Download 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
Conclusion
45
3.8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 onedimensional 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 imperfection. 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 disorderinduced 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 indispensable 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 Condensate [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 longstanding 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 nonzero 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. Another 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 disorderinduced 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 Section 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 con1 More
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
' 1.05 10−34 Js is defined via the Planck constant h, ω is the
where ~ = 2π
angular frequency of the wave and k = 2π
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 =
' kB T,
2
2m
2
(2.2)
J
where v is the average speed of the particles, kB ' 1.38 10−23 K
the Boltzmann constant, and T the absolute temperature. Combining this equation with
Eq. (2.1), we find the expression for the de Broglie wavelength λdB :
λdB = √
1
h
∝√
.
3mkB T
mT
(2.3)
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 Ångström2 . 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 Ångström, 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 In fact, for N molecules, m ' 28 g ' 4.65 10−26 kg, and thus at T = 300K, the position
2
mol
uncertainty of the center of mass is in the order of λdB ' 2.6 10−11 m = 0.26Å.
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 quantummechanical 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 wave13
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 BoseEinstein 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 H2 O, 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 intermolecular 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 usage 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 emphasized 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
by H1 to a regime in which H2 dominates. For example, let H1 = −J σix ,
i
J > 0 describe a system in which all particles (spins) tend to align along the
15
CHAPTER 2. BASIC CONCEPTS AND METHODS
x direction and H2 = −J
P
σiz σjz 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
H = −t
X
hi,ji
UX
X
[ni (ni − 1)] − µ
ni ,
b†i bj + b†j bi +
2 i
i
(2.5)
where hi, ji means that i and j are nearest neighbors, bi is the bosonic annihilation operator at site i and b†i the bosonic creation operator, ni = b†i 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. Physically, 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
V (x) = sER cos2 (kx) ,
where k =
2π
λ
is the lattice wavenumber, ER =
17
h2
2mλ2
(2.6)
is the recoil energy with m
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 Kagomé lattices.
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 superfluid 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
V (x) = s1 ER1 cos2 (k1 x) + s2 ER2 cos2 (k2 x) ,
2
(2.7)
h
where, for i = 1, 2, ki = 2π
λi is the wavenumber, ERi = 2mλ2i the recoil energy
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 result19
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. Correlation 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
V (r) =
3πc2 Γ
I(r),
2ω03 ω − ω0
(2.8)
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 V̄ and spatial extension L, VS is therefore defined by

1
VS = 2 
L
L/2
Z
 12
2 
dx V (x) − V̄  .
(2.9)
−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
d2
(2.10)
G(d) =
dx V (x)V (x + d) ∝ e− 2σ2 .
−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]. Figure 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 thickness 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 benefited from the long-standing “holy grail of disorder physics”, Anderson Localization, as well as from new experiments realizing the Bose-Hubbard Hamiltonian (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 instrumental 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: Anderson 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 atomatom 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
H = −t
X
hi,ji
UX
X
[ni (ni − 1)] −
(µ + i )ni ,
b†i bj + b†j bi +
2 i
i
23
(2.11)
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 compressibility, 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 exter27
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.
HIsing = −J
X
si sj − h
hi,ji
X
si .
(3.1)
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
system, we can then derive the sxi and syi with (sxi )2 + (syi )2 = 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
HXY = −J
X
X
sui ,
sxi sxj + syi syj − h
(3.2)
i
hi,ji
with su designating the spin component in direction u. For classical systems
with spins of unit length, sxi sxj + syi syj = 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 generalization of this model is to consider the coupling along the x and y directions
independent. The coupling term can then be written as
coupling
Hasymmetric
XY = −J
X
(1 + γ)sxi sxj + (1 − γ)syi syj ,
(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
si sj − 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−1 J. For onedimensional 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−1 J. In fact, a one-dimensional state perto
over L 1 spins costs approximately
forming a slow rotation from
2
2 π
2π
π 2
JL 1 − cos( L ) = JL 2 sin ( L )) ≈ 2JL L
= 2πL J . 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 2π 2 J. 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 2π 2 LD−2 J. 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 qualitative 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-Mottlike 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 al31
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 counterintuitively 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 nondisordered one. Finally, we propose three possible and experimentally feasible
realizations of the DIO phenomenon using ultra-cold atoms in optical lattices.
3.5
3.5.1
Disorder-induced order in the classical XY
model
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 ),
at a site i ∈ Z2 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
|i−j|=1
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 twodimensional 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 = ηi ey ,
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 = ηi ey , 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], eliminating 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 mi mj 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
P shows that the probability of such a contour is bounded above
by exp(−2β (ij) mi mj ), with β = J/kB T .
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 suffices to prove the summability of the series of the expected values. We present
two arguments for the last statement to hold.
√
c, for some
If the random variables mi are bounded away from zero, i.e. mi > P
c > 0, the moment generating function of the random variable
(ij) mi mj
satisfies
*
+
X
exp − β
mi mj
≤ exp[−cβL(γ)],
(3.9)
(ij)
with L(γ) denoting the length of the contour γ. The sum
P of the probabilities
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
P then another argument is needed. If we
assume that the terms in the sum (ij) mi mj are independent and identically
distributed, then
*
+
X
L(γ)
exp(−2β
mi mj ) = hexp(−2βmi mj )i
(3.10)
(ij)
= exp{L(γ) log hexp(−2βmi mj )i}
(3.11)
and we just need to observe that
hexp(−2βmi mj )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
mi mj )
behaves as
exp[−g(β)L(γ)]
(3.13)
(ij)
for a positive function g(β) with g(β) → ∞ as β → ∞. While mi mj 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
mi mj over subsets of Z2 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 = ηi ey in the y direction.
i ’s are independent random
√ The η√
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
m (magn. along x)
3.6. DISORDER-INDUCED ORDER IN OTHER SYSTEMS
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.75
0.7
0.65
0.6
0.6 0.65 0.7 0.75
6hy=0
6hy=0.25
6hy=0.5
0
0.2
0.4
0.6
0.8
kBT/J
1
1.2
1.4
1.6
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 behaviors 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 disorderinduced 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]:
HBBH =
X h Ub
2
j
+
X
nj (nj − 1) +
i
UB
Nj (Nj − 1) + UbB nj Nj
2
(vj nj + Vj Nj )
(3.14)
j
i X Ω
X h
j †
†
†
b Bj + h.c.
−
Jb bj bl + JB Bj Bl + h.c. −
2 j
j
hj,li
where bj and Bj are the annihilation operators for both types of Bosons in the
lattice site j, nj = b†j bj and Nj = B†j Bj are the corresponding number operators, and hj, li denote a pair of adjacent sites in the optical lattice. In Hamiltonian (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 Q
mixture. In brief, we restrict the Hilbert
space to a subspace E0 generated by { j |nj , Nj i} 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,l Bj† Bl + h.c +
Kj,l Nj Nl
hj,li
+
X
j
hj,li
Vj Nj −
X Ωj
j
41
2
Bj + h.c.
(3.15)
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 = Pb†j Bj , 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 :


Jj,l =
Kj,l = −
Jb JB 

UbB
1−
1
δj,l
UbB
2 +
1−
1
∆j,l
UbB
(3.16)

2 
2
2
/UB
2Jb2 /UbB
4JB
/UbB
4Jb2 /Ub
2JB
2 −
2
2 +
2 +
δ
δ
∆j,l
∆j,l
1 − Uj,lb
1 − Uj,l
1
−
1
−
UB
UbB
bB

Vj = Vj − vj +
X

hj,li
(3.17)

4Jb2 /Ub
1−
δj,l
Ub
2 −
Jb2 /UbB
δ
1− Uj,l
bB
J 2 /UbB
− B ∆j,l
1+ UbB
+
2
4JB
/UB
1−
∆j,l
UB

2  (3.18)
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 = b†j Bj P. In
contrast to the case of Fermi-Bose mixtures discussed in [Sanpera et al., 2004],
1 The 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 virtual 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
Sxj + iSyj = Bj and Szj = 1/2 − Nj , where Nj = Bj† Bj . It is important to
note that since
P Raman processes can convert b Bosons into B Bosons (and
conversely), j hNj i is notPfixed by the total number of Bosons of each species,
i.e. the z component of m, j hSzj 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
Heff = −J⊥
Szj Szl
Sxj Sxl + Syj Syl − Jz
hj,li
−
X
hxj Sxj
hj,li
+
hyj Syj
+
hzj Szj
(3.19)
j
where
4Jb JB
U
bB 2
2Jb
2JB2
Jb2 + JB2
Jz = 2
+
−
Ub
UB
UbB
J⊥ =
hxj = ΩR
j
; hyj = −ΩIj
; hzj = Vj − ζJz /2 ,
(3.20)
(3.21)
(3.22)
with ζ the lattice coordination number, Vj = Vj − vj + ζ[4Jb2 /Ub + 4JB2 /UB −
I
(Jb2 + JB2 )/UbB ] and Ωj = ΩR
j + 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
this way, setting vj = 0 and ΩIj = 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
with the strength determined by ΩR
j .
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,
Garcı́a-Ripoll et al., 2004, Porras and Cirac, 2004, Micheli et al., 2006]. However, 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
H=
N 2
X
p
i
i=1
2m
+ Vi
+
N
N
1X X
Uij .
2 i=1
(4.1)
i6=j=1
The factor 21 in the last term is necessary because the sums count all twopartite 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
determining states that minimize hΨ|H|Ψi
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 = |ψ1 i|ψ2 i|ψ3 i · · · |ψi i · · · |ψN i.
(4.2)
We now want to minimize the energy functional E[ψ], associating an energy
value with each wave-function ψ,
E[ψ] =
hΨ|H|Ψi
.
hΨ|Ψi
(4.3)
Our minimization approach considers each particle to be subjected to a meanfield 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):
hΨ|
Z
N
X
~2
p2i
|Ψi = N d3 r ψ ∗(r) −
∇2 ψ(r).
2m
2m
i=1
(4.4)
To solve this equation, we remember that ∇ [ψ ∗(r)∇ψ(r)] = ψ ∗(r)∇2 ψ(r) +
2
|∇ψ(r)| . Equation (4.4) can therefore be re-written as
hΨ|
Z
Z
N
X
~2
~2
p2i
2
|Ψi = N d3 r
|∇ψ(r)| − N d3 r
∇ [ψ ∗(r)∇ψ(r)] . (4.5)
2m
2m
2m
i=1
The
second term
H of this integral can be calculated: the theorem of divergence,
R
f · dS, states that the integral of the divergence of a vectorial
∇ · f dV =
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
Z
N
X
p2i
~2
2
hΨ|
|Ψi = N d3 r
|∇ψ(r)| .
2m
2m
i=1
(4.6)
The remaining terms of (4.3) are straight-forward to calculate. The term corresponding to the external – trapping and/or disordered – potential reads
hΨ|
N
X
Z
Vi |Ψi = N
d3 r ψ ∗(r)V (r)ψ(r),
i=1
48
(4.7)
4.1. THE GROSS-PITAEVSKII EQUATION
and the inter-particle interaction term reads
ZZ
N
1 X
N (N − 1)
d3 rd3 r0 ψ ∗(r)ψ ∗(r0 )U (|r−r0 |)ψ(r)ψ(r0 ). (4.8)
hΨ|
Uij |Ψi =
2
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
N d3 r δψ ∗(r) [δHψ ψ(r) − λψ(r0 )] + c.c = 0,
(4.10)
where
δHψ = −
~2 2
∇ + V (r) + (N − 1)
2m
Z
d3 r0 U (|r − r0 |) |ψ(r)|
2
(4.11)
The factor 21 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
Z
~2 2
2
3 0
0
−
∇ ψ(r) + V (r)ψ(r) + (N − 1)
d r U (|r − r |) |ψ(r)| ψ(r) = λψ(r)
2m
(4.12)
hR
i
2
3 0
0
The term (N −1) d r U (|r − r |) |ψ(r)| ψ(r) represents the effect of a meanfield potential received by a particle and created by the N − 1 other particles.
The structure of this equation is equivalent to a Schrödinger equation with a
particle of mass m that evolves in a sum of an external potential and a meanfield 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
Ug = g δ(r − r0 ) =
4π~2
a δ(r − r0 ).
m
(4.13)
Using Eq. (4.13) in Eq. (4.12) and supposing N 1 we conclude
Z
~2 2
2
2
−
∇ ψ(r) + V (r)ψ(r) + N g |ψ(r)| ψ(r) = λψ(r) ,
d3 r |ψ(r)| = 1.
2m
(4.14)
R 3
2
Quite often, the normalization d r |ψ(r)| = N is regarded to be convenient.
2
In this case, |ψ(r)| = n(r) is the density of particles at point r. In this case,
Eq. (4.14) reads
Z
~2 2
2
∇ ψ(r) + V (r)ψ(r) + g n(r)ψ(r) = λψ(r) ,
d3 r |ψ(r)| = N. (4.15)
−
2m
4.1.4
Interpretation of the Lagrange multiplier λ
R
2
If we choose normalization d3 r |ψ(r)| = 1, the above developments allow us
to write our energy functional E[ψ] of Eq. (4.3):
2
Z
Z
N (N − 1)
−~ 2
4
∇ ψ(r) + V (r)ψ(r) +
g d3 r |ψ(r)| ,
E[ψ] = N d3 r ψ ∗(r)
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 pseudopotential 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 coefficients N and N (N2−1) 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
2
Z
−~ 2
∂E[ψ]
=
d3 r ψ ∗(r)
∇ ψ(r) + V (r)ψ(r)
∂N
2m
Z
1
δE ∂ψ
4
+
N−
(4.17)
g d3 r |ψ(r)| +
2
δψ ∂N
∂ψ
The term ∂N
in Eq. (4.17) is the variation δψ of ψ resulting from varying N by
one. The functional derivative δE
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
Z
~2 2
4
λ = d3 r ψ ∗(r) −
∇ ψ(r) + V (r)ψ(r) + d3 r (N − 1)g |ψ(r)| . (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
λ ≈ ∂E[ψ]
as long as N ≈ N − 12 ≈ N − 1. By definition, the variation of
∂N
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
λ=
∂E[ψ]
= µ.
∂N
(4.19)
With this in mind, we can re-write Eq. (4.14) as the Gross-Pitaevskii Equation
~2 2
2
∇ ψ(r) + V (r)ψ(r) + N g |ψ(r)| ψ(r) = µψ(r),
2m
(4.20)
~2
2
2
|∇ψ(r)| + V (r)ψ(r) + N g |ψ(r)| ψ(r) = µψ(r).
2m
(4.21)
−
or equivalently
−
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
R
zero ( 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 equation (4.20) is invariant under any transformation ψi (r) → eiθi 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 significance 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

 − 2m |∇ψ1 | + V1 ψ1 + N1 g1 |ψ1 | ψ1 + N2 g12 |ψ2 | ψ1 = µψ1
, (4.22)


2
2
2
~2
− 2m |∇ψ2 | + V2 ψ2 + N2 g2 |ψ2 | ψ2 + N1 g12 |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
randomness would lead to a phase-difference θ ≈ ± π2 .
In close analogy with the results of Chapter 3 we are thus linking the original
XY model with uni-axial random field,
X
H = −J
σi · σj − X
hi · σi ,
(4.23)
i
|i−j|=1
to a two-spin lattice Hamiltonian system similar to
H=−
X
(σi · σj + τi · τj ) −
X
Ωi σi · τi ,
(4.24)
i
|i−j|=1
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 · τi i 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
E =
dr (~2 /2m)|∇ψ1 |2 + V (r)|ψ1 |2 + (g1 /2)|ψ1 |4
+(~2 /2m)|∇ψ2 |2 + V (r)|ψ2 |2 + (g2 /2)|ψ2 |4
+g12 |ψ1 |2 |ψ2 |2 + (~Ω(r)/2) (ψ1∗ ψ2 + ψ2∗ ψ1 ) ,
(4.25)
where V (r) is the confining potential, and gi = 4π~2 ai /m and g12 = 4π~2 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 Rfields ψ1 and ψ2 under the constraint of a
fixed total number of atoms N = dr(|ψ1 |2 + |ψ2 |2 ), in close analogy to the
52
4.3. NUMERICAL SIMULATIONS
derivation in Section 4.1,

2
2
2
~
~2

 − 2m |∇ψ1 | + V ψ1 + N1 g1 |ψ1 | ψ1 + N2 g12 |ψ2 | ψ1 + 2 Ωψ2


~2
− 2m
2
2
2
|∇ψ2 | + V ψ2 + N2 g2 |ψ2 | ψ2 + N1 g12 |ψ1 | ψ2 +
~ ∗
2 Ω ψ1
=
µψ1
.
=
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 Schrödinger Equation expresses the temporal
evolution of a system described by a stationary Hamiltonian H and reads
i~
∂
|ψ(t)i = H|ψ(t)i.
∂t
(4.27)
The eigenproblems of Hamiltonian H give important insight into the physics of
the system in consideration:
H|εi i = Ei |εi i,
(4.28)
where the eigenvalues Ei can be interpreted as energies of the associated eigenvectors |εi i. Following the Schrödinger Equation (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,
|εi (t)i = e
−iEi t
~
|εi (0)i.
(4.29)
TheP
energy eigenvectors span the full Hilbert space associated with the system,
i.e.
|εi ihεi | = 1 and, supposing non-degenerate eigenvalues Ei for readability,
i
we can therefore re-write any state |ψi as
X
X
|εi ihεi |ψi =
ci |εi i,
|ψi =
i
i
53
(4.30)
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
|ψ(t)i =
X
ci e
−iEi t
~
|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 |ε0 i. 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 −Ei τ
lim |ψ(τ )i ∝ |ε0 i.
(4.32)
|ψ(τ )i =
ci e ~ |εi i =⇒
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 |ψ0 i. Then, we apply
the imaginary time step, normalize the result and thus obtain |ψ1 i. 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
|ψ0 i 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
pwe suppose
g1 = g2 and ρ(r) = ρ1 (r) = ρ2 (r). The substitution ψi = eiθi (r) ρ(r) in the
energy functional (4.25) leads to E = E0 + ∆E where E0 is the energy for
ΩR = 0 and
2
Z
~
(∇θ)2 + ~Ω(r) cos θ
∆E =
drρ(r)
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
minimal when the sign of cos θ is opposite to that of Ω(r). For ~ΩR ~2 /2mλ2R ,
the potential term dominates and θ will vary strongly on a length scale of the
order of λR . In contrast, if ~ΩR ~2 /2mλ2R 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)Ω(r) sin θ = 0.
(4.34)
~
For the homogeneous case (ρ = const) and for slowly varying densities (neglecting 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
∇ [ρ(r)∇θ] +
δ θ̂(k) ' (2m/~)(Ω̂(k)/|k|2 ) sin θ0
55
(4.35)
CHAPTER 4. COUPLED BOSE-EINSTEIN CONDENSATES
(a)
150
0
2
2mL 1(x) / h
300
-150
1
(b)
e//
0.75
0.5
0.25
0
0
0.2
0.4
x/L
0.6
0.8
1
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
Ω(x) = −100(~/2mL2 )[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
∆E ' −mρ dk (|Ω̂(k)|2 /|k|2 ) sin2 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
(a)
0.003
0
-0.003
1
(b)
e//
h1(x) / µ
4.4. STUDIES OF THE COUPLED BECS
0.5
eœ / /
0
(c)
0.5
0
-300
-200
-100
0
x [µm]
100
200
300
Figure 4.2: DIO effect in very elongated (effectively 1D) trapped BECs. The
data corresponds to 87 Rb atoms in two different internal states in an anisotropic
harmonic trap with frequencies ωx = 2π × 10 Hz and ω⊥ = 2π × 1.8 kHz. The
total number of atoms is N = 104 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
Raman coupling ~Ω/µ for λR = 10−2 LTF and ~ΩR ' 3 × 10−3 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
R value ± standard
deviation (dashed lines). In panel (c) the solutions with 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 ThomasFermi 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
λR because ~ΩR ~2 /2mλ2R , as shown in Fig. 4.2b. However, averaging over
Rmany realizations of the random Raman coupling and keeping only those with
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
0
x [µm]
10
20
30-30
-20
-10
0
10
20
30
y [µm]
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
P are as in Fig. 4.2 and we use a quasirandom Raman coupling Ω(x, y, z) ∝ u∈{x,y,z} [sin(u/λR ) + sin(u/(1.71λR ))]
with λR = 4.68µm/2π and ~ΩR ' 5 × 10−3 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
spatial direction and with ~ΩR ' 10−2 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 twocomponent 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 87 Rb, the wave-function is
√
(4.37)
ξ ∝ (e−iφ cos2 (θ/2), 2 sin(θ/2) cos(θ/2), e+iφ cos2 (θ/2)),
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 14 Na. 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 2
H0 = −
∇ + U (r) − µ,
(5.1)
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
i
h
XZ
g
†
(r)ψ̂−σ (r)ψ̂σ (r) ,
(5.3)
Hgc =
d3 r ψ̂σ† (r)H0 ψ̂σ (r) − ψ̂σ† (r)ψ̂−σ
2
σ
where g = 4π~2 |a|/m (with s-wave scattering length a) determines strength of
the interactions and the ψ̂σ 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
theorem. For better readability, we use the notations ψ̂σ† = ψ̂σ† (r) and ψ̂σ =
ψ̂σ (r) and obtain
†
†
†
ψ̂σ† ψ̂−σ
ψ̂−σ ψ̂σ ≈hψ̂σ† ψ̂σ iψ̂−σ
ψ̂−σ + hψ̂−σ
ψ̂−σ iψ̂σ† ψ̂σ
†
†
− hψ̂σ† ψ̂−σ iψ̂−σ
ψ̂σ − hψ̂−σ
ψ̂σ iψ̂σ† ψ̂−σ
†
†
+ hψ̂σ† ψ̂−σ
iψ̂−σ ψ̂σ + hψ̂−σ ψ̂σ iψ̂σ† ψ̂−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
i
Xh
HBCS = d3 r
ψ̂σ† (r)H0 ψ̂σ (r) + W (r)ψ̂σ† (r)ψ̂σ (r)
σ∈↑↓
Z
+
h
i
d3 r ∆(r)ψ̂↑† (r)ψ̂↓† (r) + ∆∗ (r)ψ̂↓ (r)ψ̂↑ (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
W (r) = −ghψ̂↑† ψ̂↑ i = −ghψ̂↓† ψ̂↓ i
∆(r) = −ghψ̂↓ ψ̂↑ i = ghψ̂↑ ψ̂↓ i =
62
(5.6)
ghψ̂↓† ψ̂↑† i∗
(5.7)
5.1. ELEMENTS OF BCS THEORY
Equation. (5.5) can be expressed in matrix form [de Gennes, 1999]
!
Z
ψ̂↑
†
3
HBCS = d r ψ̂↑ ψ̂↓ Ω
+ C,
ψ̂↓†
with
Ω=
H0 + W (r)
∆(r)
∆∗ (r)
−H0∗ − W (r)
(5.8)
,
(5.9)
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],
σz σx Ωσx σz = −Ω∗ ,
(5.11)
where σx and σz are Pauli matrices and Ω∗ is the complex conjugate of Ω. This
symmetry implies
∗ −vn∗
un
=
σ
σ
,
(5.12)
x
z
u∗n
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
d3 r [u∗n (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 |un i −|vn∗ i
∗
∗
−hvn | hun |
hun | hvn | +
,
I=
|u∗n i
|vn i
n,En ≥0
where I denotes the identity matrix. The original Fermionic operator
can now be expressed as
!
X
ψ̂↑ (r)
un
−vn∗
†
=
,
γ̂n↑
+ γ̂n↓
u∗n
vn
ψ̂↓† (r)
(5.14)
!
ψ̂↑ (r)
ψ̂↓† (r)
(5.15)
n,En ≥0
where the projection operators are
γ̂n↑ = hun |ψ̂↑ i + hvn |ψ̂↓† i,
(5.16)
†
γ̂n↓
(5.17)
=
−hvn∗ |ψ̂↑ i
63
+
hu∗n |ψ̂↓† i,
CHAPTER 5. DISORDER-INDUCED PHASE CONTROL
fulfilling the Fermionic anti-commutation relations
{γ̂nσ , γ̂n0 σ0 } = 0,
o
n
γ̂nσ γ̂n† 0 σ0 = δnn0 δσσ0 .
(5.18)
(5.19)
In this new basis, our Hamiltonian (5.5) is diagonal and simply
X
†
†
HBCS =
En γ̂n↑
γ̂n↑ + γ̂n↓
γ̂n↓ + C̃,
(5.20)
n,En ≥0
where C̃ is an appropriate constant. In order to compute Hartree-Fock meanfield term W (r) and pairing function ∆(r) defined in (5.6) and (5.7), we use the
grand-canonical average values
†
hγ̂nσ
γ̂n0 σ i = δnn0 δσσ0 f (En ),
(5.21)
hγ̂nσ γ̂n0 σ i = 0,
(5.22)
with
1
f (En ) =
exp
En
kB T
,
(5.23)
+1
and we therefore find
W (r) = −g
X
|un (r)|2 f (En ) + |vn (r)|2 (1 − f (En )) ,
(5.24)
n
∆(r) = g
X
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
XZ
~2 2
g
†
ψ̂−σ ψ̂σ ,
(5.26)
HF =
dr ψ̂σ† −
∇ − µ ψ̂σ − ψ̂σ† ψ̂−σ
2m
2
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 consisting 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
h
i
h
i √ Z
†
†
∗ †
†
ψ̂σ† , (5.27)
dr α ψ̂d ψ̂σ ψ̂−σ + αψ̂−σ ψ̂σ ψ̂d ≈ nd dr α∗ ψ̂σ ψ̂−σ +αψ̂−σ
where α = α(r) characterizes the photo-associative-dissociative process. The
bosonic field operator ψ̂d 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
i
†
H = HF + dr Γ∗ (r)ψ̂σ ψ̂−σ + Γ(r)ψ̂−σ
ψ̂σ† ,
(5.28)
√
with Γ(r) = Γ̃(r)e−iϕΓ = nd α(r). We assume that the transfer process is
realized
so that Γ̃(r) is real, varies randomly in space and is constant in time,
R
drΓ̃(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 paring 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
system which for T below the critical temperature Tc = 8eγ e−2 π −1 TF e−π/2kF |a|
(where kB TF = εF = ~2 kF2 /2m = ~2 (3π 2 n)2/3 /2m and γ = 0.5772 the Euler
constant) reveals a transition to a superfluid phase (BCS state) which is indicated by a non-vanishing pairing function (order parameter) ∆ = ghψ̂−σ ψ̂σ i
[Sá de 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
= En
,
(5.29)
~2 ∇2
vn
vn
∆∗ + Γ ∗
+µ−W
2m
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
under global gauge transformation, i.e. ψ̂σ → eiϕ/2 ψ̂σ , which implies that
if {un , vn } are solutions of the Bogoliubov-de Gennes equations for ∆, then
{eiϕ/2 un , e−iϕ/2 vn } are the solutions corresponding to eiϕ ∆. 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 = Γ̃ 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
∆(r) ≈ ∆0 eiϕ(r) . 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
hλH1 iλ
0Z λ
Z
g
dg 0
2g 0
2
≈ −
dr |∆| +
Γ̃|∆| cos ϕ ,
(5.30)
02
g
0 g
h
PR
†
†
where we have defined H1 =
dr Γ̃ ψ̂σ ψ̂−σ + ψ̂−σ
ψ̂σ† − g ψ̂σ† ψ̂−σ
ψ̂−σ ψ̂σ
Ωs − Ω0
=
σ
[Fetter and Walecka, 2003]. According to our assumptions, |∆(r)| is constant
and cos ϕ(r) ≈ cos ϕ0 − sin ϕ0 δϕ(r), for g 0 ≈ g. Then
Z
Z
2g 0 |∆|
2g 0
Γ̃|∆| cos ϕ ≈ −|∆|2 V + sin ϕ0
drΓ̃δϕ, (5.31)
− dr |∆|2 +
g
g
R
and for drΓ̃δϕ < 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 Γ̃ 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 GinzburgLandau 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
2π ~
Tc − T
48π 2
2
2
+
Γ̃
∇ ∆ = −∇ Γ̃ −
7ζ(3)lc2 mkF g
Tc
2
2
48π Tc − T
6m
−
(5.32)
∆ + 4 2 |∆ + Γ̃|2 (∆ + Γ̃),
2
7ζ(3)lc Tc
~ kF
where lc = ~2 kF /mkB Tc . Equation (5.32) is valid for Tc − T Tc and for
Γ̃(r) that changes on a scale much larger than lc (e.g. for kF |a| = 0.5 and
n ∼ 1014 cm−3 we get lc ∼ 4 µm). For |Γ̃(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
48π 2 2π 2 ~2 Tc − T
2
2
|∆0 |∇ δϕ(r) = ∇ Γ̃(r) +
+
Γ̃(r),
(5.33)
7ζ(3)lc2 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
48π 2
2π ~
Tc − T Γ̃(k)
Γ̃(k)
−
+
,
(5.34)
δϕ(k) =
|∆0 |
7ζ(3)lc2 |∆0 | mkF g
Tc
|k|2
in the Fourier space.
Now we switch to the general case of complex Γ = Γ̃e−iϕΓ . It is easy to check
that if |∆|eiϕ corresponds to solution of the Bogoliubov-de Gennes equations
with ϕΓ = 0 then |∆|ei(ϕ−ϕΓ ) 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 fluctuations 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
the coupling constant g in ∆ = ghψ̂−σ ψ̂σ i, i.e. g → gef f , according to
!
r
√
√
E C + εF
1
mkF
1
EC
1
= − 2 2
ln √
,
(5.35)
√ −
gef f
g 2π ~
2
εF
E C − 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
Lz = 40kF−1 , L⊥ = 20kF−1 , 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
these parameters lc ∼ 100kF−1 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,
Γ0
2π
2π
Γ(r) =
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
values are of the order 10−4 ). 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
!(z)
10
-3
(a)
0
-3
"(z) / #
-10
0.55
(b)
0.5
0.45
-20
-10
0
10
z
20
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).
!"#z / $
0.6
(a)
0.55
0.5
0.45
!|%|#z / %0
0.4
1.05
(b)
1
0.95
0
0.2
0.4
T / Tc
0.6
0.8
1
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 profound 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:
Ĥ = −
N
−1
X
N
X
σ̂xi σ̂xi+1 + σ̂yi σ̂yi+1 −
hi σ̂ni ,
i=1
(6.1)
i=1
where σ̂αi are the α = x, y-Pauli spin matrices at site i, and hi is the random
field at site i. The field points along an arbitrary direction n inside the XY
plane and σ̂n = 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 valuesPof the magnetization along the parallel and orthogonal directions: m̄k = h σki i/N and
P i
m̄⊥ = h σ⊥
i/N . Typically, we also consider the local magnetization mk = hσki i
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) = −
λi log2 λi .
(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
|ψi =
χp
X
λα |ψ [1..p] i|ψ [(p+1)..N ] 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 ≤ maxp χp ≡ 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 X
1
X
X
[2..N ]
[1]
[2..N ]
1
i,
(6.5)
|ψi =
λα1 |ψ i|ψ
i=
Γ[1]i
α1 λα1 |i1 i|ψ
α1 =1
α1 =1 i1 =0
where |i1 i 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 onedimensional 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 groundstate.
6.3
6.3.1
Numerical Results
Staggered field
We begin by reviewing the case of an staggered magnetic field, hi = (−1)i 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 quan1 At
the time of writing of this thesis,
the project is called TimeEvolving 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 ) ∼
(1 − h0 /hc )β . 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 semiinfinite block diverges [Calabrese and Cardy, 2004, Calabrese and Cardy, 2009],
S∞ =
1c
log (ξ) + a.
23 2
(6.6)
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 Néel product 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
is hik = 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
near the zeros of hik . 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 oscillating field. For k = 2π
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
mi⊥ = 0 (see [McCoy, 1968]). However, when the amplitude of the field hik approaches its critical value, the local correlation length of the system, ξ ' |hik |−ν ,
i dhik can become much larger than the rate of change of the field, ` ' hk / di . In
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
an island by linearizing the magnetic field near its zero at i0 : |hik | ≈ |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-Żurek 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.
value of the parallel magnetization at the freezing point, mi⊥0 ∼ |mik0 ±R |. Using
that near the critical point |mik | ∼ |hik |1/δ [McCoy, 1968], we obtain
mi⊥0 ∼ (hkR)
1/δ
1
= (hk) δ(ν+1) .
(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 '
ν
h−0.369 , with the exponent close to the predicted value ν+1
' 0.363. For the
i0
amplitude of the magnetization of the islands we obtain m⊥ ∼ h0.092 , again in
1
' 0.089.
good agreement with our prediction δ(ν+1)
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
magnetic field hik . In (b) and (c) respectively magnetization along the field mk
and spontaneous
q magnetization m⊥ . In panel (d) we show the total magnetizai
tion |m | = (mi⊥ )2 + (mik )2 .
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 87 Rb. This is in principle feasible with all the alkaline atoms, since
they share the same fine structure. We propose to use the wavelength λ̄ 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 7 Li due to the small fine splitting. We can, however, take heavier atoms
(like 39 K, 41 K or even better 85 Rb or 87 Rb) 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.
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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. Similarly, 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 justifying this result are values of various components of magnetization (which
are directly measurable in the experimental scheme discussed above), and nonmonotone 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
[Staliūnas et al., 2006, Staliūnas et al., 2010].
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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 feasibility [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 Localization. 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. Intuitively, 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 motivated 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. Theoretical 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, disorderinduced 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 disorderinduced 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 travelled to Kraków several times to work with two groups there. Krzysztof Sacha
from the the Jagiellonian University in Kraków became 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 Kraków were 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 Rodrı́guez, and all the other members of the Institute of Theoretical Physics at the University of Hannover as well as Jakub Zakrzewski,
Roman Marcinek, Bartlomiej Oleś, Guilhem Collier, and the members of the
Kraków groups.
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, Ophélia Fornari for collaborating on this project.
On the personal side, I would like to thank Marı́a Belén Pérez-Ramı́rez, Helena,
Lukas, Astrid, and Isabelle Niederberger, as well as Viktor Käslin and his family.
Also, I would like to express my gratitude to Rodrı́go Valdivieso, Natalia Arañó,
Antón Albajes 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.
90
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