Disorder-Induced Order with Ultra-Cold Atoms

Disorder-induced order with ultra-cold atoms

Armand Niederberger

Thesis Advisor: Prof. Maciej Lewenstein

Thesis Co-advisor: Dr. Fernando Cucchietti 2 Contents

1 Introduction 5 1.1 General context ...... 6 1.2 Main results on disorder-induced order ...... 8

2 Basic concepts and methods 11 2.1 Description of classical and quantum gases ...... 11 2.2 Quantum phase transitions ...... 15 2.3 Optical potentials used with quantum gases ...... 16 2.4 Disorder eﬀects in ultracold gases ...... 23

3 General principle of disorder-induced order 27 3.1 Common models for magnetization ...... 27 3.2 The Mermin-Wagner-Hohenberg Theorem ...... 29 3.3 Studies of disordered systems ...... 31 3.4 Large eﬀects by small disorder ...... 32 3.5 Disorder-induced order in the classical XY model ...... 33 3.6 Disorder-induced order in other systems ...... 38 3.7 Towards experimental realization of disorder-induced order in ultra-cold atomic systems ...... 40 3.8 Conclusion ...... 45

4 Coupled Bose-Einstein Condensates 47 4.1 The Gross-Pitaevskii equation ...... 47 4.2 DIO in Raman-coupled BECs ...... 51 4.3 Numerical simulations ...... 53 4.4 Studies of the coupled BECs ...... 55

3 CONTENTS

4.5 Conclusions ...... 58

5 Disorder-induced phase control in superﬂuid Fermi-Bose mixtures 61 5.1 Elements of BCS Theory ...... 62 5.2 Disorder-Induced Order for BCS/BEC systems ...... 64 5.3 Theoretical study ...... 66 5.4 Numerical study ...... 67 5.5 Conclusions ...... 68

6 Disorder-induced order in quantum XY chains 71 6.1 Model description ...... 71 6.2 Numerical Methods ...... 73 6.3 Numerical Results ...... 74 6.4 Experimental realization ...... 80 6.5 Conclusions ...... 83

7 Conclusions 85

Acknowledgements 89

Bibliography 91

4 Chapter 1

Introduction

This thesis summarizes my research on ordering as a consequence of disorder in the field of ultra-cold atomic gases in the Quantum Optics Theory Group at ICFO – The Institute of Photonic Sciences. It contains a general introduction to the field, a description of the results from all relevant publications in which I was involved during my doctoral studies, as well as the conclusions from our research. After this introductory chapter, in Chapter 2, we explain some fundamentals of the physics of ultra-cold atomic gases and disordered systems. In particular, we review the development and use of the most prominent example of ultra-cold atomic systems: the Bose-Einstein Condensate (BEC). We show how BECs are used to mimic condensed matter lattice models [Lewenstein et al., 2007], thus enabling experimentalists to realize well-controlled model systems. In fact, the exquisite control over the experimental setup persists even for disordered systems and we review the most common approaches to realize disorder used nowadays [Fallani et al., 2008]. Finally, we illustrate the impact of disorder on both weakly and strongly interacting systems. Within the framework of disordered systems, chapters 3 to 6 report the results described in the articles published during my doctoral research. First, chapter 3 explains the fundamental ideas behind disorder-induced order in the case of the classical XY model [Wehr et al., 2006]. Chapter 4 shows then how we applied this idea to systems composed of a two-component BEC in which disorder-induced order sets the relative phase between the two condensates [Niederberger et al., 2008]. Then, chapter 5 explains an analogous study involving Bose-Fermi mixtures in which the phase between the pairing function and the condensate wave-function becomes fixed by disorder [Niederberger et al., 2009]. Finally, Chapter 6 shows our studies of the quantum-XY model, in which we numerically simulated one- dimensional spin chains and observed signatures of disorder-induced ordering leading to a magnetized phase [Niederberger et al., 2010]. Concluding remarks on present and future studies are given in Chapter 7.

5 CHAPTER 1. INTRODUCTION

1.1 General context

Disorder is common in many fields of physics, and often considered as an imper- fection. For example, during crystallization processes, atoms or molecules form solid structures with very precise arrangements that minimize the total energy of the system. However, thermal motion of the molecules, chemical impurities, or pressure differences, often lead to defects in the stacking of these molecules or atoms. Dislocation points, lines, or even planes are then formed, reducing the “ideal” mechanical, chemical, electronic, magnetic or optical properties of the sample. In such situations, disorder is usually treated as a perturbative effect in an otherwise perfectly ordered setting. Oftentimes, this kind of disorder has an undesirable impact: mechanical strength of airplane wings, lifetime of a solid state laser, or even the value of a diamond typically suffer if the ordering is not perfect. But disorder is far more than a dangerous lack of order: it can also induce new phases in matter. The most famous disorder-induced phase is Anderson Localization, in which particles localize in a conducting medium due to multiple scattering from impurities. After its original proposal in 1958 [Anderson, 1958], Anderson localization was soon understood to be a wave phenomenon, which was applied consequently to sound-waves [Weaver, 1990, Hu et al., 2008] as well as different electromagnetic-waves [Wiersma et al., 1997, Chabanov et al., 2000, Schwartz et al., 2007, Lahini et al., 2008, Szameit et al., 2010]. Observation in the originally considered scenario of matter waves, however, turned out to be extremely difficult and was only achieved in 2008 using ultra-cold atomic gases [Billy et al., 2008, Roati et al., 2008]. The quest for Anderson localization fueled research on ultra-cold atomic quantum gases leading to a wide range of interesting results. In recent years, a number of theoretical approaches to create controlled disorder were thus developed [Damski et al., 2003, Roth and Burnett, 2003a, Roth and Burnett, 2003b]. On the experimental side, signatures of the disordered Bose-glass state were observed in ultra-cold gases [Fallani et al., 2007]. Furthermore, a number of groups have been studying the impact of disorder on localization phenomena, particularly in Bose gases [Scalettar et al., 1991, Fort et al., 2005, Lye et al., 2005, Bilas and Pavloff, 2005, Clément et al., 2005, Schulte et al., 2005]. Another related area of intense research are spin glasses [Edwards and Anderson, 1975, Sherrington and Kirkpatrick, 1975, Fisher and Huse, 1986], where, once again, studies involving ultra-cold atomic gases turned out to be a powerful approach for advancing our understanding of disordered systems [Sanpera et al., 2004, Ahufinger et al., 2005]. This thesis studies disorder as a resource for creating order in physical systems. In fact, even if disordered quantities commonly tend to destroy perfect ordering, we have found that there exist physical settings, in which a disordered parameter can induce new, ordered phases. We termed this new effect disorder-induced order and found that it is applicable to a wide range of condensed matter systems

6 1.1. GENERAL CONTEXT and ultra-cold atoms. Our numerical and analytical studies show that disorder- induced order allows to induce spontaneous magnetization in systems that would not normally magnetize spontaneously, and to control the relative phase of certain coupled quantum systems. Also, we conclude that disorder-induced order is a highly robust effect, which works even if the disordered quantity is not truly random, but quasi-random, i.e. quasi-periodic or even periodic. In order to study disorder effects, it is hugely beneficial and often indispens- able to control the disordered quantity. In most solid states systems, however, disorder originates from noisy experimental parameters and is thus very difficult to control. In recent years, ultra-cold atomic gases have become a popular testbed for experimentally simulating condensed matter and quantum information problems. Apart from intrinsic quantum behavior, ultra-cold gases experiments are a tremendously powerful tool [Jaksch and Zoller, 2005] because of the unprecedented degree of control over experimental conditions achievable in such systems. Interactions between particles, deterministically disordered potentials and the possibility to control the dimensionality of the system have already led to seminal breakthroughs like the creation of a Bose-Einstein Con- densate [Anderson et al., 1995, Davis et al., 1995], the transition between Mott Insulator and Superfluid [Greiner et al., 2002], as well as the above-mentioned creation of Anderson Localization in matter waves. Due to the numerical and experimental complexity of the studies in atomic, molecular, and optical (AMO) physics, experimentalists and theoreticians often specialize on their respective aspects. For this thesis, we are focusing on the numerical studies of disorder-induced order, as well as on some analytical models to improve our intuitive understanding of disorder-induced order. The low-temperature behavior of the XY and Heisenberg model are long- standing topics in condensed matter and AMO physics, presenting a wealth of interesting phenomena [Petrov et al., 2004]. In 1D at T = 0, for example, neither the XY , nor the Heisenberg quantum model magnetize in the infinite system limit, because of phase fluctuations [Pitaevskii and Stringari, 1991]. In 2D, however, the same systems order at T = 0. Finite systems can order at non- zero temperatures, as long as their phase correlations are larger than the system size. At slightly higher temperatures, with phase correlations smaller than the system size, phase fluctuations destroy order of the system. However, there are no density fluctuations, and the system is thus in a quasi-condensate state up to a critical temperature. Above this temperature, the system ceases to order. An- other subtlety worth mentioning is the presence of a trap in the system, which changes the density of states of the system. Therefore, trapped 1D system is very similar to a trap-less 2D system, and a trapped 2D system similar to a trap-less 3D system. At low dimensions and non-zero temperature, low-energy excitations immediately destroy macroscopic magnetization in continuous symmetry (e.g. XY or Heisenberg) systems. Even so, the 2D XY model presents an interesting phase transition: there exists a quasi-ordered low-temperature phase with correlation functions that decrease with distance as a power law. The corresponding

7 CHAPTER 1. INTRODUCTION transition from the high-temperature regime with its exponential correlations to the quasi-ordered phase is called Berezinskii-Kosterlitz- Thouless (BKT) transition [Berezinskii, 1972, Kosterlitz and Thouless, 1973, Hadzibabic et al., 2006].

1.2 Main results on disorder-induced order

The main results obtained during this thesis are all published and reported in one chapter per publication. The fundamental mechanism of disorder-induced order is summarized in 1.2.1. Then, an overview of our application of the eﬀect to two types of ultra-cold atomic systems is given in 1.2.2. Finally, the main result of our study involving the quantum version of the XY model is given in 1.2.3.

1.2.1 The mechanism of disorder-induced order

In chapter 3, we present the basic intuition of disorder-induced order: take a system with a continuous symmetry such as the classical isotropic ferromagnetic XY model. In this model, neighboring spins tend to align, at zero temperature, between themselves within a preferential plane, the XY plane spanned by the x and y axis. For this kind of system yielding a continuous symetry (the angle of the spin-orientation), the Mermin-Wagner-Hohenberg theorem [Mermin and Wagner, 1966, Hohenberg, 1967] states that there is no spontaneous magnetization for any T > 0. The effect of an external magnetic field applied to ferromagnetic XY system is to align the spins with this field. The disordered fields used for disorder-induced order have a distribution with a different symmetry than the system. The bare XY model, without any magnetic field, has a continuous U(1) symmetry represented by the angle of the spins with respect to, for example, the x axis. Our disordered distribution must not have the same symmetry. In fact, the role of the disordered field is to break the continuous symmetry. This can be achieved by introducing a disordered field which is always in the same direction such as, for example, the x axis. The amplitude of our field is disordered and follows, for example, a normal distribution centered at zero. This disordered XY system no longer presents the U(1) symmetry but merely a mirror symmetry with respect to the x axis. This remaining symmetry is lifted by setting the boundary spins to ping in the, say, positive y direction. The resulting system will now have a tendency to spontaneously magnetize in the positive y direction, which would not be the case in absence of the disordered field. It is important to realize that the magnetization originates from the disordered field and not from the boundary conditions. Also, we argue that our arguments also hold for small temperatures, at which low-energy excitations would completely destroy magnetization of the bare XY model.

8 1.2. MAIN RESULTS ON DISORDER-INDUCED ORDER

1.2.2 Phase control ultra-cold atomic gases

In chapter 4, we show that disorder-induced order can fix the relative phase of two Raman-coupled Bose-Einstein Condensates. The mathematical description of such a system resembles the ferromagnetic XY model: the arbitrary relative phase of the two uncoupled condensates is analogous to the continuous symmetry of the bare XY spins, and a strictly real-valued Raman-coupling takes the role of the magnetic uni-axial random field. Our results show that disorder- induced order fixes the average relative phase to any desired value in one, two and three dimensions. Chapter 5 shows that the same principles holds for BCS paired fermions in presence of a reservoir of di-atomic molecules of the same atomic species in a BEC. Photo-associatve-dissociative coupling the two subsystems is taken to be strictly real-valued, which creates an analogous situation to the case of the two BECs. Our results show that disorder-induced ordering allows experimenters to arbitrarily fix the relative phase between the condensate wave-function and the pairing function of the superfluid fermions.

1.2.3 Induced magnetization in quantum XY spin chains

Our studies of quantum XY spin chains in 1D show that disorder-induced order leads to spontaneous magnetization and shows signs of a quantum phase transition. We base our conclusions on the numerical study a full range of staggered, regularly oscillating and normally distributed random fields. In all of these types of fields, we find disorder-induced ordering in the form of the appearance of spin magnetization orthogonal to the disordered field. Also, we study the block entropy for different block sizes, calculated by dividing the chain into two sub-chains at a given position. For partitions in the bulk of the chain, we see that the block entropy gradually decreases from zero disorder up to a certain disorder amplitude, and then rapidly increases. This indicates the presence of a quantum phase transition between an orthogonally ordered and an orthogonally disordered state at zero temperature.

9 CHAPTER 1. INTRODUCTION

10 Chapter 2

Basic concepts and methods

In this chapter, we explain the general context in which this thesis is situated. In Section 2.1 we illustrate the basic intuition behind ultra-cold quantum gases and how classical particles transform into quantum particles. Then, in Sec- tion 2.2, we compare classical and quantum phase transitions. In Section 2.3, we present one of the most important tools for manipulating ultra-cold gases, namely optical lattices, and explain how controlled disordered potentials can be created. Finally, in Section 2.4, we show which areas of disordered systems are mostly studied using ultra-cold gases and why these studies are attracting attention from condensed matter, quantum information and atomic, molecular and optical physics.

2.1 Description of classical and quantum gases

The standard approach to a classical gas is to describe the gas particles as tiny billiard balls. In this picture, the gas pressure originates from the impacts of the gas particles, and temperature is a direct measure of the thermal motion of the particles [Feynman et al., 1964]. A direct consequence of this description is that temperature is not expressed in an arbitrary scale such as degrees Celsius or Fahrenheit, but rather in the absolute Kelvin scale. Zero Kelvin, in this classical picture, means that the particles have no kinetic energy1. Interestingly, the intuition of temperature being a measure for the thermal motion of particles also holds for classical liquids and solids, and often serves as a tremendously helpful parameter for describing experimental systems: rather than having to consider the full microscopic description of a system, it is often enough to resort to a simpliﬁed thermodynamic picture involving only temperature, pressure and some material constants. As it turns out, however, the classical description does not cover ultra-cold phenomena, like superconductivity, or Bose-Einstein con-

1More precisely: there exists an inertial system in which all particles are at rest.

11 CHAPTER 2. BASIC CONCEPTS AND METHODS densation. A more adequate, quantum description of such systems is therefore needed. The de Broglie wavelength serves to illustrate why the classical picture does not hold at ultra-low temperatures. In quantum physics, the model of particles being represented by tiny billiard balls with perfectly defined momentum and position is not applicable [Piron, 1998, Peres, 1995]. Nonetheless, using the simplified picture of particles with uncertain position and/or speed, we can postulate that the particles have a well-defined momentum but an uncertain position [Cohen-Tannoudji et al., 1998]. The (center of mass) position uncertainty is, in this picture, described by the de Broglie wavelength. To derive it, one associates to a particle of mass m, energy E and momentum p a wave satisfying the same relations as photons, namely

E = ~ω p = ~k, (2.1)

h −34 where ~ = 2π ' 1.05 10 Js is deﬁned via the Planck constant h, ω is the 2π angular frequency of the wave and k = λ its wave vector. In this intuitive picture, we further suppose that the thermal motion of the gas particles in a three-dimensional recipient follows the classical equipartition law

1 p2 3 mv2 = ' k T, (2.2) 2 2m 2 B

−23 J where v is the average speed of the particles, kB ' 1.38 10 K the Boltz- mann constant, and T the absolute temperature. Combining this equation with Eq. (2.1), we ﬁnd the expression for the de Broglie wavelength λdB: h 1 λdB = √ ∝ √ . (2.3) 3mkBT mT

Figure 2.1 illustrates the transition between a classical and a quantum gas using the de Broglie wavelength. A Nitrogen molecule contained in air at T = 300K, for example, has a de Broglie wavelength a fraction of an Angström˚ 2. The intuitive, classical picture of a tiny billiard ball is therefore very accurate because the position uncertainty of the center of mass is smaller than the size of the atoms. For very cold particles in the order of a few Kelvin only, the de Broglie wavelength and thus the spatial spread of the particle will be in the order of several Angström,making˚ classical predictions increasingly inaccurate. As λdB becomes of the order of the interatomic distance or, more precisely speaking, the mean free path of the particles, quantum effects play a role in the description of the gas and the classical model completely breaks down. At and below these temperatures, a new quantum state of matter appears: the Bose-Einstein Condensate.

2 g −26 In fact, for N2 molecules, m ' 28 mol ' 4.65 10 kg, and thus at T = 300K, the position −11 ˚ uncertainty of the center of mass is in the order of λdB ' 2.6 10 m = 0.26A.

12 2.1. DESCRIPTION OF CLASSICAL AND QUANTUM GASES

Figure 2.1: Intuitive reason for quantum behavior in gases: At high temperatures, the particles of a weakly interacting gas can be treated as tiny billiard balls because their de Broglie wavelength is negligible. At low temperatures, the de Broglie wavelength is no longer negligible and leads to pronounced quantum- mechanical eﬀects. At ultra-low temperatures, the de Broglie wavelength is in the order of the interatomic distance and a Bose-Einstein Condensate forms. As temperature further approaches absolute zero, the thermal cloud disappears and the whole system condenses in the same quantum state. Illustration taken from [Ketterle et al., 1999].

For several decades, the Bose-Einstein Condensate (BEC) was one of the most challenging and intriguing predictions of quantum statistical physics. When Albert Einstein generalized Satyendranath Bose’s results in 1925, Bose-Einstein condensation was rather a Gedankenexperiment than an experimental proposal. Several technological breakthroughs were necessary to pave the way for the realization of Bose-Einstein condensation: the invention of lasers, transistors, integrated circuits and laser trapping of atoms – each of which were rewarded by the Nobel Prize in Phyics. Figure 2.2 therefore marks the conclusion of the quest for Bose-Einstein condensation in 1995 with Rubidium and shortly after with Sodium atoms, which earned Eric A. Cornell, Carl E. Wieman, and Wolfgang Ketterle the Nobel Prize in 2001 [Cornell and Wieman, 2002, Ketterle, 2002]. Bose-Einstein Condensates are a coherent state of matter, similar to a laser beam. All atoms in the BEC are indistinguishably described by the same wave-

13 CHAPTER 2. BASIC CONCEPTS AND METHODS

Figure 2.2: First experimental observation of a Bose-Einstein Condensate in Rubidium atoms. This false-color image shows the velocity distribution of a cloud of Rubidium atoms around the condensation temperature. Left: right before the appearance of the BEC, we see the broad velocity distribution of the atoms. Middle: right after the appearance of the BEC, the thermal cloud is still present, but many atoms have already condensed into the same motional state indicated by the asymmetric central peak. Right: at even lower temperatures we see a reduced thermal cloud and a nearly perfect BEC. Picture: University of Colorado.

function. This allows to study intrinsic quantum phenomena directly, simply by recording the probability distribution with a CCD camera. Furthermore, the fact that a BEC is in such a precisely defined quantum state also allows to perform highly accurate studies on fundamental aspects of quantum physics. And finally, the experimental techniques developed alongside studies on BECs allow for unprecedented control of experimental parameters such as interactions between particles, as well as the geometry and even dimensionality of the physical system. Today, a mere 15 years after the first experimental realization, Bose-Einstein condensates and ultra-cold atomic gases in general have already become a popular testbed for a number of quantum optics, condensed matter, and quantum information problems. The impact and complexity of these achievement can hardly be overstated and keep growing as new atomic species are Bose- Einstein condensed successfully: Lithium [Bradley et al., 1997] and Potassium [Modugno et al., 2001], followed by Caesium [Weber et al., 2003] and Chromium [Griesmaier et al., 2005], and recently joined by Calcium [Kraft et al., 2009], Strontium [Stellmer et al., 2009], Ytterbium [Fukuhara et al., 2007] and even molecules of fermionic isotopes [Jochim et al., 2003].

14 2.2. QUANTUM PHASE TRANSITIONS

2.2 Quantum phase transitions

One aspect in which ultra-cold atomic gases are of particular value as an experimental tool are quantum phase transitions. Disorder-induced order, as we show in this thesis, shows signs of a quantum phase transition between ordered magnetized states and disordered demagnetized states, even at zero temperature. In order to appreciate the differences between classical and quantum phase transitions, let us start by reminding some fundamentals of classical thermal phase transitions. Probably the best-known example of thermal phase transitions occurs in H2O, and serves as the basis for the Celsius temperature scale. At normal pressure, 1013.25 mbar, and a low temperature defined as 0◦C, crystalline ice melts and turns into liquid water. At a considerably higher temperature, water turns into gaseous water vapor, defining, at normal pressure, 100◦C. The intuitive picture explaining this behavior is that ice is solid because the thermal agitation is dominated by the hydrogen bonds between the water molecules. At 0◦C, the rigid crystalline structure can be broken by adding heat to the system, but hydrogen bonds keep playing a major role in liquid water. At 100◦C, the kinetic energy of the water molecules is sufficient to beat attractive intermolec- ular forces, and water turns into vapor. The three classical phases – solid, liquid and gas – are complemented at very high temperatures by the plasma phase, in which the molecules break apart to form a gas of free electrons and ions. Thermal phase transitions are generally categorized according to the Ehrenfest classification scheme, which highlights the lowest discontinuous derivative of the free energy. According to this scheme, solid/liquid/gas/plasma transitions are called first-order phase transitions because their density – the first derivative of the free energy with respect to the chemical potential – is discontinuous through the phase transition. Second-order phase transitions, in contrast, are continuous in the first-order derivatives of the free energy but contain a discontinuity in one of its second-order derivatives. A well-known example of a second-order transition is the transition to superconductivity. While higher-order transitions are, in principle, included in the Ehrenfest classification scheme, modern us- age typically only distinguishes between first-order and higher-order transitions [Huang, 1987]. Quantum phase transitions, in contrast, are driven by quantum fluctuations and do not require added (or extracted) heat [Sachdev, 1999]. The fact that temperature is not the driving factor in the phase transition is often empha- sized by considering a system at zero temperature, described by a two-partite Hamiltonian of the form

H(g) = H1 + gH2, (2.4) where g ∈ R controls the passage from a regime in which the system is dominated P x by H1 to a regime in which H2 dominates. For example, let H1 = −J σi , i J > 0 describe a system in which all particles (spins) tend to align along the

15 CHAPTER 2. BASIC CONCEPTS AND METHODS

P z z x direction and H2 = −J σi σj be a system in which all nearest neighbors, hi,ji hi, ji tend to have the same, maximal z component of their spin. In this case, for g = 0 the ground-state of the system (2.4) is therefore all spins aligned in direction of x, whereas the ground-state for g −→ ∞ is that all spins have to point in the z direction. The transition between these two states is not induced by temperature fluctuations, but by quantum fluctuations, that can be controlled via the parameter g; this transition occurs, strictly speaking, at the critical value gc. At non-zero temperature, quantum and thermal phase transitions often mix. While quantum phase transitions can occur at absolute zero, temperature fluctuations can support the phase transition at positive temperature. In this case, the critical value gc depends on the temperature and the theory of phase transitions in classical systems driven by thermal fluctuations can be applied as long as g ' gc up to a critical temperature Tc.

2.3 Optical potentials used with quantum gases

One powerful tool to study quantum phase transitions in ultra-cold quantum gases, are counter-propagating laser beams creating ordered or controllably disordered structures of potentials for the gas particles. These potentials are induced by the AC Stark shift and can spatially conﬁne cold atoms in diﬀerent lattice geometries and dimensions. By loading cold atoms into such an optical lattice, it is possible to experimentally study long-standing problems in condensed matter physics such as the Bose-Hubbard model [Hecker Denschlag et al., 2002, Jaksch et al., 1998]. We follow [Fallani et al., 2008] for this short review of the key experimental techniques used at present. Intuitively, the Bose-Hubbard Hamiltonian describes interacting particles moving in a lattice [Fisher et al., 1989] and reads

X U X X H = −t b†b + b†b + [n (n − 1)] − µ n , (2.5) i j j i 2 i i i hi,ji i i

where hi, ji means that i and j are nearest neighbors, bi is the bosonic annihila- † † tion operator at site i and bi the bosonic creation operator, ni = bi bi gives the number of particles at site i, and µ is the chemical potential. The parameter t > 0 is the hopping amplitude and decreases with the lattice depth, and U > 0 describes the on-site repulsion of the atoms. Generally speaking, for a deep lattice, the atoms are localized, and for a shallow lattice, they can tunnel from one site to the next. The availability of an experimental system for which the tunneling can be controlled by increasing the lattice depth, of course, boosted Bose-Hubbard studies in ultra-cold atoms.

16 2.3. OPTICAL POTENTIALS USED WITH QUANTUM GASES

2.3.1 Periodic potentials

Periodic potentials made of a single frequency are the easiest to generate. Phys- ically, they are realized by the interference of two by counter-propagating laser beams creating a spatially periodic potential [Greiner et al., 2001, Bloch, 2005]. Figure 2.3 illustrates the creation of two common types of optical lattices: a three-dimensional lattice and a two-dimensional array of potentials leading to cigar shaped BECs.

Figure 2.3: The use of lasers to create optical lattices: the top row shows the creation of a two-dimensional array of one-dimensional structures using four lasers. The bottom row shows the creation of a three-dimensional array of harmonic oscillator potentials trapping the atoms using six lasers. Picture from [Bloch, 2005].

Periodic optical potentials are the cornerstone for experiments on lattice models such as the Bose-Hubbard lattice gas described in Eq. (2.5). For simplicity, let us consider the potential of a one-dimensional lattice of wavelength λ. Two counter-propagating laser-waves create a lattice potential 2 V (x) = sER cos (kx) , (2.6) 2π h2 where k = λ is the lattice wavenumber, ER = 2mλ2 is the recoil energy with m

17 CHAPTER 2. BASIC CONCEPTS AND METHODS the mass of the individual atoms, and s is a dimensionless parameter indicating the strength of the lattice in units of the recoil energy. By arranging the lasers accordingly, it is possible to create not only 2D and 3D square lattices, but even more complex configurations like honeycomb, triangular or Kagomélattices. Major breakthroughs such as the superfluid-Mott insulator transition have been realized using optical lattices [Greiner et al., 2002]. For a weak lattice potential, the BEC is in the superfluid phase in which all atoms (i.e. their wave-functions) are spread out over the whole lattice and exhibit long-range phase coherence. For a deep lattice potential, the atoms can no longer move freely in the lattice and become localized. This insulating state is called Mott Insulator and is characterized by a well-defined number of atoms on each lattice site, with no long-range phase coherence across the lattice. The experimental realization of a reversible transition between the superfluid phase and a Mott insulator phase in a BEC proved that quantum phase transitions can be realized and studied using ultra-cold gases in optical lattices. In particular, it demonstrated that strongly correlated quantum systems can be studied using BECs in optical lattices. Therefore, it further fueled the interest of the scientific community in ultra-cold gases as a tool for studying intrinsic quantum phenomena and condensed-matter models.

2.3.2 Pseudo-random potentials

Apart from the phase-transition between the Mott Insulating state and the su- perﬂuid state, the Bose-Hubbard Hamiltonian (2.5) is very interesting for studying disordered systems. One way of studying these phenomena is to superimpose an auxiliary lattice of an incommensurate wavelength in order to create a quasiperiodic (or pseudo-random) lattice potential [Fallani et al., 2007]. Figure 2.4 illustrates the creation of a bi-chromatic pseudo-random potential. The idea behind pseudo-random potentials is that the superposition of two sine-waves of incommensurate wavelength is not a periodic function. In fact, incommensurate refers to the ratio of the two wavelengths being an irrational number. Of course, experimentalists cannot superimpose two truly incommensurate wavelengths because the laser wavelengths are not deﬁned with absolute precision. Incommensurate, in this context, therefore always means that the resulting superposition is not repeated within the size of the system being studied. Pseudo-random potentials created through the superposition of a number of incommensurate wavelengths are called multichromatic lattices. Their simplest case, the bichromatic potential, reads

2 2 V (x) = s1ER1 cos (k1x) + s2ER2 cos (k2x) , (2.7)

2π h2 where, for i = 1, 2, ki = is the wavenumber, ERi = 2 the recoil energy λi 2mλi with m the mass of the individual atoms, and si is a dimensionless parameter

18 2.3. OPTICAL POTENTIALS USED WITH QUANTUM GASES

Figure 2.4: A pseudo-random potential created by superposing two laser beams of incommensurate wavelength. The superposition of the two laser beams forms a non-trivial pattern for all the trapped atoms. The resulting potential is no longer translationally invariant. Picture from [Fallani et al., 2008]. indicating the strength of the lattice in units of the recoil energy. If one lattice is far stronger than the other, for example s1 s2, the resulting lattice is essentially a regular lattice with a rather uniform tunneling rate t. As the powers of both lasers become comparable, the resulting lattice transforms into a very irregular pattern with site-dependent, pseudo-random tunneling rates. While bichromatic lattices are frequently used or considered for studying disorder eﬀects with ultra-cold gases, it is important to realize that none of these potentials are truly random because the their spatial Fourier transform consists of a small number of discrete peaks, only. Their advantage, however, is that multichromatic lattices provide a comparably easy way to create irregular potentials with a small length scale on which irregularity occurs.

2.3.3 Speckle patterns

One very popular way for creating truly random potentials is to shine a laser through a diffuse medium, which creates a speckle pattern. Since experimenters often use a flat piece of diffusive glass, it is often referred to as the diffuse plate. The first experimental realization [Boiron et al., 1999] of these disordered potentials initiated an enormous interest in speckle patterns which lasts to this day. Figure 2.5 illustrates the transmission of light through a diffusive speckle plate. While we mostly focus on speckles created through transmission, note that speckle patterns can also be created through reflections from a rough surface [Goodman, 2006]. A speckle plate can be modeled as a transparent solid medium containing randomly distributed impurities which scatter the incident laser light. The result-

19 CHAPTER 2. BASIC CONCEPTS AND METHODS

Figure 2.5: A speckle pattern created by shining light through a diffusive plate: a) the diffusive plate transforms the laser beam into a broader, disordered beam, which can be used to generate a disordered 2D potential on a BEC. b) a typical speckle potential recorded by a CCD camera. Picture from [Fallani et al., 2008]. ing speckle pattern preserves most of the laser’s coherence because scattering is mainly a coherent process. Therefore the speckle pattern can directly be shone onto the atoms producing a static disordered potential V (r) proportional to the local intensity distribution I(r). By slightly moving the speckle plate in an orthogonal direction to the incident laser beam, experimenters can easily create a different realization of a random pattern with the same spectral and statistical properties. One great feature of speckles is therefore the ease of use and versatility, in particular the way to create alternative random patterns. Corre- lation and statistical properties of the speckle can be accurately measured using a CCD camera – usually the same camera that is used to image the BEC. In experimental setups, the laser wavelength used to produce the speckle pattern is chosen to be far detuned from any atomic resonance. In this case, absorption can be neglected and the atoms are subjected to a potential energy of the form 3πc2 Γ V (r) = 3 I(r), (2.8) 2ω0 ω − ω0 with c the speed of light, ω0 the frequency of the atomic resonance, Γ its radiative line-width, ω − ω0 the detuning of the incident laser of frequency ω from the resonance, and I(r) the resulting laser intensity at position r. It is worth noticing that the speckle pattern can be either attractive or repulsive depending on the sign of the detuning. If the frequency of the incident laser is blue detuned, i.e. ω > ω0, V (r) > 0. In this case, the atoms are repelled from high-intensity regions of the potential and gather at low-intensity regions. In the opposite case, if the incident laser is red detuned, ω < ω0. Therefore, V (r) < 0 and the atoms are attracted to high-intensity regions of the potential. Two important parameters to characterize the speckle pattern are the average speckle height VS and the autocorrelation length σ [Clément et al., 2006]. The average speckle height, on the one hand, is a measure for the intensity of the random lattice. The autocorrelation length, on the other hand, is a measure for the lengthscales beyond which the speckle can be considered to be a random

20 2.3. OPTICAL POTENTIALS USED WITH QUANTUM GASES pattern. Intuitively, σ is often referred to as the speckle grain size. The most common way to deﬁne the VS is to take the double of the standard deviation of the speckle potential [Lye et al., 2005]. Considering a one-dimensional speckle potential V (x) of mean V¯ and spatial extension L, VS is therefore deﬁned by

1  L/2  2 1 Z 2 V = 2  dx V (x) − V¯  . (2.9) S L  −L/2 The autocorrelation length σ is deﬁned as the Root-Mean-Square of the autocorrelation integral G(d) of the speckle potential [Goodman, 2006] and deﬁned by L/2 Z 2 − d G(d) = dx V (x)V (x + d) ∝ e 2σ2 . (2.10)

−L/2 It is important to note that the autocorrelation length σ depends not only on material properties of the diffuse plate and the wavelength of the incident laser light, but also on the lens system used to image the speckle onto the atoms. As experimenters typically aim for the smallest possible grain size of their speckle pattern, σ is typically determined by the diffraction limit spot size of the imaging system. While speckle-patterns are intrinsically generated in a two-dimensional plane orthogonal to the propagation direction of the laser beam, it is possible to create speckle pattern in 1D and 3D [Fallani et al., 2008]. A speckle pattern varies along the propagation direction, but the correlation length is too large to use it as a random potential. One-dimensional speckle patterns are therefore produced differently, by using cylindrical lenses that stretch the speckle pattern along one axis. Three-dimensional speckle patterns, in return, are obtained by combining two two-dimensional patterns in different directions.

2.3.4 Alternative approaches

Apart from speckle patterns and pseudo-periodic potentials, there are other ways to create random potentials for experiments with ultra-cold atomic gases. For example, it is possible to mimic disordered impurities by using two different atomic species [Gavish and Castin, 2005, Massignan and Castin, 2006]. It is then possible to spatially fix the atoms of one of the species using a deep optical lattice not affecting the other atomic species. Small concentrations of the impurity species create a well-controlled disordered potential for the atoms of the other species. Figure 2.6 illustrates this approach for a one-dimensional optical lattice. Yet another approach proposes the use of spatially controlled laser beams to draw an effective potential for the trapped atoms [Henderson et al., 2009]. Fig- ure 2.7 shows the vast range of experimentally paintable potentials ranging from

21 CHAPTER 2. BASIC CONCEPTS AND METHODS

Figure 2.6: Schematic representation of a disordered potential created by random scatterers. The scattering atoms are trapped at the nodes of a periodic optical lattice (in gray) and cooled to the vibrational ground-state. Particles from a diﬀerent atomic species (here the test particle) are insensitive to the optical lattice and only experience the delta potentials (in black) created by the scatterers. Picture from [Gavish and Castin, 2005].

circular or square patterns up to controlled disordered potentials. The size of the laser spot is in the order of 10µm. Compared to speckle patterns or quasiperiodic lattices, the spatial resolution of the potentials achievable with this method is still rather crude. Nonetheless, the enormous freedom in the design of the potential is unprecedented.

Figure 2.7: Eﬀective potentials painted by rotating laser beams. An alternative way to control the potential of the atoms is to use rapidly moving laser beams, analogous to quickly moving lasers used for entertainment purposes. The thick- ness of all laser spots is approximately 10µm. Pictures: Malcom Boshier

Disorder can not only be produced in the optical potential but also in the interactions between the particles. One possibility to achieve this is to employ magnetic field fluctuations in the proximity of a micro-trap, which stem from imperfections during the production of the chip [Gimperlein et al., 2005]. Since the scattering length can be controlled magnetically using Feshbach resonances, these fluctuations can lead to disordered scattering lengths within the experimental setup [Inouye et al., 1998].

22 2.4. DISORDER EFFECTS IN ULTRACOLD GASES

2.4 Disorder eﬀects in ultracold gases

Studies of disorder using ultra-cold gases [Fisher et al., 1989] have greatly ben- eﬁted from the long-standing “holy grail of disorder physics”, Anderson Local- ization, as well as from new experiments realizing the Bose-Hubbard Hamilto- nian (2.5) [Jaksch et al., 1998, Fallani et al., 2007]. Anderson localization is a multiple-scattering phenomena in random media and does not involve interactions between the particles. In contrast, the Bose-Hubbard Hamiltonian usually relies on repulsive on-site interactions between the particles. Also, ultra-cold atoms usually do interact between themselves, and non-interacting systems are not very easy to realize. Therefore, the Bose-Hubbard Hamiltonian was instru- mental for exploring diﬀerent regimes of disordered phenomena in ultra-cold gases.

2.4.1 Weakly interacting systems

In absence of interactions between particles, multiple scattering in a random potential can lead to the fundamental example of disordered systems: Ander- son Localization. In fact, in an infinite one-dimensional system, any amount of a truly random potential can be shown to lead to Anderson Localization [Abrahams et al., 1979]. However, in a finite system such as a trapped BEC, the localization length needs to be small. Real systems with repulsive atom- atom interactions, on top of this, further tend to spread the atoms instead of localizing them. Conditions on localization length and interactions of the disordered potential are the reason why different approaches competed for the first observation of Anderson localization in matter waves. In 2008, two groups simultaneously published their realization of Anderson localization in one-dimensional Bose-Einstein condensates: one using speckles [Billy et al., 2008] and the other using a pseudo-random potential [Roati et al., 2008] created by incommensurate lattices [Aubry and André,1980]. Figure 2.8 shows the temporal evolution of a Bose-Einstein condensate showing (from left to right) that the atoms are localized horizontally for a disordered potential.

2.4.2 Strongly correlated systems

Interactions between particles in a disordered systems can lead to the appearance of diﬀerent kinds of glass phases. As we mentioned, the interplay between interactions and tunneling, in a non-disordered Bose-Hubbard system leads to the appearance of a Mott insulating and a superﬂuid phase. If we consider a Bose-Hubbard system with a disordered chemical potential, Eq. (2.5) reads

X U X X H = −t b†b + b†b + [n (n − 1)] − (µ + )n , (2.11) i j j i 2 i i i i hi,ji i i

23 CHAPTER 2. BASIC CONCEPTS AND METHODS

Figure 2.8: Time of flight images of a BEC in presence of different disorder strengths showing Anderson localization in one dimension. In absence of disorder (left) the BEC expands freely. Increasing disorder strength ∆/J confines the horizontal expansion up to a completely localized state which does not expand at all. Source: Massimo Inguscio, LENS Florence.

where i describes the site-dependent energy accounting for the inhomogeneous external potentials taken to be bounded, i ∈ [−∆/2, ∆/2], where ∆ serves as the energy scale describing the disorder. Figure 2.9 summarizes the different phase diagrams of this system. In the absence of disorder (for ∆ = 0), the system forms the well-known Mott lobes with an integer number of atoms per lattice site inside a superfluid sea. For weak disorder, ∆ < U, the Mott lobes progressively shrink and a new Bose-Glass phase appears [Fisher et al., 1989, Damski et al., 2003]. For strong disorder, ∆ > U, the Mott insulating phase vanishes and the phase diagram is only composed of a Bose-Glass phase and a superfluid phase. Both the Mott insulating and the Bose-Glass phase are insulating states with neither long-range coherence nor a superfluid fraction. However, the Bose-Glass exhibits gapless excitations and a finite compressibil- ity, neither of which is the case for a Mott insulating phase. The (numerical) complexity such disordered systems is tremendous, which explains why quantum phase transitions in disordered systems are still being studied very actively. For example, the absence of a direct quantum phase transition between a superfluid and a Mott insulator in a Bosonic system with generic, bounded disorder, has only been proven recently [Pollet et al., 2009].

24 2.4. DISORDER EFFECTS IN ULTRACOLD GASES

Figure 2.9: Phase-diagram for disordered interacting bosons as a function of the chemical potential µ and the tunneling strength t normalized by the interaction energy U. a) In absence of disorder, there is a clear separation between the Mott Insulator (MI) state and the Superﬂuid (SF) phase. b) Weak disorder leads to an additional Bose-Glass (BG) phase separating the two original phases. c) Strong disorder makes the Mott Insulator state disappear. Adapted from [Fallani et al., 2007] after [Fisher et al., 1989].

25 CHAPTER 2. BASIC CONCEPTS AND METHODS

26 Chapter 3

General principle of disorder-induced order

This chapter reports on our first article [Wehr et al., 2006] introducing the general effect and general mechanism of disorder-induced order (DIO), also called random-field-induced order (RFIO). The paradigm example used throughout this thesis is the classical ferromagnetic XY model on a two-dimensional square lattice. We prove rigorously that the system has spontaneous magnetization at zero temperature in low dimensions, which is not the case in presence of a disordered external field with a U(1) symmetry as the XY model. Also, we present strong evidence and arguments that magnetization persists for small positive temperatures. Furthermore, we show how to generalize this fundamental effect to classical and quantum systems. Finally, this chapter shows possible realizations of this mechanism using ultra-cold atoms in an optical lattice.

3.1 Common models for magnetization

In condensed matter physics, one typically studies systems made of an enormous number of particles and thus with an unmanageable number of degrees of freedom. However, a large number of effects such as phase transitions between a liquid and a solid phase or spontaneous magnetization typically do not depend on the exact state of each particle individually. Therefore, it is possible to resort to simplified model systems, which only consider a small number of parameters to reproduce the main characteristics of the physical system. Magnetism has been a long-standing area of intense research in condensed matter physics [Ashcroft and Mermin, 1976]. Magnetic ions are localized in the lattice structure of a material, influencing the orientation of the magnetic moment of nearby peers. It is important to notice that without magnetic interactions, there would be no overall magnetization of a material in the absence of an exter-

27 CHAPTER 3. GENERAL PRINCIPLE OF DIO nal magnetic field at any temperature. Experimentally, some solids are known to have non-vanishing average vector moments up to a critical temperature Tc. The individual localized magnetic moments may or may not add up to a net magnetization density for a certain solid. If they align even in the absence of an external magnetic field, the solid exhibits an overall magnetization known as spontaneous magnetization and the corresponding magnetic state is called ferromagnetic. Even more common, however, is the case in which individual local magnetic moments sum to zero and no spontaneous magnetization indicates the presence of microscopic magnetic ordering. This magnetic ordering is called paramagnetic. Finally, if the spins align in a regular fashion to sum up to zero, the corresponding order is called antiferromagnetic. The fundamental idea behind most microscopic models of magnetization is to postulate the existence of tiny magnetic moments called spins s(r) at positions r. Depending on the model being considered, s will be a scalar, a vector, or even an operator. When considering a solid material, we expect that all particles are arranged according to a given lattice structure, typically supposed to be square, triangular or hexagonal. Therefore, we usually replace the continuous description of s(r) by a discrete set of si describing the spins on a certain lattice point i. For simplification, we typically assume that the tendency for interacting spins to align is restricted to nearest neighbors and constant over the whole lattice. Ferromagnetic and antiferromagnetic orderings are thus described by opposite signs of the nearest neighbor coupling J. A famous simplified model describing magnetization is the Ising model. It describes each particle i possesses a microscopic spin si = ±1. This magnetic spin is not considered to be a vector but merely pointing in one of two opposite directions up or down. The model assumes that neighboring spins interact with a coupling strength J and that each spin is susceptible to an external magnetic field h. X X HIsing = −J sisj − h si. (3.1) hi,ji i

For ferromagnetic and antiferromagnetic systems, J > 0 and J < 0, respectively. The notation hi, ji indicates that the corresponding sum is taken over all pairs of nearest neighbors i and j. The Ising model contains a lot of interesting physics such as the appearance of macroscopic magnetization below a critical temperature for D ≥ 2. Through its simplicity is therefore a paradigm example of a microscopic model in condensed matter systems. In order to model magnets that magnetize in a preferential plane, we can enhance the Ising model by allowing spins to be two-component vectors si. This description is called the XY model. By projection onto the x and y axis of the x y x 2 y 2 system, we can then derive the si and si with (si ) + (si ) = 1. In the following, we will mostly consider the the isotropic XY model (also called XX model)

28 3.2. THE MERMIN-WAGNER-HOHENBERG THEOREM which, in presence of a uniform external magnetic ﬁeld in a given direction u is

X x x y y X u HXY = −J si sj + si sj − h si , (3.2) hi,ji i with su designating the spin component in direction u. For classical systems x x y y with spins of unit length, si sj + si sj = cos θi,j. The most important diﬀerence between XY and Ising systems is therefore that the spin-spin coupling can take arbitrarily small values or multiples of J only, respectively. One important gen- eralization of this model is to consider the coupling along the x and y directions independent. The coupling term can then be written as

coupling X x x y y Hasymmetric XY = −J (1 + γ)si sj + (1 − γ)si sj , (3.3) hi,ji with 0 ≤ γ ≤ 1. In this way, we can study the transition between an Ising-type model (γ = 1) to an isotropic XY model (γ = 0). For materials that magnetize isotropically, we can consider three-dimensional spins si. This extension of the XY model is called the Heisenberg model and described by X X HHeisenberg = −J sisj − h si. (3.4) hi,ji i Again, J > 0 and J < 0 describe ferromagnetic and antiferromagnetic systems, respectively, and hi, ji indicates that the sum is taken for all nearest neighbors i and j on the lattice.

3.2 The Mermin-Wagner-Hohenberg Theorem

Let us study the low-temperature behavior of the Ising and XY models to determine the minimal energy necessary to destroy macroscopic magnetization. We follow an Imry-Ma-type argument [Imry and Ma, 1975] in order to illustrate the Mermin-Wagner-Hohenberg theorem [Mermin and Wagner, 1966, Hohenberg, 1967]. In absence of an external magnetic field h, there are two equivalent classical ground-states of an Ising system: the spin-spin coupling term of equation (3.1) is maximized if all spins point in the same direction, be it all up or all down. The lowest energy state that completely annihilates disorder in an Ising system is a bi-partite configuration with one part all spins up and the other part all spins down. In one dimension, for example, it would be ,..., , ,..., , where corresponds to a spin of value 1 and corresponds to a spin of value −1. The two domains of constant spin are divided by a domain wall that costs a fixed amount of energy, J. In a two-dimensional system on a lattice of L×M positions (assuming L ≤ M without loss of generality), the lowest energy

29 CHAPTER 3. GENERAL PRINCIPLE OF DIO excitation of an Ising system is formed by a domain wall in the direction of the shorter lattice axis

...... (3.5) ...... and costs an energy of LJ. Continuing in this logic, the lowest energy demagnetized state of an Ising system in a D-dimensional box with all sides at least of length L always has an energetic cost of at least LD−1J. For one- dimensional systems, this cost is L-independent and therefore, the Ising model does not present spontaneous magnetization at any positive temperature. For higher-dimensional systems, however, there exists a positive critical temperature below which there is spontaneous magnetization. In absence of an external magnetic ﬁeld, the ground-states of both the Ising and the XY model reﬂect the symmetry of their spins. Since there is no intrinsic preferred direction inside the XY plane, the only condition for minimizing the energy (3.2) is that all spins point in the same direction: the system is rotationally symmetric. The minimal excitation energy to destroy magnetization is no longer at least LD−1J. In fact, a one-dimensional state per- forming a slow rotation from to over L 1 spins costs approximately 2π 2 π π 2 2π2J JL 1 − cos( L ) = JL 2 sin ( L )) ≈ 2JL L = L . Hence, the energetic cost of such a spin-wave that completely destroys the macroscopic magnetization of the sample decreases with the length of the one-dimensional system. For a large system, there will therefore never be any macroscopic magnetization in the sample at any positive temperature. In two dimensions, we can, again, imagine a situation where only one dimension performs the turn and the other just replicates the rotation

...... (3.6)

and, assuming both dimensions to be at least L 1, this conﬁguration has an energetic cost of 2π2J. This expression, again, does not depend on the length of the system. For a D-dimensional system, the energetic cost of a spin-wave is ﬁnally 2π2LD−2J. In particular, the energetic cost of spin-waves does not increase with the size for samples of dimension two or lower. Following comparable arguments, the Mermin-Wagner-Hohenberg (MWH) theorem states that continuous symmetry systems do not present spontaneous magnetization for systems of dimension two or lower at any T > 0.

30 3.3. STUDIES OF DISORDERED SYSTEMS

3.3 Studies of disordered systems

As discussed in Chapter 2, disordered systems constitute a rather new and rapidly developing branch of the physics of ultra-cold gases. In condensed matter physics, the role of quenched (i.e. independent of time) disorder cannot be overestimated: it is present in nearly all condensed matter systems, and leads to numerous phenomena that dramatically change both the qual- itative and the quantitative behavior of these systems. This leads, for instance, to novel thermodynamical and quantum phases [Lifshits et al., 1988, Akkermans and Montambaux, 2006], as well as to novel phenomena, such as Anderson localization [Anderson, 1958, Mott and Twose, 1961, Borland, 1963, Halperin, 1968, Abrahams et al., 1979, van Tiggelen and Skipetrov, 2003]. In general, disorder can hardly be controlled in condensed matter systems. In contrast, as we discussed, quenched disorder (or pseudo-disorder) can be introduced in a controlled way in ultra-cold atomic systems [Damski et al., 2003], using optical potentials generated by multi-chromatic lattices [Guidoni et al., 1997, Roth and Burnett, 2003a, Roth and Burnett, 2003b], or by speckle radiation [Grynberg et al., 2000]. Alternative methods include impurity atoms serving as random scatterers [Gavish and Castin, 2005, Massignan and Castin, 2006], and quasi-cristalline lattices [Sanchez-Palencia and Santos, 2005]. The multitude of ways to produce controlled disordered potentials opens fantastic possibilities to investigate the effect of disorder in controlled systems (a review in the context of cold gases can be found, for example, in [Ahufinger et al., 2005]). Over the course of the past few years, several groups have initiated the experimental study of disorder with BECs [Lye et al., 2005, Fort et al., 2005, Clément et al., 2005, Clément et al., 2006, Schulte et al., 2005, Schulte et al., 2006], and strongly correlated Bose gases [Fallani et al., 2007, White et al., 2009]. In the center of interest of these studies is one of the most fundamental issues of disordered systems: the connection between Anderson localization and interactions in many body Fermi or Bose systems at low temperatures. In non-interacting atomic systems, localization is feasible experimentally [Kuhn et al., 2005], but even weak interactions can drastically change the scenario. While weak repulsive interactions tend to delocalize, strong ones in confined geometries lead to Wigner-Mott- like localization [Fisher et al., 1989, Scalettar et al., 1991]. Both, experiments as well as theoretical studies, indicate that gaseous systems with large interactions, present stronger localization effects in the excitations of a Bose-Einstein condensate [Clément et al., 2005, Clément et al., 2006, Bilas and Pavloff, 2006, Paul et al., 2007, Sanchez-Palencia, 2006], rather than on the wave-function itself. Also, particle interactions allow for the creation of solitons which can, as it turns out, also reveal Anderson Localization [Sacha et al., 2009]. In the limit of weak interactions, a Bose gas enters a Lifshits glass regime, in which several BECs in various localized single atom orbitals from the low energy tail of the spectrum coexist [Lugan et al., 2007] (for “traces” of the Lifshits glass in the mean-field theory see [Schulte et al., 2005, Schulte et al., 2006]). Finally, note that disorder in Fermi gases, or in Femi-Bose atomic mixtures, should al-

31 CHAPTER 3. GENERAL PRINCIPLE OF DIO low one to realize various fermionic disordered phases, such as a Fermi glass, a Mott-Wigner glass, “dirty” superconductors, etc. [Ahuﬁnger et al., 2005], or even quantum spin glasses [Sanpera et al., 2004].

3.4 Large eﬀects by small disorder

One of the most appealing effects of disorder is that even extremely small randomness can have dramatic consequences [Imry and Ma, 1975]. The paradigmatic example in classical physics is the Ising model for which an arbitrarily small random magnetic field with a symmetric distribution destroys magnetization even at temperature T=0 in dimension D = 2 [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990], which is not the case in D > 2 [Imbrie, 1984, Bricmont and Kupiainen, 1987]. This result has been generalized to systems with continuous symmetry in random fields distributed in accordance with this symmetry [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990]. For instance, the Heisenberg Model in a SO(3)-symmetrically distributed field does not magnetize up to D = 4. In quantum physics, the paradigmatic example of large effects induced by small disorder is provided by the above-mentioned Anderson localization which occurs in 1D and 2D in arbitrarily small random potentials [Abrahams et al., 1979]. Here, we propose an even more intriguing opposite effect, where disorder counter- intuitively favors ordering: a general mechanism of disorder-induced order (DIO) in which the presence of arbitrarily small disorder leads to a higher critical temperatures for certain spin models, provided that the disorder breaks the continuous symmetry of the system.

3.4.1 Main result of this chapter

As we have seen in Section 3.2, a consequence of the Mermin-Wagner-Hohenberg theorem [Mermin and Wagner, 1966] is that spin systems with continuous symmetry in dimensions less or equal to 2D cannot exhibit long range order. The mechanism that we propose here breaks the continuous symmetry, and in this sense acts against the Mermin-Wagner-Hohenberg no-go rule in 2D. In particular, we prove rigorously that the classical XY spin model on a 2D lattice in a uniaxial random field magnetizes spontaneously at T = 0 in the direction perpendicular to the magnetic field axis, and provide strong evidence that this is also the case at small positive temperatures. We discuss generalizations of this mechanism to classical and quantum XY and Heisenberg models in 2D and 3D. In 3D, the considered systems do exhibit long range order at finite T > 0, but in this case the critical temperature decreases with the “size” of the symmetry group: the critical temperature is largest for the Ising model (the discrete group Z2), smaller for the XY model (the continuous group U(1)), and the smallest for the Heisenberg model (the continuous group SU(2), or SO(3)). Thus, we

32 3.5. DISORDER-INDUCED ORDER IN THE CLASSICAL XY MODEL expect that our mechanism will lead to an increase of the critical temperature for the XY and Heisenberg models, and to an increase of the order parameter value at a ﬁxed temperature for the disordered system in comparison to the non- disordered one. Finally, we propose three possible and experimentally feasible realizations of the DIO phenomenon using ultra-cold atoms in optical lattices.

3.5 Disorder-induced order in the classical XY model

3.5.1 The system under study

Consider a classical spin system on the 2D square lattice Z2, in a random magnetic field, h (see Fig. 3.1). Our two-dimensional spin variable σi = (cos θi, sin θi), 2 at a site i ∈ Z is a unit vector in the XY plane (we use σi instead of si to emphasize the two-dimensional nature of the problem). Adapting Eq. (3.2), our system is therefore described by the Hamiltonian X X H = −J σi · σj − hi · σi. (3.7) |i−j|=1 i Here the first term is the standard nearest-neighbor interaction of the XY-model, and the second term represents a small random field perturbation. The hi’s are assumed to be independent, identically distributed, random 2D vectors of mean zero. For = 0, the model has no spontaneous magnetization, m, at any positive T . This was first pointed out in [Herring and Kittel, 1951], and later developed into a class of results known as the Mermin-Wagner-Hohenberg theorem [Mermin and Wagner, 1966] for various classical, as well as quantum two- dimensional spin systems with continuous symmetry. In higher dimensions the system does magnetize at low temperatures, which follows from the spin wave analysis [Zinn-Justin, 2004]. A rigorous proof of this statement is given in [Fröhlich and Spencer, 1976]. The impact of a random field on the behavior of the model was first addressed in [Imry and Ma, 1975, Aizenman and Wehr, 1989, Aizenman and Wehr, 1990], where it was argued that if the distribution of the random variables hi is invariant under rotations, there is no spontaneous magnetization at any positive T in any dimension D ≤ 4. A rigorous proof of this statement was given in [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990]. Both works use crucially the rotational invariance of the distribution of the random field variables.

Here we consider the case where hi is directed along the y-axis: hi = ηiey, where ey is the unit vector in the y direction, and ηi is a random real number. Such a random ﬁeld obviously breaks the continuous symmetry of the interaction and a question arises whether the model still has no spontaneous magnetization in two dimensions. This question has been given contradictory

33 CHAPTER 3. GENERAL PRINCIPLE OF DIO

Figure 3.1: XY model on a 2D square lattice in a random magnetic field. The magnetic field is oriented along the y axis, hi = ηiey, where ηi is a real random number. Right boundary conditions are assumed on the outer square, possibly placed at infinity.

answers in literature: [Dotsenko and Feigelman, 1981] predict that a small random field in the y-direction does not change the behavior of the model, while [Minchau and Pelcovits, 1985] argues that it leads to the presence of spontaneous magnetization, m, in the direction perpendicular to the random field axis in low (but not arbitrarily low) temperatures. Both works use renormalization group analysis, with [Minchau and Pelcovits, 1985] starting from a version of the Imry-Ma scaling argument to prove that the model magnetizes at zero temperature. The same model was subsequently studied in [Feldman, 1998], using ideas similar to the argument given here. As we argue below, however, his argument contains a gap, which is filled in the present work. We first present a complete proof that the system indeed magnetizes at T = 0, and argue that the ground state magnetization is stable under inclusion of small thermal fluctuations. For this, we use a version of the Peierls contour argument [Peierls, 1936], eliminat- ing first the possibility that Bloch walls or vortex configurations destroy the transition.

34 3.5. DISORDER-INDUCED ORDER IN THE CLASSICAL XY MODEL

3.5.2 Disorder-induced order at T = 0

Let us start by a rigorous analysis of the ground state. Consider the system in a square Λ with the “right” boundary conditions, σi = (1, 0), for the sites i on the outer boundary of Λ (see Fig. 3.1). The energy of any spin configuration decreases if we replace the x components of the spins by their absolute values and leave the y components unchanged. It follows that in the ground state, x components of all the spins are nonnegative. As the size of the system increases, we expect the x component of the ground state spins to decrease, since they feel less influence of the boundary conditions and the ground state value of each spin will converge. We thus obtain a well-defined infinite-volume ground state with the “right” boundary conditions at infinity. We emphasize that the above convergence statement is nontrivial and requires a proof, even if the conclusion may seem quite natural. A similar statement has been rigorously proven for ground states of the random field Ising model using Fortuin-Kasteleyn-Ginibre monotonicity techniques [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990, Fortuin et al., 1971].

3.5.3 Inﬁnite volume limit

A priori this infinite-volume ground state could coincide with the ground state of the random field Ising model, in which all spins have zero x component. The following argument shows that this is not the case. Suppose that the spin σi at a given site i is aligned along the y-axis, i.e. cos θi = 0. Since the derivative of the energy function with respect to θi vanishes at the minimum, we obtain X sin(θi − θj) = 0. (3.8) j:|i−j|=1 P Since cos θi = 0, this implies j:|i−j|=1 cos θj = 0. Because in the “right” ground state all spins lie in the (closed) right half-plane x ≥ 0, all terms in the above expression are nonnegative and hence have to vanish. This means that at all the nearest neighbors j of the site i, the ground state spins are directed along the y-axis as well. Repeating this argument, we conclude that the same holds for all spins of the infinite lattice, i.e. the ground state is the (unique) random field Ising model ground state. This, however, leads to a contradiction: one can construct a field configuration (occurring with a positive probability) which forces the ground state spins to have nonzero x components. To achieve this we put strong positive (ηi > 0) fields on the boundary of a square and strong negative fields on the boundary of a concentric smaller square. If the fields are very weak in the area between the two boundaries, the spins will form a Bloch wall, rotating gradually from θ = π/2 to θ = −π/2. Since such a local field configuration occurs with a positive probability, the ground state cannot have zero x components everywhere, contrary to our assumption. We would like to emphasize the logical structure of the above argument, which

35 CHAPTER 3. GENERAL PRINCIPLE OF DIO proceeds indirectly assuming that the ground state spins (or, equivalently, at least one of them) have zero x components and reach a contradiction. The initial assumption is used in an essential way to argue existence of the Bloch wall interpolating between spins with y components equal to +1 and −1. It is this part of the argument that we think is missing in [Feldman, 1998]. Note that this argument applies to strong, as well as to weak random fields, so that the ground state is never, strictly speaking, field-dominated and always exhibits magnetization in the x-direction. Moreover, the argument does not depend on the dimension of the system, applying in particular in one dimension. We argue below that in dimensions greater than one the effect still holds at small positive temperatures, the critical temperature depending on the strength of the random field (and presumably going to zero as the strength of the field increases).

3.5.4 Disorder-induced order at low positive T

To study the system at low positive T , we need to ask what are the typical low energy excitations from the ground state. For = 0, continuous symmetry allows Bloch walls, i.e. configurations in which the spins rotate gradually over a large region, for instance from left to right. The total excitation energy of a Bloch wall in 2D is of order one, and it is the presence of such walls that underlies the absence of continuous symmetry breaking. However, for > 0, a Bloch wall carries additional energy, coming from the change of the direction of the y component of the spin, which is proportional to the area of the wall (which is of the order L2 for a wall of linear size L in two dimensions), since the ground state spins are adapted to the field configuration, and hence overturning them will increase the energy per site. Similarly, vortex configurations, which are important low-energy excitations in the nonrandom XY model, are no longer energetically favored in the presence of a uniaxial random field. We are thus left, as possible excitations, with sharp domain walls, where the x component of the spin changes sign rapidly. To first approximation we consider excited configurations, in which spins take either their ground state values, or the reflections of these values in the y-axis. As in the standard Peierls argument [Peierls, 1936], in the presence of the right boundary conditions, such configurations can be described in terms of contours γ (domain walls), separating spins with positive and negative x components. If mi is the value of the x component of the spin σi in the ground state with the right boundary conditions, the energy of a domain wall is the sum of mimj over the bonds (ij) crossing the boundary of the contour. Since changing the signs of the x components of the spins does not change the magnetic field contribution to the energy, the Peierls estimate shows that the probability of such a contour is bounded above P by exp(−2β (ij) mimj), with β = J/kBT . We want to show that for a typical realization of the field, h,(i.e. with probability one), these probabilities are summable, i.e. their sum over all contours containing the origin in their interior is finite. It then follows that at a still

36 3.5. DISORDER-INDUCED ORDER IN THE CLASSICAL XY MODEL lower T , this sum is small, and the Peierls argument proves that the system magnetizes (in fact, a simple additional argument shows that summability of the contour probabilities already implies the existence of spontaneous m). To show that a series of random variables is summable with probability one, it suf- fices to prove the summability of the series of the expected values. We present two arguments for the last statement to hold. √ If the random variables mi are bounded away from zero, i.e. mi > c, for some P c > 0, the moment generating function of the random variable (ij) mimj satisfies * + X exp − β mimj ≤ exp[−cβL(γ)], (3.9) (ij) with L(γ) denoting the length of the contour γ. The sum of the probabilities P of the contours enclosing the origin is thus bounded by γ exp[−cβL(γ)]. The standard Peierls-Griffiths bound proves the desired summability. The above argument does not apply if the distribution of the ground state, m, contains zero in its support. For unbounded distribution of the random field this may very well be the case, and then another argument is needed. If we P assume that the terms in the sum (ij) mimj are independent and identically distributed, then * + X L(γ) exp(−2β mimj) = hexp(−2βmimj)i (3.10) (ij)

= exp{L(γ) log hexp(−2βmimj)i} (3.11) and we just need to observe that

hexp(−2βmimj)i → 0 as β → ∞, (3.12) since the expression under the expectation sign goes point-wise to zero and lies between 0 and 1, to conclude that * + X exp(−2β mimj) behaves as exp[−g(β)L(γ)] (3.13) (ij) for a positive function g(β) with g(β) → ∞ as β → ∞. While mimj are not, strictly speaking, independent, it is natural to assume that their dependence is weak, i.e. their correlation decays fast with the distance of the corresponding bonds (ij). The behavior of the moment generating function of their sum is then qualitatively the same, with a renormalized rate function g(β), still diverging as β → ∞. As before, this is enough to carry out the Peierls-Griﬃths estimate which implies spontaneous magnetization in the x-direction. We remark that our assumption about the fast decay of correlations implies that the sums of 2 mimj over subsets of Z satisfy a large deviation principle analogous to that for sums of independent random variables and the above argument can be restated using this fact.

37 CHAPTER 3. GENERAL PRINCIPLE OF DIO

3.5.5 Numerical Monte Carlo simulations

Based on the above discussion it is expected that the DIO effect predicted here will lead to the appearance of magnetization, m, in the x direction of order 1 at low temperatures in systems much larger than the correlation length of typical excitations. For small systems, however, the effect may be obscured by finite size effects, which, due to long-range power law decay of correlations, are particularly strong in the XY model in 2D. In particular, the 2D-XY model shows finite magnetization (m) in small systems [Bramwell and Holdsworth, 1994], so that DIO is expected to result in an increase of the magnetization. We have performed numerical Monte-Carlo simulations (done by a co-author of [Wehr et al., 2006]) using similar approaches as [Troyer et al., 1998] for the 2D-XY classical model [Hamiltonian (4.23), with = 1]. We generate a random magnetic field, hi = ηiey in the y direction.√ The η√i’s are independent random real numbers, uniformly distributed in [− 3∆hy, 3∆hy]. Note that ∆hy is thus the standard deviation of the random field hi. Boundary conditions on the outer square correspond to σi = (1, 0) [see Fig. 3.1]. The calculations were performed in 2D lattices with up to 200×200 lattice sites for various temperatures. The results are presented in Fig. 3.2. At very small temperature, the system magnetizes in the absence of disorder (m approaches 1 when T tends to 0) due to the finite size of the lattice [Bramwell and Holdsworth, 1994]. In this regime, a random field in the y direction tends to induce a small local magnetization, parallel to hi, so that the magnetization in the x direction, m, is slightly reduced. At higher temperatures (T ' 0.7J/kB in Fig. 3.2), the magnetization is significantly smaller than 1 in the absence of disorder. This is due to non-negligible spin wave excitations. In the presence of small disorder, these excitations are suppressed due to the DIO effect discussed here. We indeed find that, at T = 0.7J/kB, m increases by 1.6% in presence of the uniaxial disordered magnetic field. At larger temperatures, excitations, such as Bloch walls or vortices are important and no increase of the magnetization is found when applying a small random field in the y direction.

3.6 Disorder-induced order in other systems

The DIO eﬀect predicted above may be generalized to other spin models, in particular those that have ﬁnite correlation length. Here we list the most spectacular generalizations:

3.6.1 2D Heisenberg ferromagnet in random ﬁelds of various symmetries

Here the interaction has the same form as in the XY case, but spins take values on a unit sphere. As for the XY Hamiltonian, if the random ﬁeld

38 3.6. DISORDER-INDUCED ORDER IN OTHER SYSTEMS

1 0.9 0.75 0.8 0.7 0.7 0.65 0.6 0.6 0.5 0.6 0.65 0.7 0.75 0.4 y 0.3 6h =0

m (magn. along x) y 0.2 6h =0.25 y 0.1 6h =0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

kBT/J

Figure 3.2: Results of the Monte-Carlo simulation for the classical 2D-XY model in a 200×200 lattice. The Inset is a magnetiﬁcation of the main ﬁgure close to T = 0.7J/kB.

distribution has the same symmetry as the interaction part, i.e. if it is symmetric under rotations in three dimensions, the model has no spontaneous magnetization up to 4D (see [Imry and Ma, 1975, Aizenman and Wehr, 1989, Aizenman and Wehr, 1990]). If the random field is uniaxial, e.g. oriented along the z axis, the system still has a continuous symmetry (rotations in the xy plane), and thus cannot have spontaneous magnetization in this plane. It cannot magnetize in the z direction either, by the results of [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990]. Curiously enough, a field distribution with an intermediate symmetry may lead to symmetry breaking. Namely, arguments fully analogous to the previous ones imply that if the random field takes values in the yz plane with a distribution invariant under rotations, the system will magnetize in the x direction. We are thus faced with the possibility that a planar field distribution fully breaks the continuous symmetry, while this is broken neither by a field with a spherically symmetric distribution nor by a uniaxial one.

3.6.2 3D XY and Heisenberg models in a random ﬁeld of various symmetries

We have argued that the 2D XY model with a small uniaxial random field orders at low T . Since in the absence of the random field spontaneous magnetization occurs only at T = 0, this can be equivalently stated by saying that a small uniaxial random field raises the critical temperature Tc of the system. By analogy, one can expect that the (nonzero) Tc of the XY model in 3D becomes higher and comparable to that of the 3D Ising model, in the presence

39 CHAPTER 3. GENERAL PRINCIPLE OF DIO

of a small uniaxial field. A simple meanfield estimate suggests that Tc might increase by a factor of 2. The analogous estimates for the Heisenberg model in 3D suggest an increase of Tc by a factor 3/2 (or 3) in a small uniaxial (or planar rotationally symmetric) field respectively. These conjectures are the subject of a forthcoming project.

3.6.3 Antiferromagnetic systems

By flipping every second spin, the classical ferromagnetic models are equivalent to antiferromagnetic ones (on bipartite lattices). This equivalence persists in the presence of a random field with a distribution symmetric with respect to the origin. Thus the above discussion of the impact of random fields on continuous symmetry breaking in classical ferromagnetic models translates case by case to the antiferromagnetic case.

3.6.4 Quantum systems

All of the effects predicted above should, in principle have quantum analogs. Quantum fluctuations might, however, destroy the long-range order, so each of the discussed models should be carefully reconsidered in the quantum case. Some models, such as the quantum spin S = 1/2 Heisenberg model, for instance, have been widely studied in literature [Diep, 2004]. The Mermin-Wagner theorem [Mermin and Wagner, 1966] implies that the model has no spontaneous magnetization at positive temperatures in 2D. For D > 2 spin wave analysis [Bloch, 1930, Dyson, 1956, Ashcroft and Mermin, 1976] shows the existence of spontaneous magnetization (though a rigorous mathematical proof of this fact is still lacking). In general, one does not expect major differences between the be- haviors of the two models at T 6= 0. It thus seems plausible that the presence of a random field in the quantum case is going to have effects similar to those in the classical Heisenberg model. Similarly, one can consider the quantum Heisenberg antiferromagnet (HAF) and expect phenomena analogous to the classical case, despite the fact that unlike their classical counterparts, the quantum Heisenberg ferromagnet and Heisenberg antiferromagnet systems are no longer equivalent. We expect to observe spontaneous staggered magnetization in a random uniaxial XY model, or random planar field Heisenberg ferromagnet. A possibility that a random field in the z-direction can enhance the antiferromagnetic order in the xy plane has been pointed out in [Huscroft and Scalettar, 1997].

3.7 Towards experimental realization of disorder- induced order in ultra-cold atomic systems

Further understanding of the phenomena described here will beneﬁt from experimental realizations and investigations of the above-mentioned models. Below,

40 3.7. TOWARDS EXPERIMENTAL REALIZATION OF DIO we discuss the possibilities to design quantum simulators for these quantum spin systems using ultra-cold atoms in optical lattices.

3.7.1 Two-component lattice Bose gas

Consider a two-component Bose gas conﬁned in an optical lattice with on-site inhomogeneities. The two components correspond here to two internal states of the same atom. The low-temperature physics is captured by the Bose-Bose Hubbard model [Jaksch et al., 1998]:

X hUb UB i H = n (n − 1) + N (N − 1) + U n N BBH 2 j j 2 j j bB j j j X + (vjnj + VjNj) (3.14) j X h i X Ωj − J b†b + J B†B + h.c. − b†B + h.c. b j l B j l 2 j j hj,li j where bj and Bj are the annihilation operators for both types of Bosons in the † † lattice site j, nj = bjbj and Nj = BjBj are the corresponding number operators, and hj, li denote a pair of adjacent sites in the optical lattice. In Hamilto- nian (3.14), (i) the first term describes on-site interactions, including interaction between Bosons of different types, with energies Ub, UB and UbB; (ii) the second accounts for on-site energies; (iii) the third describes quantum tunneling between adjacent sites and (iv) the fourth transforms one Boson type into the other with a probability amplitude |Ω|/~. The last term can be implemented with an optical two-photon Raman process if the two Bosonic “species” correspond to two internal states of the same atom (see also Fig. 3.3). Possibly, both on-site energies vj, Vj and the Raman complex amplitude Ωj can be made site-dependent using speckle laser light [Lye et al., 2005, Fort et al., 2005, Clément et al., 2005, Clément et al., 2006, Schulte et al., 2005, Schulte et al., 2006].

Consider the limit of strong repulsive interactions (0 < Jb,JB and |Ωj| Ub,UB,UBb) and a total filling factor of 1 (i.e. the total number of particles equals the number of lattice sites). Proceeding as in the case of Fermi-Bose mixtures, analyzed for example in [Sanpera et al., 2004], we derive an effective Hamiltonian, Heff, for the Bose-Bose mixture. In brief, we restrict the Hilbert Q space to a subspace E0 generated by { j |nj,Nji} with nj +Nj = 1 at each lattice site, and we incorporate the tunneling terms via second-order perturbation theory as in [Sanpera et al., 2004]. We then end up with

X † X Heﬀ = − Jj,lBj Bl + h.c + Kj,lNjNl hj,li hj,li X X Ωj + V N − B + h.c. (3.15) j j 2 j j j

41 CHAPTER 3. GENERAL PRINCIPLE OF DIO

Figure 3.3: Atomic level scheme of a two-component Bose mixture in a random optical lattice used to design spin models in random magnetic ﬁelds (see text).

† † where Bj = PbjBj, P is the projector onto E0 and Nj = Bj Bj. Hamiltonian Heﬀ contains (i) a hopping term, Jj,l, (ii) an interaction term between neighbour sites, Kj,l, (iii) inhomogeneities, Vj, and (iv) a creation/annihilation term. Note that the total number of composites is not conserved except for a vanishing Ω. The coupling parameters in Hamiltonian (3.15) are 1:   J J 1 1 b B   Jj,l = 2 + 2 (3.16) U  δ ∆  bB 1 − j,l 1 − j,l UbB UbB

4J 2/U 2J 2/U 4J 2 /U 2J 2 /U K = − b b + b bB − B B + B bB (3.17) j,l 2 2 2 2 1 − δj,l 1 − δj,l 1 − ∆j,l 1 − ∆j,l Ub UbB UB UbB   2 2 2 2 X 4Jb /Ub Jb /UbB JB/UbB 4JB/UB Vj = Vj − vj +  − − +  (3.18)  2 δj,l ∆j,l 2  δj,l 1− 1+ ∆j,l hj,li 1− UbB UbB 1− Ub UB where δj,l = vj −vl and ∆j,l = Vj −Vl. Hamiltonian Heﬀ describes the dynamics † of composite particles whose annihilation operator at site j is Bj = bjBjP. In contrast to the case of Fermi-Bose mixtures discussed in [Sanpera et al., 2004],

1The coupling parameters are the same as calculated in [Sanpera et al., 2004, Ahuﬁnger et al., 2005] except for the third term in Eq. (3.17) which corresponds to a vir- tual state with two B bosons in the same lattice site—forbidden for Fermions.

42 3.7. TOWARDS EXPERIMENTAL REALIZATION OF DIO where the composites are fermions, in the present case of Bose-Bose mixtures, they are composite Schwinger Bosons made of one B boson and one b hole. † Since the commutation relations of Bj and Bj are those of Schwinger Bosons [Auerbach, 1994], we can directly turn to the spin representation by defining x y z † Sj + iSj = Bj and Sj = 1/2 − Nj, where Nj = Bj Bj. It is important to note that since Raman processes can convert b Bosons into B Bosons (and P conversely), jhNji is not fixed by the total number of Bosons of each species, P z i.e. the z component of m, jhSj i is not constrained. For small inhomogeneities (δj,l = vj −vl, ∆j,l = Vj −Vl Ub,Ub,UbB), Hamiltonian Heff is then equivalent to the anisotropic Heisenberg XXZ model [Auerbach, 1994] in a random field:

X x x y y X z z Heﬀ = −J⊥ Sj Sl + Sj Sl − Jz Sj Sl hj,li hj,li X x x y y z z − hj Sj + hj Sj + hj Sj (3.19) j where

4JbJB J⊥ = (3.20) UbB 2 2 2 2 2Jb 2JB Jb + JB Jz = 2 + − (3.21) Ub UB UbB x R y I z hj = Ωj ; hj = −Ωj ; hj = Vj − ζJz/2 , (3.22)

2 2 with ζ the lattice coordination number, Vj = Vj − vj + ζ[4Jb /Ub + 4JB/UB − 2 2 R I (Jb + JB)/UbB] and Ωj = Ωj + iΩj. In atomic systems, all these (possibly site-dependent) terms can be controlled almost at will [Sanpera et al., 2004, Lewenstein et al., 2007, Jaksch and Zoller, 2005]. In particular, by employing various possible control tools one can reach the Heisenberg ferromagnet (J⊥ = Jz) and XY (Jz = 0) cases making it possible to implement disorder-induced order.

3.7.2 Bose lattice gas embedded in a BEC

The quantum ferromagnetic XY model in a random ﬁeld may be alternatively obtained using the same Bose-Bose Hubbard model, but with strong state dependence of the optical dipole forces. One can imagine a situation in which one-component (say b) is in the strong interaction limit, so that only one b atom at a site is possible, whereas the other (B) component is in the Bose condensed state and provides only a coherent BEC “background” for the b-atoms. Mathematically speaking, this situation is described by Eq. (3.14), in which ni’s can be equal to 0 or 1 only, whereas Bi’s can be replaced by a classical complex ﬁeld (condensate wave function). In this limit the spin S = 1/2 states can be associated with the presence, or absence of a b-atom in a given site. In I this way, setting vj = 0 and Ωj = 0, one obtains the quantum version of the

43 CHAPTER 3. GENERAL PRINCIPLE OF DIO

XY model (4.23) with J = Jb and a uniaxial random ﬁeld in the x direction R with the strength determined by Ωj .

3.7.3 Two-component Fermi lattice gas

Finally, the S = 1/2 antiferromagnetic Heisenberg model may be realized with a Fermi-Fermi mixture at half filling for each component. This implementation might be of special importance for future experiments with Lithium atoms. As recently calculated [Werner et al., 2005], the critical temperature for the Néel state in a 3D cubic lattice is of the order of 30nK. It is well known that in a 3D cubic lattice the critical temperatures for the antiferromagnetic Heisenberg, the XY and the Ising models are Tc,XY ' 1.5Tc,Heis, and Tc,Ising ' 2Tc,Heis. The estimates of these critical temperatures can be, for instance, obtained applying the Curie-Weiss mean field method to the classical models. Suppose that we put the Heisenberg antiferromagnet in a uniaxial (respectively, planar) random field, created using the same methods as discussed above, i.e. we break the SU(2) symmetry and put the system into the universality class of XY (respectively, Ising) models. Mean field estimates suggest then that we should expect the increase of the critical temperature by factor 1.5 (respectively, 2), that is up to ' 45 (respectively, 90)nK. Even if these estimates are too optimistic, and the effect is two, three times smaller, one should stress, that even an increase by, say 10nK, is of great experimental relevance and could be decisive for achieving of antiferromagnetic state. We would like to stress that similar proposals, as the three discussed above, have been formulated before [Duan et al., 2003, Kuklov and Svistunov, 2003, Garc´ıa-Ripoll et al., 2004, Porras and Cirac, 2004, Micheli et al., 2006]. How- ever, none of them treat simultaneously essential aspects for the present schemes: i) disordered fields, but not bonds; ii) arbitrary directions of the fields; iii) possibility of exploring Ising, XY or Heisenberg symmetries; iv) realizing the coherent source of atoms; and v) avoiding constraints on the magnetization along the z axis. It is also worth commenting on what are the most important experimental challenges that have to be addressed in order to achieve DIO. Evidently, for the proposals involving the strong interaction limit of two-component Bose, or Fermi systems, the main issue is the temperature which has to be of order of tens of nano-Kelvins. Such temperatures are starting to be achievable with current experimental advances (for a careful discussion in the context of Fermi-Bose mixtures see [Fehrmann et al., 2004]). In recent experiments, for example, temperatures in the order of 1nK were measured directly in an optical lattice [Weld et al., 2009]. Note also that there exist several proposals for supplementary cooling of lattice gases, using laser (photons) or couplings to ultra-cold BEC (phonon cooling) that can help (for reviews see [Lewenstein et al., 2007, Jaksch and Zoller, 2005].

44 3.8. CONCLUSION

3.8 Conclusion

This chapter introduces a general mechanism of disorder-induced order (DIO) or, equivalently, random-field-induced order (RFIO), occurring in systems with continuous symmetry, placed in a random field that breaks, or reduces this symmetry. We have presented rigorous results for the case of the 2D-classical ferromagnetic XY model in a random uniaxial field, and proved that the system has spontaneous magnetization at temperature T = 0. Furthermore, this chapter presents rather strong evidence that this is also the case for small T > 0. Several generalizations of this mechanism to various classical and quantum systems are discussed in the above sections. We have presented also detailed proposals to realize DIO in experiments using two-component Bose lattice gases, one-component Bose lattice gases embedded in BECs, or two-component Fermi lattice gases. These results shed light on controversies in existing literature, and open the way to realize DIO with ultra-cold atoms in an optical lattice.

45 CHAPTER 3. GENERAL PRINCIPLE OF DIO

46 Chapter 4

Coupled Bose-Einstein Condensates

This chapter reports on our study of disorder-induced order in a system of two coupled Bose-Einstein Condensates, published in [Niederberger et al., 2008]. We start by deriving the Gross-Pitaevskii equations describing this system and show the analogy with the XY model discussed in Chapter 3. Then, we explain the basic numerical approach we used and discuss the results and the context of this thesis.

4.1 The Gross-Pitaevskii equation

Our BECs are each described by a system of two coupled Gross-Pitaevskii equations. For the derivation, we follow Cohen-Tannoudji’s lectures at the college de France [Cohen-Tannoudji, 1999]. We assume N identical bosons of mass m, each subjected to an external potential Vi = V (ri). This potential can be thought of e.g. as a harmonic trapping potential, a site-dependent random-potential or a combination of both. Taking into account the interaction potentials Uij = U(|ri − rj|) acting between the Bosons, the Hamiltonian H of the system is

N N N X p2 1 X X H = i + V + U . (4.1) 2m i 2 ij i=1 i=1 i6=j=1

1 The factor 2 in the last term is necessary because the sums count all two- partite interactions twice. The exact ground-state of a system described by (4.1) cannot be calculated in general. We are therefore using a variational approach hΨ|H|Ψi determining states that minimize hΨ|Ψi . The variational ansatz assumes that

47 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES all particles are in the same state |ψi and the system is therefore described by

|Ψi = |ψ1i|ψ2i|ψ3i · · · |ψii · · · |ψN i. (4.2)

We now want to minimize the energy functional E[ψ], associating an energy value with each wave-function ψ,

hΨ|H|Ψi E[ψ] = . (4.3) hΨ|Ψi

Our minimization approach considers each particle to be subjected to a mean- ﬁeld representing the eﬀect of the N − 1 other particles.

4.1.1 Calculating hΨ|H|Ψi

In order to calculate hΨ|H|Ψi in (4.3), we ﬁrst consider the mean value of the kinetic term of (4.1):

N X p2 Z 2 hΨ| i |Ψi = N d3r ψ∗(r) − ~ ∇2ψ(r). (4.4) 2m 2m i=1

To solve this equation, we remember that ∇ [ψ∗(r)∇ψ(r)] = ψ∗(r)∇2ψ(r) + |∇ψ(r)|2. Equation (4.4) can therefore be re-written as

N 2 Z 2 Z 2 X p 2 hΨ| i |Ψi = N d3r ~ |∇ψ(r)| − N d3r ~ ∇ [ψ∗(r)∇ψ(r)] . (4.5) 2m 2m 2m i=1 The second term of this integral can be calculated: the theorem of divergence, R ∇ · f dV = H f · dS, states that the integral of the divergence of a vectorial Ω ∂Ω function f over a volume Ω is equivalent to the integral of the function f itself taken over the delimiting surface ∂Ω of this volume. In the present case, Ω denotes all R3 and ∂Ω therefore denotes, for example, the surface of a sphere of radius r → ∞ centered at the origin. Since ψ is a normalized wave-function, we know that for |r| → ∞, ψ(r) → 0 suﬃciently fast. Thus, the second integral of (4.5) vanishes and we are left with

N 2 Z 2 X p 2 hΨ| i |Ψi = N d3r ~ |∇ψ(r)| . (4.6) 2m 2m i=1

The remaining terms of (4.3) are straight-forward to calculate. The term corresponding to the external – trapping and/or disordered – potential reads

N Z X 3 ∗ hΨ| Vi|Ψi = N d r ψ (r)V (r)ψ(r), (4.7) i=1

48 4.1. THE GROSS-PITAEVSKII EQUATION and the inter-particle interaction term reads

N 1 X N(N − 1)ZZ hΨ| U |Ψi = d3rd3r0ψ∗(r)ψ∗(r0)U(|r−r0|)ψ(r)ψ(r0). (4.8) 2 ij 2 i6=j=1

4.1.2 Variational equations to minimize the energy functional

In order to minimize Eq. (4.3) we will now minimize hΨ|H|Ψi under the constraint that hΨ|Ψi = 1 using Lagrange multipliers λ:

δhΨ|H|Ψi − λ δhΨ|Ψi = 0. (4.9)

Our calculation of δhΨ|H|Ψi and λ δhΨ|Ψi will naturally lead to integrals containing δψ and δψ∗. Since we can independently vary Re(δψ) and Im(δψ), it is possible to treat δψ and δψ∗ as independent variations. Writing Eq. (4.9) in its integral form reads Z 3 ∗ 0 N d r δψ (r)[δHψ ψ(r) − λψ(r )] + c.c = 0, (4.10) where 2 Z δH = − ~ ∇2 + V (r) + (N − 1) d3r0 U(|r − r0|) |ψ(r)|2 (4.11) ψ 2m

1 ∗ The factor 2 of Eq. (4.8) disappeared because there are two ψ (r) in that equation. Since Eq. (4.10) has to be veriﬁed for any δψ∗(r), its solution is simply

2 Z − ~ ∇2ψ(r) + V (r)ψ(r) + (N − 1) d3r0 U(|r − r0|) |ψ(r)|2 ψ(r) = λψ(r) 2m (4.12) h i The term (N −1) R d3r0 U(|r − r0|) |ψ(r)|2 ψ(r) represents the effect of a mean- field potential received by a particle and created by the N − 1 other particles. The structure of this equation is equivalent to a Schrödingerequation with a particle of mass m that evolves in a sum of an external potential and a mean- field potential created by the other particles. Since we consider N 1, we can replace N − 1 by N for better readability, and without changing the results significantly.

4.1.3 Replacing the real potential by a pseudo-potential

The variational method we are using neglects the short-range correlations between atoms. Since the atomic gas is dilute, the atoms are mostly far from each other and hence we mainly have to consider the asymptotic behavior of the

49 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES wave-functions to describe interactions. We can therefore replace the potential U(r) by a pseudo-potential Ug with 4π 2 U = g δ(r − r0) = ~ a δ(r − r0). (4.13) g m Using Eq. (4.13) in Eq. (4.12) and supposing N 1 we conclude 2 Z − ~ ∇2ψ(r) + V (r)ψ(r) + Ng |ψ(r)|2 ψ(r) = λψ(r) , d3r |ψ(r)|2 = 1. 2m (4.14) Quite often, the normalization R d3r |ψ(r)|2 = N is regarded to be convenient. In this case, |ψ(r)|2 = n(r) is the density of particles at point r. In this case, Eq. (4.14) reads 2 Z − ~ ∇2ψ(r) + V (r)ψ(r) + g n(r)ψ(r) = λψ(r) , d3r |ψ(r)|2 = N. (4.15) 2m

4.1.4 Interpretation of the Lagrange multiplier λ

If we choose normalization R d3r |ψ(r)|2 = 1, the above developments allow us to write our energy functional E[ψ] of Eq. (4.3): Z − 2 N(N − 1) Z E[ψ] = N d3r ψ∗(r) ~ ∇2ψ(r) + V (r)ψ(r) + g d3r |ψ(r)|4 , 2m 2 (4.16) where, of course, E[ψ] represents the mean values of the Hamiltonian H assuming that we can replace the interactions between the particles by a pseudopoten- tial Ug as in (4.13). The functional E[ψ] depends both explicitly and implicitly on the total number of particles N. Explicit dependence is given via the coeﬃ- N(N−1) cients N and 2 that appear in Eq. (4.16), and implicit dependence enters via ψ(r) since Eq. (4.14) involves N for deﬁning ψ(r). It is therefore revealing to calculate ∂E[ψ] Z − 2 = d3r ψ∗(r) ~ ∇2ψ(r) + V (r)ψ(r) ∂N 2m 1 Z δE ∂ψ + N − g d3r |ψ(r)|4 + (4.17) 2 δψ ∂N

∂ψ The term ∂N in Eq. (4.17) is the variation δψ of ψ resulting from varying N by δE one. The functional derivative δψ describes the variation of E[ψ] resulting from varying ψ by δψ. However, our calculation of ψ supposes that E[ψ] is stationary for any variation of ψ by δψ. Therefore, any solution ψ of Eq. (4.16), yields δE δE ∂ψ δψ = 0 and thus δψ ∂N = 0 in Eq. (4.17). By multiplying the left Eq. (4.14) by ψ∗(r) and integrating over r, we obtain Z 2 Z λ = d3r ψ∗(r) − ~ ∇2ψ(r) + V (r)ψ(r) + d3r (N − 1)g |ψ(r)|4 . (4.18) 2m

50 4.2. DIO IN RAMAN-COUPLED BECS

Comparing Eq. (4.18) to Eq. (4.17) in the limit of N 1 we conclude that ∂E[ψ] 1 λ ≈ ∂N as long as N ≈ N − 2 ≈ N − 1. By deﬁnition, the variation of the mean energy of the system when adding a particle at constant entropy S is called the chemical potential µ. The entropy is constant because we remain at T = 0, and therefore S = 0. In summary, we can therefore postulate for a system containing a large number of particles ∂E[ψ] λ = = µ. (4.19) ∂N With this in mind, we can re-write Eq. (4.14) as the Gross-Pitaevskii Equation

2 − ~ ∇2ψ(r) + V (r)ψ(r) + Ng |ψ(r)|2 ψ(r) = µψ(r), (4.20) 2m or equivalently

2 − ~ |∇ψ(r)|2 + V (r)ψ(r) + Ng |ψ(r)|2 ψ(r) = µψ(r). (4.21) 2m

4.2 DIO in Raman-coupled BECs

In this case, we consider a trapped two-component Bose gas with repulsive interactions and assume that the two components consist of the same atomic species in two different internal states, coupled via a position-dependent (random, quasi-random, or just oscillating) real-valued Raman field Ω(r) of mean zero (R Ωdr = 0). The typical amplitude and spatial variation scale of Ω(r) are denoted by ΩR and λR. At sufficiently small T , the trapped gases form BECs which can be represented by the classical fields ψ1,2(r) in the mean-field approximation.

We use a Gross-Pitaevskii-type equation to compute the wave-functions ψ1(r) and ψ2(r) of the condensates. It is important to note that the uncoupled equa- iθi tion (4.20) is invariant under any transformation ψi(r) → e ψi(r), rotating the wave function ψi(r) by a constant complex phase θi. Of course, the global phase of any quantum-physical wave-function does not have a physical signif- icance by itself. However, if we consider two condensates described by the Gross-Pitaevskii Equation, both of their global phases and thus also their relative phase θ = θ2 − θ1 is arbitrary. Such a system is described by adding an interaction term to the Gross-Pitaevskii Equation (4.20)

 2 2 2 2 − ~ |∇ψ | + V ψ + N g |ψ | ψ + N g |ψ | ψ = µψ  2m 1 1 1 1 1 1 1 2 12 2 1 1 , (4.22) 2  ~ 2 2 2 − 2m |∇ψ2| + V2ψ2 + N2g2 |ψ2| ψ2 + N1g12 |ψ1| ψ2 = µψ2 where m is the atomic mass (supposed to be equal for both condensates). The constants {gi}i∈{1,2} and g12 describe their inter- and intra-species coupling,

51 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES

respectively. For better readability, we also used the implicit notation Vi = Vi(r) denoting the external potentials acting on condensates i = 1, 2. Once we identify a physical process such as Raman coupling that lifts this symmetry in close analogy to the uni-axial random field of the XY model, we can expect the concept of disorder-induced order to be applicable to this system. In this case, the relative phase θ will take a value orthogonal to the uni-axial randomness provided by the disordered component: for example, real-valued π randomness would lead to a phase-difference θ ≈ ± 2 . In close analogy with the results of Chapter 3 we are thus linking the original XY model with uni-axial random field, X X H = −J σi · σj − hi · σi, (4.23) |i−j|=1 i to a two-spin lattice Hamiltonian system similar to X X H = − (σi · σj + τi · τj) − Ωiσi · τi, (4.24) |i−j|=1 i where Ωi are independent real-valued random couplings with (identical) symmetric distributions. As discussed in Chapter 3, it can be proven rigorously that there is no first order phase transition with the order parameter σi · τi in dimensions D ≤ 4 [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990]. More precisely, in every infinite-dimensional Gibbs state (phase), the disorder average of the thermal mean hσi · τii takes the same value. By symmetry, this value has to be zero, implying that the average cosine of the angle between σi and τi is zero. At T = 0, these results also apply [Aizenman and Wehr, 1989, Aizenman and Wehr, 1990] and are consistent, by analogy, with the relative phase π/2 of two randomly coupled BECs. The energy functional of the system reads Z 2 2 2 4 E = dr (~ /2m)|∇ψ1| + V (r)|ψ1| + (g1/2)|ψ1|

2 2 2 4 +(~ /2m)|∇ψ2| + V (r)|ψ2| + (g2/2)|ψ2| 2 2 ∗ ∗ +g12|ψ1| |ψ2| + (~Ω(r)/2) (ψ1 ψ2 + ψ2 ψ1) , (4.25)

2 2 where V (r) is the confining potential, and gi = 4π~ ai/m and g12 = 4π~ a12/m are the intra- and inter-state coupling constants, with ai and a12 the scattering lengths and m the atomic mass. The last term in Eq. (4.25) represents the Raman coupling which can change the internal state of the atoms, and breaks the relative U(1) phase symmetry. The ground state of the coupled two-component BEC system is obtained by minimizing E as a function of the fields ψ1 and ψ2 under the constraint of a R 2 2 fixed total number of atoms N = dr(|ψ1| + |ψ2| ), in close analogy to the

52 4.3. NUMERICAL SIMULATIONS derivation in Section 4.1,

 2 2 2 2 − ~ |∇ψ | + V ψ + N g |ψ | ψ + N g |ψ | ψ + ~ Ωψ = µψ  2m 1 1 1 1 1 1 2 12 2 1 2 2 1 . 2  ~ 2 2 2 ~ ∗ − 2m |∇ψ2| + V ψ2 + N2g2 |ψ2| ψ2 + N1g12 |ψ1| ψ2 + 2 Ω ψ1 = µψ2 (4.26) ~ In the following, we will consider very small Raman couplings, 2 |Ω| µ, such that Eq. (4.26) will exhibit disorder-induced ordering. Nonetheless, already an arbitrarily small Ω(r) breaks the continuous symmetry of the relative phase. If the Raman coupling were very strong, the term is expected to dominate the system and the species should choose its relative phase according to Ω. We are therefore facing a situation that is analogous to the XY model in an external random ﬁeld and can expect the relative phase to be ﬁxed by the Raman coupling.

4.3 Numerical simulations

Our numerical simulations were based on imaginary-time evolution, which is a numerical trick to converge eﬃciently and rather reliably from an initial state to the ground-state of a system. Since this approach was used in all of our projects for calculating ground states, we expose the idea for a generic stationary Hamiltonian. In fact, the linear Schr¨odingerEquation expresses the temporal evolution of a system described by a stationary Hamiltonian H and reads ∂ i |ψ(t)i = H|ψ(t)i. (4.27) ~∂t The eigenproblems of Hamiltonian H give important insight into the physics of the system in consideration:

H|εii = Ei|εii, (4.28) where the eigenvalues Ei can be interpreted as energies of the associated eigenvectors |εii. Following the Schr¨odingerEquation (4.27) we can calculate the temporal evolution of the energy eigenstates |εi(t)i, knowing the energy eigenstate |εi(0)i at a certain time t = 0,

−iEit |εi(t)i = e ~ |εi(0)i. (4.29)

53 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES

where the coeﬃcients ci = hεi|ψi are the projection of the state onto the energy eigenstates of the system.

If we know the wave-function ψ(0) = ψ0 describing our system at time t = 0, we can use this procedure to calculate the state of the system at any time t

X −iEit |ψ(t)i = cie ~ |εi(0)i, (4.31) i where we calculate the coeﬃcients ci = hεi(0)|ψ(0)i at time t = 0. Imaginary time evolution is used for calculating the ground-state of a system, i.e. the eigenstate with the lowest energy |ε0i. If E0 < E1 ≤ E2 ≤ ... we can always choose E0 = 0 and choose an imaginary time t = −iτ in Eq. (4.31). In this case, |ψ(t)i will automatically tend to the ground-state of the Hamiltonian because X −Eiτ |ψ(τ)i = cie ~ |εii =⇒ lim |ψ(τ)i ∝ |ε0i. (4.32) τ→∞ i Imaginary time evolution is highly eﬃcient because the initial state tends towards the ground-state exponentially rapidly as long as |hε0|ψi| > 0, i.e. as long as the initial state and the ground-state are not orthogonal. In simulations this requirement turns out to be always met except for pathetic cases at most, because ground-states are non-trivial and therefore it is hugely unlikely to start with an orthogonal state. Also, simulations typically involve several runs with slightly changed parameters, so even if one was unlucky enough to start with an orthogonal state at a certain run, this would become obvious as values like energy, magnetization or the-like would abruptly change at a given, isolated datapoint. During my simulations, I never ran into any problems with the condition of non-orthogonality of the initial state to the ground-state.

In our calculations, we first choose an initial, normalized |ψ0i. Then, we apply the imaginary time step, normalize the result and thus obtain |ψ1i. On this new state, we then apply an imaginary time step and normalize again, to obtain the subsequent state. After a fixed number of iterations, e.g. 1000, we always compute the energy of the state. The iterative process is continued until the relative change in energy is less than a given value, e.g. in the order of 10−6 to 10−8. In order to ensure independence of the final result, we use a variety of |ψ0i during the test-phase of our code. While the method described above presupposes a linear Hamiltonian H, it turns out that it is also able to solve the non-linear Gross-Pitaevskii equations considered here. In fact, the imaginary-time evolution of Gross-Pitaevskii equations is even the steepest-decent trajectory towards the minimum of the canonical energy functional [Dalfovo and Stringari, 1996]. Nonetheless, it is, of course, highly important to check the numerical simulation in order to ensure that the system converges towards the correct ground-state. The numerical simulations of the two coupled BECs were developed in close collaboration with Krzysztof Sacha from the Jagiellonian University in Krakow

54 4.4. STUDIES OF THE COUPLED BECS and were based on ﬁnite diﬀerence in discretized one, two, and three-dimensional space.

4.4 Studies of the coupled BECs

At equilibrium, for ΩR = 0 and g1, g2 > g12, it can be shown that the BECs are miscible [Ho and Shenoy, 1996, Timmermans, 1998]. Their phases θi are uniform, arbitrary and independent. Now, a weak Raman coupling (~|ΩR| µ) does not noticeably affect the densities. However, arbitrarily small Ω(r) breaks the continuous U(1) symmetry with respect to the relative phase of the BECs and, as discussed in Chapter 3, the relative phase can be expected to be fixed. To make this clearer, we neglect the changes of the densities when the weak Raman coupling is turned on, and analyze the phases. For simplicity we suppose iθi(r)p g1 = g2 and ρ(r) = ρ1(r) = ρ2(r). The substitution ψi = e ρ(r) in the energy functional (4.25) leads to E = E0 + ∆E where E0 is the energy for ΩR = 0 and Z 2 ∆E = drρ(r) ~ (∇θ)2 + Ω(r) cos θ 4m ~ Z 2 + drρ(r) ~ (∇Θ)2, (4.33) 4m where Θ = θ1 + θ2 and θ = θ1 − θ2. Minimizing ∆E implies Θ = const, hence the second line in Eq. (4.33) vanishes and the only remaining dynamical variable in the model is the relative phase θ between the BECs. Note that if ρ1 6= ρ2 the variables Θ and θ are coupled and one cannot consider them independent (the ρ1 6= ρ2 case is analyzed in the sequel). Equation (4.33) is equivalent to the classical field description of the spin model (4.23) in the continuous limit, where the relative phase θ(r) represents the spin angle and the Raman coupling Ω(r) plays the role of the magnetic field. Thus, we expect DIO to show up in the form cos θ ' 0 for weak random Ω(r). Let us examine Eq. (4.33) in more detail. It represents a competition between the kinetic term which is minimal for uniform θ, and the potential term which is 2 2 minimal when the sign of cos θ is opposite to that of Ω(r). For ~ΩR ~ /2mλR, the potential term dominates and θ will vary strongly on a length scale of the 2 2 order of λR. In contrast, if ~ΩR ~ /2mλR the kinetic term is important and forbids large modulations of θ on scales of λR. The Euler-Lagrange equation of the functional (4.33) is 2m ∇ [ρ(r)∇θ] + ρ(r)Ω(r) sin θ = 0. (4.34) ~ For the homogeneous case (ρ = const) and for slowly varying densities (ne- glecting the term ∇ρ), assuming small variations of the relative phase, θ(r) = θ0 + δθ(r) with |δθ| π, the solution of Eq. (4.34) reads ˆ ˆ 2 δθ(k) ' (2m/~)(Ω(k)/|k| ) sin θ0 (4.35)

55 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES

300 (a) 150

(x) / h 0 1 2 -150 2mL 1 (b) 0.75

/ 0.5 e 0.25

0 0 0.2 0.4 0.6 0.8 1 x / L

Figure 4.1: DIO effect in a 1D two-component BEC trapped in a box of length L and in a quasi-random Raman field. Panel (a): Raman coupling function 2 Ω(x) = −100(~/2mL )[sin(x/λR + 0.31) + sin(x/(2.44λR) + 1.88)] with λR = 0.00939L. Panel (b): Relative phase θ(x) = θ1(x) − θ2(x) obtained by solving Eq. (4.34) numerically (solid black line) and comparison with Eq. (4.35) (dashed red line — nearly identical to the solid black line). in Fourier space. Inserting Eq. (4.35) into Eq. (4.33), we find Z 2 2 2 ∆E ' −mρ dk (|Ω(ˆ k)| /|k| ) sin θ0. (4.36)

The energy is thus minimal for θ0 = ±π/2, i.e. cos θ0 = 0. This indicates DIO in the two-component BEC system owing to the breaking of the continuous U(1) symmetry of the coupled GPEs. For a random Raman coupling, even if the resulting ﬂuctuations of θ are not small, the average phase is locked at θ0 = ±π/2. Note that if θ(r) is a solution of Eq. (4.34), so is −θ(r). This follows ∗ ∗ from the fact that for any solution (ψ1, ψ2) of the GPEs (4.26), (ψ1 , ψ2 ) is also a solution with the same chemical potential. The sign of θ0 thus depends on the realization of the BECs and is determined by spontaneous breaking of the θ ↔ −θ symmetry.

Let us turn to numerics starting with g1 = g2. For homogeneous (ρ = const) gases, we solve Eq. (4.34). Figure 4.1 shows an example for a 1D two-component BEC, where Ω(x) is a quasi-random function chosen as a sum of two sine functions with incommensurate spatial periods. The dynamical system (4.34) is not integrable. It turns out that the solution we are interested in corresponds to a hyperbolic periodic orbit surrounded by a considerable chaotic sea. Figure 4.1 conﬁrms that θ(x) oscillates around θ0 ' ±π/2. The oscillations of θ(x) are weak and follow the prediction (4.35), which in 1D, after inverse Fourier transform, corresponds to the double integral of Ω(x).

56 4.4. STUDIES OF THE COUPLED BECS

0.003 (a) µ

0 (x) / 1 h -0.003 1

(b) /

/ 0.5 e

0 (c) /

/ 0.5 e

0 -300 -200 -100 0 100 200 300 x [µm]

Figure 4.2: DIO effect in very elongated (effectively 1D) trapped BECs. The data corresponds to 87Rb atoms in two different internal states in an anisotropic harmonic trap with frequencies ωx = 2π × 10 Hz and ω⊥ = 2π × 1.8 kHz. The 4 total number of atoms is N = 10 and the scattering lengths are a1 = 5.77 nm, a2 = 6.13 nm and a12 = 5.53 nm. Panel (a): Single realization of the random −2 −3 Raman coupling ~Ω/µ for λR = 10 LTF and ~ΩR ' 3 × 10 µ. Panel (b): Relative phase θ corresponding to Ω(x) shown in panel (a). Panel (c): θ averaged over many realizations of Ω(x) (solid line) and the averaged value ± standard deviation (dashed lines). In panel (c) the solutions with R θdx > 0 only are collected (the other class of solutions with θ → −θ is not included).

For trapped gases and for g1 6= g2 we directly solve the coupled GPEs (4.26). Figure 4.2 shows the results for a 1D two-component BEC in the Thomas- Fermi regime confined in a harmonic trap with a random Ω(x). A typical realization is shown in Fig. 4.2a. For each realization of Ω(x), the resulting relative phase θ can change significantly but only on a scale much larger than 2 2 λR because ~ΩR ~ /2mλR, as shown in Fig. 4.2b. However, averaging over many realizations of the random Raman coupling and keeping only those with R θ(x)dx > 0 (resp. < 0), we obtain hθ(x)i ≈ π/2 (resp. −π/2), with the standard deviation about 0.3π as shown in Fig. 4.2c. The dynamical stability of the solutions of the GPEs (4.26) found in the 1D trapped geometry can be tested by means of the Bogoliubov-de Gennes (BdG) theory which allows also to estimate the quantum depletion of the two BECs [Pitaevskii and Stringari, 2003]. The BdG analysis shows that the solutions of the GPEs (4.26) are indeed stable and that the BdG spectrum is not significantly affected by the Raman coupling. It implies that turning on the Raman field does not change the thermodynamical properties of the system, and the DIO effect should persist for sufficiently low T > 0. Note that the GPEs (4.26) possess also a solution with both components real. However, this solution is dynamically

57 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES

e// 0.52 1 0.51 0.75 0.5 0.5 0.49 0.25 0.48 0 30 20 10 -30 -20 0 -10 -10 y [µm] 0 10 -20 x [µm] 20 30-30

Figure 4.3: DIO effect in a 3D two-component BEC trapped in a spherically symmetric harmonic trap with frequency ω = 2π × 30Hz. The total number of atoms is N = 105, the scattering lengths are as in Fig. 4.2 and we use a quasi- P random Raman coupling Ω(x, y, z) ∝ u∈{x,y,z}[sin(u/λR) + sin(u/(1.71λR))] −3 with λR = 4.68µm/2π and ~ΩR ' 5 × 10 µ. Shown is the relative phase θ in the plane z = 0µm in units of π. unstable. In fact, there is a BdG mode associated with an imaginary eigenvalue and the corresponding BECs phases (under a small perturbation) will evolve exponentially in time. In addition, the BdG analysis shows that the quantum depletion is about 1% and can therefore be neglected. Calculations in 2D and 3D show essentially the same Disorder-Induced Ordering effect in all dimensions. For example, Fig. 4.3 shows the result for two coupled 3D BECs in a spherically symmetric harmonic trap. Here, the Raman coupling is a sum of quasi-random functions similar to that used for Fig. 4.1 in each −2 spatial direction and with ~ΩR ' 10 µ. The density modulations are found to be negligible. However, even for this low value of the Raman coupling, Fig. 4.3 shows that the relative phase is fixed around θ0 = π/2 with small fluctuations. Other calculations confirm that the sign of θ0 is random but with |θ0| = π/2 for all realizations of Ω(r) and that the weaker the Raman coupling, the smaller the modulations of θ(r) around θ0. This shows once again the enormous robustness of DIO in two-component BECs.

4.5 Conclusions

In this part of the thesis, we have shown that DIO occurs in a system of two BECs coupled via a real-valued random Raman field. We have demonstrated the effect in 1D, 2D and 3D for homogeneous or trapped BECs. The signature of DIO is a fixed relative phase between the BECs around θ0 = ±π/2. For quasi-random Raman coupling, the fluctuations can be very small (0.05π for the parameters used in Fig. 4.1). For completely random Raman coupling the

58 4.5. CONCLUSIONS

fluctuations can be larger (about 0.3π for the parameters used in Fig. 4.2). Interestingly, the two-component BEC system is continuous and DIO is stronger and more robust than in lattice spin Hamiltonians of realistic sizes discussed in Chapter 3. DIO can thus be obtained in current experiments with two- component BECs [Matthews et al., 1998, Hall et al., 1998a, Hall et al., 1998b] and observed using matterwave interferometry techniques [Hall et al., 1998b]. Apart from its fundamental importance, DIO can have applications for engineering and manipulations of quantum states by providing a simple and robust method to control phases in ultra-cold gases. We find particularly interesting applications of phase control in spinor BECs and, more generally, in ultra-cold spinor gases [Lewenstein et al., 2007]. For example, in a ferromagnetic spinor BEC with F = 1 as in 87Rb, the wave-function is √ ξ ∝ (e−iφ cos2(θ/2), 2 sin(θ/2) cos(θ/2), e+iφ cos2(θ/2)), (4.37) the components correspond to mF = 1, 0, −1 and the direction of magnetization is ~n = (sin θ cos φ, sin θ sin φ, cos θ). Applying two real-valued random Raman couplings between mF = 0 and mF = ±1, fixes φ = 0 or π, i.e. the magnetization will be in the XZ plane. By applying two random real-valued Raman couplings between mF = 0 and mF = 1 and between mF = −1 and mF = 1, we force the magnetization to be along ±Z. Similar effects occur in antiferromagnetic spinor BECs with F = 1, such as 14Na. Using Raman transitions with arbitrary phases, employing more couplings, and higher spins F offers a variety of control tools in ultra-cold spinor gases.

59 CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES

60 Chapter 5

Disorder-induced phase control in superﬂuid Fermi-Bose mixtures

The goal of this part of the thesis is to introduce the ideas of disorder-induced order to the field of superfluid Fermi gases that possess a U(1) phase symmetry. Even if we use direct analogy between this system and the case of two BECs or (a less direct) analogy to the XY model in a uniaxial field, as discussed in chapters 3 and 4, it is important to understand that the underlying physical systems and mathematical models, as well as the numerical and analytical methods of these different cases are very different. Note, that the disorder effects are hardly visible for the homogeneous XY models, but very spectacular in the case of the coupled BECs. It is by no means a priori obvious how pronounced they will be in the Fermi case. Here, we show that a Fermi superfluid coupled to a molecular BEC via a random, symmetry-breaking photo-associating-dissociating coupling undergoes relative phase ordering, so that the phase of the order parameter can be efficiently controlled by the phase of the coupling. First, we derive some elements of BCS theory, in particular the Hartee-Fock mean-field term and the pairing function. Then, we show rigorously that for small disorder the effect is large and robust, and that it occurs practically for all temperatures below the superfluid transition temperature. Finally, we present numerical results showing the presence of disorder-induced phase control realistic experimental parameters.

61 CHAPTER 5. DISORDER-INDUCED PHASE CONTROL

5.1 Elements of BCS Theory

We start by considering the single-particle Hamiltonian H0 in close analogy to chapter 4, 2 H = − ~ ∇2 + U(r) − µ, (5.1) 0 2m where m is the mass of the Fermion, U(r) an external (e.g. trapping) potential and µ the chemical potential, included in H0 for convenience. For simplicity, we suppose that interactions V (r1 − r2) between any two particles of opposite spin are only present if the particles occupy the same spatial element,

V (r1 − r2) = −gδ(r1 − r2), (5.2) where g > 0 is a coupling constant and δ is the Dirac-Delta. Having included µ in Eq. (5.1), the grand-canonical Hamiltonian Hgc [Fetter and Walecka, 2003] reads X Z h g i H = d3r ψˆ† (r)H ψˆ (r) − ψˆ† (r)ψˆ† (r)ψˆ (r)ψˆ (r) , (5.3) gc σ 0 σ 2 σ −σ −σ σ σ where g = 4π~2|a|/m (with s-wave scattering length a) determines strength of ˆ the interactions and the ψσ stand for the fermionic ﬁeld operators. In order to study this equation, we use the BCS approach: we complement the Hartree-Fock approximation to write the four-body operator as a sum of two-body operators with two additional terms, the anomalous averages in the spirit of the Wick ˆ† ˆ† ˆ theorem. For better readability, we use the notations ψσ = ψσ(r) and ψσ = ˆ ψσ(r) and obtain

ˆ† ˆ† ˆ ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ ψσψ−σψ−σψσ ≈hψσψσiψ−σψ−σ + hψ−σψ−σiψσψσ ˆ† ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ − hψσψ−σiψ−σψσ − hψ−σψσiψσψ−σ ˆ† ˆ† ˆ ˆ ˆ ˆ ˆ† ˆ† + hψσψ−σiψ−σψσ + hψ−σψσiψσψ−σ. (5.4) Due to the absence of any physical process changing the spin of the particles, the second line of Eq. (5.4) is zero. We can therefore re-write Eq. (5.3) as Z 3 X h ˆ† ˆ ˆ† ˆ i HBCS = d r ψσ(r)H0ψσ(r) + W (r)ψσ(r)ψσ(r) σ∈↑↓ Z 3 h ˆ† ˆ† ∗ ˆ ˆ i + d r ∆(r)ψ↑(r)ψ↓(r) + ∆ (r)ψ↓(r)ψ↑(r) , (5.5) describing the interactions resulting from all other particles by the Hartree-Fock mean-ﬁeld term W (r) and introducing the pairing function ∆(r), deﬁned by

ˆ† ˆ ˆ† ˆ W (r) = −ghψ↑ψ↑i = −ghψ↓ψ↓i (5.6) ˆ ˆ ˆ ˆ ˆ† ˆ† ∗ ∆(r) = −ghψ↓ψ↑i = ghψ↑ψ↓i = ghψ↓ψ↑i (5.7)

62 5.1. ELEMENTS OF BCS THEORY

Equation. (5.5) can be expressed in matrix form [de Gennes, 1999] Z ˆ ! 3 ˆ† ˆ ψ↑ HBCS = d r ψ↑ ψ↓ Ω ˆ† + C, (5.8) ψ↓ with H0 + W (r) ∆(r) Ω = ∗ ∗ , (5.9) ∆ (r) −H0 − W (r) involving the constant C to complete the transcription. To diagonalize HBCS it is now enough to study the Bogoliubov-de-Gennes equations, the eigensystem of matrix Ω, un un Ω = En , (5.10) vn vn un where are the eigenvectors of Ω corresponding to eigenvalue En. The vn structure of (5.9) yields an important symmetry [Sacha, 2004],

∗ σzσxΩσxσz = −Ω , (5.11) ∗ where σx and σz are Pauli matrices and Ω is the complex conjugate of Ω. This symmetry implies ∗ ∗ −vn un ∗ = σxσz ∗ , (5.12) un vn which allows, for every eigenvector of eigenvalue En an eigenvector corresponding to eigenvalue −En, which we will denote E−n. The orthonormality of the eigenvectors is reﬂected by Z 3 ∗ ∗ d r [un(r)un0 (r) + vn(r)vn0 (r)] = δnn0 , (5.13) where δnn0 is the Kronecker-Delta. By virtue of Eq. (5.12), the completeness relation is ∗ X |uni −|vni ∗ ∗ I = hun| hvn| + ∗ −hvn| hun| , |vni |uni n,En≥0 (5.14) ˆ ! ψ↑(r) where I denotes the identity matrix. The original Fermionic operator ˆ† ψ↓(r) can now be expressed as

ˆ ! ∗ ψ↑(r) X un † −vn ˆ† = γˆn↑ +γ ˆn↓ ∗ , (5.15) ψ↓(r) vn un n,En≥0 where the projection operators are ˆ ˆ† γˆn↑ = hun|ψ↑i + hvn|ψ↓i, (5.16) † ∗ ˆ ∗ ˆ† γˆn↓ = −hvn|ψ↑i + hun|ψ↓i, (5.17)

63 CHAPTER 5. DISORDER-INDUCED PHASE CONTROL fulﬁlling the Fermionic anti-commutation relations

{γˆnσ, γˆn0σ0 } = 0, (5.18) n † o γˆnσγˆn0σ0 = δnn0 δσσ0 . (5.19)

In this new basis, our Hamiltonian (5.5) is diagonal and simply

X † † ˜ HBCS = En γˆn↑γˆn↑ +γ ˆn↓γˆn↓ + C, (5.20) n,En≥0 where C˜ is an appropriate constant. In order to compute Hartree-Fock mean- ﬁeld term W (r) and pairing function ∆(r) deﬁned in (5.6) and (5.7), we use the grand-canonical average values

† hγˆnσγˆn0σi = δnn0 δσσ0 f(En), (5.21)

hγˆnσγˆn0σi = 0, (5.22) with 1 f(En) = , (5.23) exp En + 1 kB T and we therefore ﬁnd

X 2 2 W (r) = −g |un(r)| f(En) + |vn(r)| (1 − f(En)) , (5.24) n X ∗ ∆(r) = g un(r)vn(r) [1 − 2f(En)] . (5.25) n

5.2 Disorder-Induced Order for BCS/BEC systems

In order to achieve disorder-induced order for the BCS/BEC system, we consider a mixture of fermions in two diﬀerent internal states interacting via an attractive zero-range potential in a 3D volume V . In absence of an external potential U(r), Eq. (5.3) reads

X Z 2 g H = dr ψˆ† − ~ ∇2 − µ ψˆ − ψˆ† ψˆ† ψˆ ψˆ , (5.26) F σ 2m σ 2 σ −σ −σ σ σ where the chemical potential µ ﬁxes the average density n. In addition to our BCS Fermions, we assume the presence of a BEC of molecular dimers consist- ing of the two fermionic species and a (weak) coupling that transforms these dimers into fermion pairs and vice versa. Experimentally, we can sweep a mixture of fermions with two diﬀerent internal states over a Feshbach resonance, which leaves us with a BEC of diatomic molecules and unbound fermions. Then,

64 5.2. DISORDER-INDUCED ORDER FOR BCS/BEC SYSTEMS we approach a second Feshbach resonance which turns the previously unbound fermions into BCS pairs without affecting the molecular BEC. For example, Potassium-40 has known s-wave resonances at 202G and 224G with widths of the order of 10G [Regal and Jin, 2003, Regal et al., 2004], and might be realistic candidates for such experiments. The coupling between the molecules and the fermions can be realized through photo-association and photo-dissociation. Taking the limit of a large BEC, we do not need to consider its dynamics because the effect of the weak coupling with the fermions on BEC is negligible. Following these considerations, the Hamiltonian (5.26) has to be supplemented by one term only: the coupling between the fermions and the BEC which we approximate by Z Z h ∗ ˆ† ˆ ˆ ˆ† ˆ† ˆ i √ h ∗ ˆ ˆ ˆ† ˆ† i dr α ψdψσψ−σ + αψ−σψσψd ≈ nd dr α ψσψ−σ +αψ−σψσ , (5.27) where α = α(r) characterizes the photo-associative-dissociative process. The ˆ bosonic field operator ψd for molecules is substituted by a real-valued condensate wave-function which for a homogeneous case considered here is the square root √ of the density of dimers, nd. The full Hamiltonian therefore reads Z h ∗ ˆ ˆ ˆ† ˆ† i H = HF + dr Γ (r)ψσψ−σ + Γ(r)ψ−σψσ , (5.28)

√ −iϕΓ with Γ(r) = Γ(˜ r)e = ndα(r). We assume that the transfer process is realized so that Γ(˜ r) is real, varies randomly in space and is constant in time, R drΓ(˜ r) = 0 and ϕΓ is a real number. We will show that for ϕΓ = 0 the relative phase between the condensate wave-function of molecules and the par- ing function of the superfluid fermions is fixed to π/2 (or −π/2). Then, we show that we can control the relative phase and fix it to any value by changing a control parameter ϕΓ, and the relative phase will be different from ϕΓ by ±π/2. This is in stark contrast to the case of constant coupling of strength c, i.e. Γ(r) ≡ c, where the relative phase trivially follows the phase of Γ(r). For Γ = 0 and in the weak coupling limit, i.e. g → 0, we deal with a Fermi γ −2 −1 −π/2kF |a| system which for T below the critical temperature Tc = 8e e π TF e 2 2 2 2 2/3 (where kBTF = εF = ~ kF /2m = ~ (3π n) /2m and γ = 0.5772 the Euler constant) reveals a transition to a superfluid phase (BCS state) which is in- ˆ ˆ dicated by a non-vanishing pairing function (order parameter) ∆ = ghψ−σψσi [Sáde Melo et al., 1993]. In the general case (i.e. including non-zero Γ) the pairing function is given in terms of solutions of Bogoliubov-de Gennes equations (5.10) with

2 2 ! ~ ∇ − 2m − µ + W ∆ + Γ un un 2∇2 = En , (5.29) ∗ ∗ ~ vn vn ∆ + Γ 2m + µ − W with the energy distribution f(En), the Hartree-Fock term W (r), and the pairing function ∆(r) deﬁned by Eq. (5.23), (5.24), and (5.25), respectively.

65 CHAPTER 5. DISORDER-INDUCED PHASE CONTROL

If the transfer process is absent (i.e. Γ = 0) the system (5.28) is invariant ˆ iϕ/2 ˆ under global gauge transformation, i.e. ψσ → e ψσ, which implies that if {un, vn} are solutions of the Bogoliubov-de Gennes equations for ∆, then iϕ/2 −iϕ/2 iϕ {e un, e vn} are the solutions corresponding to e ∆. This continuous symmetry is broken when the transfer process is turned on, as can be seen from (5.28). Then the phase of the pairing function becomes relevant because it is the relative phase with respect to the (real-valued) condensate wave-function of dimers.

5.3 Theoretical study

We begin with an analysis of the ϕΓ = 0 case, i.e. for Γ = Γ˜ real. Let us assume that for Γ = 0 and for some temperature T we have a non-zero pairing function which is chosen to be real and positive, ∆0 > 0. When we turn on Γ with |Γ(r)| ∆0 one may expect that it results in a new pairing function where iϕ(r) ∆(r) ≈ ∆0e . That is, any non-zero Γ has a dramatic effect on the phase because without the transfer process the system is degenerate with respect to the choice of ϕ. On the other hand an infinitesimal Γ is not able to change |∆(r)| because this would cost energy. Moreover, we may expect that ϕ(r) oscillates around some average value ϕ0 with small amplitude because we assume that Γ(r) fluctuates around zero with infinitesimal variance. Under these assumptions we can observe that |ϕ0| = π/2. In fact, let us neglect the Hartree-Fock term W (r) (which is not essential for Fermi superfluidity) and calculate the difference of the thermodynamic potentials between the superfluid and the normal phase

Z 1 dλ Ωs − Ω0 = hλH1iλ 0 λ Z g dg0 Z 2g0 ≈ − dr |∆|2 + Γ˜|∆| cos ϕ , (5.30) 02 0 g g h P R ˜ ˆ ˆ ˆ† ˆ† ˆ† ˆ† ˆ ˆ where we have deﬁned H1 = dr Γ ψσψ−σ + ψ−σψσ − gψσψ−σψ−σψσ σ [Fetter and Walecka, 2003]. According to our assumptions, |∆(r)| is constant 0 and cos ϕ(r) ≈ cos ϕ0 − sin ϕ0δϕ(r), for g ≈ g. Then

Z 2g0 2g0|∆| Z − dr |∆|2 + Γ˜|∆| cos ϕ ≈ −|∆|2V + sin ϕ drΓ˜δϕ, (5.31) g 0 g R and for drΓ˜δϕ < 0 the thermodynamic potential is minimized when ϕ0 = π/2. With the transformation δϕ → −δϕ and ϕ0 → −π/2 we obtain another solution what reﬂects the symmetry of the system. That is, for a real Γ, if ∆(r) corresponds to solution of (5.29) then the solution of complex conjugate Bogoliubov-de Gennes equations results in a new pairing function equal ∆∗(r). In experiments the sign of ϕ0 will depend on the realization and is determined by spontaneous breaking of the ϕ → −ϕ symmetry.

66 5.4. NUMERICAL STUDY

It is important to note that Equation (5.31) shows that the disorder-induced ordering effect is present for very different types of couplings: pseudo-random and random couplings, and even regularly oscillating ones. The effect is present as long as the mean value of Γ˜ is zero and the resulting fluctuations are small, i.e. |δϕ(r)| π.

Having determined ϕ0 we would like to estimate ﬂuctuations of the phase of the pairing function δϕ(r). To this end let us employ the Ginzburg-Landau approach [Fetter and Walecka, 2003]. Adapting the Gorkov’s derivation of the Ginzburg- Landau equation [Gorkov, 1959, Baranov and Petrov, 1998] (with the standard regularization of the bare interaction g for the case of cold atomic gases) to our problem, we obtain

2 2 2 2 2 ˜ 48π 2π ~ Tc − T ˜ ∇ ∆ = −∇ Γ − 2 + Γ 7ζ(3)lc mkF g Tc 2 2 48π Tc − T 6m ˜ 2 ˜ − 2 ∆ + 4 2 |∆ + Γ| (∆ + Γ), (5.32) 7ζ(3)lc Tc ~ kF

2 where lc = ~ kF /mkBTc. Equation (5.32) is valid for Tc − T Tc and for Γ(˜ r) that changes on a scale much larger than lc (e.g. for kF |a| = 0.5 and 14 −3 n ∼ 10 cm we get lc ∼ 4 µm). For |Γ(˜ r)| much smaller than |∆0(T )|, where ∆0(T ) is the pairing function in the absence of the transfer process, we may introduce further approximations that reduce Eq. (5.32) to

2 2 2 2 2 ˜ 48π 2π ~ Tc − T ˜ |∆0|∇ δϕ(r) = ∇ Γ(r) + 2 + Γ(r), (5.33) 7ζ(3)lc mkF g Tc where |∆| ≈ |∆0| and we have chosen ϕ0 = π/2 in the expansion ϕ(r) ≈ ϕ0 + δϕ(r). The solution of (5.33) reads

˜ 2 2 2 ˜ Γ(k) 48π 2π ~ Tc − T Γ(k) δϕ(k) = − 2 + 2 , (5.34) |∆0| 7ζ(3)lc |∆0| mkF g Tc |k| in the Fourier space. Now we switch to the general case of complex Γ = Γ˜e−iϕΓ . It is easy to check that if |∆|eiϕ corresponds to solution of the Bogoliubov-de Gennes equations i(ϕ−ϕΓ) with ϕΓ = 0 then |∆|e is related to the solution for ϕΓ 6= 0. This implies that, if for ϕΓ = 0 we are able to fix the relative phase between the condensate wave-function of molecules and the pairing function of the superfluid fermions to π/2 (or −π/2), then changing ϕΓ allows us to fix it to φ0 = π/2 − ϕΓ (or φ0 = −π/2 − ϕΓ), and phase control emerges.

5.4 Numerical study

Assuming that the transfer process with small |Γ(r)| results in phase ﬂuctua- tions of ∆(r) only, we have shown that the ﬂuctuations occur around π/2 − ϕΓ

67 CHAPTER 5. DISORDER-INDUCED PHASE CONTROL

(or −π/2 − ϕΓ), and they are given by Eq. (5.34). Now we would like to switch to numerical solutions of the Bogoliubov-de Gennes equations (where, in contrast to the analytical study, we do not neglect the Hartree-Fock term W (r)) to demonstrate that indeed for |Γ| |∆0| the fluctuations are small and the predicted phase control is possible. To achieve an optimal performance, our simulation describes the system in the plane-wave basis. The actual diagonalization is performed using the standard lapack routine zheevx. Development and validation of the simulations were done in close collaboration with Krzysztof Sacha from the Jagiellonian University in Krakow. In 3D calculations we regularize ˆ ˆ the coupling constant g in ∆ = ghψ−σψσi, i.e. g → geff , according to √ √ ! 1 1 mk 1 E + ε rE F √ C F C = − 2 2 ln √ − , (5.35) geff g 2π ~ 2 EC − εF εF where the logarithmic term results from the sum over Bogoliubov modes corresponding to energy above the numerical cutoff energy EC . The sum is performed in the spirit of the local density approximation [Bruun et al., 1999, Bulgac and Yu, 2002, Grasso and Urban, 2003], which supposes that the system can locally be considered homogeneous. For the simulations we choose −1 −1 Lz = 40kF , L⊥ = 20kF , µ = 0.83εF and kF |a| = 0.4 which for Γ = 0 and the cut-off EC = 100εF leads to ∆0(T = 0) = 0.036εF and Tc = 0.019TF . Using −1 these parameters lc ∼ 100kF is larger than the system size and we are able to explore a regime beyond Ginzburg-Landau theory. We assume real Γ(r) given by a pseudo-random function that changes along the z axis only, Γ 2π 2π Γ(r) = 0 sin (9z + 8.8) + sin (13z + 3.6) . (5.36) 2 Lz Lz

In Fig. 5.1 we show the phase of the pairing function ϕ(z) in the case when Γ0 = 0.01|∆0(0)| and ϕΓ = 0 for two different temperatures, T = 0 and T = 0.9Tc. One can see that indeed the phase oscillates around π/2 with a small amplitude (standard deviation of the order 10−2). The fluctuations of the absolute value of ∆(z) are negligible (standard deviations divided by average −4 values are of the order 10 ). When T approaches Tc the average |∆| decreases and, at some T , becomes much smaller than Γ0 and we enter another regime, where the transfer term in the Hamiltonian (5.28) starts to dominate. For a very large Γ0 we may expect that a real-valued ∆, which oscillates in space with a phase approximately opposite to the one of Γ(z), minimizes the thermodynamic potential. In Fig. 5.2 we present average values and standard deviations for ϕ and |∆| versus temperature, where one can observe an increase of the fluctuations for T → Tc which is typical for critical phenomena [Herbut, 2007].

5.5 Conclusions

In this chapter, we have shown how to control the relative phase ϕ between the wave-function of a molecular condensate and the pairing function of a mixture

68 5.5. CONCLUSIONS

-3 10 (a)

(z) 0 !

-3 -10 0.55 (b) # 0.5 (z) / " 0.45 -20 -10 0 10 20 z

Figure 5.1: Panel (a) shows Γ(z) given in Eq. (5.36) for Γ0 = 0.01|∆0(0)|. Panel (b) represents the corresponding phase ϕ(z) of the pairing function for T = 0 (black solid curve) and T = 0.9Tc (red dashed curve).

0.6 (a) 0.55 $ / z 0.5 !"# 0.45

0.4 1.05 (b) 0 % / z

# 1 | % | ! 0.95 0 0.2 0.4 0.6 0.8 1 T / Tc

Figure 5.2: Panel (a) shows average value of the phase of the pairing function hϕiz versus temperature obtained for Γ as in Fig. 5.1. In panel (b) we present the corresponding average value of the modulus of the pairing function h|∆|iz divided by ∆0(T ), i.e. the pairing for the Γ = 0 case. Solid black curves are related to average values, dashed red curves to average values ± standard deviation. The ﬁgure shows simulations for temperatures up to T = 0.99Tc where ∆0(T ) = 0.16∆0(0).

69 CHAPTER 5. DISORDER-INDUCED PHASE CONTROL of fermions in the BCS state. It turns out that weak couplings of a certain class which transfer pairs of fermions into molecules and vice versa, fix this relative phase. Contrary to phase control using constant couplings, disorder-induced phase control employs spatially randomly varying or oscillating couplings; they can be realized by optical means, with a desired phase and amplitude, which allows for efficient control of ϕ. In this part of the thesis we have considered the Fermi system in a weak coupling regime but similar behavior is expected in the strong regime. In particular, translation of our results to the simplified resonant superfluidity theory (cf. [Holland et al., 2001]) is straightforward. Our results hold also for 0 < kF a 1, where the pairing function becomes a condensate wave-function of tightly bound pairs. Hence the present situation turns out to be similar to the control of the relative phase between two Bose-Einstein condensates, analyzed in Chapter 4. The problem considered here also belongs to a general class of disorder-induced order phenomena, that rely on continuous symmetry breaking and further illustrates the applicability of this ordering mechanism to ultra-cold atomic gases.

70 Chapter 6

Disorder-induced order in quantum XY chains

This chapter reports on the most difficult project of my PhD studies: numerical studies of the one-dimensional XY quantum chains. Probably the most pro- found consequence of the quantum mechanical description is that states can be entangled. In Chapter 3, we have introduced the XY model as a paradigmatic example of disorder-induced order and discussed the classical system. In this Chapter, we present our studies on the quantum version of the one-dimensional XY spin chain with an external, site-dependent uni-axial random field within the XY plane. We first describe the mathematical model describing our quantum XY spin chains, and explain the numerical methods used to perform our studies. Then, we present our results concerning different types of external fields, namely staggered fields, oscillating fields and random fields. Finally, we discuss possible experimental implementations of such a model and conclude.

6.1 Model description

We consider a ferromagnetic spin chain with N spins 1/2 in a random external magnetic ﬁeld, described by the following hamiltonian:

N−1 N ˆ X i i+1 i i+1 X i i H = − σˆxσˆx +σ ˆyσˆy − h σˆn, (6.1) i=1 i=1

i i whereσ ˆα are the α = x, y-Pauli spin matrices at site i, and h is the random ﬁeld at site i. The ﬁeld points along an arbitrary direction n inside the XY plane andσ ˆn = n·~σ. Also within the XY plane, we will distinguish observables (and measurements) aligned about the axis n with a k subscript, and observables perpendicular to n with a ⊥ subscript. As mentioned before, the second term of

71 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Eq. (6.1) does not have the same symmetry as the first term (which is invariant with respect to rotations along the Z axis). The relevant order parameters are the mean expectation values of the mag- P i netization along the parallel and orthogonal directions:m ¯ k = h σki/N and P i i m¯ ⊥ = h σ⊥i/N. Typically, we also consider the local magnetization mk = hσki i and m⊥ = hσ⊥i, indicating which regions of the chain are being discussed. Both mk and m⊥ vanish as the amplitude of the external fields approaches zero. For large field intensity, mk follows the local direction of the field. In this case, the average m⊥ is essentially zero. Entanglement is known to be a good predictor and indicator of quantum phase transitions [Osterloh et al., 2002, Amico et al., 2008, Calabrese and Cardy, 2004, Calabrese and Cardy, 2009, Eisert et al., 2010]. Although there is a variety of possibilities, the observation of a singularity in an entanglement measure most certainly implies a second order quantum phase transition. In order to measure entanglement we will use the block entropy S(p), defined as the Von Neumann entropy of the reduced density matrix obtained by tracing out the degrees of freedom of N −p spins of the chain. By means of a Schmidt decomposition, any pure state |ψi of the system can be expressed as

X 1/2 [1...p] [p+1...N] |ψi = λi |ψi i ⊗ |ψi i, (6.2) i

[1...p] [p+1...N] where {|ψi i} and {|ψi i} are orthonormal states in the Hilbert space of the first p and last N − p spins respectively. Because of the orthonormality property, in this basis it is easy to write down the reduced density matrix for P [1...p] [1...p] the first p spins, ρp = λi|ψi ihψi |. The positive numbers λi are the so i called Schmidt coefficients, and give the block entropy X S(p) = − λilog2λi. (6.3) i The value of S(p) depends on both classical and quantum correlations (such as entanglement) between the two blocks [1, . . . , p] and [(p+1),...,N] of the N-spin chain. The so called area law says that the block entropy of a ground state generally scales with the size of the boundary (area) of the system [Eisert et al., 2010] — except at criticality, where there are typically logarithmic corrections. In one dimensional systems, the boundary of a block is constant. Thus, away from the critical point, entropy saturates beyond a certain block size p0: S(p) = S(p0) for all p > p0. Let us analyze the behavior of the block entropy of our system for some limit cases. For small fields, the XY term in Eq. (6.1) dominates, and the system exhibits long range entanglement. This leads to a large number of non-zero Schmidt coefficients —and therefore large block entropy. In contrast, for large amplitudes of the field, the second sum of Eq. (6.1) dominates the behavior of the system: the ground state is a product state with only one non-zero Schmidt

72 6.2. NUMERICAL METHODS coeﬃcient (that must be equal to one because of normalization), which gives S(p) = 0.

6.2 Numerical Methods

To obtain the ground state of ﬁnite chains for arbitrary conﬁgurations of disorder we employ the Time Evolving Block Decimation algorithm with an imaginary time evolution [Vidal, 2003, Vidal, 2004, Vidal, 2007]. The algorithm is based on calculating Schmidt decompositions at all links of the spin chain, which leads to describing the quantum state through a product of matrices. In fact, any quantum chain of length N can be written as a partition of two chain segments including spins 1 to p and p + 1 to N, respectively, as

χp X [1..p] [(p+1)..N] |ψi = λα|ψ i|ψ i, (6.4) α=1 where χp is the Schmidt rank, representing a natural measure of the entanglement between the two sides of the partition. For the algorithm, one defines a global Schmidt rank χ ≤ maxpχp ≡ χmax. For χ = χmax, the decomposition (6.4) is exact. For χ < χmax, Eq. (6.4) represents a less entangled approximation of the real quantum state. In order to construct the matrix representation of the quantum state, we consider the partition [1 : 2..N] of the spin chain. This partition can be written as χ χ 1 X X X |ψi = λ |ψ[1]i|ψ[2..N]i = Γ[1]i1 λ |i i|ψ[2..N]i, (6.5) α1 α1 α1 1 α1=1 α1=1 i1=0 where |i1i denotes the base states of the first spin (qubit) of the chain. The rank of these matrices reflects the number of Schmidt coefficients that are retained for the simulations. Therefore, slightly entangled systems (in terms of the number of non-vanishing Schmidt coefficients) are described accurately by small matrices, which leads to a large computational speedup. Strongly entangled systems, in contrast, require very large matrices to be described accurately. Excessive truncation of the matrices induces a breakdown of the algorithm, although in general one can monitor the accuracy before this happens — for example, by measuring the value of the smallest retained Schmidt coefficients. We also perform additional tests to ensure that the numerical solution does not depend on the maximum number of Schmidt coefficients. The TEBD algorithm used for our simulations is particularly efficient for one- dimensional systems with on-site and nearest-neighbors interactions only, as is the case for the XY system in presence of the external random field. We implemented the algorithm for finite systems with open boundary conditions, and for infinite systems with a periodic Hamiltonian by imposing the periodicity of the solution. Due to the numerical complexity of the algorithms, we used

73 CHAPTER 6. DIO IN QUANTUM XY CHAINS a wide range of resources from desktop computers and local clusters to the Zaragoza supercomputer with up to 50 parallel processors. In order to write and test accuracy and efficiency of the numerical code, we have collaborated with several groups. Today, there are many more groups who are using similar methods and there is even an open source variant of an essentially equivalent code available1. At the beginning of this PhD thesis, when we started developing our simulation, no such code was available. Also, we wanted to optimize our programs for large systems involving many processors and/or computers working in parallel. Therefore, we decided to program in C/C++ using lapack’s zheevx diagonalization routine rather than relying on third-party general-purpose programs. Early on in the thesis, we wrote the multi-processor parts of the simulation in close collaboration with Alex Cojuhovschi from the Institute of Theoretical Physics at the Leibniz University of Hannover. He had previously developed his own version of the code, which allowed to work on code validation and performance optimization. At a later stage, key tests regarding the accuracy of the results were done by comparing results from independently written codes by Marek Rams from Jacek Djarmaga’s group at the Jagiellonian University in Krakow. This collaboration with the Krakow group ultimately strongly influenced the direction and advancement the this part of my thesis. The disorder-induced order effect is symmetric with respect to the orthogonal direction of the disordered field. Because of this, the original proposal used convenient boundary conditions in order to lift this symmetry [Wehr et al., 2006]. For our simulations, usually it turned out to be enough to impose non-symmetric initial conditions for computing the imaginary time evolution towards the ground- state.

6.3 Numerical Results

6.3.1 Staggered ﬁeld

i i We begin by reviewing the case of an staggered magnetic field, h = (−1) h0. Although it is not random, its non-uniformity will help us gain a good intuition for the random case. We observe two distinct regimes as a function of the magnetic field intensity h0 (see Fig. 6.1a). For small fields, a finite spontaneous magnetization arises in the direction orthogonal to the field. On the other extreme, for large h0, the magnetization in this direction is zero. Interestingly, we observe that the transition between the two regimes is sharp, indicating the presence of a second order quantum phase transition. Our numerical estimate of the critical point is hc = 2.915±0.001, which is in agreement with previous studies showing a quan-

1At the time of writing of this thesis, the project is called Time- Evolving Block Decimation Open Source Code, v2.0 beta and can be found at http://physics.mines.edu/downloads/software/tebd/.

74 6.3. NUMERICAL RESULTS

tum critical point for hy ≈ 2.92 [Kurmann et al., 1981, Kurmann et al., 1982, Kenzelmann et al., 2002]. As the field intensity approaches hc from below, the spontaneous magnetization decays according to a power law, m⊥(h0) ∼ β (1 − h0/hc) . Our numerical analysis gives β = 0.125 ± 0.002. In Fig. 6.1b, we see evidence that at the critical field intensity of the staggered magnetization along the direction of the field does, indeed, show a singularity in the first derivative.

In Fig. 6.1c we show the block entropy S∞ for a semi-infinite block as a function of intensity of the staggered field. Near the critical point, the entropy of a semi- infinite block diverges [Calabrese and Cardy, 2004, Calabrese and Cardy, 2009],

1 c S = log (ξ) + a. (6.6) ∞ 2 3 2 where c is the central charge of the underlying conformal field theory. The factor 1 2 in (6.6) appears because we measure entropy between two semi-infinite parts of the chain with only one boundary between them. Through a best fit to the data shown in Fig. 6.1d, we obtain a value of c = 0.53 ± 0.05 for the central charge. For small values of the field, the entropy diverges as we approach the isotropic XY critical point. For larger values of the field intensity, entropy decays to zero, which is expected as the ground state becomes a product state. As a curiosity,√ for field intensities smaller than the critical, there is a special value h0 = 2 2 of the field for which the block entropy is exactly zero, and the ground state is thus a Néelproduct state [Kurmann et al., 1981, Kurmann et al., 1982, Kenzelmann et al., 2002].

6.3.2 Oscillating ﬁelds

Next, we focus on the case of a smooth periodic field such that at site i the field i is hk = h sin(ki), where k 1 is the wave number of the periodic field. The system exhibits spontaneous perpendicular magnetization for small, non-zero values of h, whereas for large intensities h the parallel magnetization follows the oscillating field. Figure 6.2 shows the orthogonal magnetization of the individual spins for different amplitudes of the external oscillating field. Similar to the staggered field (Fig 6.1b), we observe two regimes of orthogonal magnetization: presence of orthogonal magnetization for small amplitudes up to a given value depending on k, and disappearance thereof for larger amplitudes. The sinusoidal nature of the uni-axial external magnetic field is translated to a slight variation in the strength of the orthogonal magnetization. In Fig. 6.3, we see that the parallel magnetization of the spins is dominated by external field when the latter has large amplitudes, confirming physical intuition. The amplitude region of the presumed phase transition shows a strictly

75 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Figure 6.1: Panel (a) shows the ground state magnetization in the direction orthogonal to the staggered field depending on the strength of the staggered field. In (b), we see the staggered magnetization in the direction parallel to the external field. Panel (c) represents the entropy of entanglement S∞ for a semi-infinite block. Results in panel (d) show the entropy of entanglement S∞ as a function of the logarithm of the correlation length near the critical point hc ' 2.915. monotone increase in the tendency of individual spins to align with the external magnetic field. We show the block entropy as a function of the site and disorder amplitude in Fig. 6.4. As expected, at very low fields — near the isotropic XY critical point — the entropy grows slowly with system size, while it stabilizes rapidly for larger h. Moreover, the saturation value decreases with field intensity. The entropy has a local maximum at a field value that coincides with an abrupt decrease in perpendicular magnetization. Figure 6.5 summarizes our studies of the oscillating fields with different amplitudes. Comparing the magnetization in Fig. 6.5a to the transverse magnetization shown in Fig. 6.5c, we see that spontaneous symmetry breaking appears i near the zeros of hk. For strong enough intensities h, this leads to the creation of a set of “islands” of perpendicularly magnetized spins in a sea of transverse magnetization. When h is weak enough, however, the island size R becomes greater than the distance between the islands π/k, the isolated islands merge, and there is non-zero m⊥ of definite sign everywhere (Fig. 6.5c). The periodic

76 6.3. NUMERICAL RESULTS

Figure 6.2: Orthogonal ground state magnetization in presence of a regularly os- 2π cillating field. For k = 8 , for example, we see the appearance of a regime with orthogonal magnetization for disorder amplitudes of around 1.0 and disappearance of orthogonal magnetization around disorder amplitude of about 1.5. This confirms that uni-axially oscillating magnetic fields can induce magnetization orthogonal to the oscillating direction. transition from one phase to the other can be understood in terms of quantum phase transitions in space [Zurek and Dorner, 2008, Damski and Zurek, 2009, Dziarmaga and Rams, 2009]: away from the critical points, the system follows the local value of the field adiabatically (Fig. 6.5b) and remains on the corresponding phase. In our system, this corresponds to the regions where the field is very large and therefore the local magnetization is in the symmetric phase i i m⊥ = 0 (see [McCoy, 1968]). However, when the amplitude of the field hk ap- i −ν proaches its critical value, the local correlation length of the system, ξ ' |hk| , i dhk can become much larger than the rate of change of the field, ` ' hi / . In k di this regime the system cannot heal fast enough (compared to the change in the field), and it begins to transition from one phase to the other, forming an island of broken symmetry with a random sign of m⊥. We can estimate the size R of i an island by linearizing the magnetic field near its zero at i0: |hk| ≈ |hk (i−i0)|. At the boundary of the island we have the condition ξ ' `, which writes as |hk R|−ν ' |R|. Thus, the size of an isolated island of perpendicular magnetization results − ν R ' (hk) ν+1 . (6.7)

This line of reasoning, based on the Kibble-Zurek˙ mechanism, allows us to also estimate how the amplitude of perpendicular magnetization in the islands depends on h and k. In a ﬁrst approximation, the total magnetization is constant during the transition in space (Fig. 6.5d). When the local density approximation starts to break, i.e. at i0 ± R, the system goes from an adiabatic to an impulse region, and the order parameter at this point must “freeze”. Therefore, we can estimate the amplitude of perpendicular magnetization in the island with the

77 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Figure 6.3: Parallel magnetization in the presence of a regularly oscillating field. The plot shows that the spin chain magnetizes according to the external magnetic field if this uni-axially oscillating field has a large enough amplitude. This confirms that the physical intuition trivially works for strong magnetic fields.

i0 i0±R value of the parallel magnetization at the freezing point, m⊥ ∼ |mk |. Using i i 1/δ that near the critical point |mk| ∼ |hk| [McCoy, 1968], we obtain

1 i0 1/δ δ(ν+1) m⊥ ∼ (hkR) = (hk) . (6.8)

We simulated an inﬁnite system with k = 2π/N and N=512. We compare with the critical exponents of a pure XY chain, 1/δ ' 0.14 and ν ' 0.57 [McCoy, 1968, Continentino, 1994]. The results, shown in Fig. 6.5, give R ' −0.369 ν h , with the exponent close to the predicted value ν+1 ' 0.363. For the i0 0.092 amplitude of the magnetization of the islands we obtain m⊥ ∼ h , again in 1 good agreement with our prediction δ(ν+1) ' 0.089.

6.3.3 Randomly oscillating ﬁelds

We study randomly oscillating fields produced from a normal distribution of mean zero and varying standard deviation. A large standard deviation is equivalent to a large amplitude of an oscillating field. Producing random numbers following a normal distribution is similar to the possible experimental realization of a disordered field using laser speckles (which has already been demonstrated). Figure 6.6 shows the formation of islands of magnetization, coinciding with positive, negative, or alternating regions of the parallel magnetization. Since the external random field oscillates rapidly, the parallel magnetization, shown Fig. 6.7, cannot no longer follow the field exactly, even at unit amplitude. In close analogy to the staggered and sinusoidally oscillating field, Fig. 6.8 shows a maximum of the block entropy at the disorder amplitude corresponding to the disappearance of the orthogonal magnetization. In fact, the bulk of the

78 6.3. NUMERICAL RESULTS

Figure 6.4: Ground state block entropy of a partition in the presence of a regularly oscillating field. This configuration shows a minimum for amplitudes around 1.0 corresponding to orthogonal magnetization and a maximum for amplitudes of around 1.5 corresponding to the abrupt disappearance of the orthogonal magnetization, further indicating the presence of a quantum phase transition. This plot also indicates that the boundary effects become negligible beyond 3-5 sites from the edge of the spin chain.

material no longer shows a monotone decrease of the block entropy for increasing amplitudes up to the disappearance of the orthogonal magnetization. There now appears a more complex structure dumping into a marked minimum right before the maximum at which the amplitude of the orthogonal magnetization vanishes. The contrast between these two final extremal points resembles the discontinuity observed in the staggered field and appears to support the claim of a phase-transition at the corresponding disorder amplitudes. The fact that even for uncorrelated magnetic fields there appears to be (in most cases), a region of orthogonal magnetization illustrates the robustness of the effect. Our studies show that the disorder amplitude for which the orthogonal magnetization disappears now strongly depends on the individual uni-axial random field configuration. Figure 6.9 shows the mean value over 10 disordered realizations of the perpendicular magnetization. The graph shows a clear average presence of orthogonal magnetization for small amplitudes. The mean orthogonal magnetization appears to be strongest for a disorder amplitude of approximately 30% of the XY spin-spin correlation. As expected, the randomly oscillating field presents a number of islands of different sizes because there are regions of predominantly positive, negative or oscillating random values. When averaging over several realizations, we still see a clear overall induced constant order for small amplitudes of the external magnetic uni-axial random field.

79 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Figure 6.5: Results for an infinite system simulated with a regularly oscillating field, N = 512 sites, and periodic boundary conditions. Panel (a) shows the i magnetic field hk. In (b) and (c) respectively magnetization along the field mk and spontaneous magnetization m⊥. In panel (d) we show the total magnetiza- i q i 2 i 2 tion |m | = (m⊥) + (mk) .

6.4 Experimental realization

The proposal below combines Raman coupling to realize the spin dependent lattice with radio-frequency transitions to induce the desired structure of the hopping matrices. The main idea stems from L. Mazza et al. [Mazza and Rizzi, 2010], and consists of creating a lattice that traps certain bosonic alkali atoms in all states from the lower hyperfine manifold, and atoms in a certain state of the upper hyperfine manifold in between the sites of the square lattice. The main ingredient is to use the magic wavelength, experimentally realized in systems with 87Rb. This is in principle feasible with all the alkaline atoms, since they share the same fine structure. We propose to use the wavelength λ¯ such that the contribution for trapping the state |S1/2; mS = +1/2i coming from the states |P3/2; mS = −1/2i and |P1/2; mS = −1/2i exactly cancel. In this way, the state |S1/2; mS = 1/2i feels the potential coming from the σ+ polarized light whereas the state |S1/2; mS = −1/2i that coming from the σ− polarized light (the quantization axis of mS coincides with the propagation direction of

80 6.4. EXPERIMENTAL REALIZATION

Figure 6.6: Orthogonal magnetization of a particular realization of the random external field. We see that magnetization in the orthogonal direction of the random field is non-zero up to a threshold. The sudden drop to zero at a certain amplitude (0.3 in this realization) indicates the phase-transition between the disorder-induced order state and the state primarily following the external magnetic field.

the circularly polarized light). One should avoid working with light atoms such as 7Li due to the small fine splitting. We can, however, take heavier atoms (like 39K, 41K or even better 85Rb or 87Rb) and eventually pump the atoms to the extremal Zeeman levels, that will then serve as spinless atoms. This approach will limit the lifetime to be ≤ 1s, but should suffice to observe at least some of the physics of DIO. For experimental realizations along these lines, see for example [Lin et al., 2008]. Also, a similar approach can be applied using superlattice techniques, where the problems of spin-dependent lattices do not appear [Mazza et al., 2010, Bermudez et al., 2010]. We start by confining the atoms in 1D, say along the X axis. We take two laser pulses propagating in the X direction with circularly polarized light with respect to the X direction. Denoting by I the nuclear spin, the general result of Ref. [Mazza and Rizzi, 2010] is that for appropriately designed laser fields all F = I − 1/2 states are trapped in the 1D lattice sites, with X now the natural quantization axis. The optical potential for the F = I+1/2 manifold has minima in the same lattice sites, but, interestingly, develops also minima for the states |F = I + 1/2; Fx = I + 1/2i in the middle of the links in the X directions. These states, on the one hand, have good overlaps with the F = I − 1/2 states in the basic 1D lattice sites, and obviously can serve as intermediate states for the radio-frequency transitions between the F = I − 1/2 atoms. On the other hand, the trapping potential for the states in the F = I + 1/2 manifold can be quite weak, so that tunneling effectively dominates over interactions. The trapping potential for F = I − 1/2 atoms can be strong enough to put them in the Mott insulating regime.

81 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Figure 6.7: Parallel magnetization for a given random ﬁeld. Note that no abrupt change occurs in the amplitude region of the phase-transition (here 0.3). At unit amplitude, the parallel magnetization is no longer exactly following the quick variations of the randomly oscillating external ﬁeld.

In the case of Rubidium I = 3/2, one should prepare a large condensate in the F = 2 manifold, and then pump some atoms to the F = 1,Fx = −1 states. In the strong repulsion regime (hard bosons regime), the Hamiltonian for F = 1 atoms reduces to that of the XY model. The uni-axial random field in the XY plane can be easily realized using Raman (optical or RF) transitions with fixed phases and random strengths, as in the proposal of Ref. [Niederberger et al., 2008]. Interestingly, the same scheme can be generalized to 2D in a square, and even 3D in a simple cubic lattice. In 2D for instance, we start by confining the atoms in 2D, say in the XY plane. We take two laser pulses propagating in the X direction with circularly polarized light with respect to the X direction. Simi- larly, we apply two laser pulses in the Y direction with the circular polarizations corresponding to propagation axis Y . Again, the general result is that for appropriately designed laser fields all F = I − 1/2 states are trapped in the square lattice sites. The natural quantization axis for them is now a 45 degrees axis between X and Y . The optical potential for the F = I + 1/2 manifold has minima in the same lattice sites, but, interestingly, develops also minima for the states |F = I + 1/2; Fx = I + 1/2i and |F = I + 1/2; Fy = I + 1/2i in the middle of the links in the X and Y directions, respectively. These states have good overlaps with the F = I − 1/2 states in the basic square lattice sites, and obviously can serve as intermediate states for the radio-frequency transitions between the F = I − 1/2 atoms. The remaining ingredient of the proposal are the same as in 1D.

82 6.5. CONCLUSIONS

Figure 6.8: Block-entropy for a particular realization of the random external ﬁeld. Note the marked minimum shortly before, and the maximum at the disorder-amplitude for which the orthogonal magnetization disappears (0.3 in this realization of the disorder). This apparent discontinuity in the block entropy also indicates the presence of a quantum phase transition between an orthogonally magnetized and a not magnetized state.

6.5 Conclusions

After the effect of disorder-induced order has been shown in classical systems, we have presented numerical evidence that this effect also exists in quantum systems such as quantum XY chains. The effect consists in the appearance of magnetization in the direction orthogonal to a spatially disordered external magnetic field for small amplitudes of this field. The key ingredients for jus- tifying this result are values of various components of magnetization (which are directly measurable in the experimental scheme discussed above), and non- monotone block entropy. The latter cannot be measured directly, but its properties can be inferred from the measurements of density-density correlations using Bragg spectroscopy, noise interferometry, and/or spin polarization spectroscopy. Finally, let us mention that recently an analogue of DIO in the time domain (i.e. with time dependent perturbations) has been proposed and termed rocking [Stali¯unaset al., 2006, Stali¯unaset al., 2010].

83 CHAPTER 6. DIO IN QUANTUM XY CHAINS

Figure 6.9: Average orthogonal magnetization in the bulk of the spin chain. Our results show the existence of a range of disorder amplitudes, for which disorder-induced order occurs. The maximum average orthogonal magnetization is obtained for disorder strengths of approximately 0.3; the average is taken over 10 realizations of random external ﬁelds.

84 Chapter 7

Conclusions

Ultra-cold atomic gases by themselves and in combination with disorder are a highly active and innovative field of research. Since the first experimental realization of Bose-Einstein condensation in 1995, ultra-cold quantum gases have become a powerful tool to study condensed matter, quantum optics, and quantum information problems. Disordered systems have traditionally been extremely difficult to study, but recent advances of ultra-cold experiments have brought about new tools and approaches to tackle these problems as well. In particular, ultra-cold quantum gases can be subject to controlled spatially disordered potentials. Most notably, this has allowed for the seminal observation of Anderson Localization in 2008. Such achievements illustrate the powerful control of quantum systems experimentally that is achievable today. Nonetheless, there are still many ambitious ideas waiting to be realized, especially quantum simulators or even general purpose quantum computers allowing to mimic entangled, and other tremendously complex systems. In order to get there, however, we still need to learn a lot about the control and manipulation of quantum systems. While multi-chromatic lattices and speckle patterns are frequently used today to generate controlled random fields, a number of new, challenging approaches to creating controlled disorder:

• Rapidly moving laser beams can create an eﬀective optical potential, similar to ﬁgures created by a laser-show. This “drawing” of optical potentials allows for essentially arbitrary two-dimensional patterns and was already experimentally demonstrated [Henderson et al., 2009].

• Particles of a different species than the (primary) ultra-cold gas can be fixed in small concentrations inside an optical lattice. If this optical lattice is transparent to the (primary) ultra-cold atoms, this approach produces randomly placed scatterers fixed in space, interacting with the (primary) ultra-cold gas. This approach was discussed, for example, by [Massignan and Castin, 2006] and is not yet realized experimentally.

85 CHAPTER 7. CONCLUSIONS

• Holographic masks can be used to project an optical potential onto the ultra-cold atoms. This approach provides tailor-made, static potentials and can produce arbitrary potentials, limited only by lithographic feasi- bility [Newell et al., 2003] and [Bakr et al., 2009].

The physics of disordered systems is still a field with many open problems. In fact, the experimental realization of Anderson Localization in 2008, fueled the interest of the scientific community in disorder phenomena: The interplay between interactions and localization in different physical systems, for example, is a very active field of research:

• In non-interacting Bose gases, disorder can lead to Anderson Localiza- tion. This effect is rooted in multiple scattering of the quantum particles at the disordered potential. Weakly repulsive interactions between particles reduce the localization created by the disorder. Nonetheless, weakly interacting systems usually allow for a perturbative treatment of the interactions and approach the non-interacting system rather well [Damski et al., 2003]. • Strongly interacting disordered Bose gases are often forming glass phases in which the system takes a metastable (deep local minimum) state. Intu- itively, the interactions between the particles make it impossible to minimize the energy locally only. Therefore, the system is “trapped” in a metastable glass configuration [Fallani et al., 2007]. • Disordered interacting Fermi gases have traditionally attracted significant interest because of transport phenomena in imperfect conductors and mo- tivated the proposal of Anderson Localization. One active field of research using optical lattices is the crossover between the Mott insulating phase, in which strong repulsive interactions lead to the localization of the trapped gas, and the Fermi glass phase, in which the (almos) non-interacting particles Anderson localize in different sites because of the Pauli principle [Sanpera et al., 2004]. • Spin glasses are frustrated systems because of competing, usually randomly distributed, ferromagnetic and antiferromagnetic interactions. The exact nature of spin glass ordering is a still highly debated issue. The- oretical and experimental studies using ultra-cold atoms are expected to shed new light on this issue; for example [Edwards and Anderson, 1975, Sherrington and Kirkpatrick, 1975, Sanpera et al., 2004].

This thesis reports on the applicability and robustness of a new phenomenon called disorder-induced order. In essence, it is the process by which certain systems order when a speciﬁc type of controlled disorder is applied. The systems we consider present, in the absence of disorder, a continuous symmetry, e.g. a rotational U(1) symmetry as in the ferromagnetic XY model. If we apply a homogenous magnetic ﬁeld along a given direction of the XY plane, all spins

86 will align (roughly) in this direction. If we replace this homogenous field by a randomly oscillating field in the same direction, the system tends to avoid this external field and orders in a direction orthogonal to the field, inside the XY plane at low temperature. Therefore, it is important to understand that disorder, in this context, does not mean “lack of control” but rather “spatially varying”. Interestingly, the effect turns out to be robust enough to occur with regularly oscillating fields (i.e. staggered fields, sinusoidally oscillating fields), pseudo-random fields experimentally realizable by a superposition of optical lattices, and random fields as created by speckle plates. However, we insist that these disordered fields must not be distributed according to the symmetry of the continuous symmetry of the system but restricted to a subspace: e.g. in the case of the XY model, the spins can orient in any direction of the XY plane and if the disordered field is also orienting with random angles, the system will not magnetize at all. The effect only occurs if the disorder is always along a given direction with random orientation (and possibly real-valued amplitude). Disorder-induced order leads, for example, to spontaneous magnetization in a system that would not magnetize without disorder. Equivalently, disorder- induced order allows to fix the otherwise arbitrary relative phase between two coupled physical (sub)systems. Thus, this new phenomenon offers a different and very elegant way of controlling experimental parameters and is, we believe, within reach of present-day technology. In particular, this thesis considers the experimental implementation of

• two coupled Bose-Einstein condensates, where the relative phase between the two condensates can be controlled by disorder-induced order, • a superﬂuid (BCS) Fermi system coupled to a reservoir, in which case the phase of the BCS pairing function can be controlled by disorder-induced order, • and a one-dimensional quantum XY chain, where our results show the appearance of orthogonal magnetization through disorder-induced order.

A number of other systems are good candidates for experiments of disorder- induced order. In particular, we think that spinor condensates, that can serve as a good model system for disorder-induced ordering, coupling different internal degrees of freedom. Also, there are other systems, not considered by us, which present continuous symmetries and the possibility for ordering because of disorder. At the time of writing, there are reports of randomness-induced XY ordering in a graphene quantum Hall ferromagnet [Abanin et al., 2007] as well as disorder-induced ordering of superfluid 3HE-A in aerogel [Volovik, 2006, Volovik, 2008]. Synthetic gauge fields are recently attracting a lot of attention, because they can be used to mimic the physics of e.g. many condensed matter systems [Lewenstein et al., 2007]. With gauge fields, it is, for example, possible to control the matrix elements of the hopping term of a lattice gas, i.e. the phase of

87 CHAPTER 7. CONCLUSIONS the tunneling. Therefore, we see the possibility to study various new kinds of disorder phenomena and in particular disorder-induced order with a different kind of controlled disorder: neither potential disorder nor interaction disorder, but phase disorder. For this reason, we are confident that our findings contribute to the understanding of disorder and its possible usefulness for controlling quantum and classical systems. And hopefully, these findings will prove to be helpful on the way to developing new concepts and technologies for quantum control and engineering.

88 Acknowledgements

I have the great pleasure to thank all the people supporting my research over the past years. First and foremost, I thank Maciej Lewenstein for his constant support and mentoring. Furthermore, I thank Fernando Cucchietti and all other past and present members of ICFO’s Quantum Optics Theory Group for their friendship and their scientific support during all of my stay at ICFO. Also, I would like to thank ICFO’s administrative staff who have enormously facilitated my stay at ICFO as well as ICFO’s IT-Support, who have worked hard to support my numerical simulations. Through Maciek, I had the privilege to collaborate with several international groups: from the beginning, I had the pleasure of working with Jan Wehr from Arizona on a variety of aspects related to disorder-induced order. Early on, I then visited Alain Aspect’s group to work with Laurent Sanchez-Palencia in Orsay (at the time). Later I visited Eric Jeckelmann’s group to work with Alex Cojuhovschi in Hannover. Finally, I trav- elled to Krakówseveral times to work with two groups there. Krzysztof Sacha from the the Jagiellonian University in Krakówbecame not only a collaborator and teacher but also a constant source of physical intuition and motivation for me. The other important pillar of my stay in Krakówwere the discussions and the collaboration with Jacek Dziarmaga and Marek Rams who’s patience and hospitality I also greatly appreciate. In addition, I thank Thomas Schulte and Boris Malomed, with who I collaborated at ICFO, Henning Fehrmann, Arturo Argüelles,Karen Rodr´ıguez,and all the other members of the Institute of The- oretical Physics at the University of Hannover as well as Jakub Zakrzewski, Roman Marcinek, Bartlomiej Ole´s,Guilhem Collier, and the members of the Krakówgroups. In addition to my scientific collaborators, I thank Lluis Torner for making it possible for us students to build up a student organization and for creating a very supportive environment for our initiatives. I am very happy that I have been given the opportunity of working with Giovanni Volpe and Giorgio Volpe on several of our shared visions and dreams. In relation with our projects, I would like to thank also Silvia Carrasco, Dolors Mateu, and, once again, ICFO’s administrative staff for their great support. Our student organization would not have been such a great experience without Lars Neumann, Osamu Takayama, Gajendra Patrap Singh, Manoj Mathew Zhiyong Xu and all the other past

89 CHAPTER 7. ACKNOWLEDGEMENTS and present members of “the chapter” as well as Jens Biegert, Valerio Pruneri and Romain Quidant who served as student advisors. For the support of the IONS Project, I would also like to thank all of our co-founders, participants and organizers, Thomas Baer, the Optical Society of America, KiKi L’Italien, April Zack, as well as Liz Rogan and all the other members of OSA’s staff. Finally, I also thank the Maciej Kolwas, David Lee, and the European Physical Society for giving us the possibility to start EPS Young Minds, and Antigone Marino, Sylvie Loskill, OphéliaFornari for collaborating on this project. On the personal side, I would like to thank Mar´ıaBelénPérez-Ram´ırez,Helena, Lukas, Astrid, and Isabelle Niederberger, as well as Viktor Käslinand his family. Also, I would like to express my gratitude to Rodr´ıgoValdivieso, Natalia Arañó, AntónAlbajes Eizagirre, Toni Rios Rodriguez, Lara Andreu Gallego, Miguel Fernandez, Danail Obreschkow,Lukasz Zawitkowski, Bartosz Chmura, Giulia Ferrini, Alberto Curto, Rafael Betancur, Sergio Di Finizio, Zoran Djordjevic, Oscar Cordero, Christoph Schwitter, Nicolas Gauger, Markus Hennrich, Ana Predojevich, Elke Hager, Larissa, Jennifer and Melissa Yepez, Karla Brugal, Pedro Guzmán, and all of my other good friends from around the world who have made my stay in Barcelona such a fantastic experience.

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