmetries only depend on rotational and gauge invariance and are exact. So the energy is an even and periodic function of ξ, with the period equal to π/3. For an analytic function with these symmetries, ξ = 0, π/6 and π/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 ξ and at ξ + π/3 are equivalent.  3.4  Uniaxial and Biaxial Spin Nematic States  We now examine the topology of the manifold of fluctuation-induced spin nematics. Without loss of generality, we √ first consider the √ uniaxial nematic state at ξ = 0, i.e., χxx = χyy = −1/2 3 and χzz = 1/ 3; in this case, the nematic easy axis specified by a unit vector e is pointing along the zdirection (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 2 simply ZS2 ⊗ S1 ; here S 2 /Z2 is where the nematic director lives and S 1 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 ξ = π/2, χxx = −χyy = 1/2 and χzz = 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 x±y = 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 π. The manifold therefore is [SO(3) × S 1 ]/Dih4 , where the Dihedral elements in the denominator contain the rotations mentioned plus the π-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 fluctuations.  3.5  Discussion and Conclusion  The perturbative calculation carried out here is valid when fluctuations of the order parameter χ are small. Here we estimate the fluctuations along each of the five directions ν = x, y, z, t, p. Using Eq. 3.16, we evaluate the amplitude of local fluctuations in an orthogonal mode, A1 (ν) = † ⟨0|θk,ν θk,ν |0⟩. Taking into account the expression for vν , we have the following estimate for the relative amplitude of fluctuations, A1 (ν) a ˜ν aL + cL 3/2 ∼ M 1/2 ( ) . M a ˜p tL  (3.22)  a ˜ν is the effective scattering length of each mode which is a linear combination of scattering lengths in F = 0, 2, 4 channels: a ˜α = a ˜t = a ˜p =  2 − Gαα 1 (a2 − a4 ) − (a0 − a4 ) , 7 5 2 1 (a2 − a4 ) − (a0 − a4 ) , 7 5 1 1 (2a2 + 5a4 ) + (a0 − a4 ) . 7 5  (3.23) (3.24) (3.25) 46  3.5. Discussion and Conclusion For 87 Rb atoms in optical lattices, a ˜α /˜ ap ∼ 10−2 , aL ∼ 50nk and when tL ∼ 300nk, the relative amplitude of fluctuations 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 87 Rb). One of experimental signatures of quantum fluctuations is number fluctuations in spin populations. The microscopic origin of spin-population fluctuation is the two-body spin-changing scattering processes. For example atoms in the uniaxial nematic condensate(|2, 0⟩) are possible to be scattered into other spins via these channels |2, 0⟩ + |2, 0⟩ |2, +2⟩ + |2, −2⟩ and |2, 0⟩ + |2, 0⟩ |2, +1⟩ + |2, −1⟩. The relative amplitude of the spinpopulation fluctuations is directly related to the expression in Eq.3.22. In this chapter, we have found that zero-point quantum fluctuations 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 hyperfine spin-two manifold, bL is between −3cL and −10cL . According to the above analysis, the ground state should be a uniaxial spin nematic. The fluctuation-induced energy landscape can be experimentally mapped out by investigating the macroscopic quantum dynamics of condensates prepared in certain initial states. A condensate of rubidium atoms initially prepared at state √12 (|2, 2⟩ + |2, −2⟩) (corresponding to ξ = π/2 point), because of the fluctuation-induced potential shown√in Fig.3.2, could evolve towards a condensate with atoms at state 1 √3 2 |2, 0⟩ + 2 2 (|2, 2⟩ + |2, −2⟩) (corresponding to ξ = π/3 point). This leads to a temporal oscillation of the population of atoms at state |2, 0⟩, 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 usually driven by mean field potentials, either due to scattering between atoms or due to coupling to external fields; 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 fluctuations of spin collective coordinates. Unlike the usual mean-field coherent dynamics, quantum fluctuation-controlled spin dynamics or QFCSD studied here are very sensitive to variation of quantum fluctuations and the corresponding driving potentials induced by zero-point motions can be tuned by four to five orders of magnitude using optical lattices. These dynamics have unique dependence on optical lattice potential depths and quadratic Zeeman fields. We also find that thermal fluctuations generally can further enhance the induced 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 fluctuations and investigate correlated fluctuations 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 spinrelaxation collisions were first 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 field interactions or external fields were later demonstrated in condensates of hyperfine spin-two rubid48  4.1. Introduction ium atoms [39], and hyperfine spin-one rubidium atoms[87, 92]. Ordering in spinor gases is usually induced by hyperfine spin dependent two-body scattering[107, 108]. Coherent spin dynamics observed in experiments are related to the coherent quantum dynamics explored in solid state superconductors, 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 interferometers. Remarkably, coherent dynamics also provide a unique direct measure of interaction energies or scattering lengths as emphasized before [152, 39, 87]. Quantum fluctuation-controlled spin dynamics or QFCSD we are going to study in this chapter on the other hand are a direct measure of quantum fluctuations; they can be potentially used to calibrate quantum fluctuations and investigate correlated fluctuations near quantum critical points or universal scaling properties. Furthermore, QFCSD of cold atoms can be designed to simulate many other quantum-fluctuation-induced phenomena 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 (87 Rb) in hyperfine spin-two (F = 2) states. The twobody 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 different two-spin states have also been used to measure spin dependent interactions[96]. However it is quite challenging to probe QFCSD in traps without optical lattices, if not impossible. For dilute gases in the absence of optical lattices, the effective driving potential induced by quantum fluctuations is about 10−5 pK per particle (see section III for more discussions) because the relative amplitude of fluctuations is very small. The corresponding dynamics 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 experiments. In addition, a tiny external magnetic field of strength 1mG can result in a quadratic Zeeman coupling of order of 6 × 10−3 pk 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 effective quadratic Zeeman coupling can be controlled only up to an uncertainty that is equivalent to a magnetic field of an order of 1mG. This further complicates future experimental studies of QFCSD . To resolve these difficulties, we propose to enhance the effect of QFCSD using optical lattices. To vary the amplitude of quantum fluctuations and optimize the effect of fluctua49  4.2. Microscopic Hamiltonian for F = 2  87 Rb  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 87 Rb atoms in optical lattices and discuss the range of parameters we have used to investigate this phenomenon. In Sec.4.4, we briefly discuss the structure of spin nematic states. In Sec.4.5, we derive an effective Hamiltonian for QFCSD by introducing low energy collective coordinates and their conjugate momentum. In Sec.4.6 and Sec.4.7, we present our main numerical results on quantumfluctuation 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 effects of thermal fluctuations and analyze the potential-depth dependence of thermal enhancement of induced driving potential. In Sec.4.9, we further investigate effects of spin exchange losses and propose how to observe quantum-fluctuation-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 Optical Lattices  87  Rb in  We are considering the following Hamiltonian for F = 2 atoms in optical lattices: ) ) ∑ ( aL ( ) bL ( 2 † 2 ˆ H = ρˆk − ρˆk + Fk − 6ˆ ρk + 5cL Dk Dk 2 2 k ) ∑( † ∑ ∑ − tL ψk,αβ ψl,βα + h.c. − µ ρˆk + qB Qk,zz . (4.1) <kl>  k  k  Here k is the lattice site index and < kl > are the nearest neighbor sites, µ 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 ψ † that was † introduced previously in Sec.2.4.2 of Chap.2. Components ψαβ , α, β = † x, y, z are linear superpositions of five spin-two creation operators, ψm F,  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 construction of rotationally invariant operators. We use it to analyze collective spin modes that correspond to small rotations around various axes. The † number operator ρˆk , the dimer or singlet pair(creation ) ) operator Dk(, the total spin operator Fˆk,α are defined as ρˆk = 1 Tr ψ † ψk , D† = √1 Tr ψ † ψ † , 2  k  k  40  k k  † Fˆk,α = −iϵαβγ ψk,βη ψk,ηγ (See Chap.2 for details). The quadratic Zeeman operator Qk,zz is defined as ( ) Qk,zz = Tr ψk† Qψk   0 0 0 Q =  0 0 0 . (4.2) 0 0 1  4.3  Range of Interaction Parameters for the Lattice Hamiltonian  Spin correlations between hyperfine spin-two rubidium atoms are determined by three two-body s-wave scattering lengths aF , F = 0, 2, 4. In optical lattices, local spin-dependent interactions contain two contributions as shown in Eq.(4.1); one is, bL Fk2 /2, the energy of having total hyperfine spin Fk at site k, and the other is the energy of creating spin singlet pairs (dimers), 5cL Dk† Dk where Dk is the dimer creation operator. The usual contact interaction at site k is of the form aL (ρ2k − ρk )/2, where ρk is the number of atoms. Three effective coupling constants aL , bL , cL which characterize various interactions are functions of two-body scattering lengths aF , F = 0, 2, 4 and on-site orbitals ψ0 (r), ∫ 4π 2 dr (ψ0∗ (r)ψ0 (r))2 . (4.3) aL (bL , cL ) = a(b, c) m Here a = (4a2 + 3a4 )/7, b = (a4 − a2 )/7, c = (7a0 − 10a2 + 3a4 )/35  (4.4)  are three effective scattering lengths; ψ0 is the localized Wannier function obtained by solving the Schrodinger equation for an atom in periodical potentials. The band structures and Wannier functions are calculated through 51  4.3. Range of Interaction Parameters for the Lattice Hamiltonian  80  tL[nK]  aL[nK]  16  4  6  8  0.0 2  10  0 2 0  4  8  10  8 6 V[ER]  10  6  cL[nK]  bL[nK]  0 2 1.0  4  8 6 V[ER]  -0.4 10 2  4  Figure 4.1: (Color online) Coupling parameters aL , bL , cL and hoping integral 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 λ = 850nm lasers. standard methods outlined in the Appendix. aL , bL , cL can then be calculated 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 = h2 /2mλ2 , where λ is the wavelength of lasers.) Below 2ER , quantum fluctuations 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 find that quantum depletion is usually small, less than twenty percent and our lowest order calculation should suffice. Although quantum fluctuations can be further enhanced above 10ER and a superfluid-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 first 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 first 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 filling factors due to pauli exclusion principles that the existence or absence of a band gap can make a crucial difference. For F = 2 rubidium atoms, the quadratic Zeeman coupling qB is related to a uniform external magnetic field B via qB = 3(µB B/2)2 /∆, ∆ = 6.8GHz (see Appendix). Here µB is the Bohr magneton and ∆ is the hyperfine splitting. For the purpose of studying QFCSD, we set the range of B to be 70mG > B > 1mG where effects of quantum fluctuations are most visible; beyond 70mG the dynamics are mainly driven by the mean field quadratic coupling qB (see section VI for more discussions). The corresponding range for the quadratic Zeeman coupling is then from 10−2 pK to 100pK. Here we assume that the optical lattice potential does not dependent on hyperfine spin levels. Practically, the optical dipole potential has a week hyperfine spin dependence for alkali atoms in many cases because dipole transitions depend on which hyperfine Zeeman levels are involved. For 87 Rb, the hyperfine spin dependent dipole interaction up to the leading order of mF is given by [158, 159, 160]   ∆1/2 − ∆3/2 ∑ I (r) q  (4.5) V (r) = V0 1 + gF mF q ∆1/2 + 2∆3/2 I(r) q=−1,0,1  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 hyperfine spin dependent term is ∆ 1/2+2∆3/2 1/2  3/2  which is about 0.07 for 87 Rb loaded in optical lattices operating at a wavelength of 850nm. By choosing a suitable polarization of the laser beams[160], one can make the first order term vanish, resulting in an even weaker hyperfine 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 10−1 at maximum of the spin-dependent two-body interaction. 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 definition of qB given in the previous paragraph. For these two reasons, we will neglect the effects 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 χ, ˜ i.e. the expectation value of matrix operator ψ † . The corresponding equation of χ ˜ is ˜ i ∂χ 2 ∂t ∂Hsc ∂χ ˜∗  = = +  ∂HQ ∂χ ˜∗  ∂Hsc ∂HQ ∂Hqf + + ; ∂χ ˜∗ ∂χ ˜∗ ∂χ ˜∗ cL ∗ aL χTr( ˜ χ ˜∗ χ) ˜ + χ ˜ Tr(χ ˜χ) ˜ 4 4 bL [χ, ˜ [χ ˜∗ , χ]] ˜ − (ztL + µ/2)χ, ˜ 2  = qB Qχ. ˜  (4.6)  Here Hsc is the mean field (semiclassical) Hamiltonian (of matrix χ, ˜ χ) ˜ 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: aL cL Tr(χ ˜∗ χ)Tr( ˜ χ ˜∗ χ) ˜ + Tr(χ ˜∗ χ ˜∗ )Tr(χ ˜χ) ˜ 8 8 bL + Tr[χ ˜∗ , χ] ˜ 2 − (ztL + µ/2)Tr(χ ˜∗ χ); ˜ 4 = qB Tr(χ ˜∗ Qχ). ˜  Hsc =  HQ  (4.7) (4.8)  z is the coodination number of optical lattices. For a field along the zdirection, matrix Q is defined as Qαβ = δαz δβz . And Hqf is the quantumfluctuation-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. Effective Hamiltonian for QFCSD quadratic Zeeman coupling, ground states are spin nematics characterized by real and symmetric tensor wavefunctions χ ˜ (up to an overall phase). Any condensate that is initially prepared in this submanifold has no mean field dynamics because the potential gradient ∂Hsc /∂ χ ˜∗ vanishes.  4.5  Effective Hamiltonian for QFCSD  To investigate the kinetic energy and quantum fluctuation-induced potential energy for dynamics of a condensate initially prepared in this submanifold, † we expand tensor ψαβ about a reference condensate wavefunction χ(ξ) in terms of five collective coordinates Xν introduced above and their conjugate operators Pˆν . We furthermore separate the macroscopic dynamics of condensates (q = 0-mode) from the microscopic zero-point quantum fluctuations (q ̸= 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 arbitrary condensate amplitude χ ˜ in the spin nematic submanifold. It can be parameterized using an SO(3) rotation, R and a U (1) phase shift ϕ, and a real diagonal traceless matrix χ; i.e., √ χ(X ˜ x , Xy , Xz , ξ, ϕ) = 4M eiϕ RT χ(ξ)R, R(Xx , Xy , Xz ) = exp(T x Xx + T y Xy + T z Xz )  (4.9)  where M is the number density or average number of atoms per lattice site. R is an SO(3) rotation matrix defined by three spin angles Xα , α = x, y, z, α = −ϵ and antisymmetric generators Tβγ αβγ . χ(ξ), ξ ∈ [0, 2π], are normalized real diagonal traceless matrices that form a family of solutions specified by a single parameter ξ[124]; χαα =  sin(ξ − ξα ) √ , 3  (4.10)  and ξx = π/6, ξy = 5π/6 and ξz = 3π/2. Nematics with different ξ exhibit different spin configurations. Following the definition of tensor operator in Eq.(2.34), one can easily show that these solutions represent condensates of spin-two atoms specified by five-component wavefunctions ψ T = (ψ2 , ψ1 , ψ0 , ψ−1 , ψ−2 ) and √ sin ξ sin ξ ψ T = M ( √ , 0, cos ξ, 0, √ ). (4.11) 2 2 55  4.5. Effective Hamiltonian for QFCSD  Figure 4.2: (Color online) An artistic view of the five-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 ξ. We have also plotted the nematic wavefunctions ψS (θ, ϕ) = αβ χαβ nα nβ as a function θ, ϕ in a spherical coordinate. nα , is the αth component of a unit vector n(θ, ϕ) and colors indicate phases of wavefunctions. So in a more conventional representation, these states labeled by value ξ correspond to condensates where all atoms occupy a particular spin-two state, sin ξ |ξ >= cos ξ|2, 0⟩ + √ (|2, 2⟩ + |2, −2⟩). 2  (4.12)  Therefore, the nematic submanifold is effectively a five-dimension space that is characterized by five collective coordinates Xν , ν = x, y, z, ξ, p: three spin rotational angles Xα (α = x, y, z), one spin deformation ξ-angle Xξ (or ξ) specifying spin configurations and one phase angle Xp (= ϕ). Furthermore, it is easy to verify that up to an SO(3) rotation and a phase factor, states at ξ and at ξ + π/3 are equivalent and the fundamental period of this characterization is π/3. When the system is rotationally invariant (i.e. no external fields), we also find that E(ξ) = E(ξ + π/3) andE(ξ) = E(−ξ)(See previous discussion in Chap.3). E(ξ) is the energy of a state defined by ξ. 56  4.5. Effective Hamiltonian for QFCSD  4.5.2  Collective Coordinates and Conjugate Momentum  To study QFCSD, we expand ψ † about a reference condensate χ, √ † ψˆk,αβ = 4M χαβ (ξ) ∑ 1 ∑ ν ˆ† + √ Lαβ (θν (0) + eiq·rk θˆν† (q)). NT ν q̸=0  (4.13)  Here NT is the number of lattice sites and M is the number of atoms per site. The superscripts (or subscripts) ν = x, y, z, ξ, p specify condensate motion along the five orthogonal directions of the submanifold. θˆν (q) (θˆν† (q)) are annihilation (creation) operators of the collective excitations of mode ν with lattice crystal momentum q. One can further include bilinear terms in the expansion. The motion along these five orthogonal directions is studied by introducing five mutually orthogonal matrices Lν , i.e. Tr(Lµ Lν ) = 2δµν . 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 decomposition introduced in Eq.(4.13), we obtain an effective Hamiltonian that is bilinear in terms of θ† and θ. It is convenient to introduce five collective ˜ ν defined in terms of θν† (0), θν (0) for q = 0 modes, Hermitian operators X ˆp = X ˆt = X ˆx = X  θp† − θp √ , 2i N θt† + θt √ , 2 N θx† + θx √ , 4 N (χyy − χzz )  ˆy = X  θy† + θy √ , 4 N (χzz − χxx )  ˆz = X  θz† + θz 4 N (χxx − χyy ) √  (4.14)  where N = NT M is the total number of atoms in a lattice with NT sites. By comparing Eq.(4.13) and the expression for χ ˜ in terms of five spin coordinates Xν (see discussions before Eq.(4.10)), one identifies the semiclassical collective coordinates Xν as the expectation value of harmonic oscillator opˆ ν . In the following, we do not distinguish between X ˆ ν and Xν . erators X 57  4.5. Effective Hamiltonian for QFCSD The conjugate momentum operators Pˆν can be introduced accordingly so ˆ ν , Pˆµ ] = iδνµ . that the usual commutating relations are obeyed, i.e [X ) √ ( † Pˆp = N θˆp + θˆp , ) √ ( Pˆt = −i N θˆt − θˆt† , ) ( √ † ˆ ˆ ˆ Px = −2i N (χyy − χzz ) θx − θx , ) ( √ Pˆy = −2i N (χzz − χxx ) θˆy − θˆy† , ( ) √ Pˆz = −2i N (χxx − χyy ) θˆz − θˆz† . (4.15)  4.5.3  Spectra of Collective Modes  Five modes of zero-point motions (microscopic) around χ(ξ) can also be labeled by the same set of indices. For collective excitations of mode-ν with lattice momentum q(̸= 0), creation operators are θ]ν† (q). These operators [ † obey the bosonic commutation relations, θk,µ , θl,ν = δkl δµν . Note that only the properties of three rotation modes and deformation mode (ξ-mode) depend on the value of ξ. The collective mode dispersion around ξ = 0 or ξ = π/2 has a very simple form, √ Eν,q (ξ, Xξ , qB ) = ϵν,q [2mBN vν2 (ξ + Xξ ) + ϵν,q ]. (4.16) And in the expression for Eν,q , ∑ ϵν,q = 4tL (1 − cos qα dL ) + κν qB  (4.17)  α  is the energy of an atom with crystal quasi-momentum q = (qx , qy , qz ); dL is the lattice constant. mBN = 1/(4tL d2L ) is the effective band mass. κν , ν = x, y, z, ξ are coefficients introduced in Eq.(4.19),(4.20). vν is the sound velocity of the ν-mode in the small-|q| limit, vα2 (ξ) =  M (4bL Gαα (ξ) − cL ) , α = x, y, z, t mBN  Gαα (ξ) and κα are a function    Gxx  Gyy G    zz Gtt  of ξ = = = =  ( sin2 (ξ − sin2 ξ + sin2 (ξ) 0  (4.18)  )  2π 3 ) 2π 3  58  4.5. Effective Hamiltonian for QFCSD  κx    κy  κz   κt  = = = =  χ2xx − χ2zz χ2yy − χ2zz 0 4(χ˙ 2zz − χ2zz )  Here χ˙ zz = dχzz /dξ. 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 ξ or Xξ . And among four gapless spin modes, only the x-,y- and z- spin rotational modes contribute to the ξ-dependence of the quantum fluctuation-induced potential Vqf . Substituting Eq.(4.16) into Eq.(4.19), one obtains the quantum-fluctuation-induced potential Vqf . When the quadratic Zeeman coupling is present, we find that for ξ = 0, κx,y,z,t are not positive defined and collective spin modes can be unstable. To calculate induced-potential Vqf , we only include stable modes (q ̸= 0) which contribute to the renormalization of adiabatic condensate dynamics. At ξ = π/2, all collective modes are stable.  4.5.4  Effective 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 Xν , Pˆν and obtain an effective 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 ̸= 0. The Hamiltonian for the q = 0 sector generates the kinetic energy needed for the dynamics along five orthogonal directions. The corresponding effective 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 field potential VQ . This potential VQ as illustrated below always favors a biaxial nematic with ξ = π/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 ̸= 0 collective modes contains zero-point energies of those modes. These energies in general depend on spin configurations or the values of ξ. So quantum fluctuations of collective coordinates effectively induce a ξdependent 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. Effective Hamiltonian for QFCSD Zeeman coupling. Turner et al also pointed out that thermal fluctuations further enhance the amplitude of induced potentials and this order-fromdisorder phenomenon is robust against finite temperatures[161]. We now study the dynamical consequences of both quadratic Zeeman coupling HQ and zero-point quantum fluctuations Hqf . For simplicity, we are mainly focused on dynamics around a) a uniaxial nematic at ξ = 0 or a condensate with rubidium atoms occupying hyperfine spin state |2, 0⟩; b) a biaxial nematic at ξ = π/2 or a condensate with rubidium atoms √ occupying hyperfine spin state (|2, 2⟩ + |2, −2⟩)/ 2. The resultant Hamiltonian for oscillations around a state ξ (ξ = 0 or π/2) can be cast in the following form, Hqf  =  ∑  Vqf (ξ, Xξ ) =  Pˆν2 + NT (Vqf (ξ, Xξ ) + VQ (ξ, Xν )), 2NT mν ν 1 ∑∑ Eν,q (ξ, Xξ , qB ), 2NT ν q̸=0  VQ (ξ, Xν ) =  ∑ 4M qB cos2 (ξ + Xξ ) + 4M qB κα Xα2 . 3 α  (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 five directions are calculated to be mp = mt = mα =  1 , aL + cL 2M , qB κξ − 2M cL 8M Gαα . 8M bL Gαα − 2M cL + κα qB G−1 αα  (4.20)  κt is a function of ξ and is −4/3 when ξ = 0, and 4/3 when ξ = π/2. Eν,q (ξ, Xξ , qB ) is the energy of a mode-ν (ν = x, y, z, t, p) collective excitation with crystal momentum q, and is a function of parameters ξ, Xξ , and quadratic Zeeman coupling qB . Only spin modes with ν = x, y, z, ξ contribute to the ξ-dependence of potential Vqf . Fluctuations of phase modes are independent of parameter ξ 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 ξ, qB and Xξ ; and VQ is a function of ξ, qB and Xν , ν = x, y, z, ξ. 60  4.6. Potentials Induced by Quantum Fluctuations  4.6  Potentials Induced by Quantum Fluctuations  We first consider a situation where the quadratic Zeeman coupling is absent and the potential VQ vanishes. As argued before, generally speaking Vqf (ξ, 0) = Vqf (ξ + π/3, 0) and Vqf (ξ, 0) = Vqf (−ξ, 0). The explicit form of Vqf calculated here indeed is consistent with this general requirement. We also find that the main contribution to Vqf is from fluctuations of wavelength 1/(mBN vα ) that is much longer than the lattice distance dL . More specifically, the ξ-dependent induced potential can be written as a sum of ∑ d+2 polynomials of spin-wave velocities, i.e. v for d-dimension optical α α lattices. Spin-wave velocities are functions of cL , bL and tL . We numerically integrate over all wavelengths and obtain the ξ-dependence of Vqf as shown in the inset of Fig.4.3. Consequently, we find that in d-dimension lattices (d = 2, 3), the barrier height Bqf which is defined as Vqf ( π6 , 0) − Vqf (0, 0), the energy difference between ξ = π/6 and ξ = 0 satisfies the following simple scaling function, ( ) d+2 bL M |cL | |M cL | 2 Bqf = gd , . (4.21) d cL tL tL2 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 find that the ξ-dependent energy potential is proportional to α vα5 /(d5L t4L ), dL is the lattice constant and tL is the hopping integral. the ξ-dependent ∑ In 42d lattices, 4 3 energy potential Vqf (ξ) is proportional to α vα /(dL tL ) ln tL dL /vα . Eq.4.21 suggests that one can increase the quantum fluctuation-induced potential by increasing cL . This is consistent with our intuition that stronger two-body interactions lead to higher quantum fluctuations. As pointed out in Sec.3.3, the microscopic origin of the zero-point spin fluctuations in condensates is the two-body spin-changing scattering, of which the energy scale is the spin-dependent coupling |cL | in the optical lattices. Alternatively, one can enhance the quantum fluctuations by increasing the ratio ctLL , 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 ρa3s , in which ρ is the density and as is the scattering length. The “gas √ parameter” for spin dependent interaction strength in optical lattices is M |c|3L /t3L . One striking consequence of Eq.4.21 is that one can enhance the quantum fluctuation induced potential very effectively by increasing the optical 61  4.6. Potentials Induced by Quantum Fluctuations  10  Bqf[pK]  1  M=1.0 M=2.0 M=3.0  0.1 Vqf[pK]  0.8  0.01  0 0  0.001 2.0  0.4  4.0  6.0 V[ER]  π/3 ξ 2π/3  8.0  π  10.0  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 ξ-dependence of Vqf , a potential induced by quantum fluctuations when V = 6ER and M = 2.0. potential depth V . As shown in Fig.4.1 |cL | increases and tL decreases as V increases, both of the changes are enhancing fluctuations. When V increases from 2ER to 10ER , |cL | increases by 4 times, and tL becomes one-fifth, 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 find 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 10−5 to 10−4 pK and is negligible in experiments. When the optical lattice potential depth V is varied, the barrier height typically increases by four or five orders of magnitude. Particularly, as V increases from 2ER to 10ER , mean field 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 10−3 pK to a few pK. The energy shift between the uniaxial state at ξ = 0 and biaxial state at ξ = π/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 effective potential Vef f = 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  (a) 0.01  Veff[nK]  0  -0.01  -0.02 0  0.2π  0.4π  ξ (b)  0.6π  V=2ER V=5ER V=8ER V=9ER V=11ER π 0.8π  0.7π ξ 0.5π  0.3π 2  4  6 V[ER]  8  10  Figure 4.4: (Color online) (a) Effective potential Vef f = VQ + Vqf (in units of nK) as a function of ξ 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 ξ 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  (a)  24  M=1.0 M=2.0 M=3.0  8  0 2  M=1.0 M=2.0 M=3.0  16 f0[Hz]  f0[Hz]  16  (b)  24  8  4  8 6 V[ER]  10  0 2  4  8 6 V[ER]  10  Figure 4.5: (Color online) (a),(b), frequencies f0 = Ω0 /(2π) for population oscillations around |2, 0⟩ state are shown as a function of potential depth V in (a) 2D and (b) 3D optical lattices for different atom number density M ; here the quadratic Zeeman coupling is absent. biaxial nematic with ξ = π/2 is stable and as V > Vc , it becomes locally unstable. For 87 Rb 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 ξ = 0. Values of ξ at which global potential minima are found are shown as a function of Vef f 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 ξ = 0 that is selected out by the fluctuation-induced potential Vqf or the biaxial nematic state at ξ = π/2 that is favored by the quadratic Zeeman potential VQ . Particularly we are interested in oscillations along the Xt -direction around state ξ = 0 or π/2. These motions correspond to the following time evolution of condensates, |t⟩ ≈ (cos ξ − sin ξX0 F (t)) |2, 0⟩ + 64  4.7. Oscillations Induced by Quantum Fluctuations 1 √ (sin ξ + cos ξX0 F (t))(|2, 2⟩ + |2, −2⟩); 2 mξ Ω Ω Ω F (t) = cos t + i sin t 2 4M 2 (4.22) where X0 is the initial deviation from state ξ = 0 or π/2 and mξ is the effective mass given in Eq.(4.20). By solving the equation of motion in Eq.(4.19) including the quadratic Zeeman effects, we derive a general expression for oscillation frequencies Ω(= 2πf ). The oscillation frequency Ω(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 ξ = 0 or π/2, we obtain the following frequencies √ 2qB 8M qB ′′ + Vqf )(σ + |cL |); (4.23) Ω = 2 (σ 3 3M ′′  here σ = ∓1 for ξ = 0 and π/2 respectively. Vqf is the curvature of potentials at ξ = 0 or π/2. Note that both the real and imaginary part of F (t) oscillate as a function mξ Ω is exactly equal to one which only occurs when of time; thus unless 4M atoms are noninteracting or the quadratic Zeeman coupling qB is infinite, the magnitude of F (t) oscillates as the time varies. Practically, in most cases we examine below, mξ Ω turns out to be much less than unity because both ′′ the curvature Vqf and the Zeeman coupling qB are smaller than the inverse of effective mass mξ , or the spin interaction energy cL ; thus oscillations of the imaginary part of F (t) can be neglected. Oscillations along the direction of Xξ therefore always lead to temporal oscillations in population of rubidium atoms in either |2, 0⟩ or |2, ±2⟩ states that can be observed in experiments. We first examine the cases when the quadratic Zeeman coupling is absent i.e. qB = 0. The frequency in this limit √ is a direct measure of quantum ′′ fluctuations and Ω0 = Ω(qB = 0, V ) = 2 Vqf |cL |. In the absence of lattice potentials, or in traps we find that oscillation frequencies are about 10−3 Hz. At finite temperatures, thermal fluctuations do enhance oscillation frequencies by a factor of two to four; however, this is far from sufficient for the experimental study of QFCSD within the life time of these isotopes[39]. In optical lattices, we find that the enhancement of spin-dependent interactions 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 Ω0 is closely related to that of Bqf . Taking into account the expression for effective mass mξ , we find Ω0 ∼  M  d+2 4  |cL | d 4  tL  d+4 4  1  gd2 (  bL M |cL | , ). cL tL  (4.24)  In Fig. 4.5, we show the plot of f0 = Ω0 /2π, the oscillation frequency versus V in the absence of Zeeman coupling qB . Experimental studies of QFCSD can also be carried out by investigating dynamics of rubidium atoms in the presence of finite quadratic Zeeman coupling qB . The submanifold dynamics now are driven by both quadratic Zeeman coupling and quantum fluctuations. There are three main modifications to the effective Hamiltonian Hef f . Firstly, according to Eq.(4.20), effective masses now depend on the quadratic Zeeman coupling qB . Secondly, following discussions in the appendix, among four spin collective modes (ν = x, y, z, t) and one phase mode (ν = p), two of spin modes, x- and y-mode, can be gapped and one, z-mode, always remains gapless. Details of spectra of ξ-mode depend on the value of ξ; excitations are unstable when κξ is negative and are gapped when κξ are positive. The energy gap of stable ξ-mode excitations depends on ξ and becomes maximal at ξ = π/2. Finally, a mean field quadratic Zeeman potential that depends on Xξ and Xα , α = x, y, z, is also present and modifies the spin dynamics. Oscillation frequencies Ω 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 ξ = 0, or |2, 0⟩ state. For rubidium atoms, this is the ground state when the quadratic Zeeman coupling is absent, and remains to be locally stable along the direction of Xt up to a finite qc1 . When the quadratic Zeeman coupling qB is much smaller than qc1 , the effective potential Vef f is mainly due to the quantum-fluctuation induced one, Vqf . Above qc1 , a dynamical instability occurs and a perturbation along the Xξ -direction around |2, 0⟩ starts to grow exponentially. We find that in traps where V = 0, qc1 is about 10−2 pK; in optical lattices when V varies between 2ER to 10ER , the value of qc1 increases from 0.1pK up to about 0.1nK. And as qB approaches zero, the oscillation frequency Ω(qB , V ) saturates at the value Ω0 shown in Fig.4.5. The qB -dependence of frequencies for oscillations around ξ = 0 is shown in √ Fig.4.6; near qB = qc1 , the frequencies scale as qc1 − qB . Alternatively, one can also study oscillations around states at which global potential minima 66  4.7. Oscillations Induced by Quantum Fluctuations  (a)  f[Hz]  20  M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER  M=1.0 M=2.0 M=3.0  qC1[nK]  30  (b) 0.15  10 0 0.00  0.05 0.10 qB[nK]  0.00 0.15 2  4  8 6 V[ER]  10  (c) M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER  30 f[Hz]  20 10 0 0.00  0.10 qB[nK]  0.20  Figure 4.6: (Color online) Frequencies f = Ω/(2π) for population oscillations as a function of quadratic Zeeman coupling qB (in units of nK) for various potential depth V in 2D optical lattices with M atoms per lattice site. (a) for oscillations along the Xt -direction around state |2, 0⟩; 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 ξ = ξm (< π/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  M=1.0 V=6ER M=1.0 V=10ER M=3.0 V=6ER M=3.0 V=10ER  20  0 0.0 0.2  10 0 0.0  0.1 0.2 qB[nK]  (b)  100  qC2[nK]  f[Hz]  30  (a)  f[Hz]  40  0.0 2  qB[nK] (c)  1.5  M=1.0 M=2.0 M=3.0  4  8 6 V[ER]  10  Figure 4.7: (Color online) (a), (b) Frequencies f = Ω/(2π) for oscillations along the Xt -direction around biaxial nematic state |2, 2⟩ + |2, −2⟩ 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  10 8  qC1 qC2  Uniaxial  V[ER] 6 Biaxial 4 2 0.00  0.05  qB[nK]  0.10  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 |2, 0⟩ (biaxial nematic or state |2, 2⟩ + |2, −2⟩) are locally stable along the Xξ -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 Vef f 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 ξ = π/2 point or (|2, 2⟩ + |2, −2⟩)/ 2 state. This state becomes stable when qB is larger than qc2 and oscillations are well defined only in this limit. When qB is smaller than this critical value, the dynamics are mainly driven by quantum fluctuations and oscillations are unstable. In the vicinity of qc2 , √ the oscillation frequency again scales as qB − qc2 . When qB is much bigger √ than qc2 but smaller than cL , the frequency is proportional to qB ; in this limit, the potential Vef f is already dominated by the quadratic Zeeman term, or VQ , but the effective 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 external coupling and the frequency Ω approaches 8qB /3, which is equal to the quadratic Zeeman splitting between state |2, 0⟩ and |2, ±2⟩. In this limit, scattering between atoms during a short period of 2π/Ω becomes  69  4.8. Effects of Thermal Fluctuations negligible; the population oscillations due to scattering or interactions are therefore significantly suppressed as here hyperfine spin two atoms are effectively noninteracting. Indeed, following the expression for F (t) in Eq.(4.22) and Eq.(4.20),(4.23), one finds that the amplitude of population oscillations scales as 1/qB at large qB limit and F (t) becomes exp(i4qB t/3) when qB approaches infinite, 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 fluctuations at zero temperature. In the following section, we analyze the effect of temperatures, or thermal fluctuations.  4.8  Effects of Thermal Fluctuations  We now turn to the effect of temperatures and focus on three-dimension optical lattices. At a finite temperature, collective modes including spin waves are thermally excited. The occupation number of αth spin-wave excitations with momentum q and energy Eα,q is 1  n(α, q) = e  Eα,q kT  −1  .  (4.25)  The free energy density (or per lattice site) of a condensate characterized by ξ is ( ) Eα,q 1 ∑ Eα,q − kT + kT ln(1 − e F (ξ, T, V ) = ) . (4.26) NT α,q 2 Just as in the zero temperature case because of the ξ-dependence of spin wave velocities, the free energy density also depends on the values of ξ and the main contribution to the ξ-dependence of free energy density is again from fluctuations of wavelength 1/(mBN vα ), or of a characteristic energy T∗ that is of an order of |cL | (or bL ). Thermal fluctuations become more important than quantum ones when temperatures are much higher than T∗ . The other temperature effect is from the temperature dependence of spin wave velocities vα . Because of the thermal depletion, spin wave velocities vα (α = x, y, z) decrease as temperatures increase and become zero when the BEC temperature TBEC is approached. Therefore the temperature 70  4.8. Effects 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 contributions from these thermal excitations can be calculated by substituting quasiparticle spectra Eα,q approximately with phonon-like spectra vα q. One then obtains the following expression for the free energy per lattice site (up to a constant) F (ξ, T, V ) = Vqf (1 − ( −  T  3  5  )2 )2 TBEC ( ) π 2 kT ∑ dL kT 3 . 90 α vα  (4.27)  The barrier height, Bth (T, V ) = F (ξ = π6 , T, V ) − F (ξ = 0, T, V ), can be calculated accordingly and the asymptotics at low temperatures is Bth (T, V ) − Bth (0, V ) ∼  |cL |5/2 kT 4 ( ) . 3/2 |cL | t  (4.28)  L  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 important at very low temperatures that are of an order of |cL |(|cL |/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 thermally occupied. For modes of energy T∗ , we can approximate n(α, q) with a classical result n(α, q) = kT /Eα,q . Our scaling analysis of the free energy per lattice site shows that in this limit, F (ξ, T, V ) = −  3 3 T 2kT ∑ vα 3 ( ) (1 − ( )2 )2 . 3π α 4dL tL TBEC  (4.29)  The barrier height, Bth (T, V ) = F (ξ = π6 , T, V ) − F (ξ = 0, T, V ), can be calculated accordingly and asymptotics at high temperatures are 5/2  3 3 cL kT T Bth (T, V ) ∼ 3/2 (1 − ( )2 )2 . c T L BEC t  (4.30)  L  71  4.9. Effects of Spin Exchange Losses The above result shows that at relatively high temperatures thermal spin wave fluctuations enhance induced potentials and this enhancement is characterized by a linear function of kT /|cL | or kT /T∗ . However, because of the thermal depletion of condensates at finite temperatures, there is an overall suppression given by a factor 1 − (T /TBEC )3/2 . The competition between these two effects 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 thermal depletion becomes significant. 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 dependent 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 |cL |. This ratio and therefore the enhancement is quite sensitive to the optical lattice potential depth V . We find that the enhancement in the absence of lattice potentials is substantial and thermal fluctuations increase the induced potential by about a factor of fifty. 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 significant. At V = 10ER , we find that the thermal effect only increases the potential by a few tens of percent and the potential can be predominately due to quantum fluctuations.  4.9  Effects 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 effect on the time evolution studied here. Spin exchange losses put a very serve constraint on possible observations of full coherent oscillations driven by quantum fluctuations at frequencies well below 5Hz. To overcome this difficulty, we suggest to apply a quadratic Zeeman coupling close to qc1 (qc2 ) and study time evolution of state |2, 0⟩ or √12 (|2, 2⟩ + |2, −2⟩). 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. Effects of Spin Exchange Losses  (a)  Bqf[pK]  10  V=2ER V=4ER V=6ER V=8ER V=10ER  1  0.1  0.01 0  10  20  30 T[nK] (b)  40  50  60  4  Bqf[pK]  3 2 V=6ER High T asymptotics  1 0 0  5  10  15  20  25  T[nK] Figure 4.9: (Color online) Induced free energy barrier height as a function of temperature for different 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 significant in deep optical lattices.  73  4.9. Effects of Spin Exchange Losses  (a) relative pop. in mF=0  1.0 0.8 0.6 0.4 0.2 0.0 0  1mG 17mG 35mG 43mG 64mG  100  200  300  400  500  300  400  500  200 300 time[ms]  400  500  (b)  relative pop. in mF=0  1.0 0.8 0.6 0.4 0.2 0.0 0  1mG 17mG 35mG 43mG 64mG  100  200  (c)  relative pop. in mF=0  1.0 0.9 0.8  0.6  1mG 5.8mG 9.8mG 12mG 19mG  0.5 0  100  0.7  Figure 4.10: (Color online) Estimated time evolution of rubidium population at |2, 0⟩ hyperfine spin state when the quadratic Zeeman coupling or Zeeman field 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 finite loss rate 1/τ = (200ms)−1 . In (a),(b),(c), magnetic fields which yield dynamical critical quadratic Zeeman coupling qc1 (∼ qc2 ) are 35mG, 35mG and 10mG respectively. 74  4.9. Effects of Spin Exchange Losses Dynamical stabilities √ of |2, 0⟩ state below the threshold qc1 , or instabilities of (|2, 2⟩ + |2, −2⟩)/ 2 state below threshold qc2 are directly induced by quantum fluctuations. Coherent quantum dynamics around state |2, 0⟩ √ ((|2, 2⟩ + |2, −2⟩)/ 2) below and above dynamical critical coupling qc1 (qc2 ) are qualitatively different and it is therefore plausible to probe these dynamics that are driven mainly by quantum fluctuations at relative low frequencies. To explore this possibility, we have studied strongly damped population oscillations around a condensate |2, 0⟩ or a uniaxial state, at different 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 inhomogeneity of rubidium atoms in traps. Main results are shown in Fig.4.10 for different magnetic fields; at a given field, the corresponding quadratic Zeeman coupling is given as qB = 3(µB B/2)2 /(6.8GHz). Our major findings are three-folded and outlined below. Firstly, for a condensate with small deviations from |2, 0⟩ state, because of fast spin exchange losses complete coherent oscillations around |2, 0⟩ are hard to observe. Secondly, dynamics can be dramatically modified by a finite quadratic Zeeman coupling. When the coupling is much less than qc1 (defined 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 |2, 0⟩ always grows initially at short tome scales when the quadratic Zeeman coupling is zero or smaller than qc1 , as a direct consequence of dynamical stabilities of a |2, 0⟩-condensate. By contrast, beyond qc1 the population of atoms at state |2, 0⟩ always decreases initially at short time scales as a result of dynamical instabilities in this limit. These two distinct shorttime behaviors of population at state |2, 0⟩ are signatures of a transition from quantum-fluctuation 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 effective way to probe manybody quantum fluctuations in condensates we have considered. At very large quadratic Zeeman coupling, we have found expected rapid oscillations corresponding to mean field coherent dynamics and we don’t show those results here. Finally, we also estimate the time-dependence of population oscillations at finite temperatures using thermal-fluctuation induced potentials discussed in a previous section. Thermal fluctuations usually speed up the coherent dynamics around state |2, 0⟩ when the quadratic Zeeman cou75  4.9. Effects of Spin Exchange Losses  (a) relative pop. in mF=0  1.00 0.95 0.90 0.85 0.80 0.75 0  T=0 T=0.2TBEC T=0.4TBEC T=0.6TBEC T=0.8TBEC  100  200 300 time[ms]  400  500  400  500  (b) relative pop. in mF=0  1.00 0.80 0.60  0.20  T=0 T=0.2TBEC T=0.4TBEC T=0.6TBEC T=0.8TBEC  0.00 0  100  0.40  200 300 time[ms]  Figure 4.11: (Color online) Estimated time evolution of population at |2, 0⟩ state at different temperatures. (a) is for a magnetic field of 13mG that is below the dynamical critical field corresponding to qc1 and (b) is for 38mG field 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 sufficiently weak (smaller than qc1 ); on the other hand, they can also slow down coherent dynamics around state |2, 2⟩ + |2, −2⟩ 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 dynamical analysis into a regime where oscillation amplitude is substantial. So strictly speaking, results for 35mG, 44mG, 65mG magnetic fields 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 finite temperatures. An important reason for such an estimation is to verify the validity of perturbative calculation that we have carried out in the previous section. The perturbative calculation or Bogoliubov 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 < EVR < 10 the quantum depletion is smaller than 15 , and our perturbative calculation should suffice. Another reason to estimate the condensation fraction is to understand how the fluctuation-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 effective Hamiltonian. Five collective modes on top of a nematic state are described by the following: ( ) ∑ 1 † H= Eν,q θ˜ν,q θ˜ν,q + . (4.31) 2 ν,q † Here θ˜ν,q and θ˜ν,q are bosonic creation and annihilation operators for the ν-th mode. These operators can be obtained by Bogoliubov transformations: † θ˜ν,q = uν,q θν,q + vν,q θν,−q , † ∗ † ∗ θ˜ν,q = uν,q θν,q + vν,q θν,−q .  The coefficients uν,q and vν,q are given: (√ )1/2 2 mBN vν,q 1 1+ +1 , uν,q = √ Eν,q 2 (√ )1/2 2 mBN vν,q 1 vν,q = √ 1+ −1 . Eν,q 2  (4.32) (4.33)  (4.34)  (4.35) 77  4.10. Estimates of Condensate Fraction  1.00 M=1.0 M=2.0 M=3.0  Condensate Fraction  0.95 0.90  1.0 0.8  0.80  C.F.  0.85  0.6 0.4 0.2  0.75 0.70 2.0  0.0 0  5  10 15 T[nK]  4.0  20  6.0 V[ER]  25  8.0  10.0  Figure 4.12: (Color online) Condensation fraction versus optical lattice potential depth V (in units of ER ). The inset shows the condensate fraction at different temperatures when V is equal to 6ER . Condensate deletions from the ν-th mode with crystal momentum q are ( ) † † 2 2 (4.36) ⟨θν,q θν,q ⟩ = u2ν,q + vν,q ⟨θ˜ν,q θ˜ν,q ⟩ + vν,q . † ˜ Here ⟨θ˜ν,q θν,q ⟩ is the occupation number of the νth mode and it is given by the Bose-Einstein statistics: † ˜ ⟨θ˜ν,q θν,q ⟩ =  1 . exp (Eν,q /kT ) − 1  (4.37)  Putting all these together, we reach the final equation to determine the condensate density: ( ) 2 u 1 ∑ ν,q 2 M − M0 (T ) = (4.38) vν,q + Eν,q . NT e kT − 1 ν,q̸=0  M is the atom number density or number of atoms per lattice site and M0 is the condensed number density. The first 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 2 which At zero temperature, quantum depletion is given by the term vν,q captures the effect of interactions or two body scattering processes. Our estimate shows that ( ) ∑ 1 mBN vν 3 M0 1− ∼ M M tL d L ν=p,ξ,x,y,z ) ( 1 M aL 3/2 . (4.39) ∼ M tL  We have noticed that major contributions are from the phase fluctuations since aL ≫ |bL |, |cL |. In optical lattices when the potential depth V increases, the bandwith tL decreases exponentially as a function of V and aL increases; as a result, quantum depletion grows as ( atLL )3/2 . At finite temperatures, M0 decreases as temperature T increases and becomes zero at the transition temperature TBEC . In the weakly interacting limit (tL >> aL , bL , |cL |), the condensate fraction is approximately equal to M0 (T ) =1− M0 (T = 0)  (  T TBEC  )3 2  ,  (4.40)  2/3  and TBEC ∼ 5M0 tL in optical lattices.  4.11  Conclusion  In conclusion, we have demonstrated that unlike the usual mean-field-interaction driven coherent dynamics, QFCSD is a novel class of coherent dynamics fully driven by quantum fluctuations. 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 fluctuations have both distinct lattice-potential dependence and quadratic Zeeman-coupling dependence that can be studied experimentally. For rubidium atoms, QFCSD can be directly probed either at frequencies of a few Hz or when the quadratic Zeeman coupling is about 10−1 nK. One of potential applications of QFCSD perhaps is the possibility of performing precise measurements of strongly correlated quantum fluctuations and critical exponents 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-fluctuation controlled dynamics might 79  4.11. Conclusion be an alternative and promising path towards probing critical correlations. Finally, we have also investigated the effects of finite temperatures, spin exchange losses and studied time evolution of condensates mainly driven by quantum fluctuations.  80  Chapter 5  Half Quantum Vortex (HQV) in F = 1 Superfluids 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 conventional superfluid, 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 (1) [125] in the cyclic phase of F = 2 spinor condensates such as SO(3)×U T 1 [123] in nematic phase of F = 1 spinor sysmentioned in Chap.3 or S2Z×S 2 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 superfluid 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 fractionalization phenomena due to strong spin fluctuations in two-dimension superfluids. Quantum number fractionalization has been one of the most fundamental and exciting concepts studied in modern many-body physics and topological field theories[172, 173, 174, 175]. In more recent theoretical studies of strongly correlated two-dimension electrons [176, 177, 178, 179, 180, 181], spin-charge separated fractionalized excitations in spin liquids and the underlying topological order have also been focuses of many investigations although their experimental realization is still under exten81  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 carrying 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 superfluids of ultracold sodium atoms which can potentially be studied in experiments[38]. Particularly, we investigate fluctuation-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 first part of this chapter I describe the structure and interactions of HQVs in the spinor condensates using mean field theory. In the following part I study the HQVs with strong fluctuations in 2D spinor superfluids. Finally I study the nucleation of the HQVs in 2D spinor superfluids 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 BoseEinstein 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 Γ around a vortex line ∫ (5.1) C= dl · vs Γ  is quantized in units of 2π 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 circulation C = ±2, ±3, ... spontaneously splits into a few elementary ones which interact via long-range repulsive potentials. A configuration 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 twodimension configuration, where the condensate phase angle Φ(r, θ) rotates slowly and uniformly by 180◦ in the r − θ plane around a vortex center but jumps from π to 2π when the polar angle θ is equal to 2π. The πphase jump here effectively induces a singular cut in the wavefunction. The corresponding circulating velocity field is simply vs (r, θ) =  eθ (5.2) 2mr , leading to C = 1/2 that is one-half of an elementary value. The energy of a cut per unit length along θ = 2π line where the phase jumps is finite 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 confines 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 confinement of fractionalized vortices. Hyperfine-spin degrees of freedom can drastically change the above arguments about elementary vortices. In condensates of sodium (23 Na) or rubidium (87 Rb) atoms in optical traps [38], hyperfine spins of cold atoms are correlated because of condensation. A pure spin defect (or a spin disclination in the case of 23 Na) where spins of cold atoms slowly rotate but no supercurrents flow, can carry a cut, i.e. a line along which a π-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) configuration, which consequently leads to a linear confining potential between the spin defect and HQV(see Fig.5.1(c) and (d)). For instance, a HQV with C = 1/2 confined 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 briefly 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 lo83  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 ∫ ∫ ∫ − 2 ∇2 c2 c0 † 2 ˆ H = drψα (r) ψα (r) + drS (r) + drˆ ρ2 (r). (5.3) 2m 2 2 Here ψα† (ψα ), α = x, y, z are creation (annihilation) operators for sodium atoms in hyperfine states |α⟩; they are defined as linear superpositions of creation operators for three spin-one states, |1, mF ⟩, mF = 0, ±1. ψx† = √ √ † † (ψ1† − ψ−1 )/ 2, ψy† = (ψ1† + ψ−1 )/i 2 and ψz† = ψ0† . c0,2 are interaction parameters that depend on two-body s-wave scattering lengths a0,2 for total spin 0,2: c0 = 4π 2 (a0 + 2a2 )/3m and c2 = 4π 2 (a2 − a0 )/3m. For condensates of 23 Na atoms, a0 ≃ 50 aB and a2 ≃ 55 aB (aB is the Bohr raˆ α = −iϵαβγ Ψ∗ Ψγ and ρˆ = ψα† ψα are local spin-density and density dius). S β operators(see Chap.2). Let’s first find out the mean field ground state of the system. Generally, a condensate wavefunction Ψα = ⟨ψα† ⟩  (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 Ψ = exp (iΦ)  √  ρ n.  (5.5)  Here n is a unit vector with three components nα , and ρ is the number density of sodium atoms. This spin nematic state is invariant under an inversion of n and a π-phase shift [123] n → −n  (5.6)  Φ → Φ+π  (5.7)  The spinor condensate specified in Eq.5.5 is a spontaneously symmetry breaking state which breaks both the gauge symmetry and rotational symˆ metry. Although local spin densities ⟨S(r)⟩ in a nematic state vanish, a nematic condensate carries a spin quadrupole moment defined as ˆ αS ˆ β ⟩ − 1 δαβ ⟨S ˆ 2 ⟩, Qαβ (r) = ⟨S 3  (5.8)  85  5.2. HQV in Spinor Bose-Einstein Condensates  (a) ρ−1 ρ+1 ρ+1−ρ−1  16 interaction energy [µ]  density  0.8  (b)  0.6 0.4 0.2  8 4  0.0  -1.0 -40 -20  0 20 r [ξ]  40  Vpm Vpp  0  0.0 -20  12  1.0 ρ+1−ρ−1  1.0  -10  0 r [ξ]  10  20  0  10  20 d [ξ]  30  40  Figure 5.2: (color online). (a) Density (ρ±1 = |ψ±1 |2 ) and spin-density (ρ1 −ρ−1 ) profiles of an individual plus half-quantum vortex (HQV) centered at r = 0. (b) Interaction potentials (in units of chemical potential µ) 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 = 30ξ, ξ is the healing length. which can be calculated and is specified by the nematic unit director n introduced above: ( ) 1 Qαβ = ρ nα nβ − δαβ . (5.9) 3 From the experimental point of view, the nematic spin vector has a special meaning that all the atoms in the condensate are in the m′F = 0 level where m′F 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 − θ plane rotate by 180◦ forming a spin-disclination. The corresponding condensate wavefunction far way from the vortex core is ( ) ( ) iθ √ θ θ Ψ(θ, r → ∞) = exp ρ cos , sin , 0 . (5.10) 2 2 2 86  5.2. HQV in Spinor Bose-Einstein Condensates ( ) The 180◦ rotation of nematic director n(θ) = cos 2θ , sin 2θ , 0 around the vortex illustrates that a π-spin disclination where spin quadrupole moments Qαβ (θ) slowly rotate is indeed confined to this HQV. In the Zeeman basis of |1, mF ⟩ states, mF = 0, ±1, the above vortex state is equivalent to √ ρ ψ+1 = exp(iθ)f (r) , (5.11) 2 √ ρ ψ−1 = −g(r) (5.12) , 2 ψ0 = 0. (5.13) When it is far away from the vortex core, f (r → ∞) = g(r → ∞) = 1. The core structure can be studied by numerically solving the multiplecomponent Gross-Pitaevskii (GP) equation ] [ 2 − ∇2 + (c0 + c2 )ρ±1 + (c0 − c2 )ρ∓1 ψ±1 = 0, (5.14) 2m here ρmF = |ψmF |2 . The corresponding boundary conditions when r → ∞ 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 |1, ±1⟩ 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, |1, 1⟩ atoms state are completely depleted while |1, −1⟩ atoms are not. This is because supercurrents are only present in |1, 1⟩ component; in fact the density of |1, −1⟩ atoms has an additional small bump at the center of core, to further take advantage of the depletion of |1, 1⟩ atoms in the same region to minimize the overall repulsive interactions between |1, ±1⟩ atoms. So the core has a nonzero local spin ˆ z (r)⟩(= |ψ1 |2 − |ψ−1 |2 ) with excess atoms at |1, −1⟩ state and we density ⟨S define it as a minus HQV. Similarly, one can construct a plus HQV with an identical vorticity (i.e. ∇ × vs ) but with excess |1, 1⟩ 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 corresponding interaction potential Vpp(mm) is a logarithmic long-range one due to interference between coherent supercurrents. However, the interaction between 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, supercurrents flow in different components and they don’t interfere; the short-range interaction is entirely due to inter-component interactions between |1, 1⟩ and |1, −1⟩ atoms. Indeed, the amplitude of potential Vpm is proportional to, when it is small, c0 − c2 which characterizes mutual interactions between |1, ±1⟩ 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 θ 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, ∫ 2 |∇Ψ|2 (5.15) Ekin ≃ dr 2m ∫ 2ρ ( ) |∇θ|2 + |∇n|2 . ≃ dr (5.16) 2m The first 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]): C1 C2  π  2L ρ z  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 ξ 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 C˜1 and C˜2 for spin vectors. Therefore the interaction between two HQVs is ) π 2 Lz ρ R( Vint (d) ≃ log C1 C2 + C˜1 C˜2 . (5.18) 2m d For example, in Fig.5.1(c), we have C1 = C2 = C˜1 = C˜2 = 21 , 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 C˜1 = −C˜2 = 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 coupled GP equations of spin-one BEC 2 ∂ψ±1 = [− ∇2 + Vtr − µ ∓ λ − ΩLz + c0 ρ ∂t 2m + c2 (ρ±1 + ρ0 − ρ∓1 )+W± ]ψ±1 +c2 ψ02 ψ¯∓1 , 2 ∂ψ0 (i − γ) = [− ∇2 + Vtr − µ − ΩLz + c0 ρ ∂t 2m + c2 (ρ1 + ρ−1 )]ψ0 + 2c2 ψ1 ψ−1 ψ¯0 , (5.19) ∑ where ρ = mF ρmF is the total condensate density, Vtr (r) is a spin-independent confining potential of an optical trap, and W± (r) are pulsed magnetic trapping potentials which we further apply in order to create HQVs. µ and λ are the Lagrange multipliers used to preserve the total number and magnetization of atoms respectively; γ 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 λ = ω⊥ /ωz ∼ 14 which was also used in early experiments [50]. We consider a two-dimension cylindrical trap which is characterized by two dimension(a2 −a0 ) 0 +2a2 ) less parameters C0 = 8πN (a and C2 = 8πN 3L , with Lz the size of 3Lz z 6 system along the z-axis and N = 3 × 10 the total number of sodium atoms. When combined with a nonaxisymmetric dipole potential that can be created using stirring laser beams [50, 51], the optical trapping potential in a 2 {(1 + ϵ)x2 + (1 − ϵ)y 2 }/2. Here ω = rotating frame is given by Vtr (r) = mω⊥ ⊥ 2π×250 Hz, and anisotropic parameter is set to be ϵ = 0.025. We also include 2 (x2 + y 2 )/2, an additional magnetic trapping potential: W± (r) = ∓βmω⊥ which could be realized in an Ioffe-Pritchard trap via a Zeeman splitting mF gF µB B with the Land´e factor gF =−1 for sodium atoms. We start our numerical simulations with an initial state where |1, ±1⟩ are equally populated. Experimentally, it was demonstrated that when the bias field is small and the gradient field along the z-axis of the trap is almost canceled, such a state can be prepared and the |1, ±1⟩ components are completely miscible (however, with the immiscibility between |1, ±1⟩ and |1, 0⟩ components) [38]. We then study the time evolution of this initial  (i − γ)  89  5.2. HQV in Spinor Bose-Einstein Condensates  Figure 5.3: Creation of a half-quantum vortex. Density profiles of |ψ+1 |2 , |ψ−1 |2 , |ψ+1 |2+|ψ−1 |2 , and spin-density profile |ψ+1 |2 –|ψ−1 |2 (bold line) are shown. The rotating frequency is suddenly decreased from an initial value Ω= 0.65ω⊥ to Ω= 0.3ω⊥ 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 off. state using unit of length √ the Crank-Nicolson implicit scheme [166]. The −1 is ah = /2mω⊥ = 0.48 µm and the period of the trap is ω⊥ = 4 ms; the interaction parameters are C0 =500 and C2 = 450 and the damping rate is γ = 0.03. Further, we include symmetry breaking effects in our simulations by allowing the trap center to randomly jump within a region [−δ, δ]×[−δ, δ] (δ = 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 indicate that dynamic instabilities for creation of integer vortices in rotating BECs occur almost at same frequencies as for HQVs and a triangular integervortex 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 additional optical trapping potential with an oscillating trapping frequency to effectively shake them, indicating their metastability. For this reason, a time-dependent magnetic trapping potential with harmonic form is applied; and we find that when a pulsed magnetic field with β > 0.005 is applied, HQV lattices could be formed. Here we set β = 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, |1, 1⟩ component further spreads to the edge, while |1, −1⟩ component remains at the center of the trap and surfaces of equally populated |1, ±1⟩ components become mismatched with two different 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 Ω = 0.65ω⊥ and the trap anisotropy ϵ is increased rapidly from zero to its final value 0.025 in 20 ms. At t = 800 ms, only one vortex in |1, −1⟩ component appears. Afterwards, We decrease Ω to 0.3ω⊥ suddenly, and switch off the magnetic trapping potential adiabatically within 200 ms. We then find a stable single HQV formed at t = 1600 ms as shown in Fig.5.3. The 0.3ω⊥ 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 Ω = 0.7ω⊥ starts abruptly and anisotropy ϵ is set to its final 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 |1, 1⟩ component of the condensate, while no surface oscillations appear on the surface of |1, −1⟩ component. At t = 240 ms, we find that the density profile of |1, 1⟩ component is along a short-axis while the |1, −1⟩ component is along a long-axis due to repulsive interactions between two components, and the surface of |1, −1⟩ component is not always buried in the inner region of |1, 1⟩. The surfaces of two components oscillate independently and are decoupled dynamically. At t = 430 ms, we find that two plus HQVs with excess |1, 1⟩ atoms inside cores have nucleated at the center. Correspondingly, we observe two small regions near the center where |1, −1⟩ atoms are completely depleted and the density of |1, 1⟩ atoms remains high. At t = 800 ms, two components are phase separated, but the structure of HQV cores remains almost unchanged. Finally, we switch off the additional magnetic potential adiabatically within 200 ms and a HQV lattice with interlaced square configuration 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 |1, ±1⟩ 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 Ω = 0.7ω⊥ with a magnetic trapping potential on until t = 800 ms; afterwards, the magnetic trap is adiabatically switched off within 200 ms. The bottom panel shows the half-quantum vortex lattice formation at t = 1600 ms; a square lattice in the spin-density profile is clearly visible. 92  5.3. HQV in 2D Superfluid 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 bipartite 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 23 Na 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 field 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 confined to a spin-density-wave structure at relatively low frequencies; this structure can be conveniently probed by taking absorption images of ballistically expanding cold atoms in a Stern-Gerlach field [38].  5.3  HQV in 2D Superfluid of Ultracold F = 1 Atoms  In this section we will examine the fluctuation-driven vortex fractionalization and topological order in 2D nematic superfluids of cold sodium atoms. The Monte Carlo simulations suggest that a softened π-spin disclination be confined to a half-quantum vortex when spin correlations are short ranged. Both this result and direct calculations of winding-number operators indicate that a non-local topological spin order emerges simultaneously as cold atoms become a superfluid below a critical temperature. We have also obtained fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps and discussed probing these excitations in experiments. In the following we will first introduce the system and Hamiltonian, and get an insight from a gauge field formulation, then we will present the results from numerical simulations and finally the nucleation of HQVs in rotations traps.  5.3.1  Effective Hamiltonian  We employ the Hamiltonian introduced previously for F=1 sodium atoms in optical lattices[129, 128], H =  ∑ bL k  2  ρˆ2k +  cL ˆ2 S 2 k  93  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms − tL  ∑  † (ψk,α ψl,α + h.c.) −  ⟨kl⟩  ∑  µˆ ρk .  (5.20)  k  Here k is the lattice site index and ⟨kl⟩ are the nearest neighbor sites. µ is the chemical potential and tL is the one-particle hopping amplitude. Two 2 ∫ coupling constants are bL (cL ) = b(c) 4πm dr(ϕ∗w (r)ϕw (r))2 ; b, c are effective s-wave scattering lengths, ϕw is the Wannier function for atoms in a periodical potential. Operators ψα† , α = x, y, z create hyperfine spin-one atoms in √1 (|1⟩ − | − 1⟩), −i √1 (|1⟩ + | − 1⟩) and |0⟩ states respectively. The spin and 2 2 number operators are defined as Sˆα = −iϵαβγ ψ † ψγ , and ρˆ = ψα† ψα . Spin β  correlations are mainly induced by interaction cL Sˆ2 . Minimization of this † spin-dependent interaction requires that the order parameter Ψα (= ⟨ψk,α ⟩) √ be a real vector up to a global phase, i.e. Ψ = N n exp(iχ) where n is a unit director on a two-sphere, exp(iχ) represents a phase director and N is the number of atoms per site. All low energy degrees of freedom are characterized by configurations where n and χ vary slowly in space and time[128]. Low lying collective modes include spin-wave excitations with √ energy dispersion√ω(q) = vs q, vs = cL tL a and phase-wave excitations with ω(q) = vp q, vp = bL tL a (here a is the lattice constant). In one-dimensions, low energy quantum fluctuations destroy spin order leading to quantum spin disordered superfluids[123]. In two-dimensions, the amplitude of quantum spin fluctuations is of order of cL /tL and is negligible in shallow lattices as tL is order-of-magnitude bigger than cL . At finite temperatures, spin correlations are mainly driven by long wave length thermal fluctuations, analogous to quantum 1D cases. This aspect was also paid attention to previously and normal-superfluid transitions were investigated[191]. We therefore study the following Hamiltonian which effectively captures long wave length thermal fluctuations ∑ H = −J (nk · nl ) (Φk · Φl ) ; (5.21) ⟨kl⟩  here states at each site are specified by two unit directors: a nematic director, n = (cos ϕ0 sin θ0 , sin ϕ0 sin θ0 , cos θ0 ) and a phase director, Φ = (cos χ, sin χ). J = 2N tL is the effective coupling between two neighboring sites and depends on N , the number of atoms per site. The model is invariant under the following local Ising gauge transformation: ni → si ni , Φ → si Φi , and si = ±1. In the following, we present results of our simulations on 2D superfluids, especially spin structures, energetics of HQVs and nucleation of HQVs in rotating optical traps using the effective Hamiltonian in Eq.5.21. 94  5.3. HQV in 2D Superfluid 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 field model in which the softened spin disclination is represented by an excitation in the Z2 gauge fields. The Z2 lattice gauge model has also been introduced for spin singlet condensates in low dimensional optical lattices [128, 129], where the Z2 gauge field 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 (Φi · Φj )(ni · nj ) (5.22) ⟨ij⟩  Here Φi and Φj are phase vectors and their inner product is given by cos (Φi − Φj ). ni is the nematic spin vector at site i. The summation runs over all the nearest neighbor sites ⟨ij⟩. J is a positive coupling constant which measures the strength of the phase and nematic vector twisting energy. Let’s first look at the continuum limit of the system where only fluctuations of long wavelengths are concerned. In this case we can treat n and Ψ as a smoothed function of space coordinates: ( ) ( ) ∂Φ a2 (Φi − Φj ) a2 ∂Φ · Φi · Φj ≃ 1 − ≃1− (5.23) 2 a2 2 ∂x ∂x ( ) ( ) a2 (ni − nj ) a2 ∂n ∂n ni · nj ≃ 1 − ≃1− · (5.24) 2 2 a 2 ∂x ∂x The effective Hamiltonian in the continuum limit is (up to a constant) ( ) ∫ J J 2 2 2 H≃ d x |∇n| + |∇Φ| . (5.25) 2 2 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 Superfluid 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 effective theory. Although the nonlinear sigma model captures the long-wavelength physics of spin and phase fluctuations, HQVs are missing in the continuum model in which spin and phase are decoupled. To obtain a low energy effective Hamiltonian which allows for HQV structures, we can introduce a gauge field by using the Hubbard-Stratonovich transformation[192]. We introduce a real-valued auxiliary gauge field σij defined on bonds ⟨ij⟩ in the lattice. The effective Hamiltonian with the gauge field can be defined as [128]: J∑ 2 σij + (Φi · Φj )2 + (ni · nj )2 2 ⟨ij⟩ ∑ + J σij (Φi · Φj ) + σij (ni · nj )  H(σ) =  (5.26)  ⟨ij⟩  It can be shown that the gauge field Hamiltonian listed above is equivalent to the original one. In fact, an integration of the σ field in the new Hamiltonian will recover the original Hamiltonian as ∫ −βH (5.27) e ∼ D[σ]e−βH(σ) . The integration on the right hand side is performed straightforwardly by noticing that it can be decoupled to independent Gaussian integrals. We can define a local gauge transformation associated with the Z2 symmetry in the nematic representation. The Hamiltonian is invariant under the following transformation: ni → ri ni  (5.28)  Φi → ri Φi ,  (5.29)  σij  → ri rj σij  (5.30)  Here ri = ±1. ˜ includes phase vectors and gauge fields can An effective Hamiltonian H 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 field approximation. Namely the correlation between ni and nj can be replaced by a vector coupled to a effective field which is the average of all the other nematic vectors in the system. We assume that 96  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms ⟨ni ⟩ = xez , where −1 ≤ x ≤ 1. The correlation term can be approximated as: ni · nj  = ⟨ni ⟩ · nj + ⟨nj ⟩ · ni − ⟨ni ⟩ · ⟨nj ⟩.  (5.31)  For simplicity let’s first choose the gauge field to be σij = σ. The Hamiltonian of the nematic spin fields is ∑ 2Jx2 n2i,z − 4Jσxni,z − Jx4 − 2Jσx2 (5.32) Hn = i  =  ∑  ( ) −ni,z 4Jσx − 2Jx2 ni,z − . . .  (5.33)  i  Here (4Jσx − 2Jx2 ni,z ) plays a role of an effective field coupling to nematic spin vector. The value of x can be determined from the self consistent equation x = ⟨ni,z ⟩T ] ) [ ( ∫ dni,z ni,z exp βJ 4σx − 2x2 ni,z ni,z ∫ = dni,z exp [βJ (4σx − 2x2 ni,z ) ni,z ]  (5.34)  Numerical solutions of the above self consistent equation are shown in Fig.5.5. There are two different phases in this system, and a phase transition occurs when we tune the parameter to across the boundary line. A ferromagnetic phase with nz ̸= 0 appears at low temperatures or for large σ values where the thermal fluctuations are too small to destroy the spin orders and the effective molecule field is large enough to hold the spin order; a paramagnetic phase appears in the opposite limit: high temperatures and low σ field. The phase transition line separated two phases is exactly given by the following: 3 βJc σ = . 4  (5.35)  After the integration of spin degrees of freedom the free energy of the system now is F Fn  ∑ J (Φi · Φj − σ)2 + Fn (βJ, σ) 2 <ij> (∫ ) ∑ −βHn = −T ln dni,z e =  (5.36) (5.37)  i  97  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms  3.0 0 0.12 2.5  0.24 0.36 0.48  2.0  0.60 0.72  J  0.84 1.5  0.96  1.0  0.5  0.0 0.0  0.5  1.0  1.5  2.0  2.5  4  5  3.0  -2 -3  free energy  -4  βJ=0.1 βJ=0.2 βJ=0.3 βJ=0.4 βJ=0.5 βJ=0.6  -5 -6 -7 -8 -9 0  1  2  3 σ  6  Figure 5.5: (a) Solution of the self consistent equation at different temperatures βJ and for gauge field of different strength σ. (b) Free energy as a function of σ at different temperatures.  98  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms Here Fn is the free energy due to the fluctuations of the spin vectors. Given βJ and σ, one first 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 βJ and σ only. The overall free energy F as a function of βJ and σ can be obtained by minimizing with F with respect to Φi · Φj first with a given σ. The resultant free energy as a function of σ at different temperatures is plotted in Fig. 5.5. The free energy’s dependence on σ is very different at low temperatures and high temperatures. The average value of the sigma field ⟨σ⟩ 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 field and phase vectors at low temperatures. In another words, it suggest a nonzero σij (Φi · Φj ) coupling term other the usual (Φi · Φj )2 ˜ phase correlation term in the low energy effective Hamiltonian H. In the above heuristic argument we consider the uniform gauge field σij = ⟨σ⟩ which we believe to describe be the low energy states. One may expect that a gauge field with σij = ±σ is close to uniform gauge field in their free energy, since a gauge transformation can change the sign of σij . σij appear to be relevant In this case, the sign of the gauge field sij = ⟨σ⟩ low energy degrees of freedom. One can obtain the effective Hamiltonian in terms of sij and Φi as [128] ] ∑[ ∑∏ ˜ = − ˜ ij (Φi · Φj ) + K ˜ H J˜2 (Φi · Φj )2 + Js sij . (5.38) ⟨ij⟩  ij∈  ˜ J˜2 and K ˜ are renormalized coupling constants. The last term is the Here J, energy of pure gauge fields defined as the product of sij on four edges of a plaquette. In the language of the gauge field model, a HQV can be viewed as a combination of a gauge defect and a π phase winding. In Fig.5.6 we plot the phase vectors and the gauge fields around a single HQV. A gauge defect located at the center is put in the center by setting σij = −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 configuration won’t raise too much energy. In fact, the energy cost of adding a HQV in this system is approximately π J˜2 d2 log  π˜ 2 R R ˜ + Jd log + K. ξ2 4 ξ  (5.39)  Here R is the system’s size, d is the lattice constant, and ξ is the core size ˜ term is coming from the gauge defect located at the of the vortex. The K 99  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms  Figure 5.6: A HQV in the Z2 lattice gauge model. Phase vectors Φi are plotted as red arrows. Gauge fields with σij = −1 are marked in green lines and σij = 1 are not marked. center and is originating from the spin fluctuations 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 finite contribution of spin fluctuations in the 2D superfluids 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 = sij . (5.40) ⟨ij⟩∈C  Here C is a closed loop add ⟨ij⟩ are edges on the loop. For a given gauge field, 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 confinement or deconfinement of the gauge fields, 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 field confinement 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 effective gauge model Eq.5.38 [193] shows that there is indeed a gauge confinement phase when the temperature is not too high. The above heuristic arguments illustrated the central idea in this sec100  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms tion: a HQV can be stabilized by a gauge defect in two-dimension spinor superfluid. Considering the gauge defect, a quantity defined globally (such as quantum circulation), is actually coming from short-ranged spin correlations, 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 simulation 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 (θ, ρ), a HQV √ in condensates is represented by Ψ(θ, ρ) = N (ρ) exp(iθ/2)n(θ), with n = (cos(θ/2), sin(θ/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 fluctuations, we carry out Monte Carlo simulations on a square lattice of 128 × 128 sites and study spatial correlations between a HQV and a π-spin disclination, and topological order. We first identify critical temperatures of the normal-superfluid phase transition by calculating correlations and the phase rigidity. The gaugeinvariant quadrupole-quadrupole correlation functions we have studied are s,p f s,p (r1 , r2 ) = ⟨Qs,p αβ (r1 )Qαβ (r2 )⟩.  (5.41)  Here Qsαβ (r1 ) = n1,α n1,β − (1/3)δαβ , α = x, y, z; Qpαβ (r1 ) = Φ1,α Φ1,β − (1/2)δαβ , α = x, y. In simulations, we have studied these correlation functions and found that the phase correlation length for f p (r1 , r2 ) becomes divergent at a temperature 0.35J which is identified as a critical temperature Tc . We also calculate the phase rigidity or the renormalized phase coupling Jp  Jp =  ∂2F ; ∂δχ2  (5.42)  here δχ is a small phase difference applied across the opposite boundaries of the lattice and F is the corresponding free energy. We indeed find 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 find that the spin correlation length diverges only at T = 0 (see Fig.5.7a). Our simulations for 101  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms (a)  (b)  1 10  1  3  0.5 10  2  0 0  10 10  1  0.2 0.4 T/J  0.6  0  0.35  0.40 T/J (c)  0.45  0.1  0.01  1  1  0.5  0.5  0  0  -0.5  -0.5  -1  -1 0  64  128 192 loop size (e)  256  1  0  64  128 loop size (d)  192  256  0  64  128 loop size (f)  192  256  0  64  128 loop size  192  256  1 0.5 0  0  -0.5 -1  -1 0  64  128 loop size  192  256  Figure 5.7: a)Spin(circle) and phase(square) correlation lengths versus temperatures; inset is for the renormalized phase √ coupling constant Jp (in units of J). Dashed lines are the fits to exp(A/ T − Tc ) for phase correlations with Tc ≈ 0.35J and exp(B/T ) for spin correlations. b) Thermal average ⟨Ws ⟩(circle) and ⟨Wp ⟩(square) versus the loop perimeter at T = 0.33J(solid line) and T = 0.45J(dashed line). c)-f): ⟨Ws,p ⟩hv and ⟨C⟩hv (diamond) averaged over configurations with half-vortex boundary conditions. c)-d),f) are for T /J = 0.45,0.33,0.20 respectively; in e), we also show ⟨Ws,p ⟩hv normalized in terms of back ground values ⟨Ws,p ⟩bg at T = 0.33J. The background values ⟨Ws,p ⟩bg (dashed lines) are obtained by averaging over configurations with a uniform phase distribution at the boundary. 102  5.3. HQV in 2D Superfluid 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 fluctuating back ground, we introduce the following gauge invariant π-rotation checking operator, which is essentially a product of sign-checking operators ∏ sign(nk · nl ), Ws = ⟨kl⟩∈C  Wp =  ∏  sign(Φk · Φl ).  (5.43)  ⟨kl⟩∈C  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 π-spin disclination (HQV). The gauge invariant circulation of supercurrent velocity (in units of π /m) is defined as C=  1∑ sign(nk · nl ) sin(Φk − Φl ). π  (5.44)  ∈C  This quantity is equal to one in a HQV. In our simulations, we investigate the winding number ⟨Ws,p ⟩hv averaged over configurations where phase directors rotate by 1800 around the boundary of the lattice and the center plaquette. At temperatures above the normal-superfluid 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 configuration in the absence of phase rigidity. Below Tc , the circulation C is averaged to one indicating that the boundary conditions effectively project out HQV configurations. Meanwhile, we observe loopperimeter dependent ⟨Ws,p ⟩hv which can be attributed to the background fluctuations of HQV or disclination pairs. The loop-perimeter dependence of ⟨Ws,p ⟩hv here is almost identical to that for uniform boundary conditions, i.e. the back ground value. After normalizing ⟨Ws,p ⟩hv in terms of ⟨Wp ⟩hv s ⟩hv background winding numbers ⟨Ws,p ⟩bg , we find both ⟨W ⟨Ws ⟩bg and ⟨Wp ⟩bg approach −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 sufficiently short compared to the size of the lattice. At further lower temperatures, the spin correlation length becomes longer than the lattice size and fluctuations of pairs of disclination103  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms anti-disclination are strongly suppressed; ⟨Ws,p ⟩hv are equal to −1 for almost all loops, which corresponds to a mean field result. Results in Fig.5.7 indicate that a HQV is confined to a softened spin disclination and vice versa to form a fundamental excitation. Thus, πdisclinations 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 Qsαβ vanish because of strong fluctuations, an overall π-rotation of nematic directors in disclinations is still conserved because of a coupling to the superfluid component. This coupling between a HQV and disclination can also be attributed[137] to a coupling between Higgs matter and discrete gauge fields[194]. Furthermore, the absence of unbound π-disclinations in superfluids indicates a topological order, 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 verified by examining the average of product-operator Ws,p over all configurations (with open boundaries). Above the normal-superfluid transition temperature Tc , we again find that Ws,p both are averaged to zero within our numerical accuracy implying proliferation of unbound HQVs or disclinations. Below Tc , we study the loop-perimeter dependence of average winding numbers ⟨Ws,p ⟩ and find that both ln⟨Wp ⟩ and ln⟨Ws ⟩ are linear functions of loop-perimeter analogous to the Wilson-loop-product of deconfining gauge fields[195]; if there were unbound disclinations, one should expect that ln⟨Wp ⟩ 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), 0 Fh.v. (r) = Fh.v. (r) − ΩLz (r).  (5.45)  0 (r) is the free energy of a HQV located at distance r from the trap Here Fh.v. axis in the absence of rotation, Lz (r) is the angular momentum of the vortex state and Ω is the rotating frequency. In a 2D lattice without a trapping  104  5.3. HQV in 2D Superfluid of Ultracold F = 1 Atoms (a)  (b)  2000  80 0 -100 35  40  60  45 Ωc [Hz]  0  -1000  -2000 -64  D  Fh.v.[nK]  1000  0 0.08ωtr 0.23ωtr 0.32ωtr -32  32  64  0.5 0.4 0.3 0.2 0.2  0.3 T[J]  0.4  40  20  0 r  Ratio  100  0 0.1  HQV IQV  0.2  0.3  0.4  T[J]  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 ωtr = 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 ΩD c (solid line) varies from 0.32ωtr (or 38Hz) to 0.17ωtr (or 20Hz) as the temperature T increases. As a reference, we also show ΩD c versus T for integer quantum vortices (dashed line). Inset is the temperature-dependence 0 of the ratio between the HQV energy Fh.v. and the IQV energy Fv0 . 0 = (πJ /4+πJ /4) ln(L/a), with conpotential, the energy of a HQV is Fh.v. p s 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 Fv0 = πJp ln(L/a). The ratio 0 and F 0 depends on the ratio J /J or spin fluctuations; in the between Fh.v. s p v 0 /F 0 changes discontinuously from limit L approaches infinity, the ratio Fh.v. v 1/2 at T = 0 where Jp ≈ Js = J to 1/4 at finite low temperatures in 2D where Js vanishes. In simulations of a finite trap (see below), because of a finite size effect we find 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 Superfluid of Ultracold F = 1 Atoms 2 r 2 , with ω sume a nearly harmonic trapping potential V (r) = 1/2mωtr tr being the trap frequency. The average number of particles per site N (r) has a Thomas-Fermi profile; N (r) = N0 (1 − r2 /RT2 F ), here N0 is the number density at the center and RT F is the Thomas-Fermi radius. Furthermore, the optical lattice potential along the axial direction is sufficiently deep so that atoms are confined in a two-dimension xy plane; the in-plane lattice potential 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 ωtr = 120Hz, √ we find that RT F = 43a and Lh.o. = 4a where Lh.o. (= 1/ 2mωtr ) is the harmonic 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 0.08ωtr 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 spatially narrower barrier. Within the range of temperatures studied, thermal activation turns out to be insignificant within an experimental time scale (∼ 100ms) because of low attempt frequencies. So only when the spatial width of barrier becomes comparable to a hydrodynamic breakdown length[196], the barrier can no longer be felted and vortices start to penetrate into the trap. The hydrodynamic breakdown length LB is about (L4h.o. /2RT F )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 D critical frequency for vortex nucleation ΩD c ; for IQVs, the estimated Ωc is a flat function of T (see Fig.5.8b) which is qualitatively consistent with earlier 0 estimates[197, 198]. For HQVs, however, as Fh.v. depends on the amplitude D of spin fluctuations, Ωc varies from about 0.32ωtr at T = 0 where Js ≈ Jp due to a finite size effect, to about 0.17ωtr at temperatures close to Tc where Js ≈ 0. Note when Js approaches zero as in the thermodynamic limit, ΩD c (∼ 0.17ωtr ) for HQVs is about one-half of the critical frequency for IQVs (about 0.37ωtr for the trap studied here). The interaction between two HQVs with the same vorticity at a separation distance d contains two parts. One, Vcc (> 0) is from interactions between two supercurrent velocity fields which is logarithmic as a function of d; and the other, Vss is from interaction between two spin twisting fields accompanying HQVs. For a disclination-anti disclination pair, in the dilute  106  5.4. Conclusion limit one finds that Vcc ∼ −Vss resulting in a cancelation of long-range interactions. The resultant short-range repulsions lead to square vortex lattices found in numerical simulations[199]. For fluctuation-driven fractionalized vortices, Vss is almost zero and the overall interactions are always logarithmically 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 superfluids[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, superfluids 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 superfluid 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 superfluids of sodium atoms can have a non-local topological spin order associated with half quantum vortices. In rotating traps, fluctuation-driven fractionalized vortices can nucleate at a critical frequency which is about half of that for integer vortices. In the future, it is interesting to explore 2D-3D dimension transition that can be realized in optical lattices. When a superfluid of cold atoms in a 3D lattice is gradually separated into independent layers of 2D superfluids, 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 lattice transition connecting square lattices in 3D condensates and triangular lattices in 2D superfluids.  107  Chapter 6  Conclusion In conclusion, we have studied fluctuation-driven phenomena in spinor superfluids of ultracold systems. We have shown that fluctuations in spinor systems of ultracold atoms studied in the thesis have completely changed the picture suggested by mean field analysis and have given rise to several novel fluctuation-driven phenomena. In hyperfine F = 2 nematic condensates, quantum fluctuations can change the mean field prediction in magnetic correlations and spin dynamics dramatically. It is shown that zero-point quantum fluctuations completely lift the accidental continuous degeneracy that is found in mean field analysis of quantum spin nematic phases of hyperfine spin-two cold atoms. Two distinct spin nematic states with higher symmetries are selected by quantum fluctuations. In ultracold experiments, it is possible to probe quantum fluctuations directly via studying coherent spin dynamics of ultracold atoms in optical lattices. We have carried out a detailed study of fluctuationdriven spin dynamics. Unlike the mean field coherent dynamics, quantum fluctuation-controlled spin dynamics are very sensitive to the variation of quantum fluctuations. They have peculiar dependence of Zeeman fields and potential depths in optical lattices. In particular this novel type of spin coherent dynamics can be potentially explored under accessible experimental conditions. Our results not only point out the fundamental limitations of previous measurement based on mean field coherent dynamics, but also provide a new type of tool to investigate fluctuation-driven phenomena. In the future, it would be interesting to use fluctuation-driven dynamics to investigate correlated fluctuations and various universal scaling properties near quantum critical points. In another spinor system, 2D nematic superfluids of cold sodium atoms, spin structures of fluctuation-driven fractionalized vortices and topological spin order are studied. Monte Carlo simulations suggest a softened π-spin disclination structure in a half quantum vortex when spin correlations are short ranged; in addition, calculations indicate that a unique non-local topological spin order emerges simultaneously as cold atoms become a superfluid below a critical temperature. We have also proposed an experimental study 108  Chapter 6. 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Physical Review Letters 85(11), 2223–2227 Sep (2000).  123  Appendix A  Band Structure in Optical Lattices An optical lattice is made of pairs of counter propagating laser beams. The standing wave pattern formed by laser beams can provide a periodical potential for atoms loaded in optical lattices. In a simple theoretical model, the potential strength is proportional to the local intensity of the electric field which has a sinusoidal spatial dependance. For a cubic lattice in 3D, the following periodical potential is used in the calculation of the band structure. [ ( ) ( ) ( )] 2 2π 2 2π 2 2π V (r) = V0 sin x + sin y + sin z (A.1) λ λ λ Three counter-propagating laser beams are slightly mismatched in their frequencies to avoid interference between laser fields of different directions. λ is the wavelength of the laser, and V0 is the depth of optical lattice which is proportional to the intensity of laser fields. V0 is often measured in units of 2 the recoil energy associated with the laser beams as ER = 2mλ2 . 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: [ 2 2 ] ∇ − + V (r) ψ(r) = Eψ(r). (A.2) 2m 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. ψα (r) = eik·r un,k (r)  (A.3)  E = En (k)  (A.4)  Here un,k (r) is a periodic function with a period λ/2 and is normalized as ∫ (A.5) dr |u2n,k (r)| = 1. u.c.  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) = dk eik(r−R) un,k (r). (A.6) k∈BZ It can be easily shown that Wannier functions labeled by the localization center R and the band index n forms an orthogonal basis, ∫ ∗ dr wn,R (r)wn′ ,R′ (r) = δR,R′ δn,n′ . (A.7) u.c.  In the second quantization form one can also introduce creation (anni† hilation) operators ψn,R (ψn,R ) for Wannier functions ψ † (r) = † ψn,R  ∑  † wn,R (r)ψn,R  ∫nR ∗ = drwn,R (r)ψ † (r)  [ψn,R , ψn† ′ ,R′ ] = δn,n′ δR,R′  (A.8)  One can rewrite the following many-body Hamiltonian of interacting bosons by using the expansion given in Eq.A.8: [ ( 2 2 ) ∫ ∇ † H = dr ψ (r) − − µ + V (r) ψ(r) 2m ] 2π 2 as † † + ψ (r)ψ (r)ψ(r)ψ(r) m ) ∑ ( † ∑ † ≃ −tL ψR′ ψR + h.c. − µL ψR ψR ⟨RR′ ⟩  +  aL ∑ † † ψR ψR ψR ψR . 2  R  (A.9)  R  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 different 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 =  aL  ∗ (r)wn=0,R′ (r) dr wn=0,R  ∫u.c. ′ = dk En=0 (k)eik·(R−R ) ∫ 4πas 4 = dr |wn=0,R (r)|. m u.c.  (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 Wan2 w2 nier function which can reduce overlapping terms such as dr|wn,R n,R′ | that may appear in the lattice model.  126  Appendix B  The β-function of the Nonlinear Sigma Model We derive the β function of O(3) nonlinear sigma model in this section. Let’s consider a unit vector |⃗s| = 1 which has three components (s1 , s2 , s3 ). ∫ J dd x (∇⃗s(x))2 H= (B.1) 2 It is convenient to rescale the Hamiltonian by introducing a coupling con1 stant g = βJ , 1 βH = 2g  ∫ dd x (∇⃗s(x))2  (B.2)  The coupling constant has a dimension of (length)d−2 , and one can also introduce a dimensionless constant u as g = uad−2 0 ,  (B.3)  here a is the smallest length scale in the system and it is inverse proportional to the momentum cutoff Λ. Let’s consider small fluctuations around the mean field ground state s = (1, 0, 0) in terms of two small parameters s3 and θ: √ s1 = 1 − s23 cos(θ) (B.4) √ s2 = 1 − s23 sin(θ) (B.5) (B.6) The energy density H can be written in terms of θ and s3 as: ] [ 1 (s3 ∇s3 )2 2 2 2 βH(x) = (∇s3 ) + (1 − s3 )(∇θ) + 2g 1 − s23  (B.7)  127  Appendix B. The β-function of the Nonlinear Sigma Model Notice the fluctuations of the s3 field has a magnitude of ∼ can rescale the s3 field as √ gψ = s3 ,  √  g. Hence we (B.8)  and expand the Hamiltonian according to the orders of g 1 1 1 (∇ψ)2 + (∇θ)2 − ψ 2 (∇θ)2 2 2g 2 g g2 + (ψ∇ψ)2 + ψ 2 (ψ∇ψ)2 + O(g 3 ) 2 2  H(x) ≃  (B.9) (B.10)  Let’s now perform the integration over the “fast” fields or fields with momentum within the shell bΛ < |p| < Λ. To obtain the leading order contributions in terms of g, we can assume that the θ field is smooth and we only need to integrate over the ψ field within the shell: ∫ (B.11) Dψ e−βH[ψ,θ] = bΛ<|p|<Λ  ∫  − 12  Dψ(p) e  ∫  [ ] dd x (∇ψ)2 + g1 (∇θ)2 −ψ 2 (∇θ)2 +O(g)  (B.12)  bΛ<|p|<Λ  After the Fourier transformation of ψ field, we can perform a Gaussian integral and then rewrite the result in the exponential form. ∫ ∫ dd p −1 [p2 −(∇θ)2 ]|ψ(p)|2 (B.13) Dψ(p) e 2 (2π)d bΛ<|p|<Λ  ≃  ∏ bΛ<|p|<Λ  [  2π p2 − (∇θ)2  [ ∫ 1 ≃ exp 2 bΛ<|p|<Λ [ ∫ 1 ≃ exp 2 bΛ<|p|<Λ  ]1 2  ] dd p 2π log (2π)d p2 − (∇θ)2 ( )] dd p 2π (∇θ)2 log 2 + (2π)d p p2  (B.14)  (B.15) (B.16)  The coarse grained Hamiltonian H′ up to the first order of g is ∫ 1 dd p 2π 1 1 βH′ (x) = − log 2 + (∇ψ)2 − ψ 2 (∇θ)2 2 bΛ<|p|<Λ (2π)d p 2 2 [ ] ∫ 1 1 dd p 1 + − (∇θ)2 + O(g) (B.17) d p2 2 g (2π) bΛ<|p|<Λ 128  Appendix B. The β-function of the Nonlinear Sigma Model When we rescale the Hamiltonian, the new cutoff Λ′ is reduced and the new length scale a′ is increased(b < 1), indicating the system is dilated. Λ′ = bΛ, a′0 =  a0 b  (B.18)  The coupling term g ′ in the coarse grained Hamiltonian H ′ can be written as: 1 1 = d−2 − ′d−2 ′ u a0 ua0  ∫ bΛ<|p|<Λ  dd p 1 (2π)d p2  (B.19)  By differentiating the above equation with the length scale a0 , one can obtain the renormalization group function or β-function for the coupling constant u. β(u) = a0  du da0  = (2 − d)u +  u2 + O(u3 ) 2π  (B.20)  The β function of u is very different for d ≤ 2 and d > 2. In the former cases, the coupling constant goes to infinite 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 different phases separated by the fixed point uc (β(uc ) = 0). At d = 2, one can also estimate the spin correlation length ξ by solving the equation of the beta function: ( ) ( ) 2πJ 2π = a0 exp , ξ = a0 exp (B.21) u T here a0 and J are the lattice constant and the coupling constant of the lattice model.  129  


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