Master Thesis

Towards the Creation of Vortices
in a Dipolar Bose-Einstein
Condensate
Sandra Brandstetter, BSc
supervised by
Prof. Dr. Francesca Ferlaino
Dr. Manfred Mark

Thesis submitted in fulfillment of the requirements
for the degree of Master of Science in Physics
at theUniversity of Innsbruck,
Faculty of Mathematics, Computer Science and Physics,
Institute for Experimental Physics
Innsbruck, January 2020

‘What’re quantum mechanics?’‘I don’t know. People who repair quantums, I suppose.’- Sir Terry Pratchett, Eric

Abstract
In this thesis, I describe and characterize a setup for the production of vortices ina Bose-Einstein Condensate with dipolar interactions. The formation of vortices
as a response to rotation is one of the key features of a superfluid, which has been
observed in contact interacting Bose-Einstein condensates and Fermi gases. The
long-range, anisotropic dipole-dipole interaction present in Bose-Einstein condensatesof dipolar atoms is expected to alter the properties of the vortices as well as theinteraction between them. Although many theoretical predictions on vortices in
dipolar quantum gases exist, they have not yet been observed experimentally. The
aim of this thesis is to built and design a setup for the creation of vortices in a dipolarcondensate of Erbium and/ or Dysprosium atoms.
The first part of the setup consists of a one-dimensional lattice with variable spacing
and confinement, a so-called accordion lattice. It is formed by the interference of twolaser beams in the focal point of an aspherical lens. The confinement in this lattice
can be altered by changing the angle of interference, allowing us to trap the atoms
in a configuration with weak confinement and high lattice spacing. As the initiallattice spacing is larger than the spatial extent of the condensate, we can trap all
atoms within a single node of the lattice. By increasing the separation of the beams,
the vertical confinement can be increased, while the confinement in the horizontal
plane remains weak, leading to the desired trap geometry for the creation of vortices.
To find the appropriate parameters, we programmed a numerical simulation of theaccordion lattice. Based on the results of these computations, the experimental setup
for the accordion lattice was planned and set up. We characterize the stability as
well as the compression of the accordion lattice.
The second part of the experimental setup consists of a blue detuned variable
trap, which is achieved by modulating the intensity distribution of a laser beamwith a wavelength of λ = 370 nm using a Digital micromirror device (DMD). This
wavelength is blue detuned for both Erbium and Dysprosium. We design an opticalsetup to image the pattern displayed on the DMD onto the atomic plane.

Zusammenfassung
Diese Arbeit beschreibt einen Aufbau zur Erzeugung von Wirbeln in einem BoseEinstein-Kondensat mit dipolaren Wechselwirkungen. Die Bildung von Wirbeln als
Reaktion auf Rotation ist eines der Schlüsselmerkmale einer Supraflüssigkeit, welches
sowohl in Bose-Einstein-Kondensaten als auch in Fermigasen mit Kontaktwechselwirkung beobachtet wurde. Es wird erwartet, dass die anisotrope, langreichweitige
Dipol-Dipol-Wechselwirkung die Eigenschaften der Wirbel sowie die Wechselwirkungzwischen ihnen verändert. Obwohl viele theoretische Vorhersagen über Wirbel in
dipolaren Quantengasen existieren, wurden sie noch nicht experimentell beobachtet.
Ziel dieser Arbeit ist es, einen Aufbau für die Erzeugung von Wirbeln in einem
dipolaren Kondensat aus Erbium- und/ oder Dysprosium Atomen zu entwickeln.
Der erste Teil des Aufbaus besteht aus einem eindimensionalen Gitteraufbau mit
variablem Gitterabstand und -einschluss, dem sogenannten Akkordeongitter. Dies
wird durch die Interferenz zweier parallel verlaufender Strahlen im Fokuspunkt einer
asphärischen Linse erreicht. Die Fallenfrequenzen sind abhängig vom Winkel und
können daher durch eine Veränderung des Abstands der beiden Strahlen variiert
werden. Zu Beginn werden die Atome in einem Gitter mit niedrigen Fallenfrequenzen und hohem Gitterabstand gefangen, so dass sie nur eine einzige Gitterebene
einnehmen. Durch Erhöhen des Strahlabstands in vertikaler Richtung können wir die
Fallenfrequenzen in dieser Richtung erhöhen, während sie in der horizontalen Ebene
schwach bleiben. Um die gewünschten Parameter festzulegen programmieren wir eine
numerische Simulation des Interferenzmusters. Basierend auf den Ergebnissen der
Berechnungen haben wir den Versuchsaufbau für das Akkordeongitter erstellt. Wir
charakterisieren die Stabilität sowie die Kompression des Akkordeongitters.
Der zweite Teil des Versuchsaufbaus besteht aus einer blau verstimmten variablen
Falle. Das variable Potential wird erreicht durch die Projektion des Musters auf einem
Mikrospiegelaktor auf die atomare Ebene erreicht. Der Mikrospiegelaktor moduliert
die Intensitätsverteilung eines Laserstrahls mit λ = 370 nm, eine Wellenlänge diesowohl für Erbium als auch für Dysprosium blau verstimmt ist. Wir entwerfen einen
optischen Aufbau zur Projektion dieses Intensitätsmuster auf die Atome.

Contents
Acknowledgemenents

13

1. Introduction
15
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2. Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Dipolar Quantum Gases
19
2.1. Interactions in quantum gases . . . . . . . . . . . . . . . . . . . . . . 19
2.2. Optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3. Vortices in Bose-Einstein Condensates
3.1. Superfluid Properties of a Bose-Einstein Condensate . . . . . . . . . .
3.2. Quantized vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Density profile of a quantized vortex . . . . . . . . . . . . . . . . . .
3.4. Multiple vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Creation of Vortices in Bose-Einstein Condensates . . . . . . . . . . .

29
29
30
32
35
36

4. Accordion lattice set-up
4.1. Desired properties of the trapping potential . . . . . . . . . . . . . .
4.2. Accordion lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Numerical calculation of trap parameters . . . . . . . . . . . . . . . .
4.4. Experimental implementation . . . . . . . . . . . . . . . . . . . . . .
4.5. Characteristics of the accordion lattice . . . . . . . . . . . . . . . . .
4.6. Experimental control . . . . . . . . . . . . . . . . . . . . . . . . . . .

39
39
41
44
47
58
65

5. Variable blue detuned potential
5.1. Digital Micromirror Device . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Alignment strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
68
71
73

6. Conclusion and Outlook

77

A. Matlab code
A.1. Numerical simulation of the accordion lattice . . . . . . . . . . . . . .
A.2. Calculation of beam displacement by two rotating parallel mirrors . .
A.3. Calculation of trapdepth created by DMD . . . . . . . . . . . . . . .

79
79
83
86

Eidesstattliche Erklärung
Ich erkläre hiermit an Eides statt durch meine eigenhändige Unterschrift, dass ich
die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen
Quellen und Hilfsmittel verwendet habe. Alle Stellen, die wörtlich oder inhaltlichden angegebenen Quellen entnommen wurden, sind als solche kenntlich gemacht.

Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als
Magister-/Master-/Diplomarbeit/Dissertation eingereicht.

Innsbruck, 31. Januar 2020,
Sandra Brandstetter, BSc

11Acknowledgements
First of all I want to thank Francesca for giving me the opportunity to work in
the RARE team. Further, I want to thank her for encouraging me to excel and forsupporting me in making the decisions about my scientific future. Moreover I am
happy that I could participate in a PhD meeting as a fresh master student, allowing
me to gain deeper insights into the field of Dipolar Quantum gases.
I am grateful that Manfred gave me the opportunity to work on his ESQ discovery
project. Thanks for letting me work independently but always having my backand nudging me when I needed it. I enjoyed our discussions about science, optics,
electronics and programming. Moreover I want to thank you for initially sparking
my interest in the field of ultracold quantum gases during the work on my bachelor’s
thesis.
Thanks to all of the members of the ERBIUM, the RARE as well as the T-REQS
team, for helping me whenever I had questions, and for the physics and non-physics
related discussions during lunch time and coffee breaks. I really enjoyed being a partof the team. Special thanks goes to Philipp for teaching me many of the experimental
techniques (e.g. coupling a fiber efficiently), helping me to get the Msquared laser
working again and answering all of my questions concerning SolidWorks! I also wantto thank the people in the other ultracold groups for creating such a pleasant working
environment and the exchange of knowledge!
None of our work would be possible without the people working in administration
as well as in the mechanical workshop. Special thanks goes to our administrative
assistant Silvia for taking care of all of my orders and to Bernhard from the mechanicalworkshop of the IQOQI. Thank you for your fast, precise work, for your help with
my designs and for the introduction to turning and milling.
I also want to thank the people who survived the Bachelor together with me.
Thanks to Sebi, Flo, Kathi, Medi, Steve, Raphi, Josh et al. for study sessions in the
Victor Franz Hess house, coffee breaks, legendary lectures (sometimes even includinga Frisbee), wonderful trips to CERN and Kasan, an occasional beer (or two) in the 9b,
the Wohnzimmer or elsewhere, skiing sessions, game nights, Glühwein, watching the
Tagebuchslam in Treibhaus, ... . You really helped to make these years in Innsbruckto some of the most amazing years so far.
Thanks to my friends outside of University, especially those in scouting, for
distracting me from physics whenever I needed it. Thanks especially to Oliver,
Katha, Elli, Ingrid and Hanna among many others. Thanks also to the guys fromthe ’Physikalische Übungen’ for discovering physics and nerding out together back inHigh school.
I also want to thank my high school teachers Mag. Jutta Wirth and Prof. Helmut

Sitter for encouraging me to pursue a career in physics even though the focus of myschool was on languages. I would not be here today without you!
Lastly, but most importantly I want to thank my family. Danke Mama und Papa,
dafür, dass ihr mir stets die Freiheit gelassen habt meinen eigenen Weg zu gehen, und
all meine Pläne unterstützt. Danke dafür, dass ich mich immer auf euch verlassenkann und dass ihr immer für mich da seid! Danke Mama, für die endlosen Gesprächeüber Gott und die Welt, beim Wandern und Spazierengehen und Danke, dass dumir stets vorgelebt hast, dass man als Frau locker in einem ’Männerberuf’ bestehen
kann. Danke Papa, für die gemeinsamen Stunden in der Luft (ob über Funk oder
im gleichen Flieger) und danke für die gelegentlichen Erinnerungen dass Pausen
durchaus sinnvoll sind. Danke auch an meine Schwester Susi und meinen Schwager
Stefan, für eure Unterstützung, eure Urlaubseinladungen und euer Vertrauen. Dankean meine Patenkinder Martin und Miriam, dafür, dass ihr mich immer daran erinnert,
dass Physik vielleicht nicht das Wichtigste im Leben ist.

1. Introduction
1.1. Motivation
In 1924 Satyendra Nath Bose developed a derivation of Plancks quantum radiation
law that was, contrary to earlier derivations, not based on classical electrodynamics
but instead on statistical quantum mechanics [1]. Einstein extended this theory tomatter, leading him to the prediction of a novel state of matter, the so called BoseEinstein Condensate (BEC). This state manifests itself in a macroscopic occupation of
one state, which can be described by a single wave function. Thus the BEC behaves
as a macroscopic quantum object, which facilitates the exploration of quantum
phenomena.
The first experimental observation of a BEC followed in 1995, more than 70 years
after the initial theoretical prediction. This delay can be explained by the fact thatthe experimental realization of BECs is directly linked to the development of thelaser, a key ingredient for cooling and trapping of neutral atoms. The first BEC wasproduced by the group of Carl Wieman and Eric Cornell at the NIST-JILA lab in avapour of 87 Rb atoms [2], followed shortly after by the group of Wolfgang Ketterlewith a BEC in a vapour of Na atoms [3].
Different atomic species followed after the initial discovery. Bose-Einstein condensation has been achieved with 13 different atomic species, molecules [4] and photons[5].
• Alkali metals: Li [6], Na [3], K [7], Rb [2] and Cs [8]• Alkaline Earth metals: Ca [9] and Sr [10]• Lanthanides: Dy [11], Er [12] and Yb [13]• Transition metals: Cr [14]• Nobel gases: Metastable He [15]• Others: H [16]Owing to their different properties, the various atomic species can be used to
explore different physical phenomena. A special property of Chromium as well as
the two condensed lanthanide atoms, Erbium and Dysprosium, is their large intrinsic
magnetic dipole moment. While the interaction of atoms is normally governed by thecontact interaction, these atoms exhibit an additional interaction - the dipole-dipole15

1. Introduction
interaction (DDI). The DDI is long range and anisotropic, thus enabling the studyof a whole new class of physical phenomena. Other systems exhibiting large dipole
moments are hetero-nuclear molecules [17] and Rydberg atoms [18].
The phenomenon of Bose-Einstein condensation is closely linked to superfluidity, aphenomenon first observed in liquid helium in January 1938 by Pyotr Kapitza [19],and John F. Allen and Don Misener [20]. They studied the flow of liquid Helium II
through long capillaries. These measurements gave an upper limit to the viscosity ofliquid Helium II that is 4 orders of magnitude smaller than the viscosity of Hydrogengas. This observation let Pyotr Kapitza to the intuition that ‘(...), by analogy withsuperconductors, that the helium below the λ-point enters a special state whichmight be called a superfluid ’ [19]. As interacting BECs are superfluid, they provide
an ideal test bed for the phenomenon of superfluidity due to the tunability of key
parameters such as the density and the interaction of particles. This tunability ishardly possible with superfluid He.
One of the distinguishing features of a superfluid is its response to rotation. As
the velocity of a superfluid is proportional to the gradient of the phase, the fluid is
irrotational, signifying that it can not rotate as a normal fluid. Instead the superfluid
forms vortices as a response to rotation. These vortices differ from vortices appearing
in classical systems such as drained bathtubs and at the wing-tips of aeroplanes,
due to the irrotationality of the superfluid and the fact that it does not rotate as arigid body. In addition to this, the angular momentum of vortices in a superfluid isquantized. The occurrence of vortices in a quantum system is thus a hallmark forsuperfluidity. Vortices have been observed in both BECs [21] and strongly interacting
Fermi gases [22] as well as superfluid He. The study of vortices can lead to a deeper
understanding of the properties of novel states of matter and phase transitions
between them (e.g. the Berezinskii–Kosterlitz–Thouless transition [23])Up to now, the study of vortices has been limited to contact-interacting BECs.
The dipole-dipole interaction is expected to add new effects to both the formationand properties of single vortices as well as the interaction of multiple vortices. The
RARE experiment is the first mixture experiment combining two highly magnetic
atomic species, Erbium and Dysprosium [24]. It allows both single and dual species
operation with multiple isotopes. The aim of this thesis is to design and test thecomponents necessary for the creation of vortices in a dipolar quantum gas of Er
and/or Dy atoms.

1.2. Thesis Overview
This thesis describes an experimental setup designed for the production of vortices
in a Bose-Einstein Condensate of Erbium and Dysprosium atoms. It is structured asfollows:
Chapter 2 introduces the physics of quantum gases with contact and dipole-dipole
interactions and the concept of optical dipole traps. First, it gives an overview of
the interactions in quantum gases, discussing both the contact and the dipole-dipole

161.2. Thesis Overview
interaction. Moreover we discuss the Gross-Pitaevskii equation for both cases, which
can be used to calculate the wave function describing the Bose-Einstein condensate.
The Section on optical dipole traps describes the general concept of such traps aswell as the polarizability of both Erbium and Dysprosium.
Chapter 3 discusses the physics of vortices in Bose-Einstein condensates with bothcontact and dipole-dipole interactions. It starts with the superfluid properties of
Bose-Einstein condensates and then goes on to discuss the properties of quantizedvortices as well as their interactions.
Chapter 4 describes the accordion lattice setup. It is divided into different Sections
describing the prerequisites, the calculations performed for design considerations, aswell as the experimental implementation and its characteristics.
Chapter 5 describes the variable blue detuned potential which will be used tocreate vortices in the Bose-Einstein condensate. We discuss the digital micromirror
device used in our experimental scheme and the optical set-up which we designed toproject the pattern displayed on the DMD onto the atoms.
Finally I conclude this thesis with an outlook in Chapter 6.

17

2. Dipolar Quantum Gases
2.1. Interactions in quantum gases
2.1.1. Contact interactions
Usually, atoms interact via the van der Waals (vdW) force, resulting in a potential
C6
UvdW = − 6 ,
r

(2.1)

where C6 is the vdW coefficient and r is the inter-particle distance. For low temperatures (i.e. low collision energies) the scattering of two particles is dominated bys-wave scattering. In this limit, the scattering problem can be fully described by thescattering length a, which is determined by the phase shift δ0 between the incoming
and the scattered wave
δ0 = ka,(2.2)
where k is the wave number of the incoming wave. The scattering length can bedependent on the magnetic field, in cases when the scattering potentials can betuned via the Zeeman effect. This is due to the different magnetic moments of boundmolecular states and the scattering atomic state, allowing a tunability of the resulting
scattering length via an external magnetic field. Feshbach resonances occur when
the atomic scattering state couples to a bound molecular state. The dependence ofthe scattering length a on the external magnetic field B can typically be written as
∆a(B) = aBg 1 −,(2.3)
B − B0where aBg is the background scattering length, B0 is the position of the Feshbachresonance and ∆ is the width of the resonance [25].
In the case of low energy scattering, the contact interaction potential can be
written as
4π~2 aUmf (r) =δ(r) = gδ(r),(2.4)
mwhere m is the mass of the atoms and g is the coupling constant characterizing theinteraction. The short-range nature of this interaction becomes apparent considering
the dependence of the interaction potential on the Dirac delta function. This
interaction is thus referred to as contact interaction.

19

2. Dipolar Quantum Gases

2.1.2. Dipole-Dipole interactions
Atoms also have a permanent magnetic dipole moment, the strength of which isdependent on their internal structure and quantum numbers. The projection of themagnetic moment of an atom along the quantization axis is given byµ = mJ gJ µB ,(2.5)

where mJ is the magnetic quantum number for the total angular momentum of theelectron, giving the projection of the total angular momentum J on the quantization
axis (i.e. the magnetic field direction) and gJ is the Landé g-factor for the total
electronic angular momentum. In the case of LS coupling, it can be written as
gJ = 1 + (gS − 1)

J(J + 1) + S(S + 1) − L(L + 1),
2J(J + 1)(2.6)

where gS ≈ 2.002 is the Landé g-factor of the electron and S = 1 and L denotespin and angular momentum quantum number, respectively. For lanthanide atoms,
several corrections of this term have to be taken into account, which stem from thedeviation from perfect LS-coupling as well as relativistic effects [26]. In addition tothese corrections, jj-coupling has to be included. To estimate the magnetic momentof Er and Dy, we consider only LS-coupling without any additional corrections. For
the ground state of Erbium J = 6 and for Dysprosium J = 8, giving gJ ≈ 1.167
for Er and gJ ≈ 1.223 for Dy. Experiments give a Landé factor of gJ = 1.163 801(1)for Er and gJ = 1.241 66(7) for Dy [27]. The energetically lowest magnetic states
are mJ = −6 (Er) and mJ = −8 (Dy). The resulting magnetic moments (using theexperimentally obtained Landé g- factors from [27]) are equal to µ = −6.983 µB for
Erbium and µ = −9.933 µB for Dysprosium.
The above derivation is only valid for the bosonic isotopes, which do not feature a
hyperfine splitting (i.e. I = 0, I being the nuclear spin quantum number). For thefermionic isotopes I 6= 0, thus one has to account for the magnetic moment of thedifferent hyperfine levels. The magnetic moment is then given byµ = mF gF µB ,(2.7)

where mF is the magnetic quantum number for the total angular momentum of theatom F . The Landé g-factor of the atoms total angular momentum gF is given by
gF = gJ

F (F + 1) − I(I + 1) + J(J + 1).
2F (F + 1)

(2.8)

For the energetically lowest hyperfine state of 167 Er with an nuclear spin of I = 72 ,F = 19, with the lowest magnetic state being mF = − 19
. For 161 Dy with an nuclear
22
spin I = 52 , the energetically lowest state has a total angular momentum quantum
number F = 21
, with the lowest magnetic state being mF = − 21
. This results in2
2
a magnetic moment µ = −6.98 µB for 167 Er and µ = −9.93 µB for 161 Dy, which are20

2.1. Interactions in quantum gases
both equal to the magnetic moments of the bosonic isotopes [28]. In comparison,
the magnetic dipole moment of Alkali atoms such as Rb is significantly smaller(µRb = 1 µB ).
Atoms with a permanent magnetic dipole moment feature an additional interaction– the dipole-dipole interaction (DDI). This interaction between two dipoles with adipole moment µ, which are orientated along e1 and e2 can be written as [29]µ0 µ2 (e1 · e2 )r2 − 3(e1 · r)(e1 · r)
Udd (r) =
,
4π
r5

(2.9)

where r is their relative position. In the presence of a magnetic field these atoms
orientate along the field, leading to a polarized sample where all dipoles point in thesame direction. The above expression then simplifies to [28]Udd (r) =Cdd 1 − 3 cos2 θ,4π
r3(2.10)

where Cdd = µ0 µ2 denotes the dipolar interaction strength and θ gives the angle
between the quantization axis and the interatomic axis of the two dipoles (see Fig.
2.1). The potential is long range (∝ r−3 ) and anisotropic (∝ (1 − 3 cos2 θ)).

Figure 2.1.: Illustration of the parameters governing the interaction between two particles
interacting via the dipole-dipole interaction.

The angular dependence of the DDI can be used to tune the interaction by altering
the direction of the magnetic field. The factor (1 − 3 cos2 θ) changes from -2 to
1 when altering θ from 0° to 90°. Thus, the DDI can be attractive or repulsive,
depending on θ. Two atoms that are arranged in an angle θ = 0° (’side-by-side’
configuration) will repel each other, while atoms arranged in an angle θ =90° (’headto-tail’ configuration) attract each other. This is illustrated in Fig. 2.2. Additionally,
Udd = 0 for θ = 54.74°.

21

2. Dipolar Quantum Gases

Figure 2.2.: Illustration of the DDI between two particles. The dipoles are illustrated
by red and green arrows, the direction of the force is represented by black arrows. The
side-by-side configuration is illustrated in a), the head-to-tail configuration is shown in b).

The total interaction potential for dipolar atoms, accounting for both contact and
DDI is given by
4π~2 aCdd 1 − 3 cos2 θU (r) =
δ(r) +.
(2.11)m
4π
r3
To compare the strength of the DDI to that of the contact interaction, we introduce
the dipolar length [29]µ0 µ2 madd B,
(2.12)
12π~2which is equal to add = 66 a0 for 168 Er and add = 130 a0 for 164 Dy. In comparison, for85
Rb the dipolar scattering length is equal to add = 0.69 a0 . The dipolar length gives
the absolute strength of the DDI. The relative strength of the DDI compared to thecontact interaction is given by [29]
dd =

add
Cdd
=
as
3g

(2.13)

where as is the contact interaction scattering length. The relative weight of theinteraction can thus be tuned by changing as via the strength of the magnetic field.
The DDI dominates over the contact interaction for dd > 1.

2.1.3. Gross Pitaevskii equation
To derive an equation for the wave function describing a BEC of N bosons, we utilize
the mean field approximation. In this approximation, the wave function describing
the system Ψ can be written as a product of the individual particle wave functions ψ,Ψ(r1 , r2 , ..., rN ) =NY

ψ(ri ),(2.14)

i=1

where ri is the coordinate of the i-th boson. This approximation is only valid fora  n−1/3 (dilute quantum gas) or a ≥ n−1/3 (strongly interacting regime), where n

22

2.1. Interactions in quantum gasesis the particle density. In this approximation the effective interaction potential canbe replaced with a pseudo potential
Umf = gn(r).

(2.15)The Hamiltonian for this system can be written as

H=

N  2
X
p
i

i=1

2m


+ V (ri ) + gX

δ(ri − rj ),(2.16)

i<jwhere V (ri ) is the external (trapping) potential.
The many body wavefunction Ψ0 (r) describing a system with the aforementioned
Hamiltonian can be derived using the Gross Pitaevskii equation (GPE), that was
independently derived by Gross [30] and Pitaevskii [31] in 1961. For a contact
interacting BEC the time-independent GPE reads



~2 ∇2
2
−
+ Vext (r) + g |Ψ0 (r)| Ψ0 (r) = µ0 Ψ0 (r),2m

(2.17)

where Vext (r) is the external potential and µ0 is the chemical potential. The GPE
has the form of a non-linear Schrödinger equation, where the potential is the sum of
the external trapping potential and the mean field produced by the other bosons.
The non-linearity stems from the interaction of the particles.
The DDI can be accounted for by adding another term Φdd to the mean fieldpotential so that
 2 2

~∇
2
−
+ Vext (r) + g |Ψ0 (r)| + Φdd Ψ0 (r) = µ0 Ψ0 (r),2m

(2.18)

leading to the GPE describing BECs of atoms with strong magnetic dipole moment(dBEC). The term Φdd readsZΦdd (r, t) =|Ψ(r, t)|2 Udd (r − r0 )d3 r0 ,(2.19)and is non-local due to the long range character of the interaction. It depends on theinteraction of the majority of the dipoles and is therefore dependent on the dipole
orientation and the shape of the BEC. In oblate traps, where the majority of thedipoles are aligned side-by-side (see Fig. 2.3b), the repulsive interaction dominates,
leading to Φdd (0) > 0. In prolate traps (see Fig. 2.3a) the opposite is true, leadingto Φdd (0) < 0.

23

2. Dipolar Quantum Gases

Figure 2.3.: Illustration of different trap geometries. Subfigure a shows the prolate trap
where the majority of dipoles are aligned head-to-tail, b shows the oblate trap, where thedipoles are aligned side-by-side. Figure adapted from [32]2.2. Optical dipole traps
Optical dipole trapping is the very basis for many modern cold atom experiments.
While radiation pressure traps are used for the initial cooling of the atoms, optical
dipole traps are used to cool the atoms to degeneracy and to manipulate the condensate. Compared to magnetic traps, dipole traps offer a greater geometric flexibility
as they are not restricted by the arrangement of coils. This flexibility is important toachieve the ideal confinement (described in Chapter 4) of the BEC for the productionof vortices, as well as the introduction of rotation into the system.

2.2.1. Oscillator model
The aforementioned external potential Vext can be either created by laser light ormagnetic fields. This thesis will focus on optical dipole traps, which are used in our
experiment. Optical dipole traps are based on the interaction of the electric field
E of the laser with the induced atomic dipole moment p. The negatively charged
electrons are separated from the positively charged core due to the electric field,
leading to an induced dipole moment. [33] In an isotropic medium the amplitude ofthe dipole moment p is directly proportional to the amplitude of the electric field E
p = αE,(2.20)

where α is the complex polarizability, governing the strength of the interaction
between the medium and the electric field. It can be derived by modelling the motionof the electron with an harmonic oscillator with eigenfrequency ω0 . The damping of
this motion stems from the spontaneous decay of the excited state population. In an
ideal two-level system with ground state |gi and excited state |ei the damping rateof the motion can therefore be written asω0ˆ |2 ,| he|d|gi(2.21)
Γ=
3π0 ~c324

2.2. Optical dipole traps
where dˆ = −er̂ is the electric dipole moment operator. The polarizability can becalculated using the equation of motion ẍ + Γẋ + ω0 x = −eE(r, t)/me , whereE(r, t) = eE(r) exp(−iωt) + c.c(2.22)

denotes the oscillating electric field. The polarizability α equates to
α=

1
e2
.
2
me ω0 − ω 2 − iωΓ(2.23)

This leads to two effects: First the potential energy of an atom in an oscillating
electric field E(r, t)is given by
Udip = −1
Re(α)I,20 c(2.24)

where Re(α) is the real part of the polarizability describing the in-phase componentof the dipole oscillation and I is the intensity of the field. This potential leads to aconservative force
Fdip (r) = −∇Udip (r) =1Re(α)∇I(r).
20 c(2.25)

Secondly, the scattering rate is given byΓsc (r) =1
Pabs=
Im(α)I(r)~ω~0 c(2.26)

where Pabs is the power absorbed by the atom and Im(α) is the imaginary polarizability
describing the out of phase component of the dipole oscillation.
In an ideal two level system, given that the detuning is ∆ B ω − ω0  ω0 , thepotential energy and scattering rate can be rewritten to
Udip (r) =3πc2 Γ
I(r)2ω03 ∆(2.27)and
3πc2
Γsc (r) =
2~ω03



Γ
∆

2
I(r)

(2.28)

using Eq. (2.23). This clearly shows that the sign of the detuning governs theinteraction between the atom and the light field. For negative detuning (red detuned)the potential is negative and thus attracts atoms to the intensity maximum. For
positive detuning (blue detuned) the atoms are repelled by high intensities. [33]25

2. Dipolar Quantum Gases

2.2.2. Multilevel atoms
We use a different picture to derive the effect of laser light on multilevel atoms. While
the same results can be obtained when using a state dependent polarizability in theharmonic oscillator model, it is more intuitive to treat the effect of the far detuned
laser light as a second order perturbation. As this perturbation of the energy levels
is proportional to |E(r, t)|2 , it is linear in the intensity I(r, t).
The second order perturbation of the state |ii by the Hamiltonian Ĥ1 is given by
2

∆Ei =

X hj| Ĥ1 |ii
i6=j

Ei − E j,(2.29)

where Ei is the unperturbed energy of the i-th state. The Hamiltonian Ĥ1 is given byˆĤ1 = dE.

(2.30)

To derive the energy of |ii and |ji, we utilize the dressed state picture. Setting
the internal energy of the ground state to zero, the energy of the ground state |ii issolely given by the energy of the photons in the light field
Ei = n~ω,(2.31)

where n gives the number of photons. The energy of the state |ji is given by
Ej = ~ω0 + (n − 1)~ω(2.32)

where ~ω0 is the energy difference between |ii and |ji. For a two level atom thisresults in a light shift of the ground state |gi ofhe| dˆ|gi∆E = ±∆

2|E|2 .

(2.33)

This simply equates to the AC-Stark shift. The dipole matrix element he| dˆ|gi
relates to the spontaneous decay rate asΓ=ω03
he| dˆ|gi .
3π0 ~c3(2.34)

Consequently the light shift is dependent only on the detuning ∆, the spontaneous
decay rate Γ and the intensity of the light field I(r, t) ∝ |E|2 .
In the case of multilevel atoms, we have to consider all of the allowed transitions
from a ground state |gi i to an excited state |ei i. In a first order approximation, only
dipole allowed transitions are taken into account. The dipole allowed transitions are
dependent on the angular momentum difference ∆J of the ground and excited states(∆J = 0, ±1), as well as the parity of the wave function. In addition to this, the26

2.2. Optical dipole traps
polarization of the light field has to be considered, as different transitions can beaddressed with different polarizations.
The specific transition element dij can be written asˆ
dij = cij ||d||,(2.35)

ˆ is the reduced matrix element and cij is the real transition coefficient.
where ||d||The reduced matrix element relates to the spontaneous decay rate according to Eq.
(2.34). The coefficients cij depend, as mentioned above on the angular momentum
difference ∆J as well as the polarization of the laser light.
We can now write the energy shift of the ground state as
3πc2 Γ X c2ij
∆Ei =
I
,
2ω03
∆ij
j

(2.36)

where ∆ij is the detuning of the laser beam to the transition frequency of |ii → |ji.
Consequently the contribution of all coupled excited levels has to be considered to
calculate the shift of the ground state [33]. This makes it exceedingly difficult to
calculate the light shift for an increasing number of levels.

2.2.3. Anisotropic polarizability
The discussion above is based on the assumption that the medium is isotropic, i.e.
that the polarizability of the atoms is independent of the direction of the electricfield. Due to the non-zero orbital momentum quantum number L of Erbium andDysprosium, both have to be considered as an anisotropic medium. Thus the atomic
polarizability has to be written as a 3x3 tensor and the potential energy is modified
to [34]

U (ω) = −h1mJ
I(r) αs (ω) + |u∗ × u| cos θkαv (ω) +202J
3m2J − J(J + 1) 3 cos2 θp − 1
αt (ω)
J(2J − 1)
2(2.37)

where αs is the scalar polarizability coefficient (diagonal elements of the polarizability
tensor), αv is the vectorial polarizability coefficient (antisymmetric parts of the offdiagonal elements) and αt is the tensorial polarizability coefficient (symmetric partsof the off-diagonal elements). The polarization vector is denoted by u, and θk and θpdenote the angle between the propagation/polarization axis and the quantization
axis which is set by the external magnetic field. For linear polarization the middle
term of Eq. (2.37) vanishes as |u∗ × u| = 0. [35]27

2. Dipolar Quantum Gases

2.2.4. Polarizability of Erbium and Dysprosium
Figure 2.4 shows the calculated atomic polarizability for Erbium and Dysprosium
under the assumption that θk = θp = π/2. This plot allows us to identify the
wavelength regions which we can use to trap the atoms. [36, 37]
The wavelength used for the accordion lattice should be red detuned for both Er
and Dy. Figure 2.4 clearly shows a red detuned region for wavelengths greater than
700 nm, as well as around 500 nm. Thus, both the fundamental (1064 nm) and thesecond harmonic (532 nm) of a high power Nd:YAG laser can be used for red detuned
traps.
Blue detuned traps for both Er and Dy are more difficult to realize, as both species
have many transition lines in the ultraviolet and blue region. Nonetheless, there isa blue detuned region for both species with not to many lines close by, at around370 nm. This allows us to realize a blue detuned trap, giving us the possibility tocreate flat bottom potentials.

Figure 2.4.: Dynamic polarizability of Erbium and Dysprosium. The polarizability of
Erbium is displayed in red, that of Dysprosium is displayed in blue. The finite height of
the narrower lines comes from the finite step size of the calculation. [36, 37]28

3. Vortices in Bose-Einstein
Condensates
3.1. Superfluid Properties of a Bose-Einstein
Condensate
Superfluidity has first been demonstrated independently by Pyotr Kapitza [19] andJohn F. Allen and Don Misener [20] in 1938. They observed the frictionless flowof Helium II through narrow capillaries. The theory behind this phenomenon haslater been developed by L. Landau in 1941 [38]. He formulated a general argument
to show when a fluid can transport mass without friction, based on the idea that
an elementary excitation of the system can only occur when such an excitation isenergetically favourable.
A single excitation with momentum p can change the energy of the fluid by(p) + p · v, where (p) denotes the dispersion relation. This process is only
energetically favourable for (p) + p · v < 0. This is the case for v > (p). Thus thepcritical velocity vc is given by(p)vc = min
,
(3.1)
p
p
where the minimum is calculated over all values of p. Hence, excitations with v < vc
cannot couple to the fluid [39].
In a interacting BEC, the dispersion relation is given by the Bogoliubov dispersion
law [40]" 2 2 #1/2
p
(p) = c2 p2 +
,
(3.2)
2m
where c is the speed of sound in the BEC and p is the momentum. The Bogoliubov
excitation spectrum for a weakly contact interacting Bose-Einstein condensate isillustrated in Fig. 3.1a). For small momenta p  mc, the excitation spectrum
is linear and follows a phonon like dispersion relation ((p) = cp). In the highp2
momentum limit p  mc the dispersion relation becomes (p) = 2m
+ mc2 , thus
resembling that of a free particle with an additional interaction energy c2 m. The
critical velocity of the weakly contact interacting BEC is given by the speed of soundc, as illustrated in Fig. 3.1a). Consequently, the BEC can be considered superfluidfor c 6= 0.

29

3. Vortices in Bose-Einstein Condensates

Figure 3.1.: Schematic graph of the excitation spectrum of a Bose-Einstein condensatea) without and b) with a roton minimum. The blue line shows the excitation spectrum,
the dashed red line illustrates the critical velocity of the BEC. The green dotted line in
subfigure b) marks the speed of sound.

For a strongly interacting quantum fluid, such as superfluid He, the excitation
spectrum features a local minimum at finite momentum, which is referred to as theroton minimum. It has been observed in superfluid He in 1961 [41]. Here, the roton
momentum is inversely proportional to the interatomic distance. In a trapped dipolar
BEC (dBEC) the roton occurs despite the absence of strong interactions, due to theanisotropic long range character of the DDI, for dd & 1. Here, the roton occurs at
a momentum which is inversely proportional to the harmonic oscillator length inz-direction. The occurrence of the roton in dBECs has recently been experimentallyobserved by Chomaz et al. in 2018 [42]. The roton minimum changes the criticalvelocity, as illustrated in Fig. 3.1b). Thus the critical velocity of a dBEC is notnecessarily equal to the speed of sound. [43]3.2. Quantized vortices
An ordinary fluid is described by a many-body wavefunction depending on thecoordinates of all particles, a superfluid on the other hand, can be described by awavefunction that requires only a single position coordinate r. In the hydrodynamic
description, the wavefunction Ψ(r, t) depends on two real fields - the superfluid
density field n(r, t) and the phase field S(r, t), such that [44]pΨ(r, t) = n(r, t)eiS(r,t) .
(3.3)
The velocity field of this wave function can be written as~∇S(r, t).
(3.4)
m
As the velocity field is equal to the gradient of the phase field it is irrotational i.e.
∇ × v(r, t) = 0. Hence the superfluid cannot react to rotational excitations in thev(r, t) =30

3.2. Quantized vortices
same way as a normal fluid. When the superfluid is placed in a rotating container
it will remain at rest for rotation frequencies below a critical rotation frequency.
Nonetheless at some point this will become energetically unfavourable. The energyof the superfluid Er with angular momentum L and energy E in a container rotatingwith angular frequency Ω is given by
Er = E − Ω · L,

(3.5)

in the rotating frame. At high angular frequencies a state with Ω · L > 0 becomesenergetically favourable to the L = 0 state. The system then starts to develop
vortices, i.e. regions around which the fluid rotates.
The circulation κ of a vortex is given byIκ=
dr · v(r, t)(3.6)
C

where C denotes a closed contour around the vortex core. Using the definition of thevelocity field (3.4), the circulation is
I
~
~
(3.7)
κ=
dr · ∇S(r, t) = ∆S,m Cmwhere ∆S is the change in phase when going from the starting to the end point of
the closed contour. As the wave function has to be single-valued, ∆S = 2πqν withqν ∈ Z. This giveshκ = qν ,(3.8)
m
where qν denotes the vortex charge. The circulation is therefore quantized in multiples
of h/m. [39, 44] The sign of the vortex charge gives the direction of rotation. For
sgn qν = +1, we refer to the object as a vortex, for sgn qν = −1 it is called anti-vortex.
The only point where the phase is not well defined is the centre of circulation (r = 0).
To have a defined wave function Ψ(r, t) at r = 0, the density n(r = 0, t) = 0, meaningthat there is a density hole in the superfluid at the centre of rotation. The phaseprofile of a vortex as well as the velocity field is illustrated in Fig. 3.2.
The angular momentum per particle is given byZmlz =dr(r × v)z n(r) = ~,(3.9)
N
where N is the total number of particles in the condensate. The creation of a vortex
with energy Eν and vortex charge qν becomes energetically favourable forEν < Ω · L → Ω >

Eν
,
N hqν

(3.10)

with Lz = Lz = lz N and Ω = Ωz . For a trapped BEC in the Thomas-Fermi regime
the critical angular velocity Ωcr can be calculated to be [45]
5 ~0.67R⊥Ωcr =ln,(3.11)
2
2 mR⊥
ξ

31

3. Vortices in Bose-Einstein Condensates

Figure 3.2.: Schematic sketch of the phase profile and velocity field of a quantized vortex
with qν = 1. The colour scale gives the phase of the wave function, the black arrows
represent the velocity field. The hole in the centre illustrates the vortex core.

where R⊥ is the condensate size in the radial plane and ξ is the healing length(the derivation of ξ can be found in Subsection 3.3). This formula is, with somemodifications, also valid for the dBEC. Even though the healing length is alteredby the DDI, this should not alter Ωcr profoundly as Ωcr ∝ ln(ξ −1 ). The dominantalteration of Ωcr is expected to be due to the rescaling of the radial size R⊥ due tothe DDI [46]. According to theory [46] the main change of ΩC in a dBEC is due tothe alteration of the radius of the BEC by the DDI. The critical rotation frequency
thus decreases in an oblate trap and increases in a prolate trap.

3.3. Density profile of a quantized vortex
The reaction of a BEC to the hole in the centre of the vortex can be described using
the GPE (see Subsection 2.1.3). For a contact interacting BEC, the GPE (see also
equation (2.17)) reads
 2 2

~∇
2
−
+ Vext (r) + g |Ψ0 (r)| Ψ0 (r) = µΨ0 (r).
2m

(3.12)

To obtain the density profile of a vortex, we consider the example of a 1D BEC
confined by a potential Vext = 0 from 0 < x < ∞ with a hard wall potential(Vext = ∞) at 0 > x > −∞. This leads to the boundary condition Ψ0 (x = 0) = 0.
As x → ∞ the homogeneity (particle density n) of the BEC is restored, leading toµ = gn. The characteristic length scale at which changes in the density and phase of
the condensate are removed by the interactions, is given by ξ = ~/p0 . Here p0 is themomentum at which the transition from the phonon to the free particle regime takes

32

3.3. Density profile of a quantized vortex
place (see Subsection 3.1). It is given byp02= mc2 ,
2m
thus
p0 =

√

2mc.

(3.13)(3.14)We refer to ξ as the healing length which can be written as [39, 47]
1 ~
ξ=√
.
2 mc

(3.15)√
Using Ψ̃0 = Ψ0 / n and introducing z̃ = z/ξ gives a dimensionless form of the
GPE equation:d2− 2 Ψ̃0 (z̃) + Ψ̃0 (z̃)3 = Ψ̃0 (z̃)(3.16)dz̃
which can be solved by Ψ̃0 (z̃) = tanh √x2ξ . The healing length thus gives thelengthscale over which the BEC recovers from a disturbance of the density. [39, 44]
This is illustrated in Fig. 3.3. As a vortex creates a hole in the density profile of the
BEC, the healing length governs the size of the vortex core.

Figure 3.3.: Reaction of the wave function of a weakly interacting BEC to a hard wall
blue detuned potential. The hard-walled potential is illustrated in blue, the wave function
is displayed in black. The wave function returns to its original value on the order of thehealing length ξ

The parameter governing the healing length of a BEC is the speed of sound c,which is strongly dependent on the interactions present in the system. For a weaklyinteracting BEC with only contact interactions, the speed of sound is given byrgn.
(3.17)c=m33

3. Vortices in Bose-Einstein Condensates
Equation (3.15) can thus be rewritten as
ξ=√

1
.
8πna

(3.18)In BECs with DDI, the speed of sound is anisotropic. For a 3D uniform dipolarcondensate, the speed of sound is given by [48]rgn
[1 + dd (3 cos2 θ − 1)],
(3.19)c(θ) =
mwith θ denoting the angle between the direction of the excitation momentum and thedipole axis in momentum space. This formula is valid for the phonon modes of the3D uniform dipolar condensate. Thus the healing length of the dBEC is anisotropic,
leading to an asymmetric density profile of the vortex. The numerically calculated
vortex profiles for different dd and α are illustrated in Fig. 3.4. The numericalsimulation [49] assumes a two-dimensional dBEC, where α gives the angle between
the z-axis and the dipole orientation. The first picture shows a vortex in the purely
contact interacting case (dd = 0). The second part displays the density profile of avortex in a BEC with DDI, where the dipoles are tilted.

Figure 3.4.: Numerically calculated density profiles of vortices with different dipole
orientations and dipolar interaction strengths. The calculation assumes a BEC with asingle plane of dipoles. Here α is the angle between the z-axis and the orientation of thedipoles and dd gives the relative strength of the DDI compared to the contact interaction.
Figure adapted from [49]In the case of trapped dBECs, the roton minimum can alter the density profile ofthe vortex core. When the energy of the roton is close to zero, the roton mode ispopulated due to the perturbation of the potential by the vortex. This leads to theemergence of density ripples around the vortex core with a wavelength related to thewavelength of the roton. These ripples align in the direction of the attractive dipolarinteractions and decay with increasing distance from the vortex core. [48, 49, 50]
When the roton minimum approaches zero (i.e. roton softening) a new state ofmatter, the supersolid, emerges [51, 52, 53]. The formation and properties of vortices

34

3.4. Multiple vortices
in a supersolid go beyond the scope of this thesis, but will be the topic of further
investigations.

3.4. Multiple vortices
As discussed in Section 3.2, the formation of a single vortex is energetically favourable
for BECs rotating with a frequency Ω ≥ Ωcr . As can be seen in Eq. (3.10) Ωcr
is directly proportional to the circulation. For angular momenta Ω ≥ 2 · Ωcr it isenergetically favourable for the system to have a total circulation of κ = 2 · h/m,
which can, in principle, either be supported by one vortex with qν = 2 or two vortices
with qν = 1 each. To see which of these options is energetically favourable, one has
to consider the kinetic energy of a vortex. Assuming a cylindrical condensate withheight L and radius R, where ξ  R, the kinetic energy associated with the velocity
field of a vortex with vortex charge qν can be written as [48]Ekin =

Lπn~2 2 R
q ln .
m ν ξ(3.20)

Due to the quadratic dependence on the vortex charge qν , two vortices with qν = 1 areenergetically favourable to a doubly charged vortex, when neglecting the interactions
between two vortices.
The interaction of two vortices with vortex charges q1 and q2 can be understoodwhen looking at the interaction of their velocity fields. For two vortices rotating inthe same direction (sgn q1 = sgn q2 ) the velocity fields have opposite directions in
the region between the vortices. This leads to a slower flow of the fluid in between
the vortices than outside. Following Bernoulli’s principle, the pressure is lower in theoutside region, leading to a repulsive interaction between the vortices. The opposite
is true for a vortex anti vortex pair (sgn q1 = − sgn q2 ).
The same conclusion can be drawn from looking at the total kinetic energy of thevortices across the system. For a cylindrical condensate with the same geometry as
assumed for (3.20) and two vortices in a distance d with vortex charges q1 and q2
the total kinetic energy is equal to [48]Lπn~2 2 R
R
R
2Ekin =
q1 ln + q2 ln + 2q1 q2 ln
,
m
ξ
ξ
d

(3.21)

where the first two terms are the energy of the single vortices and the last termis their interaction energy. For vortex-vortex pairs this term is positive, thus thetotal energy is minimized for large separations d. The opposite is true for vortex anti-vortex pairs.

353. Vortices in Bose-Einstein Condensates
n

n

n

=
r

+
r

r

Figure 3.5.: Illustration of the interaction of two vortices in a dBEC. The orientation of
the dipoles is symbolized by the direction of the arrows. Figure adapted from [48].

In dipolar condensates the interaction of vortices cannot be described by theinteraction of their velocity fields alone. They exhibit an additional interaction, as
the vortices can be considered as anti-dipoles in empty space. This is illustrated inFig. 3.5. Consequently the interaction becomes dependent on the orientation of thedipoles relative to the plane of rotation [48]. For large separations d, the logarithmichydrodynamic interaction dominates the DDI. The interaction of a vortex pair isthus most significantly modified for short to mid range separations (d < 10ξ) [54].

3.5. Creation of Vortices in Bose-Einstein
Condensates

Figure 3.6.: Transverse absorption image of a BEC stirred with a laser beam, a) below
and b) above the critical rotation frequency. Figure adapted from [55].

Numerous experimental methods have been used to create vortices in BECs. The
first successful creation of vortices in a BEC has been performed by Matthews et
al. in 1999 [21], where they formed the desired vortex wave function directly via
coherent transitions in a two component BEC. Here one component (i.e. one spinstate of 87 Rb) forms a vortex while the other component fills the vortex core.
An alternative approach is the direct rotation of the BEC using stirring potentials.
This approach is more flexible as it does not require resonant interactions with the
BEC. It has first been demonstrated by Madison et al. in 1999 [55]. They utilize a
red-detuned laser beam, superimposed with the potential of a Ioffe-Pritchard trap tocreate a cigar shaped harmonic trap with an anisotropic transverse potential. By
rotating the laser beam, they can shift this anisotropy, similar to rotating a bucket of36

3.5. Creation of Vortices in Bose-Einstein Condensates
fluid to create a vortex. Here the anisotropy of the potential replaces the roughnessof the bucket walls. The transverse absorption image of the BEC at different rotation
frequencies is shown in Fig. 3.6.
Recent developments in spatial light modulation technology have lead to new,more flexible possibilities to induce rotation in a BEC. A DMD (Digital Micromirror
Device) (see Section 5.1) can be used to alter the profile of a laser beam in a (nearly)
arbitrary way. This can be used to create vortices either via the rotating bucket
approach, where an anisotropic trap is rotated or by moving obstacles through the
Bose-Einstein condensate. This approach has been demonstrated by Gauthier et al.
[56] in 2019. They create giant vortex clusters by stirring the BEC with two pedal
potentials.

374. Accordion lattice set-up
The aim of this master thesis is to built an optical setup to create vortex excitations
in a dBEC of Erbium and Dysprosium. To this end, we developed two setups,
one to trap the atoms in an appropriate geometry and the other to create thevortex excitations. In this chapter we will review the setup designed for creating anappropriate trapping potential. The desired properties of this potential are discussed
in Section 4.1. We planned the necessary optical setup (see Sections 4.3 and 4.2),
constructed a test setup (see Section 4.4) and characterized the resulting trap (seeSection 4.5).

4.1. Desired properties of the trapping potential
We aim to trap the BEC in an optical dipole trap with an appropriate geometry forthe creation of vortices. We assume a harmonic trap withm 2 2
Vext (r) =ωx x + ωy2 y 2 + ωz2 z 2 ,(4.1)
2
where ωi denotes the trap frequency in i−direction. The vortex line is orientatedalong the axial (z)-direction. The trap as well as the orientation of the vortex isillustrated in Fig. 4.1. The fluid rotates around the vortex core in the x-y-plane.

Figure 4.1.: Illustration of the trap geometry used for the creation of vortices as well as
the orientation of the vortex line.

4.1.1. Axial confinement
Vortices are topologically stable excitations of a superfluid BEC. Nonetheless, forfinite temperature systems, vortices can decay due to the presence of the thermal

39

4. Accordion lattice set-up
cloud, which provides a source of dissipation. In finite temperature systems, longwavelength transverse helicoidal deformations of the vortex line known as Kelvinwaves can be thermally excited. These deformations cause the vortex to quiver andemit acoustic radiation, enabling the vortex to dissipate energy. [57]In a dBEC the three dimensional character of the vortex is expected to be evenmore significant due to the long range character of the DDI. Different parts of thevortex line can interact due to the non-local DDI, leading to an alteration of thestability [58]. As these excitations are three-dimensional, they can be suppressedby reducing the size of the condensate in direction of the vortex line [57]. This canbe achieved by tightening the confinement along the vortex line (z-direction) such that
r
~
lz =
≈ ξ,
(4.2)
mωz
where lz is the harmonic oscillator length in z-direction. As the vortex line cannot
support deformations in the transversal direction, the Kelvin mode contribution isessentially frozen out. When the above criterion is fulfilled, the vortex is considered
to be two-dimensional. It is important to note while the vortex dynamics are 2D,
the condition~ωz  kB T
(4.3)
for a two dimensional BEC is not necessarily fulfilled.
We estimate the healing length of Er and Dy considering only the influence ofthe contact interaction as there is no analytic formula for the anisotropic healing
length of a dBEC. This estimate gives us a healing length (according to Eq. (3.15))
on the order of 1 µm for both Er and Dy, based on parameters typically found in
our experiments. The healing length corresponds to a trapping frequency of ωz ≈2π·2 kHz. As this is only a crude estimate, we will design the system such that it
is possible to reach ωz ≈ 2π· 4 kHz to be able to compensate for influences of thedipole-dipole interaction on the healing length.

4.1.2. Radial confinement
The trapping potential in the radial (x-y-) plane governs the motion of the vortex
core. Similar to objects in a classical fluid, the vortex core moves due to a density
gradient force and the Magnus effect. As the density in the vortex core vanishes,
the core experiences a force towards regions with low densities. This resembles
the classical buoyant force, which causes a bubble within a fluid to experience a
force which is anti-parallel to the pressure gradient. The Magnus effect causes a
rotating cylinder to drift perpendicular to an applied force. Thus the vortex core
moves perpendicular to both its quantization axis and the density gradient force.
Consequently the vortex core precesses around the trap centre in an inhomogeneous,axisymmetric (ωx = ωy := ωr ) BEC [59]. The precession frequency φ0 is given by [60]  0
2~ωr2
ωr2
2µ
φ0 =
3 + 2 ln
(4.4)
228µ(1 − r0 /R⊥ )
5ωz
~ωr

40

4.2. Accordion latticewhere ωr and ωz are the radial and axial trapping frequencies respectively. The radial
coordinate of the vortex is given by r0 , µ0 gives the chemical potential and R⊥ is theradial Thomas-Fermi radius of the BEC. We introduce the aspect ratio AR, defined
as
ωz
.
(4.5)
AR = √
ωx ωyAs the precession frequency is inversely proportional to AR, the motion of the vortices
can be minimized by either maximizing AR for a given ωz or reducing the distancer0 from the trap centre. We minimize φ0 , to rule out complications caused by the
unknown influence of the DDI on the precession of the vortex.
We aim to ensure a low precession frequency by maximizing the aspect ratio.
While it would also be possible to minimize r0 , we want to be able to produce
multiple vortices to study their interaction without additional influences of the trap.
As the trapping frequencies in the radial plane are too low to hold the BEC, theconfinement in the radial plane has to be provided by an additional potential. This
will be achieved using a blue detuned beam modulated by a DMD, which is also used
to introduce rotation in the system (see Chapter 5). Additionally the DMD could
be used to compensate the remaining radial confinement, so that the Bose-Einsteincondensate is uniform. This can be done by choosing the pattern on the DMD
such that the resulting intensity distribution has the opposite curvature as the trap,
creating an effective flat-bottom potential.
The discussion above assumes the case of a axisymmetric trap (ωx = ωy ). In
non-axisymmetric traps (ωx 6= ωy ) the rotating walls induce an irrotational flow inthe system, which is also present in the absence of vortices. This additional velocity
field influences the critical rotation frequency of the condensate. Moreover the vortex
precesses around the centre of a non-axisymmetric trap in an elliptical trajectory[61]. We aim to avoid these additional effects by choosing the trap parameters suchthat the confinement in the x-y-plane is symmetrical.
Summing up, the three requirements for the trap are a strong axial confinement,
making the vortices two dimensional, ensured by ωz ≈ 2π · 4 kHz, a high aspect ratio√
to minimize the precession of the vortices ωz  ωx ωy and an axi-symmetric radialconfinement ωx ≈ ωy . This trap geometry is commonly referred to as a pancake trap.

4.2. Accordion lattice
According to the discussion above, a pancake like trapping potential with tight axial
confinement is ideal for the creation of vortices. Such a confinement cannot be
achieved with a crossed beam optical trap, where ωz is on the order of a few hundred
Hertz for typical laser powers. Alternative approaches include a gravito-opticalsurface trap [62], based on an repulsive evanescent wave against which the atoms arepushed by gravity. This concept however requires placing a prism inside the vacuum
chamber, which limits the optical access for future experiments. Another possibility
would be to focus a red-detuned beam using a cylindrical lens, leading to a tight41

4. Accordion lattice set-up

Figure 4.2.: Schematic illustration of the accordion lattice setup. A change in the heightof the incoming beam results in a change of the distance D of the beams reflected by thepolarizing beam splitter (PBS) cubes. This leads to a change in their angle of interference φ,which alters the spacing d of the interference pattern. The interference pattern is sketchedin the inset, with the red plane marking the central fringe of the lattice.

focus in one direction, which results in a high trap frequency in this direction [63].
Considering the limitations set by the minimal focal lengths, which is constrainedby the size of our vacuum chamber, this approach is not feasible as it leads to a
non-axisymmetric trapping potential.
The desired geometry can also be achieved using a one-dimensional lattice, createdby the interference of two counter-propagating laser beams. The 1D lattice consistsof a linear array of pancake shaped traps with a lattice spacing of λ/2. For atypical trapping wavelength of λ = 1064 nm, a BEC with a Thomas Fermi radiusof RTF ≈ 10 µm (commonly found in our experiments) occupies around 20 latticesides. This is problematic as the linear array is orientated along the imaging axis. As
the individual lattice sites in an array cannot be resolved with a standard imaging
technique, we would need a lattice with a spacing of ≈ 10 µm to trap the condensate
in a single lattice site. This would require a wavelength of 20 µm. Thus an alternative
approach is needed.
A one-dimensional optical lattice with tunable lattice spacing and confinement,(commonly known as accordion lattice) allows us to achieve the desired trapping
parameters, while enabling us to trap the atoms in a single lattice side. Insteadof two counter propagating beams, the accordion lattice is formed by two beams

42

4.2. Accordion latticeinterfering at a shallow angle. This is illustrated in Fig. 4.2. The lattice spacing dcan be altered by changing the angle of interference φ. Thus we can reach d = 10 µm
allowing us to trap the entire condensate in a single lattice site. The axial trapping
frequency in a lattice with a d ≈ 10 µm is on the order of a few hundred Hertz
for typical laser powers. As the axial trapping frequency is indirectly proportional
to the d, we decrease the lattice spacing after trapping the condensate, until ωz isat ≈ 2π · 4 kHz. Owing to the fact that the tunnelling rates for the final latticeconfiguration are small (≈ 1 · 10−11 s−1 for our setup), we can ignore the influence ofthe lattice structure on the system. This setup was first proposed by Li et al. [64] in2008. The first experimental implementation of this optical accordion lattice setup
with 87 Rb was achieved by Ville et al. [65] in 2017.
The two beams used to form the accordion lattice are created by splitting a single
laser beam using the setup shown in Fig. 4.2. The polarization of the incoming
beam is chosen such that half of the intensity is reflected at the first PBS cube. The
transmitted beam passes a second PBS cube and a quarter-wave plate (λ/4). It isthen reflected by a mirror, passing the λ/4 plate a second time. After passing theλ/4 plate twice, the polarization of the beam has been rotated by 90° such that it isreflected by the second PBS cube. The two beams interfere in the focal point of alens with focal length f . The half-angle φ between the two beams is equal toφ = arctan

D
,
2f

(4.6)

where D is the distance between the two beams. This distance can be altered bychanging the height of the incoming beam on the first mirror.
The lattice spacing can be derived using Fourier optics. The field in the focal
point of the lens U0 (x, y, f ) is the Fourier transform of the incoming field on the lensU (x, y, 0). Consequently, for two beams with a distance of ±D/2 from the centre of
the lens, the fields can be written as


πD
±D/2
U0
(x, y, f ) = exp ±i
x U (x, y, 0)(4.7)
λf
resulting in an intensity
2πDI(x, y, f ) = 2 cos
x + 1 I(x, y, 0).
λf

(4.8)

The spacing of the resulting interference pattern d is thus
d=

λf
.
D

(4.9)

Hence, real time control of the height of the incoming beam leads to real time controlof D, allowing the tuning of d during the experimental sequence. This enables us
to load the entire condensate into a single lattice side and then decrease the latticespacing adiabatically.

43

4. Accordion lattice set-up

4.3. Numerical calculation of trap parameters

Figure 4.3.: Numerically calculated interference of two Gaussian Beams in the x-z-plane.
The beams propagate along the x-direction.

The trapping frequencies (ωx , ωy , ωz ) of the accordion lattice depend on the wavelength
λ of the laser beam, the power P , the beam waists in the focal point (wy , wz ) as wellas the half-angle φ at which the beams interfere. To find the right parameters sothat the lattice fulfils the desired properties discussed in Section 4.1, we calculate thetrap frequencies numerically. The Matlab script can be found in Appendix A.1. We
calculate the values of the electric fields E1,2 (r) for discrete positions on a grid. Thisis then used to derive the intensity pattern I(r) around the point at which the beams
interfere (i.e. the focal point of the lens), as well as the resulting trap frequencies.
The trap frequencies in x-direction are given byrkx∂ 2 U (x, 0, 0)
ωx = 2π ·with kx =,(4.10)m∂x2
x=0
where U (x, y, z) is the trapping potential, derived using Eq. (2.24). To find the
optimum parameters, we vary wy from 100 µm to 3000 µm in steps of 100 µm and wz
from 20 µm to 500 µm in steps of 10 µm. The half-angle φ is kept constant at the
maximum achievable angle φ = 5.711°, which is limited by the optical access of the
viewport of the vacuum chamber. We use the maximum achievable φ to get the themaximal AR and ωz . The obtained intensity in the x-z plane is shown in Fig. 4.3.
There are two main wavelengths which we consider for the accordion lattice:
532 nm and 1064 nm. Both wavelengths are red detuned for Erbium and Dysprosium(see Section 2.2) with similar polarizabilities for both species. At 532 nm, thepolarizabilities of Er and Dy are equal to 317 a.u. and 350 a.u. respectively, at 1064 nm,
they equate to 176 a.u. (Er) and 184 a.u. (Dy) [35, 66]. High power commercial

44

4.3. Numerical calculation of trap parameters

Figure 4.4.: Comparison of the aspect ratio in dependence of the axial trapping frequency
for wavelengths 532 nm and 1064 nm. The data points for 532 nm are marked in green, theones for 1064 nm are marked in red. The data was obtained using a numerical calculationof the trap parameters, for fixed power and angle of interference.

laser sources at both wavelengths are readily available. In order to find out which
wavelength is more suitable for our experimental setup, we calculate the trapping
potential for both and compare the parameters of the trap. We do this at a fixed
power of P = 4.4 W for 532 nm and P = 35 W for 1064 nm. These powers are limitedby the maximum available laser powers as well as the losses and limits arising fromadditional optical elements such as an acousto-optical modulator (AOM) used forthe intensity stabilization and an optical fiber used for mode cleaning.
The trapping potentials for 532 nm and 1064 nm differ mainly in the achievable
aspect ratio AR for a given ωz . We calculate AR and ωz for each pair wy , wz . All
combinations give a functional form between AR and ωz , displayed in Fig. 4.4. The
aspect ratio grows for decreasing ωz . At a given ωz the AR is significantly higher for
532 nm than for 1064 nm. As a high AR is favourable for the production of vortices(see Section 4.1.2), we choose λ =532 nm for the accordion lattice setup.
Another key parameter is the symmetry of the lattice in the axial plane. Thissymmetry can be altered by changing the ratio of the beam waists wy and wz . To
find the optimum parameters, we introduce the symmetry factor
S=

ωx
−1
ωy

(4.11)

which can take values between 0 and 1. The trap is symmetric in the radial plane
when S = 0. The graph in Fig. 4.5a depicts both the AR and the symmetry factor
as a function of the beam waists (wy , wz ). A symmetry factor of S = 0 is possible fornearly all aspect ratios.
As seen in Fig. 4.5a, the symmetry condition is fulfilled for a fixed ratio wy /wz .
To determine the desired ratio of the beam waists, we only consider the beam waists

45

4. Accordion lattice set-upa)b)

Figure 4.5.: Graphs to determine the conditions for S = 0. A three-dimensional plot of
the aspect ratio AR in dependence of the beam waists is shown in a). The colour scale
marks the symmetry factor S. The beam waists for which S < 0.01 are plotted in b). The
data points are displayed in blue, a linear fit of the curve is displayed in red.

for which S < 0.01. Due to the discreteness of the step size of the calculation, we do
not obtain S = 0. Figure 4.5b) shows the beam waists wy , wz for which S < 0.01.
We approximate the dataset using a linear equation to determine the ratio of thewaists. The slope k of the resulting line is equal to 10. Consequently we design thesystem such that wy = 10 · wz .

Figure 4.6.: Aspect ratio AR and vertical trapping frequency ωz in dependence of thevertical beam waist. The beam size in the horizontal direction is fixed to wy = 10 · wz tofulfil the requirement S < 0.01. The vertical trapping frequency ωz is marked with blue
dots, the AR is marked with red squares.

To choose wz we have to find a compromise between the axial trapping frequency
ωz and the aspect ratio AR. Increasing the beam size at a given power decreases theintensity, causing a decrease of the trapping frequencies ωy and ωz . At the same time,
the aspect ratio increases. This is shown in Fig. 4.6, where the aspect ratio AR andthe trapping frequency ωz is plotted in relation to wz , for pairs of beam waists (wy , wz )fulfilling the symmetry condition wy = 10 · wz . As discussed in Section 4.1.1, we want46

4.4. Experimental implementationto achieve ωz ≈ 2π · 4 kHz. Hence we choose the beam waists at the focal point to be(wy , wz ) = (900, 90) µm, resulting in trap frequencies of (ωx , ωy , ωz )= 2π·(5.56, 5.56,
3898) Hz (AR ≈ 701.1), for the maximum achievable half-angle φ = 5.71°.

4.4. Experimental implementation
As discussed in Section 4.2, the key feature of the accordion lattice is the ability
to change the trapping frequencies and lattice spacing by altering the angle ofinterference of two beams. This is accomplished using the setup depicted in Fig.
4.2. The distance of two parallel beams can be changed by varying the height of theincoming beam. The beams interfere in the focal point of a lens with f = 200 mm.
The focal length was chosen so that the lens can be placed outside the vacuum
chamber. The optical setup includes a part to produce two parallel beams (see 4.4.1),
a setup to alter the height of the incoming beam (see 4.4.2), as well as the focusing
optics used to interfere the two beams (Subsection 4.4.3) and additional opticalelements (Subsection 4.4.4).

4.4.1. Beam separation
We designed a housing for the optical components (excluding
the lens), shown in Fig. 4.2, to minimize disturbances causedby air turbulences and to achieve a good passive mechanicalstability of the optical components relative to each other.
The housing is shown in Fig. 4.7. It consists of 6 anodized
aluminium plates with a mean thickness of 6.5 mm andwas machined in-house. A 2 inch mirror is mounted undera 45° angle to direct the light upwards. Two 1 inch PBS
cubes are placed in a milling pocket to ensure the correctorientation. They are glued to the side of the tower using
two-components-epoxide resin glue. The parallelism of thetwo cubes is vital for the setup. We ensured it by checking
that the two outgoing beams are parallel before fixing the
cubes. This was done by verifying that the separation of
the two beams remains constant for different distances fromthe tower. This test was repeated after fixing the cubes.
A λ/4-plate in a rotational mount and a 1 inch mirror are Figure 4.7.: Rendered
mounted above the PBS cubes. The tower is designed so image of the self dethat all optical elements are centred on the PBS cubes. The signed tower used to creadjustment screws of the mirror mount are accessible outside ate two parallel beams
the tower, so that the alignment can be done without opening with tunable spacing.
the tower. There are openings in the aluminium plates onboth sides of the PBS cubes, allowing the light used for the experiment as well as
the stray light, which is transmitted by the first PBS cube and then reflected by the47

4. Accordion lattice set-up
second cube, to leave the tower. This should reduce heating of the tower caused bystray light.

Figure 4.8.: Illustration of the accordion lattice setup for minimum and maximum beam
separations D. The beams with minimum (maximum) separation are displayed in blue
(green).

The maximum half-angle of interference is equal to φmax = 5.711°, as discussed in
Section 4.2. For a lens with f = 200 mm, this equates to a beam distance Dmax =40 mm. The limiting factor for the minimum angle is the free aperture of the λ/4plate, which is smaller than the free aperture of the other optical elements. We
use a 1 inch zero-order quarter-wave plate from Thorlabs (WPQ10M-532), which is
specified to have a clear aperture of 22.6 mm. As the wave plate is centred on the
cubes, the smallest possible beam separation is equal to Dmin =4.8 mm, which is equal
to an angle of φmin = 0.57°. The beams with maximum and minimum separation are
illustrated in Fig. 4.8. The lattice spacing d can be varied from 2.66 µm to 22.2 µm
and the trapping frequencies (ωx , ωy , ωz ) can be altered from 2π·(0.55, 5.56, 422)
Hz (φ = 0.57°) to 2π·(5.56, 5.56, 3898) (φ = 5.71°) at a laser power of 4.4 W. The
numerically calculated trap frequencies ωx and ωz for different φ are shown in Fig.
4.9.

48

4.4. Experimental implementation

Figure 4.9.: Numerically obtained trap frequencies ωx and ωz for different angles φ
at P = 4.4 W. The trap frequencies in z(x)-direction are marked by the blue(orange)squares(circles). The beam size was fixed at (wy , wz )=(900 µm, 90 µm)

The minimum beam separation could be decreased by changing the position of the
wave plate. As the distance of the beams at maximum separation to the edge of the
wave plate is larger than the distance at the minimum separation, the wave plate
could be shifted from being centred on the cubes towards the beam position at theminimum beam separation by ≈ 2 mm. This modification would allow for a smaller
minimum angle of interference, leading to a higher maximum lattice spacing, while
the minimum lattice spacing remains the same.

4.4.2. Beam displacement
To go from the maximum (Dmax ) to the minimum (Dmin ) beam separation, theincoming beam has to be shifted by a distance h in z-direction (see Fig. 4.8). The
distance h can be determined geometrically to beh=Dmax − Dmin
= 17.6 mm.
2(4.12)

There are multiple possibilities to do this. We looked more closely into the following
methods:• Moving the beam in the optical plane using a mirror mounted on a linear
translation stage• Altering the angular position of the beam with an acousto-optical deflector
(AOD)

49

4. Accordion lattice set-up• Changing the height of the beam by rotating two parallel mirrors
Moving the beam using a mirror mounted on a linear translation stage requires aslight alteration of the experimental setup (see Fig. 4.2 and 4.7). Instead of mounting
the 45° mirror parallel to the reflecting planes of the PBS cubes, it has to be turnedby 90° around the z-axis. Then a translation of the beam in the optical plane will
result in the desired shift in the distance of the two output beams. Commerciallyavailable linear translation stages with a travel range of 20 mm are specified to have avelocity of 200 mm/s. Consequently we can shift from the maximum to the minimum
beam separation in ≈ 0.1 s. This provides only a lower bound for the time-scale, as
it does not account for the time needed to accelerate and decelerate the stage. While
this time scale is appropriate for an adiabatic compression of the condensate, it could
be a limiting factor for possible future experiments (e.g. quench experiments). Thus,
we consider other, faster options.
Another possibility is to alter the angle using an AOD, which utilizes the acoustooptic effect to spatially control the beam. The diffraction angle of the AOD isdependent on the frequency of the sound-wave propagating inside the crystal. Thus,by altering the frequency, the beam can be deflected to different angular positions.
A lens is needed to be able to convert the angular shift of the beam into a positional
shift. Typical commercial AODs offer a scan angle of ≈ 2°. A lens with a focal lengthf = 350 mm is required to convert this angular shift into a positional shift of 2 cm.
This results in a strongly focused beam inside the AOD, which further limits the
maximum power that can be used, owing to the damage threshold of the AOD as
well as thermal effects [67].

Figure 4.10.: Schematic sketch of the beam displacement using parallel mirrors. Twomirrors are placed in a distance ∆ parallel to each other. The angle between the mirrors
and the beam is given by α. After the two reflections the outgoing beam is parallel to theincoming beam. The height of the outgoing beam is dependent on the angle α. Thus byaltering α, the beam can be displaced parallely.

The third options is to change the height of the beam by altering the angle of twoparallel mirrors relative to the beam. This concept is illustrated in Fig. 4.10. The

50

4.4. Experimental implementation
beam is reflected by the first mirror (M1) and by the second mirror (M2), which isplaced parallel to M1 in a distance ∆. This double reflection leads to a shift in theheight of the beam. By changing the angle α between the mirrors and the incoming
beam, the resulting beam height can be changed.
To find the distance ∆, the length of the mirrors and the angular shift resulting
in the desired height displacement, we wrote a Matlab code (see Appendix A.2) tocalculate the displacement of the beams analytically. The beam and the mirrors areboth expressed as vectors. We calculate the intersection points and angles and use
this to propagate the beam using geometric optics. The calculation was performed
with different centres of rotation, which does not have an influence on the beamdisplacement. Thus, we chose the centre of rotation to be in the centre of the twomirrors. According to our calculation, a change in angle of 20° results in a verticaldisplacement of 18.6 mm. We use a DRTM 40-SM rotary stage from OWIS with amaximum angular velocity of 2500 °/s. This angular velocity corresponds (according
to our simulations) to a velocity of approximately 2300 mm/s, which is a factor of 10
larger than the velocity of the linear translation stage.

Figure 4.11.: Mounting of the two mirrors on the rotational stage. The first mirror ismounted on a kinematic mirror mount so that it is possible to alter its tilt in both directions.
The mount was designed such that the centre of mass is at the centre of rotation.

The geometry of the mount was determined using the code in Appendix A.2. The
first mirror is chosen to be smaller than M2, so that it does not cut off the beam
after the reflection by the second mirror. The displacement of the beam position on
M2 when going from the maximum angle to the minimum angle, is large comparedto the displacement on M1. Hence M2 is longer than the first one. The second mirror
is horizontally displaced with respect to M1 to be able to utilize its entire size. Both51

4. Accordion lattice set-up
mirrors are rectangular with lengths 25.4 mm (M1) and 35 mm (M2). As the heightof M2 is larger than that of the first, M1 is mounted on a base plate such that thebeam hits the centre of both mirrors. They are mounted in a distance ∆ = 35.4 mm.
Several points need to be considered to ensure the optimal performance of thesystem. First, to avoid a rotating unbalance, the mount was designed such that thecentre of mass coincides with the centre of rotation. We have also tested a setup
where the centre of mass is not equal to the centre of rotation. This decreases therepeatability of the system. Second, the parallelism of the two mirrors is essential
for the beam translation. When the two mirrors are not parallel, the beam is not
displaced parallely when changing α. This effect cannot be compensated for by
mirrors after the rotation stage, leading to a shift of the interference pattern whenchanging the confinement of the accordion lattice. Hence the first mirror is mounted
on a small kinematic mirror mount, enabling fine-tuning of the vertical and horizontal
tilt. The final setup is shown in Fig. 4.11. The parallelism of the mirrors can bechecked by looking at the vertical and horizontal position of the beam after the twomirrors for different α. The shift of the beam in vertical and horizontal position was
minimized by altering the angle of mirror 1 in relation to mirror 2 as well as theangle of the incoming beam on the mirrors.

4.4.3. Focusing Optics
The two parallel beams interfere in the focal point of a lens. It is critical for theaccordion lattice that the position of the focus does not change when altering the angle
of interference. The shift of the focal point was obtained with Zemax OpticStudio,
using a large incoming beam to simulate different beam separations. The spread of
the focal position of the large beam is equivalent to the shift of the focal point during
the compression of the lattice. When using a standard cylindrical lens, the positionof the focal point shifts by 2 mm during the compression (see Fig. 4.12a,b). This
shift is caused by spherical aberrations, causing the focal point to be dependent on
the distance of the centre of the lens. This can be avoided by using an aspherical lens,
which has a form that deviates from the standard spherical form. For an asphericallens the shift reduces to ≈ 2 µm (see Fig. 4.12c,d)
As discussed in Section 4.3 the optimum beam size is (wy , wz )=(900 µm, 90 µm).
Owing to the fact that a beam with a waist of 900 µm has a Rayleigh length of
4800 mm, it is effectively collimated. Thus the incoming beam has to be divergent toobtain the desired beam waist after the lens. This can be achieved by inserting acylindrical lens in front of the aspherical lens so that the incoming beam is divergent
in y-direction. An alternative approach would be to shape the beam such that it iscollimated with wy = 900 µm prior to the last lens. The beam can than be focused
only in z-direction using an acylindrical lens. As an acylindrical lens with a focal
length of f =200 mm is not a standard component, we decided to use a planoconcave
cylindrical lens to expand the beam in y-direction in front of the aspherical lens.
This setup is illustrated in Fig. 4.12e. The distance between the two lenses is chosensuch that the beam is collimated in y-direction after the pair of lenses. We use

52

4.4. Experimental implementationa cylindrical lens with a focal length f = −75 mm that is placed in a distance of
125 mm from the aspherical lens. Back propagating the desired waist in the focal
point through the optical system yields an input beam waist of wy = wz = 337.5 µm.

Figure 4.12.: Comparison of the shifts caused by spherical aberrations of different opticalsetups. The waist of the incoming beams w = 50 mm. The optical setup with a single
cylindrical lens (f=200 mm) is shown in a), a zoom-in on the focal point is shown in b).
Subfigure c) shows the optical setup consisting of a cylindrical lens with (f=−75 mm) ina distance of 125 mm from an aspherical lens (f=200 mm). Subfigure e) depicts the same
optical setup as c) tilted by 90° around the x-axis. The graphs were made using Zemax
OpticStudio.

The choice of a two-lens setup offers another great advantage. By removing thecylindrical lens, the waist in y-direction is decreased by a factor of 10, while theposition of the interference point is not significantly altered. Beam waists wy = wz =53

4. Accordion lattice set-up
90 µm result in trapping frequencies (ωx , ωy , ωz )= 2π·(20 Hz, 196 Hz, 14 000 Hz) fora power of 5 W. While this trap does not provide the symmetry in the x-y planedesired for the creation of vortices, ωz is high enough to fulfil the requirement for thequasi 2D confinement for a gas of distinguishable particles 1~ωz  kB T,(4.13)

for typical temperatures of T = 100 nK. This means that the motion in z-direction
is essentially frozen out and the BEC is quasi two-dimensional. Thus, by placing thecylindrical lens on a flip mount, we have the possibility to create quasi-2D dipolar
Bose-Einstein condensates.

4.4.4. Additional optical elements
Up to now, all calculations have assumed a perfect interference of the beams, leading
to a full contrast of the lattice. This is only true when the beams have the samepolarization and the power in both beams is equal. When the two beams do nothave the same polarization, the minima of the interference patterns will not be equal
to zero. We utilize Thorlabs PBS25-532-HP PBS cubes, which are specified to havethe highest extinction ratio (Tp : Ts > 2000 : 1) when used in transmission, but are
not specified in reflection. As we use the beams reflected by the two PBS cubes inthe tower, we insert two additional cubes after the tower. These cubes are rotatedby 90° around the x-axis relative to the cubes in the tower. The s-polarized lightreflected by the PBS cube in the tower is thus p-polarized with respect to the PBS
cube outside and is transmitted, leading to a clean interference of the two beams. We
were able to observe a significant increase in the contrast after inserting the cubes.

Figure 4.13.: Comparison of lattice before(blue triangles) and after (red squares) insertingthe cubes and controlling the power balance. The line serves as a guide to the eye.
1

In a Bose-Einstein Condensate this condition is weakened due to the phenomenon of transverse
condensation. However, the discussion of this phenomenon goes beyond the scope of this thesis.

544.4. Experimental implementation
The power ratio of the two beams has a similar influence as the polarization.
If the power is unequally distributed between the two beams, the minima of theinterference pattern will not be at zero intensity. The ratio can be controlled usinga λ/2 waveplate in front of the tower. The polarization of the incoming beam ischosen such that half of the intensity is reflected at the first PBS cube and half istransmitted. The comparison of the interference pattern before and after insertingthe cubes and controlling the power balance is shown in Fig. 4.13. The pictures weretaken using an AV U-503B camera with a pixel size of 2.2 µm. It is important to notethat this camera has a higher pixel size than the camera used for the characterizationof the lattice (see Sec. 4.5). The fact that the minima of the interference pattern
are not at zero intensity after inserting the cubes and controlling the power can be
explained by the finite size of the pixels.

4.4.5. Experimental setup

Figure 4.14.: Accordion lattice setup. The laser light is polarization cleaned using a
PBS cube and shaped using two telescopes. We use the first order of an AOM which isintensity stabilized using a PID to control the RF signal for the AOM. The beams interfere
in the focus point of a lens with f = 200 mm. The setup used to characterize the lattice
did not include the photonic crystal fiber. When implementing the setup in the main
experiment, the atoms will be at the position at which the camera was positioned for thecharacterization. For further information see text.

55

4. Accordion lattice set-up
The final experimental setup used for testing the accordion lattice is shown in Fig.
4.14. We use a Coherent Verdi V18 laser with a maximum output power of 18 W
and an output beam waist of 1360 µm. We insert a λ/2 waveplate and a PBS cube
immediately after the laser output port, to be able to reduce the laser power. The
intensity of the beam is stabilized using an AOM. Owing to the fact that an aperture
has to be ≈ 1.5× larger than the beam diameter to limit the losses to below 1 %, thelasers’ output beam waist is slightly too large for the active aperture of the AOM(3 mm × 6 mm). Consequently, we insert a telescope (consisting of a f=−50 mm anda f=100 mm lens) before the AOM, to decrease the beam waist by a factor of two(1:2 telescope).
A quartz AOM (M115 - FS80L -3), chosen due to its high damage threshold, isused to stabilize the power. We measure the light transmitted by one of the mirrors
using a photodiode. This signal is then processed using a sample and hold PID
controller, which controls a variable attenuator that alters the amplitude of the RF
signal going to the AOM. This allows us to stabilize the power in the first order ofthe AOM.
The beam is subsequently shaped by another 1:2 telescope to achieve the desired
beamshape (wy = wz ' 337.5 µm (see Section 4.4.3)). The height of the beam isaltered using the rotational stage (see Section 4.4.2) before it enters the tower setup.
The resulting parallel beams pass through a f = −75 mm cylindrical lens and are
directed onto the f = 200 mm aspherical lens using two mirrors. The final interference
pattern is observed using a mvBlueFOX camera,with a microscope objective with
tenfold magnification to expand the interference pattern.

4.4.6. Alignment procedure

Figure 4.15.: The back reflections of a planoconvex lens. The beams reflected by theplane side of the lens can be used to adjust the tilt of the lens by assuring that they havethe same distance to the incoming beams. For further information see text.

The setup shown in Fig. 4.14 was used to optimize and determine the characteristicsof the accordion lattice. We aim to minimize the movement of the position of the
focal point during the compression, as it can lead to a displacement of the central
fringe of the lattice as well as an alteration of the trap depth caused by a change inthe overlap of the two beams. The position of the focal point slightly varies when56

4.4. Experimental implementation
changing the separation of the two beams. To determine this variation, we measure
the vertical position of the individual beams for different separations. The camera
is fixed to the focal point (i.e. the point where upper and lower beam have thesame vertical position) at a given separation. There are two main effects on thechange of the vertical beam position for different beam separations. Firstly, whenthe mirrors on the rotation stage are not parallel, the outgoing beam is reflectedunder an angle that is dependent on the angle α between the incoming beam andthe mirrors on the rotation stage. Thus the beam direction changes for different α,
causing a change of the focal position for different beam separations. Second, thefocal position at different separations is influenced by the tilt of the aspherical lens,
which changes the vertical position of the upper and lower beam in the focal point.
Moreover, the tilt of the lens alters the position of the central fringe when ramping
the confinement. We expect this to be due to an alteration in the phase difference ofthe two beams, caused by a change in the optical path length difference for different
beam separations, due to the tilt of the lens. Consequently we have to ensure thatthe tilt of the aspherical lens is minimized and that the beams exiting the tower are
parallel for all beam separations, which is equivalent to ensuring that the directionof the beams remains constant when changing α.
The parallelism of the outgoing beams at different separations was checked without
the f = 200 mm lens. First we ensure that the two outgoing beams are parallel for
one separation. The mirror on top of the tower was aligned such that the reflectedbeam coincides with the incoming beam. Second, the last mirror in front of thetower was adjusted such that the outcoming beams are parallel. This was ensured
by checking that their separation remains constant at three different positions after
the tower. Afterwards we alter the beam separation and verify that the beams arestill parallel. If this is not the case, the tilt of mirror M1 on the rotational stage isadjusted. This process is repeated until the beams at the maximum and minimum
beam separation are parallel.
A rough adjustment of the tilt of the lens can be made by ensuring that the backreflections of the plain side of the lens (illustrated in Fig. 4.15) are symmetric withrespect to the incoming beams. The angle of the reflection by the plain side depends
on the angle between the lens and the incoming beam. Thus, by ensuring that thetwo back reflected beams are in the same distance of the incoming beams for a given
distance from the lens, one can roughly adjust the tilt of the lens.
The fine adjustment can be done by looking at the central peak position of thetwo beams in the focal point. This was done by blocking one of the two beams andrecording the position of the other beam on the camera. The camera is fixed at the
focal point (i.e. the point where the beams have the same vertical position) for agiven beam separation. Due to the fact that the aspherical lens is not perfect, thefocal point does not remain constant for different beam separations, leading to adisplacement of the beam position relative to the focal point in z-direction for a fixedcamera position. When the lens is orthogonal to the incoming beams, the change inthe vertical position of the upper and the lower beam for different beam separations
is symmetric.

57

4. Accordion lattice set-up

4.5. Characteristics of the accordion lattice
We test the the experimental set-up shown in Fig. 4.14 (without the photonic
crystal fiber) offline, observing the lattice using the aforementioned camera. We firstcharacterize the change in beam separation for different angles α (angle between the
mirrors on the rotation stage and the incoming beam) and the change in the verticalbeam position in the focal point when varying the separation of the beams. Second,
we measure the lattice spacing for different beam separations as well as the dynamic
stability of the lattice during compression. Furthermore, we measured the temporalstability of the lattice for different laser powers.

4.5.1. Beam separation

Figure 4.16.: Plot of the beam separations for different angles α. The black line marks
the beam separation, for all α where the beam hits both mirrors. The measured data
points are marked in blue. A linear fit of the data points is shown in cyan.

We measure the separation of the beams in dependence of the angle α, which isaltered in steps of 1°, going from the maximum to the minimum beam separation.
The black line shows the beam separation obtained using the analytic calculation(see Appendix A.2), for all α, where the beam is reflected by both mirrors. The
experimentally obtained data points are within 1σ of the results of the numericalcalculation. The measured data points are approximated using a linear fit function,
to obtain a calibration of the beam separation as a function of α. Although thenumerically obtained separations do not have a linear dependence on α, a linear fit of
the measured points has χ2ν = 0.3211. As χ2ν < 1 the model is over fitting the data,
meaning that our estimate for the errors is too large. Nonetheless χ2ν is in the 95 %confidence interval, signifying that a linear equation sufficiently describes the dataset

58

4.5. Characteristics of the accordion latticein the region of α used for the experiment. The linear equation used to convert α tothe corresponding beam separation is given byD(α) = α · 2.14(2) mm/° + 148(1) mm.

(4.14)

4.5.2. Vertical beam displacement in focal point

Figure 4.17.: Vertical central peak position of the upper and lower beam (defined according
to their relative heights when exiting the tower) in dependence of the beam separation.
The red squares (blue dots) show the position of the upper (lower) beam. The displayed
points are the mean of three sets of measurements taken successively. The error bars
represent the standard error of these three measurements and are smaller than the sizeof the data points. The lines connecting the data points serve as a guide to the eye. The
beam diameter in the vertical direction is indicated by the shaded region.

The setup was aligned as discussed in Subsection 4.4.6. We observe the change of theindividual vertical beam positions in the focal point when varying the separation of
the beams, to ensure that their overlap does not change significantly when altering
the confinement of the lattice. This is important as a variation in the trap depth
could induce heating. We block one of the beams and measure the vertical positionof the other beam. The position of the camera is fixed at the point where thebeams have the same central peak position for a separation of 3 cm. A central cut
through the obtained image was fit with a Gaussian function to obtain the central
peak position in the vertical direction. Figure 4.17 shows the resulting data. The
vertical position of the beams changes by a maximum of 24 µm. As this shift is small
compared to the individual beam waist of wz = 90 µm (indicated by the shaded areain the plot), the overlap of the two beams does not alter significantly during thecompression. The central fringe intensity changes by ≈ 2 % during the compression,59

4. Accordion lattice set-up
owing to the change in the overlap. Comparing our results to those specified by Ville
et al. [65] shows that the displacement of the beams (≈ 20 µm) specified in the paper
is roughly the same as the displacement that we measured. As we use a larger beam
waist, the variation of the trap depth will be smaller than that observed in [65].

4.5.3. Fitting Routine

Figure 4.18.: Raw picture and cut of the lattice, obtained at a beam separation D = 11 mm.
Subfigure a shows the raw picture obtained using a mvBlueFox camera with a microscope
objective (tenfold magnification). The red line marks a cut through the picture. The
cut was chosen to be off-centre, due to the aberrations (caused by dust on the camera
or objective) at the centre. Subfigure b) shows the intensity pattern of this central cut(displayed in black), approximated using function Eq.4.15 (shown in red).

An exemplary picture of the lattice is shown in Fig. 4.18a. The lattice was evaluatedby fitting a central cut of the interference pattern with the function
 


2
x−o
x−c
y(x) = A · sin 2π,(4.15)+d· exp −
pe
which is the convolution of a sinusoidal and a Gaussian function. The parameter
A gives the amplitude, o is a constant offset, p is the period of the lattice and dis the phase of the sinusoidal. The parameter c gives the central position of theGaussian and e describes the width of the Gaussian. The intensity pattern of suchan individual row, including the fit is shown in Fig. 4.18b.

4.5.4. Lattice spacing
The fit parameter p gives the lattice spacing. As the accuracy of the fit function isstrongly dependent on the starting value chosen for p, we programmed a peak-finder,that determines the maxima in the signal by comparing the intensity at a given pixel
to that of the neighbouring pixels. The mean distance between adjacent maxima isused as a starting value for p. The measured lattice spacing in dependence of thebeam separation is shown in Fig. 4.19. We approximate the curve using Eq. (4.9),

60

4.5. Characteristics of the accordion lattice

Figure 4.19.: Lattice spacing in dependence of beam separation. The green data points
mark the mean of three subsequent measurements. The error bars are smaller than thesymbol size. The blue curve marks a fit with Eq. (4.9) to the data points.

giving a focal length f = 192(4) mm, which is within 2σ of the focal length specified
by the manufacturer (f = 200 mm).

4.5.5. Dynamic stability
The movement of the central lattice fringe is vital for the performance of the accordion
lattice. Changes in the central fringe position during compression could lead tounwanted heating effects. In addition to this, the atoms might move out of the focusof the imaging objective during the compression when the central fringe positionchanges too much. We measure the displacement of the central beam by taking a
picture of the lattice for different beam separations (changing the angle α in stepsof 1°). The camera is placed at a fixed distance from the lens. We visualize themovement of the central fringe, by cutting an area with a width of 15 pixels fromthe centre of each picture and putting them after each other. The resulting pictureis shown in Fig. 4.20a. We determine the maximum of the central fringe using a
Gaussian fit, limited to a small region around the central peak. The obtained central
peak positions are represented by the black dots. Figure 4.20b shows a zoom-in on
the central peak positions. The y-axis is shifted such that the central peak positionat the highest beam separation is equal to zero. The limits of the y-axis reflect thedepth of focus of the vertical imaging objective (' ±3 µm), given that its focal point
is at the position of the central fringe for the maximum beam separation. The central
peak moves by 1.6(1) µm during the compression (see Fig. 4.20b). This movement
stems from residual imperfections in the alignment (e.g. tilt of the lens, parallelism of
mirrors on rotation stage). As can be seen in Fig. 4.20b, the position of the central
peak stays safely within the depth of focus of the vertical imaging objective.

61

4. Accordion lattice set-up

Figure 4.20.: Dynamic stability of the lattice. Subfigure a) shows cuts of the central
interference pattern with a width of 15px put after each other. The black data points markthe centre of a Gaussian fit with which the central peak was approximated. The error bars
are smaller than the symbol size. The line serves as a guide to the eye. A zoom-in on these
positions is shown in b). The y-axis is shifted such that the central peak position at thehighest beam separation is at zero. The limits of the y-axis reflect the depth of focus of
the imaging objective (' ±3 µm). The error bars are obtained from the fit of the central
fringe with a Gaussian function.

4.5.6. Temporal stability
In addition to the shift of the central peak position during compression, the positionof the central fringe is dependent on the stability of the phase of the interference
pattern. This phase is characterized by the parameter d in Eq. (4.15). To be able todetermine d, we introduce a constant offset o into the equation, which is manuallychosen to be at the centre of the lattice. This offset provides a constant reference
point for the phase.
The phase of the lattice is determined by the phase difference of the two beams.
The beams have a path difference of L =111 mm in the tower, as the upper beam
passes through the second cube and wave plate twice. Outside the tower, the beampaths are symmetric around the optical axis of the lens. While it is important toshield this part from external influences (e.g. air currents), we assume that the62

4.5. Characteristics of the accordion latticechange in the phase of the beams occurs mainly in the tower. The change in phaseof the upper laser beam relative to the lower one is given by Φ, where
Φ=

2π
L.
λ

(4.16)

A change in Φ can thus occur when either λ or L changes. A change in Φ, which isdifficult to observe with our experimental setup, will cause a change in the phase of
the lattice d, which we directly obtain from a fit of the interference pattern using Eg.
(4.15).
We evaluate the stability of the lattice by observing the change in d over time. The
first measurement was performed at low power (P < 0.1 W), directly after switchingon the laser. A plot of the phase of the lattice is shown in the inset in Fig. 4.21.
Multiple phase jumps occurred during the measurement. We speculate that these
jumps stem from frequency jumps i.e. mode hops of the laser during the heat-up
period, leading to a change in Φ, caused by a slight change in λ. The change infrequency necessary for one of those phase jumps can be calculated using Eq. (4.16).
The average change in the lattice phase is equal to 0.43(3) · 2π corresponding to afrequency difference of 1.16(8) GHz.

Figure 4.21.: Phase of the lattice measured at low laser power (P < 0.1 W) after a heat-up
period (2.5 h). The inset shows the phase of the lattice measured directly after turning on
the laser. The data points are marked in red, the line serves as a guide to the eye.

We repeat the measurement after a heat-up period (2.5 h). During the heat-up
period, the laser is blocked immediately after the output port, to ensure that thephase jumps are not caused by another element with a non-linear thermal behaviour.
The results are plotted in Fig. 4.21. As no jumps occur after the heat up period,
we assume that the phase-jumps are caused by mode-hops occurring while the laser
is not at thermal equilibrium. The change in phase is getting smaller over time,consequently we can speculate that it is caused by a thermal expansion of the tower,63

4. Accordion lattice set-up
leading to a change in L, which decreases as thermal equilibrium is approached.
However, the influence of a change in λ on Φ must not be overlooked, especially forexperiments requiring a high stability of the lattice. Here it might be advisable tolock the laser to an external reference.
We repeat the measurement at higher laser powers after a heat-up period. During
the measurement we rapidly change the power from 0.5 W to 4.5 W, to observe theinfluence of the laser power on the stability. We change the power by altering the RF
power of the AOM signal. This ensures that the laser can remain at a high output
power, so that the shifts are not caused by fluctuations in the frequency which weobserved when the laser is not at thermal equilibrium. Figure 4.22 shows the stabilityof the interference pattern over time at high laser powers. We observe a change inphase per time ∆p = 2π·0.0016(1)/s at 4.5 W, which is a factor of 2 larger than thechange at 0.5 W (∆p = 2π·0.0008(1)/s).
Such a behaviour could be explained by an increase in the temperature of thetower caused by stray light. To obtain the change in path length due to an increasein temperature, we calculate the thermal expansion of the aluminium housing. The
linear thermal expansion coefficient of Aluminium c = 24 · 10−6 1/K. Consequently
the path difference (L = 111 mm) changes by 2.64 µm/K. A change in temperatureof 0.2 K will thus alter the path length difference by one wavelength (λ = 532 nm),leading to a phase change of 2π. The energy E needed to change the temperature ofAluminium with mass m by a temperature difference T is given by
E = cmT,(4.17)

where c = 0.9 kJ/(kg K) is the specific heat capacity of aluminium. For our estimate
we use m = 200 g which corresponds to the mass of a single aluminium plate. Thisestimate is justified as the tower consists of anodized aluminium plates that arescrewed to each other, leading to a relatively weak thermal contact between them.
The change in phase per time ∆p at 4.5 W (∆p = 2π·0.0016(1)/s) can be caused bystray light with a total power of P ≈ 60 mW which corresponds to 1.33 % of theinitial power. While this percentage is higher than expected (the transmission of
the top mirror was measured to be 0.3(1) %), we can still assume that the change inphase is at least partially caused by heating due to stray light.
This drift could be minimized by a better thermal contacting of the individualaluminium plates that make up the tower. At the same stray power, the change
in phase per unit time would decrease by a factor of 5 for the perfectly thermallycontacted tower. Another possibility would be to stabilize the temperature of thetower. Even though the power used during the experimental sequence is 4.5 W,
the average power will be below that, due to the fact that the laser is only turned
for a short time during the experimental sequence. Therefore, the drift during one
experimental cycle should be negligible. Nonetheless we need to be able to control
the phase of the lattice.

64

4.6. Experimental control

Figure 4.22.: Phase of the lattice over time. The red dots mark the data acquired ata laser power P = 0.5 W at the focus position, the blue dots mark the data acquired at
P = 4.5 W. The error bars are obtained from the uncertainty of the fit of the individualinterference patterns.

4.6. Experimental control
We need to be able to control the phase of the lattice, in order to repeat theexperiment in a controlled manner. This is done by actively changing the path length
difference L using a piezo stack placed under the top mirror in the tower. We use
a Round PICMA® Chip Actuator from Physical Instruments (PD160.3x), with atravel range of 2.2 µm at 0 V to 100 V. This allows us to change path difference bya single wavelength (λ = 532 nm), which results in a change of phase of 2π, while
operating the piezo in the range of 0 V to 24 V, which is provided by a standard lab
power supply. We measure the interference pattern for different piezo voltages. The
resulting plot of the phase as a function of the piezo voltage is shown in Fig. 4.23.
We did not implement the long term phase stabilization during the course of thisthesis. The proposed sequence for stabilization the phase is to measure the phaseat a given lattice separation before the experimental sequence. This information isused to adjust the piezo voltage so that the phase is set to a fixed value. The piezo
voltage is then kept constant during the experimental sequence.
The phase of the lattice can be measured by recording the interference pattern ontoa camera and obtaining the phase using the same fitting routine used for the stability
measurements. Naturally we cannot place the camera at the position of the atoms.
One possibility would be to record the interference pattern of the back reflections of
the viewports. This is possible as we enter the viewport under an angle, meaningthat the back reflections do not coincide with the incoming beams. Nonethelessthis can be challenging due to space constraints in the experimental setup. Anotherpossibility would be to use the beams that are reflected by the two cubes used to65

4. Accordion lattice set-up

Figure 4.23.: Lattice phase as a function of the piezo voltage. The data points represent
the mean of 3 subsequent measurements. The error bars represent the standard error ofthese measurements. The line serves as a guide to the eye.

clean the polarization of the beams and letting them interfere in the focal point ofa lens. This lens must not necessarily be aspheric as only one beam separation isused. Even though the beams reflected by the cubes do not have exactly the samepolarization, the contrast should still be high enough to observe the interference
pattern. As the cubes are farther away from the vacuum chamber, there is moreavailable space to insert the lens and camera. The drawback of this proposal is thatthe change in phase difference after the cubes can not be observed. Although thechange in phase difference after the tower is expected to be much smaller than thatin the tower, this reduces the quality of the stabilization. The third possibility is to
directly image the lattice onto a camera by placing a lens on the other side of thevacuum chamber. This method is again constrained by the space we have in theexperiment, but should account for all drifts in the phase difference. Nonetheless
it is possible that the beams acquire an additional phase difference in between theatoms and the camera. This would lead to a stabilization to an incorrect phase.

665. Variable blue detuned potential
We aim to implement a blue detuned, variable potential, directed onto the atoms
along the vertical (z-)axis, to create vortex excitations. We choose the wavelength
λ = 370 nm, which is blue detuned for both Erbium and Dysprosium (see Section
2.2). The spatial intensity profile of the laser beam is modulated using a digital
micromirror device (DMD) and projected onto the atoms via the vertical imaging
objective, creating a variable blue detuned potential. This potential provides boththe in-plane confinement and the possibility to introduce rotation as well as moving
obstacles into the system. During the course of this thesis we planned the opticalsetup and optimized it using Zemax OpticStudio.

5.1. Digital Micromirror Device
The digital micromirror device (DMD) consists of an array of individually addressable
mirrors. We utilize a Texas Instruments DLP9500 consisting of 1920x1080 mirrors
with a size of 10.8 µm and a maximum pattern rate of 23 148 Hz.
A pattern can be displayed on the DMD by changing the tilt of the individual
micromirrors. The mirrors of the DMD have three possible states: ON, IDLE and
OFF. When the DMD is turned off, the mirrors are in the IDLE state, meaningthat they are parallel to the plane at which they are mounted. In the ON and OFF
configuration the mirrors are tilted by ±12° relative to the IDLE state. When lightis send onto the DMD, it is reflected by the micromirror array, under an angle givenby the tilt of the individual mirrors. This is used to selectively ’cut out’ parts of thelaser beam and thus modulate its spatial intensity profile.
The array of individually tilted micromirrors can be seen as a blazed grating witha spacing d equal to the micromirror spacing. This is illustrated in Fig. 5.1. The
relationship between the angle α and the angle β can be expressed by the grating
equationmλ = d (sin α + sin β) ,(5.1)
where m ∈ Z is the diffraction order. The grating equation can be fulfilled for any
value of m for which− 2d < mλ < 2d.
(5.2)
Thus, multiple diffraction orders exist. [68] As only a single order is used for spatial
light modulation, the diffraction efficiency (i.e. ratio of the power in the zeroth orderto the total power) is an important parameter governing the total efficiency of the
DMD.

67

5. Variable blue detuned potential
The diffraction efficiency can be maximized by satisfying the blaze condition
mλ = 2d sin θB ,(5.3)

where θB is the tilt of the mirrors. When this condition is satisfied, the incident anddiffracted ray behave according to the law of reflection, when viewed from the facetof the mirrors [68]. This is to say thatα − θB = β − θB .

(5.4)

We use equations (5.1) and (5.3) to calculate the angle of incidence α, for which
the outgoing beam exits perpendicular to the DMD (β = 0°). This condition yields
α = 24.57°. For mirrors in the OFF state (θB = −12°), the beam exits in an angle
βOFF = −56.27°. The idle DMD behaves as a mirror with βIDLE = −α = −24.57°.
This allows us to separate the beams, letting only the light reflected by the mirrors in
the ON state pass on to the experimental setup, while light reflected by the mirrors
in the OFF and IDLE state is blocked.
The efficiency of the DMD is limited by several factors. As discussed above,multiple
diffraction orders exist, leading to a decrease in the usable power as only a single
order is used. Moreover the efficiency is limited by the array fill factor (the ratio of
the area of all individual micromirrors to the total area of the micromirror array), thetransmission of the window in front of the micromirror array as well as the reflectivityof the micromirrors. All of these factors lead to a nominal diffraction efficiency of
70 % when all mirrors are in the ON state. The power is decreased further whenswitching mirrors to the OFF-state.

Figure 5.1.: Illustration of the reflection by the mirrors of the DMD. The individuallytilted mirrors make up a blazed grating with spacing d and grating angle θB .

5.2. Optical setup
We aim to implement the DMD such that the pattern displayed on the DMD isdirectly imaged onto the atoms. We illuminate the DMD with a laser beam operating685.2. Optical setup
at a wavelength of λ = 370 nm, which is blue detuned for both Er and Dy. At 370 nm
the calculated atomic polarizability is 1200 a.u. for Er, while the polarizability of
Dy is unknown. We are planning to use a diode laser with a power of ≈ 30 mW togenerate the light at 370 nm.
The first step in designing the optical setup for imprinting the DMD pattern onto
the atoms, was choosing the diameter of the incoming beam and the demagnification.
The field of view covered by the DMD should be large allowing us to vary the size ofthe trap. We aim to have an incoming beam waist of 3.6 mm and a demagnifaction
factor of 28.82, allowing us to cover a field of view with a waist of 125 µm.
We aim to achieve the demagnification factor of 28.82 using a combination of
two telescopes, similar to the setup described by Gauthier et al. [69]. This allows
us to use commercially available lenses instead of a single telescope consisting ofa lens with f = 1875 mm and the vertical imaging objective with f = 65 mm. An
intermediate image of the pattern on the DMD with a magnification factor of 0.4
is created using a telescope consisting of a f = 250 mm and a f = 100 mm lens.
The pattern is then imaged onto the atomic plane by a telescope consisting of af = 750 mm lens and the vertical imaging objective with f = 65 mm, resulting in
the total desired demagnification factor.

5.2.1. Optimization of optical setup with Zemax OpticStudio
The optical setup needs to fulfil two criteria. On the one hand, a collimated beamthat is reflected by the DMD should result in a collimated beam on the atoms (seeFig. 5.2, beams displayed in blue), to obtain the desired demagnification of the beam.
This can be achieved by optimizing the distance between the two lenses making up
a telescope (∆T1 and ∆T2 ). On the other hand, a single point on the DMD shouldbe imaged onto a single point on the atomic plane, to achieve the highest possible
resolution. A focused beam emerging from the DMD plane should thus result into a
focused beam on the atomic plane (see Fig. 5.2, beams displayed in red). We ensure
this by optimizing the distance between the DMD and the first telescope (∆DMD ) as
well as the distance between the first and the second telescope (∆1 ) using Zemax
OpticStudio.
We simulate the optical setup, including the dichroic mirror placed in a 45°
angle (used to separate the imaging light (401 nm and 421 nm) from the dipole trap(1064 nm)) using Zemax OpticStudio. The afocal image space is used to optimize
the demagnification of the beam. We optimize the distances ∆T1 and ∆T2 such thatthe diameter of the outgoing beam at three different points around the plane of
the atoms is equal to the expected beam diameter after the demagnification. The
optimization at three different points ensures the collimation of the beam.
The imaging of a single point on the DMD onto the atomic plane was optimized
by minimizing the radial spot size at the desired focal point. Here it is important tonote that the microscope objective was not designed for a wavelength of 370 nm. The
focal point of the objective at the design wavelengths (λ = 401 nm and λ = 421 nm)
will not be at the same distance from the objective as the focal point at 370 nm. We

69

5. Variable blue detuned potential

Figure 5.2.: Illustration of the DMD projection system. The beam illustrated in blueshows the demagnification of a collimated beam by the optical setup. The red beam shows
the imaging of the DMD plane onto the atomic plane. The distances ∆ that have to be
optimized for the demagnification of a collimated beam are illustrated in blue, the distancesthat are optimized for the optimum resolution are displayed in red. The central fringe ofthe accordion lattice (i.e. the atomic plane) is shown in green. Image not to scale.

can compensate for this by having a slightly focused beam before the objective. Thus
∆1 is chosen such that the focal point at 370 nm is at the same position as the one
for the design wavelengths. This is important as it allows us to create vortices in the
focal point of the imaging light. The dichroic beam splitter results in a different focus
position for the x- and y- direction. Consequently we optimized the system such thatthe total radial size is minimized at the position of the atoms. This optimization
yields ∆DMD = 256 mm, ∆T1 = 328 mm, ∆1 = 1213 mm and ∆T2 = 806 mm.
According to the demagnification factor of the imaging system, the image of asingle mirror of the DMD, which has a width and height of 10.8 µm would have awidth/height of 347.8 nm on the atoms. This exceeds the theoretical resolution limitof the objective (587.4 nm Airy disk radius at 370 nm). Consequently the image of asingle mirror on the atomic plane approximates the point spread function (PSF) of
the optical setup. The theoretical Huygens PSF cross section of the optical setup(2 telescopes + dichroic) along the x- and y-direction is displayed in Fig. 5.3. The
calculated airy disk radius is 1.457 µm, which is increased greatly by the dichroic
beam splitter mounted in an angle of 45°. Back propagating this value through theoptical setup gives a spot size of 41.98 µm at the DMD, resulting in blocks of roughly
4x4 mirrors contributing to one single spot at the atomic plane. Thus the intensity
can be altered by turning a certain number of micromirrors contributing to the same
point to the OFF state and we can achieve ≈ 16 different intensity levels. Note thatthe point spread function only provides a lower estimate of the final resolution as it
does not take all aberrations present in the real system into account.
The planned optical setup is shown in Fig. 5.4. A half-wave plate and a PBS cubeare placed immediately after the output port of the laser, to be able to manuallycontrol the power. The beam is intensity stabilized with an AOM and mode cleaned
using a polarization maintaining single mode fiber. The intensity stabilization isachieved with a PID (for detailed description see Subsection 4.4.5). Subsequently,
the beam is collimated with a FC M40L fiber coupler from Schäfter Kirchhoff, giving
us the desired beam waist of 3.6 mm. We insert a half- and a quarter-waveplate infront of the DMD to have full control over the polarization. The beam is directed
onto the DMD in an angle of 24.57°, so that the ON beam exits perpendicular to70

5.3. Alignment strategy

Figure 5.3.: Huygens PSF along the x and y axis of the optical setup calculated with
Zemax OpticStudio.

the micromirror array. The ON beam is imaged onto the atomic plane using thedescribed optical setup.
There are several limiting factors in this optical setup. The diode laser has a powerof 30 mW. The quartz AOM used for the intensity stabilization has a diffraction
efficiency of ≈ 85 % and the single mode fiber used for mode cleaning has a coupling
efficiency of ≈ 90 %. The light is then modulated by the DMD which has a nominal
diffraction efficiency of 70 % (see Section 5.1). To enter the vertical imaging objective,
the beam is transmitted through a dichroic mirror, which is used to separate theimaging light (at 401 nm and 412 nm) from the optical dipole trap light (1064 nm).
At 370 nm, 88(1) % are transmitted through the dichroic mirror. The vertical imaging
objective has a transmission of ≈ 78(1) % and the viewport of the vacuum chamber
has a transmission of 82(1) %. As these transmissions were not specified by the
manufacturer, we determined them experimentally.

5.3. Alignment strategy
The theoretical distances are only a rough guideline for the real optical setup, owing
to fabrication tolerances. In addition to this, the correct adjustment according totheoretical distances is difficult to achieve. Consequently we planned a strategy toadjust the individual lenses relative to each other.
The DMD has to be placed in the focal point of the first lens. The optical setup
used to adjust this is illustrated in Fig. 5.5 (1). The beam is focused onto the
IDLE DMD with the first lens. As the DMD is IDLE, the reflected beam exits
perpendicular to the micromirror array and is separated from the incoming light bya beam splitter cube. When ∆DM D equates the focal length of the lens, the reflectedbeam is collimated.
A focused beam on the DMD is required to align the distance between the twotelescopes (∆2 ). Thus another lens is inserted in front of the DMD (see Fig. 5.5 (2)).
This lens is mounted on a flip mount, allowing us to direct both a collimated anda focused beam onto the DMD. To place the lens in the correct distance ∆ from71

5. Variable blue detuned potential

Figure 5.4.: Sketch of the experimental setup used to project the pattern displayed on the
DMD onto the atomic plane. The polarization of the laser light is cleaned with a PBS cube
and intensity stabilized using an AOM. It is then coupled into a single mode polarization
maintaining fiber. We use a FC M40L fiber coupler to achieve a waist of 3.6 mm after thefiber. The beam is directed onto the DMD to modulate its spatial intensity pattern and isthen imaged onto the atomic plane with two telescopes. For further information see text.

the DMD, all micromirrors are turned to the ON state. The incoming beam entersperpendicular to the DMD and is reflected under 24.57°. The distance ∆ is chosensuch that the beam is collimated after the lens.
The setup used to align ∆T1 is shown in Fig. 5.5 (3). A collimated beam is directed
onto the DMD under an angle of 24.57°. The distance ∆T1 is altered to achieve acollimated beam after the second lens. To align ∆1 , the lens inserted in step 2) isused to focus the beam onto the DMD. The distance ∆1 is chosen such that thebeam is collimated after the third lens. The third lens should be mounted on a linear
translation stage to be able to change ∆1 , so that the beam can be slightly focused
afterwards to compensate for the different focal length of the objective at 370 nm.
An optical setup that allows to change ∆1 and ∆T2 independently is vital to ensurethat a collimated incoming beam is collimated at the position of the atoms and thatthe focal point at 370 nm coincides with the focal point at the imaging wavelengths.
This could for example be done by inserting a mirror on a linear translation stage

72

5.4. Application
between the objective and the last lens to be able to alter ∆T2 .

Figure 5.5.: Experimental setups used to align the two telescopes imaging the patterndisplayed on the DMD onto the atoms.

5.4. Application
The planned optical setup can be used to create a variable blue-detuned potential,
which can provide both in-plane confinement and can be used to create vortices. The
versatility of the DMD allows the creation of vortices via both the rotating-bucket andthe optical-spoon approach (see Section 3.5). The DMD is used to selectively ’cut-out’part of the incoming blue detuned Gaussian beam, by setting the mirrors within acircle to the OFF position, setting the intensity in this region to zero, hence resulting
in a flat bottom potential. This process is illustrated in Fig. 5.6. The intensity
profiles were calculated using the code in Appendix A.3. The image displayed on
the DMD is shown in Fig. 5.6a. Here all mirrors within a circle with a radius of
1.5 mm at the centre of the DMD are turned to the OFF position. The intensity
profile of the incoming Gaussian beam is shown in Fig. 5.6b. The DMD removes
the central part of the Gaussian beam (see Fig. 5.6c), significantly reducing the
maximum intensity. Projecting this beam onto the atomic plane with the described73

5. Variable blue detuned potential
optical setup leads to a potential with a trap depth of 165 nK on the atoms (see Fig.
5.6d) . A radius of 1.5 mm on the DMD was chosen for illustrative purposes.

Figure 5.6.: Illustration of the simulation of the creation of a flat bottom trap using a
DMD. Subfigure a) shows the image displayed on the DMD, where the mirrors in the ON(OFF) position are shown in white(black). The intensity profile of the incoming gaussian
beam is shown in Subfigure b). The central part of the beam is removed by the DMD,
with the resulting intensity profile shown in c). Subfigure d) shows a central cut of thecalculated potential on the atoms.

As discussed in Section 4.1, the radial confinement of the accordion lattice (≈2π · 5 Hz) is not high enough to trap the BEC. Thus the confinement has to beprovided by an additional potential. The proposed experimental sequence is to first
evaporate in a crossed beam optical dipole trap [24]. After evaporative cooling, theaccordion lattice is turned on at the maximum lattice spacing, while the dipole trap
remains on to provide the in-plane confinement, resulting in trapping frequencies of(ωx , ωy , ωz ) = 2π · (50, 30, 422)Hz. The variable blue detuned potential is turned on
while the power of the dipole trap is decreased, loading the atoms in the blue detuned
potential with trap frequencies (ωx , ωy , ωz ) = 2π · (0.5, 5.56, 422)Hz. Afterwards theaccordion lattice is compressed, resulting in the final trap with trapping frequencies(ωx , ωy , ωz ) = 2π · (5.56, 5.56, 3898)Hz. We have written a code, allowing us tocalculate the potential created using the optical setup, for different parameters (seeAppendix A.3), considering all of the aforementioned losses.
The trap depth of the blue detuned potential is strongly dependent on the size ofthe area removed by the DMD (i.e. the trap size). This is shown in Figure 5.7. To
estimate the required size of the trap, we calculate the Thomas Fermi radius of the
BEC in the crossed beam optical dipole trap after evaporation. The trap frequencies

74

5.4. Application

Figure 5.7.: Calculated trap depth in dependence of the radius of the circular area
removed by the DMD.

in this trap are equal to (ωx , ωy , ωz ) = 2π · (50, 30, 144)Hz [24]. The Thomas-Fermi
radius rTi F in direction i = x, y, z is given by [70]rTi F =

2µ
mωi

~ω̃
with µ =2

15N as
li

 25
,

(5.5)

where µ is the chemical potential, li gives the harmonic oscillator length (see Eq. (4.2))√
and ω̃ = 3 ωx ωy ωz . For this estimate we do not take the dipole-dipole interaction into
account. In the crossed beam optical dipole trap the condensate has a Thomas-Fermi
radius of (rTx F , rTy F , rTz F ) ≈ (12, 7, 2.5)µm. Consequently we plan to use a trap radiusof 15 µm.
A trap with a radius of 15 µm on the atomic plane has a trap depth Utrap =kB 210 nK. A gas with a temperature T
T =

Utrap
kB η

with η ≈ 8(5.6)

can be trapped in a trap with a trap depth Utrap [71]. Consequently a gas with atemperature of up to ≈ 26.25 nK can be trapped in the blue detuned potential. The
critical temperature TC in an harmonic potential is given by
TC = 0.94

~ω̃ √
3
N
kB

(5.7)

where N is the number of particles. Thus, the critical temperature in the accordion
lattice at maximum lattice spacing ( (ωx , ωy , ωz ) = 2π ·(0.5, 5.56, 422)Hz)) at a typicalvalue of N ≈ 105 , is equal to TC ≈ 23 nK. Thus, a gas with a temperature below thecritical temperature, can be trapped in the blue detuned potential. For a gas witha temperature larger than the critical temperature, the gas will evaporatively cool,75

5. Variable blue detuned potential
resulting in a decrease in temperature, as well as in atom number. Consequently,
one should aim for an adiabatic transition between the crossed beam optical dipole
trap and the blue detuned potential, such that the phase space density is not alteredsignificantly and the atoms are still condensed after the transfer.
The DMD can also be used to introduce rotation into the system. The concept issimilar to that shown in Fig. 5.6, but instead of a fixed potential, we use the DMD
to create a variable potential. The DMD can be used for both the rotating bucket
and the optical spoon approach (see Section 3.5). For the rotating bucket approach
the pattern displayed on the DMD has to be slightly elliptical. The BEC is trappedin this elliptical potential and vortices are created by rotating the ellipse around its
centre.
An alternative approach is to use optical spoons (or stirring potentials), i.e. small
repulsive barriers moving through the BEC. The size of the optical spoon is limitedby the resolution limit of the optical setup, leading to a minimal radius of 1.457 µm,
predicted by our simulations. Here the proposed sequence would be to trap the
BEC in a circular trap. Subsequently, small repulsive potentials are swept throughthe BEC, creating rotational excitations. Experimentally, this can be achieved byswitching a few mirrors within the circular trap to the ON state and moving theregions of high intensity through the trap. Grey scaling allows us to vary the depthof the stirring potentials.

76

6. Conclusion and Outlook
The goal of my master thesis was to built an experimental setup for the creation ofvortices in a dipolar Bose-Einstein condensate. It consists of an optical accordion
setup to create the appropriate trapping geometry as well as a variable blue detuned
potential used to create the vortex excitations in the BEC. Both setups were designed
to be used for both Erbium and Dysprosium, the two atomic species used in the
RARE experiment at the Institute of Quantum Optics and Quantum Information(IQOQI) in Innsbruck.
The optical accordion lattice is based on the interference of two Gaussian beams
under a shallow angle. The trap frequencies can be altered by changing the angle
of interference. After a literature study of vortices in a BEC, we programmed anumerical simulation of the interference of two Gaussian beams. This simulation wasused to determine the parameters for the optical accordion such that the trapping
potential is appropriate for the creation of vortices.
We built an optical setup according to the determined specifications. The opticalsetup divides the incoming laser beam into two parallel beams, whose separation isgoverned by the height of the incoming beam. We control the height of the incoming
beam using two parallel mirrors mounted on a rotation stage. Finally, the twoparallel beams interfere in the focal point of an aspherical lens, creating the desiredinterference pattern. We characterized the temporal and dynamic stability of thissystem as well as the change in lattice spacing when altering the separation of thebeams. Furthermore we have implemented a Piezo Stack in the optical setup whichallows us to alter the phase difference of the beams. This enables us to control thephase of the resulting lattice.
We designed the optical setup used to create a variable blue detuned potential
using a DMD and simulate the resulting potential. The optical setup consists of twotelescopes to project the image displayed on the DMD onto the atomic plane. The
first telescope is formed by two standard lenses, while the second one consists of alens and the existing vertical imaging objective. This combination allows us to have ahigh demagnification without using a lens with large focal length. We optimized this
optical setup using Zemax OpticStudion and discuss a strategy to align it. As theimage of a single mirror on the atomic plane is smaller than the theoretical resolutionof the system, more than one mirror contributes to the image in a single point. This
allows grey scaling of the potential by turning on or off mirrors contributing to thepotential in one point. This setup can be used for both the rotating bucket and theoptical spoon scheme to create vortices as well as to provide in-plane confinement.
The next step is to built and characterize the experimental setup.
The setup presented in this thesis can be used to study both single and multiple

77

6. Conclusion and Outlook
vortices in a dBEC, opening the door to experimentally unexplored physics. We canstudy the change in the density profile of the vortex for different trap geometries
as well as the emergence of density ripples near the roton instability. Moreover themodulation of the interaction of two vortices by the DDI can be studied. If theaxial confinement is decreased we can, given that the horizontal imaging system isgood enough, also study the effect of the DDI on the deformation of the vortex line.
Increasing the radial confinement would give us the possibility to study the influenceof the DDI on the precision of a vortex in an inhomogeneous trap. Moreover, byremoving the cylindrical lens in the accordion lattice setup, we can increase the axial
confinement such that the motion of the BEC in this direction is frozen out leading
to a quasi-2D dBEC.

78

A. Matlab code
A.1. Numerical simulation of the accordion lattice
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34

clear allclose all% Elementary constantc = 299792458;% Speed of lighth = 6.626070040 e -34;
% Planck constanthbar = h /2/ pi ;% reduced Planck constantu0 = 4* pi *1 e -7;% Vacuum permeability
e0 = 1/ u0 /( c ^2) ;% Vacuum permittivity
qe = 1.6021766208 e -19;
% Elementary charge
G = 6.67408 e -11;
% Gravitational constantkB = 1.38064852 e -23;
% Boltzmann constantme = 9.10938356 e -31;
% Electron mass
mp = 1.672621898 e -27;
% Proton massuB = qe * hbar /2/ me ;% Bohr magnetonuN = qe * hbar /2/ mp ;% Nuclear magnetonalpha = u0 * qe ^2* c /(2* h ) ;% Finestructure constantu = 1.660539040 e -27;
% Atomic mass unit
a0 = 4* pi * e0 * hbar ^2/ me / qe ^2; % Bohr Radii
Ry = 13.605693009;% Rydberg energy in eValpha = 1/137;% Finestructure constantmEr = 166* u ;% Mass Erbium% Polarizability
Cfaktor1064 = (186/1.06) * c * u0 *1.65 e -41/2; % 5e 37
Cfaktor532 = 350* c * u0 *1.65 e -41/2; % 350 for Dy , 317 for Er
Cfaktor1570 = 150* c * u0 *1.65 e -41/2;
Cfaktor370 = -1200* c * u0 *1.65 e -41/2;
Cfaktor373 = -1558* c * u0 *1.65 e -41/2;% System parametersf = 20; % Focal length ( cm )d = 2; % Beam seperation ( cm )79

A. Matlab code
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55

% Definition Coordinate System
x = [ -150:0.2:150]*1 e -6;
y = [ -1:0.5:1]*1 e -6;z = [ -150:0.2:150]*1 e -6;[X ,Y , Z ] = ndgrid (x ,y , z ) ;% Definition of Beams
lambda = 532 e -9; % Wavelength ( m )k = 2* pi / lambda ;% Wavenumber k (1/ m )Cfaktor = Cfaktor532 ; % Polarizability
P0 = 5; % Laser Power ( W )

w1a
w1b
w2a
w2b

=
=
=
=

90 e -6; % Waist x Beam 1 ( m )90 e -6; % Waist y Beam 1 ( m )w1a ; % Waist x Beam 2 ( m )w1b ; % Waist y Beam 2 ( m )theta = atan ( d / f ) ; % Half - angle between beams 1 & 2
E10 = sqrt (2* P0 /( pi * w1a * w1b * c * e0 ) ) ; % electric field
amplitude of beam 1 at origin at time 056 E20 = E10 ; % electric field amplitude of beam 2 at origin
at time 0
57
58 % Rotation of Beam 1 around y axis in an angle of theta
59 Xn = X .* cos ( theta ) + Z .* sin ( theta ) ;60 Yn = Y ;61 Zn = -X .* sin ( theta ) + Z .* cos ( theta ) ;62
63 % Electric field of Beam 1
64 w10 = sqrt ( w1a * w1b ) ; % minimum of waist radius ( m )65 z1 = pi * w10 ^2/ lambda ; % Rayleigh length ( m )66 z1a = pi * w1a ^2/ lambda ; % Rayleigh length in x - direction ( m)67 z1b = pi * w1b ^2/ lambda ; % Rayleigh length in y - direction ( m)
68 w1z = w10 * sqrt (1+( Xn / z1 ) .^2) ; % Beam waist for all z
values69 w1az = w1a * sqrt (1+( Xn / z1a ) .^2) ; % Beam waist in x direction for all z values
70 w1bz = w1b * sqrt (1+( Xn / z1b ) .^2) ; % Beam waist in y direction for all z values
71 R1 = Xn .*(1+( z1 ./ Xn ) .^2) ; % Radius of curvature of the80

A.1. Numerical simulation of the accordion lattice72
73

beam ' s wavefronts
R1a = Xn .*(1+( z1a ./ Xn ) .^2) ; % Radius of curvature of thebeam ' s wavefronts in x - direction
R1b = Xn .*(1+( z1b ./ Xn ) .^2) ; % Radius of curvature of thebeam ' s wavefronts in y - direction
R1 ( isnan ( R1 ) ) = inf ; % Sets R1 to infinity in case of Xn =0
R1a ( isnan ( R1a ) ) = inf ;R1b ( isnan ( R1b ) ) = inf ;xsi = atan ( Xn ./ z1 ) ; % Gouy - Phase

74
75
76
77
78
79
80 E1 = E10 * w10 ./ w1z .* exp ( -( Zn .^2./( w1az .^2) + Yn .^2./( w1bz.^2) ) ) .* exp ( -1 i * k *( Zn .^2./(2* R1a ) + Yn .^2./(2* R1b ) ) ) .*exp ( -1 i *( k * Xn - xsi ) ) ; %E - field beam 1
81
82 % Rotation of Beam 2 around y axis in an angle of - theta
83 theta = - theta ;84 Xn = X .* cos ( theta ) + Z .* sin ( theta ) ;85 Yn = Y ;86 Zn = -X .* sin ( theta ) + Z .* cos ( theta ) ;87
88 % Electric field of Beam 2
89 w20 = sqrt ( w2a * w2b ) ; % minimum of waist radius ( m )90 z2 = pi * w20 ^2/ lambda ; % Rayleigh length ( m )91 z2a = pi * w2a ^2/ lambda ; % Rayleigh length in x - direction ( m)
92 z2b = pi * w2b ^2/ lambda ; % Rayleigh length in y - direction ( m)
93 w2z = w20 * sqrt (1+( Xn / z2 ) .^2) ; % Beam waist for all z
values
94 w2az = w2a * sqrt (1+( Xn / z2a ) .^2) ; % Beam waist in x direction for all z values
95 w2bz = w2b * sqrt (1+( Xn / z2b ) .^2) ; % Beam waist in y direction for all z values
96 R2 = Xn .*(1+( z2 ./ Xn ) .^2) ; % Radius of curvature of thebeam ' s wavefronts
97 R2a = Xn .*(1+( z2a ./ Xn ) .^2) ; % Radius of curvature of thebeam ' s wavefronts in x - direction
98 R2b = Xn .*(1+( z2b ./ Xn ) .^2) ; % Radius of curvature of thebeam ' s wavefronts in y - direction
99 R2 ( isnan ( R2 ) ) = inf ;% Sets R1 to infinity in case of
Xn =0
100 R2a ( isnan ( R2a ) ) = inf ;101 R2b ( isnan ( R2b ) ) = inf ;81

A. Matlab code
102 xsi = atan ( Xn ./ z2 ) ; % Gouy - Phase
103
104
105 E2 = E20 * w20 ./ w2z .* exp ( -( Zn .^2./( w2az .^2) + Yn .^2./( w2bz.^2) ) ) .* exp ( -1 i * k *( Zn .^2./(2* R2a ) + Yn .^2) ./(2* R2b ) ) .*exp ( -1 i *( k * Xn - xsi ) ) ; %E - field beam 2
106
107 % Intensity of the two beams
108 I =( c * e0 * abs ( E1 + E2 ) .^2) /2; % Combined intensity of the twointersecting Gaussian beams
109
110 Ix = Cfaktor * squeeze ( I (: , floor ( size (I ,2) /2) +1 , floor ( size (I ,3) /2) +1) ) ; % Intensity in x direction
111 Iy = Cfaktor * squeeze ( I ( floor ( size (I ,1) /2) +1 ,: , floor ( size (I ,3) /2) +1) ) ; % Intensity in y direction
112 Iz = Cfaktor * squeeze ( I ( floor ( size (I ,1) /2) +1 , floor ( size (I,2) /2) +1 ,:) ) ; % Intensity in z direction
113
114 Idx = diff ( Ix ) / mean ( diff ( x ) ) ; % Numerical first derivativeof the intensity in x direction
115 kx = diff ( Idx ) / mean ( diff ( x ) ) ;% Numerical second
derivative of the intensity in x direction
116 wx = sqrt ( - kx ( z ==0) / mEr ) /(2* pi ) ; % Trap frequency in x direction
117
118 Idy = diff ( Iy ) / mean ( diff ( y ) ) ; % Numerical first derivativeof the intensity in y direction
119 ky = diff ( Idy ) / mean ( diff ( y ) ) ;% Numerical second
derivative of the intensity in y direction
120 wy = sqrt ( - ky ( y ==0) / mEr ) /(2* pi ) ; % Trap frequency in y direction
121
122 Idz = diff ( Iz ) / mean ( diff ( z ) ) ; % Numerical first derivativeof the intensity in z direction
123 kz = diff ( Idz ) / mean ( diff ( z ) ) ;% Numerical second
derivative of the intensity in z direction
124 wz = sqrt ( - kz ( x ==0) / mEr ) /(2* pi ) ; % Trap frequency in z direction82

A.2. Calculation of beam displacement by two rotating parallel mirrors

A.2. Calculation of beam displacement by tworotating parallel mirrors
1
2
3
4
5
6
7
8

clear allclose alliter1 =1;d = 3.5; % distance of mirrors M1 and M2
l1 = 2.54; % length of mirror 1
l2 = 3.5; % length of mirror 2
p1 =[ -10; -2.5]; % starting point of vector describing thebeam
9 p2 =[10; -2.5]; % end point of vector describing the beam
10
11 % Define start - and end - points of M1 and M2 :12 S1a = [ - l1 /2 -1.25; -d /2]; % startpoint M1
13 S1e = [ l1 /2 -1.25; -d /2]; % endpoint M1
14 S2a = [ - l2 /2+1.25; d /2]; % startpoint M2
15 S2e = [ l2 /2+1.25; d /2];
% endpoint M2
16
17 for theta = [40 60];
18
19 rot = [ cosd ( theta ) - sind ( theta ) ; sind ( theta ) cosd ( theta )]; % define rotation matrix
20
21 % rotating Mirrors around origin by theta
22 S1ar = rot * S1a ;23 S2ar = rot * S2a ;24 S1er = rot * S1e ;25 S2er = rot * S2e ;26
27 % Define vectors describing the mirrors
28 t1l = linspace ( -0.5 ,0.5 ,1000) ;29 b1 = 0.5*( S1ar + S1er ) ;30 a1 = ( S1ar - S1er ) ;31 S1 =@( t1l ) b1 + t1l .* a1 ;32 Mirror1 = S1 ( t1l ) ;33
34
35 t2l = linspace ( -0.5 ,0.5 ,1000) ;36 b2 = 0.5*( S2ar + S2er ) ;37 a2 = ( S2ar - S2er ) ;38 S2 =@( t1l ) b2 + t2l .* a2 ;83

A. Matlab code
39
40
41
42
43
44
45
46
47
48
49

Mirror2 = S2 ( t2l ) ;% Define vector describing the beam :xL = linspace ( -20 ,20 ,1000) ;bL = 0.5*( p1 + p2 ) ;aL = ( p1 - p2 ) ;g =@( xL ) bL + xL .* aL ;% Angle between laser beam and mirror 1 (= angle between x- axis and Mirror 1)
50 CosTheta1 = dot (( p2 - p1 ) ,( S1er - S1ar ) ) /( norm (( p2 - p1 ) ) * norm(( S1er - S1ar ) ) ) ;51 theta1 = acosd ( CosTheta1 ) ;52
53 % Angle between laser beam and mirror 2 (= angle between x- axis and Mirror 2)
54
55 CosTheta2 = dot (( p2 - p1 ) ,( S2er - S2ar ) ) /( norm (( p2 - p1 ) ) * norm(( S2er - S2ar ) ) ) ;56 theta2 = acosd ( CosTheta2 ) ;57
58 xL1 = linspace (0 ,20 ,1000) ;59
60 % Find intersection point of mirror 1 and beam
61 [ xp , yp ]= findintersection ([ S1ar (1) , S1ar (2) ; S1er (1) , S1er (2)] ,[ p1 (1) , p1 (2) ; p2 (1) , p2 (2) ]) ;62
63 % Check if the beam hits mirror 1
64
if
S1ar (1) < xp && xp < S1er (1) && S1ar (2) < yp &&yp < S1er (2)65disp ( ' Laser beam hits mirror 1 ' ) ;
66
else
67
disp ( ' Laser beam misses mirror 1 ' ) ;68
end
69 % Find intersection point of mirror 2 and beam reflectedby mirror 1
70 [ xp1 , yp1 ]= findintersection ([ S2ar (1) , S2ar (2) ; S2er (1) , S2er(2) ] ,[ xp , yp ; xp + cosd ( theta1 + theta1 ) , yp + sind ( theta1 +theta1 ) ]) ;71
72 % Check if the beam hits mirror 2
73
if S2ar (1) < xp1 && xp1 < S2er (1) && S2ar (2) < yp184

A.2. Calculation of beam displacement by two rotating parallel mirrors&& yp1 < S2er (2)disp ( ' Laser beam hits mirror 2 ' ) ;else
disp ( ' Laser beam misses mirror 2 ' ) ;end% Save height of beam and theta to A
A ( iter1 ,:) = [ theta , yp1 ];iter1 = iter1 +1;
end

74
75
76
77
78
79
80
81
82
83 % Calculate distance between the beams for the two angles.
84 distance = A (1 ,2) -A (2 ,2) ;85

A. Matlab code

A.3. Calculation of trapdepth created by DMD
1
2
3
4
5
6
7
8
9
10
11
12
13
14

close allclear all% Constantsc = 299792458; % Lichtgeschwindigkeit
u = 1.660538921 e -27;
% Atomic mass unitm = 166* u ;uB = 9.27400915 e -24; % Bohr Magneton
hbar = 1.05457 e -34;
kb = 1.3806503 e -23; % kBoltzmann
g = 9.80553; % gravitation
a0 = 0.5291772083 e -10; % Bohr ' s radius in [ m ]mu0 = 4* pi *1 e -7;% magnetic constant
Cfaktor370 = 1200* c * mu0 *1.65 e -41/2; % polarizability of Er
at 370 nm
15 Cfaktor401 = 1696* c * mu0 *1.65 e -41/2; % polarizability of DY
at 401 nm
16
17 % Variables
18 P = 30 e -3;% Input power in W
19 magnificationfactor = 0.0347;
20 wmm = 3.6;
% Waist of gaussian beam on DMD
in mm
21 innerradiusdmd = 1.5;
% Radius of circle cut with DMD
in mm
22 Cfaktor = Cfaktor370 ;% Polarizability
23
24
25 % DMD specifications
26 height = 1080;% Height of the micromirror arrayin pixel
27 width = 1920;% Length of the micromirror arrayin pixel
28 lengthpixel = 20.736/ width ;29
30 % Beam on DMD
31 Z = zeros ( width , height ) ;32 [X , Y ] = meshgrid (1: width ,1: height ) ;33 w = wmm / lengthpixel ;% Waist ofGaussian beam in pixels
34 Imax = 2* P /( pi * wmm ^2) *10^2;% Maximum86A.3. Calculation of trapdepth created by DMD
intensity of the Gaussian beam in W / cm ^2
35 R = sqrt (( X - width /2) .^2+( Y - height /2) .^2) ;% Radius inpixel
36 Z = Imax .* exp ( -(2* R .^2./( w .^2) ) ) ;% Spatial
intensity distribution of gaussian beam
37
38
39 % Beam on atoms
40 R1 = sqrt (( X - width /2) .^2+( Y - height /2) .^2) *magnificationfactor * lengthpixel ; % Radius of thecircle cut out by the DMD in px reduced bymagnificationfactor
41 w1 = wmm * magnificationfactor ;% Waistof gaussian beam in mm reduced by magnificationfactor
42 Iatom_max = 2* P *0.82*0.7/( pi * w1 ^2) ;% Maximum
intensity of Beam considering losses due to viewport
and DMD ( W / mm ^2)
43 Trapdepth = Cfaktor *2* Iatom_max *1 e15 / kb ;% Trapdepth
corresponding to the maximum intensity in nK
44 innerradius = innerradiusdmd * magnificationfactor ;% Radius of the circle cut
out by the DMD on the atoms
45
46 Zmean = Trapdepth .* exp ( -(2* R1 .^2./( w1 .^2) ) ) ;% Potential on theatoms due to Gaussian beam
47 Zmean ( R1 < innerradius ) = 0;%
Setting the intensity of the area cut out by the DMD
to zero48
49 % Smoothing the potential
50 [ xf , yf ] = ndgrid ([ -150:150] ,[ -150:150]) ;51 radblur = 1e -6/(( lengthpixel * magnificationfactor ) *1 e -3) ;52 filtmat = exp ( -( xf / radblur ) .^2 -( yf / radblur ) .^2) ;53 filtmat = filtmat /( sum ( sum ( filtmat ) ) ) ;54 Zmean2 = filter2 ( filtmat , Zmean , ' same ' ) ;55
56 figure (2)57 ellipse = surf ( X * lengthpixel * magnificationfactor *1000 , Y *lengthpixel * magnificationfactor *1000 , Zmean , ' Edgecolor '87

A. Matlab code
, ' none ' ) ;
58
59
60
61
62
63

hcb = colorbar ;hcb . Label . String = ' Potential / kB ( nK ) ' ;set ( hcb , ' fontsize ' ,12)
xlim ([ min ( min ( X * lengthpixel * magnificationfactor *1000) )max ( max ( X * lengthpixel * magnificationfactor *1000) ) ]) ;ylim ([ min ( min ( Y * lengthpixel * magnificationfactor *1000) )max ( max ( Y * lengthpixel * magnificationfactor *1000) ) ]) ;xlabl = xlabel ( ' x (\ mum ) ' ) ;ylabl = ylabel ( ' y (\ mum ) ' ) ;zlabl = zlabel ( ' Potential / k_B ( nK ) ' ) ;ylabl . FontName = ' Myriad Pro ' ;xlabl . FontName = ' Myriad Pro ' ;zlabl . FontName = ' Myriad Pro ' ;set ( xlabl , ' fontsize ' ,30)
set ( ylabl , ' fontsize ' ,30)
set ( zlabl , ' fontsize ' ,30)
ax = gca ;ax . FontName = ' Myriad Pro ' ;set ( ax , ' fontsize ' ,26)

64
65
66
67
68
69
70
71
72
73
74
75
76
77 figure (3)78
79 cut = Zmean2 ( height /2 ,:) ;80 k = plot ( X * lengthpixel * magnificationfactor *1000 , cut , ' b- ' , ' LineWidth ' , 3 ) ;81
82 xlabl = xlabel ( ' x (\ mum ) ' ) ;83 ylabl = ylabel ( ' Potential / k_B ( nK ) ' ) ;84 ax = gca ;85 set ( ax , ' fontsize ' ,26)
86 set ( xlabl , ' fontsize ' ,30)
87 set ( ylabl , ' fontsize ' ,30)
88 xlim = ([0 90]) ;89
90
91 disp ([ ' Input power : ' , num2str ( P *1000) , ' mW ' ]) ;92 disp ([ ' Maximum intensity on DMD : ' num2str ( Imax *1000) , 'mW / cm ^2 ' ]) ;93 disp ([ ' Beam Waist on Atoms : ' , num2str ( w1 *1000) , ' um ' ]) ;94 disp ([ ' Radius of circle cut by DMD : ' , num2str (innerradius *1000) , ' um ' ]) ;95 disp ([ ' Trapdepth : ' , num2str ( max ( max ( Zmean2 ) ) ) , ' nK ' ]) ;88

Bibliography[1] Satyendra Bose. Plancks Gesetz und Lichtquantenhypothese. Zeitschrift für
Physik, 26(1):178–181, 1924.
[2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor.
Science, 269(5221):198–201, 1995.
[3] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Bose-Einstein Condensation in a Gas of Sodium
Atoms. Phys. Rev. Lett., 75:3969–3973, 1995.
[4] S. Jochim. Bose-Einstein Condensation of Molecules. PhD thesis, University of
Innsbruck, 2004.
[5] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz. Bose–Einstein condensationof photons in an optical microcavity. Nature, 468, 2010.
[6] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of BoseEinstein Condensation in an Atomic Gas with Attractive Interactions. Phys.
Rev. Lett., 75:1687–1690, 1995.
[7] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio. Bose-Einstein Condensation of Potassium Atoms by Sympathetic Cooling.
Science, 294(5545):1320–1322, 2001.
[8] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm. Bose-Einstein
Condensation of Cesium. Science, 299(5604):232–235, 2003.
[9] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr. Bose-Einstein Condensationof Alkaline Earth Atoms: 40 Ca. Phys. Rev. Lett., 103:130401, 2009.
[10] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck. Bose-Einstein
Condensation of Strontium. Phys. Rev. Lett., 103:200401, 2009.
[11] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev. Strongly Dipolar Bose-Einstein
Condensate of Dysprosium. Phys. Rev. Lett., 107:190401, 2011.
[12] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino.
Bose-Einstein Condensation of Erbium. Phys. Rev. Lett., 108:210401, 2012.

89

Bibliography[13] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura,
T. Yabuzaki, and Y. Takahashi. Spin-Singlet Bose-Einstein Condensation of
Two-Electron Atoms. Phys. Rev. Lett., 91:040404, 2003.
[14] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau. Bose-Einstein
Condensation of Chromium. Phys. Rev. Lett., 94:160401, 2005.
[15] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I.
Westbrook, and A. Aspect. A Bose-Einstein Condensate of Metastable Atoms.
Science, 292(5516):461–464, 2001.
[16] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner,and T. J. Greytak. Bose-Einstein Condensation of Atomic Hydrogen. Phys. Rev.
Lett., 81:3811–3814, 1998.
[17] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel,
S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye. A High Phase-Space-Density
Gas of Polar Molecules. Science, 322(5899):231–235, 2008.
[18] W. R. Anderson, J. R. Veale, and T. F. Gallagher. Resonant Dipole-Dipole
Energy Transfer in a Nearly Frozen Rydberg Gas. Phys. Rev. Lett., 80:249–252,
1998.
[19] Pyotr Kapitsa. Viscosity of Liquid Helium below the λ-Point. Nature, 1938.
[20] John F Allen. Flow of Liquid Helium II . Nature, 1938.
[21] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman,and E. A. Cornell. Vortices in a Bose-Einstein Condensate. Phys. Rev. Lett.,
83:2498–2501, 1999.
[22] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle. Vortices and superfluidity in a strongly interacting Fermi gas. Nature,
435:1047 – 1051, 2005.
[23] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and Dalibard J. Berezinskii–kosterlitz–thouless crossover in a trapped atomic gas. Nature, 441:1118–1121,
2006.
[24] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M. Sohmen, M. J. Mark,and F. Ferlaino. Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms.
Phys. Rev. Lett., 121:213601, 2018.
[25] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. Feshbach resonances inultracold gases. Rev. Mod. Phys., 82:1225–1286, 2010.
[26] B. R. Judd and I. Lindgren. Theory of zeeman effect in the ground multipletsof rare-earth atoms. Phys. Rev., 122:1802–1812, 1961.

90

Bibliography[27] J. G. Conway and B. G. Wybourne. Low-lying energy levels of lanthanide atoms
and intermediate coupling. Phys. Rev., 130:2325–2332, 1963.
[28] Albert Frisch. Dipolar Quantum Gases of Erbium. PhD thesis, University of
Innsbruck, 2014.
[29] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau. The physics ofdipolar bosonic quantum gases. Rep. Prog. Phys., 72(12):126401, 2009.
[30] E. P. Gross. Structure of a quantized vortex in boson systems. Il Nuovo Cimento(1955-1965), 20(3):454–477, 1961.
[31] L. P. Pitaevskii. Vortex lines in an imperfect Bose gas. Soviet Physics Jetp,
1961.
[32] T. Langen and M. J. Mark. Ultrakalt magnetisiert. Physik Journal, 2018.
[33] R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov. Optical Dipole traps for
Neutral Atoms. Adv. At., Mol., Opt. Phys., 42:95–170, 1999.
[34] H. Li, J.-F. Wyart, O. Dulieu, and M. Lepers. Anisotropic optical trapping as amanifestation of the complex electronic structure of ultracold lanthanide atoms:
The example of holmium. Phys. Rev. A, 95:062508, 2017.
[35] J. H. Becher, S. Baier, K. Aikawa, M. Lepers, J.-F. Wyart, O. Dulieu, and
F. Ferlaino. Anisotropic polarizability of Erbium atoms. Phys. Rev. A, 97:012509,
2018.
[36] M. Lepers, J.-F. Wyart, and O. Dulieu. Anisotropic optical trapping of ultracold
erbium atoms. Phys. Rev. A, 89:022505, 2014.
[37] H. Li, J.-F. Wyart, O. Dulieu, and M. Lepers. Optical trapping of ultracold
dysprosium atoms: transition probabilities, dynamic dipole polarizabilities andvan der Waals C 6 coefficients. J. Phys. B: At. Mol. Opt. Phys, 50:014005, 2017.
[38] L. Landau. Theory of the Superfluidity of Helium II. Phys. Rev., 60:356–358,
1941.
[39] Pitaevskii L. and Stringari S. Bose Einstein Condensation and Superfluidity.
Oxford University Press, 2016.
[40] N. Bogolubov. On the theory of Superfluidity. Journal of Physics, 1947.
[41] D. G. Henshaw and A. D. B. Woods. Modes of atomic motions in liquid heliumby inelastic scattering of neutrons. Phys. Rev., 121:1266–1274, Mar 1961.
[42] L. Chomaz, van Bijnen R. M. W., D. Petter, G. Faraoni, S. Baier, J. H. Becher,
M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino. Observation of roton mode
population in a dipolar quantum gas. Nature Physics, 14:442–446, 2018.

91

Bibliography[43] Ryan M. Wilson, Shai Ronen, and John L. Bohn. Critical superfluid velocity ina trapped dipolar gas. Phys. Rev. Lett., 104:094501, Mar 2010.
[44] N. Verhelst and J. Tempère. Vortex Structures in Ultra-Cold Atomic Gases.
2017.
[45] E. Lundh, C. J. Pethick, and H. Smith. Zero-temperature properties of a trapped
Bose-condensed gas: Beyond the Thomas-Fermi approximation. Phys. Rev. A,
55:2126–2131, 1997.
[46] D. H. J. O’Dell and C. Eberlein. Vortex in a trapped Bose-Einstein condensate
with dipole-dipole interactions. Phys. Rev. A, 75:013604, 2007.
[47] Yoshihisa Yamamoto. Bose-Einstein Condensation and Matter-Wave Lasers.
Japan Science and Technology Agency, 2018.
[48] Martin A., Marchant N.G., O’Dell D. H.J., and Parker N. G. Vortices andVortex lattices in quantum ferrofluids. J. Phys.: Condens. Matter, 2017.
[49] B. C. Mulkerin, D. H. J. O'Dell, A. M. Martin, and N. G. Parker. Vortices inthe two-dimensional dipolar Bose gas. J. Phys. Conf. Ser., 497:012025, 2014.
[50] R. M. Wilson, S. Ronen, J. L. Bohn, and H. Pu. Manifestations of the Roton
Mode in Dipolar Bose-Einstein Condensates. Phys. Rev. Lett., 100:245302, 2008.
[51] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and
F. Ferlaino. Long-lived and transient supersolid behaviors in dipolar quantum
gases. Phys. Rev. X, 9:021012, Apr 2019.
[52] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N.
Bisset, L. Santos, and G. Modugno. Observation of a dipolar quantum gas withmetastable supersolid properties. Phys. Rev. Lett., 122:130405, Apr 2019.
[53] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, andT. Pfau. Transient supersolid properties in an array of dipolar quantum droplets.
Phys. Rev. X, 9:011051, Mar 2019.
[54] B. C. Mulkerin, R. M. W. van Bijnen, D. H. J. O’Dell, A. M. Martin, andN. G. Parker. Anisotropic and long-range vortex interactions in two-dimensional
dipolar bose gases. Phys. Rev. Lett., 111:170402, 2013.
[55] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard. Vortex Formation
in a Stirred Bose-Einstein Condensate. Phys. Rev. Lett., 84:806–809, 2000.
[56] G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. A. Baker, T. A. Bell,H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely. Giant vortex clusters ina two-dimensional quantum fluid. Science, 364(6447):1264–1267, 2019.

92

Bibliography[57] S. J. Rooney, P. B. Blakie, B. P. Anderson, and A. S. Bradley. Suppression of
Kelvon-induced decay of quantized vortices in oblate Bose-Einstein condensates.
Phys. Rev. A, 84:023637, 2011.
[58] M. Klawunn, R. Nath, P. Pedri, and L. Santos. Transverse Instability of Straight
Vortex Lines in Dipolar Bose-Einstein Condensates. Phys. Rev. Lett., 100, 2008.
[59] B. P. Anderson, P. C. Haljan, C. E. Wieman, and E. A. Cornell. Vortex
precession in bose-einstein condensates: Observations with filled and empty
cores. Phys. Rev. Lett., 85:2857–2860, Oct 2000.
[60] D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin, and D. S. Hall. RealTime Dynamics of Single Vortex Lines and Vortex Dipoles in a Bose-Einstein
Condensate. Science, 329(5996):1182–1185, 2010.
[61] A. L. Fetter and A. A. Svidzinsky. Vortices in a trapped dilute bose-einsteincondensate. J.Phys.: Condens. Matter, 2001.
[62] D. Rychtarik, B. Engeser, H.-C. Nägerl, and R. Grimm. Two-Dimensional BoseEinstein Condensate in an Optical Surface Trap. Phys. Rev. Lett., 92:173003,
2004.
[63] P. Cladé, C. Ryu, A. Ramanathan, K. Helmerson, and W. D. Phillips. Observation of a 2D Bose Gas: From Thermal to Quasicondensate to Superfluid. Phys.
Rev. Lett., 102:170401, 2009.
[64] T. C. Li and H. Kelkar and D. Medellin and M. G. Raizen. Real-time controlof the periodicity of a standing wave: an optical accordion. Opt. Express,
16(8):5465–5470, 2008.
[65] J. L. Ville, T. Bienaimé, R. Saint-Jalm, L. Corman, M. Aidelsburger, L. Chomaz,
K. Kleinlein, D. Perconte, S. Nascimbène, J. Dalibard, and J. Beugnon. Loading
and compression of a single two-dimensional Bose gas in an optical accordion.
Phys. Rev. A, 95:013632, 2017.
[66] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, S. Tzanova, E. Kirilov,
and R. Grimm. Accurate Determination of the Dynamical Polarizability of
Dysprosium. Phys. Rev. Lett., 120:223001, 2018.
[67] X. Zhang, Y. Chen, J. Fang, T. Wang, J. Li, and L. Luo. Beam pointing
stabilization of an acousto-optic modulator with thermal control. Opt. Express,
27(8):11503–11509, 2019.
[68] C. Palmer and Loewen E. Diffraction Grating Handbook. Newport Cooperation,
2005.

93

Bibliography[69] G. Gauthier, I. Lenton, N. McKay Parry, M. Baker, M. J. Davis, H. RubinszteinDunlop, and T. W. Neely. Direct imaging of a digital-micromirror device for
configurable microscopic optical potentials. Optica, 3(10):1136–1143, 2016.
[70] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari. Theoryof bose-einstein condensation in trapped gases. Rev. Mod. Phys., 71:463–512,
Apr 1999.
[71] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven. Kinetic theory of theevaporative cooling of a trapped gas. Phys. Rev. A, 53:381–389, Jan 1996.

94