論文 / 著書情報
Article / Book Information
題目(和文)

イッテルビウム量子気体顕微鏡

Title(English)

Quantum gas microscope for ytterbium atoms

著者(和文)

ﾐﾗﾝﾀﾞﾏﾙﾃｨﾝ

Author(English)

Martin Miranda

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第10063号,
授与年月日:2016年3月26日,
学位の種別:課程博士,
審査員:上妻 幹旺,金森 英人,古賀 昌久,西田 祐介,相川 清隆

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第10063号,
Conferred date:2016/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)Quantum gas microscope forytterbium atoms
Martin Santiago Miranda

A thesis presented for the degree ofDoctor of Philosophy

Department of Physics
Tokyo Institute of Technology
JapanAbstract
In this thesis I report a microscope system for fluorescence imaging of ultra-cold ytterbium
atoms trapped in a two-dimensional optical lattice with single-site resolution.
Having a rich variety of isotopes, ytterbium atoms trapped in two-dimensional optical lattices are a promising tool to study many-body quantum phenomena resulting from stronglyinteracting systems. In particular, quantum simulators using fermionic atoms are useful forstudying the Fermi-Hubbard model, which is expected to be the key to elucidate the mechanism of high temperature superconductors.
In order to observe atoms trapped in an optical lattice, a large number of photons is required to obtain a well resolved image. In contrast with to the conventional method of lasercooling the atoms while observing their fluorescence, in this experiment a deep potential was
created using a combination of a shallow ground-state and a deep excited-state potentials,
which confines the heated atoms during the imaging process. The resulting quantum gas microscope was able to resolve individual lattice sites in an optical lattice with a 544 nm spacing.

1
And I’m not happy with all the analyses
that go with just the classical theory,
because nature isn’t classical, dammit,
and if you want to make a simulation of nature,
you’d better make it quantum mechanical,
and by golly it’s a wonderful problem,
because it doesn’t look so easy.
Richard Feynman

Contents
1 Introduction

5

1.1 Historical background of laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2 Quantum simulation and high temperature superconductors . . . . . . . . . . . . . .

6

1.3 Purpose of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Ytterbium

10

2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Energy structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Scattering lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Quantum gas microscope15

3.1 Sub-Doppler cooling methods in alkali atoms . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1

Polarization gradient cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2

Side-band cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3

Sub-doppler cooling in Yb atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Doppler cooling in ytterbium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1

Magic-wavelength potential and 1 S0 -3 P1 molasses . . . . . . . . . . . . . . . . . 22

3.3.2

Estimation of the magic-wavelength for the 1 S0 -3 P1 transition . . . . . . . . 25

3.3.3

Deviation from magic-wavelength for the 1 S0 -3 P1 transition . . . . . . . . . . 27

3.3.4

Magic-wavelength potential and 1 S0 -1 P1 molasses . . . . . . . . . . . . . . . . . 28

3.3.5

Estimation of the magic-wavelength for the 1 S0 -1 P1 transition . . . . . . . . 29

3.4 The deep potential approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Lifetime and limitations of the deep-potential approach . . . . . . . . . . . . . . . . . . 35
3.5.1

Excitation beam radiative force effects . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Solid immersion lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2

CONTENTS

3

4 Experiment: Transport of atoms to the solid immersion lens surface

43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 The vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Laser cooling of Yb atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.1

Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.2

Magneto optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Transport using optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.1

The optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5.2

Horizontal ODT: Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5.3

Horizontal ODT: Lifetime and temperature . . . . . . . . . . . . . . . . . . . . . . 53

4.5.4

Horizontal ODT: Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.5

Vertical ODT: Evaporative cooling using a cross ODT . . . . . . . . . . . . . . . 58

4.5.6

Vertical ODT: Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Optical accordion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1

System for changing the accordion angle . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6.2

Optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.3

Accordion lens design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.4

Removal of amplified spontaneous emission . . . . . . . . . . . . . . . . . . . . . 69

4.6.5

Alignment of the accordion lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6.6

Loading of atoms into the optical accordion . . . . . . . . . . . . . . . . . . . . . 74

4.6.7

Bose-Einstein condensate: Single accordion . . . . . . . . . . . . . . . . . . . . . 75

4.6.8

Bose-Einstein condensate: Double accordion . . . . . . . . . . . . . . . . . . . . 76

4.6.9

Compression of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6.10 Density profile of the BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6.11 Stability of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Experiment: Fluorescence imaging

85

5.1 Loading of atoms into the 2D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Imaging of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Analysis of the microscope performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.1

Microscope resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4.2

Lattice reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4.3

Lifetime analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4.4

Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5 Light shift in a 5-level diamond system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

CONTENTS

46 Extension to fermionic isotopes

104

6.1 Deep potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1

AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.1.2

Hyperfine splitting interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1.3

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Feasibility of the extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7 Conclusions112

7.1 Experiment summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2 Current quantum gas microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.5 Future experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A Laser cooling and trapping

116

A.1 Optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.1.1 Optical dipole trap generated by a Gaussian beam . . . . . . . . . . . . . . . . . 117
A.1.2 Standing Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 Potential under the solid immersion lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.3 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.3.1 Efficiency of evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.3.2 Speed of evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction
1.1 Historical background of laser coolingLaser cooling was proposed independently by Hänsch and Schawlow [1] and Wineland andDehmelt [2] in 1975 and first realized experimentally in 1978 using magnesium ions [3]. The
laser cooling techniques were soon extended to neutral atoms in 1981, when an atomic beamof sodium atoms was slowed down to 1.5 K using a resonant laser beam. This was later improved in 1982, when the Zeeman slower technique was successfully used to reduce the temperature of a sodium atomic beam to less than a few tens of mK [4, 5].
One of the most important applications of laser cooling at that time was the possibility ofincreasing the accuracy of spectroscopy experiments and atomic clocks, which were mainly
limited by Doppler effects and time-dilatation shifts [6]. The current definition of a second is
the duration of 9192631770 cycles of radiation corresponding to the transition between thetwo hyperfine levels of the ground state of the cesium-133 atom. Since the realization of laser
cooling, the uncertainty of these cesium atomic clocks improved ∼100 times to reach a valueof nearly ∼3 × 10−16 . That is, the clock will not gain nor lose a second in more than 100 million years. As big as this number may seem, a new emerging technology called “optical latticeclocks” [7, 8] further improved this number by two orders of magnitude [9, 10] by exploiting
optical transitions of ultra-cold neutral atoms, which have frequencies five order of magnitude larger than hyperfine transitions. Today, laser cooling continues to be a fundamental
tool for maintaining frequency standards which support our daily lives through global positioning systems (GPS).
In recent years, laser cooling has also become a state-of-the-art tool to investigate manybody physics in the condensed matter field. When atoms are laser cooled to very low temperatures, their spatial extent of the wave packet, which is determined by the thermal de Broglier2πħ
h2, increases. When the inter-atomic distance is comparable with the therlength λd B = mkBTmal de Broglie length, the wave packets of each atom start to overlap and quantum effects

5CHAPTER 1. INTRODUCTION

6

become important. In the case of bosonic particles, the atomic cloud undergoes a quantummechanical phase transition and form a Bose-Einstein condensate (BEC) at n0 λ3d B ≈ 2.6,
where n0 is the peak atomic number density.
The history of BEC reaches back to 1924, when Satyendra Nath Bose and Albert Einstein
predicted the existence of a new state of matter [11, 12]. In 1937, Kaptisa, Allen and Misener
reduced the temperature of 4 He and observed that the liquid helium flowed with no viscosity
at temperatures below 2.17 K [13, 14]. One year later, theoretical work by Fritz London showedthat the zero viscosity phonema observed in liquid helium was evidence of a new superfluid
state of matter, which is intimately related to the BEC of the bosonic 4 He [15]. The existence
and the size of the BEC fraction in superfluid remained controversial for many years, mainly
hindered by presence of strong interactions between the atoms in the liquid. In 1968, a ∼10%
condensate fraction was observed using neutron scattering techniques [16, 17, 18].
With the aid of laser cooling and evaporative cooling [19] techniques, the first realization of a “pure” BEC in sodium and rubidium was reported by Nobel prize winners Wieman,Cornell and Ketterle [20, 21] in 1995. In contrast with the BEC of helium where the atoms
in the liquid are strongly interacting, the condensates formed with ultra-cold gases allowed
researchers to study the properties of weakly interacting systems. In the last 20 years the research on BEC has grown explosively, resulting in the observation of many fascinating physics
such as condensate interference [22], Anderson localization [23, 24], vortex lattices [25, 26],
Berezinskii-Kosterlitz-Thouless transition [27], Feschbach resonances [28, 29], Efimov quantum states [30] and the BCS-BEC crossover [31, 32, 33, 34, 35, 36, 37].

1.2 Quantum simulation and high temperature superconductors
Realization of everyday practical applications is one of the major tasks for laser cooled condensed matter. Ultra-cold atoms trapped in periodical potentials created by light interference (i.e. optical lattices) demonstrated to be a novel tool for studying interacting many-bodyquantum systems and creating quantum simulators [38, 39], which serve as a tool to understand the physics of solids. Fermionic atoms trapped in optical lattices, in particular, are optimal for simulating the Fermi-Hubbard model, which is believed to be the key to elucidate themechanism of cuprates high temperature superconductors [40, 41, 42, 43, 44]. Understanding
such mechanism could lead to the discovery of room-temperature superconductors, producing a huge impact in our daily lives through the improvement on generation and distributionof electric power, transportation (magnetic levitation), medicine (MRI) and the production offaster and more efficient microprocessors.
Why do we need quantum simulators to study cuprate superconductors? Thirty years
have passed since Nobel prize winners Bednorz and Muller first discovered cuprate high tem-

CHAPTER 1. INTRODUCTION

7

perature superconductors in 1986, but the mechanism responsible of generating these material is still not completely understood. Solid-state physics phenomena resulting in cuprate
superconductors is characterized by the movement of electrons inside a crystalline structure.
As the electrons are strongly correlated to each other due to the Coulomb interaction between
them, a complete understanding of a single electron state requires the knowledge of the stateof all the remaining electrons. In order to simulate strongly correlated quantum system likethis using a classical computer, the required number of bits would scale exponentially withthe size of the system. Quantum Monte Carlo (QMC) methods, which are effective for solving the Bose-Hubbard Model, notoriously suffer from the so-called “minus sign problem” [45]when solving the Fermi-Hubbard Model. Using ultra-cold neutral atoms trapped in opticallattice can circumvent this problem, as the quantum simulation of the Fermi-Hubbard model
can be realized in an experimental way.
In recent years, Markus Greiner team in Harvard University successfully implemented aquantum gas microscope with a resolution of 600 nm capable to address single atoms trappedin a two-dimensional optical lattice [46]. By using a system comprised by a solid immersion
lens (SIL) and a high numerical aperture objective lens (NA=0.55), they successfully detected
single rubidium atoms in a two-dimensional optical lattice. Later in 2010, Immanuel Blochteam also implemented a high resolution microscope capable of addressing individual Rb
atoms using a very high resolution objective lens (NA=0.68) [47]. Several demonstrations of aquantum simulators were realized with the assistance of quantum gas microscopes, including the simulation of antiferromagnetic spin chains [48], quantum walks [49], and Ising quantum magnets [50]. Since the creation of these magnificent tools used to observe and manipulate rubidium atoms with single-site resolution, the number of researchers trying to develop
quantum gas microscopes for other species, which would allow them to simulate a wider diversity of systems such as strongly correlated Fermi-Hubbard systems, have increased. It was
not until 2014 that this technology was successfully extended to ytterbium atoms (this work[51]), and in 2015, to lithium [52, 53, 54], potassium [55, 56, 57] and also ytterbium on a different way [58].

1.3 Purpose of this thesis
The final objective of our research is to realize the quantum simulation of the Fermi-Hubbard
model. One of the biggest challenges to overcome is that the temperature required to reachd-wave superfluidity is below the Neel temperature, which is in the order of Tn /TF ≈ 0.01,
where TF is the Fermi temperature [59]. Currently, the minimum temperature obtained withfermionic species is in the order of T /TF ≈ 0.1, which is one order of magnitude higher than
the required temperature. Another difficulty is that temperatures below T /TF ≈ 0.1 cannot beCHAPTER 1. INTRODUCTION

8

measured using the time-of-flight technique [59]. One idea to reduce the temperature of theatoms is to realize spatial filtering [60], which consists in tailor the lattice potential to createhigh entropy regions than can be later removed. To measure the temperature, instead of measuring the momentum distribution using the time-of-flight technique it is possible to obtainthe temperature from Quantum Monte Carlo fittings of the in-situ distributions. Both of thesetechniques, however, require that atoms are resolved with single-site resolution. Although aquantum gas microscope is capable of measuring and the optical lattice with single-site resolution, when I started my research the technology was only available for rubidium atoms
which have only bosonic isotopes.
The purpose of this thesis is to realize a quantum gas microscope of ytterbium atoms,
which have a fermionic isotope (173 Yb). One of the advantages of Yb is that the ground stateis 1 S0 , and thus, there is no electronic spin in the ground state resulting in low decoherence times due to magnetic-field fluctuations. Also, the absence of total angular momentum(J = 0) results in SU(N) symmetry which prevents spin exchanging collisions. Additionally, Yb
have two metastable states with lifetime of several tens of seconds, which are useful for highresolution spectroscopy [61] and for cooling fermionic atoms in a optical lattice through spacial filtering [60]. The 3 P2 state can also be used to to tune inter-atomic interactions throughFeshbach resonances [62].
To test the performance of a quantum gas microscope it is convenient to start with the174

Yb bosonic isotope, which have simpler energy structure due to the lack of hyperfine split-ting. The quantum gas microscope presented in this work uses the bosonic isotope, but thesame method can be applied to the fermionic isotope without major modifications of thesystem, as explained in the last chapter.

1.4 Thesis overview
This thesis is organized as follows:
Chapter 2: Ytterbium
A briefly description of the properties of Yb atoms and the most important energy levels.
Chapter 3: Quantum gas microscopeThis chapter centers in explaining the requirements of a quantum gas microscope andanalyzing the possible fluorescence imaging strategies applicable to Ytterbium atoms.
The “deep potential” strategy used to obtain a large number of photons during imaging
is explained in detail, together with simulations to test the feasibility of the method.
Chapter 4: Experiment: Transport of atoms to the SIL surface
To create a two-dimensional optical lattice with trapped Yb atoms, atoms are first re-CHAPTER 1. INTRODUCTION

9quired to be confined in a pancake-shaped region which is thinner than the depth offield of the objective lens. This chapter focuses in the experimental method used totransport ultra-cold Yb atoms under the surface of the solid immersion lens. The “optical accordion” technique used to create a Bose-Einstein condensate and compress theatoms into a thin layer is also explained in detail.
Chapter 5: Experiment: Fluorescence imaging
In this chapter, the experimental method to load the thin condensate of atoms under
the surface of the solid immersion lens into a two-dimensional optical lattice is first
explained, followed by an analysis of the fluorescence images that were obtained with
the quantum gas microscope using the “deep potential” strategy.
Chapter 6: Extension to fermionic isotopes
This chapter focuses in the requirements to extend the quantum gas microscope of ytterbium atoms to the 173 Yb fermionic isotope.

1.5 List of publications
The most relevant parts of this thesis have been summarized in the following journal articles:• M. Miranda, A. Nakamoto, Y. Okuyama, A. Noguchi, M. Ueda, and M. Kozuma, Alloptical transport and compression of ytterbium atoms into the surface of a solid immersion lens, Physical Review A 86, 063615 (2012).
• Martin Miranda, Ryotaro Inoue, Yuki Okuyama, Akimasa Nakamoto, and Mikio Kozuma,
Site-resolved imaging of ytterbium atoms in a two-dimensional optical lattice, PhysicalReview A 91, 063414 (2015).
The following journal article is also related to this thesis:• Toshiyuki Hosoya, Martin Miranda, Ryotaro Inoue, and Mikio Kozuma, Injection locking
of a high power ultraviolet laser diode for laser cooling of ytterbium atoms, Review of
Scientific Instruments 86, 073110 (2015).

Chapter 2

Ytterbium
Ytterbium is a rare earth element and member of the lanthanoid group with atomic number
70 and chemical symbol Yb. With only two electrons in the outermost shell resulting in theabsence of electronic spin in the ground state, the Yb atom have a similar energy structurecompared with other alkaline-earth-metal-like such as Ca, Sr, and Hg.

2.1 General properties
Naturally occurring ytterbium is composed of seven stable isotopes, five of which are fermionsand two are bosons. The natural abundance and nuclear spin of each isotope is summarizedin Table 2.1. Compared to other elements, the abundance is more homogeneously distributed
between the isotopes.
Mass number

Natural abundance

Nuclear Spin

Type

168

0.13

0

Boson

170

3.05

0

Boson

171

14.3

1/2

Fermion

172

21.9

0

Boson

173

16.12

5/2

Fermion

174

31.8

0

Boson

176

12.7

0

Boson

Table 2.1: Stable isotopes of Yb [63]At room temperature, Yb is a solid metal with silvery color, having an atomic mass of
173.04, a density of 6.97 g/cm3 , and melting point of 1097 K. Ytterbium is mostly non reactive, reacting very slowly to oxygen and water. Due to the high melting point, the saturated
vapor pressure at room temperature is very low, thus requiring heating up to 700 K in order toobtain sufficient gas for the experiments.
10CHAPTER 2. YTTERBIUM

11

2.2 Energy structure
Ytterbium have two valence electrons in the outermost shell resulting in a 1S0 ground statewith no electronic spin. Similar to other alkaline-earth-metal-like species the excited states
are singlet (S = 0) or triplet (S = 1) states. The energy levels up to 45000 cm−1 are summarized
in Fig. 2.1. All energy levels were represented by the term symbol 2S +1 L J , where S is the total
electronic spin (2S + 1 is the multiplicity), L is the total orbital angular momentum of all theelectrons in the system (represented by the letters S , P , D for L = 0, . . . , 2), and J is the total
angular momentum of the system. The important transitions are briefly described bellow.
1

S0 ↔ 1 P1 (399 nm)
An electric dipoled (E1) allowed transition resulting in a strong linewidth of Γ /2π =27.9 MHz. The strong radiative pressure of this transition is useful for Zeeman slowingof the atomic beam. In this experiment, this transition was used for fluorescence imaging the atoms in the two-dimensional optical lattice. Additionally, absorption imagingof the atoms is realized using this transition. This transition is essentially cyclic, with avery small branching ratio of roughly 10−7 to the 3 D1 and 3 D2 states [64].

1S0 ↔ 3 P1 (556 nm)
This singlet to triplet transition is prohibited by the LS coupling selection laws. For
heavy atoms as Ytterbium, JJ coupling between the orbital and spin motion in the twovalence electrons results in the relaxation of the selection law, resulting in this intercombination transition. With a linewidth of Γ /2π = 181 kHz, this cyclic transition havea low Doppler cooling limit and is ideal for laser cooling in the magneto optical trap.

1

S0 ↔ 3 P0 (578 nm)
In this double forbidden transition, the 3 P0 state is very weakly coupled to the 1 S0 groundstate with decay rates in the order of tens of millihertz. The resulting ultra-narrow transition is called clock-transition. In contrast with the 1 S0 ↔ 3 P2 ultra-narrow transition, the absence of electronic angular momentum in the excited state reduces the effects of magnetic fields, making this transition ideal for the creation of a new frequency
standard[65, 66]. Additionally, this transition was implemented for the study of twoorbital SU(N) systems [67], and proposed for the implementation in quantum computation [68] and side-band cooling [69].

1S0 ↔ 3 P2 (507 nm)
This quadruple magnetic (M2) transition is also an ultra-narrow transition with a linewidthof a few tens of millihertz. In contrast with the 1 S0 ↔ 3 P0 clock transition, the excitedstate have a large magnetic dipole moment of 3µB , which is useful to tune dipole-dipole
interactions between atoms in an optical lattice [70]. In recent experiments, the narrow

CHAPTER 2. YTTERBIUM

12

linewidth advantage of this transition was effectively utilized to optical-spectral andmagnetic-resonant imaging of atoms trapped in optical lattices [61, 71].
1

S0 ↔ 3 D2 (404 nm)
This quadrupole dipole (E2) transition with a linewidth of Γ /2π = 350 kHz is useful toexcite atoms to the 3 P2 and 3 P0 states through optical pumping. Atoms excited to the3

D2 state decay into the 3 P1,2 states with a ratio of 20:3. As the atoms in the 3 P1 state

decay back to the ground state, after a few cycles all the atoms can be pumped into
the 3 P2 state [72]. Additionally, by combining the 1 S0 ↔ 3 D2 transition with the dipole
allowed 3 P2 ↔ 3S1 transition, it is possible realize optical pumping to the 3 P0 state.
1

P1 ↔ (6s7s)1 S0 (1077 nm)
This dipole allowed transition from the excited 1 P1 have a linewidth of Γ /2π = 3.0 MHz
was used in this experiment to produce a deep potential in the excited 1 P1 state. Atoms
excited to the (6s 7s )1 S0 state also weakly decay to the 3 P1 state with a decay rate of 2π ×
0.2 MHz [73].

2.3 Scattering lengths
The scattering length of each isotope was estimated in previous experiments using two-color
photoassociation spectroscopy [74]. The values for the bosonic 174 Yb isotope and the fermionicisotopes (171 Yb and 173 Yb) were summarized in Table 2.2.

171
173
174

171

173

174

-0.15(19)-30.6(3.2)

22.7(7)

10.55(11)7.34(8)5.55(8)

Table 2.2: Calculated s-wave scattering lengths in nm for the 171 Yb, 173 Yb and 174 Yb isotopes[74].

CHAPTER 2. YTTERBIUM

13

Ionization Energy (50443.2 cm-1)

44760.37

41939.90(6s8s) 1S0(6s8s) 3S1

41615.0443805.42(6p2) 3P2
(6p2) 3P1

42436.91(6p2) 3P0

(6s7p) 3P 2 38551.93(6s7p) 3P 1 38174.17(6s7p) 3P 0 38090.71
34350.65(6s7s) 1S0

32694.692(6s7s) 3S1
z)
MH
.3
z)
(3
kH
)
nm
10
Hz
77
M
(2
10
.9
m
z)
5
(
1n
MH
61
nm
.5
0
(1
77
m
9n
64

(5d6s) 1D2 27677.665

25068.222(6s6p) 1P1(6s6p) 3P2

17992.007
17288.439(6s6p) 3P1(6s6p) 3P0

nm
4

507

555

.8 n

40

m(
181
nm
kHz
(11
)
mH
z)

(7
nm

m(

27.

mH
z)

9M

Hz)

(3
50

kH

z)

19710.388

578

.9 n
398

532 nm ODT(6s6d) 1D2 40061.51(6s6d) 3D3 39966.09(6s6d) 3D2 39838.04(6s6d) 3D1 39808.72(6s2 ) 1S0

Figure 2.1: Ytterbium energy levels.

(5d6s) 3D3 25270.902(5d6s) 3D2 24751.948(5d6s) 3D1 24489.102

CHAPTER 2. YTTERBIUM

14

Atomic Mass

Isotope Shift (MHz)Frequency (MHz)

176

-509

751 525 478.5

173(5/2-5/2)-253

751 525 734.3

174

0

751 525 987.7

173(5/2-3/2)

516

751 526 503.7

172

533

751 526 521.1

173(5/2-7/2)

588

751 526 575.7

171(1/2-3/2)

832

751 526 820.2

171(1/2-1/2)

1154

751 527 141.5

170

1192

751 527 180.2

168

1887

751 527 875.2

Table 2.3: Isotope shifts and absolute frequencies of the 1 S0 → 1 P1 transition [75].

Atomic Mass

Isotope Shift (MHz)Frequency (MHz)

173(5/2-7/2)-2386.7

539 384 174.1

171(1/2-1/2)-2132.0539 384 428.8

176-954.8539 385 606

174

0

539 386 560.8

1721000.0539 387 560.8

170

2286.3539 388 847.2

173(5/2-5/2)

2311.4539 382 339.8

168

3655.1

539 380 996.1

171(1/2-3/2)

3804.6539 380 846.0

173(5/2-3/2)

3805.7539 380 843.9

Table 2.4: Isotope shifts and absolute frequencies of the 1 S0 → 3 P1 transition [76, 77].

Chapter 3

Quantum gas microscope

emCCD
CameraCCD Camera Image

Eyepiece1000 photons
per atom

One site image

High NA
objective lens

Solid
ImmersionLens}

1 pixel = 90 nm

Fluorescencefrom atoms

1 site = 6x6 pixels
Excitation
light

Figure 3.1: Schematic of a quantum gas microscope. The fluorescence from the atoms
trapped in a two-dimensional are captured by a high-numerical-aperture microscope andimaged with a CCD camera.
Quantum gas microscopes are high-resolution fluorescence imaging devices capable ofresolving individual atoms trapped in a two-dimensional optical lattice [47, 46]. The conventional method to observe atoms trapped in an optical lattice is to excite the atoms andobserve the resulting fluorescence using a high-numerical-aperture microscope. Due to thelarge number of photons required to obtain a well-resolved image, a deep lattice potential isnecessary to keep the heated atoms trapped during the imaging process. In the case of rubidium atoms, the polarization- gradient-cooling technique [46, 47, 78] worked effectively
15

CHAPTER 3. QUANTUM GAS MICROSCOPE

16

to overcome this difficulty, as it could cool down the atoms to sub-Doppler temperatures
while the resultant fluorescence was observed. Unfortunately, this technique is not effective
for all the atomic species. In the case of bosonic Yb atoms, for example, the lack of hyperfine splitting makes impossible the application of sub-Doppler cooling mechanisms such aspolarization-gradient-cooling, Raman cooling [79, 80, 81, 56, 52], and electromagnetically induced transparency cooling [82, 83, 55].

3.1 Sub-Doppler cooling methods in alkali atoms
In QGM experiments using alkali atoms such as rubidium, lithium and potassium, sub-Doppler
techniques are utilized to cool down the atoms while the resultant fluorescence is collected.
Temperatures lower than the Doppler cooling limit resulted in trap lifetimes much larger thanthe exposure(irradiation) time, which is a required condition to obtain a high fidelity in the
fluorescence measurement.

3.1.1 Polarization gradient coolingIn the case of rubidium, the polarization gradient cooling technique was utilized. This technique is schematized in Fig. 3.2. First, two counter-propagating beams having linear polarization, which each polarization orthogonal to each other, are used to create a polarization

pumping

gradient field.

Figure 3.2: Polarization gradient cooling. Two counter-propagating beams with linear andorthogonal polarization create a polarization gradient field. An atom traveling trough thisfield is repetitively repumped ’downhill’ resulting in Sisyphus cooling.
To understand why this field is generated, each of the beams can be written as a superpo-

CHAPTER 3. QUANTUM GAS MICROSCOPE

17

sition of σ+ and σ− polarization beams:
 



1 1
E+ = p   e i (k x −ωt )21

1 i 
E− = p   e i (−k x −ωt ) .
2
−i

,

(3.1)

The interference of this two beams produces two independent standing waves for the circularpolarizations σ+ and σ− , having a relative phase of π/2:2sin (k x − π/4)
|E+ + E− |2 ∝ sin2 (k x + π/4)

(3.2)

Next, consider a system having states |↑〉 and |↓〉 which only couples to the σ− and σ+ polarization, respectively. Suppose, as in Fig. 3.2 that the atom initially has a state |↑〉 where thepotential is minimum. As the atom moves climbing the potential wall, the intensity of the σ+
standing wave decreases while the intensity of the σ− one increases. Around the potential
maxima, the σ− light repumps the atom to the |↓〉 state. In this state, the potential is again aminimum, and the atom will continue moving in the same direction climbing the potential
wall, until the σ− standing wave intensity decreases again and the atom is repumped backto the |↑〉 state. As the atom repetitiously climb the potential wall, the velocity of the atom
can be reduced. This cooling mechanism was named Sisyphus cooling in honor to the kingof Ephyra in the Greek mythology, who was punished to roll a heavy ball up to a hill and to
throw it back down, repeating this process indefinitely.
The resultant temperature using this cooling method can reach much lower values thanthe Doppler limit:
T≈

ħ Ω2
h
8∆

(3.3)

as the temperature can be reduced by increasing the detuning ∆ of the cooling beams while
decreasing their Rabi frequency (laser intensity) Ω2 . Using a large detuning ∆ is also an important feature of this cooling method, as it makes it robust against the inhomogeneities ofthe light shift in the space produced by the AC Stark shift of the optical lattice. Notice, however, that the trade-off for having low atom temperatures is a reduced photon emission rate,
and a lower velocity trap range.

3.1.2 Side-band coolingSide-band cooling is a technique originally used to cool down ions tightly trapped in a harmonic potential. Consider a two level system with ground state g and an excited state |e 〉with a transition frequency ω0 . When the oscillator quantum vibrational separation ω is suffi-

CHAPTER 3. QUANTUM GAS MICROSCOPE

18

ciently large, the system can be well described by the eigenstates g , n , |e , m 〉, where n and m
represent the vibrational energy. A laser light with frequency ωL couples each of the groundstates g , n with the excited states |e , m 〉 at the transition frequencies ωL = ω0 + (m − n )ω.
When the natural linewidth of the g → |e 〉 transition Γ is sufficiently smaller than ω, it ispossible to resolve each of the resonances g , n → |e , m 〉 in the spectrum. The transition
corresponding to ωL = ω0 where the vibrational energy remains the same ( g , n → |e , n〉) is
called carrier transition, while the transitions to different vibrational energy g , n → |e , n〉 aredenominated side-band transitions.
The side-band cooling technique consists in using a cooling beam that reduces the vibrational level by 1 in each excitation |e , n 〉 → g , n − 1 . After the atom is excited, it spontaneously decay with a high probability to the same vibrational level in the ground state |e , n − 1〉 →g , n − 1 , completing one cycle and reducing the energy of the atom by ħh ω. This process is repeated until the atom is in the motional ground state g , 0 . The condition that the atom falls
with high probability to the same vibrational level in the ground state is satisfied by an atomin the so called Lamb-Dicke regime, where the recoil energy of one photon ħh ωR is smaller
than the quantum vibrational separation ħh ω. This condition is often written as:sη=v
ωR t ħ
hk2
=
≪1
ω
2mω

(3.4)

where η is the Lamb-Dicke parameter, k is the wavenumber of the cooling light, and m is themass of the atom.
In the case of 40 Ca+ ions trapped in a Paul trap, the trap frequencies are often in the orderof ω/2π ≈ 1 MHz, while the recoil energy of a 729 nm beam is approximately ωR /2π ≈ 10 kHz,
resulting in a Lamb-Dicke parameter of η ≈ 0.1. The ultra narrow transition at 729 nm have alinewidth of less than 1 Hz, which is enough to resolve the side-band transitions.
Summarizing the above, two conditions are necessary to realize side-band cooling: 1) A
cooling laser with a line-width much smaller than the vibrational energy (Γ ≪ ω) in order toresolve the side-bands, and 2) A system satisfying the Lamb-Dicke condition, where the recoil
energy is much smaller than the vibrational energy (ωR ≪ ω).
In the case of neutral atoms trapped in two-dimensional optical lattices, the trap frequencies are often in the order of ω/2π ≈ 100 kHz and both conditions become more difficult tobe satisfied. For example, the narrow intercombination transition of Ytterbium atoms have alinewidth of Γ /2π = 180 kHz and a recoil energy of ωR /2π = 3.7 kHz, resulting in η = 0.14 andΓ /ω = 1.8. Although the Lamb-Dicke condition is somehow satisfied, the line-width of thecooling laser is too broad to resolve the side-band transitions.
Two different cooling methods can be used to effectively reduce the linewidth of the cooling laser, namely, Raman cooling and EIT cooling. Both methods were successfully applied toCHAPTER 3. QUANTUM GAS MICROSCOPE

19cool neutral atoms trapped in two-dimensional optical lattice [79], and also in quantum gas

Cooling laser absorption spectrum (arb. units)

cooling

laser

Repump beam

an b
Ram

g laser(a) Raman cooling. Two non-resonant beams
are used to drive the g 1 , n → g 2 , n − 1 Raman
transition in a three-level atom. A repump beamis then used to return the atom to the g 1 , n − 1state, completing one cycle of side-band cooling.

ncouplieam

microscopes in recent experiments.

(b) EIT cooling. A strong coupling laser and aweak cooling laser in a Λ-configuration produce
a Fano-like profile in the absorption spectrum(right graph). The detuning and Rabi frequencyof each beam can be choosen in such a way thatthe blue side-band excitation is suppressed, resulting in side-band cooling.

Figure 3.3: Two different sub-Doppler cooling methods used to cool down alkali atoms in an
optical lattice.
The Raman cooling method consists in using a three-level system to drive a Raman transition, as shown in Fig. 3.3a. A three-level lambda system is usually used for Raman cooling, where two of the energy levels (|g 1 〉 and |g 2 〉) are stable or metastable states, but a laddersystem can be also utilized. First, two non-resonant laser beams detuned by ∆ are used todrive a Raman transition the |g 1 〉 → |g 2 〉 transition through an intermediate transition |e 〉.
The linewidth of the Raman transition is given by the formulaΓraman =Ωg e 1 Ωg e 2
∆

(3.5)

where Ωg e 1 and Ωg e 2 are the Rabi frequencies driving the |g 1 〉 → |e 〉 and |g 2 〉 → |e 〉 transitions, respectively. It is clear in this equation that the linewidth of the Raman transition
can be reduced by increasing the detuning ∆, which is useful to resolve the side-band transition g 1 , n → g 2 , n − 1 necessary for side-band cooling. After the atom is transferred to theg 2 , n − 1 state, a new beam resonant to the |g 2 〉 → |e 〉 transition is used to optically pump
the atom back to the g 1 transition. If the Lamb-Dicke condition is satisfied, the atom will be
returned g 1 , n − 1 to the state, completing one cycle of side-band cooling.
The electromagnetically induced transparency (EIT) cooling utilizes a strong coupling
laser detuned by ∆r to the |r 〉 → |e 〉 transition to tailor the absorption spectrum of a coupling

CHAPTER 3. QUANTUM GAS MICROSCOPE

20

laser detuned to the |g 〉 → |e 〉 transition (see Fig. 3.3b). The resultant absorption spectrum
when ∆r > 0 is a Fano-like profile having a zero at ∆s = ∆r corresponding to the |g , n〉 → |e , n〉carrier transition. The idea to produce effective side-band cooling is to choose the coupling
laser detuning ∆r and a Rabi frequency Ωr such that it satisfies the equation:ÆΓ=∆r 2 + Γr 2 − ∆r
.
2

(3.6)

In this case, the narrow peak of the Fano-like profile coincides with the red-sideband transition |g , n 〉 → |e , n − 1〉. This method thus enhances the red side-band transition while itcancels the carrier excitation, which is an ideal condition for side-band cooling.

3.1.3 Sub-doppler cooling in Yb atoms
In the case of the bosonic isotopes of Yb, the lack of hyperfine and magnetic sub-levels in
the ground state impedes the use of sub-doppler techniques such as Polarization gradientcooling, Raman cooling and EIT cooling. One idea for realizing sub-doppler cooling is by using two-photon transitions such as the 1 S0 → 3 P1 → (6s 7s )1 S0 transition having linewidthsof Γ /2π = 181 kHz and Γ /2π = 210 kHz. Fermionic isotopes have a nuclear spin and consequently hyperfine structure in the excited levels. Raman cooling could be possible in principle
by using different magnetic sub-levels of the ground state. Also, by using the metastable state3

P2 as a cooling basis (which have hyperfine splitting in the fermionic case) it is possible torealize polarization gradient cooling, Raman cooling or EIT cooling.

3.2 Simulation
For the following sections, a computer simulation will be used to test the Doppler cooling anddeep potential methods. The simulation realized is semi-classical with the following properties:
Optical lattice The optical potential utilized for the simulations is given by:
V (x , y , z ) = V0 sin2 (k z z ) sin2 (k x x ) + sin2 (k y y ) .

(3.7)

2π
2π
where k x = k y = λlat cos
θacc and k z = λlat sin θacc . All simulations were realized for the caseθacc = 6◦ . Note that V0 is the lattice depth in the x y plane, while the depth in the z
direction is 2V0 . Due to the geometry of this potential, the hoppings in the z direction
will be negligible compared with the ones in the x y plane.
Movement The mechanics of the particle is considered to be classical. For a particle withmass m and position r(t ), the differential equation that governs the movement is given

CHAPTER 3. QUANTUM GAS MICROSCOPE

21

by:m r̈(t ) = F (r, t )(3.8)

where F is the force at the position r and time t . Using the Euler method, the positionof a particle is calculated by the following equations.

F (r, t )
 ṙ(t + δt ) = ṙ(t ) +δtmr(t + δt ) = r(t ) + ṙ(t )δt(3.9)

Wannier functions are not considered for the simulation. The initial position and velocity of the atom is always r(t ) = 0 and ṙ(t ) = 0, respectively.
Photons absorption and emission The probability of absorbing a photon between t and t +δt is given by the formula:p (t ) =

Γ
s0

 δt
2 1 + s + 4 ∆(r)+k·ṙ(t ) 2
0(3.10)

Γ

where k · ṙ represents the Doppler effect and ∆(r) is the detuning of the laser including
Stark shift effects at the position r. In the case of a magic-wavelength, the detuning ishomogeneous and ∆(r) is constant. When an atom absorbs a photon, the velocity ofthe particle is immediately increased by ħhmk . Absorption is immediately followed by aphoton emission in a random direction.
Population and Rabi oscillations For a multi-level atom, the population in each state |i 〉 isgiven by ρi i (t , r). The force experienced by the atom is given by:
F (r, t ) = −∑ρi i (t , r)∇Vi (r)(3.11)iwhere Vi (r) is the optical lattice potential on the the state |i 〉. Note that ρi i depends on
the time, as Rabi oscillations are considered instead of using time averages. For statesthat are not coupled by a light beam, spontaneus emission rates resulting in transition
from a excited state |i 〉 to a state with a lower energy | j 〉 are calculated using the formulaρ i i Γi j .
Each simulation is stopped when the atom hops to a neighboring site (|ki ri (t )| > π2 ). The
lifetime is calculated averaging the obtained photons after 500 repetitions.

CHAPTER 3. QUANTUM GAS MICROSCOPE

22

3.3 Doppler cooling in ytterbiumThis section is dedicated to study the possibility of using Doppler cooling to cool down theatoms in the QGM scheme. As the Doppler cooling strongly depends on the detuning of thecooling laser beam ∆, it is important to consider the light shift inhomogeneities that arise
from having different AC Stark shift in the ground and excited states. The first part of thissection is dedicated to study the simplest Doppler cooling setup, where the lattice wavelength
λlat is set to the magic-wavelength λmagic one. In this setup, the wavelength of the lattice ODTis chosen such that the result AC Stark shift in the ground and excited states are exactly thesame for every position in space, which produces an homogenous light shift in the space anddetermines a constant detuning for every point in space ∆(x , y , z ) = ∆(0, 0, 0).

3.3.1 Magic-wavelength potential and 1 S0 -3 P1 molasses
The 1 S0 -3 P1 cooling transition in Yb, having a low Doppler cooling limit of 4.4 µK, could in
principle allow one to obtain long lifetimes even with low potential depths. Simulation results
shown in Fig. 3.4a shows the lifetime (in photons) dependency on the potential depth 2V0 .
For this simulation a six beam molasses setup is used, with each beam having a saturation
intensity of s0 = 1 and a detuning of ∆/Γ = −0.8.

(a) Lifetime dependency on the potential depth
2V0 for s0 = 1 and ∆/Γ = −0.8.

(b) Lifetime dependency on the detuning ∆ fordifferent saturation intensities s0 . The potential
depth is fixed at 90 µK

Figure 3.4: Lifetimes obtained from the simulation for a magic-wavelength potential using
the 1 S0 − 3 P1 transition.
Note that the vertical axis in the figure is in the logarithmic scale. By increasing the potential depth by 30 µK the lifetime can be increased one order of magnitude. This is compatible
with the Doppler theory, in which atoms have a Gaussian velocity distribution.
To clarify this point, consider the velocity distribution of an atom during Doppler coolingCHAPTER 3. QUANTUM GAS MICROSCOPE

23

Figure 3.5: Velocity distribution of an atom during Doppler cooling. The velocity unit is inrecoil units vR = ħ
h k /m . The solid line corresponds to a Gaussian fitting.
shown in Fig. 3.5. The profile was obtained from a simple one-dimensional simulation in thefree space, for a detuning of ∆ = −Γ /2 and a saturation intensity of s0 = 10−3 . The temperature
obtained from the width of the Gaussian profile2

f1D (v ) ∝ e

mv
− 2k
T

(3.12)

B

resulted in T = 4.5 µK, which is the Doppler cooling limit. In the three-dimensional case, thelattice depth in the z direction is 2V0 while the lattice depth in the x and y directions is V0 .
Consequently, the atom will mostly escape in the x and y directions and the problem can beconsidered as two-dimensional. The velocity distribution corresponding to the 2D case is:2

f2D (v ) ∝ v e

mv
− 2k
T

(3.13)

B

An atom escapes the lattice site when its kinetic energy is larger than the lattice depth V0 . The
probability of having an atom with the escape velocity or larger is:

p v≥

v

t 2V
0

m

∫∞
=
v=

p

f2D (v )d v ∝ e

V− k 0TB(3.14)

2V0 /mwhich decreases exponentially with the potential depth. The lifetime dependency on thelaser-cooling detuning ∆ and saturation s0 is shown in Fig. 3.4b, where the potential depth
was fixed to 2V0 = 90 µK. For s0 = 0.1 the optimal saturation was obtained at ∆/Γ = −0.6. Notethat this result also follows the standard Doppler cooling theory where the Doppler coolingCHAPTER 3. QUANTUM GAS MICROSCOPE

24temperature isħ Γ 1 + s0 +h
kB T =
2∆
4
Γ



2∆ 2
Γ

(3.15)and the Doppler cooling limit is determined by
kB T =

ħΓ p
h
1 + s0
2

when

∆=−

Γp
1 + s0 .
2(3.16)For higher laser cooling saturation, the temperature increases and the optimal detuningp∆ increases by a ratio of 1 + s0 , which produces a significant reduction in the lifetime. The
photon emission rate is also an important factor in the QGC, as the lifetime of the atomic
cloud is limited by other factors such as background collisions and lattice light absorptions.
Consequently, a saturation of at least s0 = 1 is desired for the experiment. For the following
subsections, the analysis will be limited to a saturation of s0 = 1.
Six beams 3D molasses

Four beams 2D molasses

Four beams 3D molasses

Figure 3.6: Three different cooling schemes. (left) 3D molasses cooling using 6 beams in the±x ,±y and ±z directions. (center) 2D molasses using 4 beams in the ±x and ±y directions.
(right) 4 beams in the Brewster angle directions allows cooling in the three dimensions.

(a) 2D molasses using 4 beams in the ±x and ±y
directions.

(b) 4 beams setup as shown in the Fig. 3.6(right)
setup.

Figure 3.7: Lifetime as a function of the potential depth 2V0 for s0 = 1 and ∆/Γ = −0.8 for two
different molasses setup.
Another important point to consider is that in the QGM setup the objective lens is placed

CHAPTER 3. QUANTUM GAS MICROSCOPE

25

over the solid immersion lens along the z direction (Fig. 3.1). This means that the molasses
beams in the +z direction would enter directly into the camera CCD sensor. One idea to avoid
this problem is to use cooling only in the ±x and ±y directions (Fig. 3.6). Simulation resultsin Fig. 3.6 shows that, in the case of a 2D cooling, the heating along the z direction causes a
significant reduction in the lifetime. In contrast to the potential depth of 2V0 = 90 µK required
to obtain a lifetime of 10000 photons in the 3D molasses case, a potential 2V0 = 450 µK would
be required to obtain the same lifetime for the 2D molasses approach. Moreover, in this casethe lifetime only increases linearly with the potential depth, so a further increment in the lifetime would require a very large potential. Another possible idea is to use four cooling beams
in thek̂1 = (cos θB , 0, − sin θB )k̂2 = (− cos θB , 0, − sin θB )k̂3 = (0, cos θB , + sin θB )k̂4 = (0, cos θB , sin θB )directions, where θB is the Brewster angle. It can be easily seen that with this setup cooling ispossible in all directions. Simulation results in Fig. 3.7b show that this setup produces equal
performance as the obtained in the six beams molasses.

3.3.2 Estimation of the magic-wavelength for the 1 S0 -3 P1 transition

Figure 3.8: Polarizability in the ground 1 S0 state and excited 3 P1 state as a function of the optical dipole trap wavelength. Magic-wavelengths were found 424 nm, 463 nm, 551 nm, 612 nm,
800 nm and 1.53 µm.
The magic-wavelength for the 1 S0 -3 P1 transition is estimated using all the energy levels
up to 42000 cm−1 . For the calculations, a linear polarized light was used and only the m = 0
sublevel in both the ground and excited states were considered. In the case of the groundCHAPTER 3. QUANTUM GAS MICROSCOPE(6s8s) 3S1

26(41615.04)(39838.04)(39808.72)(6s7s) 3S1(32694.69)

3.3(34350.65)

42(6s7s) 1S0

0.1(24751.948)

nm

9
147

n
5.8
55

nm

45

.3

nm

8.3

45

7.7

nm

nm

61
1

68

(6s2) 1S0

(6s6d) 3D2(6s6d) 3D1

nm
9 nm
153

(17288.439)(6s6p) 3P1

Wavelength

Linewidth

Refs.

(24489.102)(5d6s) 3D2(5d6s) 3D1m(0.00)

FromTo(6s6p) P1(5d6s) D1

1539 nm

16 kHz[84](6s6p)3 P1(5d6s)3 D21479 nm

320 kHz[84](6s6p) P1(6s7s) S1

680.1 nm

4.3 MHz[84](6s6p)3 P1(6s7s)1 S0

611.3 nm

210 kHz[85](6s6p)3 P1(6s6d)3 D1

457.7 nm

2.6 MHz[86, 87](6s6p) P1(6s6d) D2

458.3 nm

2.4 MHz[86, 87](6s6p)3 P1(6s8s)3 S1

423.3 nm

2.7 MHz[86, 87](6s ) S0(6s6p) P1

555.8 nm

182 kHz

3

3

3

2 1

3

3

3

3

Figure 3.9: Transitions considered for the excited 3 P1 state, and their respective wavelengthand line-widths.
state, the 1 S0 -1 P1 , 1 S0 -3 P1 and 1 S0 -3 D1 transitions were used. The transitions from the excitedstate were summarized in the Fig. 3.9. Although only the transition information for the lower
levels is known [84, 85], the lifetime and relative intensities of the upper transitions could be
used to roughly estimate the transition linewidths [86, 87] of the remaining transitions.
The calculation results are shown in Fig. 3.8. Six different magic wavelengths with negative polarizability where found, whose wavelengths are 424 nm, 463 nm, 551 nm, 612 nm,
800 nm and 1.53 µm. Only the magic-wavelength at 800 nm is far detuned, which makes it
a good candidate for creating the two-dimensional optical lattice. Laser sources at 800 nm
are also readily available, for example using a Ti:sapphire laser or a tapered amplifier.

CHAPTER 3. QUANTUM GAS MICROSCOPE

27

3.3.3 Deviation from magic-wavelength for the 1 S0 -3 P1 transition
What would happen if the lattice wavelength is different from the magic-wavelength one?.
When the atom energy increases due to heating, the atom moves inside the trap, which changes
the detuning of the laser cooling and consequently the cooling condition is altered. If thelight-shift deviation δ defined by the change of detuning ∆(x , y , z ) when the atom moves half
the lattice spacingδ = ∆ (π/2k x , 0, 0) − ∆ (0, 0, 0)(3.17)is positive, then the detuning absolute value increases when the atom moves [see Fig. 3.10(right)],
reducing the efficiency of the Doppler cooling. Similarly, when δ < 0 the detuning absolutevalue decreases when the atom moves. Eventually, if the deviation is large, the sign of thecoolingcoolingcoolingheating

detuning changes producing heating [see Fig. 3.10(left)].

Figure 3.10: (left) If the polarizability in the excited state is lower than the one in the groundstate, the detuning of the cooling beams is reduced until eventually heating occurs. (right)
A higher polarizability in the excited state do not produce heating, but the efficiency of the
Doppler cooling is reduced.
Figure 3.11a shows the simulation results when the lattice wavelength is shifted from themagic wavelength. For the simulation the potential depth was fixed to 2V0 /k B = 90 µK. The
asymmetry of the lifetime dependency on the light-shift deviation δ is in agreement to thesurmise that a negative δ results in heating and consequently a fast reduction in the lifetime,
while a positive δ only reduces the efficiency of cooling and the lifetime is not greatly affected.
The light-shift deviation δ as a function of lattice wavelength λlat at a fixel potential depth
2V0 /k B = 90 µK is shown in Fig. 3.11b. The 3 P1 -3S1 transition at 680 nm causes a rapid increment in δ for wavelengths shorter than the magic wavelength. For λlat = 700 nm the light-shift

CHAPTER 3. QUANTUM GAS MICROSCOPE

28

deviation is δ ≈ 20Γ which reduces the lifetime to ∼ 8000 photons. On the other hand, thechange in deviation is smaller (δ ≈ −3Γ for λlat = 900 nm) but it results in a similar reduction
in lifetime.
In conclusion, simulation results shows that it is possible to obtain ∼104 photons by using
a potential depth of 2V0 /k B = 90 µK in a magic-wavelength potential at λlat = 800 nm. Thispotential can be realized using 100 mW of power per beam in the retro-reflected accordion
setup, or using 450 mW of power per beam if the standing wave is created by interferenceof six individual beams. Also, even if the lattice wavelength is deviated by 100 nm from themagic-wavelength, lifetimes of ∼8000 photons can be still obtained.
The disadvantages of using this setup is that long exposure times (∼100 ms) would require
making the system robust against mechanical vibrations. Also, the maximum power obtainable at this wavelength is currently limited to 5W using a Ti:sapphire laser pumped by 20 W oflight. Additionally, short wavelengths for the lattice result in increased heating due to spontaneous emission, which is a critical factor to consider when realizing quantum simulations
on the Fermi-Hubbard model.

(a) Simulation results of the lifetime dependency on light-shift deviation δ.

(b) Light-shift deviation δ as a function of lattice
wavelength λlat .

Figure 3.11: Simulation and wavelength calculations for deviations from the magicwavelength. In both cases the potential depth was fixed to 2V0 /k B = 90 µK.

3.3.4 Magic-wavelength potential and 1 S0 -1 P1 molasses
Despite the fact that the 1 S0 -1 P1 transition have a high Doppler cooling limit and, in consequence, it is not suitable for cooling, it is interesting to study the lifetime of atoms when thiscooling transition is used. Simulation results in Fig. 3.12a shows the lifetime dependency
on the potential depth 2V0 . For this simulation a six beam molasses setup is used, with eachCHAPTER 3. QUANTUM GAS MICROSCOPE

29(a) Lifetime dependency on the potential depth
2V0 for s0 = 1 and ∆/Γ = −0.8.

(b) Lifetime dependency on the detuning ∆ fordifferent saturation intensities s0 . The potential
depth is 4 mK.

Figure 3.12: Lifetime in photons as a function of the potential depth 2V0 for a magicwavelength potential using the 1 S0 − 1 P1 transition.
beam having a saturation intensity of s0 = 1 and a detuning of ∆/Γ = −0.8. For large potential
depths, the lifetime still increase exponentially respect to potential depth 2V0 .
The lifetime dependency on the laser-cooling detuning ∆ and saturation s0 is shown in
Fig. 3.12b, where the potential depth was fixed to 2V0 = 4 mK. For s0 = 0.1 the optimal saturation was obtained at ∆/Γ = −0.6. One important difference compared with the cooling withthe 1 S0 -3 P1 transition is the lifetime baseline at ∼5500 photons, which appears even for zerodetuning and high saturation intensities.
The presence of a baseline in the lifetime is related to the fact that atoms in the simulation
start with zero velocity and require a finite time to thermalize. Consider the one-dimensionalcase of an atom that scatters photons in random directions. The resultant heating after N
scatterings is N E r 399 , where E r 399 is the one photon recoil energy ħ
h k /2m. When the amountof heating equals the lattice depth V0 the atom escapes the trap and is lost. The lifetime of theatom is thenN≈

V0
E r 399=2mV0
ħ 2k 2h≈ 5800 photons(3.18)

for a potential of V0 /k B = 2 mK. This value is noticeably close to the photons baseline in Fig.
3.12b.

3.3.5 Estimation of the magic-wavelength for the 1 S0 -1 P1 transition
The magic-wavelength for the 1 S0 -1 P1 transition was estimated using the transitions summarized in Fig. 3.14. In the case of the ground state, the 1 S0 -1 P1 , 1 S0 -3 P1 and 1 S0 -3 D1 transitions were used. The transition linewidth information was obtained from [88, 86, 89, 90]. The

CHAPTER 3. QUANTUM GAS MICROSCOPE

30

Figure 3.13: Polarizability in the ground 1 S0 state and excited 1 P1 state as a function of the
optical dipole trap wavelength. Two magic-wavelengths were found at λlat = 775 nm andλlat = 1.30 µm.
(6s6d)3 D states also decay slowly into the 1 P1 state but were omitted from the calculations as
their transition strength is negligible compared with the nearby 1 P1 -(6s6d)1 D2 transition.
Two magic-wavelengths were found at λlat = 775 nm and λlat = 1.30 µm. Creation of apotential depth of 2V0 /k B = 4 mK at λlat = 775 nm would require 10 W of power per beam, and
13 W of power per beam in the case of λlat = 1.30 µm.

(34350.65)(6s7s) 3S1(32694.69)

66
6.8(6s7s) 1S0(6s6d) 1D2

nm(40061.51)

10
13

m

nm

(25068.22)

39

9n

m

11

77
n

(6s2) 1S0

(0.00)From

To

Wavelength

Linewidth

Refs.

(6s6p) P1(6s7s) S1

1311 nm

25 kHz[89](6s6p)1 P1(6s7s)1 S01077 nm

3.0 MHz[90, 88]1(6s6p) P1(6s6d) D2

667.0 nm

4.5 MHz[86](6s2 )1 S0(6s6p)1 P1

398.9 nm27.9 MHz1

3

1

Figure 3.14: Transitions considered for the 1 P1 state with their respective wavelength andlinewidth.

CHAPTER 3. QUANTUM GAS MICROSCOPE

31

3.4 The deep potential approach
In Sec. 3.3.4 the lifetime of atoms using a magic-wavelength potential and the 1 S0 -1 P1 molasses was studied. It was found that even for zero detunings and large saturation intensitiesof the laser cooling beam, a condition in which laser cooling has no effect, nearly ∼5500 photons could be obtained. Atoms starting at zero velocity require finite time to heat resulting ina finite number of photons obtained before the atom escapes from the trap. If a deep enoughpotential is prepared, would it be possible to attain a large number of photons without theuse of any laser cooling technique? The answer is yes, and it is the strategy selected in thiswork for creating a quantum gas microscope of ytterbium atoms.
Without the Doppler cooling requirement, utilization of magic-wavelength potentials is
no longer needed. For the excitation beam the preferable choice is to use the 1 S0 − 1 P1 as itis less prone to be affected by Doppler effects and additionally because it produces a higher
microscope resolution.

Lattice ODT~ 400 nm

399 nmexcitation(6s6p)1P1(6s2)1S0

Figure 3.15: Creation of a deep potential using a lattice near-resonant to the 1 S0 − 1 P1 transition.
The conventional method for creating deep potentials is choosing a lattice wavelengththat is near-resonant to a transition from the ground state (see Fig. 3.15). In the case of ytterbium, the natural choice is to use a lattice wavelength near-resonant to the strong 1 S0 − 1 P1transition. One problem of using this transition is that high-power laser sources are not readily available in the NUV region. If the available power per beam is 100 mW, a lattice wavelength detuned by only 1 nm (λlat = 400 nm) is necessary to create a potential of 2V0 /k B =4 mK. In this case, even for very small lattice powers the scattering rates in the ground state
are on the order of 10s −1 for potential depths of a few µK, which is not a good condition for
creating a Bose-Einstein condensate (BEC) and a Mott insulator. Another problem of using
this transition is that the wavelengths of the lattice light and the excitation light are similar
and it results difficult to reduce the effects of stray lights.
In this work, instead of creating a deep potential in the ground state, the optical latticeCHAPTER 3. QUANTUM GAS MICROSCOPE

32(6s7s)1S0
Lattice ODT1077 nm

399 nm
Lattice ODTexcitation(6s6p)1P1(6s2)1S0
Figure 3.16: Creation of a deep potential using a lattice near-resonant to the 1 P1 − (6s7s)1 S0transition.
wavelength was selected such that the potential in the excited state is deep and the one in
the ground state is shallow. The selected wavelength was of λlat = 1082 nm, which is red detuned by only ∆lat /2π = 1.2 THz to the upper 1 P1 − 1 S0 transition (wavelength 1077.2 nm,
natural linewidth 3MHz) which creates a deep potential in the excited state that is 200 times
larger than that in the ground state. Other possible choices would be using a wavelength
near-resonant to the 1 P1 − (6s6d)1 D2 transition (wavelength 667.0 nm) or the 1 P1 − (6s8s)1 S0transition (wavelength 592.7 nm).
To see the advantages of this system, consider first the case in which the excitation lightis not present in the system. As the lattice light is far off resonant from any of the transitions from the ground state, the shallow potential results in small scattering rates (10−3 s−1 for
V0 /k B = 2 µK) ideal for preparing Hubbard systems in their ground state. During the imaging
process, the resonant excitation light illuminates the atoms and couples the lower 1 S0 and excited 1 P1 states. Atoms then experience an average potential between the deep excited-state
potential and the shallow ground-state one. This deep potential is useful to trap the atoms
during the imaging process. Note that for the average approximation to be true, the Rabi frequency should be much larger than the trap frequency in the optical lattice, which is normallysatisfied by the strong 1 S0 − 1 P1 transition with large saturated intensities.
A computer simulation is used to test the deep potential approach and to determine theinitial experimental parameters such as lattice detuning ∆lat , lattice power Plat and the saturation intensity of the excitation beam s0 . As optical molasses is no longer needed for Doppler
cooling only one excitation beam in the +x direction was used in the simulation. Simulation results are shown in Fig. 3.17, where the lifetime dependency on lattice power Plat for
3 different saturation intensities (s0 = 100, 1000 and 10000) and 10 different lattice detunings (∆lat /2π = 0.1, 0.2, . . . , 1.0 THz) were tested. In general, large saturation intensities pro-CHAPTER 3. QUANTUM GAS MICROSCOPE

33

duce longer lifetimes because the resultant power broadening is useful to reduce the effectsof Doppler shifts and light-shift inhomogeneities; i.e. large saturation intensities maintainthe population in the 1 P1 state as high as possible.

CHAPTER 3. QUANTUM GAS MICROSCOPE

34(a) Excitation beam saturation intensity s0 = 100(b) Excitation beam saturation intensity s0 = 1000(c) Excitation beam saturation intensity s0 = 10000

Figure 3.17: Simulation results. Lifetime dependency on the lattice power Plat and lattice detuning ∆lat for 3 different excitation beam saturation intensities (s0 ). Left: Density plot of
the lifetime for 10 different lattice detunings and powers. Right: Dependency of lifetime fordifferent lattice powers, with each curve representing a different lattice detuning.

CHAPTER 3. QUANTUM GAS MICROSCOPE

35

3.5 Lifetime and limitations of the deep-potential approach
In the deep potential approach, the mean lifetime of an atom is limited by three factors:
Heating
When the atoms are excited to 1 P1 state by the excitation beam, the consequent scattering produces a random momentum kick which heats the atom. As there is no coolingscheme present in the system, the atom total energy will eventually exceed the potential
barrier of the lattice site, resulting in atom hopping.
Dipole trap beam absorption
As a result of the lattice beam being near resonant to the upper 1 P1 -(6s7s)1 S0 transition,
the absorption of the dipole trap beam occurs in the order of a few tens of microseconds. As the potential in the (6s 7s )1 S0 state is strongly repulsive and the decay rate iscomparable to the trap frequency, the atom is lost after a number of excitations.
Branching to the 3 P1 state
The absorption of the dipole trap beam also leads to a branching to the 3 P1 state. The
branching ratio of the (6s 7s )1 S0 state is known to be approximately 1:10 [90, 88]. As thestate 3 P1 is long lived and the potential in this state is shallow, the atom will be lost if it
falls into this state.

HeatingODT Absorption

ODT Absorption+ Branching

Figure 3.18: The three factors limiting the lifetime on the deep potential approach.
The heating due to N random photons scattering isp2
h 2 kexc
ħp2( Nħh kexc )2
∆E =
=
=N
,
2m
2m
2m

(3.19)

which is proportional to the number of scattered photons. When the kinetic energy is comparable with the site potential V0 ≈ ∆E the atom will escape from the site. The lifetime (inphotons) nheat due to heating can be written as
nheat = N = a V02
where a is a constant in the order of a ≈ 2m /ħh 2 kexc.

(3.20)

CHAPTER 3. QUANTUM GAS MICROSCOPE

36

Figure 3.19: Simulation lifetime results for different ∆lat . Each curve was fitted with the lifetime in Eq. 3.23.
The other two factors related to the dipole trap beam absorption can be considered as one,as both of these rates are proportional to the population in the (6s7s)1 S0 state. Consequently,
the lifetime (in photons) due to the dipole trap beam absorption nexc can be expressed as
nexc ∝

∆lat
1
=b
ρ55
V0

(3.21)

where b is a constant and ρ55 is the population in the (6s7s)1 S0 state. Finally, if these twoprocess are independent, the total lifetime (in photons) results in

n=

1
nheat

+

−1

1
nexc


=

1
V0
+
a V0 b ∆lat

−1.

(3.22)

This model is tested using the simulation results obtained in Fig. 3.17. To check the dependency of lifetime on lattice power Plat , Eq. can be rewritten as

n=

1
Plat
+
APlat
B

−1
.

(3.23)

where A and B are two parameters depending only in the lattice detuning ∆lat
A(∆lat ) = a

V0
1
∝
Plat
∆lat

,

B (∆lat ) = b

Plat∆lat ∝ ∆2lat
V0(3.24)

as V0 ∝ Plat /∆lat . Figure 3.19 shows the fitting result of each simulation results for different

CHAPTER 3. QUANTUM GAS MICROSCOPE

37

lattice detunings ∆lat .
The obtained parameters A and B for each of the detuning ∆lat is shown in Fig. 3.20. Eachof the parameters was fitted using a power function.

Figure 3.20: Dependency of parameters A and B on the lattice detuning ∆lat for the simulation
results. Each curve was fitted using a power function.
Knowing the parameters A and B , the maximum lifetime obtainable for a given Plat is:
nmax (Plat ) = n(Plat , ∆best ) :∂n(Plat , ∆best ) = 0.
∂ ∆lat(3.25)

Table 3.1 summarizes the results for three different saturation intensities. For high saturation intensities of the excitation beam, the lifetime model A ∝ ∆lat −1 ∧ B ∝ ∆lat 2 is in wellagreement with the fitting results.
s0

A (104 photons/W)

B (104 photons W)

nmax (104 photons)∆best (THz)104

1.63(3)∆lat −1.13(4)14.8(9)∆lat 2.25(3)1.8Plat 0.33

0.64Plat 0.59

103

1.42(1)∆lat −1.15(8)6.6(6)∆lat 1.88(6)1.3Plat 0.24

0.71Plat 0.65

102

1.11(4)∆lat −0.71(6)0.98(6)∆lat 0.87(6)0.53Plat 0.10

1.22Plat 1.26

Table 3.1: Parameters A and B, max lifetime and best detuning for different saturation intensities. The units of ∆lat and Plat are THz and W, respectively.

3.5.1 Excitation beam radiative force effects
From the simulation results in Fig. 3.17, it can be seen that for large detunings ∆lat and smalllattice powers Plat the fitting does not coincide well with the experimental data. This is due tothe effect of the excitation beam, whose radiative force tilts the lattice potential reducing theCHAPTER 3. QUANTUM GAS MICROSCOPE

38

effective trap force. The reason for not using an excitation beam in the retro-reflected setup is
only technical, as it would require to introduce expensive 3” optics with broadband coatings
capable of transmitting the accordion beam and the excitation beam.

=

Excitation
beam

Figure 3.21: As the excitation beam is not in the retro-reflection setup, the beam exerts a net
radiative force on the atoms, tilting the optical lattice potential while reducing the effective
lattice depth.
The total potential including the potential due to the radiative force of the excitation beamis: ħ
h kexc Γs0
V (x , y , z ) = −V0 sin(k z z )2 cos(k x x )2 + cos(k y y )2 −x
2 1 + 2∆ 2 + sΓ(3.26)

0Here, we consider the worst scenario, in which the saturation is large (s0 ≫ 1) and the detuning
including Doppler shift is negligible (2∆/Γ ≪ s 0). Then: ħh kexc ΓV ≈ −V0 cos(k z z )2 cos(k x x )2 + cos(k y y )2 −x
2(3.27)The potential maxima and minima for z = 0 corresponds to:∂V
h kexc Γħ= V0 k x sin(2k x x ) −=0∂x2
resulting in
x0 =
hk
ħ

1
arcsin ζ ,
2k x

x1 =

(3.28)π1−
arcsin ζ
2k x 2k x(3.29)

Γ

where ζ = V0exc
k x . Note that for ζ = 1, both points coincide (x 0 = x 1 ) and the lattice tilts completely. ζ is then an important scaling factor to determine whether the excitation beam radiation effects are negligible or not. For the experiments parameters, the minimum required
potential Vlimit is:
Vlimit =h kexc Γħ= 1.9 mK
2k x

(3.30)The resultant effective potential can be calculated replacing Eq. 3.29 in Eq. 3.27.
∆V = V (x1 , y , 0) − V (x0 , y , 0) = V0

hp

1 − ζ2 − ζπ2− arcsin ζi(3.31)

Using this equation the reduction of effective potential for different values of ζ can be calculated. These values are summarized in the following table.

CHAPTER 3. QUANTUM GAS MICROSCOPE

39

ζ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∆V
V0

100%

85%

71%

57%

45%

33%

24%

16%

9%

3%

0%

Table 3.2: Effective potential ∆V compared with V0 for different values of ζ. For a potential
depth that is 10 times larger than Vlimit , the effective potential is reduced by 15%.
Knowing the reduction of the potential due to the radiative force, the lifetime τ1 can be
rewritten as:πi
1 − ζ2 − ζ− arcsin ζ
2
v


t
2
s π
s
s
−
= APlat1−− arcsin
Plat
Plat 2Platτ1 = APlat

hp(3.32)(3.33)hkħΓAs the ζ parameter depends on the lattice potential, a new parameter s (∆lat ) = ζPlat = 2V0exck x Plat
is defined, which is independent to the lattice power, as V0 ∝ Plat . Using this equation for the
lifetime, fitting for large ∆lat can be greatly improved, as shown in Fig. 3.22.

Figure 3.22: Lifetime of atoms in simulation, correspoding to ∆lat /2π = 0.9 THz and s0 =10000. The solid lines are fittings corresponding to s = 0 (no radiative force effects) and s > 0.

s(W)A (104 photons W)

B (104 photons/W)

Without radiative force0

2.06(15)

12.3(79)

With radiative force

0.078(16)2.98(27)

4.89(85)

Table 3.3: Parameters A and B obtained from fitting the simulation results for ∆lat /2π =0.9 THz and s0 = 10000. Two different fitting models (with and without radiative forces) arecompared.
The obtained parameters for each fitting is summarized in the following table. The added
parameter tends to increase the resultant value of A and decrease the value of B. Also, fromCHAPTER 3. QUANTUM GAS MICROSCOPE

40

the obtained s it is possible to estimate the minimum required lattice power as Plat = s when
ζ = 1. For these parameters, the potential depth isV0 = 1.2(4) mK

for Plat = 0.078(16) W and ∆lat /2π = 0.9 THz(3.34)

which is in very well accordance to the potential obtained in Eq. 3.30.

3.6 Solid immersion lensObservation with a microscope system is limited by the diffraction limit:
d=

0.61λ
n sin α(3.35)

where λ is the wavelength of the incident light, n is the index of refraction of the medium
being imaged in, and α is the half-angle subtended by the optical objective lens. In orderto improve the diffraction limit of a lens system, it is necessary get closer to the object (toincrement the half-angle) or use a medium with better refraction index than the air. The oil
immersion lens works with that principle: By filling the space between the object and the lenswith oil, the diffraction limit can be improved by a factor of ∼ 1.5.

NA=0.55

NA=0.55

NA=0.80

NA=0.80

NA=0.80n=1.47

Figure 3.23: A solid immersion lens placed between the sample and the objective lens increases the numerical aperture of the system.
When observing an optical lattice, it is difficult to get closer to the lattice as the opticallattice resides inside a vacuum chamber, and techniques as the oil immersion lens cannot be
used for obvious reasons. A solid immersion lens (SIL) works in the same principle as liquid
immersion lens, with the liquid replaced by a solid lens of high refractive material. By simply
introducing a solid immersion lens between the sample and the objective lens, the resolutionof the system can be improved by a factor of the index of refraction of the hemispherical lens[91]. This technique was utilized to observe Rb atoms with resultant high numerical aperturesof ∼ 0.8[46]. Another importance of the SIL here is that it makes possible to create a stable

CHAPTER 3. QUANTUM GAS MICROSCOPE

41

optical trap by fixing the position of the lattice with respect to the SIL flat surface by using theoptical accordion technique. This technique will be explained in detail in Sec. 4.6.

NA 0.55
1.32
3.00(3.68)
2.0

15.0

24.0

r=5

11.5

r=8

Figure 3.24: Drawing of the old and new versions of the SIL.
The SIL used for this work and also for the previous experiments are schematically shownin Fig. 3.24. Both versions of the SIL are made of fused silica glass. The new version of the
SIL it is divided into three parts. The upper part is a spherical cap with a radius of 8 mm and aheight of 1.32 mm, corresponding to a numerical aperture of 0.55. The middle part consists ofa 3-mm-thick disk with a diameter of 24 mm, and the lower part is a 3.68 mm thick disk witha diameter of 11.5 mm. All three parts were optically contacted, and the lower flat and upper
spherical surfaces were super-polished to reduce stray light. The old version of the SIL wasformed by only two optically-contacted parts, consisting in 3 mm thick disk with a diameterof 15 mm and a spherical cap with a radius of 5 mm and a height of 3 mm, corresponding to anumerical aperture of 0.9.
The new version of the SIL is attached to the glass cell by placing the SIL on the cell uppersurface, were a through hole of 12 mm of diameter was opened. The SIL was finally glued to
the glass cell using a vacuum leak sealant (SPI Vacseal) compatible with ultrahigh vacuum.
The old version of the SIL, on the other hand, is located inside the cell. The lens was attachedinto a jig and glued into the upper wall of the glass cell (see Fig. 3.25).
The new version of the SIL have two important advantages respect to the old one. The
first advantage is that the 3 mm thick glass present over the old version produces sphericalaberrations that needs to be compensated with specially designed objectives lenses. TheseCHAPTER 3. QUANTUM GAS MICROSCOPE

423mm glass cell

SIL JIG

3mm glass cell
SIL

vacuumSIL

vacuum(a) New version(b) Old version

Figure 3.25: SIL attached to the glass cell.
objective lenses are not commercially available and are often high-priced. The glass also produces transmission losses that reduces the collection efficiency of the microscope system.
The second advantage is that the aperture of the spherical cap was reduced to 0.55 (which isthe same aperture of the objective lens). As shown in 3.26, a beam incident onto the flat surface of the SIL by an angle of incidence of 55.8◦ (the Brewster angle) is refracted into an angle
slightly larger than the aperture of the spherical cap. This angle of refraction ensures that thebeam will enter and exit from flat surfaces, resulting in no stray reflections that can produce
undesired interference patterns.

on

cti

e
refl

Figure 3.26: The new version of the SIL avoids reflections by reducing the aperture of thespherical cap.

Chapter 4

Experiment: Transport of atoms to thesolid immersion lens surface
4.1 Introduction
As seen in the previous chapter, the solid immersion lens is a very useful tool that can be used
to increase the numerical aperture and collection efficiency of the microscope system. Another important feature of this lens is the possibility of creating a standing wave by reflecting
a laser beam onto the flat surface of the lens, and using the resulting standing wave as an
optical trap. As the position of this standing wave is fixed with respect of the lens surface, it
is possible to trap atoms with high stability at a fixed position respect to the lens. This technique, which is denominated “optical accordion”, will be explained in detail in the following
sections.
For experiments using trapped atoms in a two-dimensional optical lattice, ultra-high vacuum and good optical access to the sample is desired. For this reason, the solid immersion
lens is allocated in a square glass cell separated from the main metallic chamber, where theultra-cold atoms are first prepared. Observation by a high resolution microscope system requires that atoms remain confined in a pancake-shaped region which is thinner than thedepth of field of the objective lens and is located sufficiently close to the surface of the lens. In
rubidium experiments, a system of multiple coils was used to create a moving magnetic trap,
which was used to transport the atoms to the surface of the lens. In the case of ytterbium, thelack of electronic spin impedes the use of magnetic fields.
This chapter focuses on the experimental procedure to transport the atoms to the surfaceof the solid immersion lens using an all-optical method. The procedure is schematized in Fig.
4.1, with the numbers representing the temperature of the atoms, and distance from the SIL
surface to them.

43

0.5m

0.5m

44

2μK

20μm

Accordion

20μK

Transport

1m

1K

MOT

Oven

700K

ZeemanSlowerCHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

0.1μK

2μm

Figure 4.1: Experimental steps to prepare a BEC under the surface of the SIL. The numbers
corresponds to the atomic cloud temperature and the distance to the surface of the SIL.

4.2 Oven
At room temperature, the saturated vapour pressure of ytterbium is roughly 3×10−21 torr andpractically no atoms cannot be obtained from a solid sample. The conventional method toincrease the number of atoms is increase the vapour pressure to ∼ 10−3 torr by heating thesample with an oven. The saturated vapour pressure of ytterbium as a function of temperature is shown in Fig. 4.2. The solid line is a theoretical estimation using the experimentallyobtained heat of sublimation and applying it to the Clausius-Clapeyron equation [92]log10 p ≈ 6.943 −8656− 0.0004647T + 0.6390 log10 T,
T

(4.1)

where p is the pressure in torr, and T is the temperature in K. The four points in the graphcorresponds to data in [93].

Figure 4.2: Saturated vapour pressure of ytterbium as a function of temperature.
In the experiment, the oven (EpiQuest STHKC-1300SH-lC) is heated to 700 K, which results
in a vapour pressure of 1 × 10−4 torr. This pressure is enough to obtain a sufficient number ofatoms in the magneto-optical trap. The heated atoms exit from a 2.6 mm diameter orifice andare collimated using a 30 mm long, 4 mm inner diameter nozzle that also serves as differential

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

45

pumping tube.

4.3 The vacuum chamber
As the oven is heated to 700 K, the vacuum in the chamber surrounding the oven degrades
to 10−8 torr. To reduce losses due to background collisions, main experiments realized in theglass cell requires a vacuum level of 10−11 torr. To reach this vacuum level, the vacuum system
is divided in three chambers separated with a nozzle that acts as differential pumping tube
and isolates the vacuum up to 1 or 2 orders of magnitude, as shown in figure 4.3.
IP (150L)

IP (150L)
TSP

IP (70L)
Zeemanslower~10-8
torr~10-10
torr

Oven
Chamber

Spectography
Chamber~10-11
torr
DPT

Main
Chamber

Figure 4.3: Schema of the vacuum chamber, showing the three main chambers. IP: Ion pump,
TSP: titanium sublimation pump, DPT: interconnected by differential pumping tubes

Figure 4.4: Photo of the vacuum chamber.
The glass cell is attached to the main chamber, where the main experiments are realized,
having an ultra high vacuum (UHV) level of ∼ 10−11 torr. The main chamber is pumped witha 150 litters ion pump (Varian VacIon Plus 150) and a titanium sublimation pump (TSP). The
spectrography chamber is between the main chamber and the oven chamber, and have a vacuum level of ∼ 10−10 torr, and is connected to the main chamber through the Zeeman slower.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

46

Spectrography experiments using the atomic beam are realized in this area. It is pumped witha 70 litters ion pump (Varian VacIon Plus 75). The oven chamber is pumped with a 150 litters
ion pump and have a vacuum level of 10−9 torr. Initially, the whole chamber was baked out to250 ◦ C for 2 weeks.

4.4 Laser cooling of Yb atoms
Ytterbium have two suitable transitions for laser cooling, the 1 S0 → 1 P1 strong transition andthe 1 S0 → 3 P1 narrow inter-combination transition. Cooling properties of both transitions are
summarized in table 4.1.
S0 ↔ 1 P1

1398.8 nm

555.8 nm1

Wavelength
Natural linewidth

2π× 27.9 MHz
2

S0 ↔ 3 P1

2π× 181 KHz

Saturation Intensity

57 mW/cm

0.14 mW/cm2

Doppler cooling limit

650 µK4.4 µK

Table 4.1: Characteristic values for lowermost transitions of 174 Yb
Initial experiments with Ytterbium were realized in 1999 using the 1 S0 → 1 P1 transition,
obtaining 106 atoms [64]. However, a branching from the 1 P1 excited state to the sub-stable
triplet states 3 P0 and 3 P2 limited the final number of trapped atoms. Afterwards, the sameteam succeeded in magneto-optical trapping of Ytterbium atoms with the intercombinationtransition, obtaining a much larger number of atoms at a temperature of 20 µK [94].
In this experiment the 1 S0 → 3 P1 transition is used for magneto-optical trapping, while
the Zeeman slower utilizes the strong 1 S0 → 1 P1 transition for slowing down the atomic beam
exiting from the oven.

4.4.1 Zeeman slower
Atoms heated by the oven have a temperature of ∼ 700 K and a RMS speed of ∼ 300 m/s. In
contrast with this velocity, the capture velocity of a magneto optical trap (MOT) using the narrow transition 1S0 → 3 P1 is less than 10 m/s, meaning that only a small portion of the atomic
beam can be captured by the MOT. In order to increase the number of trapped atoms, thetemperature of the atomic beam needs to be further reduced. The conventional method to
do this is by using a Zeeman slower device which combines the laser cooling technique with
the Zeeman shift produced by a strong magnetic field.
The Zeeman slower device used in the experiment is shown in Fig. 4.5. Atoms that exitfrom the oven are collimated into an atomic beam and travel through a long tube towards

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SILHigh current
coil
Yb atomic
beam

Erasingcoil

Low current coil

47
Main Chamber

Zeeman slowerbeam

300 mmFigure 4.5: The Zeeman slower device. The atomic beam is slowed down by a laser coolingbeam detuned to the 1S0 → 1 P1 transition. A magnetic field is used to create a Zeeman shiftthat cancels the change in the Doppler shift.
the main chamber. A laser beam detuned from the 1S0 → 1 P1 transition is shinned into theatomic beam in the opposite direction, slowing the atoms by laser cooling. To understand
why a magnetic field is used, the equation of radiative force exerted by a laser beam is:

F =

ħk
h
Γ
s0
=ħ
hk ×

∆−k
v 2
τscat
2 1+4
+s
Γ

(4.2)
0

where Γ is the natural linewidth of the 1S0 → 1 P1 transition, v is the velocity of the atom,and s0 , k and ∆ are the saturation intensity, wave-number and detuning of the cooling beam.
Note that the force is maximum when the detuning of the laser beam equals the shift due toDoppler effect (∆ = k v ). Suppose that the laser beam detuning is set such that the radiative
force is maximum for the atom initial velocity. When the atoms slow down, the change invelocity leads to a change in Doppler shift, reducing the effectiveness of the cooling process.
A magnetic field is used to create a Zeeman shift that cancels the change in Doppler shift,such that:∆ − k v (x ) − αB (x ) = 0(4.3)

for every position x , where B is the magnetic field at the position x , and α is a constant determined by the Zeeman effect. The acceleration exerted by the laser cooling beam at every
position x is then:

a=

Γ s0
F
=ħ
hk ×
.
m
2 1 + s0

(4.4)

For large saturations (s0 ≫ 1) this acceleration is roughly 5 × 105 m/s2 , which can be used
to cool down atoms travelling at a speed of 300 m/s in a few milliseconds. The distance d s
required to slow down the atoms is:
ds =

v0 2
≈ 100 mm2a(4.5)

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

48

where v0 = 300 m/s is the initial velocity. The required shape for the magnetic field B (x ) isthen

∆ − k v (x ) k v0 −B (x ) =
=
α

p



v
t
x
v0 2 − 2a x k v0
1− 1−
=
α
α
ds

(4.6)

where g is the Lande g-factor and µB is the Bohr magneton. For x = x s , the resultant peakmagnetic field is
B (x s ) =k v0 k ħ
h v0≈ 560 G.
=
α
g µB

(4.7)

The required magnetic field is created using a set of 3 coils, as shown in Fig. 4.5. The first
coil is made using a 1 mm diameter coil (low current coil), while the second (high current coil)and third (erasing coil) coils are created using a 3 mm hollow pipe which is cooled down withwater passing inside the pipe. The position of the atoms trapped in the center of the mainchamber by the MOT is highly dependent on both the offset and gradient of the magnetic
field. The third coil is used to eliminate the magnetic field at the center of the main chamberby creating a negative magnetic field. The resultant magnetic field is shown in Fig. 4.6.

Figure 4.6: Magnetic field generated by the Zeeman slower coils compared to the theoreticalrequired magnetic field.
The currents used for the low current coil, high current coil and erasing coil are 2.1 A, 67 A
and 67 A respectively. At the center of the main chamber (x = 0.345 m), the resultant bias
magnetic field is less than 0.1 G, while the gradient magnetic field is less than 0.2 G/cm. The
ideal curve in the plot was drawn using the parameters B (x s ) = 487 G and x s = 218 mm.
To test the performance of the Zeeman slower, the fluorescence spectrum of the atomic
beam is measured at the main chamber. The probe beam intersects the atomic beam at anangle of 80 degrees. Due to the Doppler shift effects, the resultant fluorescence spectrum reflects the velocity distribution of the atomic beam. The results of this experiment is shown inCHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

49

Fig. 4.7. When the Zeeman beam is activated, the fast part of the spectrum (left side of thegraph) moves to the show part (right side of the graph), indicating that the atoms are being
effectively slowed down.

Figure 4.7: Spectrography of the atomic beam showing the performance of the Zeeman
Slower.
The laser source for the Zeeman beam at 398.9 nm is generated by a frequency doubling
bow-tie ring cavity with a BiBO crystal. In the initial experiments, a PPKTP crystal in a bow-tie
cavity was used to second harmonic generation, but the conversion efficiency was low (13%for a 1 W input) and optical damage was observed in short-time operation. The fundamental light is generated by a Ti:Sapphire laser (∼ 1.6 W at 798 nm, Coherent MBR-110) which is
pumped by a green solid-state laser (10W at 532 nm, Coherent Verdi V-10). The output powerof the ring cavity is typically ∼ 300 mW with very good short-time and long-time stability.
An ultra-low expansion cavity (Finesse 500, FSR 1.5 GHz) is utilized to frequency lock theexternal cavity of the Ti:Sapphire laser. The frequency of the Zeeman Slower beam is detuned−2π × 700 Mhz. Two double pass acousto-optic modulator are used to create the light for theabsorption imaging system.

4.4.2 Magneto optical trap
After slowing down the atoms with the Zeeman Slower, atoms are trapped into the magneto
optical trap (MOT). This trap is composed by three retro-reflected, orthogonal beams that are
red-detuned to the 1S0 → 3 P1 transition and provide Doppler cooling in all directions. As thecooling beams themselves only reduce the velocity of the atoms and do not provide a restoring
force, two coils in the anti-helmholtz configuration are used to add a magnetic field gradient,which provides the required restoring force to the cooling beams. As a result, all the atoms

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

50

To absorp"onphotography
MechanicalShu!erf=150

to Zeeman
SlowerDumper

f=150

BiBO Crystal
SHG

AOM
f=-50
f=175

Verdi V-10
EOM

to Wavelength
Meter

to ULE
Isolator

MBR-110Figure 4.8: Laser system for 399 nm (1 S0 ↔ 1 P1 transition)gathers around the zero of the magnetic field.

Atomic beam

Zeeman Slower
399 nm

MOT beams
556nm
Figure 4.9: Atoms slowed down by the Zeeman slower are trapped into the magneto opticaltrap, which combines Doppler cooling and a quadrupole magnetic field formed by two coils.
The two coils that form the quadrupole magnetic field are separated by an average distance of 100 mm, with each coil having 150 turns and an average coil diameter of 65mm;.
When a current of 0.3 A flows through the coils, a gradient magnetic field of 0.5 G/cm in the
XY axis and −1.0 G/cm in Z axis is created.
Each cooling beam has a beam diameter of 1 cm and a power of 13 mW and 26 mW in the
XY and Z directions, respectively. This results in a laser intensity of I = 17 mW/cm2 = 118I s
and 33 mW/cm2 = 236I s in the XY and Z directions, respectively. The detuning is set to −2π×

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

51

Figure 4.10: Atoms trapped in the MOT after 10 seconds of loading time.
2 MHz = 11Γ . Note that the intensity and detuning used here are larger than the requiredfor optimal laser cooling, where I ≪ I s and ∆ = Γ /2. These large intensities and detunings
increase the capture velocity range, which results in an increased number of atoms in the trap.
Figure 4.10 shows the trapped atoms inside the main chamber after 10 seconds of loading.

Figure 4.11: Absorption image of the atoms trapped in the MOT after a time-of-flight of 11
ms.
To load as much as atoms as possible in the optical dipole trap, the density of the trappedCHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

52

atoms is increased (compression) while reducing their temperature. This is done by incrementing the current of the coils to 3 A during 50 ms, while decreasing the intensity and detuning of the cooling beams. The trapped atoms after compression is shown in Fig. 4.11. The
final number of atoms after compression and cooling is 3×108 . The temperature of the atoms
was 13 µK.

Fiber Laser
1111.2 nm

IsolatorGrangFiber
to MOT
Dumperf=150

Fiber toWavelength
Meter

AOM

AOM
EOM

f=150

Fiber
to ULE

Dumper

PPKTP Crystal
SHG
Figure 4.12: Laser system for the 1 S0 ↔ 3 P1 transition at 556 nm.
The 555.8 nm laser beam is generated by a frequency doubling bow-tie ring cavity witha PPKTP crystal. The fundamental light is generated by a fiber Laser (Menlo system). An
isolator and a reflective grating are used to protect the fiber laser against back-reflections.
The output power of the ring cavity is typically ∼ 200 mW at 555.8 nm. An ultra-low expansion
cavity (Finesse 5000, FSR 1.5 GHz) is utilized to frequency lock the fiber laser.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

53

4.5 Transport using optical dipole traps
Evaporative cooling using a crossed optical dipole trap (ODT) is the conventional method tofurther cool down Yb atoms after the MOT. In this method, atoms in the MOT are first loaded
into a single ODT. The second ODT is then introduced, which crosses the first ODT at thecenter of the trap. The intensity of the first ODT is gradually reduced resulting in evaporative
cooling, where the hot atoms are removed while the overall temperature of the remaining
atoms is reduced. In this experiment, a cross ODT is also utilized to reduce the temperature of
the atoms, but each of the ODT beams have the additional purpose of transporting the atoms
near the surface of the SIL. After loading the atoms into the first, horizontal ODT, the trap
is translated 450 mm from the center of the MOT chamber to a distance of few millimeters
away under the surface of the SIL. The second, vertical ODT is then introduced to realize
evaporative cooling. After cooling the atoms, the vertical ODT is converted into a conveyor
belt, which is useful to further transport the atoms to a distance of a few tens of micrometers
to the SIL surface.

4.5.1 The optical system
The high power optical beam required for optical trapping is derived from a continuous-wave,
frequency doubled, diode-pumped solid-state laser (Coherent Verdi V18) with an output of
18 W at a wavelength of 532 nm. The optical system is shown in Fig. 4.13. The same opticalsource is used to generate the horizontal and vertical ODT beams. Details of the system will
be explained in the following sections.

4.5.2 Horizontal ODT: Loading
Atoms trapped in the MOT are first loaded into the horizontal ODT by focusing the ODT beam
into the center of the compressed atomic cloud. The focused beam is created by a pair oflenses situated on top a moving air-bearing stage (Aerotech ABL15050). The air-bearing stage
can move up to 500 mm and it ensures a smooth movement during the transport process
reducing the heating effects produced by mechanical vibrations. The beam profile at the waistof the horizontal ODT is a circular beam with a radius of 29 µm resulting in a potential depthof 560 µK. The absorption images before and after loading the atoms in the ODT are shownin Fig. 4.14. Initially 3.5 × 107 atoms at a temperature of 50 µK are loaded into the horizontal
ODT.

4.5.3 Horizontal ODT: Lifetime and temperature
The number and temperature of atoms as a function of trap holding time is shown in Fig. 4.15.
The number of atoms is shown in a semi-logarithmic graph and it was fitted using a double

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

54

DumperPBS

Motorized
HWP

HWP
Verdi V-18

Fast & slow
shutter system

PBS

AOM

Moving lenseOptical fib

rGlan-laser
prism

Isolator

HWP

MOT Chamberrtvehorizontal
ODT

l
ica

T
OD

Glass cell
3’’ mirror
with piezo actuator
PBS

Motorized Moving
QWP
mirrorConveyor belt system

Figure 4.13: Optical system for the horizontal and vertical ODT derived from a 18 W,
frequency-doubled DPSS laser.

Figure 4.14: Absorption images of atoms after loading into the horizontal ODT.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

55

Figure 4.15: Number and temperature of atoms after loading as a function of holding time.
exponential curve
N (t ) = N1 e −t /τ1 + N2 e −t /τ2(4.8)

resulting in decay times of τ1 = 5 s and τ2 = 24 s. The first decay time is due to two-bodycollisions that results in evaporative cooling of atoms in the trap. The evaporative cooling
results in a decrease of temperature, as seen in the temperature plot in Fig. 4.15. The second
decay time is due to one-body losses such as collisions with background gas and scatterings
due to the optical-dipole trap.
slow
shutter

Slow shutter
Rise time 1 ms

lens

fast
shutterlens

Fast shutter
Rise time 40 μs

Figure 4.16: Fast and slow shutter system used for temperature measurement.
Note that there are two mechanical shutters for the horizontal ODT in the optical system(Fig. 4.13). To measure the temperature of trapped atoms the conventional method is tosuddenly release the trap and measure the thermal expansion of the atoms. This rapid release

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

56

condition is satisfied when the release time is faster comparable to the inverse of the trap
frequency of the dipole trap. Such fast release speeds are realized using a “fast mechanicalshutter” that shuts the beam at the point where its focused, as shown in Fig. 4.16. As thehigh power density at the beam waist may damage the mechanical shutter, a second ”slowmechanical shutter“ is activated at almost the the same time, which shuts down the beam a
few milliseconds after the fast shutter. This ensures that the high power density only impacts
the fast shutter for very short times.

4.5.4 Horizontal ODT: Transport

Figure 4.17: Absorption imaging of the atoms in the horizontal ODT after transport.

Figure 4.18: Potential depth along the propagating axis of the horizontal ODT for different
stage accelerations.
Atoms trapped in the horizontal ODT are mechanically transported by translating the position of the optical trap. The pair of lenses that focuses the horizontal ODT are located on

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

57

top of a translating air-bearing stage. During transport, the position of the stage moves by
450 mm to transport the atoms a few millimeters below the surface of the SIL. The absorption
imaging images of the atoms after transport are shown in Fig. 4.17.
The stage movement consists in an uniformly accelerated and decelerated motion with anacceleration of 0.3 m s2 , taking 2.5 s to move a distance of 450 mm. Although the air-bearing
stage used in this experiment is capable of accelerations up to 1 m/s2 , such fast accelerations
result in spilling due the inertial force experimented by the atoms. Figure 4.18 shows the potential depth including the inertial force of the stage along the propagation axis of the horizontal ODT for different stage accelerations. The curves where calculated for a beam with apower of 16 W and a beam waist of 29 µm. For the maximum acceleration of 1 m/s2 the potential depth is greatly reduced, resulting in spilling of atoms.

Figure 4.19: Number of atoms after transport as a relation to the stage acceleration.

Figure 4.20: Number and temperature of atoms after transport as a function of holding time.
The relation between number of atoms after transport and the acceleration of the stageCHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

58

was experimentally checked. The results are shown in Fig. 4.19. For an acceleration of 1 m s2
all the atoms were lost. Low accelerations resulted in a larger number of atoms after transport.
The number and temperature of the transported atoms as a function of trap holding timeis shown in Fig. 4.20. The number of atoms is shown in a semi-logarithmic graph and it was
fitted using an exponential curve. The resultant lifetime was of 6 s. The temperature of theatoms also is reduced during the first seconds of holding time, which is an indication that
evaporative cooling continues after transport.
The initial number of atoms and temperature was 1.4 × 107 and 17 µK, respectively. Notethat the number of atoms and temperature are lower compared to that in the MOT chamber
after a holding time of 2.5 s (equivalent to the transport time), which means that the inertial
force experimented during transport accelerate the evaporative cooling process.

4.5.5 Vertical ODT: Evaporative cooling using a cross ODTAfter the first stage of transport using the horizontal ODT, atoms are positioned 2.7 mm under the surface of the solid immersion lens. At this point, a second beam (vertical ODT) is
introduced, intersecting the horizontal ODT at the beam waist to form a crossed ODT trap.
The evaporative cooling procedure is schematized in Fig. 4.21. In order to eliminate reflections from the surface of the SIL, the beam incidence angle is at Brewster angle respect to theplane surface of the lens. The polarization was carefully selected using a Gran-laser prism(see Fig. 4.13). Note that the Brewster angle for fused silica at 532 nm is of 34.4◦ , which is 1◦
larger than the aperture of the spherical cap of the SIL (a numerical aperture of 0.55 is equivalent to 33.4◦ ). This ensures that both the incident beam and the retro-reflected beam (see thevertical transport section) will both enter and exit from flat surfaces, thus avoiding any straylight interference that may complicate the transport process.
To align the horizontal ODT with the vertical one, a 12 mm motorized actuator (Thorlabs Z812) was attached to the mirror mount that reflects the horizontal ODT right before theviewport. The motorized actuator is attached to the horizontal knob of the mirror, and canbe moved with 0.2 µm precision. As the distance between the two knobs is approximately
60 mm, and the distance from the mirror to the center of the solid immersion lens is 600 mm,
by moving the actuator by 0.2 µm the beam position at the bottom of the solid immersion
lens moves approximately 4 µm. For a vertical beam with a beam waist of 50 µm, it suffices tomove the horizontal beam at intervals of 40 µm. A distance of ±1 mm can be covered within 50
measurements, which takes less than 20 minutes for this experiment speed. To verify that thetwo beams are crossed, the horizontal beam power is reduced to 1 W. When the two beams
are correctly crossed, evaporative cooling occurs and the atoms shrink to the intersection of
the two crossed ODT.
To control the intensity of the horizontal ODT, the common approach is to use an acousto-

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

59

SIL
air

glass cell

Horizontal
ODT

vacuum sideYb(a) Atoms are originally trapped inside the horizontal ODT positioned
2.7 mm under the solid immersion lens.

NA 0.55

VerticalODTHorizontal
ODTYb(b) The vertical ODT beam is introduced, intersecting the horizontal
ODT beam at the beam waist. Evaporative cooling is realized by decreasing the intensity of the horizontal ODT.

Figure 4.21: Cooling of atoms by evaporated cooling.
optical modulator (AOM). The intensity of the diffracted beam can be controlled by changing
the amplitude of the input signal. One of the disadvantages of this method is that the intensityof the first order diffraction is usually around 80%, and consequently, not all of the power canbe used for the optical trap. Also, when the amplitude of the input signal changes, the pointing
stability of the diffracted beam deteriorates. In this experiment, instead of using an AOM,a motorized half-wave-plate in conjunction with a polarized-beam splitter (PBS) is used tocontrol the power of the horizontal ODT, as shown in Fig. 4.22. By using this method nearly
all the power can be used in the optical trap. Additionally, when the power of the horizontal
ODT is reduced by moving the half-wave plate, instead of dumping the remaining power it
can be recycled and used back for the vertical ODT.
The utilized motor is a stepping motor (Plexmotion PMSB-B56D1DHW) with a hollow
axis with inner diameter of 16 mm. The micro-stepping driver (PDSA-BA) is able to move themotor in 16 micro-steps, with one micro-step representing a rotation of π/1600. The timing
for the movement is controlled by a micro-controller Arduino Nano. The driver have 5 TTL inputs: CW and CCW move the motor in clockwise or counterclockwise direction, respectively,
when the corresponding TTL input changes its value (step-down or step-up). ENABLE input
enables or disables the current to the motor, STEP activates or deactivates micro-stepping,and AUTO_CURRENT manages whatever the motor current stops automatically after 100
ms of input inactivity. For our experiments we change only CW and CCW inputs, which are

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

to horizontal
ODT

PBS

Motorized
HWP

60

HWP
Verdi V-18
PBS

to vertical
ODT

CW
CCW
ENABLESTEP
AUTO_CURRENT

analog
input

B
B
A
A

Arduino Nano
Photodetector

TTL input 1
digital
input

B
B
A
A

motor output

PMSB-B56D1DHW

digital output

PDSA-BA(microstepping driver)

TTL input 2

24V

Hollow Stepping
Motor

Microstepping
DriverArduino Nano

Figure 4.22: System used to control the intensity of the horizontal ODT beam consisting in a
motorized wave-plate and polarized beam splitter.
controlled by the digital output of the micro-controller Arduino Nano. Initially, the motor
is initialized such as the transmittance of the PBS after the motor-controlled wave-plate is
maximum. To do this, the transmitted power is measured with a photo-detector and feedbacked to the micro-controller. When the TTL input 1 is activated, the motor rotates 360
degrees (3200 microsteps) and finds the optimal initial position. The TTL input 2 controlsãČżthe evaporated cooling: When the input is activated, the motor rotates and the intensityof the horizontal ODT is gradually reduced, and when the input is off, the motor goes back to
the initial position.
The vertical ODT beam have a beam waist of 40 µm and a final power after evaporation of
1 W . After 4 s of evaporative cooling, 1.0 × 106 atoms at a temperature of 2 µK were obtained.
Further cooling of the atoms is possible, but excessively cooled atoms are more susceptible toheating and loss during the vertical transportation stage. Absorption images of atoms before
and after evaporation are shown in Fig. 4.23.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

61

Horizontal ODT (before evaporation)

Cross ODT (after evaporation)

Figure 4.23: Absorption images of atoms before and after evaporative cooling. The final number of atoms is 1.0 × 106 atoms at a temperature of 2 µK.

4.5.6 Vertical ODT: Transport
The second stage of transport consists in using the vertical ODT as conveyor belt to transport
the atoms vertically. Details of the transport process are schematized in Fig. 4.24. To adiabatically transform the vertical ODT into a conveyor belt, a motorized quarter-wave plate inconjunction with a PBS is used to gradually increase the intensity of the retro-reflected beam(see the conveyor belt system in Fig. 4.13) in 100 ms. After the vertical ODT is converted into a
conveyor belt, atoms are tightly confined into the standing wave of the vertical ODT, and thehorizontal ODT can be removed completely. Finally, the retro-reflected mirror that is situatedin top of a air-bearing stage is moved, and the atoms are transported to the surface of the SIL.
When the atoms are too close to the surface, absorption images such as in Fig. 4.17 cannot
be used to observe the atoms due to the scatterings by the edge of the surface of the SIL.
To observe atoms near the surface, a different absorption imaging approach was used. Thissystem was schematized in Fig. 4.25. The procedure consists in using a probe beam that isincident to the center of the SIL by a shallow angle. As the probe beam reflects into the surfaceof the SIL, the surface acts as a mirror, resulting in two symmetrical absorption images of theatoms.
The distance between the atoms and the surface of the SIL d is related to the distance D
between the center of the two symmetrical images byd =M

D
2 cos θ(4.9)

where M is the magnification ratio and θ is the incidence angle of the probe beam. The
absorption images during transport using this imaging method are shown in Fig. 4.26. After the vertical transport, 1.0 × 106 atoms at a temperature of 2 µK were obtained. The final
distance between the atoms and the surface was of 20 µm.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

62

Vertical
ODT

SIL

air

glass cell
vacuum side

Horizontal
ODT
atoms

(a) The vertical ODT is retro-reflected to form an optical conveyor belt.

VerticalODT(b) The horizontal ODT is completely removed.

Vertical
ODTveMo(c) By moving the retro-reflecting mirror, the atoms are transported to the surface of the SIL.

Figure 4.24: Vertical transport of atoms using an optical conveyor belt.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

Imag
e

63

f = 60
am

be
ging
Ima

Figure 4.25: Absorption imaging system using the surface of the SIL as a mirror. The probe
beam is reflected by the surface of the SIL, generating two symmetrical images of the atomic
cloud. The incindence angle θ and the distance D between the two images can be used toprecisely calculate the distance d from the atoms to the surface of the SIL.

SIL surface
1

50 μm
0

Figure 4.26: Absorption images of atoms during transport. In each image the air-bearing stage
moves by 10 µm along the vertical ODT propagating axis.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

64

4.6 Optical accordion
Observation of atoms by a high resolution microscope system requires that atoms remain
confined in a pancake-shaped region which is thinner than the depth of field of the objective
lens. In previous experiments with rubidium atoms using a solid immersion lens, a thin layerof atoms was prepared by compressing an atomic cloud between the gradient potential of amagnetic field and a repulsive potential created by a blue-shifted evanescent wave [95]. In
the case of ytterbium atoms having a very small magnetic moment, magnetic fields cannot
be used to trap the atoms.
SIL

io
ord
Acc m
bea

n

glass cell

Figure 4.27: The optical accordion method. An accordion beam reflected into the surface of
the SIL creates a standing wave which spacing can be manipulated by changing the angle ofincidence.
To compress ytterbium atoms into a thin layer, an all-optical approach is mandatory. One
idea is to compress the atoms using an “optical accordion”. An optical accordion is an opticallattice with manipulable spacing. The first realization of an optical accordion used a lens tofocus two parallel beams [96]. When the two beams cross at the focal point, an optical latticewith spacing that depends on the separation of the two beams and the focal length of the lensis created. The spacing of this optical lattice was manipulated by changing the separationof the two beams. This method, however, cannot be applied directly to this experiment as
the thin layer of atoms should be positioned directly under the surface of the SIL where theoptical access is limited. Another disadvantage of using this method here is that the positionof the standing wave is determined by the relative phase of the two beams which is normallyunstable due to fluctuations in the surrounding air temperature and density.
Here, a novel method to create an optical accordion near the surface of the solid immersion lens is presented. As shown in Fig. 4.27, an optical accordion can be created when an
incident beam (accordion beam) is reflected into the surface of the SIL. As the incident beamand reflected beam interfere, an standing wave is produced with a spacing that only depends
on the incidence angle of the accordion beam. The lattice spacing a z can be manipulated by

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

65changing the incidence angle, related by the equation
az =

λ
.
2 sin θacc

(4.10)Th optical accordion created by surface reflection have two important advantages over
the two-beams approach. First, as the relative phase between the incident and reflected beam
is determined by the boundary condition of the surface, the position of the lattice is always“locked” respect to the surface. This ensures a stable trap that is useful when using a highresolution microscope system that have very limited depth of field. Second, as the standing
wave is created by a single beam and its reflection, only half of the power is required comparedto that of the two-beam approach.

lens

atoms

n beamaccordio

atoms

Figure 4.28: Potential of the accordion beam during loading and compression. The potential
plot was calculated for a single accordion beam with a power of 1 W.
After the vertical transport, atoms are located 20 µm away from the surface. By introducing the accordion beam at a initial angle of θacc = 0.7, all the atoms can be loaded into thefirst layer of the standing wave, as shown in Fig. 4.28. The intensity of the accordion potential is then decreased to realize evaporative cooling and create a Bose-Einstein condensate.
Finally, the accordion beam incidence angle is increased to compress the condensate into athin layer.

4.6.1 System for changing the accordion angle
The apparatus used to change the incident angle of the accordion beam consists of a lens(accordion lens) and a moving mirror, as shown in Fig. 4.29. The focal point of the lens is

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

66

Fi

xe
d

m

irr
or

Accordion lens(aspherical f150)

Moving mirror

Figure 4.29: System used to change the angle of incidence of the accordion beam. When the
mirror moves, the accordion beam displaces laterally. The accordion lens focal point is set tothe center of the SIL, resulting in a change in the angle of incidence.
aligned to the center of the SIL. When the mirror moves the beam displaces laterally, resulting in a change of incidence angle. The accordion lens is tilted by −6◦ respect to the z axis.
The mirror is placed on top of a air-bearing mechanical stage to avoid mechanical vibrations(Aerotech ABL10050, see Fig. 4.30). As the mechanical stage moves a maximum distance of

Fi

xe
d

m

irr
or

50 mm, the accordion angle can be changed from θacc = 0◦ to 15◦ .

Accordion beamMoving mirror

50mm air stage

Figure 4.30: Photo of the accordion optical setup schematized in Fig. 4.29
The photo of the optical setup used in the experiment is shown in Fig. 4.30. The moving
mirror placed in top of the air-bearing stage is a 1′′ mirror, while the fixed one is a 2′′ by 3′′
elliptical mirror. Both mirrors are tilted by 45◦ respect to the z axis, resulting in a z displacement of the accordion beam when the stage moves. Note that only one air-bearing stage isused to move the mirrors of two different accordion beams.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

67

4.6.2 Optical system
The accordion beam is derived from a fiber amplifier (ALS-IR-1SP-SF) with 15 W of output
power. It is seeded by a 50 mW laser generated by a external-cavity laser diode (ECLD) usinga Toptica fabry-perot laser-diode chip (LD-1080-0300-1). The optical system after the outputof the fiber amplifier is schematized in Fig. 4.31. Although the fiber amplifier includes an output isolator in the laser head, an extra isolator was used to protect the fiber amplifier against
strong reflections such as the retro-reflected accordion beam. A transmission grating is used
to remove the amplified stimulated emissions of the laser amplifier, as it will be explained indetail in the next chapter. The 1st index refracted beam is then divided into two beams thatform each of the accordion beams. The amplitude and frequency of each accordion beam iscontrolled by acousto-optical modulators (AOM).
After compression, all the atoms are located into a thin layer close to the surface of the SIL.
To maximize the surface of the optical trap and at the same time maintain its potential deep
enough, the cross section of the accordion beam at the waist was selected to be an ellipse witha waist of 27 µm and 41 µm along the vertical (z ) and horizontal (x , y ) directions, respectively.
The beam profile of the accordion beam before inciding the accordion lens have a beam
diameter of 25 mm and 40 mm (see Fig. 4.32) in the vertical and horizontal directions, respectively. To create such a beam shape, two pairs of beam shaping lenses and one anamorphic
prism were utilized.

4.6.3 Accordion lens design
As the accordion angle is changed from shallow angles (θacc ≈ 1◦ ) during loading to steep
angles during compression (θacc ≈ 12◦ ), the beam is vertically displaced by tens of millimetersby the moving mirror. The optical accordion technique requires that every displaced beam
always focus into the center of the SIL. For this condition to be satisfied it is important toensure that spherical aberrations of the accordion lens are kept to a minimum. The opticalaberrations where simulated using a lens design software (Zemax) for two different types ofcommercially available lenses, as shown in Fig. 4.33.
The simulation includes the aberration generated by the glass cell, that have a thickness of
3 mm and index of refraction of 1.45. As the accordion lens is tilted by −6◦ respect to the glass
cell, the glass cell in the simulation was tilted by 6◦ respect to the lens. Figure 4.33 shows thesimulation result for two different lenses, a spherical (Thorlabs LA1002-C) and an asphericallens (Thorlabs AL75150-C). The ray fan plot shown in the figure displays how much the beam
moves on the screen when the incident beam displaces vertically (transverse ray aberration).
Note the striking difference between the aberration of the two lenses, even when the verticalscale in the aspherical lens plot is one order of magnitude smaller. The aspherical lens per-

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

68

Dumper
Seed
50mW

Transmission
grating

Isolator HWP

Fiber amplifier
PBS
HWP
QWP

AOM

QWP

re
t

ro
re
fle

ct

io

n

Optical fiber

Anamorphic
prism

n
io
rd
co am
Ac be

PBS

La
tti
c

e

HWP

Moving mirrors
and fixed mirrors
SIL
Glass cell
Accordion lens(aspherical f150)

S

PB
S

PB

d
ize
or P
ot
M HW
Figure 4.31: Laser system for the accordion beams.

forms even better when the glass cell is not present, meaning that the spherical aberration
here is limited mainly by the glass cell. For the experiments, the Thorlabs AL75150-C aspherical lens was selected. Although it is possible to correct the glass cell aberration by using acustom ordered aspheric lens, the commercially available lens selected here was sufficient tosuccessfully compress the atoms up to θacc = 12◦ . For the final fluorescence imaging experiments, an angle of incidence of only θacc = 6◦ was used. Consequently, the system designed
here have a large design margin sufficient for the experiments.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

69Figure 4.32: Cross section of the accordion beam before inciding the accordion lens. The
vertical and horizontal axes represent the z and x , y directions, respectively.

±100 μm

screen3mm glassThorlabs LA1002-C(Spherical Φ75 f150)±10 μmscreen3mm glassThorlabs AL75150-C(Aspherical Φ75 f150)±15.8 mmFigure 4.33: Spherical aberration simulation using the software Zemax. Two different lenses
are compared.

4.6.4 Removal of amplified spontaneous emission
The ytterbium-doped fiber amplifier used as light source for the accordion beam have thedisadvantage inherent to all the fiber amplifiers, which is the production of amplified spontaneous emission (ASE). One of the problems of using this source here is that the accordion
beam wavelength was selected to be near resonant to the 1 P1 ↔ 1 S0 at 1077 nm in order tocreate the deep potential in the 1 P1 state required by the fluorescence imaging experiment.
If the ASE at the transition wavelength is strong enough to excite the atom into the (6s7s)1 S0 ,

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

70

the lifetime of the atom during the imaging process can be considerably reduced. First, an
optical spectrum analyzer was used to measure the ASE of the laser system for a pump lightat 1081 nm, as shown in Fig. 4.34. In the figure is also shown the spectrum after adding along-pass filter (Semrock BLP01-1064) and a transmission grating (Ibsen photonics FSTGPCG-1250-10XX-986) directly before the fiber coupling device.

Figure 4.34: Spectrum of the fiber amplifier output. The input power is 60 mW and the resolution was set to 0.1 nm. The plot at right shows a detailed measurement nearby the resonancewavelength (1077 nm). Note that the baseline at −80 dBm is the detection limit of the instrument.
The maximum allowed ASE (laser power at the resonance wavelength) for the experimentcan be calculated with the formula
1
PASE = I s s0 w̄02 π = 0.5 pW
2(4.11)

where I s = 0.3 mW/cm2 is the saturation intensity of the 1 P1 ↔ 1 S0 transition, s0 = I /I s = 10−4is the ratio of the light intensity I to the saturation intensity, and w̄0 is the average radius of thebeam waist. Note that s0 = 10−4 here is the inverse of the number of photons required for the
fluorescence imaging. In terms of power spectral density (PSD), this value can be rewritten
as :δPASE (ω) = PASERω
= −53 dBm/nm
Γ(4.12)

where Rω is the resolution in terms of angular frequency (0.1 nm represents roughly 2π ×25 GHz), and Γ /2π = 3.0 MHz is the linewidth of the transition. Looking back to the plot at Fig.
4.34, the PSD of the ASE at the resonance wavelength was −45 dBm/nm for an input powerof 60 mW. For the lattice power used in the experiment during the fluorescence imaging, the
PSD increases by a factor of 25 dB to −20 dBm/nm, which is 33 dB stronger than the estimatedmaximum δPASE .

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

71

To reduce the effects of the ASE a transmission grating (Ibsen photonics FSTG-PCG-125010XX-986) was introduced directly after the output of the laser amplifier (see Fig. 4.31). The
measurement using the optical spectrum analyzer shows that the ASE is reduced by at least−25 dB at the resonance wavelength (see in Fig. 4.34).
The filtering capability of a transmission grating can be numerically estimated. The diffraction angle θm of the ray with index m in the case of a transmission grating can be calculatedusing the formulad (sin θm − sin θi ) = m λ(4.13)

where d = 800 nm is the grating period, θi is the angle of incidence, and λ is the wavelengthof the input beam. For the experiment m = −1 was used, resulting insin θ−1 = sin θi −λ.
d(4.14)The dispersion of the grating results from the derivative of the previous equation,
∂ θ−1
1
=−
= −0.098◦ /nm∂λd cos θ−1

(4.15)for λ = 1081 nm and θi = 41.7◦ (the angle of incidence according to the specification sheet).
To calculate how much of a dispersed beam couples into a fiber, the idea is to calculate
the power coupling coefficient of two Gaussian beams ψa and ψb at a plane S [97], as∫∫Ka b =

2ψa ψb d S .

(4.16)In the case of two Gaussian beams with the same beam waist w0 tilted by an angle θ respect
to each other, Eq. 4.16 reduces to  πθ w0 2
K a b = exp −.
λ(4.17)

Finally, replacing Eq. 4.15 into the transmission grating Eq. 4.17, the fiber coupling efficiency  π(λ − λd )w0 2.
T (λd ) = exp −d cos θ−1 λ(4.18)

of a dispersed beam with wavelength λd is obtained. Figure 4.35 shows the transmission for
λ = 1081 nm and w0 = 500 µm. For a change of wavelength of 1 nm, the transmission of thefiber coupling is reduced by −30 dB. Figure 4.34 shows that for a change of wavelength of 1 nm
the PSD is reduced by roughly −15 dB by using a transmission grating, which is much less thanthe expected theoretical reduction. It is possible to think that detection limit of the spectrum
analyzer limits the peak width of the carrier signal. Two facts reinforced this hypothesis: 1) a

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

72

Figure 4.35: Transmission of the grating including fiber coupling, for a central wavelength of
λ = 1081 nm.
measure of the spectrum of a single-mode Ti:S output spectrum, which should be extremelly
sharp, resulted in a similar profile. 2) When a long pass filter was used in combination to
the transmission grating, the transmission profile remained unchanged. If the long pass filter
reduces the ASE at 1077 nm by 20dB and the grating reduces it by 25dB, using a combination ofboth should result in a reduction of 45dB. Experimental results, however, showed no changes
in the spectrum when the long pass filter was added after the transmission grating.
The scattering in the optics before and including the grating also limit the extinction ratioof the ASE. Supposing that the optical elements are D = 1 m away from the fiber coupling, andonly the cross section equivalent to the beam surface (πw02 ) couples into the fiber. then only
πw02
4πD 2 = −72 dB of the scattered light will be transmitted. If only 10% of the total power scattersin the optical elements, then at least −82 dB of the ASE will always couple into the fiber for all
the wavelengths.
In conclusion, the ASE should be reduced by at least 33dB for the experiments when thenumber of photons required is ∼ 10000. The optical spectrum analyzer used to test the grating
showed a reduction of at least 25dB, which is possibly limited by the detection performance of
the apparatus. In theory, the transmission grating would reduce the ASE intensity by at least
80dB which is five order of magnitude higher than the required for the experiment. It should

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

73

be noted that our team previously tested the performance of reflective gratings for linewidth
filtering in [98], showing good agreement between the theory and the experimental results.
One way to ensure that the ASE is completely removed would be to use ultra-steep band-passor long-pass filters, which are very costly but guarantee a reduction of at least 60dB.

4.6.5 Alignment of the accordion lensAs explained in a previous section, in order to compress the atoms using the optical accordion system, the focus of the accordion lens should be precisely positioned on the center of
the SIL. The alignment of the accordion lens was realized using three axis stage where the 3”
aspherical lens holder was mounted, as shown in Fig. 4.36.

Figure 4.36: The accordion lens mounted in a three axis stage.
For a precise alignment it is convenient to observe the atoms from two different angles.
In the case of this experiment, the reflected absorption imaging technique was used to observe the atoms from one side, while the fluorescence resultant from absorption imaging was
captured by the microscope system using an objective lens with small magnification (Mitsutoyo M Plan Apo SL 20X). Initially, atoms after vertical transport were observed as shownin Fig. 4.37. To test whether the accordion beam intersects the atoms, the accordion beamis activated while the vertical ODT is suddenly cut, and the atoms are observed after 3 ms oftime-of-flight. If the accordion beam is not correctly aligned to the center of the SIL, the atoms
fall due to gravity on the z direction. On the other hand, in the case of a correct alignment
the atoms are trapped by the accordion potential and expand along the y (accordion beam
propagation) axis (bottom images of Fig. 4.37). If the accordion beam is not correctly aligned,

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

74

the position of accordion lens in the x and z direction should be modified until the atoms aretrap.
Top View (fluorescence imaging)
After vertical transport

Top View (fluorescence imaging)
Accordion ON + 3ms ToF

Lateral View (absorption imaging)
After vertical transport

Lateral View (absorption imaging)
Accordion ON + 3ms ToF

Figure 4.37: Absorption and fluorescence imaging of atoms used for the alignment of the accordion lens.
Note that there are three degrees of freedom on the position of the accordion lens, fromwhich only two can be determined using the alignment mentioned before; it is possible tocross the atoms with the accordion beam even when the focal plane of the lens is not positioned at the center of the SIL. The third degree of freedom is determined when the accordion beam crosses the atoms for two different angle of incidence. The alignment procedure
is as follows: suppose that for an accordion angle θacc = θ1 the correct position of the lens
is (x1 , y , z 1 ) and for a different angle θacc = θ2 the correct position is (x2 , y , z 2 ). The goal is tochange the position y such that both ∆x = x 1 − x 2 and ∆z = z 1 − z 2 converges to zero. In
other words, the accordion lens focal plane crosses the center of the SIL if and only if for two
different angles θacc the accordion beam traps the atoms. This procedure can be realized in asystematic way ensuring a precise alignment of the accordion lens.

4.6.6 Loading of atoms into the optical accordion
The experiment procedure to load the atoms into the accordion potential consists in gradually reducing the vertical ODT retro-reflected intensity and at the same time increasing theintensity of the accordion beam (Fig. 4.38b). After loading the atoms, a BEC is created using
evaporative cooling by reducing the intensity of the accordion beams (Fig. 4.38c). As seen in
Fig. 4.31 there are two different orthogonal accordion beams in the x and y directions. The

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SILa) Vertical ODTb) Vertical ODT + Accordion

75c) BEC

Figure 4.38: Absorption imaging of atoms. a) After vertical transport. b) After loading theatoms into the first layer of the optical accordion. c) After creating a BEC.
reason for using two accordion beams is that each of the beams can be later retro-reflected toform a two-dimensional optical lattice.

4.6.7 Bose-Einstein condensate: Single accordion
Accordionloading

Evaporative cooling

Vertical ODTStanding Wave
Vertical ODTPowerAccordion 1 (or 2)
Power
100ms

100ms

900ms

Figure 4.39: Time sequence for creating a BEC with a crossed-dipole trap using a single accordion beam.
Although there are two orthogonal accordion beams, it is possible to create a BEC using
a single accordion. For alignment and optimization purposes, it is convenient to first create the BEC using a crossed trap between the vertical ODT and each of the accordion beams
separately. Figure 4.40 shows the time sequence used to load the transported atoms into thefirst layer of the optical accordion and create a BEC. First, the intensity of the retro-reflectedvertical ODT is reduced in order to transform the optical conveyor belt back into a travelingwave. At the same time, the power of the accordion beam is increased to 2.1 W, and atoms
are loaded into the first layer of the accordion potential. To create a BEC, the intensity of thevertical ODT is maintained constant while the intensity of the accordion beam is reduced to
500 mW in an quasi-exponential way. Figure 4.40 shows the time-of-flight of the BEC for eachof the accordion beams. Due to the strong potential produced by the vertical ODT beam, theBEC expands anisotropically in a direction almost perpendicular to the vertical ODT propa-

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

76

gation axis.
Vertical ODT + Accordion 1

0 ms

Vertical ODT + Accordion 2

7 ms

0 ms

7 ms

Figure 4.40: Time-of-flight of the BEC up to 7 ms for each of the two different accordion
beams.
The number of atoms as a function of the holding time after creating the BEC is shownin Fig. 4.41. The results were fitted using a three-body and one-body loss model for a threedimensional BEC [99]:dN= −γN 9/5 − αN
dt

(4.19)hγ  4 αt γ i−5/4
N0 −4/5 +e5 −αα(4.20)

whose solution is given by:
N (t ) =resulting in an initial three-body loss ratio γN 4/5

−1= 1.4 s and τ = 23 s.

4.6.8 Bose-Einstein condensate: Double accordion
After optimizing each of the accordion beams to create a BEC, it is possible to create a 3beam-crossed ODT formed by the two combined accordion beams (double accordion) andthe vertical ODT. In this case, the time sequence to create the BEC consists in first reducing
the power of the optical accordions and later removing the vertical ODT completely, as shownin Fig. 4.42. This procedure resembles an evaporative cooling technique using a dimple trap[29]. Figure 4.43 shows the time-of-flight of the BEC in the double accordion case. The final
power of the accordion beams is 300 mW.
Figure 4.43 shows the time-of-flight of the BEC. In contrast with the time-of-flight images

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

77

Figure 4.41: Lifetime of atoms after creating the BEC.
Accordionloading

Evaporative cooling

Vertical ODTStanding Wave
Vertical ODTPowerAccordion 1+2
Power
100ms

500ms

500ms

Figure 4.42: Time sequence for creating a BEC with a crossed-dipole trap using a double accordion beam.
of the BEC when only one accordion beam was used, the absence of the vertical ODT here
results in a symmetrical trap in the x and y directions. As the trap frequency in the z direction
is higher, the cloud expands anisotropically where the z direction is slightly elongated.
The number of atoms as a function of the holding time after creating the BEC is shownin Fig. 4.44. The results were fitted using a three-body and one-body loss model, resulting in
−1
γN 4/5= 69 s and τ = 71 s.

4.6.9 Compression of atoms
After creating the BEC, atoms are compressed by moving the air-stage where the moving mirrors are situated (see Fig. 4.29). The absorption images after compression are shown in Fig.
4.45.
Note that for steep angle of incidences the atoms move very close to the surface and themirrored absorption images become superposed. In the case of Fig. 4.45, the air-stage moves

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

78

Vertical ODT + Accordion 1 + Accordion 2

0 ms

7 ms

Figure 4.43: Absorption images of the atoms for the double accordion setup. Time-of-flights
are between 0 ms and 7 ms.

Figure 4.44: Lifetime of atoms after creating the BEC in the double accordion setup.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

79

Figure 4.45: Absorption images of the atoms after compression. Time-of-flights are between
0 ms and 7 ms.

Figure 4.46: Lifetime of atoms after compression. The solid line is a fitting using a two-bodyloss model.
27.3 mm with a constant acceleration and deceleration of 1m/s2 , requiring 330 ms. The final
angle after compression was θacc = 11◦ , resulting in a strong confinement in the z axis which

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

80

produces a highly anisotropic density profile after free expansion. The number of atoms asa function of the holding time after compressing the atoms is shown in Fig. 4.46. The results
−1
was fitted using a two-body losses curve resulting in an initial loss rate of β 〈n〉r
≈ 300 ms.
A three-body model did not fit properly with the results. The three-body losses rate was es−1
Ktimated theoretically as 63 〈n 2 〉r
= 260 ms, which is comparable with the initial loss rate
obtained in the fitting. This means that the lifetime profile is probably determined by a combination of two-body and three-body losses, which cannot be solved analytically.
Note that the final experiments were realized at θacc = 6◦ . This angle is sufficient to produce a condensate gas in the 2D regime and it provides deeper potentials as the reflectivity of
the SIL decreases as θacc increases. The lifetime at this angle resulted in ≈ 2 sec.

4.6.10 Density profile of the BEC
The integrated density profile in the horizontal direction of the BEC before compression isshown in Fig. 4.47 for three different stages of evaporative cooling and 7 ms of time-of-flight.
When the accordion beams power is 170 mW the atomic cloud is a mixture of a thermal gas
and a condensate fraction resulting in a bimodal density profile (see Fig. 4.47(a)). The thermal cloud have a Gaussian profile from which a temperature of 180 nK was estimated. The
condensate part of the cloud represent 20% of the total atoms and have a Thomas-Fermi(parabolic) distribution. After further reducing the accordion beams power to 120 mW theevaporative cooling process finishes resulting in a pure BEC with not discernible thermal
component (see Fig. 4.47(c)). The estimated critical temperature is Tc ≈ 200 nK.
After compressing the atoms to θacc = 6◦ the trap frequency in the vertical(z ) direction isincreased by a factor of ∼10 (cf. Appendix A.2), while the trap frequencies in the horizontal(x ,y )directions only increase by a factor of ∼2. The potential depth is also increased by a factorof ∼8. Consequently, the resultant integrated density profile in the horizontal direction (seeFig. 4.48) have a very similar shape to the density profile before compression. The thermal
component on the atom reappears (see Fig. 4.48(top)) if the power of the accordion beams
is maintained constant during compression, possible due to an increment of three-body collisions leading to heating. The thermal component disappear completely after the power of
the accordion beam is further reduced.
The vertical integrated density profile of the compressed BEC is shown in Fig. 4.49. Noticethat the density profile does not match a Thomas-Fermi distribution but instead a Gaussian
one. This condition arises when the Thomas-Fermi approximation is no longer valid and themean field energy Emfe becomes smaller or comparable to the vibrational energy separationh ωz , as:ħ
Emfe =4πħ
h 2a n<ħh ωzm(4.21)

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

81(a) Condensate fraction ∼20%. T = 180 nK. Pacc = 170 mW.

(b) Condensate fraction ∼40%. T = 140 nK. Pacc = 140 mW.

(c) Condensate fraction ∼100%. Pacc = 120 mW.

Figure 4.47: Horizontal integrated density profile of the BEC before compression, for threedifferent potential depths. The time-of-flight is 7 ms.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

82

BEC + thermal (250nK)

Pure BEC

Figure 4.48: Horizontal integrated density profile of the BEC after compression to θacc = 6◦ fortwo different potential depths. The time-of-flight is 7 ms.
where a is the scattering length of ytterbium (see Table 2.2) and n is the peak number densityof atoms. Eq. 4.21 can be rewritten as:

N<

v
t

v
u
32ħ
h t ω3z≈ 10000
225ma 2 ω4x y(4.22)

where ωz /2π = 1.7 kHz and ω x y /2π = 0.06 kHz was used for the calculations.
For the number of atoms after compression, the mean field energy is comparable with thevibrational energy separation ħh ωz resulting in a Gaussian profile in the z direction. For thequantum gas microscope experiments, the number of atoms is usually reduced to ∼ 1000 orless by further reducing the potential depth. In that case, the condition of Eq. 4.22 is satisfied
and the condensate behaves as a 2D quantum gas.

4.6.11 Stability of the system
The short-term stability of the system is shown in Fig. 4.50. Considering the complexity ofthe transport and compression process, the number of atoms in the compressed condensate
is very stable. Even for compression up to θacc = 10◦ the fluctuation in the number of atoms

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL
VerticalVerticalHorizontal

83

Horizontal

Figure 4.49: Integrated density profile along the horizontal and vertical axis. The horizontal
profile match a Thomas-Fermi distribution while the vertical distribution is Gaussian.
is less than 10%. It is not clear what determines the stability of the system at this point.

CHAPTER 4. EXPERIMENT: TRANSPORT OF ATOMS TO THE SIL

84(a) Single accordion beam (x axis)(b) Single accordion beam (y axis)(c) Double accordion beam (x + y axis)(d) Double accordion beam after compression toθacc = 10◦

Figure 4.50: Stability of the system during 60 ∼ 80 shots.

Chapter 5

Experiment: Fluorescence imaging
In the previous chapter a BEC of atoms was compressed into a thin pancaked-shape condensate near the surface of the SIL using the optical accordion technique. Such an atomic cloud
was a required condition for observing atoms using a high numerical aperture microscope
system as the atoms need to be confined in a region thinner than the depth of field of theobjective lens. This chapter centers in the experimental setup used to load the compressed
atoms into a two-dimensional optical lattice and the imaging process to realize fluorescenceimaging of the atoms with single-site resolution.
emCCD camera
Andor iXon888

objective lensMitutoyo
M Plan APO
SL100X
NA 0.55

Accordion retrore(2D lattic
e

)

PBS

motorid
half-waze
ve p

late

flection

PBS

opticalaccordion
1082 nmshutter

lensesexcitation
beam

atoms
fluorescenceglass cell

Figure 5.1: High numerical aperture system used for observing atoms fluorescence.
The experimental apparatus used to trap and image atoms in a two-dimensional opticallattice is shown in Fig. 5.1. The two-dimensional optical lattice is created by retro-reflection of
the optical accordion beams, as it will be explained in detail in Sec. 5.1. Section 5.3 focuses on
the imaging process using a deep potential in the excited state. This is followed by an analysis
85

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

86

of the resolution of the system in Sec. 5.4.1. Finally, the lattice reconstruction algorithm andthe lifetime of the system are discussed in sections 5.4.2 and 5.4.3.

5.1 Loading of atoms into the 2D optical lattice
Atoms trapped in the first layer of the accordion lattice are loaded into the two-dimensional
optical lattice by retro-reflecting the optical accordions. The retro-reflection system shownin Fig. 5.1 is comprised by a mechanical shutter, two polarized beam splitters and a halfwave plate mounted on a stepping motor with the same properties as the one explained inSec. 4.5.5. After opening the mechanical shutter, the stepping motor rotates 400 micro-steps(equivalent to 45◦ ) with a constant acceleration and deceleration of 2 × 105 steps/s2 . As twopolarized beam splitters are used, the retro-reflected intensity is proportional to sin4 ( 400 π2 ).
step

The potential depth during loading can be roughly estimated with the equation:


p 
p
p
V (x , y , z ) = Vacc (x , y , z ) 1 + R − 2 R + 4 R cos2 (k x x ) + 4 R cos2 (k y y )(5.1)

where R (t ) is the intensity of the retro-reflected beam divided by the accordion beam intensity, Vacc (x , y , z ) is the accordion potential, and k x = k y = 2π cos θacc /λlat are the wavenumbers in the x and y direction, respectively. For an optical accordion with an incident angle ofθacc = 6◦ , the resultant lattice spacing is of 544 nm. Figure 5.2 shows the transmittance of thedifferent optical elements used for retro-reflection and the reflectance of the surface of the

SIL reflection
R=67%

half-wave plate
T=99%
PBS T=
98%

98%
PBS T=

opticalaccordion
1082 nmdichroic
T=99% mirror

SIL for an incident angle of θacc = 6◦ .

Glass cell
T=96% per surface

Figure 5.2: Optical losses of the optical accordion and retro-reflection system.
Considering the losses of all these optical elements, only 50% of the total power can be
retro-reflected. The reflected intensity can be written as:
R = 0.50 × cos

4
step(t ) π
.
400 2

(5.2)

Figure 5.3 shows the variation in the potential relative to the accordion potential V (0, 0, z )/Vacc (0, 0, z )CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

87pduring loading time. The curves represent the relative offset 1+R −2 R and the relative latticepdepth 4 R .

Figure 5.3: Change in the offset potential and lattice depth during loading. The potential units
are relative to the initial accordion potential.
Initially, all the atoms are located in the first layer of the optical accordion at z 0 = 2.6 µm.
Using Eq. 5.1, the lattice depth isp
V0 = 4 R Vacc (0, 0, z 0 ) = 2.8Vacc (0, 0, z 0 )(5.3)

nearby the central region of the trap. After loading, the estimated depth was V0 /k B = 1.9 µK
or V0 /E r = 40, and the trap frequencies were (ω x , ω y , ωz )/2π = (12, 12, 1.8) kHz.

5.2 Laser system
The laser system used to derive the excitation beam is shown in Fig. 5.4. A Ti:sapphire laser
was frequency-doubled using a bow-tie cavity and a BiBO non-linear crystal. The output of
the cavity was fed into an acousto-optical modulator (AOM) in the double-pass configuration,that serves as optical shutter and frequency shifter. The output of the AOM was coupled into
an optical fiber, obtaining 30 mW of power at the fiber output. Due to the high output powerof the excitation light and the limited extinction ratio of the AOM (−60 dB in the best case), it
is important to use a mechanical shutter to avoid heating during transport and compression.
The exposure pulse used for imaging is generated immediately after the mechanical shutter
opens, which is detected by a photo-detector placed after the shutter. The frequency of the
Ti:S laser was locked using a frequency-offset locking system, which consists in using the beatnote produced by the interference of the excitation and Zeeman slowing laser system as afeed-back signal.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

88

HWP

glass cell
SIL

fiber

QWP
Dumper
HWP
PBS
HWP

Mechanicalshutter

AOM
PD

SHG@399nm
using BiBO crystal

on
di n
or io
cc ct
l a efle
ica -rpt ro
O ret

Excitation DM
beam

Beam
shaping

MBR-110@798nmEOM

Isolator
PD

Figure 5.4: Laser system for the excitation beam

5.3 Imaging of atoms
The strategy to obtain a sufficient number of atoms, previously discussed in Sec. 3.4, consists
in creating a deep potential in the excited state and using a resonant excitation beam to couple
the ground state with the excited one. The resultant deep effective potential is capable ofcontaining the atoms that are heated during fluorescence imaging.

Optical lattice1077 nm

399 nmexcitation

Optical lattice

Figure 5.5: Deep potential method utilized during fluorescence imaging.
The imaging process starts by increasing the power of the accordion (lattice) beams to
Plat = 2 W in 10 ms. For this lattice power and a lattice detuning of ∆lat /2π = 1.0THz, theresultant lattice depth in the ground and excited state are Ug /k B = 280 µK and Ue /k B = 62 mK,
respectively. In the case of an atom experiencing an average lattice depth U0 = (U e + U g )/2,
the corresponding trap frequencies are (ω x y , ωz )/2π = (1.6, 0.23) MHz.
The excitation light, which is aligned on the same optical axis as the optical accordion, isinserted from the retro-reflection side using a longpass dichroic mirror (Thorlabs DML650L).
The beam waist was set to 100 µm, corresponding to a peak intensity of 95 W/cm2 or 1600I s ,CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

89

where I s is the saturation intensity of the 1 S0 -1 P1 transition. The excitation beam is resonant

al on
tic rdi m
op co nac 082
1

ac op
10 cor tic
82 dio al
nm nal
tic ionop ord tionc cac efle
-r

tro
re

o
a pt
re cco ical
tro rdei
39 xcit -refl on
9 at
nm io ectinlig onht

with the shifted transition, which determines ∆0 = (Ue − Ug )/k B .

Figure 5.6: Direction and polarization of each of the accordion beams and excitationbeamãĂĂduring fluorescence imaging, as seen from the CCD camera.
Figure 5.6 shows the direction and polarization of each of the accordion beams and theexcitation beam as seen from the microscope system. Note that the excitation light is shinned
from only one direction and it is not retro-reflected, resulting in a net radiative force. Details
and a quantitative estimation of this force were explained in detail in Sec. 3.5.1. Note alsothat as the optical lattice beams are symmetrical in each direction the excitation beam could
be in principle either inserted from either the x direction or the y direction (or both at thesame time). Experimentally, however, there was a noticeable degradation in the fluorescenceintensity when the excitation beam was inserted from the y direction. It was later discoveredthat the small frequency shift used to avoid interference of the two accordion beams broke
the apparent symmetry of the system uniquely determining a “good direction” for excitation.
Details of this asymmetry are studied in Appendix 5.5.
A single-shot image of the Yb atoms trapped in a two-dimensional optical lattice after
100 µs of exposure time is shown in Fig. 5.7. The solid angle of the imaging system implies
that only 21% of the total photons can be collected by the objective lens, which results in a
total photon collection efficiency of ∼5% for the microscope system. This value includes thetransmittance of the SIL (87%), the objective lens (56%) and the low-pass filters (92%), andalso the quantum efficiency of the CCD camera (∼0.5 e − /photon at 399 nm).
An average of 100 photons per atom was collected (∼2000 photons emitted) by the CCD

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

90

1

2 μm
0

Figure 5.7: Single-shot image of the Yb atoms trapped in a two-dimensional optical lattice
after 100 µs of exposure timecamera in the 100 µs of exposure time. This number of photons is one order of magnitude
smaller than the expected from the simulations. The definition of the image is, however, notoriously sharper than the expected. Two factors contributed to the increase in definition: 1)
The resolution of this microscope system is less than half compared with that of previous experiments with rubidium (cf. Sec. 5.4.1 ). This increases the fluorescence density by a factorof ∼4. 2) The extremely short exposure times used in this experiment (5 order of magnitude
smaller than the rubidium experiments) result in a dramatic reduction in the backgrounddark noise (cf. Fig. 5.17). Both effects lead to an increment in the S/N ratio of the obtained
image.

5.4 Analysis of the microscope performanceThis section is focused in studying the performance of the microscope, which includes ananalysis of resolution, lifetime and fidelity of the system. For this purpose of study it is convenient to use very sparsely populated lattices (filling < 0.05) in order to reduce the effectsof photo-association (PA) and hopping. Photo-association effects in multiply occupied sites
result in the formation of molecules that lead to the loss of atoms in times of the order of a
few to tens of microseconds. As the short imaging times in the system are comparable withthese times, PA losses modify the shape of the histogram and affect the lifetime estimation.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

91

5.4.1 Microscope resolution
The point spread function (PSF) is obtained by averaging the fluorescence images of different
isolated sites, as shown in Fig. 5.8. In total, 277 individual sites over 32 different images wereused for the average, resulting in the PSF in Fig.5.9.

Selected sites

Figure 5.8: Example of selected isolated sites in a single fluorescence image.
The resolution of the microscope system is obtained by fitting the intensity profile alongeach axis using a Gaussian function. The resultant FWHM average resolution was 319(2) and317(1) nm in the x and y directions, respectively.
FWHM
319(1) nm

FWHM
317(1) nm

300 nmFigure 5.9: Averaged fluorescence intensity in isolated sites. The PSF along the x and y axis
were fitted using Gaussian functions.
For a microscope system with a NA of 0.81 and an imaging wavelength of λ = 399 nm, theminimum possible resolution based on the Rayleigh criterion is 0.51λ/NA = 250 nm, which isonly 20% smaller than the obtained average resolution.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

92

5.4.2 Lattice reconstruction
Original image

After deconvolution

Figure 5.10: Deconvolution of a sparse image using the PSF obtained in Fig. 5.9
The reconstruction of the original distribution of atoms from the fluorescence images isdiscussed here. The algorithm used for reconstruction was created with the Mathematica
software, and consists of three steps: 1) Deconvolution, 2) Lattice fitting, and 3) Binning.
1) Deconvolution
Using a Gaussian matrix with the same size that the obtained from the averaged PSF in
Fig. 5.9, the image is deconvoluted using the “TotalVariation” algorithm. The resultant
deconvoluted image for a typical sparse image is shown in Fig. 5.10.
2) Lattice fitting
The direction, spacing and phase of the two-dimensional optical lattice is determinedby maximizing the total fluorescence of each site when only the central 3 × 3 pixels ofeach site is considered. As this process takes a considerable amount of processing time,the direction and spacing is obtained from a single image and later fixed for the rest of
the images. Figure 5.11 (c) shows a typical result. The determined lattice directions
where 43.55(2◦ and 135.51(2)◦ for the x and y directions, respectively. The lattice resultant spacing was 6.205(5) pixels, from where the magnification M of the microscope
system can be obtained:
M = 6.205P ×

2 cos θlat
= 148.3
λacc

(5.4)

where P = 13 µm is the pixel size of the CCD camera. As the refractive index for fusedsilica at 399 nm is n = 1.47 and the magnification of the objective lens is 100X, the expected magnification ratio was 1.47 × 100 = 147, which is very close to the obtained

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

93

value. The effective pixel size in the object plane is P /M ≈ 88 nm.

a)

b)

c)

d)

Figure 5.11: Reconstructed process for a sparse image and a exposure time of 40 µs. a) Original
image. b) Deconvoluted image. c) Lattice fitting. d) Binning and atom distribution detection.

3) Binning
The binning process consists in adding the fluorescence of the central 3×3 pixels of eachsite to determine the total atom fluorescence. A site is determined to be occupied if thetotal number of photons (in the central 3 × 3 binned part) is greater than a threshold L .
Figure 5.11 (d) shows the final distribution of atoms for each site, when a threshold of
L = 130 photons is used.
The photons per site distribution is shown in Fig. 5.12 in the case of the single image in
Fig. 5.11 and in the case of 171 images.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

94(a) Histogram for the single image shown in Fig. 5.11(b) Total histogram over 171 images.

Figure 5.12: Histogram of total fluorescence per site. left) Log scale, center) Magnificatedfluorescence peak in linear scale. right) Magnificated background peak in linear scale.

Figure 5.13: Lifetime obtained from histogram by exponential fitting for Plat = 1.1 W and∆lat /2π = 1.0 THz. The exposure time was 300 µs

5.4.3 Lifetime analysis
During the imaging process, atoms absorb the excitation light and are heated up due to random light scattering. After a number of scatterings, the atom escapes from the site and is lost.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING95

The lifetime of the atoms was determined using the histogram of the total intensity in eachsite. Assuming that the number of photons emitted by each atom is proportional to the lifetime, and the number of atoms decreases exponentially, the shape of the histogram is also
expected to decay exponentially for exposure times much longer than the lifetime. Figure5.13 shows a typical histogram result for Plat = 1.1 W and ∆lat /2π = 1.0 THz, where an exposure time of 300 µs was used.
For the analysis in this section, the lifetime of atoms will be expressed always in photons.
Although using units of time for the lifetime is the natural physical choice, the lifetime inseconds is not a good quantity for the analysis here. As the emission rate depends on the potential depth and saturation intensity, a longer lifetime in seconds does not necessary meanthat the atom is emitting more photons. Consider for example the case where the the saturation intensity is very low (s0 ≪ 1). As the emission rate is very low, the lifetime in seconds
is expected to be large. However, the number of mean photons emitted will be reduced dueto the low population of atoms in the 1 P1 state, which is not beneficial to the “deep potential”
method. Also, the lifetime in seconds cannot be directly measured, as the number of total
emitted photons is measurable but the number of atoms is undetermined. The method tomeasure lifetime using the histogram exponential decay, on the other hand, does not dependon the number of atoms. For a given potential depth and saturation it is possible to estimate the emission rate knowing the lifetime in photons and the histogram of fluorescenceper site. The expected emission rate obtained in the simulations is in the range between 60
to 84 photons/µs. The maximum emission rate 2/Γ ≈ 88 photons/µs.

Figure 5.14: Lifetime dependency on lattice power Plat . The solid curves are fitting using Eq.
3.23.
The lifetime analysis here is essentially the same as the one realized for the simulation
data in Sec. 3.5. The dependence of the lifetime (in photons) n on the lattice power Plat forCHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

96

five different lattice detunings ∆lat /2π = 0.4, 0.6, 0.8, 1.0 and 1.2 THz is fitted with the Eq.
3.23. Figure 5.14 shows the measured lifetimes as a function of the lattice power Plat for two
different lattice detunings ∆lat /2π = 0.4 THz and ∆lat /2π = 1.2 THz.

Figure 5.15: Lattice detuning ∆lat /2π dependency of the parameters A and B. The solid curves
are fittings using a power equation.
From the fitting results different parameters A and B are obtained for each lattice detuning∆lat . These parameters were plot in Fig. 5.15 and fitted using power equations, resulting in:A=

1480 ± 40(∆lat /2π)

1.17±0.08

W−1,

B = (9400 ± 1100)(∆lat /2π)2.21±0.12 W(5.5)

where ∆lat /2π is in the units of THz. Replacing the equations for the parameters A and B
into the Eq. 3.23 and calculating the derivative by ∆lat , the maximum lifetime and optimal
detuning for a given Plat results in:n(Plat ) ≈ 1470Plat 0.31

at∆lat /2π ≈ 0.70Plat 0.59

(5.6)

where the units of Plat and ∆lat /2π are in W and THz, respectively. For the maximum lattice
power Plat = 2.1 W currently available in this system, the optimal lifetime is of ∼1850 photons
at ∆lat /2π ≈ 1.1 THz. The estimated photon emission rate is 25 photons/µm corresponding
to a maximum lifetime of 74 µS.
Equation 5.6 plot is shown in Fig. 5.16. Upgrading the fiber amplifier system to achieve an
output of Plat = 10 W would increase the lifetime to ∼3000 photons at ∆lat /2π ≈ 2.7 THz (λlat ≈1088 nm). Fortunately, high-power laser sources are readily available for this wavelength.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

97

Figure 5.16: Maximum obtainable lifetime for a given power Plat and the optimal lattice detuning ∆lat /2π.

5.4.4 Fidelity
The fidelity of the quantum gas microscope is determined by three main factors: 1) background noise, 2) losses and 3) hopping.
In the case of the background signal, the fidelity is reduced when the reconstruction algorithm wrongly evaluates an empty site as a filled one. The probability of making such anerror can be estimated by processing 34 empty images with the reconstruction algorithm andanalyzing the shape of the resultant histogram. Fig. 5.17 shows the total histogram for theprocessed images. For a threshold set in 150 photons, only 18 sites of the 9702 total sites resulted in a signal of more than the threshold. This means that there is a < 0.2% probability ofwrongly evaluating a empty site as an occupied one.
The width of the background signal is determined by dark noise, readout noise and lightscattering background. One of the advantages of this microscope system is that the dark noise
is negligibly small due to the very short exposure times. In the case of this system, light scattering of the excitation beam limits the width of the background signal.
The losses can be estimated knowing that the atoms have a lifetime of 1850 photons for alattice power of Plat = 2.1 W. If the atom density decay exponentially, then 1−exp(−150/1850) ≈8% of the atoms are lost before emitting 150 photons. The reconstruction algorithm wrongly
determines these filled sites as empty sites.
The usual method to evaluate hopping effects is to take multiple images from the samelattice and verify whether each atom position have changed or not. In this experimental
setup the lifetime and exposure time are very short and it is technically difficult to utilize
this method. However, hopping effects are greatly reduced because the atoms that escapefrom the site and hop to neighboring sites have a high kinetic energy and cannot be cooled by

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

98

Figure 5.17: Histogram as a probability density function of the background signal. A total of
34 empty images were processed.
the excitation beam. Such atoms are expected to hop indefinitely until they are lost, emitting a reduced number of photons in the neighboring sites. As an alternative way to estimate hopping effects we measured the intensity correlation function g (2) (R ) = 〈I (r )I (r +R )〉/〈I (r )2 〉 of occupied sites in 200 sparse lattice images. This resulted in correlations ofp(1.03(18), 1.03(24), 1.03(19)) for inter-atomic distances R = (1, 2, 2) × 544 nm. These correlation values indicates that hopping events do not produce enough photons to be counted as
an occupied site.

5.5 Light shift in a 5-level diamond system
The AC Stark shift generated by two orthogonal optical accordion (lattice) beams with orthogonal polarizations is studied here (see Fig. 5.6). For the calculations the quantization axis is
set in the y axis. The accordion beam propagating in the x direction have π polarization,
while the accordion beam propagating in the y direction is in the p12 (σ+ + σ− ) polarization.
The ground state 1 S0 and the excited state (6s 6p )1 S0 are labeled as |1〉 and |5〉, respectively.
The excited state 1 P1 have three sublevels labeled as |2〉 = |m J′ = −1〉, |3〉 = |m J′ = 0〉 and |4〉 =CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

99

Figure 5.18: “Deep potential” as a five energy-level system in diamond shape.
|m J′ = +1〉. The accordion beam propagating in the x direction couples the energy levels |3〉
and |5〉. Correspondingly, the optical accordion propagating in the y direction couples |2〉 and|4〉 with |5〉 (see Fig. 5.18).
Consider the case in Fig. 5.6 where there is only one excitation beam propagating in thex direction and with π polarization. As the atom is excited into the |3〉 state, the atom experiences a deep potential due to the coupling of the states |3〉 and |5〉. As the optical accordion
beam in the y direction only couples the states |2〉 and |4〉 with |5〉, the atom will not experience
this potential, and consequently, it will be only trapped along the “deep potential” generated
by the optical accordion in the x direction. If this is the case, the deep potential method would
only apply in one direction when one excitation beam is used.
Two things were noticed and later theoretically understood during the realization of thisexperiment: 1) Only one excitation beam is necessary to produce a two-dimensional “deeppotential” and 2) The “deep potential” is not symmetrical with respect to the x and y directions. There is always a unique “good direction” for excitation that produces brighter andsharper fluorescence images.
Both of this phenomena can be explained by the theory that follows. Consider the Hamiltonian for the unperturbed system H0 and the interaction terms:
H0 = ħ
h ω2 |2〉 〈2| + ħ
h ω3 |3〉 〈3| + ħ
h ω4 |4〉 〈4| + ħ
h ω5 |5〉 〈5|H25 = −h Ωyħe −i ω y t + e i ω y t (|2〉 〈5| + |5〉 〈2|)2h Ω x −i ωx t
ħ
H35 = −e
+ e i ωx t (|3〉 〈5| + |5〉 〈3|)2h Ω y −i ω tħye+ e i ω y t (|4〉 〈5| + |5〉 〈4|)H45 = −
2

(5.7)

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

100

which results in a total Hamiltonian given by:





H = H0 + H25 + H35 + H45 = ħ
h







Ω
− 2y e i ω y t 

ω2

0

0

0

ω3

0

0

0

ω4

− 2y e −i ω y t− 2x e −i ωx t

− 2y e −i ω y t

Ω

Ω

Ω



Ω
− 2x e i ωx t .
Ωy i ω t − 2 e y 

ω5

(5.8)

Here the rotating wave approximation, where the terms representing a photon emission
with excitation (e.g. e −i ωx t |5〉 〈3|) or a photon absorption with de-excitation (e.g. e i ωx t |3〉 〈5|)are neglected, was used. It is convenient also to introduce the effective Hamiltonian H̃ associated with the rotating-frame effective field:
H̃ = U † H U − i ħ
hU †

∂U
∂t

(5.9)

where the unitary matrix U is defined as:i ωy t

e


 0
U =

 0


0

0

0

e i ωx t

00i ωy t

0

e

0

0


0
.

0


1

(5.10)

Equation 5.9 results in:ω2 + ω y0H̃ = ħh0−Ω y /200

ω3 + ω x00

ω4 + ω y−Ω x /2−Ω y /2−Ω y /2−Ω x /2−Ω y /2

ω5(5.11)and knowing that ωe = ω2 = ω3 = ω4 , ω y = ω5 − ωe + ∆ y and ω x = ω5 − ωe + ∆ x , it can be
rewritten to:





0
0
−Ω y /2
 ∆y




∆x
0
−Ω x /2 0+ħ
H̃ = ħh h I ω5 .
 0

0
∆
−Ω
/2
y
y




−Ω y /2 −Ω x /2 −Ω y /2
0(5.12)

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING

Figure 5.19: Light shift variation in a lattice site for different ∆.

101

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING102

The eigenvalues of the effective Hamiltonian are found by solving|H̃ − λI | = 0(5.13)

for different detunings ∆ x , ∆ y and different Rabi frequencies Ω x , Ω y . In the case of a twodimensional optical lattice where each of the lattice beams have the same power Plat and different detuning ∆ x ̸= ∆ y , the parameters are rewritten as:Ω x = Ω cos(k x )1Ω y = p Ω cos(k y )2(5.14)∆ y = ∆ x + δdiffwhere the Rabi frequency is Ω/2π = 77 GHz for Plat = 2.1 W. The resultant eigenstates arecomposed by one “deep” eigenstate which couples the state |5〉 with a linear combination of|2〉, |3〉, |4〉, and two eigenstates that are not coupled with |5〉 and produce flat potentials. The
presence of such an eigenstate explains why is possible to realize the “deep potential” methodwith only one excitation beam. This is produced by the Raman coupling between the two
accordion beams, that couples the state |3〉 with |2〉 and |4〉. Fig. 5.19 shows the dependency
on the “deep” eigenvalue when moving along the x and y directions of the optical lattice. An
asymmetry in the light shift profile results from large frequency differences δdiff /2π ∼ 1 GHz
between the two lattice beams. The asymmetry disappears for frequency differences smaller
than 100 MHz.

(a) “Bad direction” illumination(b) “Good direction” illumination

Figure 5.20: Fluorescence images of a tightly filled two-dimensional optical lattice forδdiff /2π = 440 MHz for two different directions of excitation.

CHAPTER 5. EXPERIMENT: FLUORESCENCE IMAGING103

The sign of δdiff breaks the symmetry of the system. and a “good direction” for excitation
exists. Fig. 5.20 shows the fluorescence images of a two-dimensional lattice for two different
directions of excitation. The total fluorescence increases by a factor of two in the “good direction” case. The “good direction” and “bad direction” become reversed when the frequency
offset sign changed (δdiff → −δdiff ).
Note that a non-zero frequency difference δdiff is necessary in the experiment to avoid
light interferences between the two lattice beams. For the final experiments a frequency difference δdiff /2π = 80 MHz was used.

Chapter 6

Extension to fermionic isotopes
This section focuses in the requirements to extend the quantum gas microscope of ytterbium
atoms to the 173 Yb fermionic isotope. The s-wave scattering length of this fermionic isotope is
a = 10.55(11), which is positive and roughly two times bigger compared to that of the bosonic174

Yb. Fermi degeneracy was experimentally demonstrated in 2007 for a six-spin-component

atomic gas [100] where temperatures of T /TF = 0.37 were obtained via evaporative cooling. A
two-spin-component Fermi degenerate can be generated by pumping the atoms to the |m F | =5/2 states using a π polarized light [101, 102]. The lowest temperatures reported to the date
are T /TF = 0.14[101].
The process to create a two-dimensional condensate under the surface of the solid immersion lens is all-optical and it does not depend on the hyperfine structure of the excitedstate. Consequently, it can be directly be applied to the 173 Yb fermionic isotope with only
minor changes in the optical system:
Zeeman Slower The 1 S0 → 1 P1 transition from F = 5/2 to F ′ = 7/2 is used to slow the atoms.
The isotope shift from 174 Yb is 588 MHz (cf. Table 2.3). To shift the frequency it suffices
to modify the AOMs such that the fundamental wave shifts by 147 MHz (the AOMs arein the double-pass configuration).
Horizontal and Vertical transport No changes required.
Magneto Optical Trap The 1 S0 → 3 P1 transition from F = 5/2 to F ′ = 7/2 is used to slow theatoms. The isotope shift from 174 Yb is 3805.7 MHz (cf. Table 2.4). The wavelength isshifted by using an adjacent transmission peak in the reference cavity (ULE, FSR 3.0
GHz), and changing the frequency of the AOMs by 805 MHz.
Optical Accordion After loading the atoms into the optical accordion, a π polarized lighttuned to the 1 S0 → 3 P1 (F = 5/2 → F ′ = 3/2) is used to pump the atoms to the |m F | = 5/2
states. The evaporation and compression procedure that follows requires no changes.

104

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES

105

6.1 Deep potential
The deep potential method used to trap the bosonic 174 Yb during the fluorescence imaging
process involves a 5-energy level system that was studied in detail in Appendix 5.5 and experimentally in Chapter 5. In the case of the fermionic isotope 173 Yb, the presence of nuclear
spin I = 5/2 results in hyperfine splitting of the 1 P1 state, as shown in Fig. 6.1.
F’’=5/2

72 MHzF’=7/2F’=3/2769 MHzF’=5/2

F=5/2

Figure 6.1: Hyperfine splitting of 173 Yb.
In the presence of a nuclear spin, the Hamiltonian can be written as H = Hhfs + Hstark .
Here, Hhfs is the hyperfine splitting interaction given by:
Hhfs = A 173 I · J + B173

6(I · J)2 + 3I · J − 2I2 J2.
2I (2I − 1)2 J (2J − 1)

(6.1)

where A 173 = 57.682(29) MHz and B173 = 609.065(98) MHz are the hyperfine splitting constantsof the 1 P1 state in 173 Yb [75]. For this section ħ
h will be abbreviated for the sake of convenience.
When there are no external fields present, the Hamiltonian of the system is simply H = Hhfs
which is diagonal in the basis of |F, m F 〉 where F is the total angular momentum related to theoperator F = J + I. The diagonal matrix elements are:
3
G (G + 1) − 2I (I + 1)2 J (J + 1)
1
ħ A 173G + ħ
h B173 2〈F, m F | Hhfs |F, m F 〉 = h
2
2I (2I − 1)2 J (2 J − 1)

(6.2)

where G = F (F + 1) − I (I + 1) − J ( J + 1). F takes the values F = | J − I | . . . | J + I | = 3/2, 5/2, 7/2
for J = 1 and I = 5/2 resulting in:〈3/2, m F | Hhfs |3/2, m F 〉 /2π = 224 MHz〈5/2, m F | Hhfs |5/2, m F 〉 /2π = −545 MHz〈7/2, m F | Hhfs |7/2, m F 〉 /2π = 296 MHz(6.3)

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES

106

which corresponds to the hyperfine splitting shown in Fig. 6.1. Hstark is the stark shift interaction described in Appendix A.1. When the hyperfine splitting interaction is not present, theHamiltonian H = Hhfs can be approximated as being diagonal in the basis of |m J , m I 〉 withresulting energy shifts that are only dependent on m J . In this case, the resultant diagonal
matrix elements are exactly the same as the obtained for the bosonic 174 Yb (cf. Appendix 5.5).
When both interactions are present, the Hamiltonian H = Hhfs + Hstark is in general not
diagonal in neither the |F, m F 〉 basis nor the |m J , m I 〉 basis. The eigenvalues and eigenstates
can be only calculated by numerical diagonalization of the total Hamiltonian. Theoretical
calculations and experimental verifications of the Stark shift for strong fields were realized
for sodium and lithium in the case of DC fields [103, 104], and for rubidium in the case of
far-detuned ODTs [105, 106]. The calculations in this thesis follows a similar procedure, withthe difference that there are two strong fields corresponding to each of the accordion (lattice)beams. The two accordion beams have orthogonal direction and polarization, resulting in astrong Raman coupling between different m J sublevels.

6.1.1 AC Stark shift
The AC stark shift between a 1 P1 sublevel |m J , m I 〉 and a (6s7s)1 S0 sublevel |m J′ , m I′ 〉 is related
to the term (cf. Sec. A.1):〈m J , m I | e r̂ |m J′ , m I′ 〉 =

 

∑
∑
|F ′ , m F′ 〉 〈F ′ , m F′ | |m J′ , m I′ 〉 =〈m J , m I | |F, m F 〉 〈F, m F | e r̂ 
∑ ∑F,m F F ′ ,m F′

F,m F

F ′ ,m F′(6.4)〈m J , m I |F, m F 〉 〈F, m F | e r̂ |F ′ , m F′ 〉 〈F ′ , m F′ |m J′ , m I′ 〉where 〈m J , m I |F, m F 〉 is the Clebsch-Gordan coefficient defined by:〈m J , m I |F, m F 〉 = (−1) J −I +m Fp
 J
2F + 1 
mJ


I
mI

F 

−m F

(6.5)


where the notation mj11 mj22 mj represents the Wigner 3j-symbol. Using the Wigner-Eckart theorem, the matrix element 〈F, m F | e r̂ |F ′ , m F′ 〉 can be rewritten as the product of the reduced
matrix element 〈F ||e r ||F ′ 〉 and the Clebsch-Gordan coefficient 〈F ′ , 1, m F′ , q |F, m F 〉
′〈F, m F | e r̂ |F ′ , m F′ 〉 = 〈F ||e r ||F ′ 〉 (−1)F −1+m F′
p
F
2F + 1 
m F′


1
q

F 
.
−m F

(6.6)

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES

107

Here q = −1, 0, 1 represents the σ− , π and σ+ polarizations of the light field, respectively.
The term vanishes when m F′ ̸= m F + q (total m conservation). Using the Wigner 6j-symbol
¦
©
j j j
represented by j14 j25 j36 , the reduced matrix term is further factorized into′

〈F ||e r ||F ′ 〉 = 〈J ||e r ||J ′ 〉 (−1)F + J +I +1

p(2F ′ + 1)(2J + 1)



F ′

1



F


J

I


J ′

.

(6.7)

Finally, the reduced term 〈J ||e r ||J ′ 〉 is related to the transition probability by the formula:ħ 2 Γ 2 2J ′ + 1h| 〈 J ||e r || J 〉 | = 2c ε0
8Is 2J + 1
′

2

(6.8)

where Γ and Isat is the linewidth and saturation intensity of the 1 P1 ↔ (6s7s)1 S0 transition,
respectively. Figure 6.2 shows the relative transition probabilities between different hyperfine
sublevels of the 1 P1 and (6s7s)1 S0 states.
1 63

1 21

⁄

2 21

10 63

⁄

13

⁄

10 63

⁄

4 21

⁄

5 21

⁄

2 21

⁄

4 21

⁄

10 63

⁄

F’’=5/2

2 21

⁄

1 21

⁄

13

⁄

10 63

⁄

2 21

⁄

5 21

⁄

⁄

1 63

⁄

⁄

F’=7/2′′′(a) Transition probabilities between F = 7/2 and F = 5/2.

2 21

16 105

⁄

5 21

3 35

⁄

2 21

⁄

16 105

⁄

1 105

6 35

⁄

⁄

3 35

⁄

5 21

⁄

16 105⁄F’’=5/2

2 21

⁄

⁄

1 105

⁄

16 105

6 35

⁄

⁄

2 21

⁄

⁄

′

F’=5/2′′(b) Transition probabilities between F = 7/2 and F = 5/2.

29

2 15

⁄

1 15

⁄

4 45

2 15

⁄

⁄

⁄

2 15

⁄

1 45

F’’=5/2

1 45

⁄

4 45

⁄

1 15

⁄

⁄

2 15

29

⁄

⁄

F’=3/2′(c) Transition probabilities between F = 7/2 and F ′′ = 5/2.

Figure 6.2: Transition probabilities between the different hyperfine sublevels of the 1 P1 states(F ′ = 3/2, 5/2, 7/2) and the (6s7s)1 S0 state (F ′′ = 5/2).

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES108

6.1.2 Hyperfine splitting interaction
Here, the matrix elements 〈m J , m I |Hhfs |m J′ , m I′ 〉 corresponding to the hyperfine splitting interaction are calculated. Equation 6.1 is rewritten as:′
Hhfs = A 173 I · J + B173



2 2 2
2
I · J + 2(I · J) − I J .
3

(6.9)

′
173
where B173
≡ 2I (2I −1)2J(2J −1) . To calculate the action of the operator I · J on the eigenstates, it is
3B

convenient to introduce the ladder operators J± , I±
J± = J x ± i J y,

I± = I x ± i I ywhich satisfy
I · J = J x I x + J y I y + Jz I z = Jz I z +J− I+ + J+ I−.
2(6.10)(6.11)In a similar way, (I · J)2 results in:
J− I + + J+ I − 2(I · J)2 = Jz I z +2
1
1
1= Jz2 I z2 + Jz I z (J− I+ + J+ I− ) + (J− I+ + J+ I− ) Jz I z + (J− I+ + J+ I− )2
2
2
4
1
1
= Jz2 I z2 + Jz I z (J− I+ + J+ I− ) + (J− I+ + J+ I− ) Jz I z
2
2
1 2 2
2 2
+ (J− I+ + J+ I− + J− J+ I+ I− + J+ J− I− I+ ).
4(6.12)

The action of the different operators on the eigenstates |m J , m I 〉 are:I z |m J , m I 〉 = m I |m J , m I 〉Jz |m J , m I 〉 = m J |m J , m I 〉I2 |m J , m I 〉 = I˜ |m J , m I 〉J2 |m J , m I 〉 = J˜ |m J , m I 〉Æ
I± |m J , m I 〉 = I˜ ± m I (m I + 1) |m J , m I ± 1〉q
J± |m J , m I 〉 = J˜ ± m J (m J + 1) |m J ± 1, m I 〉(6.13)

where I˜ = I (I + 1) and J˜ = J (J + 1). The diagonal matrix elements results in:′〈m J , m I | Hhfs |m J , m I 〉 = A 173 m J m I + B173



2
2 2
2
2 ˜
˜
˜
˜
[ J − m J ][I − m I ] + 2m J m I − I J
3(6.14)

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES109

The non-diagonal matrix elements are:〈m J ± 1, m I ∓ 1| Hhfs |m J , m I 〉 =

Ç
A 173
′
(2m J m I ± m J ∓ m I − 1/2)
− B173( J˜ − m J2 ± m J )(I˜ − m I2 ∓ m I )2(6.15)〈m J ± 2, m I ∓ 2| Hhfs |m J , m I 〉 =′ q
B173
J˜ − m J (m J ∓ 1)][ J˜ − (m J ∓ 1)(m J ∓ 2)][I˜ − m I (m I ± 1)][I˜ − (m I ± 1)(m I ± 2)]2(6.16)

6.1.3 Numerical results
Using the matrix elements obtained in Sec. 6.1.1 and Sec. 6.1.1 and applying the effective
Hamiltonian procedure shown in Sec. 5.5 it is possible to calculate the eigenvalues numerically. The resultant eigenvalues for different light intensities are shown in Fig. 6.3 in the caseof ∆lat /2π = 1 THz.

Figure 6.3: Eigenvalues of the total hamiltonian H = Hhfs + Hstark as a function of the lightintensity. The dotted lines in red are eigenvalues obtained for the bosonic 174 Yb.

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES110

For low light intensities (Ilat < 100 kW/cm2 , ∆stark /2ħ
h < 200 MHz), F is a good quantum
number and the resultant eigenvalues split in three groups corresponding to F = 3/2, 5/2 and7/2 (inset in Fig. 6.3). The AC stark effect can be treated as a second-order perturbation,
resulting in light shifts that only depend on the m F values and change linearly with the lightintensity. For high intensities (∆stark /2ħ
h = 1 GHz), the eigenvalues split according to m J andresults in a light shift similar to the bosonic case (red dotted lines in Fig. 6.3). Due to thehyperfine splitting interaction, each m J group split in different sublevels which depends on
m F . This effect is analog to the Paschen-Back effect produced by strong magnetic fields. The
results in this calculations are consistent with the results obtained with sodium, lithium andrubidium [103, 104, 105, 106].
Figure 6.4 shows the eigenvalues for two different lattice detuning ∆lat /2π = 0.1, 10 THz.
Note that the horizontal axis is scaled to the same ratio as ∆lat /2π for the purpose of comparison. The resultant shapes are exactly the same as the obtained in Fig. 6.3. This means thatthe shape is only dependent on the interaction energies Hhfs and Hstark ∝ Ilat /∆lat .

Figure 6.4: Eigenvalues of the total hamiltonian H = Hhfs + Hstark as a function of the lightintensity for ∆lat /2π = 0.1 and 10 THz.

6.2 Feasibility of the extension
The quantum gas microscope schema presented in this thesis can be easily extended to the173

Yb fermionic isotope. The process to create a pancake-shaped condensate of atoms un-

der the surface of the solid immersion lens requires only a few changes, which includes themodification of the cooling beams frequency and the insertion of an additional beam used tospin-polarize the atomic cloud before evaporative cooling. The deep potential method wasanalyzed in the influence of the hyperfine splitting interaction. For very large light shifts as

CHAPTER 6. EXTENSION TO FERMIONIC ISOTOPES111

the used in the deep potential experiment (∆stark /2π ≈ 3 GHz) the hyperfine splitting interaction (∆hfs /2π ≈ 300 MHz) is one order of magnitude smaller, resulting in the breaking of
hyperfine splitting analog to the Paschen-Back effect in strong magnetic fields. As the resultant light shift has the same gradient that in the case of the bosonic 174 Yb, which was tested
experimentally in this work, it is possible to affirm that the procedure will also work for the173

Yb fermionic isotope. Additionally, the hyperfine splitting interaction split each of the m J

eigenvalues into multiple m F sublevels. This splitting might be beneficial to the deep potential method, as the resultant linewidth-broadening of the 1 S0 ↔ 1 P1 transition would intheory reduce the effects of light shift inhomogeneities during fluorescence imaging.

Chapter 7

Conclusions
Quantum simulators using fermionic atoms are useful for studying the Fermi-Hubbard model,
which is expected to be the key to elucidate the mechanism of high temperature superconductors. One of the biggest challenges to investigate the Fermi-Hubbard model is to reduce
the temperature of the system, which is a requirement for the observation of anti-ferromagnetic
order and d-wave super-fluidity.
The work presented in this thesis demonstrates the realization of a quantum gas microscope for ytterbium atoms, a tool capable of measuring and reducing the temperature of a
Fermi-Hubbard system. Two big challenges were present in the development of a QGM: 1)
How to prepare a thin sample of condensate atoms under the surface the solid immersion
lens and 2) How to obtain a sufficient large number of atoms to realize high fidelity fluorescence imaging of atoms trapped in a two-dimensional optical lattice.

7.1 Experiment summary
In the first part of this thesis I presented a novel all-optical approach to load ultra-cold Yb
atoms into a single layer of a standing wave directly below the surface of solid immersion lens(SIL). After loading the atoms into a magneto-optical-trap, two optical dipole trap beams wereused to transport the atoms directly under the surface of the SIL. The new method realized
here to create an “optical accordion” consists in reflecting a laser beam into the surface of
the SIL. The resultant accordion potential was used to create a Bose-Einstein condensate andcompress the atoms into a thin layer, positioned 2.6 µm below the surface of the lens. The alloptical scheme presented here can be applied to all atomic species regardless of the magnetic
moment or energy structure. In fact, a similar schema was recently used to realize a quantumgas microscope of Lithium atoms [52].
The second part of this thesis focuses on how to load the disk-shaped condensate of atoms
into a two-dimensional optical lattice and how to realize high fidelity fluorescence imag-

112

CHAPTER 7. CONCLUSIONS

113

ing of the trapped atoms. To load the atoms into the optical lattice, the accordion systemused to compress the atoms was transformed into a two-dimensional optical lattice by retroreflecting each of the accordion beams. For single-site addressing of atoms in a two-dimensional
optical lattice the conventional method until now was to laser cool the atoms while observing the resultant fluorescence. In contrast with this approach, the innovative idea presented
here is to use a deep potential without cooling to contain the heated atoms during imaging.
To create such a deep potential, the wavelength of the accordion beams the wavelength of
the accordion beams was selected to be λ = 1081 nm, which is detuned by only 4 nm to the1P1 − 1 S0 transition (wavelength 1077 nm). The lifetime limitations of this system were studied

in detail showing a good agreement with a simple model consisting in two losses. Finally, theperformance of the quantum gas microscope was analyzed, resulting in a fidelity of 92% on
the reconstruction algorithm mainly limited by atom losses due to the short lifetime. By using
the ultraviolet strong transition combined with a solid immersion lens and high-resolution
optics, the system presented here was able to resolve individual sites in an optical lattice witha 544-nm spacing, with a FHWM resolution of 318 nm.

7.2 Current quantum gas microscopes
Following the realization of two different quantum gas microscope for rubidium atoms in2010 and 2011, the work presented here is the first realization of a quantum gas microscopefor a species different than rubidium. At the time of this thesis and to the extent of my knowledge, a total of nine different quantum gas microscopes exist. The specifications for each of
the quantum gas microscope including resolution, lattice spacing, exposure time, loss andhopping rates, and utilized method is summarized in Table. 7.1.
Laboratory

Resol.

Spacing

Time

L

H

MethodRef.

Rb

Harvard

700 nm532 nm

∼1 s

2%

0%

PGC

[46]

Rb

MPQ

600 nm

640 nm

0.9 s

1%

0.5%

PGC

[47]

Yb

Titech318 nm544 nm

40 µs

8%

0%

Deep potential

K

Strathclyde

630 nm532 nm

1.5 s

9%

8%

EIT cooling

[56]

K

MIT

640 nm532 nm

1s

4.8%

1.2%

Raman cooling

[55]

K

Toronto600 nm

527 nm

∼5 s

2%

5%

EIT cooling

[57]

Li

Harvard520 nm569 nm

1.9 s

2.6%

2.3%

Raman cooling

[52]

Li

MPQ

900 nm1150 nm

1s

2.5%

5%

Raman cooling

[53]

Yb

Kyoto

364 nm

266 nm

0.4 s

6.5%

6.7%

Sideband cooling[58]

Table 7.1: Current quantum gas microscopes. The highlighted row corresponds to this work.
The quantum gas microscope presented in this work is characterized by:

CHAPTER 7. CONCLUSIONS

114

High resolution The resolution of 318 nm obtained in this work is the highest resolution
available for quantum gas microscopes up to the date. A high resolution is beneficial
to reduce the number of required photons, as the fluorescence from each atom is distributed in less pixels.
Fast imaging time The fast imaging times 40 µs are five order of magnitude smaller compared with other experiments. The advantage of having such short imaging times isthat the system is robust against mechanical instabilities in the optical system.
No hopping effects Due to the lack of a cooling method and the presence of a radiative force,atoms are unlikely to hop to a neighbor site and stay pinned. This is also an advantage
as hopping effects are difficult to compensate. In contrast, loss effects can be compensated by data averaging.
The “deep potential” schema presented here does not depend on the energy level structure and is completely all optical, and thus, can be adapted in theory to any atomic species.
In practice, however, this method is well suited for heavy species having small recoil energies(ħh k )2 /2m, such as europium, erbium, dysprosium, holmium, ytterbium and mercury.

7.3 Possible improvements
The following technical modifications would improve the fidelity of the quantum gas microscope for ytterbium atoms:
Accordion power Increasing the power of the optical lattice lattice power from Plat = 2.1 W
to Plat = 10 W would increase the lifetime to ∼3000 photons. The loss ratio in this casedecreases from 8% to 5%.
Objective Lens and CCD camera The transmission of the objective lens can be improved fromthe actual 54% to ∼80% by using custom-made objective lenses specially designed for
NUV. By using a different CCD camera, the quantum-efficiency can be increased fromto 0.5e − /photon to ∼0.7e − /photon. Both of these upgrades would result in a total collection efficiency of 10%, which is two times larger than the current value. Doubling
the collection efficiency allows one to half the exposure time to 20µs and threshold to75 photons, further decreasing the loss ratio from 5% to ∼2.5%.
Retro-reflected excitation beam A retro-reflected excitation beam would in theory increase
the lifetime by ∼10% in the current setup (Plat = 2.1 W). Note that if the lattice power isincreased to Plat = 10 W, the effect of this modification becomes negligible.

CHAPTER 7. CONCLUSIONS

115

7.4 Tasks
A remaining task is the evaluation of photo-association (PA) effects. The characteristic timefor PA inelastic collisions resulting in the loss of a pair of atoms is in the order of a few to tens of
microseconds [107, 108]. In quantum gas microscope experiments using cooling techniquessuch as PGC, Raman Cooling and EIT Cooling, the exposure time is in the order of seconds(cf. Table 7.1), which is five orders of magnitude larger than the PA characteristic time. Photoassociation losses in those experiments can be considered instantaneous, and the measured
fluorescence reflects the parity projection of the original atom-number distribution. For thequantum gas microscope presented in this work the exposure time is 40 µs, which is comparable to the characteristic PA time. The fluorescence is then expected to increase in case ofmultiple occupied sites, as pair of atoms emit a number of photons before being lost. In order
to investigate the effects of PA it is necessary to study a system where the number of atoms
is well determined, such as a Mott insulator. Current experiments focus in the observationof the Superfluid-to-Mott-insulator transition at single atom level, which would be useful toelucidate the effects of PA.
Note that PA effects will not affect the fluorescence in the case of the 173 Yb fermionic isotope. Multiple occupancy of fermions in a single-site is limited to atoms with different spin,
as the occupation of atoms with the same spin is forbidden by the Pauli exclusion principle.
Employing spin-selective excitations mediated by the 1 S0 ↔ 3 P2 transition, it is possible tocircumvent PA effects by measuring each spin separately.

7.5 Future experiments
Future experiments will focus on the study of the Fermi-Hubbard model using the 173 Yb fermionicisotope and the quantum gas microscope. The current quantum gas microscope schema canbe applied to any ytterbium isotope without major changes in the system (cf. Section 6).
One of the biggest challenges to determine the phase diagram of the Fermi-Hubbard model
will be the reduction of the temperature of the system, which is a requirement for the observation of the anti-ferromagnetic phase and the d-wave super-fluidity. The ultra-narrow transition in ytterbium combined with the high-resolution of the quantum gas microscope will
be a key feature to reduce the temperature of the system using theoretically proposed “spatial
filtering” techniques [59, 60].

Appendix A

Laser cooling and trapping
A.1 Optical dipole trap
When an electric field produced by a laser beam interacts with an atom, the energy levels of
the atom shift. The interaction Hamiltonian Hint for the dipole interaction is defined byHint = −µ · E(A.1)

where µ and E represents the dipole momentum and electric field, respectively. Using the
perturbation theory, the energy shift δEi induced in the energy level Ei is given by:
δEa =∑ |〈a |H |b 〉|2
int
E
−
Eb
a
a ̸=b

(A.2)

where |i 〉 is the eigenstate of the unperturbed Hamiltonian. Using the second quantization
representation, the electric field for a laser beam with frequency ω, can be written as
E(r) =

v
t ħ
hω
2V ε0



ê â e i k·r + â † e −i k·r(A.3)

where V is the volume of the system, and e is the polarization of the light. Substituting thisequation into Eq. A.1, the dipole interaction can be represented as
v
t ħ


hω
Hint = −ê · µ â e i k·r + â † e −i k·r .
2V ε0

(A.4)

To solve 〈a |Hint |b 〉, let’s consider the dressed states 〈a , n| and 〈b , n ′ |. Knowing that theppannihilation and creation operator â satisfies â |n 〉 = n|n − 1〉 and â † |n〉 = n + 1|n + 1〉:
2〈a , n |Hint |b , n ′ 〉 =


ħω
h
2
ê · 〈a |µ|b 〉 (n + 1)δ(n, n ′ − 1) + n δ(n , n ′ + 1)
2V ε0

116

(A.5)

APPENDIX A. LASER COOLING AND TRAPPING

117

furthermore, for a very large number of photons n ≈ n + 1 and thus,
′
′
∑ nħ
hω
2 δ(n, n − 1) + δ(n, n + 1)ê · 〈a |µ|b 〉2V ε0
Ea ,n − E b ,n ′b ̸=a ,n ′


∑ nħ
hω
1
1
2
ê · 〈a |µ|b 〉
=−
+
2V ε0
h ωb a + ħħhω ħh ωb a − ħ
hω
b ̸=a


∑ nħ
h ωc
1
1
1
2
ê · 〈a |µ|b 〉
=−
+
V 2ħ
h ε0 cωb a + ω ωb a − ωb ̸=aδEa ,n =(A.6)

for ωb a = ωb − ωa . The spontaneous decay rate Γa b , the laser beam intensity I (r), and thesaturation intensity Isat are defined by:Γa b =

ω3b a
3πε0 ħ
hc 3〈a |µ|b 〉2,

I (r) =nħh ωcVIsat =,h Γa b ω3b a
ħ
12πc 2(A.7)

Finally, considering a linear polarization, Eq. A.6 can be rewritten as∑ħh Γa2b I (r)11
δEa (r) = −+8 Isat ωb a + ω ωb a − ω
b ̸=a

(A.8)

and for near resonant frequencies ω ≈ ωa b in a two-level atom, the expression can be approximated to:δE (r) = −ħ Γ 2 I (r) 1h8 Isat ∆(A.9)

In the case of a far-detuned laser in a two-level system, the scattering rate is related to thelight shift by the equationh Γsc =ħΓδE (r).
∆(A.10)

Note that for a constant light shift δE , the scattering rate can be reduced by using large detunings ∆.

A.1.1 Optical dipole trap generated by a Gaussian beam
A red-tuned focused Gaussian laser beam is the simplest way to create a dipole trap providing
three-dimensional confinement. This type of confinement is often referred as optical tweezers. The spatial intensity distribution of a focused Gaussian beam with power P propagating
along the z axis is given by


r2
exp
−2
I (r, z ) =w 2 (z )πw (z )22P


(A.11)

APPENDIX A. LASER COOLING AND TRAPPING

118

where r denotes the radial coordinate, and w (z ) depends on the axial coordinate z viaw (z ) = w0

v
u
t


1+

z
z0

2
(A.12)

where w0 is the beam waist, and z 0 = k w02 /2 is the Rayleigh length. Substituting this equationin Eq. A.8, the dipole potential Udip generated by a Gaussian beam results in:


P
r2
ωb a
Udip (r) = −exp −222
2
4Isat πw (z )w (z ) ωb a − ω2
b ̸=a
∑ħ
hΓ 2

ab

(A.13)For two-energy level system a red-detuned laser beam (ω > ω0 ) will produce a attractive
potential in the ground state and a repulsive potential in the excited state, as shown in Fig.
A.1.

|ei

|ei

|gi

|gi

Red-shifted gaussian
optical dipole trap

Blue-shifted gaussian
optical dipole trap

Figure A.1: Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem.

A.1.2 Standing Wave
When two counter-propagating with the same frequency interfere as in Fig. A.2, a standing
wave with periodicity of half the wavelength is formed. By combining the standing wave generated by a red-detuned laser beam in one or more directions an optical lattice is created.

Figure A.2: Standing wave resulting from the interference of two counter-propagating waves

APPENDIX A. LASER COOLING AND TRAPPING

119

The total electric field generated by two counter-propagating beams is given by:
E = E0 cos(k x − ωt ) + E0 cos(−k x − ωt )(A.14)

in the case that the two beams have the same amplitude E0 . The light intensity is proportional
to the square of the electric fieldI ∝ |E |2 = |E0 |2 [cos(k x − ωt ) + cos(−k x − ωt )]2
2|E0 |2  i (k x −ωt )e+ e −i (k x −ωt ) + e i (−k x −ωt ) + e −i (−k x −ωt )42|E0 |2  i ωte
+ e −i ωt e i k x + e −i k x=
4== 4|E0 |2 cos2 (ωt ) cos2 (k x )(A.15)

resulting in a standing wave in the x direction, having a peak intensity four times bigger than
the intensity of a single propagating-wave. In the case of two counter-propagating beams
with two different amplitudes
E = E0 cos(k x − ωt ) + E1 cos(−k x − ωt )(A.16)

it is still possible to calculate the intensity with a more complicated calculation, resulting in:|E |2 = |E1 |2 (1 − α + α2 ) + 4α cos2 (k x ) cos2 (ωt )(A.17)

where α = E2 /E1 is the ratio between the two amplitudes. When α ̸= 1, the total intensity isformed by the sum of a standing wave and a propagating wave (constant intensity, also called
offset intensity). Note that Eq. A.17 reverts to Eq. A.15 for α = 1.
By combining a three different standing waves in the three directions, an optical latticeUlattice = V0 cos2 (k x ) + cos2 (k y ) + cos2 (k z )(A.18)

is formed.
The periodicity of the standing wave can also be changed by crossing two different beams
as shown in Fig. A.3. For two waves propagating in the directions kˆ1 = (cos θ , sin θ ) and kˆ1 =(cos θ , − sin θ ), the resultant standing wave intensity is given by:2
I ∝ |E |2 = |E0 |2 cos(k x cos θ + k y sin θ − ωt ) + cos(k x cos θ − k y sin θ − ωt )= 4|E0 |2 cos2 (k y sin θ ) cos2 (k x cos θ − ωt )which has a periodicity of λ/2 sin θ .

(A.19)

APPENDIX A. LASER COOLING AND TRAPPING

120

Figure A.3: Standing wave from the interference of two crossing propagating-waves.

A.2 Potential under the solid immersion lens
SIL

Figure A.4: Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem.
The electric field of a Gaussian propagating in the x ′ axis




v
v
t w0z t w0y
z ′2
y ′2
Egauss (x , y , z ) = E0
exp
exp
cos(k x ′ − ωt ) (A.20)
w z (x ′ ) w y (x ′ )w z (x ′ )2w y (x ′ )2′′′

where the complex term related to the radius of curvature and the Gouy phase was neglected.
The Gaussian beam in terms of the coordinates (x , y , z ) can be obtained by a rotation of coordinates given by
x ′ = x cos θ + z sin θ



y′= y



 z ′ = −x sin θ + z cos θ(A.21)

resulting inEθ (x , y , z , θ ) ≡ Egauss (x cos θ + z sin θ , y , −x sin θ + z cos θ )(A.22)

APPENDIX A. LASER COOLING AND TRAPPING

121

where θ is the angle of incidence. The sum of the incident beam and reflected beam can be
written as:
Eacc (x , y , z , θ ) = Eθ (x , y , z , θ ) + r (θ )Eθ (x , y , z , −θ )(A.23)

where r (θ ) is the reflectivity (including phase) of the SIL surface for a s-polarized lightr (θ ) =sin θ − nsin θ + nqq

θ
1 − cosn2
2(A.24)

θ1 − cosn22according to the Fresnel equations. For two optical accordion orthogonal to each other, thetotal relative intensity is calculated by the sum of each accordion separately:
Iacc (x , y , z , θ ) =

1
E0

2

〈|Eacc (x , y , z , θ )|2 + |Eacc (y , x , z , θ )|2 〉(A.25)

where 〈· · · 〉 denotes the time average.
Finally, the potential depth can be calculated multiplying the relative intensity by the Eq.
A.8
Uacc (x , y , z ) = −Iacc (x , y , z , θacc )

∑ħ
hΓ 2

Plat
2Isat πw0z w0y
b ̸=a
ab



1
1
+
ωb a + ωlat ωb a − ωlat
(A.26)

Fig. A.5 shows the potential in the ground state for two different angles of incidence andinput powers.
The trap parameters such as potential depth in the z direction and trap frequencies weresummarized in Table A.1. For deep angles of incidences both trap frequencies are roughlyp
proportional to Plat and the vertical trap frequency ωz is proportional to sin θacc .
Plat (W)θacc (◦ )z 0 (µm)

Uz (µK)ωz /2π (kHz)ωxy /2π (kHz)ωz /ωxy

0.095

0.7

17.5

1.04

0.27

0.078

3.5

1.1

0.7

15.0

47.4

1.1

0.28

4.0

2.1

0.7

14.9

95.0

1.5

0.38

4.0

0.095

1

13.5

3.4

0.45

0.096

4.7

1.1

1

12.4

71

1.6

0.33

4.9

2.1

1

12.4

139

2.2

0.45

4.9

0.095

6

2.60

8.5

2.9

0.17

25

1.1

6

2.57

105

9.7

0.40

25

2.1

6

2.57

200

13

0.55

25

Table A.1: Optical accordion: Trap parameters for different angles of incidence and input
powers.
The potential for a retro-reflected optical accordion that forms an optical lattice can be

APPENDIX A. LASER COOLING AND TRAPPING

122

Figure A.5: Potential generated by an optical accordion for two different incident angles andinput power.
calculated using Eq. A.23 by the formulapElat (x , y , z , θ ) = Eacc (x , y , z , θ ) + r (θ ) T Eacc (x , y , z , π + θ )(A.27)

where T is the transmission of the optical elements used for retro-reflection (see Fig. 5.2).
The relative intensity is defined in a similar way as in the accordion potential case:Ilat (x , y , z , θ ) =1
E0 2〈|Elat (x , y , z , θ )|2 + |Elat (y , x , z , θ )|2 〉.

(A.28)The resultant potential profile in the z direction and x direction is shown in Fig. A.6 for anpangle of incidence of θacc = 6◦ . The potential depth is roughly 4 0.5 = 2.8 times bigger thanthe potential of the optical accordion as only 50% of the total power can be retro-reflected
due to optical losses (see Fig. 5.2).
The potential depth in the z direction Vz is roughly two times bigger than the potential
barrier in the x y direction (lattice depth). For simple calculations purposes and for the sim-

APPENDIX A. LASER COOLING AND TRAPPING

123

Figure A.6: Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem.
ulations in Chapter 3 the potential can be approximated to:
V (x , y , z ) = −V0 sin2 (k z z ) cos2 (k x x ) + cos2 (k y y )

(A.29)by neglecting the effects of the harmonic envelope in the x y directions and the Gaussian
profile in the z direction. The important parameters for the optical lattice trap were summarized in Table A.2. The trap frequency of the harmonic envelope in the x , y directions ωhar
was included as it represents an important parameter to calculate the atomic distribution of
a Mott-insulator.
Plat (W)z 0 (µm)

Vz (µK)

V0 (µK)

V0 /E rωz
2π (kHz)ωx y
2π (kHz)ωh a r2π (Hz)

0.007

2.71

1.4

0.95

20

1.3

8.6

52

1.1

2.57

305

150

3185

17

109

655

2.1

2.57

584

286

6080

23

150

905

Table A.2: Optical lattice: Potential and trap parameters for θacc = 6◦ and different input powers.

A.3 Evaporative Cooling
The local number density of atoms at the position r in a trap is n(r). As long as trap volume andtemperature do not change and no new atoms are loaded into the trap, this density evolves
in time according toAPPENDIX A. LASER COOLING AND TRAPPING124

d n (r, t )= −αn (r, t ) − β n(r, t )2 − γn(r, t )3 − O (n 4 )dt(A.30)

where α, β , γ are rate coefficients for different kinds of loss mechanism, e.g. backgroundgas collisions (α), two-body inelastic collisions as radiative escape or photoassociation (β ),or three-body collisions (γ). Integrating over all the volume gives an expression related to thetotal number of atoms N
d N (t )= −αN (t ) − β
dt

∫

∫
2

n(r, t )3 d rn(r) d r − γ
V(A.31)V

The single-particle loss coefficient α takes into account collisions with the backgroundgas in the vacuum apparatus. When the density is small enough that the two-body collisions
and three-body collisions can be disregarded, eq. A.31 can be easily solved:
N (t ) = N0 e −αt(A.32)

where n0 is the initial density of atoms. α is then equal to the inverse of the 1/e lifetime τ =1/α.
The two-body loss coefficient β describes trap loss due to ultra-cold binary inelastic collisions. Inelastic collisions occurs when two atoms collide and the kinetic energy is not conserved. In general, such trap loss becomes important if the colliding atoms are not in their absolute ground state. Consequently, this type of loss can occur in a MOT due to light-assisted
binary collisions involving atoms in the excited state, but it is negligibly small in an optical
dipole trap.
Three-body losses, as described by the coefficient γ in Eq. A.31, become relevant only at
extremely high densities, far exceeding the conditions of a MOT. In a collision of three atoms,
a bound dimer can be formed and the third atom takes up the released energy, so that all
three atoms are lost from the trap. As a far-detuned dipole trap allows one to completely
suppress binary collision losses by putting the atoms into the absolute internal ground state,
it represents an interesting tool for measurements on three-body collisions. The three-bodycollision rate for 174 Yb was experimentally calculated as γ = (4.2 ± 1.5) × 10−29 cm6 s−1 [109].
Evaporative cooling in a optical dipole trap is directly related to two-body elastic collisions. When two atoms collide in an elastic way it results in two atoms having different kinetic energy. When the resultant high energy atom has enough energy to escape the trap,
evaporation occurs and the overall temperature in the trap is decreased after thermalization.
Evaporation efficiency can be technically controlled by limiting the depth of the potential
to ηk B T . In principle, there is no upper bound for the efficiency of evaporative cooling. Thiscan be demonstrated by the following extreme example: With an extremely large η, one just

APPENDIX A. LASER COOLING AND TRAPPING

125

has to wait for the event that one particle has all the energy of the system. Evaporating asingle particle then cools the whole system to zero temperature. Unfortunately, this coolingstrategy would take an almost infinite amount of time. It is thus important to considerate
both the efficiency and cooling speed of the cooling process.

A.3.1 Efficiency of evaporation
Evaporative cooling happens on a exponential scale, which means that the characteristic
quantities for the evaporation process are therefore logarithmic derivatives. The efficiencyof the evaporation process is defined as:ζ=

d (log T ) T˙ /T=d log N ) Ṅ /N(A.33)

where T is the average temperature on the system, and N is the total number of atoms. Thisdimensionless quantity represents how much temperature the system decreases comparedto the decrease in the number of trapped atoms.

A.3.2 Speed of evaporation
We consider particles at density n0 in a box potential, and we assume that η is large enough.
For an untruncated Maxwell-Boltzmann distribution, almost all the atoms with energy higher
than ηk B T that involve a collision will be removed from the trap. Thus, to calculate the rate it
suffices to divide the number of atoms Nh with energy larger than ηk B T by the collision time
τel .
Nh
dN
=−
dt
τel

(A.34)For η large enough, the number of atoms with energy larger than ηk B T can be approximated as:
Nh = 2N e −ηsηπ

(A.35)

The collision time can be expressed as1= n0 σ vhτel(A.36)

where σ is the elastic collision cross section and v is the velocity of the particles. On theother hand, the velocity of atoms with energy higher than ηk B T isvh =v
t 2ηk T
B

m

p
πηv̄
=
2

(A.37)

APPENDIX A. LASER COOLING AND TRAPPING

126

where v̄ is the mean thermal velocity. Thus, replacing these equations the rate of evaporation results in:dN
Nh= −2N e −η
=−
dt
τel

s

η
n0 σvh = −2N n0 σηe −η .
π

(A.38)

List of Figures
2.1 Ytterbium energy levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Schematic of a quantum gas microscope. The fluorescence from the atoms
trapped in a two-dimensional are captured by a high-numerical-aperture microscope and imaged with a CCD camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Polarization gradient cooling. Two counter-propagating beams with linear andorthogonal polarization create a polarization gradient field. An atom travelingtrough this field is repetitively repumped ’downhill’ resulting in Sisyphus cooling. 16
3.3 Two different sub-Doppler cooling methods used to cool down alkali atoms inan optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Lifetimes obtained from the simulation for a magic-wavelength potential using
the 1 S0 − 3 P1 transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Velocity distribution of an atom during Doppler cooling. The velocity unit is inrecoil units vR = ħ
h k /m. The solid line corresponds to a Gaussian fitting. . . . . . 23
3.6 Three different cooling schemes. (left) 3D molasses cooling using 6 beams in
the ±x ,±y and ±z directions. (center) 2D molasses using 4 beams in the ±x
and ±y directions. (right) 4 beams in the Brewster angle directions allows cooling in the three dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Lifetime as a function of the potential depth 2V0 for s0 = 1 and ∆/Γ = −0.8 fortwo different molasses setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.8 Polarizability in the ground 1 S0 state and excited 3 P1 state as a function of the
optical dipole trap wavelength. Magic-wavelengths were found 424 nm, 463 nm,
551 nm, 612 nm, 800 nm and 1.53 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.9 Transitions considered for the excited 3 P1 state, and their respective wavelengthand line-widths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.10 (left) If the polarizability in the excited state is lower than the one in the groundstate, the detuning of the cooling beams is reduced until eventually heating occurs. (right) A higher polarizability in the excited state do not produce heating,but the efficiency of the Doppler cooling is reduced. . . . . . . . . . . . . . . . . . . . . . 27
127

LIST OF FIGURES

128

3.11 Simulation and wavelength calculations for deviations from the magic-wavelength.
In both cases the potential depth was fixed to 2V0 /k B = 90 µK. . . . . . . . . . . . . . 28
3.12 Lifetime in photons as a function of the potential depth 2V0 for a magic-wavelength
potential using the 1 S0 − 1 P1 transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.13 Polarizability in the ground 1 S0 state and excited 1 P1 state as a function of the
optical dipole trap wavelength. Two magic-wavelengths were found at λlat =775 nm and λlat = 1.30 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.14 Transitions considered for the 1 P1 state with their respective wavelength andlinewidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.15 Creation of a deep potential using a lattice near-resonant to the 1 S0 − 1 P1 transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.16 Creation of a deep potential using a lattice near-resonant to the 1 P1 − (6s7s)1 S0transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.17 Simulation results. Lifetime dependency on the lattice power Plat and lattice
detuning ∆lat for 3 different excitation beam saturation intensities (s0 ). Left:
Density plot of the lifetime for 10 different lattice detunings and powers. Right:
Dependency of lifetime for different lattice powers, with each curve representing a different lattice detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.18 The three factors limiting the lifetime on the deep potential approach. . . . . . . . 35
3.19 Simulation lifetime results for different ∆lat . Each curve was fitted with the lifetime in Eq. 3.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.20 Dependency of parameters A and B on the lattice detuning ∆lat for the simulation results. Each curve was fitted using a power function. . . . . . . . . . . . . . . . . 37
3.21 As the excitation beam is not in the retro-reflection setup, the beam exerts a net
radiative force on the atoms, tilting the optical lattice potential while reducing
the effective lattice depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.22 Lifetime of atoms in simulation, correspoding to ∆lat /2π = 0.9 THz and s0 =10000. The solid lines are fittings corresponding to s = 0 (no radiative force
effects) and s > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.23 A solid immersion lens placed between the sample and the objective lens increases the numerical aperture of the system. . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.24 Drawing of the old and new versions of the SIL. . . . . . . . . . . . . . . . . . . . . . . . . 41
3.25 SIL attached to the glass cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.26 The new version of the SIL avoids reflections by reducing the aperture of thespherical cap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

LIST OF FIGURES

129

4.1 Experimental steps to prepare a BEC under the surface of the SIL. The numbers
corresponds to the atomic cloud temperature and the distance to the surfaceof the SIL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Saturated vapour pressure of ytterbium as a function of temperature.

. . . . . . . 44

4.3 Schema of the vacuum chamber, showing the three main chambers. IP: Ion
pump, TSP: titanium sublimation pump, DPT: interconnected by differential
pumping tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Photo of the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 The Zeeman slower device. The atomic beam is slowed down by a laser coolingbeam detuned to the 1S0 → 1 P1 transition. A magnetic field is used to create a
Zeeman shift that cancels the change in the Doppler shift. . . . . . . . . . . . . . . . . 47
4.6 Magnetic field generated by the Zeeman slower coils compared to the theoretical required magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.7 Spectrography of the atomic beam showing the performance of the Zeeman
Slower.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.8 Laser system for 399 nm (1 S0 ↔ 1 P1 transition) . . . . . . . . . . . . . . . . . . . . . . . . 50
4.9 Atoms slowed down by the Zeeman slower are trapped into the magneto opticaltrap, which combines Doppler cooling and a quadrupole magnetic field formedby two coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.10 Atoms trapped in the MOT after 10 seconds of loading time. . . . . . . . . . . . . . . . 51
4.11 Absorption image of the atoms trapped in the MOT after a time-of-flight of 11
ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.12 Laser system for the 1 S0 ↔ 3 P1 transition at 556 nm. . . . . . . . . . . . . . . . . . . . . . 52
4.13 Optical system for the horizontal and vertical ODT derived from a 18 W, frequencydoubled DPSS laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.14 Absorption images of atoms after loading into the horizontal ODT. . . . . . . . . . . 54
4.15 Number and temperature of atoms after loading as a function of holding time. . 55
4.16 Fast and slow shutter system used for temperature measurement. . . . . . . . . . . 55
4.17 Absorption imaging of the atoms in the horizontal ODT after transport. . . . . . . 56
4.18 Potential depth along the propagating axis of the horizontal ODT for different
stage accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.19 Number of atoms after transport as a relation to the stage acceleration. . . . . . . . 57
4.20 Number and temperature of atoms after transport as a function of holding time. 57
4.21 Cooling of atoms by evaporated cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.22 System used to control the intensity of the horizontal ODT beam consisting ina motorized wave-plate and polarized beam splitter. . . . . . . . . . . . . . . . . . . . . 60

LIST OF FIGURES

130

4.23 Absorption images of atoms before and after evaporative cooling. The final
number of atoms is 1.0 × 106 atoms at a temperature of 2 µK. . . . . . . . . . . . . . . 61
4.24 Vertical transport of atoms using an optical conveyor belt. . . . . . . . . . . . . . . . . 62
4.25 Absorption imaging system using the surface of the SIL as a mirror. The probe
beam is reflected by the surface of the SIL, generating two symmetrical imagesof the atomic cloud. The incindence angle θ and the distance D between thetwo images can be used to precisely calculate the distance d from the atoms to
the surface of the SIL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.26 Absorption images of atoms during transport. In each image the air-bearing
stage moves by 10 µm along the vertical ODT propagating axis. . . . . . . . . . . . . . 63
4.27 The optical accordion method. An accordion beam reflected into the surface of
the SIL creates a standing wave which spacing can be manipulated by changing
the angle of incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.28 Potential of the accordion beam during loading and compression. The potential plot was calculated for a single accordion beam with a power of 1 W. . . . . . 65
4.29 System used to change the angle of incidence of the accordion beam. Whenthe mirror moves, the accordion beam displaces laterally. The accordion lens
focal point is set to the center of the SIL, resulting in a change in the angle ofincidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.30 Photo of the accordion optical setup schematized in Fig. 4.29 . . . . . . . . . . . . . . 66
4.31 Laser system for the accordion beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.32 Cross section of the accordion beam before inciding the accordion lens. The
vertical and horizontal axes represent the z and x , y directions, respectively. . . 69
4.33 Spherical aberration simulation using the software Zemax. Two different lenses
are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.34 Spectrum of the fiber amplifier output. The input power is 60 mW and the resolution was set to 0.1 nm. The plot at right shows a detailed measurement nearby
the resonance wavelength (1077 nm). Note that the baseline at −80 dBm is thedetection limit of the instrument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.35 Transmission of the grating including fiber coupling, for a central wavelengthof λ = 1081 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.36 The accordion lens mounted in a three axis stage. . . . . . . . . . . . . . . . . . . . . . . 73
4.37 Absorption and fluorescence imaging of atoms used for the alignment of theaccordion lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.38 Absorption imaging of atoms. a) After vertical transport. b) After loading theatoms into the first layer of the optical accordion. c) After creating a BEC. . . . . . 75

LIST OF FIGURES

131

4.39 Time sequence for creating a BEC with a crossed-dipole trap using a single accordion beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.40 Time-of-flight of the BEC up to 7 ms for each of the two different accordion beams. 76
4.41 Lifetime of atoms after creating the BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.42 Time sequence for creating a BEC with a crossed-dipole trap using a double
accordion beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.43 Absorption images of the atoms for the double accordion setup. Time-of-flights
are between 0 ms and 7 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.44 Lifetime of atoms after creating the BEC in the double accordion setup. . . . . . . 78
4.45 Absorption images of the atoms after compression. Time-of-flights are between
0 ms and 7 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.46 Lifetime of atoms after compression. The solid line is a fitting using a two-bodyloss model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.47 Horizontal integrated density profile of the BEC before compression, for threedifferent potential depths. The time-of-flight is 7 ms. . . . . . . . . . . . . . . . . . . . . 81
4.48 Horizontal integrated density profile of the BEC after compression to θacc = 6◦
for two different potential depths. The time-of-flight is 7 ms. . . . . . . . . . . . . . . 82
4.49 Integrated density profile along the horizontal and vertical axis. The horizontal profile match a Thomas-Fermi distribution while the vertical distribution is
Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.50 Stability of the system during 60 ∼ 80 shots. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 High numerical aperture system used for observing atoms fluorescence. . . . . . . 85
5.2 Optical losses of the optical accordion and retro-reflection system. . . . . . . . . . . 86
5.3 Change in the offset potential and lattice depth during loading. The potential
units are relative to the initial accordion potential. . . . . . . . . . . . . . . . . . . . . . . 87
5.4 Laser system for the excitation beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 Deep potential method utilized during fluorescence imaging. . . . . . . . . . . . . . . 88
5.6 Direction and polarization of each of the accordion beams and excitation beamãĂĂduring fluorescence imaging, as seen from the CCD camera. . . . . . . . . . . . . . . 89
5.7 Single-shot image of the Yb atoms trapped in a two-dimensional optical lattice
after 100 µs of exposure time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.8 Example of selected isolated sites in a single fluorescence image. . . . . . . . . . . . 91
5.9 Averaged fluorescence intensity in isolated sites. The PSF along the x and yaxis were fitted using Gaussian functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.10 Deconvolution of a sparse image using the PSF obtained in Fig. 5.9 . . . . . . . . . 92

LIST OF FIGURES

132

5.11 Reconstructed process for a sparse image and a exposure time of 40 µs. a) Original image. b) Deconvoluted image. c) Lattice fitting. d) Binning and atom
distribution detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.12 Histogram of total fluorescence per site. left) Log scale, center) Magnificatedfluorescence peak in linear scale. right) Magnificated background peak in linear scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.13 Lifetime obtained from histogram by exponential fitting for Plat = 1.1 W and∆lat /2π = 1.0 THz. The exposure time was 300 µs . . . . . . . . . . . . . . . . . . . . . . . 94
5.14 Lifetime dependency on lattice power Plat . The solid curves are fitting using Eq.
3.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.15 Lattice detuning ∆lat /2π dependency of the parameters A and B. The solid curves
are fittings using a power equation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.16 Maximum obtainable lifetime for a given power Plat and the optimal lattice detuning ∆lat /2π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.17 Histogram as a probability density function of the background signal. A totalof 34 empty images were processed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.18 “Deep potential” as a five energy-level system in diamond shape. . . . . . . . . . . . 99
5.19 Light shift variation in a lattice site for different ∆. . . . . . . . . . . . . . . . . . . . . . . 101
5.20 Fluorescence images of a tightly filled two-dimensional optical lattice for δdiff /2π =440 MHz for two different directions of excitation. . . . . . . . . . . . . . . . . . . . . . . 102
6.1 Hyperfine splitting of 173 Yb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Transition probabilities between the different hyperfine sublevels of the 1 P1states (F ′ = 3/2, 5/2, 7/2) and the (6s7s)1 S0 state (F ′′ = 5/2). . . . . . . . . . . . . . . . 107
6.3 Eigenvalues of the total hamiltonian H = Hhfs + Hstark as a function of the lightintensity. The dotted lines in red are eigenvalues obtained for the bosonic 174 Yb. 109
6.4 Eigenvalues of the total hamiltonian H = Hhfs + Hstark as a function of the lightintensity for ∆lat /2π = 0.1 and 10 THz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.1 Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 Standing wave resulting from the interference of two counter-propagating waves
118
A.3 Standing wave from the interference of two crossing propagating-waves. . . . . . 120
A.4 Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.5 Potential generated by an optical accordion for two different incident angles
and input power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

LIST OF FIGURES

133

A.6 Potential energy generated by a Gaussian beam in the case of a two-energy levelsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

List of Tables
2.1 Stable isotopes of Yb [63] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Calculated s-wave scattering lengths in nm for the 171 Yb, 173 Yb and 174 Yb isotopes [74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Isotope shifts and absolute frequencies of the 1 S0 → 1 P1 transition [75]. . . . . . . . 14
2.4 Isotope shifts and absolute frequencies of the 1 S0 → 3 P1 transition [76, 77]. . . . . 14
3.1 Parameters A and B, max lifetime and best detuning for different saturation intensities. The units of ∆lat and Plat are THz and W, respectively. . . . . . . . . . . . . . 37
3.2 Effective potential ∆V compared with V0 for different values of ζ. For a potential depth that is 10 times larger than Vlimit , the effective potential is reduced by
15%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Parameters A and B obtained from fitting the simulation results for ∆lat /2π =0.9 THz and s0 = 10000. Two different fitting models (with and without radiative
forces) are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Characteristic values for lowermost transitions of 174 Yb . . . . . . . . . . . . . . . . . . 46
7.1 Current quantum gas microscopes. The highlighted row corresponds to thiswork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.1 Optical accordion: Trap parameters for different angles of incidence and input
powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Optical lattice: Potential and trap parameters for θacc = 6◦ and different input
powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

134

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Acknowledgements
First and foremost I would like to thank my advisor Prof. Mikio Kozuma. Since I started thisresearch under his advice in my fourth year of undergraduate, he taught me all the theoreticaland experimental aspects of ultracold atoms. During the six years of research in this laboratory, he encouraged my research and allowed me to grow as a research scientist.
I would also like to express my gratitude to the members of the examination committee,
Kiyotaka Aikawa, Akihisa Koga, Hideto Kanamori, Mikio Kozuma and Yusuke Nishida, who
dedicated their time and interest and provided helpful comments.
I acknowledge all the people who collaborated experimentally and theoretically in the realization of this project: Atsushi Noguchi, Yujiro Eto, Akimasa Nakamoto, Hitomi Ono, Yuki
Okuyama, Ryusuke Hisatomi, Izumi Shomura, Naoki Tambo, Takahiro Ogino, Honda Kazuhito,
Ryotaro Inoue and Mikio Kozuma. I would also like to thank to the former members of the
ERATO Project: Shin Inouye, Tetsuo Kishimoto, Takashi Mukaiyama and Masahito Ueda who
provided insightful ideas and helpful comments, specially during the first stages of this project.
Finally, this project would have not been possible without all the generous financial support. Everyday financial support for my Ph.D. course was provided by the Japan Society for
the Promotion of Science.

144