Study of a dipolar gas confined in a
one-dimensional lattice
andconstruction of a new imaging
setup for the ERBIUM experiment.
– MASTER THESIS –byLouis Lafforgue, BSc
submitted to the Faculty of Mathematics, Computer Science,
and Physics of the University of Innsbruckin partial fulfillment of the requirements
for the degree of Master of science (MSc)
advisor:Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInnsbruck, July 2022

Abstract
This thesis describes the work performed during my master thesis in the ERBIUM experiment. The first part explains the steps toward the observation of Bloch oscillations
with a dipolar gas of 166 Er. We studied Bloch oscillations with ultracold dipolar erbium
atoms confined in a one-dimensional lattice. We performed measurement at different
scattering lengths, from the contact-dominated regime to the dipolar-dominated one.
By combined theoretical and experimental efforts, we emphasize the effect of quantum
fluctuations in the dephasing time of Bloch oscillations. In the regime of small meanfield interactions, we observed that the dipole-dipole interactions play an importantrole in determining the number of occupations of the lattice sites. Special emphasis
is given to describe the experimental realization and the method used to analyse theraw data.
The second part of this thesis describes, characterizes, and details the implementationof a sub-micron resolution imaging system. Imaging systems are essential components
of ultracold gases experiments, and achieving a high-resolution imaging system enables
to probe in-situ spatial density distribution. We performed tests on an objective
produced by Special Optics. We detail the way to obtain the point-spread function
and found an experimental resolution of 0.85(2) µm. We designed a mechanical holder
including all degrees of freedom needed to align the objective with the atoms. Finally,
we highlight a very flexible optical path using four different optical wavelengths thatwill enable us to use the objective not merely to image the density distribution but
also to apply arbitrary potentials to the atoms.

iContents
Abstracti

1. Introduction
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3

2. Properties of erbium
2.1. Erbium at glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Electron configuration and energy level structure . . . . . . . . . . . .
2.3. Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Interactions between erbium atoms . . . . . . . . . . . . . . . . . . . .

5
5
6
6
7

3. Theory: quantum gas in a one-dimensional lattice11
3.1. One-dimensional optical lattices . . . . . . . . . . . . . . . . . . . . . . 11
3.2. Non-interacting quantum gas in a one-dimensional lattice . . . . . . . . 13
3.3. Bloch oscillations with a Bose-Einstein condensate . . . . . . . . . . . . 16
3.4. Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4. Observation of Bloch oscillations and ground state properties in a lattice 23
4.1. Producing a Bose-Einstein condensate of erbium . . . . . . . . . . . . . 23
4.2. Preparation of the Bose-Einstein condensate in a 1D-lattice . . . . . . . 24
4.3. Observation of Bloch oscillations . . . . . . . . . . . . . . . . . . . . . 25
4.4. Analysing the interaction-induced dephasing . . . . . . . . . . . . . . . 27
4.5. Ground state: localization in a single lattice plane . . . . . . . . . . . . 31
4.6. Publication: Strongly dipolar gases in a one-dimensional lattice: Blochoscillations and matter-wave localization . . . . . . . . . . . . . . . . . 35
5. New imaging setup for the Erbium experiment49
5.1. Motivation and challenges . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2. Objective design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3. Testing the performances of the objective . . . . . . . . . . . . . . . . . 57
5.4. Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . 63
6. Conclusion
69
6.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2. Further understanding of dipolar BEC in optical lattices . . . . . . . . 70iii

6.3. Next step for the implementation of the objective . . . . . . . . . . . .

70

A. Code analyse PSF
71
A.1. main.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2. create ROI.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.3. PSF 1D.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.4. resolution.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.5. norma.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B. Technical documents of imaging setup

77

Bibliography81

Acknowledgments

89

iv

Chapter1

Introduction

1.1. Motivation
The beginning of the 20th century saw the emergence of quantum mechanics to describe
the microscopic world. One core concept of wave-particle duality was proposed by
Louis de-Broglie [dB25], stating that a microscopic object, such that electrons or
atoms, can behave like a wave or a particle at the same, depending on the experimental
environment. Quantum mechanics appears to be the key to describe atoms, subatomic
particles and light-matter interaction, and succefully describe, e.g., the photo electriceffect or the electron structure of atoms.
Nevertheless, one question remained: how to describe macroscopic objects like a metal,
composed of 1023 atoms and electrons, all interacting with each others? The intrinsic mathematical construction of quantum mechanics causes classical computer to bevery inefficient to simulate those systems, and systems with large number of particles
are impossible to solve exactly even with supercomputers. In 1982, Richard Feynman
proposed to use an analogue quantum system where it is possible to isolate, control
and detect quantum states to simulate a more complex system [Fey82]. The conceptof a quantum simulator was born. In the past decades, many platforms, capable ofperforming qunatum simulation, have been developed, from trapped ions to superconducting circuits and photonics systems [Tra12].
In 1995, remarkable progress in the field of experimental quantum physics, quantum
optics, and lasers enabled the first realization of a Bose-Einstein condensate (BEC)[Dav95] [And95] [Bra95]. Eric A. Cornell, Carl E. Wieman and Wolfgang Ketterle
were rewarded for this breakthrough with the Nobel price in 2001 [Kru01]. A BEC
is a macroscopic metastable state of matter formed when a gas of bosons condenses
to the lowest state of energy. A pure BEC is zero entropy and then fully governedby quantum physics. The phase coherence guarantees that a BEC can be describedby a single wave function. The early studies of this new state of matter were focused
12

1. Introduction

on macroscopic quantum phenomena, such as matter-wave interference [And97] orvortices [Mat99].
But rapidly, ultracold gases have revealed themselves to meet all the requirementsfor quantum simulations [Blo12]. The system is very well isolated from its environment, it is possible to control the interactions, e.g., via a magnetic Feshbach resonance[Chi10], and direct imaging techniques have been developed, pushing the resolution toa single atom [Bak09]. Furthermore, early theoretical [Jak98] and experimental works[Hem93] proposed to create crystals of light to mimic the periodic structure of solids,
so-called optical lattices. They enabled the first experimental of the Bose-Hubbard
model [Hub63] and the first observation of the Mott insulator phase [Gre02].
At the early stage of BEC, most of the experiments were working with alkali atoms
and only contact interaction was present in the systems. The last decade marked aturning point with the production of the first BEC of lanthnides atoms, dysprosium[Lu11] and erbium [Aik12]. Those magnetic atoms add a new interaction to the ultracold toolbox, the dipole-dipole interaction [Lah09] [Bar08]. The long-range andanisotropy characters allow accessing new quantum phases, e.g., macrodroplets or supersolid [Cho22]. Furthermore, magnetic atoms combined with optical lattices form aparadigmatic platform for the study of strongly correlated systems [Bar12] [Bai16].
Until now, most of the theory of ultracold gases uses the framework of mean field approximation. More recently, beyond mean-field contributions have been the key to theunderstanding of stabilization mechanisms in dipolar gas, e.g with the macrodroplet[Cho16]. Ref. [Lim12] showed that beyond mean-field contributions have to be taken
into account with the so-called Lee-Huang-Yang (LHY) term. If in the theory, beyond
mean field effects are now well understood and their important role undisputed, theexact mathematical expression is still subject to intense debate, and a quantitative
experimental measure of its exact form is still missing.

1.2. Thesis outline

3

1.2. Thesis outline
This thesis is divided in two parts essential on ultracold experiments, controlling andimaging. The first part deals with the study of a dipolar gas of erbium confined ina one-dimensional lattice. In particular on the use of Bloch oscillations (BOs) as aninterferometric tool to asses the role of beyond mean-field contributions. The second
part relates the implementation of a new high-resolution imaging system for erbium.

Chapter 2 reviews the relevant properties of erbium used during this thesis.

Chapter 3 is devoted to the physics of ultracold gases loaded in a 1-dimensional lattice. We review in detail the emergence of the band structure and BOs in presenceof an externalforce. We present the effect of interactions on BOs and introduce thetheoretical model used for this project.

Chapter 4 is the heart of this project and addresses the experimental procedure toobserve BOs with a dipolar gas of erbium as well as the method to analysis to row
data.

Chapter 5 presents the new imaging setup which will be implemented in the ERBIUM
experiment. In the first section, we briefly review the diffraction theory to understandhow imaging systems are described. In the second section, we present the opticaldesign and the test realized on the new objective. We provide to the reader details on
the numerical analysis of the point spread function. The last section deals with theimplementation of the objective in the experiment as well as the planification of thelaser distribution.

Chapter2

Properties of erbium

2.1. Erbium at glance
Erbium(Er) is a rare-earth element belonging to the lanthanide family. In its solid
form, it is a soft and silvery metal. It is used in the industry mix with other material
to increase the workability of an alloy, for example in reducing the stiffness of thematerial [Gup05]. Erbium has a high melting point at 1529° and a boiling point at
2900°. Its mostly known form, Er3+ , finds applications in doped fiber and glass where
it is used, for instance, to amplify laser light [Bec99].
Er has an atomic number of Z=68 and presents five bosonic isotopes and one fermionicisotope. Tab. 2.1 gives the number of nucleons, the abundance of each isotope, andthe statistic associated.

Isotopes

Natural
dance

162

0.14
1.61
33.6
23.0
26.8
15.0

Er
Er
166
Er
167
Er
168
Er
170
Er
164

Abun-

Statistics
Boson
Boson
Boson
Fermion
Boson
Boson

Table 2.1.: six stable isotopes of erbium, five bosonic and one fermionic. For each isotope,
the abundance is reported. Table extracted from [M11a]

All the work of this thesis has been realized with 166 Er , presenting advantageous
scattering properties and a substantial abundance.
56

2. Properties of erbium

2.2. Electron configuration and energy level structure
The electronic shell of erbium atom is composed of 68 electrons, filling the atomic
orbitals following the Madelung rule. Among these electrons, 14 are valence electrons,
giving rise to the electronic configuration in the ground state,[Xe]4f 12 6s2 ,2.1

where [Xe] refers to the electron configuration of Xenon.
For neutral erbium in the ground state, the 6s shell is completely filled before the 4f .
This partially filled inner shell 4f gives a structure called submerged-shell. The twovacant electrons of the 4f shell results in large orbital anisotropy and a large orbitaland spin quantum number.
The submerged-shell structure also gives rise to a very rich energy spectrum in the
optical range. Indeed, the electrons of the 4f shell can be excited easily to the unfilled
positions within the optical transition range. [Kra18] reports 672 transitions from theground state with J ranging from 2 to 12. Figure 2.1 reports all the energy levels aswell as the transitions used in our experiment; see Chap. 4 or [Fri14a].
The broad 401 nm transition in combination with the one at 583 nm makes erbium
suitable to be laser cooled. More details are presented in Chap. 4. Furthermore, it
offers transitions with narrow linewidth, particularly advantageous for coherent lightmanipulation.

2.3. Magnetic moment
Another important property of erbium is the large magnetic moment µ in the groundstate. It is a direct consequence of the submerged-shell structure, as the electron spinorbit coupling leads to a large total angular momentum J. The magnetic quantum
number mj can range from −J to +J and is related to the magnetic moment asfollow:µ = mj µB gJ ,2.2
where µB is the Bohr magneton and gJ the landé factor. In the case of only spin-orbit
coupling gJ can be written:
gJ = 1 + (1 − gS )J(J + 1) + S(S + 1) − L(L + 1),
2J(J + 1)

2.3

with gS ≈ 2.002. L and S are respectively the orbital momentum number and thetotal spin number. For erbium, jj-cpoupling between electrons has to be taken into

2.4. Interactions between erbium atoms

7

104
2.5

2

1.5

1

0.5

0

Figure 2.1.: Energy level spectrum of erbium. Energy levels are represented in the relevant
range for trapping and cooling. Even and odd parity are shown in blue and red, respectively.
401 nm is used for the Zeeman slower, the transversal cooling and imaging. The narrow
line at 583 nm corresponds to the magneto-optical trap transition. The transition at 1299nm
features an ultra-narrow linewidth of 2 Hz, making it suitable for coherent spin manipulation.
The light at 1064 nm and 532 nm is used to create far detuned optical potentials [Gri00]

consideration and one finds gJ =1.163801(1) [Con63]. One finds that for 166 Er in its
ground state (mj = −6), the magnetic moment is,µ = −6.982806(6)µB .

2.4

We can notice that when the spin state is changing, the magnetic moment changes
accordingly.

2.4. Interactions between erbium atoms
2.4.1. Contact interaction
Despite the very dilute character of ultracold gases, atoms can interact with each other.
In quantum mechanics, the interaction between particles is covered by the scattering
theory. When two atoms are close to each other, they interact by the so-called Van
der Waals (vdW) interaction. This interaction can be described in the first order by8

2. Properties of erbium

1
0.8
0.6
0.4
0.2
0

0.5

1

1.5

2

2.5

3

150
120
90
60
30
0
3.5

Figure 2.2.: Feshbach resonance around 1.9 G. Green circles: Atom number versusmagnetic field. Blue diamond: scattering length obtained by cross-thermalization. Black triangle: Scattering length obtained by lattice modulation spectroscopy. Black solid line:fir ofas considering 5 resonances. Figure adapted from [Cho16].

a potential with the form,
C6
,
2.5
r6
with C6 is the vdW coefficient and r is the relative distance between the atoms. Thispotential, responsible for the contact interaction at very low temperature, is shortrange and isotropic. For low temperature (i.e. low collision energy) scattered by ashort-range scatterer cannot be resolved the exact structure of the potential since the
De Broglie wavelength is very long. The scattering of two particles is fully characterized
by the s-wave scattering length as . The interaction potential can be rewritten,
UV dW (r) =U (r) =4πℏ2 asδ(r) = gδ(r),m

2.6

with m the mass.
Addtionnaly, when two atoms collide, a twol-atom level can be resonnantly coupled
to a dimer level, this phenomenon is called Feshbach resonance [Chi10]. It is observed
experimentally by a sudden loss of atoms. More precisely, this happens when thebound energy of a molecular potential coincides with the bound energy of the scattering
potential. The difference in energy can be controlled by an external magnetic field,
resulting in a modification of the scattering length as . Typically, this dependence canbe discribed in term of simple formula, which for isolated resonances reads,
as (B) = abg (1 −∆),B − B02.7

With abg the background scattering length, ∆ the width of the resonance and B0 the
position of the Feshbach resonance.
Due to the orbital anisotropy making the contact scattering chaotics, lanthanides
atoms and in particular 166 Er exhibit many Feshbach resonances [Fri14b]. In presence2.4. Interactions between erbium atoms

9of many resonances, the depedance of as in B is more complex,as (B) = aBg

Y
(1 −
i

∆i
),
B − Bi2.8

with i labelling a resonance.
During this thesis, we use a comparatively broad Feshbach resonance around 1.9 G;
see Fig. 2.2. This allows to tune the scattering length from 55 a0 to 100 a0 without
suffering from too much atom loss.

2.4.2. Dipole-dipole interaction
The strong magnetic moment of erbium induces a dipole-dipole interaction (DDI)
between the atoms. With an external polarizing magnetic field, this interaction canbe described by a potential with the folowing mathematical expression,µ0 µ2 1 − 3 cos2 (θ)
,
2.9
Udd =
4π
r3
with µ0 the vacuum permeability, µ the magnetic moment, r the relative distance
between the two atoms and θ the angle between the magnetic field direction anddirection of r. The notations are resumed in Fig. 2.3. The two striking propertiesof the dipole-dipole potential are the long-range character in 3 dimensions and theanisotropy.
Furthermore, by utilizing an external field, it is possible to go from an attractive
potential in the head-to-tail configuration (θ = 0) , to a repulsive interaction when the
dipoles are aligned side-by-side (θ = π/2).
As the DDI is anysotropic, the many-body properties of an ensemble of erbium atoms
are drastically affected by the shape of the trapping potential. For an elongated trapwith the weak axis aligned with B, the dipoles are mostly head-to-tail and the system
is attractive. On the other hand, for a ”pancake” trap, where the strong confinement
is align with B, the dipoles sit side-by-side and the system is mainly repulsive.
To quantify the absolute strength of dipole-dipole interaction, it is usual to define thedipolar length,
add =
2

0µ
with Cdd = µ4π
.

Cdd
,
12πℏ2

2.10

10

2. Properties of erbium

θ

r

𝑩

Figure 2.3.: Schematic representation of dipole-dipole interaction. r is the distance between
the two atoms and θ the angle with the magnetic field direction. The external magnetic field
B is used to control the orientation of the dipoles.

The dipolar length can be directly compared with the s-wave scattering length and we
introduce the ratio
ϵdd =

add
as

2.11

When ϵdd << 1, the contact interaction is much stronger and DDI doesn’t play asignificant role in the properties of the system. On the other hand, for ϵdd > 1, DDI
dominates and exotic behaviours appear. Tuning the s-wave scattering length throughFeshbach resonance makes it possible to study the crossover at ϵdd ≈ 1. For 166 Er the
dipolar length is fixed at add = 66.5 a0 and with the Feshbach resonance in Fig. 2.2,
ϵdd can be varied from approximately 1.2 to 0.7.

Chapter

3

Theory: quantum gas in a
one-dimensional lattice

Already before the creation of the first Bose-Einstein condensate, physicists proposed
to engineer periodic potentials and study the emerging properties with a system ofultracold atoms [Hem93]. Loading a BEC in a periodic potential represents a very
robust system to tackle solid-state and condensed matter physics problems [Blo05].
Indeed, those systems feature a very high degree of controllability, in particular, thepotential landscape and the interactions between particles.
For instance, ultracold gases permitted the observations of Bloch oscillations (BO).
BO were originally predicted by Felix Bloch in the framework of solid-state physics[Blo29], they describe the movement of a non-interacting electron confined in a periodic potential and subject to an external force. Because of the very high oscillation
frequencies and the difficulty to image the electron motion, physicists had to wait for
the arrival of BEC to observe the first signs of BO [Niu96] [Wil96].
In this Chapter, we review the theoretical tools necessary to understand how BO arise
with a BEC and the equations used to describe BO with a gas of 166 Er .

3.1. One-dimensional optical lattices→−
In presence of an optical field (typically laser beams) described by E (ω, r), with ω
the frequency, the atoms experience two forces, the radiation pressure, dissipative
and used to cool, and the dipole potential. While the radiation pressure is related
to light scattering and is therefore a dissipative process, used for laser cooling, thedipole potential is created by the energy shift of the atomic state in the light field, theso-called AC Stark shift [Gri00].
11

12

3. Theory: quantum gas in a one-dimensional lattice

When induced by far-detuned light, the latter creates an external potential for theatoms. For alkali atoms, the potential created scales as follows,→−
V (r) ∝ Re(α(ω))| E |2

3.12

→
−
where α(ω) is a complex number called the polarizability and | E |2 the intensity of the
field at a position r. However, due to the orbital anisotropy erbium has a tensorial
polarizability ᾱ. The dipolar force depends on the polarization of the light and advancetheoritical methods are needed to predict ᾱ [Li17] [Bec18]. For more details about
dipolar potential in general, the reader can refer to [Gri00].
To engineer a stationary periodic potential, so-called optical lattice, we can counterpropagate two laser beams with the same frequency and create a standing wave. The
strength of the potential, or potential depth, can be controlled by the power of thelasers. In the ERBIUM experiment, the laser beams are propagating along the gravity
axis z, which results in a periodic potential along z, see Fig. 3.1.

g
𝜔, 𝑘

𝑉

𝜔, 𝑘

λ ⁄2

𝑧
Figure 3.1.: 1D optical potential created by 2 counter-propagating beams with the same
frequency. The atoms will experience a periodic potential Vlatt with period λ/2. ω is thefrequency and k the wave vector

The potential created has the mathematical expression,
Vlatt (z) = sErec sin2 (

2πz
).
λ

3.13

Here, we express the potential depth as a multiple s of the recoil energy Erec =ℏ2 k 2 /(2m), with k = 2π/λ, the recoil momentum, or the momentum gain after absorbing one photon and m the mass of the particle. λ is the wavelength of the lattice.
The minima of potential are often called lattice sites, they are separated by a distanced = λ/2.
In the experiments, this creates an array of two-dimensional BECs. Additionaly, thelaser beams used to create the standing waves have a transverse gaussian profile,3.2. Non-interacting quantum gas in a one-dimensional lattice13

creating harmonic confinement along the transverse directions x and y. It is very
weak in comparison with standard dipole traps (only a few hertz) but has to be taken
into consideration during the numerical simulations.

3.2. Non-interacting quantum gas in a one-dimensionallattice
3.2.1. Single particle in an optical lattice
In order to understand the physics of a BEC loaded into an optical lattice, we firsthave look at the effect of a periodic potential on the wavefunction of a single particle.
In the most simple case, we ignore the two transversal directions, which is supposed
to give only a small correction. For non-interacting systems, the ground-state andthe energy associated can be calculated by solving the 1D stationnary Schrödinger
equation,

 2 2
−ℏ d
+ Vlatt (z) ψn (z) = En ψn (z).
3.14
2m dz 2
En and ψn (z) correspond respectively to the eigenenergies and eigenfunctions thathave to be determined.
To find the ground state, we apply the Bloch band theory [Blo29], developed in thefield of solid-state physics and described in detail in [Asc76] and [Kit05]. In presence ofa periodic potential, the Bloch theorem states that we can find solutions of Eq. (3.14)
with the form,
ψn,q (z) = eiqz un (z, q).
3.15
Here, q is a continuous variable called quasimomentum and un (z, q) an unknown periodic function with the same periodicity as the external potential. Thanks to theperiodicity, we can without loss of generality restrict q ∈ [−k, k], called the Brillouinzone(BZ).
As un (z, q) is a periodic function, we can further decompose it in a sum of planewaves,X
un (z, q) =cn,j e−iqjz ,3.16
jwith the complex coefficients cn,j . This corresponds to the Fourier transform.
By inserting Eq. (3.15) and Eq. (3.16) in Eq. (3.14), we can derive the following discrete

14

3. Theory: quantum gas in a one-dimensional lattice

equation,
h
q
si
s
En
(2j + )2 +cn,j − (cn,j−1 + cn,j+1 ) =
.
k
2
4
Erec

3.17

This is a basic eigenvalue problem which can be solved numerically for an arbitraryq.
For any value of q, we retrieve the different energy levels En (q) and the coefficient cj
of the Bloch function associated Ψn,q . For a given n, En (q) is a continuous function
and is often refered as a band. Figure 3.2 shows En (q) for n = 1 − 3 ans s=1,3 and 8.
10

12

14

8

10

12

8

10

6

8

4

6

2

4

6
4
2
0
-1

0

1

0
-1

0

1

2
-1

0

1

Figure 3.2.: Energy bands from 3 different values of s, from left to right, s = 1, s = 3 ands = 8. The first 3 bands are represented with different colours, in blue E0 (q), in yellow E1 (q)and in red E2 (q). the gap between the different bands increases with the potential depth.

The dispersion relation En (q) is an essential characteristic of particles confined in aperiodic potential. For instance, in solid-state physics, this band structure explains
the conduction of electrons in a metal.
The Bloch functions form a complete orthogonal set for Eq. (3.14) and are completely
delocalized over the lattice sites. For a deep lattice, it is more convenient to introduce
the Wannier-functions as a basis [Wan37]. They have the property to be localised ata single lattice site and are mathematically defined as the inverse Fourier transform of
the Bloch functions,wn (z − zj ) ∝Z ℏkdqe−iqzj ψn,q (z),−ℏkwhere zj corresponds to the position of the j th lattice site.

3.18

3.2. Non-interacting quantum gas in a one-dimensional lattice15

The concept of localize particle in a lattice site leads to the interpretation of a delocalized state as a state with particles tunnelling between neighbouring lattice sites at
a rate J,Z ∞wn (z − zj )Ĥwn (z − zj+1 )dz.
3.19
J =−
−∞

When the lattice is deep enough(Vlatt > 5Erec ), we can considere the lattice as an arrayof of harmonic potential and use the so-called tight biding approximation. We thenfind (ref. [Mor06]),√4
J = √ Erec s3/4 exp(−2 s)/h.
π

3.20

3.2.2. Bloch oscillations
When confined in a periodic potential, a particle subject to an external Force will
start to oscillate, this phenomenon is known as Bloch oscillations. The period of theoscillations can be calculated by a simple semi-classical approach that we will detail.
In presence of an external force F , the time dependence of the quasimomentum canbe written,
q(t) = q0 + F t/ℏ,

3.21

with q0 the initial quasimomentum. Due to the k periodicity of the BZ, after a time,
TBO = 2kℏ/F,

3.22

the states are again equivalent and the wave functions are the same, besides a global
phase factor.
A more rigorous approach to the BO is the so-called Wannier-stark ladders. We will
now discuss briefly this approach which will be useful in presence of interactions. As
seen in the previous section, it is possible to express the wave function as a superposition of Wannier states.
√ X
ψ(z, t) = N
cj (t) w(z − zj ) ,3.23
jwith cj (t) complex coefficients and w(z − zj ) a function localized around each latticesite labelled by zj = jd, with d = λ/2, the distance between each lattice site. Whenthe tunnelling rate J between neighbouring lattice site is small enough in comparison
with the external force, the on-site occupation number ∥cj ∥2 can be approximatedconstant and we can write cj (t) = cj (0)e−iϕj (t) , where ϕj (t) corresponds to a phase.

16

3. Theory: quantum gas in a one-dimensional lattice

All the physics is encapsulated in the phase difference between the Wannier functions.
In presence of gravity, the phase evolution is governed by the Schrödinger equation
with an additional term accounting for the gravity. The Hamiltonian associated is,p̂2+ Vlatt (z) + mgz,
3.24
2m
With g the gravity acceleration and m the mass of the particle. Solving the Schrödinger
equation gives us,Ĥ =ϕj (t) = (q − mgt)jd/ℏ.

3.25

We see that gravity has the effect to translate with time the wave function in thequasimomentum space, and due to the periodicity of the quasimomentum, the wave
function will undergo oscillations in momentum space. When the wave function reaches
one edge of the BZ, it reappears at the other edge by a sudden jump. This sawtooth
behaviour, in momentum space, is characteristic of BO.
The first observation of BO with ultracold gases has been done in 1996, in the groupof Christoph Salomon [BD96] and Mark Raizen [Niu96] [Wil96]. Nevertheless, theyobserved damping of the oscillations, which is not included in Eq. (3.25) and prevented
the of BO for precesion measurement. As we will see below, this behaviour is due tothe actual interactions between particles in a real BEC.
In presence of an external force Fext , a particle initially in the lowest band has aprobability to jump to a higher band. The jump probability between E0 and E1 isgiven by [Mor06],
s2 Erec π 2
,
3.26
d 32
with d = λ/2 , the distance between 2 lattice sites. If this probability is too high,the atoms will get excited and lost. For typical ultracold experiments with s > 1 andatoms subject to gravity, Pt << 1 and the excitation to a higher band can be omitted
for the rest of the thesis.
Pt = e−Ft /Fext , Ft =3.3. Bloch oscillations with a Bose-Einstein condensate
3.3.1. Gross-Pitaevskii equation
A BEC is a gas of bosons in the same quantum states. Therefore, it is possible todescribe them by a single complex wave function Ψ(r, t), product of the individual
wave functions ψi (r, t),YΨ(r, t) =ψi (r, t).
3.27
i

3.3. Bloch oscillations with a Bose-Einstein condensate

17Here, r is the 3D position vector, r = (r1 , r2 , r3 ) = (x, y, z). This wave function
obeys the so-called Gross-Pitaevskii equation(GPE) Eq. (3.28), derived independently
by Gross and Pitaevskii in 1961. The GPE makes use of the Hartree-Fock and pseudopotential approximation to describe the ground state of a system of identical bosons.
In the case of only contact interacting particles, the time-dependant GPE reads,
iℏ

h ℏ2
i
∂
Ψ(r, t) =∇2 + Vext (r) + g|Ψ(r, t)|2 Ψ(r, t).
∂t
2m

3.28

The first term of the right part accounts for the kinetic energy, Vext (r) corresponds2
to the external potential and g = 4πℏm as the contact interaction, with the m mass of
166
Er and as the s-wave scattering length. Eq. (3.28) represents the Schrödinger equation counterpart for many-body bosons and all the results presented in Sec. 3.2.1 canbe derived for an assembly of bosons. We consider them unchanged if the interaction
term g = 0. However, as we will see, neglecting the interaction is in most cases a toostrong approximation to describe Bloch oscillations with a BEC.

3.3.2. Interaction-induced dephasing
As we saw in Sec. 3.2.2, in absence of interactions between particles, the phase evolution
will only be governed by the external potential, i.e gravity, and Bloch oscillations will
go on forever. But, when the atoms interact with each other, a dephasing between thelattice sites occurs, resulting in a broadening of the momentum cloud and a limited
observation time of the BO. This phenomenon can be formally understood with the
Wannier-stark Ladder presented before. When the atoms interact, an additional term
needs to be added to the phase Eq. (3.25) and we now have,
ϕj = (q − F t)jd/ℏ + gµj t,

3.29

dϕj
= F jd/ℏ + gµj .
dt

3.30

R
with µj = g|Ψ|4 , the chemical potential at lattice site j. As the phase depends now
on the local chemical potential, the different Wannier components dephase from eachother. This is clear when considering the phase velocity,

While without interactions the phase velocity between neighbouring lattice sites is aconstant, with interactions it depends on the local chemical potential. From 3.30, we
can expect the minimum of dephasing arises when the variance of the phase velocity
is the minimum, which corresponds to minimising g.
Groups of Florence managed to overtake this dephasing limitation with non-interacting
fermions [Roa04] in 2004 and with bosonic 88 Sr [Fer06] in 2006. They used BO as aninterferometric tool to measure gravity acceleration. 88 Sr presents an s-wave scattering
length substantially close to 0, allowing the observation of up to 5000 oscillations.

18

3. Theory: quantum gas in a one-dimensional lattice

Following the proposal in Ref. [Car05], they could deduce the value of g with a precisionof 6 × 10−6 .
However, this sensitivity to interactions allows to use BO as an interferometric tool.
Aditionnaly with BEC, Feshbach resonance gives the ability to control the contact
interaction, which, together with BO, to probe interactions effects. By tunning thescattering length with cesium atoms, in Ref. [Gus08] they could study the transition
from non-interacting to interacting BEC. They observed up to 20000 oscillations andassociate this minimum of dephasing with the zero-crossing of the Feshbach resonance.
Furthermore, they observed a revival of the BO after a certain time, indicating thatthe broadening and the apparent loss of oscillations is a purely dephasing effect.

3.3.3. Dipole-dipole interaction
It is possible to imagine that the attractive character of DDI can be used to compensate
for the dephasing induced by the repulsive contact interactions. But, due to the longrange property, the DDI contributes to both on-site and intersite. Although the on-site
attractive DDI can fully compensate for the on-site contact interaction and shift theminimum of dephasing to a higher scattering length, the intersite DDI becomes thelimiting factor to the BO. The DDI effect has been first highlighted in Ref. [Fat08],with a weakly dipolar gas of 39 K (µB = 0.96). This experiment points for the first timeto the effect of intersite DDI in optical lattices and addresses the problem of stabilityof a dipolar gas in a lattice, further studied in [M1̈1b].
From a naive point of view, we can imagine that when the attractive DDI is stronger
than the total contributions, from the trap and the contact interaction, the BEC will
shrink in size and after a certain time collapse. However, it has been shown recently
that strongly dipolar gas can remain stable and form self-bound ensembles of ultracold
atoms, so-called macro droplets [Sch16] [Cho16]. BO in this regime has not yet beenstudied and calls upon numerical methods.

3.4. Numerical model
To compare our experimental data, we developed a quasi 1-dimensional numerical
scheme. The following paragraph details the equations used to describe our BEC and
how we include dynamical effects created by gravity.

3.4. Numerical model

19

3.4.1. Extended Gross-Pitaevskii equation
Even though the GPE equation has been proven to be a very powerful tool to accurately describe the contact interacting BEC, when considering dipolar 166 Er , one has
to add some terms taking into account the dipolar nature of 166 Er . The full extended
Gross-Pitaevskii equation (eGPE) used in this work reads,

iℏ

h
ℏ2 2
∂
Ψ(r, t) = −∇ + Vharm (r)∂t2m+ Vlatt (z) − Fext z + g|Ψ(r, t)|2Z
+ d3 r′ Udd (r − r′ )|Ψ(r′ , t)|2i
+ γQF |Ψ(r, t)|3 Ψ(r, t) .

3.31

Vext (r) has been separated into 3Pdifferent contributions, the 3-dimensional harmonic1
external potential, Vharm (r) =i=1,2,3 2 ri ωi , the potential created by the lattice,
Vlatt (z) (Eq. (3.13)), and the potential created by the gravity Fext = mg. The third line
accounts for the long-range DDI, with Udd (r) = 3ℏ2 add /m (1 − 3 cos2 θ) /|r|3 , where θis the angle between the polarization axis (y-axis) and r. The last line is the secondorder correction to the mean-field approximation,
Lee-Huang-Yang term.
q the so-called

a3s
32
3 2
In the local density approximation, γQF = 3 g π 1 + 2 εdd [Lim11].

3.4.2. Our 1-D model
Due to the non-linear behaviour of the eGPE (Eq. 3.31) one has to use numerical methods. Even with a very powerful computer, solving Eq. (3.31) in its more general form
is hardly achievable. In this context, we developed a quasi-1-dimensional approachbased on Ref. [Bla20], allowing us to probe the dynamic and scan a large range ofparameters at a reduced cost.
We start by reducing the wave function to one dimension (1D) by applying theansatz,1222
Ψ(r, t) = Φ(x, y, l, η)ψ(z, t) ≡ √ e−(ηx +y /η)/2l ψ(z, t) ,
πl

3.32

where l and η are variational parameters corresponding respectively to the width andthe anisotropy of the radial wave function. We now insert the above ansatz into the(3.31) and obtain a quasi 1-dimension eGPE.

20

3. Theory: quantum gas in a one-dimensional lattice

To solve numerically this equation, we need to discretize it over the lattice sites. For
that, we decompose the longitudinal wave function in Wannier functions over every
lattice sites.
√ X
cj (t) w(z − zj ) ,3.33
ψ(z, t) = N
jwith the complex amplitudes cj , and the positions of lattice minima zj = (λ/2)j.
When the lattice is deep enough, it is possible to approximate the Wannier function√−1/4 −z 2 /2l22
−1
4
latt , where we defined l
by Gaussian of the form w(z) = (πllatt)elatt = (k s) .
Those manipulations lead us to the discrete equationiℏ∂cj= − J(cj+1 + cj−1 )∂tn+ − Fext zj + Vharm (z) + g 1D N |cj |2oX
dd2
1D 3/2
3+N
U|j−k| |ck | + γQF N γQF |cj | cj ,

3.34

k

Here, J is the tunnelling rate between two neighbouring lattice sites and the reduced
1D
parameters, γQF= 23/2 /(5π 3/2 l2 llatt )3/2 γQF and g 1D = g/((2π)3/2 l2 llatt ). The DDI in
1D can be rewritten,

dd
U|j−k|(l, η) =Znd3⃗x |Ψ0 (⃗x − z|j−k| êz , l, η)|2Zod3⃗x′ Udd (⃗x − ⃗x′ )|Ψ0 (⃗x′ , l, η)|2 ,3.35

with Ψ0 (⃗x′ , l, η) = Φ(x, y, l, η)w(z)The computation of the wave function Ψ(r, t) is done as followed . First, Eq. (3.34) is
evolved in imaginary time without the gravity potential to find the ground state together with the minimization of the energy functional. After each step of the imaginary
time evolution, l and η are found by a variational approach and the new parameters
are used to run the next iteration of imaginary-time evolution. Secondly, the groundstate wave-function ψ(z, t) computed is evolved in real-time with Eq. (3.34). Typicalresults from the simulations are presented in Fig. 3.3.

0.8

0

0.6
-2
2

0.4
0

0.2

-2
0.5

1.5

2.5

3.5

4.5

normalized density ( 10-3 )

1

2

0

Figure 3.3.: Real time evolution of ∥ψ(q, t)∥2 for two different scattering lengths. The
upper (resp. lower) plot corresponds to 60 a0 (resp. 83 a0 ) with a total atom number of20000. In both cases observe a saw tooth behaviour. At 83 a0 , ∥ψ(q, t)∥2 is broadened due
to interactions. The parameters of those simulations are ωx,y,z = 2π × (240(3), 30(3), 217(1))
Hz, s = 8 and with the dipoles align along the weak axis of the trap.

21

Chapter4

Observation of Blochoscillations and groundstate properties in a lattice

Given the essential theory to understand the phenomenon of BO in dipolar gases, we
detail, in this Chapter, the experimental aspects to observe BO with a dipolar gas of
166
Er .

4.1. Producing a Bose-Einstein condensate of erbium
The ERBIUM experiment is a fully operating set up to study ultracold gases of srbium.
Designed and built from 2010 to 2012, it aims at producing BEC of bosonic 168 Er
isotopes [Fri12] and degenerate Fermi gas [Aik14].
The main elements are, the high temperature oven, the Zeeman slower (ZS), highvacuum chamber, narrow line magneto-optical trap (MOT), the optical dipole trap
(ODT) and the absorption imaging. We will now resume the process to reach a dipolar
BEC of erbium. For an extensive description, the reader can refer to [Fri14a].
The high temperature oven is composed of 2 cells, operating at respectively 1100 °C and1200 °C. Those temperatures are high enough to create an atomic flux with a velocityof 450 m s−1 . The first steps to slow down the atoms are the transversale coolingand the ZS, both operating on the broad transition at 401 nm. After the ZS, theatoms have a velocity of 5 m s−1 and are slow enough to be captured in the MOT. The
MOT uses the narrow transition at 583 nm, corresponding to a doppler temperatureof 4.6 µK. The MOT stage prepares aroud 2 × 107 atoms, in their lowest zeeman
spin states, at a temperature of 10 µK. To further reach the quantum degeneracy and
BEC regime, a last step is necessary. Subsequently to MOT cooling, the atoms are23

24

4. Observation of Bloch oscillations and ground state properties in a latticeloaded in the ODT. By lowering the trapping potential progressively, it is possible toperform evaporation cooling and reach BEC regime [Ket96]. An optimized sequence
takes approximately 17 s and produce 105 atoms of 166 Er with ≈ 60% of BEC fraction.
During the evaporation, the contact interaction is set to as =90 a0 by an external
magnetic field B. The orientation of B polarizes the dipoles along z. After obtaining
the BEC we change the magnetic field along y and go to the final scattering length
value in 50 ms.
To image the atomic cloud, we perform absorption imaging. Absorption imaging is astandard technique that consists in shining resonant light and recording the ”shadow”
created by the atoms on a charge-coupled device (CCD) [Smi11]. In the Erbium
experiment, we use 401 nm resonant light. By comparing two pictures, one with atoms
and one without, we can deduce the intensity absorbed and map it to an atomic density.
The total atom number is obtained by summing the atomic density over all the pixelsof the CCD. If we want to extract the number of atoms condensed, we utilize a double
gauss fit to separate the thermal part of the condensed part.
Due to the high density, BEC can only be imaged after ballistic expansion, refers astime-of-flight (TOF). This also allows to directly access the BEC in the momentum
space. In the Erbium experiment, we use a TOF of 30 ms. In addition, after 15 ms wechange the direction of B toward the camera axis, this modifies the polarizability andincreases the scattering of incoming photons.

4.2. Preparation of the Bose-Einstein condensate in a
1D-lattice
We adiabatically load the BEC in the vertical lattice by exponentially ramping up for
20 ms the lattice beams. The lattice is created by a retro-reflected 1064 nm laser beamwith a waist of 30 µm, resulting in a spacing of 532 nm between the lattice sites.For
166
Er in the ground state and a lineary polarized 1064 nm light we have a polarizability
Re(α) = 176 a.u. (atomic units) [Bec18], corresponding to a potential depth of Vlatt =527 kHz W−1 . The beam is provided by an high power and narrow line-width MOPA
laser, also used for the horizontal dipole trap. With the available power, we can reacha lattice depth of 80 Erec . With 166 Er , we estimated Erec = h × 10.5 kHz. For our
experiment, we fixed the lattice depth to 8 Erec . This corresponds to a tunnelling rate
(Eq. 3.20) of J = h × 33 Hz. In the remaining part, the tunnelling can be omitted
because of the small value compared to the gravity. The resulting system is presented
in Fig. 4.1(a).
When all the laser beams are switched off, the clouds confined in each lattice site
will start to expand and interfere. This gives rise to matter-wave interference pat-4.3. Observation of Bloch oscillations

25

B

Figure 4.1.: (a) Schematic representation of the final system. A 1D optical lattice is loadedwith an Erbium BEC, confined in a dipole trap with frequency from an optical dipole trapwith trapping frequencies ωx,y,z = 2π × (240(3), 30(3), 217(1)) Hz. Gravity acts along z. The
dipoles are polarized along y by an external magnetic field. (b) absorption imaging after
30 ms TOF for as =58.8 a0 . The two interference peaks sits at a distance ℏk from the centralpeak.

terns, represented in Fig. 4.1(b). In absence of interactions during the expansion, themomentum distribution directly represents the Fourier transform of the initial wave
function in the lattice. The first-order diffraction appears at a distance of ℏk from thecentral peak. The occupation of the diffracted peak gives information on lattice depth
and the extension of the ground state. It can be used to calibrate the lattice depth,with for instance the Kapitza-Dirac method [Gad09].

4.3. Observation of Bloch oscillations
To initialize the Bloch oscillations, we switch-off abruptly the dipole trap and let thesystem evolve for a defined time th . The entire sequence is resumed in Fig. 4.2. thelattice beam creates a residual lateral confinement along x and y of 4 Hz. However,
our system presents an angle of 9(1) degrees between the gravity axis and the lattice,which is too important for the atom to be laterally confined.
We then hold for a variable time th and perform TOF imaging. We take 4 to 6
pictures for many th , separated by 0.1 ms time step. Figure. 4.3 shows typical pictures
obtained for as =83 a0 and 58.8 a0 . We observed the sawtooth behaviour, a featureof BO. When the contact interaction is small (58.8 a0 ), no dephasing occurs and the26

4. Observation of Bloch oscillations and ground state properties in a latticeTime

Figure 4.2.: Graphic representation of the parameters of the experiment as a function oftime. In grey, the lateral harmonic confinement. The lattice induces a latteral confinementof 4 Hz, remaining during the BO. In blue, the optical lattice depth. In red, the verticalconfinement and in green the scattering length. th is a variable time and TOF corresponds
to 30 ms.

width of the momentum distribution remains the same. On contrary, with strong
contact interactions (83 a0 ), the oscillation dephases, indicated by the broadening of
the momentum distribution.
To analyse the BO more quantitatively, we start by integrating the density profile
along x to obtain a 1D momentum distribution that we named for the following n(qz ).
First, we looked at the position of the maximum of n(qz ), qmax as a function of theholding time th , represented in Fig. 4.4 for as =58.8 a0 and 83 a0 . Due to the angle
between the gravity and the lattice, the atoms slide on the side and eventually fallout of the lattice potential and this limits our observation time to 12 ms. The pictures
are recentered by fitting a 2nd order polynomial function to qmax and removing theoffset.
To ensure that our observations are indeed BO, we fit the maximum positions qmax (th )at as =58.8 a0 with the sawtooth function as done in Ref. [Roa04],
f (t) = −1 + 2mod(

t−O+ C, 1),T

4.36

where O, T and C are the fitting parameters. T directly corresponds to the period of
the oscillation and yields T = TBO = 0.469(4) ms, in agreement with Eq. 3.22 yielding

4.4. Analysing the interaction-induced dephasing

27

Time
Figure 4.3.: Absorption imaging after TOF for different holding times in the lattice, for
as =58.8 a0 (a) and 83 a0 (b)theoryTBO= 0.469 ms.

In the contact-dominated case (83 a0 , Fig. 4.4(a)), the amplitude of the oscillations ofqmax decays rapidly and no oscillation is observed after a few periods. We further tried
to include a decay in Eq. (4.36) but the fit didn’t converge to acceptable results as too
many fit variables are needed.
This first analysis already points out the clear effect of interactions in the dephasing of
BO. However, a more robust method is needed to have a quantitative understanding.

4.4. Analysing the interaction-induced dephasing
One of the main challenges of this thesis was to find and implement a method toquantify the dephasing of BO as a function of as . As seen in Chap. 3, when BO
dephases, the width of the momentum distribution increases. By studying the width,
we could systematically quantify the dephasing of both theoretical and experimental
data. This method relies on finding the right observable for the width. In this section,
we review step by step, how the analysis of the dephasing was done and the main
results obtained. We show example for two as .

28

4. Observation of Bloch oscillations and ground state properties in a lattice

q max (hk)

1
0.5
0
-0.5
-1
0

2

4

6

0

2

4

6

8

10

21

23

25

8

10

21

23

25

q max (hk)

1
0.5
0
-0.5
-1

hold time t h ( TBO )Figure 4.4.: Evolution of the peak position for as =83 a0 (a) and 58.8 a0 (b). In grey, sawtooth fit. The error bars correspond to the standard error on the mean over 4-6 repetitions.
0.5

0.5

0.45

0.45

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25
0

2

4

6

8

10

0

2

4

6

8

10

Figure 4.5.: ⟨|qz |⟩ as a function of time th for as =58.8 a0 (a) and 83 a0 (b). With blue edges,
the points selected for the fit. The blue line corresponds to the fit result A(t − τ ) + 0.5

We start with the 1D distribution n(qz ), from it we extract the quantity ⟨|qz |⟩ , givenby the mathematical expression,X⟨|qz |⟩ =n(qz ) |qz − qzmax | ,4.37qzwith qmax , the position of the maximum. This quantity is proportional to the width ofthe distribution and is less sensitive to noise than the conventional ⟨qz2 ⟩.We consider

4.4. Analysing the interaction-induced dephasing

29

that BO are fully dephased when the cloud is spread among all the Brillouin zone,
this corresponds to ⟨|qz |⟩ = 0.5 ℏk. We can easily convince ourselves by taking a fullydephased normalized wave function in theory and calculate ⟨|qz |⟩ .
To evaluate the spreading of ⟨|qz |⟩ , we fit the first 8 ms with the non-linear model,g(t) = A(t − τ ) + 0.5,

4.38

with A and τ the fitting parameters. We define the dephasing time τ it takes to g(t) toreach 0.5 ℏk. Thus, the above parametrization allows to directly extract the dephasing
as a fitting parameter. At the edge of the Brillouin zone, the cloud is split in two and⟨|qz |⟩ is artificially increased which will lead to underestimation of the dephasing time.
Furthermore, we don’t want to consider, for the fit, the points when ⟨|qz |⟩ saturates.
To take into account all those constraints we select only particular points, following
the condition,
qmax ∈ [−0.2 ℏk, 0.2 ℏk]and⟨|qz |⟩ < 0.9 ⟨|qz |⟩ 5 ,4.39

where ⟨|qz |⟩ 5 is the average value of the last five ⟨|qz |⟩ for a given scattering length.
⟨|qz |⟩ and the fit associated for two scattering lengths are presented in Fig. 4.5.
1
0.8
0.6
0.4
0.2
0

3

10 -3

3

2

2

1

1

0

10 -3

0
12

13

14

15

16

17

2.4

2.6

2.8

3

3.2

3.4

Figure 4.6.: (a,b) χ2 as a function of the fitting parameters A and τ . (c,d) Probability
distribution of τ for as =58.8 a0 and as =83 a0

30

4. Observation of Bloch oscillations and ground state properties in a lattice

To find the best fitting values, we used the maximum likelihood method presented
in Ref. [Ifa10]. Close to the standard least-squares fitting, this method is based on
the minimization of the quantity,
χ2 =X (yi − g(xi ))2
i

αi2

,

4.40

with the data points labelled by (xi , yi ) and αi the uncertainty associated. To beconsistent with the theoretical data, we fix αi = α by assuming equal statisticalfluctuation around the fitting model and use the expected value ⟨χ2 ⟩ = Ndata − 2.
This implies,sP
2i (yi − g(xi ))4.41
α=Ndata − 2
χ2 (A, τ ) is calculated numerically for many values of A and τ and shown in Fig. 4.6 for
as =58.8 a0 and as =83 a0 . In both cases, χ2 shows a clear asymmetric shape due to thenon-linear fitting model. one consequence of the shape of χ2 is that the uncertainties
associated with τ will also be asymmetric and the linear approximation to calculate theuncertainties can’t be applied. A general method to extract uncertainties on multivariable fits consists in integrating, over all the parameters but one, the function
exp(−χ2 /2), and after normalization, we obtain a probability distribution. Thus, we
get the probability distribution for τ ,R
2dAe−χ (A,τ )/2A.
4.42
Pχ2 (τ ) = R
dτ dAe−χ2 (A,τ )/2A,τ
We define the uncertainties of τ as the 68% confidence interval. We then convert thedephasing time τ into a dephasing rate γ = 1/τ .
To compare the experimental data with the theory predictions, we run the same analysis with the theory data. Due to a variable atom number at different scattering
lengths, we vary accordingly the atom number used for the theory.
In the experiment, we estimate the atom number as a function of as with a linear fit.
We account for the fluctuation by taking an interval of ±20% of the experimental BEC
atoms number for the theory. For every theory data, we determine the upper (resp.
lower) value of τ in the ±20% range. We expand those values with upper (resp. lower)
uncertainties on the parameter τ . Thus, we obtain an upper and lower bound of τ asa function of as , that we use to create the shaded area in Fig. 4.7.
We also analysed the theory when the Lee-Huang-Yang (Eq. 3.31) is turned to 0 and
we obtain the blue dash line in Fig. 4.7.

4.5. Ground state: localization in a single lattice plane31(ms-1)

10-1

dephasing rate

(a)

10-2

10-3

10-4
55

60

65

70

75

scattering length as (a0)Figure 4.7.: Dephasing rate as a function of as . In grey, experimental points with uncertainties. The green line represents the numerical simulations results and the shaded area
the uncertainties associated. In blue dashed line, theory prediction without quanum fluctuations.

We observe a minimum of dephasing in agreement with the theory around 60 a0 . It
is shifted in comparison with the expected mean-field cancellation at add ≈ 66 a0 .
This can be explained by first, the LHY contribution which adds a repulsive termand secondly by the inter-site contribution of the DDI. In fact, we understood thatthe minimum of dephasing occurs when the variance of chemical potential between
each lattice site is the smallest. Nevertheless, if we expect a purely interaction-driven
dephasing we would expect a symmetric shape in the dephasing rate, but at a low
scattering length (<57 a0 ) the BO disappear suddenly. This behaviour can’t be explained by the effect of the interactions and necissitate a more precise analysis of theinitial momentum distribution.

4.5. Ground state: localization in a single lattice plane
As seen previously, for high as , the lattice induces two sidepeaks at a position ±k.
Nevertheless, when as crosses 57 a0 the interference patterns desappear suddenly. Thisbehaviour indicates a structural change of the ground state wave function, which canbe explained by the localization of the BEC to one lattice site. Figure 4.8 shows
the transitions around 57 a0 . For as =56.7 a0 and 51.7 a0 , the ground state alreadyoccupies the full Brillouin zone and no BO can be observed. To confirm the localization
hypothesis we analyse the interference patterns at th = 0 as a function of as . We
associate the absence of sidepeaks with the localization to a single lattice plane.

32

4. Observation of Bloch oscillations and ground state properties in a lattice

51.7

56.7

57.4

58.2

58.9

Figure 4.8.: TOF images for 5 scattering lengths around the transition as =57 a0

To quantify the intensity of the sidepeaks, for each picture, we perform a Fouriertransform (FT) of the density profile n(qz ). This is done numerically by the fast
Fourier transform algorithm (fft). Figure 4.9 (a,b) shows two density profiles for as =56.7 a0 and 58.9 a0 and Figure 4.9 (c,d) the Fourier transform associated. fft[n(qz )]shows many peaks at separated by a distance of z ∗ = λlatt/2 . We define the contrastC,C=

fft[n(qz )](0)fft[n(qz )](z ∗ )4.43

When no interference is observed, Fig. 4.9 (b,d), C drops to zero. To compare theresults with the theory, we evolve the extended Gross-Pitaevskii equation Eq.(3.31)
in imaginary time and apply twice the Fourier transform on the ground state wave
function obtained. We then calculate the contrast C in the same way and multiply it
by 0.7 to account for the thermal part only present in the experiment.

15

4
3

10

2
5
0
-150

1
0
-100

-50

0

50

100

150 -150

400

-100

-50

0

50

100

150

-20

-10

0

10

20

30

300

300

200

200
100

100
0
-30

-20

-10

0

10

20

30

0
-30

Figure 4.9.: (a,b) density profile n(qz ) for as =56.7 a0 and 51.7 a0 . (c,d) fourier transformof the density profile n(qz ) for as =56.7 a0 and 51.7 a0 . The green circle represents the intensity
peaks fft[n(qz )](0) and fft[n(qz )](z ∗ ) used to calculate C.

33

4.6. Publication: Strongly dipolar gases in a one-dimensional lattice: Blochoscillations and matter-wave localization

35

4.6. Publication: ”Strongly dipolar gases in a
one-dimensional lattice: Bloch oscillations andmatter-wave localization.”
Submitted to Communication Physics (2022)1
submitted 17 May 2022;
DOI: arXiv:2205.03280
G. Natale,2 T. Bland,1 S. Gschwendtner,1 L. Lafforgue,1 D. S. Grün,1 M. J. Mark,1,2
F. Ferlaino,1,2
1 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
2 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissen-

schaften, 6020 Innsbruck, Austria

1

The author of the present thesis contributed to the developement of the experimental procedures
together with G. N., A. P., and M. N., performed the measurements together with G. N., D. S. G.,
S. G., and A. P., analyzed the data with G. N. and contributed in writing the manuscript. The
theoretical calculations of this publication have been contributed by S. G, G. N, and T. B..

Strongly dipolar gases in a one-dimensional lattice: Bloch oscillations andmatter-wave localization
G. Natale,1, 2 T. Bland,2 S. Gschwendtner,2 L. Lafforgue,2
D. S. Grün,2 A. Patscheider,2 M. J. Mark,1, 2 and F. Ferlaino1, 2
1

arXiv:2205.03280v1 [cond-mat.quant-gas] 6 May 2022

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria(Dated: May 9, 2022)
Three-dimensional quantum gases of strongly dipolar atoms can undergo a crossover from a dilute
gas to a dense macrodroplet, stabilized by quantum fluctuations. Adding a one-dimensional opticallattice creates a platform where quantum fluctuations are still unexplored, and a rich variety of newphases may be observable. We employ Bloch oscillations as an interferometric tool to assess the rolequantum fluctuations play in an array of quasi-two-dimensional Bose-Einstein condensates. Longlived oscillations are observed when the chemical potential is balanced between sites, in a region
where a macrodroplet is extended over several lattice sites. Further, we observe a transition to a statethat is localized to a single lattice plane–driven purely by interactions–marked by the disappearanceof the interference pattern in the momentum distribution. To describe our observations, we develop
a discrete one-dimensional extended Gross-Pitaevskii theory, including quantum fluctuations anda variational approach for the on-site wavefunction. This model is in quantitative agreement withthe experiment, revealing the existence of single and multisite macrodroplets, and signatures of a
two-dimensional bright soliton.

The dipole-dipole interaction (DDI) between magnetic
atoms in an ultracold quantum gas has been key tothe discovery of supersolids [1–3] and macrodroplets[4, 5], new states of matter with extremely intriguingand counter-intuitive properties [6, 7]. Macrodroplets
are macroscopic quantum states that behave in many
ways like liquid droplets [4, 5, 8, 9]. They are at least
an order of magnitude denser than normal Bose-Einsteincondensates (BECs), and can be self-bound. They exist
in a parameter regime in which mean-field theories predict the collapse of the entire system when the attractive dipolar interactions overcome the repulsive contact
interactions. Instead, the system remains surprisingly
stable thanks to the so-called quantum fluctuations, thus
providing one of the rare examples where beyond-meanfield interactions substantially change the ground stateof the system [10, 11]. Although the functional form of
the beyond-mean-field term, otherwise known as the LeeHuang-Yang (LHY) correction [12], is still subject to intense study and debate [13, 14], its importance is now undoubted. Isolating beyond-mean-field effects may be crucial to settle disputes on its validity, particularly in dipole
dominated systems; however, it is very difficult to haveaccess to individual interaction contributions. Though,the differing atom number scaling between mean-fieldand LHY contributions provide a promising method todifferentiate between them.
Optical lattices enable powerful interferometric approaches to, e.g., measure with high precision the zerocrossing of the scattering length or of the mean-field interaction with the so-called Bloch oscillation (BO) technique [15–18], and to achieve an accurate determination of the background scattering length via lattice spectroscopy in Hubbard models [4, 19, 20]. Moreover, thepresence of the lattice itself may change completely thephase diagram of the system, as shown in seminal experiments with contact interacting gases [21–23]. Unique
phenomena are predicted with the addition of long-range
DDIs [24, 25]. Experiments with lattice-confined atomic
dipolar gases have already shown important results, e.g.,
the realization of extended Bose-Hubbard models [26]and spin models [27–30] in three-dimensional (3D) lattices. In 2D lattices, forming quasi-1D tubes, suppressionof dipolar relaxation [31] and the controlled breakdownof integrability [32] have been observed. Instead, up tonow, 1D lattices, forming an array of quasi-2D layers,
have been used with large wavelengths to load a single
pancake trap [33], or multi-layer traps to study the role of
DDI in the stability against collapse [34]. Further, theoretical proposals have suggested that the DDI between
layers not only can lead to modifications within eachlayer [35–38] but also to inter-layer bound states [39–41].
Other works predict the existence of bright-soliton structures along the lattice [42] or anisotropic on-site solitons[43, 44]. However, those proposals lack the important stabilization mechanism given by the LHY term, which is
known to provide many new phases in continuous systems(e.g. harmonically trapped), opening up many questions:
What is the ground state of an attractive dipolar gas ina 1D lattice potential? Can droplets be delocalized overmany lattice planes? Will solitonic solutions continue toexist?
In the present work, we study an erbium dipolar gas
in a 1D optical lattice with dominantly attractive DDI.
We employ BOs as an interferometric tool to probe theinteraction contributions of the system, and to isolate
the role of beyond-mean-field effects. We find long-lived
oscillations, associated with a minimum in the dephasing rate, close to the cancellation point between meanfield and beyond-mean-field interactions, and at scatter-

2
(a)

(b)

z
y
B

x

qmax (hk)(c)

qmax (hk)

(d)

//

1
0
-1
10

2

4

6

//
10 // 21

23

25

2

4

//
6
8
10
21
hold time th ( T BO )

23

25

8

0
-1
0

FIG. 1. Bloch oscillations of a dipolar BEC in a onedimensional optical lattice. (a) Sketch of our experiment,
consisting of a 1D optical lattice in the z-direction, loadedwith an erbium BEC from an optical dipole trap with trapping
frequencies ωx,y,z = 2π × (240(3), 30(3), 217(1)) Hz. Gravity
acts along z. (b) Absorption images after TOF showing the
momentum distributions during one Bloch cycle. (c,d) Evolution of the peak position of the momentum distribution for
as = (71.6(1.0), 59.8(1.0)) a0 , respectively. A sawtooth fit(solid grey) to the data yields TBO = 0.469(4) ms, consistent
with the expected value TBO = 2k/(mggrav ). The error bars
represent the standard error on the mean over 4-6 repetitions.

ing lengths significantly shifted from the expected meanfield result. We develop a discrete effective 1D extended
Gross-Pitaevskii equation (eGPE) with variational transverse widths [45, 46]. We find that this minimum occurs
when the chemical potentials on each site are equal, notthe energies–as has been employed successfully in contact interaction dominated systems [17, 18]–due to thedifference in density scaling between the interactions.
The close correspondence between theory and experimentshows the validity of the LHY prediction, even while
highly inhomogeneous densities are expected to break the
local density approximation [12]. Moreover, we see thatfor low scattering lengths the system undergoes a structural transition to a single localized 2D plane, signifying
an important new way to generate systems in reduced geometries through varying the interactions alone. Finally,using our theoretical model we produce a full phase diagram of the system, revealing the impact of the LHY
contribution to the predicted 2D anisotropic soliton state[43], which is instead morphed into a droplet solution athigh atom numbers. Though, promisingly, we still findsoliton-like solutions exist.
In the experiment, we prepare a degenerate dipolar gasof erbium atoms in a one-dimensional optical lattice asfollows. We start with a dipolar quantum gas of 5 × 104
spin-polarized 166 Er atoms confined in a cigar-shaped optical dipole trap [47] elongated along y. Typical BEC
fractions range from 60% to 80%. The dipolar length for
166
Er is fixed at add = 66.5 a0 , where a0 is the Bohr radius. We tune the contact interaction between atoms andtherefore the s-wave scattering length, as , via Feshbachresonances [4, 48–50] by changing the absolute value ofa bias magnetic field |B|. We fix the orientation of B to
be along the weak axis (y) of the trap, making the DDI
dominantly attractive [4, 7].
Once the harmonically-trapped cloud is prepared at
the desired as , we switch on a 1D optical lattice, aligned
along the gravity direction (z); see Fig. 1(a). The vertical lattice is created by retro-reflecting a λ = 1064 nm
laser beam. We load the planes by exponentially increasing the lattice depth V0 to 8 Erec in 20 ms, where
Erec = ~2 k 2 /2m = h × 10.5 kHz. Here, ~ = h/2π is the
reduced Planck’s constant (h), m is the mass of 166 Er
atoms and k = 2π/λ is the wave-vector of the lattice.
The 1D lattice forms an array of tightly confined quasi2D planes with a trap frequency along the tight direction
ωz ' 2π × 6 kHz, corresponding to an harmonic oscillator length zho = 100 nm. The tunnelling rate, J, between
planes is about h × 33 Hz. For these 1D lattice parameters, ~ωz > kB T and the system is kinematically 2D[51].
We first aim at inducing Bloch oscillations to interferometrically assess the role of beyond-mean-field effects
and test the validity of the LHY term. We thus suddenly switch off the dipole trap and let the system evolvein the combined lattice and gravitational potential for avariable hold time th . Finally, using standard absorption
imaging after 30 ms of time-of-flight (TOF), we record
the evolution of the momentum distribution and extract
the position of the main peak, qmax , as a function of th .
Figure 1(b) shows an exemplary set of absorption images
during a single Bloch period TBO . We observe the key
paradigm of BOs, i.e. the linear increase of the mean momentum due to the acceleration and the Bragg reflection
occurring at the border of the Brillouin zone [52], welldescribed by fitting a sawtooth function to qmax .
The high sensitivity of BOs to interactions [17, 53]
clearly appears by tracing the evolution for two different
as (see Fig. 1(c,d)), as the interaction dependence is encoded into the dephasing rate. For a contact-dominatedgas (add < as = 90 a0 , Fig. 1(c)), we see that the BOs
vanish within a few TBO . On the contrary, decreasing as , and thereby going into the regime where contact interactions and DDI nearly compensate each other(as = 60 a0 , Fig. 1(d)), we observe persisting oscillations
for more than 25 Bloch cycles, set by our limited observation time [50]. To systematically study this effect,
we repeat the BO measurements for different values ofas , and extract the corresponding dephasing rate γ [50].
As shown in Fig. 2(a), we observe a resonant-type behavior with γ showing a pronounced dip with a minimum at

3(a)dephasing rate

(ms-1 )

10-1

10-2

10-3

(1)

10-4
55

60

(b)

/h (Hz)

(2)

(1)

(3)

(4)

65
70
scattering length as (a 0)

(2)

(3)

75

(4)

100
0

-2

0

site j2-20site j2-20site j

2

-2

0

2

site jFIG. 2. Dephasing rate and chemical potential distributions. (a) Experimental dephasing rate γ (circles) as afunction of scattering length as . The green solid line shows
the theory result, with an uncertainty region (shaded area)
accounting for 20% atom number variation. The blue dashedline shows the theory expectation without LHY. The gray dotdashed line gives the prediction of the semi-analytic approximation for γ. Error bars show the 68% confidence interval[50]. (b) Chemical potential per lattice site µj extracted fromthe discrete model for as = (59, 60, 65.5, 70) a0 (1, 2, 3, 4).
The green area depicts the LHY contribution to µj .

as = 61 a0 . This minimum is clearly different to the pointas ≈ add , where the variance of the mean-field energies
across different lattice sites cancel [18], which would be
expected from previous observations [17, 53].
To get further insight on the origin of the minimum,
we develop a discrete effective 1D eGPE, inspired by theclose correspondence between predictions from discretemodels and experimental observations in non-dipolar[54, 55] and weakly dipolar [18] BECs. We separate
the 3D wavefunction into radial and axial contributions,allowing for a variational anisotropic radial width andthus maintaining the 3D character [45]. Along the lattice direction (z), we further decompose the wavefunction, ψ(z, t), as a sum of Wannier functions w(z) of
the lowest energy band over all lattice sites: ψ(z, t) =√P
N j cj (t)w(z − zj ), where N is the atom number andcj (t) the complex wavefunction amplitude on lattice sitej, leading to a set of discrete effective 1D eGPEs, each
including mean-field and beyond mean-field interactions.
For the beyond-mean-field interaction, the 3D form of the
LHY still fully applies since the contact interaction energy exceed the confinement energy scale [50, 56]. However, our system may also open to further studies on the
2D to 3D crossover of the LHY. We solve these equations

coupled to a minimization of the energy functional withrespect to the variational parameters to determine theground states, benchmarking them against the full 3D
theory. We then perform dynamic simulations of the expected time evolution [50], giving an accurate dephasingrate (solid line) in Fig. 2(a) without free parameters.
In previous studies, the point of minimum dephasing
was found to occur when the mean-field interaction energies vanish or cancel. We isolate the mean-field contribution by removing beyond-mean-field effects from oursimulations (dashed line in Fig. 2(a)), predicting a minimum at as ≈ add . However, this is in clear contradiction with our experimental observations by a shiftof 6a0 and a different overall shape due to the different scaling of the LHY term with the density. Withj
out LHY, the cancellation of mean-field energies, EMF,
is equivalent to the cancellation of onsite chemical poj
tentials, given by µj = 2EMF/|cj |2 . Note, µj dictates
the wavefunction phase winding on each site throughcj = |cj |e−iµj t/~ . Reintroducing quantum fluctuations,jj
we obtain µj = (2EMF+ 5/2EBMF)/|cj |2 , where the 5/2
5appears due to the |cj | density scaling in the beyondj
mean-field energy (EBMF). Figure 2(b) shows µj fromthe ground state calculation for four scattering lengths,
additionally indicating the contribution of the LHY correction.
We observe that the point of minimal dephasing in theexperiment is close to the point where the variance ofµj is minimized [57]. Indeed, within a semi-analytic approximation (see Ref. [50] for details), we find a direct
relationship between γ and µj , which reads γ ∝ |µ1 − µ0 |when 3 lattice sites (j = −1, 0, 1) are occupied. Thismodel can be extended to 5 lattice sites, giving the dotdashed line (Fig. 2(a)) which reproduces very well thesystem behaviour [50]. Interestingly, measuring the dephasing rate through the chemical potential is ubiquitousto systems with arbitrary interaction potentials.
Surprisingly, by further decreasing the scattering
length below 57 a0 , no BOs nor interference peaks arevisible anymore. We observe at the initial instant (th =0 ms) that the momentum distribution is already spreadover the entire first Brillouin zone. To quantify this, westudy the contrast, C, of the interference pattern of theinitial momentum distribution as a function of as , see
Figure 3(a). We extract C, defined as the amplitude of
the momentum peaks at ±2~k relative to the zero momentum peak, from the Fourier analysis of the TOF images [50]. For large as , we observe the typical matterwave interference pattern, as expected from a coherent
state populating several lattice planes (see inset) [55]. As
we lower as , C first remains fairly constant. For as below
a certain critical value a∗s ≈ 57 a0 , we observe a sudden
loss of the interference pattern with a sharp decrease of
C to almost zero.
Remarkably, we observe that this interaction-driven
process is reversible. To test the restoring of the interference pattern, we employ the following protocol [50]: In
brief, we first prepare the system in the lattice at constant

4

0.2
(1)

contrast C

(b)

50

0.6

(1)

55

(1)

60
65
70
scattering length as (a 0)
(2)

75

(3)
(c)

(2)
0

10

20
30
hold time th (ms)40

50

FIG. 3. Interaction-induced localization. (a) Contrastof the interference pattern after loading the lattice at different as . The green dot-dashed (black solid) line represents theresult of the 1D discrete model (3D eGPE) multiplied by 0.7.
The insets show the respective density distributions along zof the 1D discrete model (bars) and 3D eGPE (lines) and corresponding experimental averaged interference patterns after
TOF expansion (1,2). (b) Dynamic evolution of the contrast
quenching back (filled circles) or holding as (open circles); seetext. The error bars represent the standard error on the meanover 4-6 repetitions.

and large as (as = 69(2) a0 ). We then ramp down as below a∗s (as = 56(2) a0 ) in 20 ms and wait until C stabilizes
to a small value; see Fig. 3(b). Note that the interference
pattern disappears after about 10 ms, which is on the order of the tunneling time h/J between two neighboring
lattice sites. At this point, we quench as back to its initialvalue and probe the time evolution of the system towards
its new equilibrium state. On a similar timescale, we observe the reappearance of the interference pattern withan increase of C, which then saturates to about 60% of
its initial value [58]. For comparison, we also show thedata without inverting the field ramp.
The observed broad distribution in reciprocal spacesuggests that the system ground state has undergone a
structural change, with the macroscopic wavefunction localized in one lattice plane. To verify this interpretation,
we calculate the ground state of the system as a functionof as . When the repulsive contact interaction dominates(as > add ), we find an array of BECs occupying approximately three to five lattice planes; see insets Fig. 3(a).
In contrast, when the relative strength of the attractive
dipolar interaction with respect to the other terms in the
Hamiltonian is increased, the system reaches a criticalpoint. Here, it undergoes a phase transition to a quasi-2D2

Soliton5560

1
2
l x (x ho )12l x (x ho )

65

70

75

1

2

scattering length as (a0)
(d)

(b)

0.2

4

0.5

50

0.4

0

1

80

(3)

BEC

4

1.1

2

l x (x ho )045

Dropletl y (y ho )atom number N (10 4)

(2)

1.5

l y (y ho )contrast C

(1)
0.4

2

(a)

(2)

0.6

l y (y ho )

(a)

4

1
0.9

2
1
2
l x (x ho )12l x (x ho )0.8
0atom number N (10 4)

FIG. 4. Phase diagram and energy landscapes.
(a) Phase diagram as a function of as and atom number. The
white region denotes a trap-bound BEC extended over several lattice sites. The colored regions denote quasi-2D selfbound solutions: a droplet (green), a soliton (blue), each either extended over several lattice sites (lighter shade) or localized (darker shade, >95% of the atoms are localized in
the central lattice plane). Circles show our experimental
data points from Fig. 3(a). Inset (a), (b-c) Energy landscapes as a function of the radial widths lx and ly , in unitsof the radial harmonic oscillator lengths xho = 0.50(1) µm
and yho = 1.42(1) µm, respectively, with (left) and without(right) the radial harmonic trap, for (inset (a)) BEC (as , N ) =(70 a0 , 1.5 × 104 ), (b) droplet (as , N ) = (65 a0 , 1.5 × 104 ) and(c) soliton (as , N ) = (51.5 a0 , 0.4 × 104 ) regimes, with darker
shading at the minima. (d) Radial width lx versus N for
as = 51.5 a0 . The dashed line indicates the soliton-to-droplet
transition point, and the circles indicate the position of (b-c).

state, in which all atoms are localized into a single latticeplane to minimize their energy. This purely interactiondriven phase transition–somewhat reminiscent of a continuous version of a superfluid to Mott insulator transition [59]–is stabilized by quantum fluctuations (LHY),
preventing the subsequent collapse of the system [42, 60].
The predicted critical point occurs exactly where we observe the disappearance of the interference pattern in theexperiments. We find an overall excellent agreement between the measured and the calculated C from both thediscrete 1D model and the 3D theory without any free
fitting parameters, except for a rescaling factor to thecontrast amplitude to account for the thermal atoms in
the experiment.
The observation of this phase transition to a quasi-2D
localized state driven by interactions points to the existence of a rich variety of phases. The importance of

5
the LHY correction and its peculiar density scaling motivate us to investigate the properties of the ground state
as a function of as and atom number to identify distinct
phases in this unique setting. For this, we employ our discrete model to derive a full phase diagram; see Fig. 4(a).
To investigate the boundness of the states, we assess theimpact of the radial harmonic trap on the minimum of
the variational energy, which is a function of the radialwidths lx and ly . At large scattering lengths, as expected,
we find a stable delocalized BEC phase, where the totalinteraction energy (mean-field + LHY) is positive. The
state is trap-bound, meaning that there is no energy minimum without the radial harmonic confinement; inset of
Fig. 4(a).
Reducing as , we find an energy minimum even without
the radial harmonic trap (colored region in the phase diagram). These quasi-2D self-bound solutions (the lattice
still provides axial confinement) are either extended over
several sites (lighter color) or localized to a single plane(darker color). In the literature, there are two paradigmatic examples of self-bound objects with attractive
mean-field energy: droplets and solitons. Droplets can
exist in one, two or three dimensions and are stabilized
through the LHY correction [7]. Stable bright solitons
only exist in quasi-1D systems with attractive contact
interactions and are stabilized against collapse purely bykinetic energy. In the search for solitons in higher dimensions, theoretical studies have suggested that the DDI
could stabilize such 2D solutions [43, 44]. To the best of
our knowledge, there have been no studies on the effectthe LHY correction has on this prediction, nor experimental observation. In the present case, where many
interactions and kinetic energy compete, a classificationof self-bound solutions is much less straightforward. Asa crucial distinction between a soliton and a droplet, we
use the scaling of the system width with atom number.
The soliton width (along the collapse direction) scales inversely with increasing atom number [61], while in contrast, the droplet size increases in all directions with N[62], as predicted in a quasi-1D setting [63]. We use this
distinction to draw a boundary between the two phases,
observing a phase transition at around 5000 atoms, forboth single-site and multi-site solitons. The overlayingof our measurements (Fig.3(a)) onto the phase diagramsuggests that the experiments have already reached theinteresting regimes of both 2D self-bound droplet anddipolar solitons. This opens the door to future experimental investigation on the self-bound nature and properties of these new 2D phases.
In conclusion, we theoretically and experimentally investigate the behavior of a strongly dipolar quantum gas
in a 1D optical lattice. We employ BOs and characterize their dephasing rate as a function of as . We observea minimum in the dephasing shifted 6 a0 away from thepurely mean-field prediction, providing an interferometric measure of the beyond-mean-field contribution. For
low enough as , the system enters into a quasi-2D state
which is localized onto a single lattice plane, providing a
genuine interaction-driven path to reach reduced dimensions in dipolar gases. Using our developed discrete theory model, we derive a full phase diagram which confirms
the observed localization transition. This also reveals signatures of quasi-2D self-bound dipolar droplet solutions,and the long sought-after 2D anisotropic dipolar soliton,
first predicted in Ref. [43] (see also [44, 64]). Our workpaves the way for future studies of the soliton-to-droplet
crossover in a dipolar gas, as observed in a Bose-Bose gas[65], and of the “solitonic” nature [66] of dipolar solitary
waves [67–71].

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ACKNOWLEDGMENTS

We thank R. N. Bisset, A. Houwman, L. Lavoine,
R. Grimm, and L. Tarruell for stimulating discussions,and B. Yang for his support in the early stage of the experiment. This work is financially supported throughan ERC Consolidator Grant (RARE, no. 681432) and a
DFG/FWF (FOR 2247/I4317-N36). We also acknowledge the Innsbruck Laser Core Facility, financed by theAustrian Federal Ministry of Science, Research and Economy. Part of the computational results presented havebeen achieved using the HPC infrastructure LEO of the
University of Innsbruck.
* Correspondence and requests for materials should be
addressed to Francesca.Ferlaino@uibk.ac.at.

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8

Supplemental materials: Strongly dipolar gases in a one-dimensional lattice: Blochoscillations and matter-wave localization
G. Natale, T. Bland, S. Gschwendtner, L. Lafforgue, D. S. Grün, A. Patscheider, M. J. Mark, and F. Ferlaino
THEORETICAL MODEL

In this work, we use an extended Gross-Pitaevskii theory for direct comparison to our experimental results. We
employ both the standard three-dimensional form of theextended Gross-Pitaevskii equation (eGPE) and derivea discrete effective one-dimensional eGPE. Starting withthe three-dimensional case, our system can be describedby the 3D eGPE of the form [4, 72–74]
i~

h
∂
~2 2
Ψ(~x, t) = −∇ + Vharm (~x)∂t2m
+ Vlatt (z) − Fext z + g|Ψ(~x, t)|2Z
+ d3 ~x0 Udd (~x − ~x0 )|Ψ(~x0 , t)|2i
+ γQF |Ψ(~x, t)|3 Ψ(~x, t) ,(S1)

where the wavefunction
R Ψ is normalized to the total
atom number N = d3 ~x |Ψ|2 . The atoms are conP
fined in a harmonicpotential Vharm = ξ=x,y,z 12 mωξ2 ξ 2with single particle mass m and trap frequencies ωξ , together with the lattice potential Vlatt = sErec sin2 (kz)where s is the tunable lattice depth in multiples of therecoil energy Erec and k = 2π/λ is the lattice spacing in reciprocal space. The mean-field interaction contributions are g = 4π~2 as /m for the contact interaction, governed by the s-wave scattering length as , andthe long-ranged anisotropic dipolar interaction potential Udd (~x) = 3~2 add /m 1 − 3 cos2 θ /|~x|3 , where add =µ0 µ2m m/12π~2 with magnetic moment µm and θ is theangle between the polarization axis (y-axis) and the vector between neighboring atoms. We also include beyondmean-field effects
through the quantum fluctuations term
q

a3s
32
γQF = 3 g π 1 + 32 ε2dd [12], which depends on therelative strength between the dipolar and short-ranged
interactions εdd = add /as . Finally, Fext = ggrav m de-∂cji~= −J(cj+1 + cj−1 ) +∂t−Fext zj + Vharm (z) + g

1D

with the reduced effective one-dimensional parame1D
ters γQF= 23/2 /(5π 3/2 l2 llatt )3/2 γQF and g 1D =g/((2π)3/2 l2 llatt ). Here, J denotes the tunneling rate between two neighboring lattice sites. The dipolar interaction coefficients between lattice sites j and k depend
both on the separation |j − k|, and non-trivially on thenotes the external force exerted on the system by gravity.
In this work, we employ the imaginary time-evolution
technique on Eq. (S1) in order to find stationary solutions
for the wavefunction in the lattice, without gravity. For
various atom numbers and scattering lengths, we use anumerical grid of lengths (Lx , Ly , Lz ) = (6, 33.3, 6) µm,
with corresponding grid points 128 × 256 × 128. The
dipolar term is efficiently calculated in momentum space,
and we use a cylindrical cut-off in order to negate theeffects of aliasing from the Fourier transforms [75].
To derive the effective one-dimensional model, we follow Ref. [45] by assuming a wavefunction decomposition2221
Ψ(~x, t) = Φ(x, y, l, η)ψ(z, t) ≡ √ e−(ηx +y /η)/2l ψ(z, t) ,πl
(S2)with variational parameters l and η representing thewidth of the radial wavefunction and the anisotropy ofthe state, respectively. Integrating out the transverse directions (x, y) in Eq. (S1) upon substitution of the ansatz
above gives the continuous quasi-one-dimensional eGPE,
which when combined with a variational minimization of
the energy functional to find (l, η) gives close agreement
to the full 3D eGPE [45]. We further decompose thelongitudinal wave function ψ(z, t) into a sum of Wannierfunctions w(z) of the lowest energy band over all lattice
sites
√ X
cj (t) w(z − zj ) ,(S3)ψ(z, t) = N

jfor complex amplitudes cj , and positions of lattice minima zj = (λ/2)j. For deep enough lattices, the Wannierfunctions are well approximated by Gaussians of the form
−1/4 −z2 /2l2√2
−1
4
latt , with lw(z) = πllatte. Aflatt = (k s)∗ter multiplying on the left by cj and integrating over z,
we obtain a set of discrete effective one-dimensional eGPEs

2

N |cj | + NXkdd
1D 3/2
U|j−k||ck |2 + γQF
N γQF |cj |3

!

cj ,

(S4)

size l and anisotropy η of the transverse cloud. For thevariational minimization, we generate an interpolating
function for a sensible range of (l, η) and separations up

9

(S5)

where Ψ0 (~x0 , l, η) = Φ(x, y, l, η)w(z) [see Eqs. (S2) and(S3)]. This allows us to simply look up the values ofdd
U|j−k|
without having to recalculate for every time stepduring the energy minimization. We note that the energy contribution rapidly declines for separations larger
than 2 sites, and find that 6 is more than sufficient toquantitatively describe the physics.
To find the stationary state solution of Eq. (S4) (without gravity) we employ an imaginary time-evolution in
combination with an optimization scheme, aiming to findthe state which minimizes the total energy functional
E[c; l, η] = E⊥ [l, η] + Ek [c; l, η] ,200

E int < 0

DD offsite

E int

DD onsite

QF

Contact

100
0
-100
-200
-300
50

55

60
65
scattering length a s (a 0)7075FIG. S1. Interaction energy contributions. Scattering
length dependency of the individual interaction contributionsof the ground state solutions from the 1D model, calculated
for N = 104 atoms.

(S6)

where c = (c1 , c2 , . . . , cn ) for n total lattice sites. Here,E⊥ [l, η] gives the energy contribution from the transverse
variational wave function, which reads




~2
1
ml2 ωx2
2
E⊥ [l, η] =
η
+
+
+
ηω
. (S7)y2ml2
η4ηThe latter term of Eq. (S6) gives the discrete energy functional for the amplitudes cj , which includes the tunnelingand all interaction terms
X
Ek [c; l, η] = −J(cj+1 + cj−1 )cj
jX
1 X dd1|cj |4 + N
U|j−k| |ck |2 |cj |2+ N g 1D
2
2
j
j,k

X
2
1D
+ N 3/2 γQF|cj |5 .
5j300

Energy (Hz)to |j − k| = 6 viaZndd
U|j−k|(l, η) = d3 ~x |Ψ0 (~x − z|j−k| êz , l, η)|2Zod3 ~x0 Udd (~x − ~x0 )|Ψ0 (~x0 , l, η)|2 ,(S8)Starting from an initial distribution of the amplitudes
cj we first determine the variational parameters (l, η),which is done via an optimization scheme minimizing
Eq. (S8). Subsequently, we evolve the amplitudes inimaginary time using Eq. (S4) and repeat this processuntil we find the minimum of the total energy function
Eq. (S6).
In Fig. S1 we assess the different interaction energy
contributions to Eq. (S6) for a range of scattering lengths.
For as > add the total interaction energy is positive, andit corresponds to a dilute BEC. Following as to smaller
values all interaction contributions are almost constant,until at around as = 60 a0 there is a phase transition
from the BEC to droplet state, as identified in Fig. 4of the main text. This sharp gradient ceases at aroundas = 55 a0 , where the atoms are localized to a single
lattice plane. Note that although the DDI offsite energy is typically only 10% of the onsite counterpart, it

constitutes a significant contribution to the total interaction energy in the system, shifting the BEC to droplet
crossover and localization transitions by a few a0 .
Once we have the ground state of the system, we employ the discrete effective one-dimensional eGPE in realtime to simulate the Bloch oscillations in the presence ofgravity.
2D TO 3D CROSSOVER

The dimensionality of the system is known to highly
influence the size and even the sign of the beyond-meanfield contribution, in both Bose-Bose [76–78] and dipolar [56, 79, 80] gases. Here, we assess the validity of
employing the full 3D LHY correction to our system.
Following Ref. [56], we define the dimensionless parameter ξ = gn/0 –dependent on the contact interactionsg, peak 3D density n, and the confinement energy scale20 = ~2 π 2 /2mzho–that indicates which dimensionality
regime our system is in. If ξ & 1 we are safe to use the3D LHY term, whereas if ξ  1 the 2D solution deviatesfrom the 3D one. Deep in the localized droplet regime,
where the peak density is on the order of 1022 m−3 , wefind ξ ≈ 2, and the 3D LHY as used throughout this workis valid. Even at large scattering lengths, where the peakdensity is closer to 5 × 1020 m−3 , we find ξ ≈ 0.5, which
introduces an error of less than 5% between the 2D and3D LHY terms [56]. In this limit, the 2D LHY term may
be more appropriate, however in the dilute BEC phase
the impact of the LHY is minimal.
ANALYTIC MODEL OF DEPHASING

Starting from the discrete 1D eGPE we decompose
the coefficients cj into amplitude and phase as cj =

10
|cj | exp(−iφj ), and then integrate Eq. (S4) in time to giveφj (t) =− Fext zj + g 1D N |cj |2 + NXkdd|ck |2
U|j−k|

1D 3/2+ γQF
N γQF |cj |3≡ (−Fext zj + µj )

!

t
,
~

t
~
(S9)

with onsite chemical potentials µj , and where we havealso assumed that Fext d  J such that the amplitudes|cj | are frozen.
Following Ref. [81], we write the Fourier transform of
the quasi-1D wavefunction asψ(k, t) = w(k)Xj|cj | exp[−i(kzj + φj (t))] = w(k)C̃(k, t) ,(S10)

where w(k) is the momentum space Wannier function,and phases φj are given above. If all interactions are
set to zero this function is initially a delta function situated at k = 0 and moves in k-space as k̃ = k − Fext t/~.
Interactions broaden C̃(k, t), leading to a dephasing ofcoefficients cj . Fig. S2(a) depicts |C̃(k, t)|2 as a functionof k at different times t, normalized to |C̃(0, 0)|2 .
We extract an analytic approximation to the dephasing time by considering the temporal behaviour of thepoint |C̃(0, t)|2 , i.e. at k̃ = 0. During dephasing this point
rapidly decreases through interference between neighboring sites. This quantity is plotted in Fig. S2(b) for a
few example scattering lengths. It reaches the threshold α/C̃(0, 0) = 0.5 at the dephasing time t = td , where
many k-modes are now highly occupied. This time canbe found through the smallest positive solution of


α= 

X
j

|cj | cos






 2

 2
X
µj td 
µ
t
j
d
 .
+
|cj | sin
~
~
j
(S11)

Exact solutions to |C̃(0, td )|2 = α can be only found inlimiting cases. For the three lattice site case, with j =−1, 0, 1 and noting the symmetry of |cj | = |c−j | we obtaintd = arccos



4|c0 c1 |α + |c0 |2 − 2~,(µ1 − µ0 )FIG. S2. Analytic dephasing rate. (a) Evolution of
the function C̃, with k̃ normalized to the Brillouin zone in
the moving frame, and as = 60.5 a0 . Here, the solution of
Eq. (S12) is td = 0.59s. (b) Time evolution of the central
point of C̃, showing when |C̃|2 crosses α = 0.5. The functionC̃ is scaled to the value at C̃(0, 0). (c) Analytic dephasingrate (γ = 1/td ) obtained for the 3 lattice site approximation Eq. (S12) and the 5 lattice site approximation Eq. (S13),
compared to the numerically obtained value from a real-time
simulation of the discrete model, Eq. (S4).

scendental equation(µ0 − µ1 )td
α = 2 − |c0 |2 + 4|c0 c1 | cos~(µ0 − µ2 )td+ 4|c0 c2 | cos~(µ1 − µ2 )td+ 8|c1 c2 | cos
,
(S13)
~
for the smallest non-zero root td . We compare the resultsfrom Eqs. (S12) and (S13) to the numerically obtaineddephasing rate, γ = 1/td , in Fig. S2(c), as presented inFig. 2 of the main text, and find excellent agreement.

(S12)This relation is expected to give an accurate prediction of
the dephasing time for all states where only 3 lattice sites
are dominant. From this equation, one can see how thedephasing time tends to infinity in the limit of equallydistributed chemical potentials, as observed in Fig. 2 ofthe main text. We can extend this to 5 sites, but it is
not as trivial. One needs to numerically solve the tran-

EXPERIMENTAL PROTOCOL

We prepare a 166 Er spin-polarized BEC similar toRef. [4]. The magnetic field during the evaporation isalong the z-axis with absolute value |B| = Bz = 1.9 G(as = 80(1) a0 ), see Fig. 1(a). The B-to-as conversion
has been precisely mapped out in previous experiments[4, 20]. Before loading the lattice, we rotate the magnetic

11
0.45

17

(a)

18

19

20

21

22

0.4

A (ħk.ms -1 )

0.9

0.3
0.25

0

2

4
6
hold time th ( T BO )8

10

FIG. S3. Evolution of the h|qz |i . In the figure h|qz |i asa function of time for as =58.8 a0 . The points with red edges
are the one selected for the fit. The red line corresponds tothe fit result A(t − τ ) + 0.5

0.8
0.7

(b)

3

2

1

0

field direction along the y-axis in 50 ms and change its absolute value to set the scattering length. At this step, we
typically achieve 5×104 atoms with more than 60 % condensed fraction in a cigar shape dipole trap with trapping
frequencies ωx,y,z = 2π (240(3), 30(3), 217(1)) Hz. For
our experiments, the atoms are then loaded in a 1D lattice by a 20 ms exponential ramp of the lattice depth.
This is the experimental protocol used in Fig 1, 2, and3(a).
To study the reversibility of the interaction-induced
transition to a single lattice site (3(b)), i.e. the evolutionof the contrast due to a change of the scattering length,
we employ a different protocol from the one above. In
fact, in our experiment, the magnetic field along the ydirection can be changed on a timescale of ' 20 ms, whichis slower compared to the z-direction (' 1 ms). For thisdataset, we prepare the BEC with B = (0, 0.25, 1) G andthen we load the lattice as described above. We thenlinearly ramp the field in 20 ms to B = (0, 0.25, 0) G andrecord the time evolution. In Fig. 3(b), we study thecontrast evolution after the ramp. For the black dataset,the magnetic field is quenched back to the initial value
after 10 ms.
For Fig 4, we extract the atom number condensed in
the lattice by releasing the cloud from the combined
ODT-lattice trap and by performing an absorption imaging after 30 ms of TOF. We integrate the density along
the lattice axis and use a double Gaussian fit on the integrated density profile. We repeat the sequence 4-8 times
for every scattering length. At low scattering lengths, wefind a decreased number of condensed atoms, see Fig.4.
We attribute this to an increase of three-body loss in thevicinity of a Feshbach resonance [4] and the increased
density of the groundstate.

10 -3

P( )

0.35

12

13

14
15
16
dephasing time (ms)17

FIG. S4. Uncertainties χ2 analysis. (a) χ2 as a functionof A and τ . The black dash lines correspond to the value
χ2min + 1. Below, (b) probability distribution of τ , given byR2numerical integration of A dAe−χ (A,τ )/2 and normalization
to 1. as =58.8 a0 . The dashed line indicates the fit result andthe shaded area the 68% confidence interval.

ANALYSIS OF MOMENTUM DISTRIBUTION
DURING BLOCH OSCILLATION

When the Bloch oscillation dephases, the width of the
momentum distribution increases with time [82]. To evaluate the dephasing rate we analyze the 1D momentum
distribution along z, n(qz ), as a function of the holding
time. Because of our limited vertical optical access, the1D lattice is not perfectly aligned with the z (gravity) direction. We measure a tilt of 9(1)°. Such a tilt effectively
weakens the radial trapping strength, limiting our observation time to 12 ms, which anyhow allows us to observe
up to 25 BO period.
From n(qz ), we can extract the maximum position(qzmax ) and the quantity h|qz |i , given byX
h|qz |i =n(qz ) |qz − qzmax | .
qzThis quantity is proportional to the width of the distribution. In Fig. S3, we report h|qz |i for as = 65.7(1.0) a0 .
To quantify the dephasing rate γ, we apply a linear fit toh|qz |i . For the fit, we select only the points at the centerof the Brillouin zone, up to the time when h|qz |i is reaching the fully dephased configuration, 0.5~k. Indeed, when
the cloud is at the edge of the Brillouin zone, h|qz |i is artificially increased and it does not represent the dephasing,as shown in Fig. S3. We define the dephasing rate γ as
the inverse of the time τ that the fitted function needs to12reach the value 0.5 ~k. Thus, using the fit parametrization A(t−τ )+0.5, where A and τ are the fitting variables
and t is the time, we can directly extract τ and its inverseγ.
To determine the uncertainties with our non-linear
parametrization, we analyze the χ2 (A, τ ). We estimate
the uncertainties on our data points by assuming equal
statistical fluctuations around our fitting model and using the expected value χ2 = Ndata − 2. Figure S4
shows a clear asymmetric shape for χ2 , indicating asymmetric uncertainties on our fit parameters. As we are
only interested
R in the 2 uncertainties on τ , we considerPχ2 (τ ) = N1 A dAe−χ (A,τ )/2 , with N a normalization
constant. Pχ2 (τ ) corresponds to the probability distribution of τ for our fitting model. Finally, from Pχ2 (τ ),we define the 68% confident interval of our dephasingrate γ shown in Fig. S4.
In order to compare our experimental data with thetheoretical predictions, we repeat the same analysis withthe data from the 1D discrete model. Since in the experiment the condensed atom number changes with thescattering length, see Fig. 4, the atom number consid-

ered in the theoretical simulations varies accordingly. In
Fig. 2, we account for the experimental fluctuations bytaking an interval of ±20% of the BEC atoms number.
For each scattering length, we determine the extreme values of γ in the ±20% range, which we use to create theshaded area.
CONTRAST OF THE INTERFERENCEPATTERN

The density modulation that usually characterizes a
BEC loaded into a 1D lattice can be experimentally extracted from the matter-wave interferometry after a TOF
expansion [59]. To study the transition to one single
occupied lattice site, we record the density distribution
as a function of as . In more details, for each picture
we perform a Fourier transform (FT) of the integrated
momentum distribution, n(qz ). In the contact dominated regime, the lattice induces two sidepeaks at ±qz∗
in n(qz ). Consequently, in the FT analysis the peaks areat z ∗ ' λlattice . The visibility of the interference patternis then estimated as nFT (|z ∗ |)/nFT (0).

Chapter5

New imaging setup for theErbium experiment5.1. Motivation and challenges
In 2019, a new state of matter has been observed featuring the density modulationof solid and the phase coherence of a superfluid, called supersolid [Cho19] [Tan19][B1̈9]. The density modulations have typically a spacial period of λ∗ = 2.5 µm. To
distinguish the modulation maxima and minima it is thus needed to have an imaging
with a resolution smaller than λ∗ , but typical absorption imaging does not fulfil sucha requirement. This is the reason why, in our lab (the ERBIUM experiment), we have
. In the last two
rather probed the momentum distribution showing peaks at k = 2πλ∗years, supersolid has been subject to intense interest and the importance of in-situ
imaging became increasingly large. More recently, theorists predicted the presence ofa domain supersolid (DSS) in a spin mixture of 166 Er with a mj = 0 and mj = −6
[Bla22]. As the domain supersolid is long-lived and the spacial extension is predicted
to be of more than 40 µm for 3 × 104 atoms, observing it experimentally would allow
gaining further understanding of supersolid states in general. Moreover, The ERBIUM
experiment has also the ability to use the narrow transition at 1299 nm [Pat21] for state
preparation of DSS.
Part of my master project was to build a new imaging setup allowing to reach thetarget high-resolution of 532 nm. The core of such setup is an objective, customizedby the company Special Optics, it allows to use all the numerical aperture (NA) givenby the current vacuum chamber. Thanks to the 5 beams MOT [Ilz18], the objective
can be implemented without tremendous effort.
An objective brings also new optical engineering solutions. The rich spectrum of
erbium gives many tools to control the atoms. It is in principle possible to focus a1299 nm laser beam to locally address atoms confined in a 1064 nm optical lattice.
49

50

5. New imaging setup for the Erbium experiment

Combined with a digital micro-mirror device (DMD) this will allow the creation of
local perturbations in the spatial plane or the realization of box potentials.
Implementing a new high-resolution imaging system in the erbium experiment presents
many opportunities and challenges that will be discussed in this chapter. The firstpart is devoted to the design and the testing of a high-resolution objective designed
to work with a 401 nm imaging light. In the second part, we describe the challenges
regarding the implementation in the experiment and the future projects to use theobjective.

5.2. Objective design
Let us first briefly review the key formulas and useful notions to understand the basisof the imaging theory. We place ourselves in the framework of diffraction theory andstudy the optical field after an aperture. This permits to set the definitions of thenumerical aperture (NA), the resolution and the point-spread function (PSF). We will
discuss briefly aberrations in optical systems and how to characterize them via the
Intensity Strehl ratio. Finally, we present the objective designed and produced by thecompany Special optics.

5.2.1. Basics of optical design
Numerical aperture
In general, an object emit light in every direction. Nevertheless, any practical opticalsystem features a finite size can only collect a limited amount of light. The angle overwhich a system can collect light is quantified by the numerical aperture (NA), definedas,NA = n sin(θ),

5.44with n the refractive index of the medium and θ the angle of the last incoming beamwith the optical axis. The notations are presented in Fig. 5.1. The presence of n assuresthat the NA is conserved when the light is propagating through different materials.
This is an important property since most ultracold experiments present high vacuum
setups and the light is travelling through a viewport before being collected by theimaging system. Equivalently, one can introduce the numerical aperture on the image
plane and define the magnification M as,
NAo = |M |NAi .

5.45

5.2. Objective design

51

Figure 5.1.: . Light emited by a point source through an aperture. θ is the angle of thelast ray with the optical axis and n the refractive index.

In rays optics, the magnification is often considered as the scaling factor between theobject and the image.

Diffraction theory
We are now interested in what happens when light travels through an aperture. If
one considers light as rays and records the image on a screen, only a shadow withthe same shape as the aperture will be observed. However, in practice, depending on
the wavelength of the incident light or the size of the aperture, more exotic patterns
appear. In the scalarpapproximation, the light field at a position r is described by acomplex field u(r) = I(r)eiϕ(r) . A plane formed by ϕ(r) =const. is called wavefront.
The field distribution obeys the scalar wave equation or Helmholtz equation,(∇2 + k 2 )u = 0,

5.46

, λ the wavelength of the light and n the refractive index ofwith the wave vector k = n2π
λthe medium. Equation (5.46) is linear and if u1 and u2 are solutions, then u1 +u2 is alsoa solution. We will use this property later to calculate the field distribution after anaperture with any shape. In the following, we give a brief review of diffraction theory,
based on Ref. [Goo05] and Ref. [Tei07]. The simplest theory of diffraction is basedon the assumption that an aperture lets a wave travel through without modification
through and reduces it completely outside. An aperture is therefore described by the52

5. New imaging setup for the Erbium experiment

Figure 5.2.: Sketch of the coordinate system. x, y (resp. x2 , y2 ) correspond to the diffraction or object plane (resp. the observation or image plane). z is the distance between thetwo planes and r is the relative distance between P1 and P2 , the field distributions at (x, y)and (x2 , y2 )function A(x, y), where,A(x, y)

1 inside the aperture
0 outside the aperture

5.47

We consider in the following, P1 (x, y) the incoming complex wave in the aperture andP2 (x, y) the outgoing complex wave at a distance z from the aperture plane. P1 andP2 are also called light field distribution. The notations are resumed in Fig. 5.2.
According to the Huyghens-Fresnel principle and the linearity of Eq. (5.46), the field
P2 (x, y) at a distance z can be calculated by summing over all diverging sphericalwaves coming from the aperture,
Z Zi
exp(−ikr(x, y))
P2 (x2 , y2 ) =dxdyP1 (x, y),
5.48
λr(x, y)where r is the distance between P1 and P2 .
x2 /λ, y 2 /λ << z), Eq. (5.48) becomes,

In the Fraunhofer approximation (5.2. Objective design53
ie−ikzk
P2 (x2 , y2 ) =exp(−i x22 + y22 )
zλ
z

Z Z

2k

dxdyP1 (x, y)e−i z [x2 x+y2 y]

5.49

It is now very easy to calculate the outcoming field as the Fourier transform of theyx
and fx = λz.
aperture distribution evaluated at the frequencies fx = λz
Lets now consider a particular example of the diffraction intensity pattern from acircular aperture, described by A(x, y) = circ(x0 , y0 , R), the circle of centre (x0 , y0 )with a half diameter R. If we insert P1 in Eq. (5.49) we obtain,
i exp(−ikz)
ikzr2
P2 (r) =exp()B[P1 ]|r ,
5.50
zλ
2z
p
where we used the r =x22 + y22 and the circular symmetry to replace the twodimensional integral by the Fourier-Bessel transform B[...]. The Fourier-Bessel transform of a circular function is given by,
B[circ(R)] = AJ1 (2πRρ),
πRρ

where we introduced the reduce variable ρ =order Bessel function.

5.51

p 2
fx + fy2 and J1 represents the first-

Experimentaly, only the intensity of the diffracted light can be observed. The intensity
is defined as the modulus squared of the field and we thus obtain,
I(r) = (1 2
J1 (kRr/z) 2
) A(2
).
λz
kRr/z

5.52

This intensity pattern is known as the Airy pattern or Airy disk and is schown in
Fig. 5.3. The first minimum occurs at,
r0 = 0.61

λz
.
R

5.53

This points out a major difference between ray optics and wave optics. Indeed, an
image of an infinitely small point by an aberration-free system is of finite size. As
consequence, two points at a distance smaller than r0 will not be differentiable bycommon means. This gives the diffraction limit of resolution for an optical system.
By approximating R/z ≈ N A we can write,
r0 = 0.61

λ
.
NA

5.54

We define the diffraction-limited resolution of an optical system as the distance r0 .

54

5. New imaging setup for the Erbium experiment
1

-10

0.8

-5

0.6
0
0.4
5

10
-10

0.2

-5

0

5

10

0
-10

-5

0

5

10

Figure 5.3.: Airy distribution. Left, 2D Airy distribution. Right, cut at y = 0 of the Airy
distribution. the dashed line correspond to the first minimum

Image formation and point-spread function
Thanks to the linearity of the wave equation, one can use the image of a point source tocalculate the light field distribution of a more complex object. Using the supperposition
principle we find,Z Z
P2 (x2 , y2 ) =dxdy h(x, y, x2 , y2 )P1 (x, y),5.55
where h(x, y, x2 , y2 ) represents the image at (x2 , y1 ) of a point source at (x, y) in theobject plane. Using the isoplanatism property in the object plane we can write,
h(x, y, x2 , y2 ) = h(x2 − x, y2 − y)and reduce Eq. (5.55) to,Z Z
P2 (x2 , y2 ) =dxdy h(x2 − x, y2 − y)P1g (x, y) = h(x, y) ∗ P1g (x, y),

5.56

5.57

with P1g (x, y) = P1 (x, y)/∥M ∥ the geometric image of the P1 (x, y) and ∗ stands forthe convolution product. One can apply the same procedure to calculate the intensity
distribution,
I2 (x2 , y2 ) = ∥h(x, y)∥2 ∗ I1g (x, y).

5.58

The function ∥h(x, y)∥2 is referred to as the point spread function(PSF) and represents
the impulse response of an optical system. For an ideal system ∥h(x, y)∥2 is an Airyfunction (Fig. 5.3).

5.2. Objective design

55

Aberrations and Strehl ratio
In practice no imaging system is perfect and aberrations occur. Aberrations can have
many origins, misalignments of optics or surface imperfection of the optical components. They are characterized by the deformation of the wavefront from the reference
spherical wave [Mah98]. The most common ones are the five Seidel aberrations [Smi07]:
The tilt, the defocus, the coma, the astigmatism and spherical aberrations.
To quantify the aberrations, one method consists in reconstructing the wavefront as
a superposition of the Zernike polynomials [Mah13]. This method is very efficient to
get an expression of the aberrations as each polynomial can be associated with one
type of aberration. Nevertheless, accessing experimentally the wavefront is difficult.
It is either asking for many images of PSF at different distances to scan through the
focal point of the optical system [Öt19] [Mar17], or using a Shack-Hartmann wavefront
sensor like the WFS20-5C/M from thorlabs [Kle17]. In the work of this thesis, thesize of the beam going out of the objective has a diameter of 60 mm and a standart
wavefront sensor could not be used, as it would only probe a very small area of thebeam. With the setup employed in the thesis, collecting many PSF at different distanceis also imposible.
Another method to quantify aberrations is to calculate the so-called intensity Strehl
ratio S. In presence of aberrations, the intensity maximum of the Airy peak is reduced
since the spot is bigger than the diffraction-limited one. S is then defined as the ratio
between the peak intensity between the PSF measured and the peak intensity of thecorresponding perfect PSF. This gives a value to S between 0 and 1, where S = 1
corresponds to an ideal system. According to the Marechal criterion [Goo05], we
considere that an optical system is diffraction-limited if S > 0.8.

5.2.2. Design considerations
As seen previously, the resolution of an imaging system depends on the NA and thewavelength λ used. In our experiment, the imaging wavelength λ is 401 nm [Fri14a],which corresponds to a strong transition and allow enough scattering events withatoms. No transition at lower wavelength is available in the optical range for erbium
and therefore λ can’t be reduced to gain further resolution. One other solution to
improve the resolution is to increase the NA. Here we are ultimately limited by the
opening angle given by the size of the viewport, which is fixed for the existing vacuum
chamber. Due to the relatively large distance of the first possible position of theobjective to the object (the atoms themselve), the diameter of the objective has to belarge enough to capture the full angle available. Nevertheless, the bigger the objective
is, the bigger the aberrations and this necessitates a complex optical design.

1

0.8

0.6

0.4

0.2

0
-1.5

-1

-0.5

0

0.5

1

1.5

Figure 5.4.: Design of the objective with the software OpticsStudio. (a) Schematic of thedesign from the company Special Optics.(b) Ray spot diagram for a wavelength of 401 nm
for different incidence angles. (c) Theoretical PSF, featuring a resolution of 532 nm

56

5.3. Testing the performances of the objective

57

We finally chose the design proposed by the compagny Special Optics, which offers
an objective with NA= 0.46, with an effective focal length (EFL) of 60.1 nm andan aperture of 60 mm. This corresponds to a resolution of Res = 532 nm. The
characteristics of the objective are shown in Fig. 5.4. The design produced includes
a compensation for the spherical aberrations introduced by the viewport. The spots
diagram in Fig. 5.4(b) shows the sensibility of the objective to angle of the incomming
beam.
Additionally, the design from Special Optics features an interchangeable front glass
plate, allowing to change the reflection coating of the objective according to the purpose of the current project. Extra documents on the objective are presented in Appendix. B.

5.3. Testing the performances of the objective
5.3.1. Test target and Magnification
25.4× Magnification
CCD

MirrorL

Iris

USAF1951

401nm
Objective
Figure 5.5.: Test target imaging setup. A test target USAG 1952 is illuminated by a collimated beam. The collimated light after the objective is focused by 1524 mm FL Achromaticlens to CCD camera.

Before setting up a new objective in an experiment, we have carried tests of theperformance. We experimentally determined the characteristics of the objective andcompared them to the specifications sold by the company Special Optics.
To get a first measurement on the resolution we try to image the smallest object
possible. For that, we use the standard test target USAF 1951 and try to resolve thesmallest feature possible. Images of groups 7,8 and 9 are given in Fig. 5.6(a,b). We canresolve easily the bars up to group 8, separated by a distance of 1.1 µm. The group 9
presents a maximum distance of 0.98 µm between the bars. At this scale, aberrations
arise and it is not possible to clearly differentiate the bars. This first test indicatesthat the resolution of the objective is inferior to 1 µm but an exact value cannot be
extracted.

58

5. New imaging setup for the Erbium experiment

2500
2000
1500
1000
500
0

20

40

60

80

0
20
40
60

0

10

20

30

40

50

60

Figure 5.6.: Image of a test target UASF1951. (a)(b) group 7-8-9. the red square indicatesthe object used to determine the magnification (b) in green, the position of the square in
the image plane versus position in the object plane. In blue, linear fit.

Measure on the test target also permits to determine the magnification of an opticalsetup. Indeed, we know the size of the camera pixel and the one of a bar and we. This gives a good approximation of M , but to have a more precise
find M = Object
Imagevalue, one method consists in moving an object of a known distance and recording
how much the image has moved. We can directly extract M with the slope of a linear
fit. We applied this method with the square in Fig. 5.6(c) and obtained the result in
Fig. 5.6(b). We found a magnification of,
M = 24.4(3).

5.59

A single pixel of our CCD used has a size of 2.2 µm × 2.2 µm, which corresponds to aregion of ≈ 0.09 µm × 0.09 µm on the object.

5.3.2. Determination of the point spread function
Once we know the magnification of the experimental setup, we can try to image thepoint spread function of the objective. This is not an easy task since we need to createa point source smaller than 532 nm, the resolution of the objective.
The easiest method consists of shining light through a small pinhole [Kle17]. However,for our objective we would need a pinhole with a diameter smaller than 500 nm and5.3. Testing the performances of the objective

59

CCD
Single‐mode fiberൎ25× Magnification

401nm
Iris
MirrorL

Objective

Figure 5.7.: Sketch of the testing setup. A 401 nm light beam is emitted from a single
mode fiber with a mode-field diameter of 3.6 µm and expands freely up to a collimation lens
(L). The collimated beam is focussed through the objective and reflected back by a mirror.
The light is focused by L and sent to a CCD camera recording the PSF

very advanced engineering technics not present at the University of Innsbruck areneeded. Other methods can be used like, imaging gold nanoparticles or light emitted
from a SNOM fiber [Kle17].
We modified the setup of Fig. 5.5 to the one presented Fig. 5.7. Here, the point source
is self-created by the objective, retro-reflected by a mirror and imaged by the objective.
More in detail, a 401 nm beam goes out a single-mode fiber with a mode-field-diameterof 3.6 µm and expands for 1524 mm. A 4-inch achromatic lens is used to create a 60 mm
collimated beam. The beam diameter can be controlled by an iris. The beam is then,focused by the objective onto a mirror and retro-reflected until a 50:50 plate beam
splitter, sending one part of the light to a CCD, where the PSF is imaged.
In this configuration, the objective is probed twice and every aberrations is multipliedby two. Therefore, it is a good way to get an upper bound for the resolution but it
is very hard to reach the specifications delivered by Special Optics. The point spreadfunction obtained is presented in Fig. 5.8.

5.3.3. Analysis of the point spread function
Once we obtained the PSF, we developed a Matlab routine to extract the resolution
and the Strehl ratio. We start by recentering with a 2D gaussian the PSF obtained on
the CCD (Fig.5.8(a)). As seen with Eq. (5.52), the image of a point source presents
a circular symmetry and to reduce the local disturbance we integrate radially the 2D
PSF. This is done by summing over all the pixels of (Fig.5.8(a)) and, calculating the

60

5. New imaging setup for the Erbium experiment

-2

150

-1
100

0
1

50

2
0
-2

-1

0

1

2

0

0.5

1

1.5

2

Figure 5.8.: Experimental point spread function of the objective. (a) 2D centred PSF. (b)In grey, experimental integrated PSF. A fit ∝ sinc2 is represented in blue.

distance between the centre of the pixel and the centre of image, ρobj . We convert
the distance from the image plane to the object plane with the magnification M andobtain the radial PSF presented in Fig. 5.8(b).
To determine the resolution of the objective, we extract the first zero crossing by fitting
a sinc2 function to the data in Fig. 5.8(b). The exact mathematical expression of the
fit is given by,
I0 (sin(Aρ) 2
),
Aρ

5.60

where I0 and A are the fitting parameters. The result of the fit is represented by theblue line in Fig.5.8(b). The resolution is simply given by Res=1/A and we find,Res = 0.85(2) µm
The error corresponds to the uncertainties on the fit parameter A. The resolution issatisfying for future projects and will allow resolving density modulations of a domainsupersolid.
Before calculating the Strehl ratio, the data are transferred to bins with a width of
0.05 µm and shown in Fig. 5.9 in blue. This reduces the local fluctuations and thestatistical uncertainties on the maximum. To determine the Strehl ratio, one needs tonormalize the PSF by the area of the Airy disk (Eq. (5.52)). Mathematically, this isequivalent to write,
Inorm = C × IP SF ,

5.61

with,
RR
A(x, y)dxdy.
C= RR
IP SF dxdy

5.62

5.3. Testing the performances of the objective

61

1.5

100

1

50

0.5

0
0

0.5

1

0
1.5

Figure 5.9.: Binarized and normalized PSF. In blue, binarized point spread function withright axis. In green, normalized PSF, left axis and in red perfect PSF. The error bars
correspond to the standart deviation over the bins.

A(x, y) is the airy function with a radius set to the experimental resolution. The
denominator corresponds to the sum over all the pixels of the local intensity multipliedby the pixel area. We find C = 5.9(3) × 10−3 and dispaly the normalized signal PSF
in green in Fig. 5.9.
It is now possible to directly read the Strehl ratio as the peak intensity of Inorm andwe find,S = 0.88(4).

5.63

The error is evaluated by the propagation of uncertainties with the peak intensity,
the calculated resolution and the magnification. S is over 0.8, indicating that inprinciple the imagining is diffraction limited. Nevertheless, we don’t reach the expected
resolution, meaning that the full aperture is not used. This can come from a slight
defocus of the reflecting mirror and therefore creating a point source bigger than
532 nm. Furthermore, if the aberations are coming from the tilt of the objective,
the spherical asymetry of the PSF is reduced during the spherical integration and byconsequence will have smaller weight in the calculation of S.

5.3.4. Aperture
As we saw, the resolution of an optical system is directly proportional to the NA,
which is proportionnal to the aperture size of the objective. To experimentaly study
how the PSF changes with the aperture size, we vary the aperture size with theiris in Fig. 5.7 and analyse the PSF. The results obtained are presented in Fig. 5.10.
The theoretical resolution as the function of the iris diameter D can be expressed by62

5. New imaging setup for the Erbium experiment

2.5
0.9
0.8

2

0.7
0.6
1.5
0.5
0.4
1

0.3
0.2

0.5

0.1
20

40

60

20

40

60

Figure 5.10.: (a) Resolution with the aperture diameter. In green, the data points. The
blue line represents the theory calculation based on Eq. (5.54). The dark line represents
the aperture limitation due to the mirror size. The dashed line indicates the point where
we expect the theory to differe from the experiment due to aberrations. The errorbars are
smaller than the symbole size. (b) Strehl ratio S versus the aperture. In green, S calculatedwith the resolution measured and in blue with the theoretical resolution. The dashed line
indicates the diffraction limit at S = 0.8.

inversing Eq. (5.54) and yields,
Res(R) =0.61λ.
R
sin(atan( EFL
))

5.64

For small apertures, the experimental data are in good agreement with the theory but
from, D ≈ 40 mm start to deviate. As the maximum resolution obtained is 0.85 µm,
we expect the resolution to saturate from D = 38 mm, corresponding to a resolutionof 0.85 µm in Eq. (5.64). This is represented by the dashed line in Fig. 5.10(a). The
black line corresponds to the limitation of D due to the plastic mount and the sizeof the objective. We also see that the Strehl ratio S decreases with the size of theaperture. This is caused by an increased sensitivity to aberrations. We calculate
S for the measured resolution and theoretical resolution. From D = 38 mm the two
calculations diverge drastically, indicating that the actual aperture is somehow limited
to 38 mm. This can come from the testing setup where the alignment is crucial, anda defocus of less than a µm will cause this deviation.
Another characteristic essential to know is the depth of focus (DOF) as a functionof the aperture. Indeed, during TOF, the atomic cloud expands in 3 dimensions andone part will then be out of focus. To the price of losing resolution, by reducing theaperture, it is possible to increase the DOF and image the cloud. The theoretical
calculations are presented in Fig. 5.11. Unfortunately, it is impossible with the current

5.4. Experimental implementation

63

20

15

10

5

0
20

Figure 5.11.:
diameter D.

30

40

50

60

70

Depth of focus (blue line) and resolution (red) as a function of the iris

testing setup to see the influence of a defocus in the resolution and the Strehl ratio.

5.4. Experimental implementation
The last challenge of this thesis was to implement the new objective in the ERBIUM
experiment. In the current setup, additionaly to the imaging beam, two laser beams
are travelling in the vertical directions, a 401 nm beam for Bragg spectroscopy and1064 nm for the vertical dipole trap. The new objective would allow to focus a 1299 nm
laser beam and use a DMD directly in the image plane. In this section, we present the
new design to implement the objective and the plan for the distribution of the laser
beams.

64

5. New imaging setup for the Erbium experiment

Dichroic mirror D1
Step motor
Z‐stage
XY‐translation stage
XY‐tilt stage
Objective holder
Objective

Figure 5.12.: Drawing of the holding tower of the objective realized with Solidworks 2020.
Design adapted from Maximilian Sohmen thesis [Soh21].

5.4. Experimental implementation

65

5.4.1. Implementing the imaging setup
High-resolution imaging setups are very sensitive to misalignment on the micrometerscale. The imaging beam pointing on the atoms must be perfectly centred and alignedwith the objective and the viewports if one wants to reach the 532 nm resolution
specified by the company Special Optics.
Our Er-Dy experiment developed a support for the objective which we modify toobtain the one presented in Fig. 5.12. It features all the degrees of freedom needed toalign the objective with the atoms. The angle alignment of the imaging beam with theobjective is tuned via a self-made XY-tilt stage. An XY-translation stage KT150 fromOwis is used to centre the beam with the objective. In the current configuration, it
offers a travel range of 20 mm. Finally, the height of the objective is controlled by a Ztranslation stage MT60 also from Owis, featuring a travel range of 50 mm. To be able
to change the height of the objective in a controllable way, we have mounted on the
Z-translation stage a high-resolution linear actuator M-228 from Physical Instrumentwith a 25 mm range. We found that the M-228 presents a resolution of 1 µm with aspeed of 0.1 mm s−1 over a 10 mm range (corresponding to the range between in-situ
to TOF ) but features a bigger error when going back to its inital in-situ position.
We, therefore suggest, when going from TOF to in-situ, to go further and come backto the in-situ position. In that way, the error for both, TOF to in-situ and in-situ
to TOF is of 1 µm, which is low enough to be used in a reliable way. The objectiveis screwed on a self-made holder out of fiber-enforced plastics PAS-PEEK-GF30 from
Faigle Gmbh with a thermal expansion coefficient of 22 µm/(Km). The support alsopresents a holder for a 100 mm diameter dichroic mirror which can be rotated by ±20◦ .
A picture of the objective and the holder mounted in the experiment is presented inFig. 5.15. The breadboards are standard 3/4 of inch breadboards from thorlabs thatwere cut by the University workshop.

5.4.2. Laser distribution for further implementations
This last section is devoted to the future plans for the ERBIUM experiment. As we
mentioned in Sec. 5.1, we want to use the objective for different purposes other thanimaging. Using lasers with different wavelengths is always a challenge as most of the
optical elements are made to work with a small range of wavelengths. Furthermore,
one needs elements to split the laser in different paths also called laser distribution.
The two applications we would like to use the objective for are, the creation of lightpatterns with a DMD setup [Pet15] working with 401 nm and the use of a focused1299 nm. For further projects, we also want to keep the possibility to use 631 nm and532 nm in transmission through the objective.

66

5. New imaging setup for the Erbium experiment

Figure 5.13.: Implementation of the objective in the ERBIUM experiment

To keep as much possible of flexibility, we designed optical paths allowing to use simultaneously light at 401 nm, 1299 nm, 532 nm and eventually 631 nm. The distribution
is presented in Fig. 5.14. We decided, to first split the wavelengths 532 nm and 631 nm
thanks to a 100 mm dichroic mirror (D1 in Fig. 5.14); see Appendix. B. The light fromthe imaging is planned to travel through a 50:50 75 mm cube or plate BS and then,be focused by L1 to a CCD camera Andor Luca. The 50:50 BS splits the light comingfrom the DMD to the light of the imaging. Finally, a second dichroic mirror (D2)overlaps the light at 1299 nm with the one at 401 nm.
A first draft of the spatial organisation was made and is presented in Fig. 5.15.

BS

Iris L
CCD

𝐷

𝐷
DMD

Objective

Figure 5.14.: Schematic side view of the implementation of the objective in the experiment.
The grey arrows indicate the sens of propagation.

67

DMD

1299nmopticsFigure 5.15.: breadboard configuration of the second floor of the ERBIUM experiment.
In blue, the space used for the two 401 nm light splitting. In red, the overlapping with the1299 nm light. In yellow and green are additional spaces for optical implementations.

68

Chapter6

Conclusion

6.1. Summary
The first purpose of this thesis was to participate in the experimental study of dipolargas confined in a 1D optical lattice, creating an array of quasi-2D dipolar gases. We
employed Bloch oscillations to understand the role quantum fluctuations play in anarray of quasi-two-dimensional BEC. We developed a method to quantify the dephasing rate. We observed long-live oscillations when the repulsive short range interaction
and quantum fluctuations balance the attractive long-range dipole-dipole interactions.
A minimum of dephasing occurs at 60 a0 , corresponding to the minimization of thevariance of chemical potential over the lattice sites. By further decreasing the scattering length, we observed the disappearance of the interference pattern in momentum
space, a sign of transition to a localized state in a single lattice plane. To quantify
the transition region, we performed a Fourier transform to the atomic cloud and defined the contrast as the ratio between the main and second order components. The
observations were compared with a discrete quasi-one-dimensional theoretical model
developed in our group. Based on the extended Gross-Pitaevskii theory, it shows agreat quantitative agreement.
The second purpose of this thesis was to design, characterize and implement an highresolution imaging setup, allowing to reach in-situ density distribution of dipolar BEC
of erbium directly in the trap. We imaged the point-spread function of an objective
designed and produced by Special Optics. We found a resolution of 0.85(2) µm, withan intensity Strehl ratio of S = 0.88(4). This is sufficient for the purpose of theexperiment and would allow us to observe in-situ supersolid density modulation. Toalign the objective with the atoms, we designed a support for the objective including
5 precision adjustable degrees of freedom, the XY-tilt, the XY-translation and Ztranslation. The Z-translation can be controlled by a stepper motor to change from
TOF to in-situ imaging reliably. Finally, we draw a first planing for the future laser
implementations of the ERBIUM experiment. This new imaging setup represents a
69

70

6. Conclusion

major step for the actual experiment toward quantum simulation and offers many newopportunities to observe exotic states of matter with dipolar erbium.

6.2. Further understanding of dipolar BEC in opticallattices
Our study on Bloch oscillations opened the door to a deeper questioning. As alreadyemphasized in Ref. [Kum19], dipolar gases in optical lattices represent a unique opportunity to control and experimentally access the effects of quantum fluctuation. In
our experiment we probed BO for a lattice depth of 8 Erec . In order to understandbetter the role played by LHY in distributing the chemical potential over the latticesites without suffering from atom loss, it would be interesting to find a way to shift
the minimum of dephasing to higher as .
We investigated only dipoles aligned along the elongated axes, with principaly meanfield attractive interactions. Early results for B aligned along the gravity axis indicatedthat the behaviour is different. In this configuration DDI, is repulsive and only few
Bloch cycles were observed and further studies are needed.

6.3. Next step for the implementation of the objective
The objective has been set up recently in the experiment, but to reach a sub-micron
resolution, alignment between the atoms, the viewport and the objective must be
perfect. So far, we aligned the imaging beam perpendicular to the viewport, giving
us a reference to align the objective. The next step will include the setting of theimaging optical path and the camera Andor Luka. Once done, we will perform thelast fine adjustment of the objective by trying to resolve the density modulation of asupersolid.

Appendix

A

Code analyse PSF

A.1. main.m
1%% p a r a m e t e r s
ROI cut =50;
4
p i x e l c a m e r a = 2 . 2 ; %i n microns
5
magnification =24.4;
6
e r r o r m a g n i f i c a t i o n =0.4;
7
r e s c o e f f=p i x e l c a m e r a / m a g n i f i c a t i o n ; %r e s c a l i n g c o e f f i c i e n t
8
e r r r e s c o e f f= p i x e l c a m e r a * e r r o r m a g n i f i c a t i o n / m a g n i f i c a t i o n
ˆ2;9p i x a r e a=r e s c o e f f * r e s c o e f f ;10
e r r p i x a r e a=r e s c o e f f * 2 * e r r r e s c o e f f ;2
3

11%% a n a l y s i s%l o a d t h e p i c t u r e s
14
f i l e n a m e =[ ’%%f i l e n a m e%%’ , ’mm. bmp ’ ] ;
12
13

15
16

p s f=imread ( f i l e n a m e ) ;17
18

ROI=c r e a t e R O I ( p s f , ROI cut , 1 ) ;%1D i n t e g r a t i o n o f t h e p s f
21[ rho , p s f i n t ]= PSF 1D (ROI , ROI cut , 1 ) ;22 %c a l c u l a t i o n
o f t he r e s o l u t i o n
23[ r e s , e r r r e s , I0 , e r r I 0 ]= r e s o l u t i o n ( rho , p s f i n t , r e s c o e f f ,e r r r e s c o e f f , 1) ;24 %e x t r a c t i o n
o f th e n o r m a l i s a t i o n c o e f f i c i e n t
25[ C res , e r r C r e s ]=norma (ROI , r e s , e r r r e s , p i x a r e a ,err pix area ) ;
19
20

71

72

A. Code analyse PSF

26%c a l c u l a t i o n o f s t r e h l r a t i o
28
s t r e h l= C r e s * I 0 ;29
e r r s t r e h l=s t r e h l ( inda ) * s q r t ( ( e r r C r e s / C r e s ) ˆ2+( e r r I 0 / I 0 )
ˆ2) ;
27

A.2. create ROI.m
f u n c t i o n [ ROI]= c r e a t e R O I ( data , ROI cut , show p )2 %% c r e a t e
a ROI out o f a 2D p s f f u n c t i o n
3 % showp =1 t o show a
plot

1

4
5

%f i n d t h e c e n t e r
7[ maxv , maxpx]=max( data ) ;8[ maxvv , maxpy]=max( maxv ) ;9
maxpx=maxpx ( maxpy ) ;610%Cut t h e ROI and c e n t e r
12 ROI=do uble ( data ( maxpx=ROI cut : maxpx+ROI cut , maxpy=ROI cut :maxpy+ROI cut ) ) ;13 %removet he background
14 ROI=ROI=mean ( mean ( data ( 1 : 3 0 , 1 : 3 0 ) ) ) ;
11

15
16
17
18

i f show p==1
figure (10)imagesc (ROI)

19
20

end

A.3. PSF 1D.m
1f u n c t i o n [ rho , c f i t ]= PSF 1D (ROI , ROI cut , show p )2% r e t u r n th e c i r c u l a r y i n t e g r a t e d PSF
4 % ROI => 2D a r r a y
5 % Magn => m a g n i f i c a t i o ncoefficient
6 % px sizeo f t he camera
7 % show p => show t h e
plots
3

8
9

% t h e a n a l y s i s i s based on a c i r c u l a r i n t e g r a t i o n o f th e PSF
and f i t t i n g

A.3. PSF 1D.m

1073

% with a s i n c ˆ2

11
12
13
14
15
16
17
18
19
20
21
22
23

[ s1 , s 2 ]= s i z e (ROI) ;x f i t=z e r o s ( 1 , s1 * s2 ) ;y f i t=z e r o s ( 1 , s1 * s2 ) ;c f i t =z e r o s ( 1 , s1 * s2 ) ;f o r i nd1 =1: s1
f o r ind 2 =1: s2
x f i t ( ( ind1 = 1) * s1+i nd2 )==ROI cut+ind1 = 1;
y f i t ( ( ind1 = 1) * s1+i nd2 )==ROI cut+ind2 = 1;c f i t ( ( ind1 = 1) * s1+i nd2 )=ROI( ind1 , ind2 ) ;
end
end

24
25

%f i t a g a u s s i a n t o g e t th e c e n t e r p o s i t i o n27maxv=max( c f i t ) ;28
myfit gau=f i t t y p e ( ’ I * exp ( = 2 * (x=A) . ˆ 2 /Bˆ 2 ) * exp ( = 2 * ( t =C) . ˆ 2 /Dˆ 2 ) ’, ’ indep ’ , { ’ x ’ , ’ t ’ } , ’ c o e f ’ , { ’A ’ , ’B ’ , ’C ’ , ’D ’ , ’ I ’ }) ;29
[ mdl1 , g o f 1 ]= f i t ( [ x f i t ’ , y f i t ’ ] , c f i t ’ , myfitgau , ’ s t a r t ’, [ 0 , 3 , 0 , 3 , maxv ] ) ;26

30%r e c e n t e r th e p s f
c e n t e r x=mdl1 .A;
33
c e n t e r y=mdl1 .C;

31
32

34

%c i r c u l a r n u m e r i c a l i n t e g r a t i o n36
rho=z e r o s ( 1 , l e n g t h ( c f i t ) ) ;37
f o r i nd1 =1: l e n g t h ( c f i t )38
rho ( ind1 )=s q r t ( ( x f i t ( ind1 )= c e n t e r x ) ˆ2+( y f i t ( ind1 )=c e n t e r y ) ˆ2 ) ;
39
end
35

40
41
42
43
44

i f show p==1
figure (1)imagesc (ROI)45

figure (2)h old on ;p l o t ( rho , c f i t , ’ o ’ , ’ M a r k e r s i z e ’ , 1 )

46
47
48
49

end

74

A. Code analyse PSF

A.4. resolution.mf u n c t i o n [ r e s , e r r r e s , I0 , e r r I 0 ]= r e s o l u t i o n ( rho , psf 1D ,r e s c o e f f , e r r r e s c o e f f , show p )2 %c a l c u l a t e
th e r e s o l u t i o n with a s i n c ˆ2 f i t
3 % rho => X a x i s
o f t he p s f i n p i x e l
p s f , 1D a r r a y
4 % 1D ROI= 1 D i n t e g r a t e d
5 % p x s i z e => s i z eo f t he p i x e l o f camera
6 % magn => m a g n i f i c a t i o n
o f t he o p t i c a l system
on t he m a g n e f i c a t i o n
7 % err magn => e r r o r
see plots
8 % show p=1 t o
1910

maxv=max( p s f 1 D ) ;12 %c a l c u l a t e
th e r e s o l u t i o n with a s i n c ˆ2 f i t
11

13
14

15

16
17
18

19

m y f i t=f i t t y p e ( ’ I * s i n c (A* x ) ˆ2 ’ , ’ indep ’ , { ’ x ’ } , ’ c o e f ’ , { ’A ’ , ’ I ’ });f i t p o l y=f i t t y p e ( ’A+(x=B) ˆ3+C* x ˆ2 ’ , ’ indep ’ , { ’ x ’ } , ’ c o e f ’ , { ’A ’ ,’B ’ , ’C ’ }) ;f i t f u n c=@( I , A, x ) I * s i n c (A* x ) . ˆ 2 ;[ mdl , g o f ]= f i t ( rho ’ , psf 1D ’ , myfit , ’ s t a r t ’ , [ 0 . 1 , maxv ] ) ;[ mdl poly , g o f ]= f i t ( rho ’ , psf 1D ’ , f i t p o l y , ’ s t a r t ’ , [ maxv
, 0 , 1 ] , ’ lower ’ ,[0 ,0 , =1000]) ;r e s =1/mdl .A* r e s c o e f f ; %r e s o l t u i o n i n um

20
21

%r e s o l u t i o n = 0 . 6 2 7 ;22%c a l c u l e r r o r on th e r e s o l u t i o n c a l c u l a t i o n
a lp h a = 0 . 6 8 ;25
c i = c o n f i n t ( mdl , alpha ) ;26
e r r A =( c i ( 2 )= c i ( 1 ) ) / 2 ;27
e r r I =( c i ( end )= c i ( end = 1) ) / 2 ;
23
24

28

%g i v e t h e e r r o r on t he r e s o l u t i o n30
e r r r e s=r e s * s q r t ( ( e r r A /mdl .A) ˆ2+( e r r r e s c o e f f / r e s c o e f f ) ˆ2 ) ;31
I 0=mdl . I ;32
e r r I 0=e r r I ;33
i f show p==134x p l o t=l i n s p a c e ( 0 , 3 0 , 1 0 0 0 ) ;35
figure (20) ;36
h o l d on ;37
p l o t ( rho * r e s c o e f f , psf 1D , ’ o ’ , ’ M a r k e r s i z e ’ , 1 ) ;38
p l o t ( x p l o t * r e s c o e f f , mdl ( x p l o t ) ) ;29

A.5. norma.m

39
40
41

x l a b e l ( ’ d i s t a n c e t o c e n t e r (\mu m) ’ ) ;ylabel ( ’ intensity (a . u) ’ ) ;xlim ( [ 0 , 2 ] ) ;end
43 %p l o t ( x p l o t * 0 . 1 , mdl poly ( x p l o t ) ) ;42

A.5. norma.m
f u n c t i o n [ C 2D , err C 2D ]= norma ( psf 2D , r e s , e r r r e s ,pix area , err pix area )2 % renormalizet h e wave f u n c t i o n
3 % returnth e r e n o r m a l i z a t i o n c o e f f i c i e n t and t he e r r o ra s s o c i a t e d to i t
14%p s f 2 D => 2D a r r a x o f th e p s f%r e s => r e s o l u t i o n o f th e o b j e c t i v e
7 %e r r r e s => e r r o ro f t he r e s o l u t i o n
8 %p i x a r e ae q u i v a l e n t o b j e t o f t he a r e a o f one p i x e l
e r r o r o f t he p i x e l a r e a
9 %e r r p i x a r e a
5

6

10

%% e x p e r i m e n t a l d e t e r m i n a t i o n o f t he i n t e g r a l
12
i n t 2 d=sum ( sum ( p sf 2 D ) ) * p i x a r e a ;13
e r r i n t 2 d= sum ( sum ( p s f 2 D ) ) * e r r p i x a r e a ;
11

14
15

%% 2D p e r f e c t b e s s e l f u n c t i o n
17 X= [ ] ;Y= [ ] ;1618x = =10:0.2:10;
20
y = =10:0.2:10;
21 %s c a l i n gof x , y22
r e s c a l=r e s / 3 . 8 3 1 7 ; %r e s c a l i n g f a c t o r
23r e s c a l m a x =( r e s+e r r r e s ) / 3 . 8 3 1 7 ; %r e s c a l i n g f a c t o r
24 X=meshgrid ( x ) ;25 Y=meshgrid ( y ) ’ ;26
Rho=s q r t (X.ˆ2+Y. ˆ 2 ) ;27
I0 = 1;28
I = I 0 * (2 * b e s s e l j ( 1 , Rho ) . / ( Rho ) ) . ˆ 2 ;29
I ( i s n a n ( I ) ) =1;30
figure (1)31 X=r e s c a l *X;
32 Y=r e s c a l *Y;
33
s u r f a c e (X, Y, I , ’ E d g e c o l o r ’ , ’ none ’ )
19

75

76

A. Code analyse PSF

34
35
36

t h e o r y p i x a r e a =(0.2 * r e s c a l ) ˆ 2 ;t h e o r y p i x a r e a m a x =(0.2 * r e s c a l m a x ) ˆ 2 ;37%i n t e g r a l o f I
39
i n t 2 D t h e o r y=sum ( sum ( I ) ) * t h e o r y p i x a r e a ;40
i n t 2 D t h e o r y m a x=sum ( sum ( I ) ) * t h e o r y p i x a r e a m a x ; %upperbound f o r t he i n t e g r a t i o n o f t h e a i r y p a t t e r n .
41
e r r i n t 2 D t h e o r y=i n t 2 D t h e o r y m a x = i n t 2 D t h e o r y ;
38

42
43

C 2D=i n t 2 D t h e o r y / i n t 2 d ;45
err C 2D=C 2D * s q r t ( ( e r r i n t 2 D t h e o r y / i n t 2 D t h e o r y ) ˆ2+(e r r i n t 2 d / i n t 2 d ) ˆ2 ) ;46 %y l a b e l ( ’ Normalized PSF Amplitude ’ )47 %x l a b e l ( ’ R a d i a l
Coordinates ’ )44Appendix

B

Technical documents ofimaging setup

This chapter presents additionnal documents related to the new imaging setup.

77

Figure B.1.: Technical drawing of the objective produced by SpecialOptics

78

A

B

C

D

7

6

8

2X

7

99.00 TYP

14.53

75.00

6

ALL MECHANICALS PARTS: ULTEM 2300 PEI-30
FINISH: NATURAL

8

5

66.93
(CENTER OF MASS)

5

120.41

4

THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF
SPECIAL OPTICS. ANY REPRODUCTION
IN PART OR AS A WHOLE WITHOUT
THE WRITTEN PERMISSION OF SPECIAL
OPTICS IS PROHIBITED.

PROPRIETARY AND CONFIDENTIAL

4
ZONE

13.07

20°

3

DO NOT SCALE DRAWINGCAD PART N0. 291691

THIRD ANGLEPROJECTION

INTERPRET GEOMETRIC TOLERANCING
PER: ASME Y14.5M, 1994

66.00

FINISH:PROJECT #MATERIAL:

AS NOTEDAS NOTED

2

AFP

E4235

DATE
7/6/2021

NAME

OUTLINE

28.60(VACUUM)

3.30(SILICA)UNLESS OTHERWISE SPECIFIEDUNTOLERANCED DIMENSIONS ARE INMILLIMETERS, ARE BASIC, AND INCLUDE
CHEMICALLY APPLIED OR PLATEDFINISHES±0.5º
ANGULAR:±0.25
ONE DECIMAL PLACE:DRAWN BY
±0.13
TWO DECIMAL PLACE:CHECKED
THREE PLACE DECIMAL: ±0.013
0.4
MAX SURF. ROUGHNESS:

28.20
(AIR)

2.21

SCALE: 1:1

B

SIZE

TITLE:1

ASSEMBLY

07/06/2021

DATE

1

WEIGHT:2.47lb

PART NUMBER

1-291691

APPROVEDSHEET 1 OF 1

REVSpecial Optics Inc.
3 Stewart Court
Denville, NJ 07885

57-60.1-65 @ 401nm

REVISIONS

89.00

PROTOTYPE-

2
DESCRIPTION

REV.

2.70
(REMOVABLE WINDOW)

3

B

C

D

Figure B.2.: , Coating of the dichroic mirror D1 produced by OptoSigma79

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Acknowledgements
First, I would like to thank my supervisor Francesca Ferlaino for giving me the opportunity to work in the ERBIUM experiment. I’m thankful for the support she gave me
all along this journey, up to the very last minute before handing in this thesis.
I’m very grateful to Manfred for all the time he accorded me during this project andfor his joy to share technical knowledge. Thank you also for the last-minute corrections
to the thesis.
Je remercie également l’école des Mines de Saint-Etienne qui m’a permis de réaliser ce
double master avec l’université d’Innsbruck, en particulier Patrick Ganster pour avoirété mon tuteur durant cet échange. Un remerciement va aussi à mes professeurs declasse préparatoire qui m’ont donné le goût pour les sciences fondamentales et m’ont
orienté dans la voie de la recherche.
Thanks to all the members of the ERBIUM, the ER-DY and T-REQS experiments
as well as the members of the theory group, including Russel, and Wyatt with whom
I share my office for the last months. Special thanks go to, Gabriele for introducing
me to the very rich and exact science of coffee, Alexander for sharing with me all its
technical expertise and Arfor for the time spent in the lab with a very good mood andnice music. Thanks also to Daniel, Simon, Sarah and Théo to have shared time withme in the lab.
A deep thank goes to our administrative assistant Silvia, who dealt with all my orders
and was always providing me with administrative helps in the fastest delay. This thesis
would not have been possible without the help of the mechanical workshop from boththe university and IQOQI, thank you.
I also want to thank the people outside the university, starting with my incredible
WG, who always gave me the chance to talk about all and nothing after long daysin the lab. Ready at any time to go skiing, climbing or simply watching a movie. I
should also not forget to thank our group climbing and diner with who I spent many
hours whether outside or inside climbing.
Finalement, et non le moins important, je voudrais remercier ma famille. Merci à mon
Papa et ma pitite moman de m’avoir donné l’opportunité d’aller étudier à Innsbruck
et toujours encouragé dans mes choix d’étude. Merci à mes frères, Emilien et Anatole,
pour me faire rire et rêver avec vos idées extravagantes. Enfin, un grand merci àNadine pour me donner un socle si solide dans ma vie, je t’aime vvvvraiment.

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 inhaltlich den
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.

Datum

Unterschrift