adilet imambekov- strongly correlated phenomena with ultracold atomic gases

8/3/2019 Adilet Imambekov- Strongly Correlated Phenomena with Ultracold Atomic Gases

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Strongly Correlated Phenomena with Ultracold Atomic

Gases

A dissertation presented

by

Adilet Imambekov

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

June 2007


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c2007 - Adilet Imambekov

All rights reserved.


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Thesis advisor AuthorEugene Demler Adilet Imambekov

Strongly Correlated Phenomena with Ultracold Atomic Gases

Abstract

In this thesis we investigate strongly correlated phenomena in the field of ultra-cold atomic gases. Chapter 2 addresses a question of the insulating phases of cold spin-onebosonic particles with antiferromagnetic interactions, such as 23N a, in optical lattices. Mag-netic properties of the ground state in the insulating regime are studied using various tech-niques. Chapter 3 considers a one dimensional interacting Bose-Fermi mixture with equalmasses of bosons and fermions, and with equal repulsive interactions between Bose-Fermiand Bose-Bose particles. Properties of such mixture are studied using exact Bethe-ansatztechniques. Chapter 4 deals with certain phenomena which appear in the experiments with

imbalanced fermionic mixtures in strongly anisotropic traps. Chapter 5 gives a comprehen-sive review of interference phenomena, analyzing effects which contribute to the reductionof the interference fringe contrast in matter interferometers.

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Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivCitations to Previously Published Work . . . . . . . . . . . . . . . . . . . . . . . vii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

2 Spin-one bosons in optical lattices 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Derivation of Bose-Hubbard model for spin-one particles . . . . . . . . . . . 11

2.3 Insulating state with an odd number of atoms . . . . . . . . . . . . . . . . . 142.3.1 Effective spin Hamiltonian for small t . . . . . . . . . . . . . . . . . 14

2.3.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Insulating states with two atoms per site . . . . . . . . . . . . . . . . . . . 192.4.1 Two site problem: exact solution . . . . . . . . . . . . . . . . . . . . 192.4.2 Effective spin Hamiltonian for an optical lattice . . . . . . . . . . . . 21

2.4.3 Phase diagram from the mean-field calculation . . . . . . . . . . . . 22

2.4.4 Quantum fluctuations corrections for the spin singlet state . . . . . . 272.4.5 Spin wave excitations in the nematic phase . . . . . . . . . . . . . . 29

2.4.6 Magnetic field effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Large number of particles per site . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.1 Mean field solution without magnetic field . . . . . . . . . . . . . . . 372.5.2 Magnetization plateaus . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Global phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 Two and three dimensional lattices . . . . . . . . . . . . . . . . . . . 432.6.2 One dimensional lattices . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.7 Detection of spin order in insulating phases . . . . . . . . . . . . . . . . . . 462.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Exactly solvable case of a one-dimensional Bose-Fermi mixture 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Bethe ansatz solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Numerical solution and analysis of instabilities . . . . . . . . . . . . . . . . 59

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Contents v

3.4 Local density approximation and collective modes . . . . . . . . . . . . . . . 623.5 Zero-temperature correlation functions in Tonks-Girardeau regime . . . . . 70

3.5.1 Factorization of spin and orbital degrees of freedom . . . . . . . . 71

3.5.2 Bose-Bose correlation function . . . . . . . . . . . . . . . . . . . . . 723.5.3 Fermi-Fermi correlation function . . . . . . . . . . . . . . . . . . . . 793.5.4 Numerical evaluation of correlation functions and Luttinger parameters 82

3.6 Tonks-Girardeau regime at low temperatures . . . . . . . . . . . . . . . . . 843.6.1 Low energy excitations in Tonks-Girardeau regime. . . . . . . . . . . 863.6.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.6.3 Fermi-Fermi correlations . . . . . . . . . . . . . . . . . . . . . . . . . 893.6.4 Bose-Bose correlation function . . . . . . . . . . . . . . . . . . . . . 94

3.7 Experimental considerations and conclusions . . . . . . . . . . . . . . . . . 96

4 Breakdown of the local density approximation in interacting systems of

cold fermions in strongly anisotropic traps 102

5 Fundamental noise in matter interferometers 1095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1.1 Interference experiments with cold atoms . . . . . . . . . . . . . . . 1095.1.2 Fundamental sources of noise in interference experiments with matter 115

5.2 Interference of ideal condensates . . . . . . . . . . . . . . . . . . . . . . . . 1175.2.1 Interference of condensates with a well defined relative phase . . . . 1175.2.2 Interference of independent clouds . . . . . . . . . . . . . . . . . . . 123

5.3 Full counting statistics of shot noise . . . . . . . . . . . . . . . . . . . . . . 1245.3.1 Interference of two independent coherent condensates . . . . . . . . 128

5.3.2 Interference of independent clouds in number states . . . . . . . . . 1295.3.3 Clouds with a well defined relative phase . . . . . . . . . . . . . . . 131

5.4 Interference of low-dimensional gases . . . . . . . . . . . . . . . . . . . . . . 1315.4.1 Interference amplitudes: from high moments to full distribution func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4.2 Connection of the fringe visibility distribution functions to the parti-

tion functions of Sine-Gordon models . . . . . . . . . . . . . . . . . . 1405.4.3 Non perturbative solution for the general case . . . . . . . . . . . . . 143

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.5.2 Some experimental issues . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A Appendix to Chapter 2 154

A.1 Derivation of the effective magnetic Hamiltonian for insulating states withodd number of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154A.1.1 Normalization of the states . . . . . . . . . . . . . . . . . . . . . . . 155A.1.2 Calculation of0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.1.3 Calculation of1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158A.1.4 Calculation of2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


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vi Contents

A.2 Derivation of the effective magnetic Hamiltonian for the insulating state withtwo atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.3 Mean field solution for the case of two bosons per site . . . . . . . . . . . . 162A.4 Large N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B Appendix to Chapter 3 165

C Appendix to Chapter 5 167C.1 Expansion to order (1/K)2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 167C.2 General properties of (1/K)m terms, and expansion to order (1/K)5. . . . . 171C.3 Properties of the K distribution . . . . . . . . . . . . . . . . . . . . . 174C.4 D=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Bibliography 177


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Citations to Previously Published Work

Publications relating to the particular chapters are as follows:Chapter 2:

Adilet Imambekov, Mikhail Lukin, Eugene Demler,Spin-exchange interactions ofspin-one bosons in optical lattices: singlet, nematic and dimerized phases, Phys.Rev. A 68, 063602 (2003), also as cond-mat/0306204.

Adilet Imambekov, Mikhail Lukin, Eugene Demler, Magnetization plateaus for spin-one bosons in optical lattices: Stern-Gerlach experiments with strongly correlatedatoms, Phys. Rev. Lett. 93, 120405 (2004), also as cond-mat/0401526.

Chapter 3:

Adilet Imambekov, Eugene Demler, Exactly solvable case of a one-dimensional Bose-Fermi mixture, Phys. Rev. A 73, 021602(R) (2006), also as cond-mat/0505632.

Adilet Imambekov, Eugene Demler, Applications of exact solution for strongly inter-acting one dimensional Bose-Fermi mixture: low-temperature correlation functions,density profiles and collective modes, Annals of Physics 321, 2390 (2006), also ascond-mat/0510801.

Chapter 4:

Adilet Imambekov, C.J. Bolech, Mikhail Lukin and Eugene Demler, Breakdown of

the local density approximation in interacting systems of cold fermions in stronglyanisotropic traps, Phys. Rev. A 74, 053626, available as cond-mat/0604423.

Chapter 5:

Adilet Imambekov, Vladimir Gritsev, Eugene Demler, Distribution functions of in-terference contrast in low-dimensional Bose gases, submitted to Phys. Rev. Lett.,available as cond-mat/0612011.

Adilet Imambekov, Vladimir Gritsev, Eugene Demler, Fundamental noise in matterinterferometers, to be published in the Proceedings of the 2006 Enrico Fermi Sum-mer School on Ultracold Fermi gases, organized by M. Inguscio, W. Ketterle andC.Salomon (Varenna, Italy, June 2006), available as cond-mat/0703766.

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Acknowledgments

First of all, I would like to thank my advisor, Eugene Demler. His constant supportand encouragement, as well as the freedom he gave me made my years at graduate schoolenjoyable and rewarding.

I would also like to thank Misha Lukin, who was a co-advisor on some of the

projects I did. His unique approach to formulating complicated problems in a simple lan-guage is inspiring. I am also grateful to John Doyle for accepting the burden of serving onthe thesis committee.

During the later stages of my thesis work, I enjoyed working with postdocs CarlosBolech and, especially, Vladimir Gritsev. Vladimirs attitude to science always remindsme why I chose to b e a scientist in the first place. There have been many other postdocsand fellow graduate students with whom I have b een lucky to interact. Among them areVincenzo Vitelli, Ryan Barnett, Bob Cherng, Daniel Podolsky, Gil Refael, Alexey Gorshkov,Ehud Altman, Anatoli Polkovnikov, Daw-Wei (Charles) Wang, and Anton Burkov.

I owe a special trubute to my dear friends, Dima Abanin and Itay Yavin, for beinglike a family to me during my years at graduate school. Itaychik taught me that being

able to laugh at ones shortcomings is the best way to overcome them. His example alwaysinspires me to try out new things. Dimochka was a great friend during last 10 years, andhis open-mindness is a constant source of fun. I am also thankful to Tom Hunt, for beinga great office mate during the first year; Pavel Petrov, for sharing with me his masterfulcontrol of Russian language; George Gosha Brewster, Ilya Tatar and Alexey Dynkin, fortheir company during numerous adventures outdoors and selfless driving.

Id also like to thank administrative staff of the department, especially SheilaFerguson, for making it such a great place. Her care for students and their needs makes thedepartment feel like home.

My parents, Dzhanat and Onlasyn, my brother Akniet and my sister Akbota havealways been with me in my heart, even though most of the time we were on the opposite

parts of the globe. I am grateful to my father for getting me interested in science inchildhood. Last, but not the least, I would like to thank Aigerim for adding a whole newdimension to my life.

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Dedicated to my father, Onlasyn Imambekov,

and my mother Dzhanat Imambekova.

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Chapter 1

Introduction

In this thesis we will discuss strongly correlated phenomena in the field of ultracoldatomic gases. As has been concisely formulated by J. R. Anglin and W. Ketterle in 2002[1], Our field is now at a historic turning point, in which we are moving from studyingphysics in order to learn about atom cooling to studying cold atoms in order to learn aboutphysics. The remarkable experimental progress in the field of ultracold atoms in the lastdecade has reached the stage at which interactions between dilute gases cannot be describedusing a picture of weakly interacting quasi-particles. Such regime is characteristic of thephysics of strongly correlated systems. Problems in which strongly correlated phenomenaappear are notoriously hard to treat theoretically, with high temperature superconductivitybeing a prominent example. The simplest model proposed to study high temperature su-perconductivity, single band 2D fermionic Hubbard model, is intractable analytically. Even

if a solution of this problem were available, it is clear that there is a variety of propertiesof high temperature superconductors which are not contained in 2D fermionic Hubbardmodel. Thus in a majority of traditional solid-state systems which exhibit strong correla-tions, one can at most hope to have a qualitative agreement between theory and experiment.Systems of ultracold gases, on the other hand, provide a unique example for which micro-scopic Hamiltonians are usually known from first principles, hence a detailed quantitativecomparison between experiment and theory is possible. On the top of that, the remarkabledegree of control achieved in experiments can be used to tune the interactions, thus drivingquantum phase transitions. The most outstanding achievement up to date in this directionis the observation of the superfluid-insulator transition [2] for ultracold bosons in opticallattices [3, 4], which will be illustrated below. In the rest of this introductory chapter wewill give a brief overview of current experimental situation and will provide references to

more comprehensive review articles and books.

Experiments on cooling and trapping of neutral atoms have started in 70s, andculminated in the achievement of Bose Einstein Condensate (BEC) in 1995 [5, 6, 7]. Fig. 1.1illustrates the typical energy and length scales involved in the atomic cooling. Experimentaltechniques used to cool and trap atoms at such low temperatures are nicely summarizedin a book by H. J. Metcalf and P. van der Straten [9]. The limit of quantum degeneracyis reached when atoms de Broglie wavelength dB = h/

2mkBT becomes comparable to

characteristic interparticle separation n1/3, where n is the atomic density. For non inter-

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2 Chapter 1: Introduction

Figure 1.1: Typical energy scales in atomic cooling and trapping. Reprinted with permissionfrom Macmillan Publishers Ltd: Nature ([8]), copyright (2002).


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Chapter 1: Introduction 3

Figure 1.2: Observation of Bose-Einstein condensation by absorption imaging. The data wastaken after the gas had been allowed to freely expand for several milliseconds. The imageswere taken at three different temperatures: one just above the transition temperature,one just below the transition temperature, and one well below the transition temperature.Image courtesy of W. Ketterle and Dallin S. Durfee, MIT.

acting bosons, condensate appears at a precise temperature, controlled by n3dB = 2.612 (seee.g. [10]). Most experiments reach quantum degeneracy with temperatures 500 nK 2 Kand densities 1014 1015 cm3. Fig. 1.2 illustrates density profiles measured by absorptionimaging after the gas had been allowed to freely expand for several milliseconds. Thesetime of flight images show the velocity distribution of the atoms. Above the transitiontemperature, the velocity distribution is a spherical gaussian. But as the transition line

is crossed, there is a sudden change. The distribution becomes bimodal, with two sepa-rate contributions from excited states and from the ground state. In the third picture, thetemperature is low enough so that most of the atoms are in the condensate.

For such low temperatures under non-resonant conditions, only scattering in s wave channel is important [11], and collisions are characterized by a single parameter, scat-tering length a. It can have different signs, with positive sign corresponding to repulsionand negative sign corresponding to attraction between atoms. Scattering length is sensitiveto the details of the interatomic molecular potentials, and currently numerical methods


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are available to calculate scattering lengths from first principles [12]. Dimensionless com-bination, which controls the strength of interactions, is given by so-called gas parameter,na3. The remarkable success of the theory of weakly interacting Bose gases relies on thesmallness of the gas parameter. Sufficiently below the condensation temperature, diluteBECs are essentially pure condensates, with all atoms occupying the same macroscopicstate. Static and dynamic properties of such condensates are quantitatively described by anonlinear Schrodinger equation, the celebrated Gross-Pitaevskii equation [13, 14]. In thiscase many-body physics is reduced to a single-particle description, with interactions pro-viding additional density-dependent effective potential. By adding small fluctuations onthe top of Gross-Pitaevskii state, one recovers the familiar Bogoliubov picture of weaklyinteracting quasi-particles.

Major experimental advances in the studies of weakly interacting BECs includeobservation of interference of two independent Bose clouds [15], optical trapping of spinorBECs [16, 17], cooling of Fermi gases and Bose-Fermi mixtures to quantum degeneracy

[18, 19], demonstration of long-range coherence [20] and observation of vortices and vortexlattices [21, 22]. Excellent reviews of this area of research have b een presented in reviewarticles [23, 24, 25] and in more recent books by C.J. Pethick and H. Smith [26] and L.Pitaevskii and S. Stringari [27].

In the past several years, the physics which is accessible by ultracold gases hasbeen enormously enlarged by two major developments, the ability to tune interactions bymagnetic Feshbach resonances [28], and the possibility to generate periodic potentials foratoms using optical lattices [4, 29, 30, 31, 32]. The idea of Feshbach resonances dates backto work in nuclear physics in 50s [33]. Feshbach resonance in a two particle collision ap-pears whenever a bound state in a closed channel is resonant with the energy of scatteringparticles in an open channel. Particles then are temporarily captured in the quasi-bound

state, which results in a Breit-Wigner type resonance [11] in the scattering cross-section.Feshbach resonances are particularly useful in the scattering of cold atoms, since the mag-netic moments of closed and open channels are different. Hence the position of the resonantenergy level can be controlled by uniform external magnetic field [34]. More detailed in-troduction to Feshbach resonances can be found in review articles [35, 36, 37]. Opticallattices [4, 29, 30, 31, 32] have become another standard tool to control the interactions inrecent years. The basic mechanism which produces an external potential is the polarizationof atoms due to external ac-Stark shift in off-resonant light field. Induced dipole momentinteracts with oscillating electric field, creating trapping potential

V(r) = dE(r) (L)|E(r)|2.

Here () denotes the polarizability of an atom and |E(r)|2 I(r) is proportional tothe intensity of the laser field. A periodic potential can be formed by overlaping twocounter-propagating beams. By interfering several laser beams, one can create 1D, 2D or3D lattices, as illustrated in Fig. 1.3. By varying the intensity of laser fields, one cancontrol the tunneling and interactions of atoms. The regime of strong interactions can bereached not by increasing the atom-atom scattering length, but by reducing other relevantenergy scales, such as kinetic energy of atoms. In the lattice the latter is controlled by thetunneling, and can be made small.


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Figure 1.3: Optical lattice potentials formed by superimposing two or three orthogonalstanding waves. a, For a 2D optical lattice, the atoms are confined to an array of tightlyconfining 1D p otential tubes. b, In the 3D case, the optical lattice can be approximated by

a 3D simple cubic array of tightly confining harmonic oscillator potentials at each latticesite. Reprinted with permission from Macmillan Publishers Ltd: Nature Physics ([32]),copyright (2005).


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Figure 1.4: Absorption images of multiple matter wave interference patterns. For weakoptical lattices interference peaks get sharper, while at critical lattice strength they disap-pear, signaling the superfluid - Mott insulator transition. Reprinted with permission fromMacmillan Publishers Ltd: Nature ([4]), copyright (2002).

The natural Hamiltonian which describes bosons in optical lattice is given by [3, 38]

HBH = tij

(aiaj + ajai)

i

ni +U

2

i

ni(ni 1). (1.1)

Here t is the tunneling, U is the onsite interaction and is the chemical potential. Ratiot/U may be controlled by varying the intensity of laser beams, so one can go from theregime in which kinetic energy dominates (weak periodic potential, t U), to the regimewhere interaction energy is the most important part of the Hamiltonian (strong periodicpotential, t U). For integer fillings (number of atoms per lattice site), the two regimeshave superfluid and Mott insulating ground states, respectively, as can be obtained fromthe mean-field analysis of the Bose-Hubbard Hamiltonian [3, 2]. In the superfluid phase,atoms are delocalized in the lattice, fluctuations in the number of atoms on each site arestrong, and there is a phase coherence between different sites. In the insulating state,atoms are localized, fluctuations in the particle number at each site are suppressed, andthere is a gap to all excitations. Such an insulating state represents a correlated many bodystate of bosons, where strong interactions between atoms result in a new ground state ofthe system. Quantum phase transition between these two states has been demonstrated inexperiments of Greiner et al. [4] via analysis of time of flight images, as shown in Fig. 1.4.For superfluid phase, long range coherence results in interference peaks at the reciprocallattice wave vectors. For weak optical lattices, interference peaks get sharper, since Wannierwave functions at individual lattice sites are getting more localized. However, interferencedisappears at a critical lattice strength, signaling the superfluid - Mott insulator transition.

One of the new directions opened by ultracold gases is the studies of low-dimensionalBose systems. Following the suggestion by Olshanii [39], the exactly solvable Lieb-Liniger


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model [40] of bosons was realized in 1D. In the regime of strong interactions, this model ex-hibits effective fermionization, realizing the famous Tonks-Girirdeau [41] gas of stronglyinteracting bosons [42, 43]. As suggested by Wilkin and Gunn [44], physics of quantumHall effect with bosons can be explored in fast rotating quantum gases [45]. Feshbach reso-nances with ultracold fermions can be used to experimentally study [46] the crossover frommolecular BEC of paired fermions to BCS superfluid of Cooper pairs. In particular, onecan create imbalanced Fermi mixtures [47, 48] and explore physical regimes not accessiblein traditional condensed matter systems. More comprehensive review of the current stateof affairs in the field of strongly correlated phenomena with ultracold atoms can be foundin recent reviews [49, 50].

In the rest of this thesis, we will discuss the original work by the author on severaltopics of current interest. Chapter 2 discusses the insulating phases of cold spin-one bosonicparticles with antiferromagnetic interactions, such as 23Na, in optical lattices. We studymagnetic properties of the ground state in the insulating regime. In chapter 3 we consider a

one dimensional interacting Bose-Fermi mixture with equal masses of bosons and fermions,and with equal repulsive interactions between Bose-Fermi and Bose-Bose particles. Proper-ties of such mixture are studied using exact Bethe-ansatz techniques. In chapter 4 we discusscertain phenomena which appear in the experiments with imbalanced fermionic mixturesin strongly anisotropic traps. In chapter 5 we give a comprehensive review of interferencephenomena, analyzing effects which contribute to the reduction of the interference fringecontrast in matter interferometers.


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Chapter 2

Spin-one bosons in optical lattices

In this chapter we consider insulating phases of cold spin-one bosonic particles withantiferromagnetic interactions, such as 23Na, in optical lattices. We show that spin exchangeinteractions give rise to several distinct phases, which differ in their spin correlations. In twoand three dimensional lattices, insulating phases with an odd number of particles per siteare always nematic. For insulating states with an even number of particles per site, there isalways a spin singlet phase, and there may also be a first order transition into the nematicphase. The nematic phase breaks spin rotational symmetry but preserves time reversalsymmetry, and has gapless spin wave excitations. The spin singlet phase does not breakspin symmetry and has a gap to all excitations. In the presence of magnetic field we find aseries of quantum phase transitions between states with fixed magnetization (magnetizationplateaus) and a canted nematic phase. In one dimensional lattices, insulating phases withan odd number of particles per site always have a regime where translational symmetry is

broken and the ground state is dimerized. We discuss signatures of various phases in Braggscattering and time of flight measurements.

2.1 Introduction

Modern studies of quantum magnetism in condensed matter physics go beyondexplaining the details of particular experiments on cuprate superconductors, heavy fermionmaterials, organic conductors, or related materials, and aim to develop general paradigmsfor understanding complex orders in strongly interacting many body systems [51, 52, 53,54, 55, 56, 57, 58, 59, 60]. Spinor atoms in optical lattices provide a novel realizationof quantum magnetic systems that have several advantages compared to their condensedmatter counterparts, including precise knowledge of the underlying microscopic models, thepossibility to control parameters of the effective lattice Hamiltonians, and the absence ofdisorder.

Degenerate alkali atoms are generally considered as a weakly interacting gas dueto the smallness of the scattering length compared to the interparticle separation [26]. Thesituation may change dramatically either when atomic scattering length is changed by meansof Feshbach resonances [8], or when an optical potential created by standing laser beamsconfines particles in the minima of the periodic potential and strongly enhances the effects

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Chapter 2: Spin-one bosons in optical lattices 9

of interactions. In the latter case the existence of the nontrivial Mott insulating state ofatoms in optical lattices, separated from the superfluid phase by the quantum superfluid-insulator (SI) phase transition, was demonstrated recently in experiments [30, 4, 61]. Lowenergy (temperature) properties of spinless bosonic atoms in a periodic optical potentialare well described by the Bose-Hubbard Hamiltonian [3]

HBH = tij

(aiaj + ajai)

i

ni +U02

i

ni(ni 1), (2.1)

The parameters of (2.1) may be controlled by varying the intensity of laser beams, so one cango from the regime in which kinetic energy dominates (weak p eriodic potential, t U0), tothe regime where interaction energy is the most important part of the Hamiltonian (strongperiodic potential, t U0). For integer fillings (number of atoms per lattice site), the tworegimes have superfluid and Mott insulating ground states, respectively, as can be obtained

from the mean-field analysis of the Bose-Hubbard Hamiltonian [3, 2]. In the superfluidphase, atoms are delocalized in the lattice, fluctuations in the number of atoms on each siteare strong, and there is a phase coherence between different sites. In the insulating state,atoms are localized, fluctuations in the particle number at each site are suppressed, andthere is a gap to all excitations. Such an insulating state represents a correlated many bodystate of bosons, where strong interactions between atoms result in a new ground state ofthe system.

In conventional magnetic traps, spins of atoms are frozen so effectively that theybehave like spinless particles. In contrast, optically trapped atoms have extra spin degreesof freedom which can exhibit different types of magnetic orderings. In particular, alkaliatoms have a nuclear spin I = 3/2. Lower energy hyperfine manifold has three magnetic

sublevels and a total moment S = 1. Various properties of such condensate in a single trapwere investigated [66, 63, 62, 64, 65, 67]. For example, for particles with antiferromagneticinteractions, such as 23Na, the exact ground state of an even number of particles in theabsence of a magnetic field is a spin singlet described by a rather complicated correlatedwave function [66]. However, when the number of particles in the trap is large, the energygap separating the singlet ground state from the higher energy excited states is extremelysmall, and for the experiments of Ref. [16], the precession time of the classical mean-field ground state is of the order of the trap lifetime. So, experimental observation of thequantum spin phenomena in such systems is very difficult. To amplify quantum spin effectsone would like to have a system with smaller number of particles and stronger interactionsbetween atoms. Hence it is natural to consider an idea of spin-one atoms in an opticallattice, in which one can have a small number of atoms per lattice site (in experiments ofRef. [4] this number was around 1-3) and relatively strong interactions between atoms.

In this chapter we study bosonic spin-one atoms in optical lattices with spin sym-metric confining potentials and antiferromagnetic interaction between atoms. We demon-strate that spin degrees of freedom result in a rich phase diagram by establishing the ex-istence of several distinct insulating phases, which differ from each other by their spincorrelations.

In the insulating state of bosons in an optical lattice fluctuations in the particlenumber on each site are suppressed but not frozen out completely. Virtual tunneling of


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10 Chapter 2: Spin-one bosons in optical lattices

-0.5 0.5 1 1.5 2 2.5 3

1

2

3

4

5

6

N=1

N=2

N=3

N=4

N=5

Nematic

Nematic

Nematic

Nematic

Nematic

Singlet

Singlet

POLAR SUPERFLUID

tunnelling

chemical potential

Figure 2.1: General phase diagram for spin-one bosons in 2D and 3D optical lattice. Detaileddiscussion of the phase diagram, including explicit expressions for various phase boundaries,is given in section 2.6.

atoms between neighboring lattice sites gives rise to effective spin exchange interactionsthat determine the spin structure of the insulating states (spin exchange interactions forS = 1/2 bosons in optical lattices were discussed previously in [68, 69]).

We will show that in two and three dimensional lattices, insulating states withan odd number of atoms per site are always nematic, whereas insulating states at evenfillings are either singlet or spin nematic [70], depending on the parameters of the model.In one dimensional systems, even more exotic ground states should be realized, includingthe possibility of a spin singlet dimerized phase that breaks lattice translational symmetry[71, 72]. The 2D and 3D general phase diagram, including singlet, nematic and superfluidphases, is shown in Fig. 2.1. The extended version of this diagram, including discussion ofvarious transition lines, is presented in section 2.6.

It is useful to point out that the lattice model for spin-one bosons, which we

analyze here, is very general and may also be applicable to systems other than cold atomsin optical lattices. For example, triplet superconductors in strong coupling limit may bedescribed by a similar Hamiltonian, and some of the phases discussed in this chapter maycorrespond to non-BCS states of such superconductors [73].

The chapter is organized as follows. In section 2.2 we provide a derivation of theHubbard-type Hamiltonian for spin-one bosons in optical lattices starting from microscopicinteractions between atoms, and describe some general properties of our model. In section2.3 we derive an effective spin Hamiltonian which is valid for any odd number of atoms


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per site, N, in the limit of small tunneling between sites. We demonstrate equivalencebetween our system and a Heisenberg model for S = 1 spins on a lattice with biquadraticinteractions, and argue that the ground state is a nematic in two and three dimensionsand is a dimerized singlet in 1D. In section 2.4 we derive effective spin Hamiltonian for asystem with N = 2 atoms per site, valid deep in the insulating regime, and use mean-fieldapproximation to determine the phase boundaries between singlet and nematic phases. Weconsider the phase diagram in the presence of external magnetic field. In section 2.5 wederive effective spin Hamiltonian for the limit of large number of particles per site N >> 1and small tunneling, and discuss singlet-nematic transition for even N. In the presenceof magnetic field we find a series of quantum phase transitions between states with fixedmagnetization (magnetization plateaus) and a canted nematic phase. In section 2.6 wesummarize our results and review the global phase diagram for spin-one bosons in opticallattices. Finally, in section 2.7, we discuss approaches to experimental detection of singletand nematic insulating phases of spin-one bosons. We also describe how magnetization

plateaus can be detected. The technical details of calculations are presented in AppendicesA.1-A.4.

2.2 Derivation of Bose-Hubbard model for spin-one particles

At low energies scattering between two identical alkali atoms with the hyperfinespins S = 1 is well described by the contact potential [26]

V(r1 r2) = (r1 r2)(g0P0 + g2P2), (2.2)gF = 4h

2aF/M. (2.3)

Here PF is the projection operator for the pair of atoms into the state with total spinF = {0, 2}, aS is the s-wave scattering length in the spin F channel, and M is the atomicmass. When writing Eq. (2.2) we used the fact that s-wave scattering of identical bosonsin the channel with total spin F = 1 is not allowed by the symmetry of the wave function.Interaction (2.2) can be written using spin operators as

V(r1 r2) = (r1 r2)( g0 + 2g23

+g2 g0

3S1S2). (2.4)

For example, in the case of 23N a, g2 > g0, and we find effective antiferromagnetic interac-tion, as was originally discussed in Refs. [62, 63].

Kinetic motion of ultracold atoms in the optical lattice is constrained to the lowest

Bloch band when temperature and interactions are smaller than the band gap (this is thelimit that we will consider from now on). Atoms residing on the same lattice site haveidentical orbital wave functions and their spin wave functions must be symmetric. If weintroduce creation operators, ai, for states in the lowest Bloch band localized on site i andhaving spin components = {1, 0, 1}, we can follow the approach of Ref. [3] and writethe effective lattice Hamiltonian as

H = tij,

(aiaj + ajai) +

U02

i

ni(ni 1) + U22

i

(S2i 2ni) i

ni, (2.5)


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where

ni =

aiai (2.6)

is the total number of atoms on site i, and

Si =

ai Tai (2.7)

is the total spin on site i ( T are the usual spin-one operators). The first term in Eq.(2.5) describes spin symmetric tunneling between nearest-neighbor sites, the second termdescribes Hubbard repulsion between atoms, and the third term penalizes non-zero spinconfigurations on individual lattice sites for antiferromagnetic interactions. The origin ofthis spin dependent term is the difference in scattering lengths for F = 0 and F = 2 channels

as was discussed in Ref. [66]. Finally, the fourth term in Eq. (2.5) is the chemical potentialthat controls the number of particles in the system.

Hamiltonian (2.5) carries important constrains on possible spin states of the sys-tem. The first of them derives from the fact that the total spin of a system of N spin-oneatoms cannot be larger than N, so for each lattice site we have

Si Ni. (2.8)

The second constraint is imposed by the symmetry of the spin wave function on each site

Si + Ni = even. (2.9)

Optical lattices produced by far detuned lasers with wavelength i = 2/|ki| createan optical potential V(r) = i Vi sin2 ki r, with ki being the wave vectors of laser beams.

Using various orientations of beams, one can construct different geometries of the lattice.For the simple cubic lattice, parameters of Hamiltonian (2.5) can be estimated as

U2 =22

3ER

a2 a0

x3/4,

U0 =22

3ER

a0 + 2a2

x3/4,

t =4

ERx3/4e2x

1/2,

where ER = h2k2/2M is the recoil energy and x = V0/ER. Note that the ratio U2/U0 isfixed by the ratio of scattering lengths, a2/a0, for all lattice geometries. Scattering lengthsfor 23N a given in [74] are a2 = (52 5)aB and a0 = (46 5)aB , where aB is the Bohrradius. This corresponds to 0 < a2 a0


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8 10 12 14 16

0.5

1

1.5

2

kHz

t

U2

V0

ER

Figure 2.2: U2 and t for23Na atoms in the simple cubic optical lattice created by three

perpendicular standing laser beams with = 985nm. V0

is the strength of the opticalpotential and ER = h

2k2/2M is the recoil energy. The ratio of the interaction terms in(2.5), U2/U0, is fixed by the ratio of the scattering lengths and is independent of the natureof the lattice (U2/U0 0.04 for 23N a).

U2/U0 is small enough to see the interplay between tunneling and spin dependent U2 termbefore the superfluid-insulator transitions take place. The positions of superfluid-insulatortransitions and the validity of this assumption will be discussed in detail in section 2.6. Wewill use the value U2/U0 = 0.04 to make estimates of various phase boundaries. In Fig.2.2 we show U2/h and t/h as a function of the strength of the optical potential for a threedimensional cubic lattice produced by red detuned lasers with = 985nm.

Superfluid-insulator transition is characterized by a change in fluctuations of par-ticle numbers on individual lattice sites. When the spin dependent interaction (U2) is muchsmaller than the usual Hubbard repulsion (U0), the superfluid - insulator transition is de-termined mostly by U0. The spin gap U2 term, however, is important inside the insulatingphase, where it competes with the spin exchange interactions induced by small fluctuationsin the particle number, and an interesting spin structure of the insulating states appears asa result of such competition. The spin structure of the insulating phases of spin-one bosonsin optical lattices will be the main subject explored in this chapter.

In what follows we will often find it convenient to use particle creation operatorsthat transform as vectors under spin rotations. Such representation may be constructed as

az = a0, ax =

(a)

a+

2 , ay = i(a) + a

+

2 . (2.10)Operators a{x,y,z} satisfy the usual bosonic commutation relations, and they can be used toconstruct spin operators as

Sia = ieabcaibaic, S2i = (bnm bmn)abaanam. (2.11)We can verify the transformation properties of a{x,y,z} by noting that

[Sa, ab] = [ieapcapac, ab] = ieabcac, (2.12)


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[Sa, ab] = [ieapcapac, ab] = ieabcac. (2.13)

Using these operators, we can rewrite the hopping term in the Hamiltonian (2.5)as

t

,p{x,y,z}(aipajp + a

jpaip),

and it is invariant under global spin rotations. We will use this property later to simplifycalculations and classify eigenstates of effective interactions by the total spin.

2.3 Insulating state with an odd number of atoms

2.3.1 Effective spin Hamiltonian for small t

We start with the insulating state of Hamiltonian (2.5) with an odd number (N =2n + 1) of bosons per site in the limit t = 0. The number of particles on each site is fixed,and the bosonic symmetry of the wave function requires that the spin in each site is odd.The interaction term U2 is minimized when the spins take the smallest possible value Si = 1.In this limit the energy of the system does not depend on the spin orientations on differentsites. When t is finite but small, we expect that we still have spin Si = 1 in each site, butthat boson tunneling processes induce effective interactions between these spins. In thissection we will compute such interactions in the lowest (second) order in t. We will alsodiscuss conditions for which our effective Hamiltonian provides an adequate description ofthe system.

In the second order perturbation theory in t, we generate only pairwise interactionsbetween atoms on neighboring sites, so we can write the most general spin Hamiltonian forSi = 1 that preserves spin SO(3) symmetry as

H = J0 J1ij

SiSj J2ij

(SiSj)2. (2.14)

Here ij labels near neighbor sites on the lattice. The absence of the higher order terms,such as (SiSj)

3, follows from the fact that the product of any three spin operators for S = 1can be expressed via the lower order terms.

To find the exchange constants J0,1,2, we need to consider virtual processes thatcreate a state with Ni = 2n, Nj = 2n + 2, and Ni = 2n + 2, Nj = 2n. The difference inenergy between the intermediate state and low energy Si = Sj = 1 subspace is of order U0.

Since our subspace is much lower in energy, the second order perturbation theory is valid.It is convenient to rewrite the Hamiltonian (2.14) as

H = 0ij

Pij(0) + 1ij

Pij(1) + 2ij

Pij(2), (2.15)

0 = 4J2 + 2J1 J0,1 = J2 + J1 J0,2 = J2 J1 J0. (2.16)


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Here Pij(S) is a projection operator for a pair of spins on near neighbor sites i and j intoa state with total spin Si + Sj = S (S = 0, 1, 2). The equivalence of Eqs. (2.14) and (2.15)can be proven by noting simple operator identities for two spin one particles

1 = Pij(0) + Pij(1) + Pij(2),

(Si + Sj)2 = 4 + 2SiSj = 2Pij(1) + 6Pij(2),

(Si + Sj)4 = 16 + 16SiSj + 4(SiSj)

2 = 4Pij(1) + 36Pij(2). (2.17)

Note that states |Si = 1, Sj = 1; Si+ Sj = S have only trivial degeneracy corresponding topossible projections of total spin S on a fixed quantization axis DS = 2S+ 1.

Since we know a general form of our effective Hamiltonian, we can compute 0,1,2 bycalculating the expectation values of energy for arbitrary states in the appropriate subspaces

S

=

t2 |m

| < m|(aipajp + ajpaip)|Si = 1, Sj = 1; Si + Sj = S > |2

Em E0. (2.18)

Here E0 = 2U2 is the energy of the configuration with N = 2n + 1 bosons in each of thetwo sites, and Em is the energy of the intermediate (virtual) states, |m, that have 2n and2n + 2 bosons in the two sites, respectively. Both energies should be computed in the zerothorder in t.

It is useful to note that the tunneling Hamiltonian is spin invariant; therefore,intermediate states in summation over |m in Eq. (2.18) should also have the total spin S.Another constraint on the possible states |m comes from the fact that the tunneling termcan only change the spin on each site by 1, since in a Hilbert space of each site operatorsaip, aip act as vectors, according to their transformational properties (2.12)-(2.13).

Direct calculations in Appendix A.1 give

0 = 4t2(n + 1)(2n + 3)

3(U0 2U2) 16t2n(5 + 2n)

15(U0 + 4U2), (2.19)

1 = 4t2n(5 + 2n)

5(U0 + 4U2), (2.20)

2 = 28t2n(5 + 2n)

75(U0 + 4U2) 4(15 + 20n + 8n

2)

15(U0 + U2). (2.21)

Combining Eqs. (2.15)-(2.21) we find

J0t2

=4(15 + 20n + 8n2)

45(U0 + U2) 4(1 + n)(3 + 2n)

9(U0 + 2U2)+

128(5 + 2n)

225(U0 + 4U2),

J1t2

=2(15 + 20n + 8n2)

15(U0 + U2) 16(5 + 2n)n

75(U0 + 4U2),

J2t2

=2(15 + 20n + 8n2)

45(U0 + U2)+

4(1 + n)(3 + 2n)

9(U0 2U2) +4n(5 + 2n)

225(U0 + 4U2). (2.22)


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1 2 3 4n

0.7

0.8

0.9 J1

J2

Figure 2.3: Ratio J1/J2 for the effective spin Hamiltonian (2.14) for an odd number ofbosons N = 2n + 1 per site. U

2= 0.04U

0.

It will turn out that the ratio between J1 and J2 determines magnetic groundstate, and the dependence of this ratio on n is quite strong, as shown in Fig. 2.3.

We now discuss limitations of the Hamiltonian (2.14) with parameters given by Eq.(2.22). In the insulating state with exactly one b oson per site, near neighbor interactionsalways have the form (2.14). Explicit expressions for the Js given in Eq. (2.22) onlyapply in the limit t 0), we have an additional constraint: we should be able to neglect

configurations with spins on individual sites higher than 1. Matrix elements for scatteringinto such states are of the order of (Nt)2/U0 (see Eq. (2.22)), and their energy is set byU2. Therefore, the Hamiltonian (2.14) applies only when Nt 0, it is useful to start by considering a two-site problem

H12 = J1 S1 S2 J2(S1 S2)2 (2.23)with S1 = S2 = 1. Eigenstates of Eq. (2.23) can be classified according to the value ofthe total spin Stot, and their energies may be computed using 2S1 S2 = Stot(Stot + 1) 4.Two spin one particles can combine into Stot = 0, 1, and 2. The J1 term in Eq. (2.23)favors maximizing S1 S2 by polarizing S1 along the direction of S2, so that Stot = 2. Bycontrast, the J2 term favors maximizing ( S1 S2)

2 by forming a singlet state Stot = 0 (seeTable 2.1). So, the latter term acts as an effective antiferromagnetic interaction for thisspin one system, and it dominates for J2 > J1. If we go beyond a two site problem andconsider a large lattice, we see that each pair of near neighbor sites wants to establish asinglet configuration when J2 > J1. However, because one cannot form singlets on two


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Stot S1 S2 (S1 S2)2 Energy

0 -2 4 2J1 4J21 -1 1 J1

J2

2 1 1 J1 J2

Table 2.1: Eigenstates of a two site problem (2.23).

different bonds that share the same site, some interesting spin order, whose precise naturewill depend on the lattice and dimensionality, will appear.

Phase diagram for D = 1

From the discussion above we see the conflict intrinsic to the Hamiltonian (2.14):each bond wants to have a singlet spin configuration, but singlet states on the neighboringbonds are not allowed. There are two simple ways to resolve this conflict:A) Construct a state that mixes S = 0 and S = 2 on each bond but can be repeated onneighboring bonds;B) Break translational symmetry and create singlets on every second bond.

At the mean-field level, solution of the type A is given by

|N =i

|Si = 1, mi = 0. (2.24)

This can be established by noting that for any neighboring pair of sites we indeed have a

superposition ofS = 0 and S = 2 states

|Si = 1, mi = 0|Sj = 1, mj = 0 = 13|Stot = 0 +

2

3|Stot = 2, mtot = 0. (2.25)

State (2.24) describes a nematic state that has no expectation value of any component ofthe spin Sx,y,zi = 0, but spin symmetry is broken since (Sxi )2 = (Syi )2 = 1/2 and(Szi )2 = 0. It is useful to point out the similarity between wave function (2.25) that mixessinglet and quintet states on each bond, and a classical antiferromagnetic state for spin1/2 particles that mixes spin singlets and triplets on each b ond. Colemans theorem [75](the quantum analog of Mermin-Wagner theorem) forbids the breaking of spin symmetryin D = 1, even at T = 0. However, a spin singlet gapless ground state that has a close

connection to the nematic state (2.24) has been proposed in Refs. [77, 76] for J2 close toJ1.

The simplest way to construct a solution of type B is to take

|D =i=2n

|Si = 1, Si+1 = 1, Si + Si+1 = 0. (2.26)

Such a dimerized solution has exact spin singlets for pairs of sites 2n and 2n + 1, but pairsof sites 2n and 2n 1 are in a superposition of S = 0, 1, and 2 states.


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According to the variational wave functions (2.24) and (2.26), the dimerized so-lution becomes favorable over a nematic one only for J2/J1 > 3/2 in D = 1. However,numerical simulations [78] showed that for J2 > J1, the ground state is always dimerized.It is a spin singlet and has an energy gap to all spin excitations. This means that thevariational wave function (2.26) may only be taken as a caricature of the true ground state,although it captures such key aspects of it, such as broken translational symmetry and theabsence of spin symmetry breaking.

Phase diagram for D = 2, 3

The nematic state for the Hamiltonian (2.14) in a simple cubic lattice (D = 3) forJ2 > J1 has been discussed using mean-field approximations [79], semiclassical approaches[80], and numerical methods [81]. Finally, recent work of Tanaka et.al [82] provided arigorous proof of the existence of the nematic order at least in some part of this region,

which satisfies 2.66J1 > J2 2J1. The variational state for the nematic order may againbe given by equation (2.24) and its mean field energy is EMFN = 2J2. It is important toemphasize, however, that the actual ground state is sufficiently different from its mean-fieldversion (2.24). It is possible to write down dimerized states with energy expectation lowerthan 2J2; however, numerical results [81] suggest that the ground state doesnt breaktranslational symmetry. A way to obtain a more precise ground state wave function is toinclude quantum fluctuations near the mean field state, as was done in Ref. [80]. Hence, themean-field wave function (2.24) does not provide a good approximation of the ground stateenergy of the nematic state. Nevertheless, it is useful for the discussion of order parameterand broken symmetries of the nematic state.

In the nematic state, spin space rotational group O(3) is broken, though timereversal symmetry is preserved. The order parameter for the nematic state is a tensor

Qab = SaSb ab3

S2. (2.27)

In the absence of ferromagnetic order SaSb = SbSa; hence, Qab is a traceless symmetricmatrix. The minimum energy of (2.14) is achieved for Qab that has two identical eigenvalues,which corresponds to a uniaxial nematic [83].

Then, the tensor Qab can be written using a unit vector d as

Qab = Q(dadb 13

ab). (2.28)

Vector d is defined up to the direction (i.e.

d are equivalent) and corresponds to thedirector order parameter [83]. For the mean-field state (2.24), the director d can also bedefined from the condition that locally our system is an eigenstate of the operator dS witheigenvalue zero. However, such a definition may not be applied generally.

The nematic phase behaves in many aspects as antiferromagnetic [84], the directionof d being analogous to staggered magnetization. Namely in weak magnetic fields, d alignsitself in the plane perpendicular to magnetic field, and spin-wave excitations have lineardispersion [76], with velocity

c =

2zJ2(J2 J1).


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Nematic phases for spin-one lattices have been considered before in literature [77, 82, 81,78, 80, 84, 79, 76], so we will not discuss them here more extensively.

2.4 Insulating states with two atoms per site

In this section we consider an insulating state of two bosons per site. Possible spinvalues for individual sites are S = 0 and S = 2. In the limit t = 0, the interaction part ofthe Hamiltonian (the U2 term) is minimized when S = 0. The amplitude for creating S = 2states, as well as the exchange energy of the latter, is of the order of t2/U0. So, when t isof the order of (U0U2)

1/2 or larger, we may no longer assume that we only have singletsin individual sites, and we need to include S = 2 configurations in our discussion. Thisregime is still inside the insulating phase for small enough U2/U0 (the superfluid-insulatortransition takes place for zt U0). In this section we will assume that U2/U0 is smallenough, so that S = 2 becomes important in the insulating phase, before the transitionto superfluid. More careful consideration of superfluid transition line and comparison withthe case of 23Na will be presented in section 2.6. In section 2.4.1 we exactly solve theproblem for two sites. In section 2.4.2 we derive an effective Hamiltonian that takes intoaccount competition between spin gap of individual sites, that favors S = 0 everywhere,and exchange interactions between neighboring sites that favor proliferation of S = 2 states.Mean field solution of the effective magnetic Hamiltonian is considered in section 2.4.3 andwe find first order quantum phase transition from isotropic to nematic phase. We discusscollective excitations in sections 2.4.4 and 2.4.5 and the effects of magnetic field in section2.4.6. We note that the state with N = 2 has an advantage over states with higher N froman experimental point of view since it has no three-body decays.

2.4.1 Two site problem: exact solution

To construct an effective magnetic Hamiltonian for this system, we note that inthe second order in t it can be written as a sum of interaction terms for all near neighborsites (identical for all pairs of sites). These pairwise interactions can be found by solvinga two site problem and finding the appropriate eigenvalues and eigenvectors in the secondorder in t.

The Hilbert space for two sites with two atoms at each site is given by the directsum of the following subspaces:

|E1 > = |N1 = 2, N2 = 2, S1 = S2 = 0, S1 + S2 = 0 >,|E2 > = |N1 = 2, N2 = 2, S1 = S2 = 2, S1 + S2 = 0 >,|E3 > = |N1 = 2, N2 = 2, S1 = 0, S2 = 2, S1 + S2 = 2 >,|E4 > = |N1 = 2, N2 = 2, S1 = 2, S2 = 0, S1 + S2 = 2 >,|E5 > = |N1 = 2, N2 = 2, S1 = S2 = 2, S1 + S2 = 2 >,|E6 > = |N1 = 2, N2 = 2, S1 = S2 = 2, S1 + S2 = 1 >,|E7 > = |N1 = 2, N2 = 2, S1 = S2 = 2, S1 + S2 = 3 >,|E8 > = |N1 = 2, N2 = 2, S1 = S2 = 2, S1 + S2 = 4 > . (2.29)


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0.05 0.1 0.15 0.2t

-0.4

-0.3

-0.2

-0.1

0.1

0.2

Energy

e1

e2

e3

e4

e8

e6=e7

e5

Figure 2.4: Eigenstates of the effective spin Hamiltonian for a two site problem with twoatoms per site. Energy and t are measured in units of U

0, and we assumed U

2= 0.04U

0.

The lowest energy states correspond to total spin S = 0 (e1), S = 2 (e5), and S = 4 (e8).

For U2 > 0 the lowest energy state is a total spin singlet that has some mixture ofS = 2 states on individual sites when t is nonzero. The next favorable state has total spin2, |E5. When the value of t is increasing, the ferromagnetic state |E8 becomes the thirdlow lying state. At this point, we have solved the problem for two sites, taking into accountcompetition between the hopping and the spin dependent interaction (overall Hilbert spacefor two sites is 36 dimensional).

2.4.2 Effective spin Hamiltonian for an optical lattice

In the previous subsection we used perturbation theory in tunneling t to study theproblem of two sites with two atoms at each site. If we label the two sites 1 and 2, in thesecond order in t the effective Hamiltonian can be written as

H12 = 3U2 [P(S1 = 2) + P(S2 = 2)] + | >1 | >2 J,;, < |1 < |2, (2.34)

Here P(S{1,2} = 2) are projection operators into states with spin S = 2 on sites 1 and 2 andJ,;, gives exchange interactions that arise from virtual tunneling processes into states

with particle numbers (n1 = 1, n2 = 3) and (n1 = 3, n2 = 1). The second term of (2.34)includes all initial states (|1 and |2 for sites 1 and 2, respectively) and all final states(|1 and |2).

Generalization of the effective spin Hamiltonian (2.34) for the case of optical latticeis obviously

H = 3U2i

P(Si = 2) +

| >i | >j J,;, < |i < |j , (2.35)


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This Hamiltonian is linear in U2 and, therefore, can be written as a sum of the bond terms

H = ij

Hij ,

Hij =3U2

z[P(Si = 2) + P(Sj = 2)]

+ | >i | >j J,;, < |i < |j . (2.36)

Individual terms Hij differ from (2.34) only by rescaling U2 U2z , where z is the coordina-tion number of the lattice. Here we did not give explicit expressions for J,;, in the basisof eigenstates of individual spins Si and Sj but in the basis of eigenstates of the total spinof the pair (2.29), expressions for J,;, can be obtained from eigenstates and eigenvaluesof Eq. (2.34) (see Eqs. (2.30) -(2.33) and Appendix A.2, with rescaled U2). Therefore, wecan write

Hij =,,Sz

|Eij , Sz > H < Eij , Sz |, (2.37)

where states |Eij , Sz > have b een defined in equations (2.29). Expressing |Eij , Sz > viastates |Ni = 2, Si = {0, 2}, Siz = Si...Si > using known Clebsch-Gordon coefficients

|Eij , Sz >=SizS

jz

CS,SzSi ,S

iz;S

j ,S

jz

|Ni = 2, Si , Siz > |Nj = 2, Sj , Sjz >,

we can write the Hamiltonian (2.35) as

H =

| >i | >j H,;, < |i < |j , (2.38)

where states | - | belong to the set {S = 0}, {S = 2, Sz = 2,..., 2}, and H,;, isgiven by proper rotation of H,.

2.4.3 Phase diagram from the mean-field calculation

In this section we study the phase diagram of the system described by the Hamil-tonian (2.38) using translational invariant variational wave functions. Such mean-field ap-proach gives correct ferromagnetic and antiferromagnetic states for Heisenberg Hamiltoniansin d

2, so we expect it to b e applicable in our case. We think that this approach suc-

cessfully captures the main features of the system: first order transition b etween the spingapped and the nematic phases, the nature of the order parameter in the nematic phase,and elementary excitations in both phases. However, we cannot rule out the possibilityof more exotic phases that fall outside of our variational wave functions, for example, thedimerized phase discussed in Ref. [71]. Numerical calculations are required to study if suchphases will actually be present.

As we saw in the previous section, the energy of the two-site problem is minimizedwhen total spin is 0. However, energy on all bonds cannot be minimized simultaneously, so


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we cannot solve a problem exactly for a lattice. We use a mean field approach to overcomethis difficulty, taking variational wave function

| >= i

(c0,0|Ni = 2, S = 0, Sz = 0 > + m=2,...,2

c2,m|Ni = 2, S = 2, Sz = m >), (2.39)

|c0,0|2 +m

|c2,m|2 = 1. (2.40)

Now we can evaluate expectation value of energy over variational state (2.39) and find theground state numerically.

We parameterize conditions (2.40) as

c0,0 = cos , c2,m = sin am, (2.41)

m=2,...,2 |am|

2

= 1.

In Appendix A.3 we demonstrate that for which minimizes the mean-field energy, up toSU(2) rotations [am] has the form

1,

[am] = (0, 0, 1, 0, 0)T. (2.42)

Mean field energy does not depend on d and we find in the region of interest theenergy per lattice site to be

E[] = 3U2 sin2 +

zt2

12U0(51 + 4 cos 2 + 7 cos4 8

2sin2 + 4

2sin4). (2.43)

One can immediately see that if we try to expand this expression near = 0, there is nolinear term, but second and third order terms are present. This indicates that by changingthe parameters of the Hamiltonian, we will have a first order quantum phase transition, atwhich the value of that minimizes the energy changes discontinuously. This is typical forordinary nematics [83] since in Landau expansion third order terms are not forbidden byd d symmetry. The reason why our transition is of the first order can be traced backto the fact that mean field energy has terms which mix c0,0 and c2,m in odd powers, i.e.c0,0c2,2(c2,1)

2, so overall U(1) symmetry doesnt prohibit odd powers of in functional(2.43).

Since phase transition is of the first order, the transition is characterized by severalregimes. First, when t is small, global energy minimum is at = 0 and there are no other

local minima, i.e. we have spin singlets in all individual sites. Then, when conditionzt2/(U0U2) 0.4928 is satisfied, a local minimum appears at 0.25, see Fig. 2.5. Aswe continue increasing t, the minimum at nonzero becomes deeper, and eventually atzt2c/(U0U2) = 1/2 the global minimum of functional (2.43) is reached for sin c = 1/3, (seeFig.2.6). However, there is still a local minimum at = 0. If we keep increasing t, the

1Expression (2.42) describes an eigenstate of Sz with eigenvalue zero. After an SU(2) rotation we will

have [am] that is an eigenstate of dS with zero eigenvalue. Vector d corresponds to the direction of uniaxialnematic.


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35/200


0.1 0.2 0.3 0.4 0.5 0.6

-3.332

-3.328

-3.326

-3.324

-3.322

Energy

Figure 2.6: Dependence of the energy functional (2.43) on when zt2/(U0U2) = 1/2 (energyis given per lattice site in units of U2).

0.1 0.2 0.3 0.4 0.5 0.6

-3.88

-3.86

-3.84

-3.82

-3.78

-3.76

Energy

Figure 2.7: Dependence of the energy functional (2.43) on when zt2/(U0U2) = 9/16 =0.5625 (energy is given per lattice site in units of U2).


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Increasing t

global

transition

decreasing t

0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8t

0.1

0.2

0.3

0.4

0.5

Sin

Figure 2.8: Dependence of sin in equation (2.43) on t, measured in units ofU0U2z .

t tc t+

0.5 1 1.5 2t

1.2

1.4

1.6

1.8

2

Singlet Nematic

First order phase transition

Figure 2.9: Mott phase diagram for N = 2. Chemical potential is measured in units

of U0, t is measured in units ofU0U2z . Critical tunneling tc marks the actual first order

phase transition, while t and t+ correspond to limits of metastability. Superfluid-Insulatortransition line is presented as an eye guide, and will be discussed in section 2.6.


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2.4.4 Quantum fluctuations corrections for the spin singlet state

For small enough t, mean-field analysis of the previous section predicts the singlet

ground state that does not depend on t. Now we will consider quantum fluctuations nearthis state to obtain more accurate wave function and excitation spectra. We can rewriteEq. (2.38) via Hubbard operators

Ai = |i|i. (2.44)

Here |i and |i belong to the set {S = 0}, {S = 2, Sz = 2,..., 2}. Commutation relationsbetween Ai are very simple:

[Ai , Ai ] = A

i Ai . (2.45)

Now we introduce boson operators bi, that create states with Si = 2, Siz = , and c, thatcreates a singlet on ith site. Our physical subspace is smaller than a generic Fock space ofthese bosons and should satisfy the condition

ci ci +

bibi = 1 (2.46)

on each site.

One can easily check that if we set A = bb for spin S = 2 states and similarsubstitution with c bosons when one of the states is a singlet state (which we will denote ass), then commutation relations (2.45) are satisfied. Since for small enough t only a singletstate is occupied in mean field approximation, we can resolve constraint (2.46) using an

analog of Holstein-Primakoff representation near cc = 1 state [80], which is given by

Ai = bibi, A

si = (1 bibi)1/2bi, (2.47)

Assi = 1 bibi, Asi = bi(1 bibi)1/2. (2.48)

Now we expand our initial Hamiltonian in terms of now independent operators bi up to thesecond order:

H(2) =

(H;ssbib

j + Hss;bibj + Hs;sb

ibj + Hs;sb

jbi) +

+z

2i

(Hs;sbibi + Hs;sb

ibi + Hss;ss(1

bibi

bjbj)). (2.49)

Calculation of matrices H; gives necessary matrix elements

Hs;s = Hs;s= (203

t2

U0+

3U2z

),

Hss;ss = 203

t2

U0,


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Hs;s= Hs;s = 83

t2

U0,

H;ss = Hss;= 83

t2

U0

0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0

.

We rewrite our Hamiltonian in terms of Fourier transforms of bi operators, bk, bk:

H(2) = E0+z

2

k

k(H;ssbkb

k + Hss;bkbk + Hs;sb

kbk + Hs;sb

kbk)+

+2(Hs;s Hss;ss)bkbk, (2.50)

where E0 is classical energy andk =

1

z

e

eike . (2.51)

Now we will use canonical Bogoliubov transformations to diagonalize this Hamiltonian.Since most of the terms are diagonal in , subspace, it is easy to see that requiredtransformation mixes operators {b0k, b0k}, {b1k, b1k}, {b1k, b1k}, {b2k, b2k}, and{b2k, b2k}.

Transformation that mixes the first pair is

b0k = cosh k0k + sinh k0k,

b0k = cosh k0k + sinh k0k,and its complex conjugates. Substituting this transformation into Eq. (2.50) and requiring

that terms with 0k0k and 0k0k vanish, we obtain the equation for k

tanh2k = gkfk

= 83zt2

U0k

3U2 83 zt2

U0k

,

where

gk =8

3

zt2

U0k,

fk = 3U2 8

3

zt2

U0 k.

Energy of this excitation equals

E(k) =

f2k g2k =

U2(9U2 16zt2

U0k). (2.52)

Equation (2.52) suggests that the first instability appears at k = 0 and gives thephase boundary that agrees with the metastability line t+ found in the previous subsection.


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0.2 0.4 0.6 0.8 1t

0.5

1

1.5

2

2.5

3

Gap

Figure 2.10: Dependence of excitation gap (measured in units of U2) on t, measured in units

ofU0U2

z .

However, the results of the previous subsection suggest that the phase transition is of thefirst order and takes place before the mode softening at k = 0. The first order transitionmay also be obtained with the formalism presented in this section by noting that expansionof Eq. (2.48) allows third order terms cbb + c.c..

We can use similar analysis to discuss excitations with other spin quantum num-bers. For example, excitations with Sz = {+1, 1} are diagonalized by analogous Bogoli-ubov transformations with k k, and excitations with Sz = {+2, 2} are diagonalizedwith transformations with the same k. As required by the spin symmetry of the singletstate, all of these excitations have the same energy.

Now we can discuss the approximations made while expanding over bk, bk . Whiletransformation (2.48) is the exact resolution of the constraint (2.46), expansion to the secondorder adds states with higher boson occupation numbers and changes Hilbert space (this iscompletely analogous to usual antiferromagnet spin-wave theory). However, if a posterioriwe can verify that only states with occupation numbers ni = {0, 1} are present in theground state, then expansion of the constraint (2.46) up to the second order was justified.The parameter that controls such expansion is

bibi =1

N

bkbk =

sinh2 k

ddk

(2)d.

Calculation of this quantity while the singlet state is still a global maximum for D = 3gives numerical values < 0.001; therefore, our expansion is much more precise than forHeisenberg antiferromagnet, where this quantity is not much smaller than 1 and one needsthe condition S 1 to justify the spin wave theory.

2.4.5 Spin wave excitations in the nematic phase

Now we will consider excitations for the states with nematic order. For the statesdescribed by Eqs. (2.39) - (2.42), there are no expectation values of the spin operators


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S = 0, but there is a nematic order (2.27) when = 0. For example, when d is pointingalong z we find

Qab = sin2

1 0 00 1 0

0 0 2

(2.53)

In the singlet phase, expectation values of both the spin operators are S = 0 and Qab = 0.In the singlet phase the system has a gap to all excitations of order U2, while nematicphases have gapless spin-wave excitations that originate from the breaking of the continuoussymmetry. The general form of the state with minimum energy is expressed via Euler anglesof order parameter d as

Uz()Uy()Uz(){0, 0, 1, 0, 0}T,where U() are finite angle rotation matrices. From Eqs. (2.53) and (2.28) we can expressthe nematic order parameter for such a state as

Qab = 3sin2 (dadb 13

ab).

Goldstone theorem tells us that low lying modes are given by fluctuations of the directionof d, and that there will be two degenerate modes. We can utilize the approach used inthe previous subsection to consider excitations in the nematic phase. For that we shouldmake a generalized Holstein-Primakoff expansion near the nematic state. First, we make aunitary transformation in Hilbert subspace of each site, which is given by

|0

|1|2|3|4|5

=

cos 0 0 sin 0 0

0 012 0

12 0

0 0 12

0 12

0

sin 0 0 cos 0 00 1

20 0 0 1

2

0 12

0 0 0 12

|S = 0

|Sz = 2|Sz = 1|Sz = 0|Sz = 1|Sz = 2

. (2.54)

Making appropriate transformation on H; , we can write our Hamiltonian as

H =

| >i | >j H,;, < |i < |j , (2.55)

where states | - | belong to the set {|0 |5}. After that, we proceed exactly as inthe previous subsection, expanding near |0 state. Since dependence on t2/(U0U2) isdetermined by the minimization of the energy, linear terms in bk and b

k are absent.

Quadratic terms have exactly the same form as in Eq. (2.49), and all matrices becomediagonal due to the proper basis choice (2.54). Now we can use Bogoliubov transformationto diagonalize the quadratic part. For excitations to states |1 and |2, we obtain energies

E21 (k) = E22 (k) =

1

36(162k

z2t4

U20(4cos2 +

2sin2)2+


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0.72 0.74 0.76 0.78 0.8 0.82 0.84t

2.8

2.85

2.9

2.95

3.05

3.1

Velocity

Metastable

Figure 2.11: Dependence of the spin wave velocity (measured in units of U2/

z) on t,

measured in units ofU0U2

z .

(9zt2

U0 18k zt

2

U0+ 9U2 + (2(k 1)zt

2

U0+ 9U2)cos2 7 zt

2

U0cos4

+4

2(1 k) zt2

U0sin2 4

2

zt2

U0sin4)2),

where k was defined in Eq. (2.51), and dependence of on zt2/(U0U2) is shown in Fig.

2.8. We find that for k = 0, energies of these excitations are zero, as expected for nematicwaves from Goldstone theorem. These excitations create states with Sz = 1. For small k,the energy of excitations depends linearly on

|k

|, and dependence of the spin wave velocity

on the parameters of the lattice is shown in Fig. 2.11.Let us now consider gapped excitations for the nematic phase. Excitation to the

state |3 corresponds to longitudinal fluctuations in the value of , and the energy of suchexcitations becomes zero at t since at this point fluctuations of are not suppressed.Excitations to the states |4 and |5 correspond to the creation ofSz = 2 states and theyare degenerate. For all of these excitations, energies are minimized for k = 0. Dependenceof the gap on parameters is shown in Fig. 2.12.

2.4.6 Magnetic field effects

Let us now consider the effects of a magnetic field, H U2, on our system. ForU2 in the range of kHz (see Fig. 2.2), this corresponds to magnetic fields of the order of1mG. This field is small enough that it does not change the scattering lengths due to theenergy level shifts inside of atoms (Feshbach resonances). Since all atoms have the samegyromagnetic ratio, interaction with external magnetic field depends only on the total spin,and the internal structure of the states is not important.

Let us first consider the case of small magnetic fields, H = 0), and the second order contribution depends onthe relative orientation of the nematic order parameter d and magnetic field H. Suppose


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0.72 0.73 0.74 0.75t

1

2

3

4

Gap

Metastable

longitudinal excitation

Sz=2 excitation

Figure 2.12: Energy of the gapped excitations in the nematic phase of N = 2 atoms per

site at zero wave vector. The energy gaps and t are measured in units of U2 andU0U2

z ,respectively.

d is directing along the z axis and H lies in x, z plane. In the second order perturbationtheory, energy correction to the ground state is always non positive:

E(2) = En

< 0|HS|n >< n|HS|0 >En E0 =

En

(Hx)2 < 0|Sx|n >< n|Sx|0 >

En E0 < 0;

this quantity is of order (Hx)2/t2. Since for the state with d|| H it is again zero andthe system doesnt benefit from magnetic field, energy is minimized when d lies in a plane

perpendicular to H (this is completely analogous to antiferromagnets). Using this property,one can distinguish a nematic phase from a singlet phase. One should apply a small magneticfield in z direction to fix the plane in which d lies, release the trap, and let the atoms fallin the gravitational field with some magnetic gradient (to separate the states with differentSz). Then, one should measure quantities of each spin component. These values will havea sharp change when we cross the first order phase transition line. Since we know howto express spin states via original boson operators, we can calculate expectation values ofdifferent spin components to be:

n1 = n1 =2

3cos[]2 +

2

3cos[]sin[] +

5

6sin[]2,

n0 =2

3cos[]2 22

3cos[]sin[] +

1

3sin[]2.

Using known expressions for dependence of on t, we can make mean-field predictions onoccupation numbers, shown in Fig. 2.13

An interesting question to consider now is the effect of magnetic fields of the orderofH U2 on the phase diagram. We have found out earlier that singlet - nematic transitionis weakly first ordered, and mode softening analysis is quite reliable. Thus stability regionof the singlet phase can be estimated by looking at the mode softening in the presence of


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0.5 1.5 2 2.5

t

0.2

0.4

0.6

0.8

1

n1=n-1

n0

Metastable transitions

Figure 2.13: Dependence of occupation numbers n0 and n1 = n1 on t for an insulating

state with two bosons per site, N = 2. Tunneling t is measured in units ofU0U2z .

magnetic fields. Energies of the excitations out of singlet phase have been calculated earlier,see Eq. (2.52). Out of these excitations, the mode with spin Sz = +2 becomes unstablefirst, which defines the boundary of the singlet phase as

H =1

2

U2(9U2 16zt2

U0 ).

We note that mode softening analysis is exact for t = 0.Ift = 0 and H > 3U2/2, the global ground state is ferromagnetic. Using the same

approach, we can also find the instability of the ferromagnetic phase. Ferromagnetic stateis the exact eigenstate of the effective Hamiltonian, and in the second order of Holstein-Primakoff expansion terms like b+ib

+j dont appear, so one particle excitation spectra is

exact. Calculation analogous to what was done before shows that the mode which becomesunstable first is the linear combination of {S = 0, Sz = 0} and {S = 2, Sz = 0}, with thecritical magnetic field given by

H =12

3U22

4zt2U0

+12

9U22 + 16U2

zt2

U0+ 64(

zt2

U0)2 .

Results of these calculations together with the results for small magnetic fields suggest thefollowing picture for the global ground state in the presence of magnetic fields. For zeromagnetic field, there are two phases, singlet and nematic. When small magnetic field isapplied, nematic order parameter lies in the plane perpendicular to a magnetic field, andthere is a small ferromagnetic component along the axis of the magnetic field. As magnetic


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0.5 1 1.5 2

H

U2

0.2

0.4

0.6

0.8

1

1.2

1.4

zt2

U0U2

Singlet

Canted

Nematic Ferromagnet

Figure 2.14: Magnetic phase diagram for N = 2 insulating phase. On the mean field level

singlet state is the product of S = 0 states at each site. Transition out of the singlet stateis of the first order. Transition from insulating phase to the superfluid takes place at thevalues ofy axis U0/(zU2), and for small enough U2/U0 all the features of the diagram arepresent.

field goes up, magnetization gradually increases, and saturates when the instability regionof the ferromagnet is crossed. If we start from a singlet phase, there is a region of cantednematic phase between singlet and ferromagnet, and this phase is stabilized by magneticfield. Magnetization changes continuously inside of this phase. This phase diagram is shownin Fig. 2.14.

2.5 Large number of particles per site

In this section we discuss the case N 1 for both parities ofN. We show how onecan separate variables describing angular momentum and the number of particles at eachsite [86], and derive an effective Hamiltonian which is valid under conditions U2, N t U0,which is less restrictive than in section 2.3 for N >> 1.

When we have N spin-one bosons localized in the same orbital state, their totalspin may take any value that satisfies constrains

S+ N = even, (2.56)

S

N. (2.57)

We define pure condensate wave functions as

|N, n = 1F

(nxax + nya

y + nza

z)N|0, (2.58)

which minimize the U2 interaction energy at the Gross-Pitaevskii (mean-field) level at eachgiven site [64]. Here F = [2(N 1)!]1/2 is a normalization factor, which is calculated inAppendix A.4.


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Now we can construct states as

|N =

n

(n)|N, n

, (2.59)

wheren

stands for

dn/(4).Condition (2.56) corresponds to the symmetry of the states (2.58)

|N, n = (1)N|N, n. (2.60)

Hence, we need to consider only wave functions that satisfy (n) = (1)N(n).Now we can consider how a spin rotation operator acts on the wave function (n):

eiS|N =n

(n)eiS|N, n =n

(eiSn)|N, n =n

(n+ n))|N, n. (2.61)

Expanding the last expression for small we find

L = in n

, (2.62)

where we used L rather than S to show that it acts on the wave function . Therefore,operator L is an angular momentum operator for n. If we want to construct some spin state,we should take (n) to be a usual spherical harmonic. We note that the S = 0 result inRef. [64] is just a special case of our general statement. The most general form of the stateat each site can be expanded as

(n) =

S,|m|ScS,mYSm(n),

where S satisfies conditions (2.56)-(2.57).Up to this point, what we have done is valid not only for large N, but for all

N. This representation is particularly suitable for N 1 since in this limit states thatcorrespond to different ns are orthogonal to each other (see Appendix A.4)

N, n1|N, n2 = N(n1 n2). (2.63)

The delta function is defined from the conditionn1

n2

f1N(n1)f2N(n2)N(n1 n2) =

n

f1N(n)f2N(n) (2.64)

for the functions that satisfy fN(n) = (1)NfN(n).We show in the Appendix A.4 that

a|N, n = (N + 1)1/2n|N + 1, n,a|N, n = N1/2n|N 1, n (2.65)


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after projecting into the pure condensate wave functions.

This allows us to represent a, a as products of two operators, which act in

different spaces. For each trap we define the particle creation and annihilation operatorsthat change the number of particles N but not the direction of n 2

bi |Ni, ni = (Ni + 1)1/2|Ni + 1, ni,bi|Ni, ni = N1/2i |Ni 1, ni. (2.66)

The number of particles in each trap may be expressed using b operators as

Ni = bibi. (2.67)

Hamiltonian (2.5) can now be represented as

H = t

adilet imambekov- strongly correlated phenomena with ultracold atomic gases

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