Optimisation of a spin-flip Zeeman
slower and laser locking to a broad
spectral line
A master thesis conducted at the
INSTITUTE FOR QUANTUM OPTICS AND QUANTUMINFORMATIONof the Austrian Academy of Sciences,submitted to the faculty of
MATHEMATICS, COMPUTER SCIENCE AND PHYSICSof the University of Innsbruckin partial fulfillment of the requirements
for the degree ofMaster of Science
presented by

Benedict Franz Hochreiter, BSc
Supervisors:
Univ.-Prof. Dr. Francesca Ferlaino
Dr. Manfred MarkFür meine Familie.

2Abstract
During my master thesis, I participated at the buildup of a new experimental
apparatus for the creation of Erbium-Rydberg atoms. The experiment in which thisapparatus is going to be used aims to study quantum phenomena with Rydberg
atoms trapped in arrays of optical tweezers. As every optical tweezer is filled with at
most one atom only few atoms are needed for each experimental cycle. To account forthis fact, the apparatus is optimized for fast cycle times rather than the generationof large sample sizes.
Thereby, my work has been focused mainly on the design of a Zeeman slower as wellas on the building of an optical setup used to generate an error signal in order tolock the laser needed for the Zeeman slower to a broad transition at a wavelength of
401 nm. Consequently, the first part of this thesis reports on different properties of
Erbium – the multi-electron-atom we plan to use in our new experiment – that areeither important for the Zeeman slowing or the experiment in general. Subsequently,
the necessary concepts of laser cooling and magneto-optical as well as optical dipole
trapping are explained before presenting the working principle of a Zeeman slower andits task within the experimental apparatus. The process of its design and simulation
are discussed as well as a plan on how to build it. Thereby, special emphasis lies on ahigh loading rate of the magneto-optical trap that is fed through the Zeeman slower.
Additionally, the associated impact of the increased divergence of a slow atomic beamon the flux through the capture volume of the magneto-optical trap is determined.
In the second part of this thesis concepts about spectral lines and their naturallinewidths as well as different mechanisms that lead to their broadening are explained.
The operating principle of the modulation transfer spectroscopy, that we use tocounteract these broadening mechanisms, is explained ending with the presentationof the building of the corresponding optical setup and the characterization of theemerging transmission and error signals.

3

Kurzfassung
Im Zuge meiner Masterarbeit beteiligte ich mich am Aufbau eines neuen experimentellen Apparates zur Erzeugung von Erbium-Rydberg-Atomen. Das Experiment,
in dem dieser Apparat genutzt wird, zielt darauf ab Quantenphänomene anhand vonRydberg-Atomen, die in einer Anordnung von optischen Pinzetten gehalten werden,
zu untersuchen. Da dabei jede optische Pinzette mit höchstens einem einzigen Atom
besetzt ist werden für jeden experimentellen Zyklus nur eine kleine Anzahl vonAtomen benötigt. Diesem Umstand wird Sorge getragen, indem der experimentelle
Apparat statt auf die Generation von großen Sample-Größen auf schnelle ZyklusZeiten optimiert wird.
Dabei fokussierte sich meine Arbeit hauptsächlich auf das Design eines ZeemanSlowers wie auch auf den Bau eines optischen Aufbaus zur Generierung eines Fehlersignals, welches dazu dient die Wellenlänge des für den Zeeman-Slower benötigten
Lasers auf einen breiten Übergang bei einer Wellenlänge von 401 nm zu fixieren.
Dementsprechend befasst sich der erste Teil dieser Arbeit mit verschiedenen Eigenschaften von Erbium – dem Multi-Elektron-Atom, das wir planen in unserem neuen
Experiment einzusetzen – welche entweder für den Zeeman-Slower oder generellfür das Experiment wichtig sind. Darauffolgend werden die nötigen Konzepte derLaserkühlung und magneto-optischen sowie auch optischen Dipol-Fallen erklärt,
bevor das Funktionsprinzip des Zeeman-Slowers und seine Aufgabe innerhalb unseres Experiments dargelegt werden. Der Prozess seines Designs und dessen Simulation werden diskutiert sowie auch ein Plan für dessen Bau. Dabei liegt ein
besonderer Schwerpunkt auf einer hohen Laderate der magneto-optischen Falle,
die durch den Zeeman-Slower gespeist wird. Zusätzlich wird die Auswirkung dergesteigerten Divergenz eines langsamen atomaren Strahls auf den Teilchenfluss durch
das Einfang-Volumen der magneto-optischen Falle festgestellt. Im zweiten Teil dieser
Arbeit werden Konzepte über Spektrallinien und deren natürliche Linienbreiten
sowie verschiedene Mechanismen, die zu deren Verbreiterung führen erklärt. Das
Funktionsprinzip der Modulations-Transfer-Spektroskopie, die wir einsetzen, um
diesen Verbreiterungs-Mechanismen entgegenzuwirken, wird erklärt und abschließend
werden der Bau des zugehörigen optischen Aufbaus und die Charakterisierung dererzeugten Transmissions- und Fehlersignale präsentiert.

4

Acknowledgements
First of all, I would like to thank Francesca for giving me the opportunity to begin
my master thesis working on a completely new and exciting experiment and forletting me work within such a great team - both in expertise and size.
I would also like to thank Manfred who, despite constantly being in a hurry,
always found a friendly ear for my questions and helped me with writing code formy scripts.
Also Arno always had the time to answer my questions and give me precious
advise for the design of the Zeeman slower and the spectroscopy setup for which
I would like to thank him. Additionally, he was the one who pushed me gently
down the rabbit hole of Arduinos and microcontrollers in general, which I am reallygrateful for.
I would like to thank Philipp as well, who did not grow tired of answering mytechnical questions.
Many thanks to the rest of the team as well, who warmly welcomed me and wasalways available for questions as well as for fun.
I need to thank the engineers in IQOQI’s mechanical workshop who always dotheir best to build requested parts in the best possible way and are always free fordiscussions regarding the design of said parts.
Finally, I would like to thank Elisabeth and Klaus for their help in administrative
affairs, David and Valentine for their quick help with IT related topics as well as
Mister Knabl, IQOQI’s administrative director.
However, there were also the semesters prior to starting my master thesis. I lookback fondly to the massive amount of time I spent with my friends on the campus
by studying, working on homework and reports or just having fun. I would like tothank (not in a specific order) Moritz, Johannes, Jacqueline, Simone, Patrick, Gilles,
René, Felix, Lisa, Thomas, second Thomas, Nicola, Jakob, Pascal, Alvise, Marco andsecond Marco. Apart from the time on the campus, we had great fun when goingsnowboarding, playing table football, billiard or board games, spending time in bars
or barbecuing. I will never forget this time.
Zu guter Letzt möchte ich meiner Familie, insbesondere meinen Eltern, danken.
Sie haben stets an mich geglaubt und mich immer unterstützt, sei es finanziell oderdurch wertvolle Ratschläge. Ich verdanke Euch viel!

5

Contents
Abstract3

Acknowledgements

5

Introduction

8

1 General properties of Rydberg atoms
1.1 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Rydberg blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Rydberg excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10
13
15
16

2 The multi-electron-atom erbium for Rydberg physics
2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Electronic levels . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Electronic configuration . . . . . . . . . . . . . . . . . . . . .
2.2.2 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Rydberg atoms of Erbium . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Multi-valence-electron Rydberg atoms . . . . . . . . . . . . .
2.3.2 Excitation paths . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Advantages for the experiment . . . . . . . . . . . . . . . . . .

20
20
21
22
24
24
25
25
25
27
27

3 Concepts: From laser cooling to Erbium-Rydberg-Tweezers
3.1 Atoms in external fields . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Magneto-optical trap/lens . . . . . . . . . . . . . . . . . . . .
3.2 Optical tweezers and polarizability of Erbium . . . . . . . . . . . . .

30
30
34
36
37
39

4 Design and building of a Zeeman slower for Erbium
4.1 The vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Transversal cooling chamber . . . . . . . . . . . . . . . . . . .
4.1.3 MOT chamber . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Design and simulation . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Impact of beam divergence . . . . . . . . . . . . . . . . . . . .

43
43
45
46
47
50
50
52
61

6

7

CONTENTS
4.2.4

Planned building process . . . . . . . . . . . . . . . . . . . . . 65

5 Light for the Zeeman slower
5.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Spectral lines and line broadening mechanisms . . . . . . . . .
5.1.2 Modulation transfer spectroscopy . . . . . . . . . . . . . . . .
5.2 Building and characterization . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Transmission signals . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Error signal and locking sensitivity . . . . . . . . . . . . . . .

67
67
68
69
70
72
74

6 Conclusion and outlook

76

A Matlab code79
A.1 Beam propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2 Impact of beam divergence . . . . . . . . . . . . . . . . . . . . . . . . 81
B Supplementary
85
B.1 Magneto-optical lens . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.2 Housing of spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 86
Bibliography

87

Introduction
One of the first things university students, who take a class on quantum mechanics,
learn is that only a fraction of all interesting quantum mechanical problems canbe solved analytically. In fact, considering that there is almost an infinite numberof such problems, one could say almost none of them is solvable that way. But
what is about computers? Especially super computers are famous for their stellar
performance when it comes to solving complex problems like, for example, forecasting
the weather. And it is true, they do help to some extent, but at the same timethey become overwhelmed ridiculously fast when it comes to quantum mechanicalproblems. Thereby, the problems begin with simple memory requirements and end
with the complexity of the required calculations themselves. As a simple example,
we can imagine a system that contains of N particles, each with either spin +1/2
or −1/2. Such a system would result in the requirement of storing a complex
number describing each of 2N superpositions, which would overwhelm any existingclassical computer for system sizes of N = 51 and upwards [1]. The reason is thatclassical computers do not work quantum mechanically. This means we are facing afundamental barrier that cannot be overcome by throwing more computing power at
a problem because it just makes little to no difference. Already Richard Feynman
vouched in his 1982 paper “Simulating Physics with Computers” for a quantum
mechanical approach when solving quantum mechanical problems: “...nature isn’tclassical, 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” [2]. Since then different types and platforms for quantum
simulators/computers have emerged. There is analogue quantum simulation, where
the hamiltonian of the actual system, one is interested in, is remodeled in good
approximation in another well-controlled system [3]. Another type is called Lloyd’s
digital quantum simulation, where the hamiltonian of a given problem is split into
steps which are then executed consecutively, corresponding to a set of unitary
operations [4]. One of the established platforms are trapped ions, where a chain ofionized atoms is trapped in a time-varying electric field with these ions acting as thequbits [5]. The use of superconducting qubits represents another platform. Thereby,
qubits are made by fabricating anharmonic oscillators from superconducting circuits
and operating them in the millikelvin regime [6]. However, a rather new platform wefocus on in this thesis are Rydberg atoms. The idea is to create a unity-filled arrayof Rydberg atoms, which due to their strong long-range interactions, that can becontrolled via an effect called Rydberg blockade, act as qubits [7]. Additionally, thesestrong interactions are also the reason why Rydberg atoms are such a promising
platform for quantum simulations. Interactions strengths of up to 10 MHz allow gatesto operate on time scales of under one microsecond [8]. Successful experiments using
51 Rubidium atoms in Rydberg states for quantum simulation have already been
8

9

CONTENTS

conducted. However, as Rubidium belongs to the group of alkaline metals they havea flaw. Once a Rubidium atom is promoted to a Rydberg state, it cannot any longer
be trapped within an optical tweezer due to the excited electron being almost free
and the remaining ionic core having negligible polarizability [9]. Fortunately, though,as Erbium belongs to the group of lanthanides it exhibits multiple valence electrons.
This means even if we excite one of its 14 valence electrons to a Rydberg state, there
are still 13 left. In order words, the remaining ionic core is still optically active. This
makes Erbium a promising candidate for quantum simulations as the atoms could
potentially remain within the optical tweezers during Rydberg excitation.
The aim of this master thesis was to present a design for a Zeeman slower, which will
be used in the setup for such a quantum simulator, under the constraints of maximized
MOT loading rate, paired with good water cooling performance. Additionally, amethod and optical setup for reliable laser locking to the wavelength needed for theoperation of the Zeeman slower should be presented. For this purpose chapter 1 covers
the discovery and basics of Rydberg physics, chapter 2 gives important information
on Erbium, the atomic species we haven chosen for our experiment. Chapter 3
gives details on atoms in external fields and how laser cooling, optical molasses
and magneto-optical traps work. Finally in chapter 4, the most important parts ofthe vacuum apparatus for the design of the Zeeman slower, as well as the design
itself are presented. It is shown how the design was found and how limitations andconstraints were considered. In chapter 5, a method and optical setup to obtain a
Doppler-free spectroscopy signal for laser locking, as well as the characterization
thereof is presented. In the last chapter all previous chapters are summarized andan outlook for the future of the experiment is given.

Chapter 1
General properties of Rydberg
atoms
Rydberg atoms and Rydberg physics have come a long way since their discovery
approximately 135 years ago. In Johann J. Balmer’s 1885 publication ”Notiz über
die Spectrallinien des Wasserstoffs” he presented his formula [10]
λ=h·

m2
,
m2 − n2(1.1)

which was able to successfully calculate the wavelength λ of electronic transitions in
the hydrogen atom, which are now known as the Balmer series. He assumed that
all these transitions have a common factor, which he proposed to call the groundnumber of hydrogen (originally in german: ”Grundzahl des Wasserstoffs”), which = 364.56 nm [10]. m and n are then integers of whichhappened to be h = 3645.6 A
n is fixed at n = 2 with the condition of m > n.
A comparison of the first four calculated wavelengths with the ones back thenmeasured with great precision by Anders J. Ångström showed very good agreementwith the measured values and proved Balmer’s formula to have a predictive power.
However, as the formula predicted a fifth line (as well as higher lines) which shouldstill be in the visible range but was not apparent in Ångström’s measured values,
Balmer consulted Prof. Eduard Hagenbach-Bischoff, who also encouraged Balmer tostudy the hydrogen spectrum in the first place. Confronted with the predicted fifthhydrogen line, Hagenbach-Bischoff told Balmer about additional measurements forlines in the violet and ultraviolet part of the spectrum, which showed good agreement
as well. However, as great these results have been, Balmer’s formula worked only forhydrogen and only for transitions from or to n = 2.
Five years later Johannes Robert Rydberg published in his work ”Recherches sur laconstitution des spectres d’émission des éléments chimiques” a formula of which hestated that Balmer’s formula is a special case of [11]:n = n0 −N0(m + µ)2(1.2)

n and m denote a wavenumber and its order respectively with n0 , µ and N0 being
constants. And indeed, by replacing the wavelength λ and the ground number hin Balmer’s formula with the wavenumbers n and n0 and rearranging the resulting
10

11
formulan = n0 ·m2 − 44n0 N0 =4no , µ=0
N0
,
⇔
n
=
n
−
⇐
=
=
=
=
=
=
=
⇒
n
=
n
−
0
0
m2
m2
(m + µ)2(1.3)

Rydberg showed that his formula was more universally applicable.
Now, one could tackle the problem from the other way round by keeping µ = 0,
replacing n with 1/λ and generalizing 4n0 = 22 n0 to n2 n0





 2
1
n2 n0
1
1
n
n2
1
1
2
= n0 −= n n0= R∞, (1.4)
= n0
−
−
−
λ
m2
n2 m2
n2 m2
n2 m2
where R∞ = 10 973 731.568 160(21) m−1 [12] is then called the Rydberg constantand obtain the Rydberg formula as we know it today. Back when the formula wasborn Balmer and Rydberg introduced the integers m and n out of the necessity for
their formulas to work, but as quantum physics was not developed yet they actuallydidn’t know that these two integers denote what we know today as the principal
quantum number. With that in mind it is then possible to calculate the wavelength
for transitions in hydrogen between the lower n and the higher m. In other words the
wavelength which is necessary to promote the atom from n to m by light absorption
or the wavelength emitted when the atom relaxes from the excited state m to a lower
state n.
Now, one could have the impression that equation 1.4 is quite limited, being
solely capable of calculating transition wavelengths in hydrogen. Fortunately, thisis not entirely true as so-called hydrogen-like (alkali) atoms can also be calculated,which Rydberg atoms belong to [1] and are therefore discussed in the following.

Main properties
Rydberg atoms are electronically highly excited atoms. While there is no sharp
definition of at which point an atom becomes a Rydberg atom, it is empirically
accepted to set the distinction at a principle quantum number of n = 10, as in thiscase the binding energy becomes comparable to thermal energies [13]. Although
Rydberg’s formula allowed calculations of transitions to and from high n, more than
20 years had to pass by until the importance of such high n states, due to theirgigantic scales, became obvious. At this time Niels Bohr introduced his model ofhydrogen and hydrogen-like atoms which was based on Rutherford’s theory of thestructure of atoms [14]. Thereby, an atom consists of a positively charged nucleus
which contributes the largest proportion of the atomic mass with a negatively charged
electron orbiting around it. According to Max Planck the energy emitted by anatom is sharply defined by the product of the Planck constant and the frequency ofthe emitted light, E = hν, which Bohr connected with his theory as well as withthe proposal that the angular momentum of the orbiting electron is calculated by
mvr = nh̄. These assumptions allow for the following derivation of the Rydberg
formula and the Rydberg constant, from [13].
We start with hydrogen, where the electron is on a stable orbit around the proton,
described by the centripetal force and Coulomb’s law
me ve21 e2=.
r
4π0 r2

12
me denotes the electron mass, ve its velocity around the nucleus and r its distancefrom said nucleus; e is the elementary charge and 0 the permittivity of free space.
Utilizing Bohr’s assumption about the angular momentum one then obtains anequation for the n-dependent radius of the atomr=n2 h̄2 4π0
,
me e2

(1.5)

where h̄ is the reduced Planck constant. The interesting part of this relation isthat apparently r ∝ n2 , which is one of the reasons that make Rydberg atoms sointeresting. Let’s take for example hydrogen in its ground state n = 1 and in thestate we defined above as the brink of becoming a Rydberg atom, n = 10. According
to equation 1.5 the Rydberg atom, albeit not even being excited to a very highn-level, is already by a factor of 100 larger than the ground state atom. And this canbe taken even further if we consider higher states as for example the n = 110 state
presented in [15]. Thereby, equation 1.5 gives a gigantic atom radius of r = 0.64 µm.
In order to discuss the drawbacks that come with such highly excited states, we firstcompute the total energy W stored in a hydrogen atom [13]:
W =

1 e2
1
mve2
e4 me
−
=−
2
4π r
(4π0 )2 2n2 h̄2|{z}| {z0 }

kin. energy(1.6)

pot. energy

The subtraction of two states with principal quantum numbers n1 and n2 then gives
the Rydberg formula
1 e4 me 1
111
W2 − W1 =
−
= R∞
−
(1.7)
n1 n2
(4π0 )2 2h̄2 n1 n2while also delivering a formula for the Rydberg constantR∞ =1 e4 me(4π0 )2 2h̄2

(1.8)

which was up until then only obtainable through measurements.
Now, if we take a look at equation 1.7 and let n2 go towards infinity, corresponding toan ionized atom, while keeping n1 = 10, we get an ionisation wavelength of 9.11 µm.
Calculation of the radiation maximum of a black body at an ambient temperatureof 300 K gives, according to Wien’s displacement law [16] λ = b/T , where λ is the
wavelength, T the temperature and b Wien’s displacement constant, a wavelength of
9.66 µm, which shows that such a Rydberg atom can be ionized just by black bodyradiation. Another threat are electrical fields that can easily rip the loosely boundelectron off of the nucleus, ionizing the atom in the process. On the other handelectrical fields generated by the positively charged nucleus and the distant electron
also drive strong interactions between Rydberg atoms, which will be discussed after
another scaling law important for the utilization of Rydberg atoms, especially forquantum simulation/computation has been presented – the lifetime of a Rydberg
state.
The longevity of Rydberg states stems from the small spatial overlap of theground state and the Rydberg state wave function (see Fig. 1.1).

13

1.1. INTERACTIONS

Probability amplitude |R(r)r|2

0.25

Ground state
Rydberg state

0.2

0.15
0.1

0.05
0
0

50

100 150 200 250 300
Distance r from nucleus / a0

350

400

Figure 1.1: Radial probability density of the valence electron in Rubidium 87 forthe ground state and for a Rydberg state. The black ground state curve shows the
distribution in the 5s1 state, whereas the red Rydberg curve shows it in the 15s1 state.
In the ground state the probability of presence almost follows a delta-like distribution
at the nucleus. For the Rydberg state this probability increases (until a certain
point) with the distance to the nucleus. This means there is no significant overlap ofthese spatial wave functions, which results in a suppression of the decay back to theground state. Graphs have been generated utilizing the Alkali-Rydberg-Calculator
package for python [17].
This small overlap results in turn in a small transition dipole moment d~ = e~r andtherefore in a weak coupling to the light field, which suppresses relaxation back into
the ground state. Additionally, relaxation into states close by is suppressed as well.
The reason for this behavior can be explained with Fermi’s golden rule [18]2πρ(ν)| hn|V |mi |2 ,
(1.9)
Γ=
h̄
where n denotes the initial state (in our case a Rydberg state) of the atom and m
the final state. ρ(ν) describes the density of states in the electromagnetic field and V~~
corresponds to the dipole transition operator V~ = d~E(t)= e~rE(t),which is driving~
the transition, where E(t) is a time-dependent electric field. Further, | hn|V |mi |2is then proportional to ν and the density of states ρ(ν) = 8πν 2 /c3 ∝ ν 2 [16], which
eventually yields for the transition rate a frequency dependence of Γ ∝ ν 3 . Now,
transitions to close by states would have a long transition wavelength and therefore
a small corresponding frequency. Hence, such transitions are highly unlikely evenwith a dipole moment operator scaling with n2 . Ultimately, for these reasons the
lifetime of Rydberg states scale with n3 for states with low orbital momentum andwith n5 for ones with high orbital momentum [1].

1.1Interactions

Due to their large dipole moment Rydberg atoms are very sensitive to electric fields
which makes them an appropriate means for sensing these fields. While this behavior

1.1. INTERACTIONS

14

could be used for all kinds of applications [19], for us the most important consequenceof this sensitivity is the emergence of strong interactions. These strong interactions
in Rydberg atoms are governed by two mechanism – the dipole-dipole interaction
and the van der Waals interaction. In the following these two interactions shall be
explained.

Dipole-dipole interactions
The first type of interaction emerges from the fact that atoms, once excited to a
Rydberg state, acquire a significant electric dipole moment. Hence, the interactionenergy of two electric dipoles d~1 and d~2 can be described by [16]


1
~
~
E(d~1 , d~2 , R) = −
3d
d
cos
(α
)
cos
(α
)
−
d
d
(1.10)
1 2
1
2
1 2 ,
4π0 R3
where α1 , α2 denote the angles between the z-axis and the dipoles d~1 , d~2 respectively
and R the distance between the two dipoles. In this description we assume theatoms are well separated, that is the wave functions of the two atoms do not overlap.
Further, we consider the atoms to be in the states |n1 , s1 i, |n2 = n1 − 1, p2 i which
are coupled by the dipole interaction. Now, the first one of these atoms can decay
into an adjacent state |n1 − 1, p1 i while emitting a photon. The second atom canthen be excited to a higher state |n2 , s1 i – the initial one of the first atom. An
important question is the scaling of the interaction strength regarding the principal
quantum number and the distance of the two atoms. Since two dipoles interact witheach other equation 1.10 gives a scaling ∝ R−3 and the interaction strength of two
4such transitions in total scales with ∝ n4 which results in an overall scaling of ∝ Rn 3 .

Van der Waals interactions
For the second type of interaction, we consider two atoms in the same initial states|n1 , s1 i and |n2 , s2 i. Similarly to the dipole-dipole interaction the first atom in thestate |n1 , s1 i can decay into a lower state |n1 − 1, p1 i while emitting a photon withan energy corresponding to the difference between the initial state and the one it
decayed to. The second atom can absorb this photon resulting in being promoted toa higher state |n2 + 1, p2 i if it can overcome an energetic gap δE called the Förster
defect. This energetic gap is the result of the fact that it is highly improbable thatthe energy separating the two states of the first atom exactly fits the separation of
the two states of the second atom. However, according to Heisenberg’s uncertainty
relation of energy ∆E and time ∆t [16]∆E · ∆t ≥ 2πh̄ = h(1.11)

for a short amount of time the amount of energy can be ”washed out”, that islarge enough to still allow such a process. Eventually both atoms can return backto their original state while cycling through this process in reverse. This leads
E2
to an overall interaction strength scaling ∝ δEdip[20]. Because there are now two
R6transition-absorption cycles involved, we have now analogously to the dipole-dipole
2interaction a scaling of ∝ Edip
∝ n8 and ∝ R−6 due to the dipole coupling. Further,
the Förster defect scales with ∝ n−3 [20] which ultimately results in an interaction
11
strength scaling of ∝ nR6 .

15

1.2

1.2. RYDBERG BLOCKADE

Rydberg blockade

The van der Waals interactions explained above are responsible for a phenomenon
called the Rydberg blockade. It prevents the excitation of one or more atoms to a
Rydberg state.
Energy E|r, riΩ|r, gi|g, girb

Atom displacement r

Figure 1.2: Schematic of the Rydberg excitation of two atoms and the suppression
thereof. If two atoms are well separated, i.e. their wave functions do not overlap,
excitation of both atoms to a Rydberg state is not prevented. If the distance between
these two atoms is reduced further and further the first atom can be excited to a
Rydberg state |r, gi just like before. For the second atom, however, the coupling to
the light field becomes weaker and weaker until Rydberg excitation is suppressed
altogether for this atom. This happens at a distance called the blockade radius rb
which is dependent on the Rabi frequency Ω of the light field driving the transitions.
However, this suppression can be overcome by detuning the the light field responsiblefor driving the transition in a way that it compensates for the energy shift introducedby the interaction. Inspired by [1] and [8].
To explain the process, let us consider two atoms which are both in the groundstate of which we can then form a two atom ground state |g, gi. Possible excited two
atom states are then |g, ri, |r, gi and |r, ri. If we now excite one of the atoms by alight field of frequency ν (which is not practical as Rydberg excitations are usually
performed by two-photon excitations, but more on that in the next section) and Rabi
frequency Ω, the system transitions, as expected, into one of the states |g, ri or |r, gi.
Further, assuming the two atoms are well separated, excitation of the second atomby the same light field to a Rydberg state will not be hindered. However, the closer
these two atoms are brought together the weaker the coupling of the second atom tothe light field becomes until excitation is suppressed altogether. This is the result of
the interaction strength, which increases with decreasing distance introducing a level
shift of the |r, ri state (see 1.2). The radius around the Rydberg atom within which1.3. RYDBERG EXCITATION

16

Rydberg excitation of the second atom is suppressed is called blockade radius rb andcan be calculated by equating the Rabi frequency Ω with the transition strength
C6 /R6
r
!
C6
R=rb
6 C6
,
(1.12)
=
Ω
⇐
=
=
⇒
r
=
b
R6
Ω
where C6 denotes the van der Waals coefficient. The energy shift introduced by theinteraction can, however, be compensated for by detuning the driving light field by asufficient amount.

1.3

Rydberg excitation

Above, we have seen that the small spatial overlap of the ground and the Rydberg
state and the resulting small transition dipole moment lead to an exceptional long
lifetime of the Rydberg state. While this is advantageous once one has managed toexcite the atom, the path to arrive there is not so trivial. One reason is that due
to the small transition dipole moment the laser light that would excite the atom
would have to exhibit impractically high intensities in order to end up with a Rabi
frequency capable of effectively coupling the ground and the Rydberg state.
Energy E

|ri

δL
Ω2

|eint iΩcomb.
Ω1|gi
Figure 1.3: Schematic illustration of a two-photon excitation process. |gi denotes
the ground state of a given atom, |ri a Rydberg state and |eint i an intermediate
state. The atom is excited by one photon with Rabi frequency Ω1 to the intermediate
state from which it is further excited by a second atom with Rabi frequency Ω2 toa Rydberg state. In this process the two Rabi frequencies are combined to a Rabi
frequency Ωcomb. . δL which denotes the detuning of the photons from the intermediate
state should be chosen as small as possible without disturbing the excitation processby scattering at the intermediate state.
Another reason is simply the large energy difference separating the ground statefrom high n Rydberg states. This large energy would require lasers with wavelengths
in the ultraviolet region. A way to bypass this circumstance is to use a techniquecalled two-photon excitation. Here, as the name suggests, two photons with Rabi
frequencies Ω1 and Ω2 are used to first excite the atom to an intermediate level from17

1.3. RYDBERG EXCITATION

which the atom gets excited further to its final state, that is in our case the Rydberg
state (see Fig. 1.3). In this process the two Rabi frequencies of the photons are
combined to a new Rabi frequency Ωcomb. , which is given by [21]
Ωcomb. =

Ω1 Ω2
,
δL

(1.13)

where δL denotes the detuning from the intermediate state. Naturally, a large
Rabi frequency is beneficial for the excitation process, which is why minimizing thedetuning in order to maximize the Rabi frequency would seem like a sensible choice.
However, in reality a middle ground has to be found as for too small a detuning
two many photons would scatter at the intermediate state, exciting the atom tothat state in the process instead of using it to excite the atom further. As will be
discussed in the following this process is not limited to just two-photon excitations
but can be expanded to multiple photons.
As examples for this process we will have a look at the Rydberg excitation of
two atomic species commonly used for Rydberg physics - the alkali metal Rubidium
and the alkaline earth metal Strontium.

Rubidium
As a member of the first group of the periodic table, Rubidium features one valence
electron. Since this allows treating it as a hydrogen-like atom Rubidium is an obvious
choice to study Rydberg physics. In the past multiple pathways to excite Rubidium
atoms to a Rydberg state have been discovered, including two-photon as well as
three-photon transitions. In its ground state Rubidium’s only valence electron is in
the 5S state. From there, the selection rules allow only transitions to a P state (seeFig. 1.4). Due to the commercial availability of lasers operating at this wavelength,
for the first excitation often the transition to the 5P state with a wavelength of
780 nm is chosen [23]. From there the selection rules allow only transitions to S or D
states. If one chooses a two-photon excitation scheme the atom can immediately be
excited to a Rydberg state |n, Si or |n, Di by a laser with a wavelength of 480 nm.
The advantage of this scheme is that the light for both transitions is in the visible
spectrum. However, one has to note that these wavelengths are greatly different
which is why different groups came up with three-photon excitation schemes [22], [23].
These schemes have the advantage that they allow to excite the atoms without
introducing any line broadening due to Doppler shifts or photon recoil which would
heat up an atomic sample. Two of these schemes shall here be discussed, where inboth cases the first stage is to excite the atom to the |5, P i state. Then the selection
rules limit the ways to proceed with the excitation. One way is to excite the atomwith light of 1367 nm to the |6, Si state. From there laser light of 743 nm is used toexcite the atom to a Rydberg state |n, P i.
The second successfully demonstrated path from the 5, P state involves excitation to
the |5, Di state by means of a 776 nm laser. Atoms which decay to the |6, P i state
are then excited to the Rydberg state by a third laser with a wavelength of 1016 nm.
However, despite the advantages of three-photon excitation schemes regarding
the Doppler effect and photon recoil, they are not commonly used in experiments,
since they are more technically involved – more on that in section 2.3.2.

1.3. RYDBERG EXCITATION

18

energy E|n, Si|n, P i|n, Di

1016 nm3nm|5, Dim|6, Si

77
6n

74|6, P i13
67n

480 nm

m

78

0n

m

|5, P i|5, SiFigure 1.4: Transitions used for two-photon (black arrows) and three-photon (tealand orange) excitation in Rubidium. The great advantage of these three-photon
excitation schemes is that they allow to suppress the effects of the Doppler effect orphoton recoil, canceling line broadening in the process. Data from [22] and [23].

Strontium
Another commonly used atomic species for Rydberg physics is Strontium. It is amember of the second group of the periodic table and therefore features two valence
electrons. This fact opens new possibilities for Rydberg physics, as even if one ofthe valence electrons is excited to a Rydberg state the remaining ionic core is still
optically active due to the second valence electron. Apart from other advantages,
this makes it possible to bring the atom into a doubly excited Rydberg state, withnew phenomena to be discovered. In the following two-photon excitation schemes inStrontium along with their advantages shall be explained.
The first more straightforward way, which was demonstrated in [24] is depicted
in Fig. 1.5.

19

1.3. RYDBERG EXCITATION
energy E|5snsi|5sndi
413 nm46

1n

m|5s5pi|5s2 iFigure 1.5: Transitions used for two-photon excitation in Strontium. For the firststep a broad transition with a wavelength of 461 nm is used to excite the atom toa |5s5pi state. From there it can be further excited to either the state |5snsi or|5sndi. The similar wavelengths allow to suppress the consequences of the Doppler
effect. Data from [25].
The Strontium atom is excited via a broad transition from the ground state to
the |5s5pi state by means of laser light with a wavelength of 461 nm. From thisintermediate state laser light with a wavelength of 413 nm is used to then excite theatom further to the Rydberg states |5snsi or |5sndi. The advantage of this excitation
scheme lies in the similarity of the used wavelengths. By counter propagating thetwo laser beams it is possible to suppress the Doppler effects evoked by the atomic
motion parallel to the laser beams.
Another path was chosen in [25], where laser light with a wavelength of 689 nm
is used to excite the atoms to an intermediate |5s5pi triplet state. This dipole
forbidden transition has a linewidth of Γ = 7.6 kHz [26], which is more than 4 ordersof magnitude smaller than the linewidth of the broad transition (Γ = 32 MHz) used
for the previously explained excitation scheme. Such narrow transitions are not only
useful for magneto-optical traps due to the much lower temperatures achievable inlaser cooling but also for two-photon excitations. The advantage becomes clear if
we go back and have a look at equation 1.13, where the denominator contains thedetuning from the intermediate state. A small detuning increases the combined
Rabi frequency, but a too small detuning involves the danger that too many photon
scatter at the intermediate state. However, by exploiting narrow intercombinationlines one can nevertheless choose very small detunings corresponding to large Rabi
frequencies.
From this triplet state the state with mJ = 1 is chosen to be the starting point forthe second excitation step, where UV light with wavelengths between 331 nm and318 nm is used to excite the atom to |5snsi or |5sndi triplet states.

Chapter 2
The multi-electron-atom erbium
for Rydberg physics
In our group the element Erbium is the cart horse for all our experiments. It isalready being used successfully in our two other experiments called ”RARE” and”Erbium” and will be used now for our new experiment, as it possesses several usefulproperties, of which some shall be discussed in detail in the following.

2.1

General properties

Erbium is a soft, silvery-white metal that belongs to the group of lanthanoids, whichthemselves belong to the so-called rare earths. It was first discovered by Carl Gustav
Mosander in the Swedish Ytterby mine near Stockholm. Nowadays Erbium is usedfor various applications like for example as Erbium oxide for the coloring of glass [27]or as an additive to vanadium which makes the material softer. Another use case of
Erbium is as a so-called burnable poison in RBMK-type nuclear power plants where
it acts as a neutron absorber and thereby moderates the reaction [28]. However,arguable one of the most important use cases in experimental physics is its use for
Erbium doped fiber amplifiers. There, Er3+ ions are ideally pumped by light at awavelength of 980 nm radiating in the 1.53 µm region through stimulated emission
in the process [29]. As in this wavelength region telecommunication fibers have aloss minimum these fiber amplifiers are of great interest for telecommunication as
well [30].
Erbium’s atomic mass is 167.259(3) au [31] with an atomic number of 68. Likeother members of the lanthanoids it possesses a large magnetic moment of 7 µB ,which is one of the reasons for Erbium’s growing popularity in the ultra cold quantum
gas community as due to this large magnetic moment the atoms exert long range
interactions.
The melting point is located at a temperature of 1802 K = 1529 ◦C [27] which, in
combination with its vapor pressure, is important for the Erbium preparation in theexperiment. The temperature dependent vapor pressure p(T ) can be approximatedby means of the Antoine equation [32]Bp(T ) = 10A− C+T ,(2.1)

where A, B and C denote dimensionless material parameters and T the temperature.
The exact values for Erbium can be found by fitting equation 2.1 to the corresponding20

21

2.1. GENERAL PROPERTIES

vapor pressure data found in [33]. This has already been done in [34] by a former
member of our group with the resulting values A = 7.103(4), B = 12 170(20) andC = 100(2). The Antoine equation then yields pressure values in mbar (see Fig. 2.1)
for temperature values in ◦C with vapor pressures of 0.43 mbar and 0.006 mbar attemperatures of 1529 ◦C (melting point) and 1200 ◦C (the temperature to which our
Erbium sample will be heated) respectively.
101

Pressure p(T ) / mbar

100
10−1
10−2
10−3
10−4
10−5
10−6
10−7
800

900 1000 1100 1200 1300 1400 1500 1600 1700 1800
Temperature T / ◦C

Figure 2.1: Vapor pressure curve of Erbium according to the Antoine equation withthe parameters A = 7.103(4), B = 12 170(20) and C = 100(2). The dotted verticalline indicates the melting point of Erbium, whereas the dashed vertical line shows
the temperature to which the Erbium sample will be heated. At the melting point avapor pressure of about 0.43 mbar can be expected - at a temperature of 1200 ◦C avapor pressure of 0.006 mbar. Parameters from [34].

2.1.1

Isotopes

There are 6 stable isotopes of Erbium with the ones of them with an even nucleon
number – Er162 , Er164 , Er166 , Er168 and Er170 – being bosonic and the remaining
Er167 being fermionic. The odd nuclear number in Er167 also leads to a non-zero
nuclear spin of I = 7/2 which results in a hyperfine structure with 104 hyperfine
states in its ground state. Appart from the hyperfine structure another effect has
to be kept in mind. This effect is a direct consequence of the nature of an isotope
and is called the isotope shift, which can be divided into two parts – one originatingfrom a change in mass between two isotopes and the other one from a change in theactual volume of the nucleus.
Formula 1.7 in chapter 1 calculates the transition energies of a hydrogen-like atom.
However, this formula calculates this energy assuming an infinitely heavy nucleus
which is obviously not true, albeit being a good approximation, given that the proton2.1. GENERAL PROPERTIES

22

Table 2.1: The six stable isotopes of Erbium with their corresponding masses andabundances, of which the isotopes with the nucleon numbers 164 to 170 exhibit thehighest ones and are therefore especially practical for experiments. Further, one of
the six isotopes Er167 is fermionic, while the other ones are bosonic. Masses andabundances from [31]; Nuclear spin from [35]; Isotope shifts from [36].
Isotope
162

Er
Er164
Er166
Er167
Er168
Er170

Mass / auAbundance

Nuclear spin

Isotope shift / MHz

161.9287884(20)163.9292088(20)165.9302995(22)166.9320546(22)167.9323767(22)169.9354702(26)0.00139(5)0.01601(3)0.33503(36)0.22869(9)0.26978(18)0.14910(36)

0
0
0
7/2
0
0

–
849(17)0-297(6)-840(14)-1681(14)mass is ≈ 1800 me . This inaccuracy can easily be corrected by replacing the electronmass in the definition of the Rydberg constant 1.8 by the reduced mass [37]
µ=

me mN
.
me + mN(2.2)

Given, that the reduced mass changes with the nuclear mass, it becomes apparent,
that transition energies of electronic states depend on the nuclear mass and thereby
introduce a mass shift. This means that that isotopes with heavier nuclei experiencea negative shift of the transition wavelength.
The second mechanism originates in the finite size of the nucleus and the resulting
discrepancy between the electric field of a point charge and the real charge distribution
in the nucleus. This results in a size dependence of the binding energy of the atomic
levels and therefore in turn of the transition energies. For the case of the five
most abundant isotopes of Erbium the results of these shift mechanisms have beenmeasured in [36]. Relative to Er166 , the frequency shifts are −1681(14) MHz (Er170 ),−840(14) MHz (Er168 ) and 849(17) MHz (Er164 ) for the bosonic isotopes. For thefermionic Er167 the isotope shift corresponds to the frequency of the center of gravityof the hyperfine levels with a value of −297(6) MHz [36].

2.1.2Electronic levels

Apart from its several abundant isotopes, large magnetic moment and multiple
valence electrons, Erbium is special in an additional regard. As will be discussed in
the next section, Erbium features a complex electronic ground state configuration
with a fully filled 6s shell and a so-called submerged 4f shell with two electron
vacancies. This configuration leads to a plethora of electronic levels with very
different properties (see Fig. 2.2).

23

2.1. GENERAL PROPERTIES

401 nm4f¹²(³H6 )6s6p(¹P₁°)(6,1)₇°

25 000wavenumber / cm-1

20 0004f¹²(³H6 )6s6p(³P₁°)(6,1)₇°

583 nm

15 000631 nm4f¹¹(⁴I°13/2 )5d3/2 6s²(13/2,3/2)₇°4f¹¹(⁴I°15/2 )5d5/2 6s²(15/2,5/2)₇°

841 nm

10 0004f¹¹(⁴I°15/2 )5d3/2 6s²(15/2,3/2)₇°5 000

1299 nm
4f¹²(³H₆)6s² ground state

0
4

5

6

8
7
J, atom

910

11Figure 2.2: Scheme of electronic levels in Erbium of which black ones have odd parity
and red ones even parity. Arrows show transitions which are allowed according to
the selection rules and the rule of Laporte. These transitions are utilizable for laser
cooling with the two top ones with wavelengths 401 nm and 583 nm being used inour experiment. Code from [38]; data from [39] and [40].
One major use case for such a wealth of resulting transition options is to use someof them for laser cooling and/or trapping. There are five known transitions, which canbe used for laser cooling (see table 2.2) starting from the ground state to a state with
J = 7 [39]. The ground state has even parity which is why according to the Laporte
rule [41] only transitions to states with odd parity are allowed. As parity conserving
transitions are far too weak they are not relevant for laser cooling. In our experiment
we will use cooling lasers to first slow down the atoms which are coming from ouratom source and secondly to trap and cool them. The data in table 2.2 shows adecrease in transition linewidth with increasing wavelength, which in turn results ina smaller Doppler temperature TDoppler , that is the theoretical minimal temperatureof an atomic sample after cooling. Especially the transition with a wavelength of
1299 nm provides a spectacular Doppler temperature of TDoppler = 51 pK. However,
due to their small capture velocities (see section 3.1) these transitions are not suitablefor our experiment. The strong 401 nm transition features a linewidth of 27 MHz [39]in combination with a high capture velocity of 11 m s−1 which makes it ideal for
slowing but not for cooling, due to its high Doppler temperature of 660 µK. With alow Doppler temperature of 4 µK [39], the 583 nm intercombination line is suitable
for laser cooling. Additionally, the capture velocity of 0.97 m s−1 makes it possible
to directly capture atoms previously slowed on the strong 401 nm line.

2.2. GROUND STATE

24

Table 2.2: Laser cooling wavelengths with the linewidths of the correspondingtransitions. vcap is a measure for the velocity up to which a magneto-optical trap
is able to capture an atom for a given transition wavelength. vDoppler denotes theatomic velocity at the minimal temperature TDoppler achievable via laser cooling at a
specific wavelength. Data from [39].

2.2

Wavelength

Linewidth

vcap

vDoppler

TDoppler

401 nm583 nm631 nm841 nm1299 nm27 MHz0.17 MHz
28 kHz8 kHz
2.1 Hz

11 m s−1
0.97 m s−1
18 mm s−1
6.7 mm s−1
2.8 µm s−1

0.18 m s−1
14 mm s−1
5.8 mm s−1
3.1 mm s−1
50 µm s−1

660 µK
4 µK
680 nK
190 nK
51 pK

Ground state

In our experiment all (Rydberg-) excitation schemes start from Erbium’s groundstate. This is feasible as even after heating the Erbium sample to the point of
sublimation, making them available for the experiment in the process, almost all
atoms are still in their ground states. Therefore the exact configuration of Erbium’s
electrons susceptible to light fields shall be presented in the following.

2.2.1

Electronic configuration

As Erbium’s atomic number is 68 there are consequently 68 electrons to be distributed
to the various shells. Thereby, following Hund’s rules, in the ground state every shell
up to including the 6s-shell is filled, leading to 14 valence electrons. The 4f-shell canaccommodate 14 electrons, while only being filled with 12 (see table 2.3).
Table 2.3: Schematic of the filling of the 4f shell. The f shell corresponds to anorbital angular momentum of l = 3, which results in values for the magnetic quantum
number ml ranging from −3 to +3 in increments of one. These magnetic quantum
numbers belong to corresponding sub shells which are filled according to Hund’s rules.
This yields a total spin quantum number of S = 1/2 + 1/2 = 1 and a total orbital
angular momentum quantum number L = 3 + 2 + 1 + 0 − 1 = 5 with J = L + S = 6.

ms

+3
↑↓

+2
↑↓

+1
↑↓

ml
0
↑↓

-1
↑↓

-2
↑

-3
↑

Writing out this configuration in the spectroscopic notation following the Aufbau
principle yields
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f 12 = [Xe]6s2 4f 12 ,

25

2.3. RYDBERG ATOMS OF ERBIUM

where the configuration for all shells up to the 6s2 and the 4f 12 shells can besubstituted by the electron configuration of Xenon. With the electron distribution
shown in table 2.3 one can furthermore easily derive Erbium’s ground state term
symbol. The two electron vacancies in the 4f-shell yield, according to Hund’s rules,
a total spin quantum number of S = 1/2 + 1/2 = 1 and a total orbital angularmomentum of L = 3 + 2 + 1 + 0 − 1 = 5. These two quantum numbers and the total
angular momentum quantum number J = L + S = 6 eventually result in the term
symbol:
2S+1
LJ = 3 H 6

2.2.2

Magnetic properties

Furthermore, the large total angular momentum quantum number leads in turn to alarge magnetic dipole moment [16]µ = mJ gJ µB ,(2.3)

where mJ is the projection of the total angular momentum along its quantization
axis that is, the direction of an external magnetic field. gJ denotes the dimensionless
Landé g-factor and µB the Bohr magneton. Thereby, the Landé g-factor is definedas [16]J(J + 1) + S(S + 1) − L(L + 1)
gJ = 1 +,(2.4)
2J(J + 1)
which, for the values of S, L and J calculated above, results in a g-factor ofgJ = 1.167. Equation 2.3 then yields a magnetic moment of µ = 7 µB , which is an
exceptionally high value with only Holmium (9 µB ), Dysprosium and Terbium (both
10 µB ) possessing larger magnetic moments. Although not relevant for the experimentdiscussed in this thesis, the large magnetic moments of Erbium and Dysprosium
are key ingredients in our other two experiments ”RARE” and ”ERBIUM”. The
reason for this is the rise of anisotropic long-range magnetic dipole-dipole interactions
between the atoms, which go beyond ”simple” contact interactions [42].

2.3

Rydberg atoms of Erbium

With the more general atomic properties of Erbium explained above, it is time todive into what distinguishes single-valence-electron atoms from atoms with multiple
valence-electrons with a focus on Rydberg excitation. The advantages Erbium has
to offer for our experiment will be discussed as well as what makes it attractive asa platform for Rydberg physics. Eventually, different excitation paths to Rydberg
states in Erbium will be presented.

2.3.1

Multi-valence-electron Rydberg atoms

Due to their simple ground state electronic structure alkali metal atoms have been a
popular choice for Rydberg physics. Being in the first group of the periodic table allof these atoms feature one valence electron, which then is the only electron readily
available for Rydberg excitation. The reason for this is the shielding effect whichbecomes effective when atomic orbitals are fully filled with electrons as, in this case,2.3. RYDBERG ATOMS OF ERBIUM

26

the time averaged electron density distribution becomes spherically symmetric. This
leads to the shielding of the electric field for all but one protons, creating a Coulomblike potential for the valence electron in the process. However, in alkali atoms, in the
ground state, all valence electrons are situated in an s-orbital corresponding to anorbital angular momentum of l = 0 (see Fig. 2.3). This means that there is a finite
probability for the valence electron to be inside the core or at least inside the cloudcreated by non-valence electrons, which ultimately mitigates the shielding effect. For
this very reason the line spectrum of multi-electron atoms is calculated by means of
the quantum defect theory. Within this theory Bohr’s formula 1.6 is augmented bya quantum defect δl in such a way, that the individual energy levels in the atom arecalculated with a so-called effective principal quantum number n∗ = n − δl . Naturally,
due to the mechanism explained above, the quantum defect takes on different values
depending on the l-values of the valence electrons. This leads to an energy of [37]E(n, l) = −hc

R∞.
(n − δl )2(2.5)

However, while this formula holds true for one-valence-electron atoms like the alkali
metals, this is expected not to be the case for multi-valence-electron atoms. Since
in this case the remaining non-excited valence-electron(s) do not form a sphericallysymmetric electron density no proper shielding of the ionic core takes place and themodel utilizing a Coulomb-like potential breaks down. Furthermore, excitation of
additional valence-electrons is possible. The solution is a description by means of
the multi-channel-quantum-defect theory.
z

x

z

y

x

l = 1, ml = −1zyl = 1, ml = 0
z

x

x

y
l = 1, ml = 1y
l = 0, ml = 0

Figure 2.3: Plot of the atomic orbitals given by the spherical harmonics for orbital
angular momentum quantum numbers l = 0, 1 with magnetic quantum numbers
ml = 0 and ml = −1, 0, 1 respectively. For electrons in the s-shell with l = 0, ml = 0
there is a finite probability to find an electron in the nucleus, which is not the casefor electrons in the p-shell with l = 1, ml = −1, 0, 1.

27

2.3.2

2.3. RYDBERG ATOMS OF ERBIUM

Excitation paths

Erbium’s electronic level structure provides us with many potential paths for Rydberg
excitation. For this process we make use of the two-photon excitation techniqueexplained in 1.3. Given that in the ground state the valence electrons are distributed
to the s- and f -shells, there are many possible excitation paths in terms of theangular orbital momentum (see table 2.4). The f -shell offers the most possibilities,
as from this shell the selection rules allow transitions to intermediate states withquantum numbers l = 2, 4 corresponding to d and g shells. For the second excitationtransitions to states with l = 1, 3 and l = 3, 6 corresponding to the p-, f - and h-shells
are allowed. Starting from the s-shell the only allowed shell for the intermediate
state is the p-shell with final shells s and d.
Table 2.4: The different shells to which transitions are allowed by the selection rules
in a two-photon excitation scheme. Exciting an electron initially in the s-shell the
only allowed intermediate state is in the p-shell with the Rydberg states being either
in the s- or the d-shell. Starting from an f -shell electron intermediate shells d and g
are allowed with p-, f - and h-shells being allowed Rydberg shells.
initial shellintermediate shell

final shell

s
f

p
d, g

s, d
p, f , hDespite offering fewer excitation paths one of the 6s electrons is a natural choicefor the first excitation, due to their larger distance to the core compared to the 4f
electrons. Using the 401 nm transition also used for the Zeeman slower (more onthat in section 4.2) the atom can be excited to the 4f 12 6s6p state (see Fig. 2.4) fromwhich a second photon with a wavelength of 411 nm can excite it to the Rydberg
state 4f 12 6sns/nd. The advantage of this exact scheme is the small difference inwavelengths which almost cancels the Doppler shift.
Other options include (but are not limited to) excitation of one 4f electron to
the state 4f 11 5d6s2 by a photon with a wavelength of 631 nm. The second 249 nm
photon then excites the atom to the Rydberg state 4f 11 6s2 np/nf . The first ofthese two transitions has already been presented in subsection 2.1.2 and features an
exceptionally small linewidth of 28 kHz. This ensures strong coupling to the Rydberg
state during the excitation process.
Needless to say, that these schemes are not the only possible ones. However additional
Rydberg excitation schemes have to be discovered first.

2.3.3

Advantages for the experiment

There are many reasons Erbium was chosen as a platform for our new experiment.
Certainly one of them is the plethora of electronic levels presented in the preceding
subsection. These provide an impressive degree of freedom in terms of the possible
ways to prepare atoms in Rydberg states for the subsequent use in quantum simulation/computation. Another reason is that Erbium features apart from bosonic2.3. RYDBERG ATOMS OF ERBIUM

28

9/2

11/2

13/2

17/2

19/2

21/2

4f¹²(³H₆)13/2 6pns/nd

60 0004f¹²(³H₆)13/2 6sns/nd

50 000249 nm4f¹¹(⁴I°)15/2 6s²np/nf

411 nm4f¹¹(⁴F°)5d6s²₅25 000wavenumber / cm-1

15/2

391 nm70 000Ry states

wavenumber / cm-1

J, ion

401 nm4f¹²(³H6 )6s6p(¹P₁°)(6,1)₇°

20 0004f¹²(³H6 )6s6p(³P₁°)(6,1)₇°

15 000389 nm583 nm631 nm4f¹¹(⁴I°13/2 )5d3/2 6s²(13/2,3/2)₇°4f¹¹(⁴I°15/2 )5d5/2 6s²(15/2,5/2)₇°

841 nm

10 0005 000

1299 nm

4f¹¹(⁴I°15/2 )5d3/2 6s²(15/2,3/2)₇°4f¹²(³H₆)6s² ground state

0
4

5

6

8
7
J, atom

910

11Figure 2.4: Scheme of electronic levels in Erbium of which black ones have odd parity
and red ones even parity. Arrows show transitions which are allowed according to theselection rules and the rule of Laporte. The transitions with wavelengths 1299 nm,
841 nm, 631 nm, 583 nm and 401 nm can be used for laser cooling. Additionally the
401 nm transition can be used as the first step in a two-photon excitation processwith the 411 nm transition as the second step. Another option would be to use the631 nm transition as a first step using the 249 nm transition as the second step toRydberg excitation. Code from [38]; data from [39] and [40].
ones also one fermionic isotope with a large natural abundance. Although, as will be
discussed in section 3.2, the aim is to use unity filled tweezer arrays to hold individual
Rydberg atoms, fermionic Erbium offers to use another technique to create an arrayof interacting atoms for quantum simulation, called collectively encoded qubits [43].
Thereby an array of atom ensembles is constructed with the atoms in each ensemble
being able to interact with each other as well as the different ensembles amongst
themselves. Through collective encoding it is then possible to create qubits withineach ensemble driving the total qubit count in a quantum processor up. As thenumber of qubits in an ensemble is dependent on the number of hyperfine levels of agiven atom, Er167 with its 104 hyperfine levels would be a well-equipped candidate
for this technique. In the construction of tweezer based Rydberg atom arrays animportant value is the polarizability of an atom. Since Erbium is in use since years
in our group a lot of experience has been gained with this species including its
polarizability in the ground state and one excited state. This is a good starting point
for further studies most probably necessary to successfully trap Erbium-Rydberg
atoms in tweezers. Furthermore, during these years a special type of magneto-optical29

2.3. RYDBERG ATOMS OF ERBIUM

trap (MOT) has been designed and successfully operated which will be explainedin subsection 4.1.3. To give a first glimpse, the advantage which this open-top
narrow-line MOT, how it is called, offers to the experiment is open optical accessfrom the top and very low ensemble temperatures without the application of any
additional advanced cooling techniques.

Chapter 3
Concepts: From laser cooling toErbium-Rydberg-Tweezers
In the preceding chapter I have briefly mentioned an important type of laser cooling,called Doppler cooling, and presented various transitions in Erbium where thistechnique is applicable. Laser cooling depends on the transfer of the momentum
carried by the photon to excite the atom to be slowed/cooled. The details of how it
works shall be explained in this chapter. Another form of light-matter interaction
important for the experiment is the formation of optical tweezers to trap individual
Erbium-Rydberg atoms which will be discussed as well.

3.1

Atoms in external fields

In atomic experimental physics, there are several methods to interact with atoms.
All these methods rely on the application of external fields which can be coarsely
divided into two types - electric and magnetic fields. Electro-magnetic fields are used
to excite atoms to higher states by stimulated absorption and actively relax them bystimulated emission. The presence of just electric fields leads to phenomena like theDC- and AC-Stark-effect in case of constant electric fields and ones varying in time,
respectively. This effect results in a splitting of states with total angular momentum
quantum number J into 2J + 1 substates. For magnetic fields another phenomenon
called the Zeeman effect emerges, which works analogously to the Stark effect butwith magnetic fields.
Before familiarizing us with these two phenomena let us first dive deeper into
the dynamics of the aforementioned stimulated absorption and emission as well as
the spontaneous emission. The important degrees of freedom connected to these
dynamics can be separated into two groups – internal and external ones. The external
ones affect the center-of-mass velocity of the atom as a whole, whereas the internal
degrees of freedom address the electronic state of the atom. Consequently the internal
degrees of freedom are described by the energy stored in the atom, the principal
quantum number of the highest energy electron, its total angular momentum J andthe corresponding magnetic quantum number mJ . The important question is now
how internal and external degrees of freedom are connected. To answer it, let us
consider a resting atom in its ground state |gi which is being excited to a state|ei by a photon carrying an energy h̄ωexc resonant to the two-level separation. We
additionally define some detuning −δ (red) or +δ (blue) (see Fig. 3.1).
30

31

3.1. ATOMS IN EXTERNAL FIELDS
Energy E

|ei

+δ
−δ
h̄ωexc|gi
Figure 3.1: Schematic showing the excitation of an atom from the ground state toan excited state. The frequency of the photon responsible for the excitation fits thefrequency separation ωexc with a possible detuning of −δ (red) or +δ (blue) fromthe excited state.
This photon additionally carries a momentum pp = h̄kp = h̄ (2π/λp ) = h̄ (ωp /c)which is transferred to the atom during the excitation process (see [44] for first
demonstration), causing the atom to move in a direction parallel to the photon.
However, it is very unlikely to find an atom completely at rest which makes theexcitation dynamics a little bit more complex. In the case of the resting atom we
implicitly assumed the spectator to be in the laboratory frame as in this case it
does not make a difference to distinguish between the laboratory frame and theatom frame. For a moving atom and a spectator in the laboratory frame a shift of
the resonance used for excitation in the resting atom can be observed. The reason
is the Doppler effect which makes the wavelength of the photon appear blue- or
red-detuned if the atom is moving antiparallel or parallel to the photon, respectively.
The conservation of momentum then allows us to set up the following equation
relating the initial and the final momentum of the atom pinit and pfinal as well as thephoton’s momentum pp :p~final = p~init + p~p = p~init + h̄~kp(3.1)

From an energetic standpoint the excitation process can be described by a relationof the photon energy Ep = h̄ωp , the ground and excited state energy of the atom
E|gi and E|ei and its initial and final kinetic energy Ekin,init and Ekin,final :h̄ωp + E|gi + Ekin,init = E|ei + Ekin,final(3.2)

After some transformations and use of equation 3.1 one eventually ends up with aformula for the necessary photon frequency
p~init~kp(h̄kp )2
ωp = ωexc ++.
matom 2h̄matom(3.3)

Now, one can identify (~pinit~kp )/matom = ~vinit~kp as the Doppler shift and (h̄kp )2 /(2matom )as the kinetic energy transferred to the atom called the recoil energy Erec as this isalso the kinetic energy which is being released in the event of spontaneous emission.

3.1. ATOMS IN EXTERNAL FIELDS

32

Eventually, the photon frequency needed to excite an atom to some excited state
reads
Erec
(3.4)
ωp = ωexc + ~vinit~kp +h̄with the contribution of the recoil energy becoming negative for spontaneous or
stimulated emission. The contribution of the Doppler effect leads to a phenomenon
called Doppler broadening, which causes the natural linewidths of transitions to bebroadened due to the velocity distribution in an atomic sample - more on that inchapter 5. Of course, the contribution of the Doppler effect is dependent on theatom velocity, while the contribution of the recoil energy is solely dependent on the
wavelength of the photon exciting the atom and the atomic mass. The heavier theatomic species is the smaller the frequency shift imposed by the recoil energy. As
we have seen in subsection 2.3.2, for Er166 with a mass of 165.930 299 5(22) au we
can use light with a wavelength of 401 nm to excite it to a state 4f 12 6s6p. With
these parameters the frequency ωrec = 10.68 kHz corresponding to the recoil energy
is well within the natural linewidth Γ of the corresponding transition, which is givenby the radiative lifetime τ of the excited state via Γ = 1/τ . Until now, excitations
have only been examined in view of the properties of excited states and what an
excited atom can be used for. However, the excitation of an atom can also be a tool
to communicate with the external degrees of freedom of an atom. In particular, thepossibility of changing an atom’s momentum is a powerful asset (see Fig. 3.2).
As discussed above, a photon used to excite an atom carries a momentum pp = h̄kp ,which is transferred to the atom causing it to move in the direction of the photon
in the case of an atom initially resting along the axis of photon movement. For amoving atom – for the sake of simplicity we assume the atom’s direction of movement
to be antiparallel to the one of the photon – the momentum transfer along this axis
causes the atom to be slowed down. If an appropriate excited state was chosen, fromwhich the atom will decay back to its original state, i.e. a closed cycling transition,
the atom can be excited numerous times to this state and decay back until the atom
almost comes to rest. The time span the atom is in the excited state is given by its
radiative lifetime, which ultimately also limits the deceleration rate. Now, one could
think the spontaneous emission of a photon associated with the decay hinders theslowing process. However, these photons are sent out in random directions cancelingthe net contribution to the momentum change in the process.
As mentioned above, the deceleration rate is limited by the radiative lifetime ofthe excited state as during the time the atom passes in this state it is not available
for further deceleration. This means, the maximum possible number of deceleration
cycles per second is given by Γ = 1/τ . Together with the velocity change ∆v
associated with the transferred momentum
∆v =

pp
h̄kp
=
matom
matom

the deceleration rate ∆v/∆t is then just the product of the cycles per second andthe velocity change after one cycle∆vh̄kp= adec = Γ
.
∆t
matom

(3.5)

However, a precise derivation of the radiative forces acting on an atom in [45]shows that this expression lacks an additional factor of 1/2. These forces can be33

3.1. ATOMS IN EXTERNAL FIELDS

p0atom =patom − h̄kphtn

pphtn = h̄kphtn

patom

patom = p0atom

Figure 3.2: Schematic of the process of decelerating an atom. An atom with initialmomentum patom moves along a direction antiparallel to the one of a resonant photonwith momentum pphtn = h̄kphtn . The photon excites the atom and transfers its
momentum to the atom, reducing the atom’s initial momentum by the amount h̄kphtn(depicted by shorter arrows in the sketch). After a timespan given by the radiative
lifetime of the excited state the atom decays back to its original state, sends out aphoton in the process and is available for the next excitation. The direction, thatthe photon is sent to is completely random, hence over many absorption-emission
cycles the recoil transferred back to the atom cancels out in average.
differentiated into dissipative and reactive forces and can be derived from the steadystate solutions of the optical Bloch equations. For technical reasons, we are onlyinterested in dissipative forces here. For a resting atom at ~r = 0 and therefore initial
phase φ(~r = 0) = 0, the dissipative force F~dissip than reads [45]


Γ s
~
~
Fdissip = −h̄Ωvst ∇φ(~r) with vst =,2Ω 1 + s(3.6)

with s denoting the saturation parameter [45]
s=

Ω2
1
2
2 δ + Γ2 /4

and vst the steady state component of the electric dipole moment. If we now consider
a plane wave described by its time-varying electric field~ r, t) = E~ 0 cos(ωp t − φ(~r)) = E~ 0 cos(ωp t − ~kp~r)
E(~~ 0 being a constant electric field perpendicular to the propagation direction,with E3.1. ATOMS IN EXTERNAL FIELDS

34

we can find an expression for the dissipative force acting on an atom:Ω2~ r) = h̄~kp ΓF~dissip = −h̄Ωvst ∇φ(~
4 δ 2 + Γ2 /4 + Ω2 /2| {z }~ (−~kp ~=∇r)=−~kp(3.7)

Naturally, since we want to exert the highest force possible for effective slowing/cooling of atoms, we are interested in the limit of high-intensity light, corresponding to alarge value of the electric field and therefore to a large Rabi frequency. Hence, forΩ  δ, Γ, we get a maximum force and corresponding deceleration ofΓ∆~v
Γ h̄~kp
F~max = h̄~kp =⇒= adec =.
2∆t
2 matom(3.8)

However, apart from merely accelerating and decelerating atoms, this principle canbe put to good use in creating tools particularly useful in the community of ultracold
atoms - Doppler cooling and the creation of an optical molasses, which will be
explained in the following subsection.

3.1.1

Optical molasses

If we consider atoms that are completely or nearly slowed down to vcl. = 0 the
dissipative force acting on the atoms can be approximated by the series [45]FDoppler = F0 + F (v) = F0 − αv,(3.9)

where the F0 part is not dependent on the velocity of the atoms. However, the second
part is dependent on v with a friction coefficient α [45]α = h̄kp2

s
−δΓ
.
2 2
(1 + s) δ + Γ2 /4(3.10)

Since F0 is not velocity dependent it causes all irradiated atoms to move by anamount given by equation 3.7 in the direction of the photon causing the movement.
As useful a force given by F0 is for decelerating a packet of atoms, it can not be used
to cool the atoms contained in such a packet, for which a velocity dependent force isnecessary.
The velocities of the atoms in a thermal cloud follow a Gaussian function givenby the Maxwell-Boltzmann distribution where the spread of the velocities ∆v is inturn given by the cloud’s temperature. Conveniently, due to the velocity dependent
part of equation 3.9 this spread can be reduced. In other words, the cloud can becooled. This is possible if one uses a laser red-detuned from the actual excitation
frequency to irradiate the atoms. If we assume the laser to propagate along thenegative x-axis, due to the velocity spread of the atoms there are then atoms withpositive or negative velocities along the propagation direction of the laser beam. In
the atom frame for an atom moving antiparallel to the laser the light’s frequency is
Doppler shifted back towards the resonance by the amount ~kp~v , whereas it is shifted
even further away for an atom moving in the opposite direction. The reason, why thisis only applicable for slow atoms lies in a modified version of equation 3.7, where thedetuning δ is enhanced by an additional factor given by the Doppler shift. The sum

35

3.1. ATOMS IN EXTERNAL FIELDS

of this modified function with two different ~kp having values of the same magnitude
but opposite directions yields a point-symmetric curve which is depicted in Fig. 3.3.
For small Doppler shifts, and therefore slow atoms, this curve is linear around thepoint of origin. As the atoms moving in a light configuration, like the one describedabove, act as if they where embedded in some kind of viscous liquid like molasses
or honey, such a configuration is called optical molasses and was first proposed (forneutral atoms) in [46]. Further, by constructing a configuration, where six beams (orthree, which are being retroreflected) which form three counter propagating pairsthat are oriented perpendicular to each other, one can cool an atomic sample in all
three spatial dimensions.
Force |F~ (v)|+kphtn+δ
Doppler shift ~kphtn~v−δ−kphtnFigure 3.3: The velocity dependent force acting on a moving atom. A laser beam
detuned by δ with a negative wave vector −kphtn propagating in the opposite directionof an atom moving with a positive velocity v along the x-axis is in the atom’s frame
Doppler shifted back towards its actual resonance frequency. Analogously a second
laser beam with a positive wave vector kphtn and the same detuning δ counter
propagating to the first one is Doppler shifted back to resonance for an atom moving
along the negative x-direction. The force acting on atoms in either case follow a
Lorentzian function; the sum of both functions yield a point-symmetric function
linear around the point of origin.
One could fall into the misbelief, that with this technique an atomic sample could
be cooled down indefinitely. In reality, there exists a minimum temperature which
can not be undercut by Doppler cooling, which is called the Doppler temperature [45]TDoppler =

h̄Γ
,
2kB

(3.11)

where kB denotes the Boltzmann constant. The reason for this minimal temperature
are fluctuations in the number of photons absorbed per second and momentum
fluctuations of the atoms originating from the random directions into which photon
emission takes place. These ultimately result in a random walk of the atoms which3.1. ATOMS IN EXTERNAL FIELDS

36

limits the cooling performance to a point, where heating from these sources andcooling by means of the optical molasses reach a balanced state.
The presented mechanism harnessed by the optical molasses has emerged to one of
the most important tools in atomic physics. It has, however, one major insufficiency- an optical molasses is not able to trap atoms. Hence, atoms in the molasses canbe cooled but will slowly drift out of the interaction region, limiting the usefulnessof this technique. Fortunately, another technique to bypass this problem has been
invented and used successfully ever since – the magneto-optical trap or in short
MOT. But before presenting it, another phenomenon called Zeeman effect, which isimportant for the operation of such a MOT shall be explained.

3.1.2

Zeeman effectAtoms in external magnetic fields experience a shift of their energy levels, which, forsmall values (Bext  Bint ), can be treated as a small perturbation and is proportional
to the applied magnetic field. Here, Bint is the magnetic field evoked by the electron’sorbital angular momentum in the atom. If the direction of the external magnetic
field is chosen as the quantization axis, the energy shift can then be calculated by [37]∆EZeeman = mJ gJ µB Bext

(3.12)

for all of the 2J + 1 values of mJ . While this holds true for bosonic atoms/isotopes,
equation 3.12 has to be modified in order to account for the hyperfine structure offermionic isotopes. Due to their non-zero nuclear spin I the atom’s total angularmomentum F = I + J is different from just the total angular momentum J, resulting
in 2F + 1 different values of mF corresponding to the same amount of Zeeman states.
Eventually, swapping mJ and gJ in 2.3 with mF and gF yields a formula for theenergy shift of hyperfine states∆EZeeman = mF gF µB Bext ,(3.13)

where the g-factor gF is calculated by [34]gF = gJ

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

(3.14)In the case of bosonic Erbium, the ground state has J = L + S = 6 and therefore
splits up into 2J + 1 = 13 Zeeman states with magnetic quantum numbers mJ =[−J, −J + 1, . . . , J − 1, J] with the corresponding energy shifts being depicted inFig. 3.4. For Erbium, the only stable isotope possessing a hyperfine structure Er167 ,formula 3.14 together with J = 6 and I = 7/2, resulting in a lowest state with
F = 19/2, yields a g-factor of gF = 0.737. Further, analogously to the bosonic case,this state F = 19/2 splits up into 2F + 1 = 20 Zeeman states.

37

3.1. ATOMS IN EXTERNAL FIELDS

Energy shift ∆EZeeman / MHz

80mJ = 6 mF = 19/2
m = 17/2
mJ = 5 mF = 15/2
F
mJ = 4 mF = 13/2
mF = 11/2
mJ = 3 mF = 9/2
mJ = 2 mF = 7/2
mF = 5/2
mJ = 1 mF = 3/2
F = 1/2
mJ = 0 m
mF = −1/2
mJ = −1 mF = −3/2
m = −5/2
mJ = −2 mFF = −7/2
mJ = −3 mF = −9/2
mF = −11/2
mJ = −4 mF = −13/2
mJ = −5 mF = −15/2
mF = −17/2
mJ = −6 mF = −19/2

60
40
20
0
−20
−40
−60
−80
0

2

4

6
8
10
Magnetic field Bext / G

12

14

Figure 3.4: Zeeman splitting of the ground states of bosonic Erbium with J =L + S = 6 (black) and fermionic Er167 with F = I + J = 19/2 (blue) in dependenceof an external magnetic field Bext , given by equations 3.12 and 3.13.

3.1.3

Magneto-optical trap/lensA magneto-optical trap (first demonstrated in 1987 in [47]) operates with a threedimensional laser configuration like the one explained in subsection 3.1.1. To “upgrade”
this setup to a trap though, it is necessary to add a force, which is dependent on
an atom’s position. This can be achieved by using two coils in the so-called antiHelmholtz configuration, that is with the currents flowing in opposite directions
and with a coil distance dcoil that matches the coil radius Rcoil (see Fig. 3.5). Thisconfiguration produces the magnetic quadrupole field, that is needed to create, in
conjunction with the used laser light, a position dependent force by exploiting the
Zeeman effect. A second vital ingredient for the construction of a MOT is theuse of circular polarized light with two counter propagating beams having different
polarization directions σ + and σ − . The underlying working principle in one dimension
is depicted in Fig. 3.6 and can be explained on an atomic transition from a statewith J = 0, mJ = 0 to a state with J = 1 that splits up into three Zeeman stateswith mJ = −1, 0, 1. A ground state atom moving along the positive x-axis in thepositive direction with a red-detuned, σ − polarized laser beam counter propagating
can then only be excited to a state with mJ = −1. It works similar for atoms
on the negative x-axis moving in the negative direction with the difference of σ +polarized light exciting the atom to a state with mJ = 1. The reason why this canbe used for trapping is that for red-detuned laser light of some frequency ωp there
are two positions along the x-axis (and every other laser propagation axis for thatmatter) where the Zeeman effect shifts the atomic transition towards the detuned
laser light, creating a position dependent force in the process. For the total force3.1. ATOMS IN EXTERNAL FIELDS

38

acting on an atom in a magneto-optical trap FMOT , the velocity dependent force F (v)in equation 3.9 is augmented by an additional position dependent term yielding [45]−δΓ,FMOT = −αv − κx with κ = kp µb0 s 2
δ + Γ2 /4(3.15)

where µ denotes the magnetic moment of the excited state and b0 the magnetic field
gradient.
σ+

σ+

σ−

I
~
B

σ−

I

σ+

σ−
Figure 3.5: Schematic sketch of the setup necessary to create a magneto optical
trap. It consists of a laser configuration similar to the one necessary for a threedimensional optical molasses with every beam in a pair of counter propagating beams
being differently circularly polarized - either σ + or σ − . This can be achieved by
six independent laser beams or three beams which are retro-reflected after passing
through the MOT once. The second important component for building a MOT are
two coils in the anti-Helmholtz configuration (the currents flow in different directions),
which create a magnetic quadrupole field. The magnetic field vanishes in the center
while being homogenous around it. This combination of an optical molasses which
creates a velocity dependent force together with a magnetic field that introduces anadditional position dependent force by harnessing the Zeeman effect, makes for a
powerful tool in an atom physicists toolbox.
A special form of a magneto-optical trap can be achieved by constraining an atomic
sample in only two dimensions by removing two laser beams from the design discussed
above and and depicted in Fig. 3.5. Thereby, the lower and upper beams would be anatural choice. What we then end up with is called a magneto-optical lens (MOL),
which can be used in conjunction with or as a replacement for a transversal cooling
section. The advantage of such a MOL over a traditional transversal cooling section

39

3.2. OPTICAL TWEEZERS AND POLARIZABILITY OF ERBIUM

is the fact that the latter of the two can at best collimate an atomic beam, whereas a
MOL is able to focus such a beam, which has been successfully demonstrated in [48]and [49].
mJ = −1

mJ = 1J =1mJ = 0mJ = 0mJ = −1

mJ = 1σ+mJ = 0

ωphtn

σ−

ωphtn
J =0

0

mJ = 0
Position x

Figure 3.6: One dimensional schematic of the creation of a position dependent force ina MOT by exploiting the Zeeman effect. In an exemplary electronic transition from astate with J = 0, mJ = 0 to a state with J = 1, mJ = −1, 0, 1 a magnetic quadrupole
field splits the latter state into three Zeeman states. Thereby, the magnitude of
the splitting is determined by the distance from the center of the MOT, where themagnetic field vanishes. An atom moving along the positive x-axis in the positive
direction with a red-detuned, σ − polarized, counter propagating laser beam withsome frequency ωphtn can only be excited to the Zeeman state with mJ = −1. As themagnitude of the splitting increases with the distance from the MOT center, at somepoint the Zeeman effect compensates for the detuning of the laser beam and allows
an excitation. The same applies for atoms moving along the negative x-axis and in
the negative direction, but in this case with a σ + polarized laser beam exciting theatom to a Zeeman state with mJ = 1. Eventually, this creates a position dependent
force around the MOT center.

3.2

Optical tweezers and polarizability of Erbium

In the previous subsections we have discussed ways of cooling and trapping atoms.
However, some of the real fun begins when the interaction and movement of individual
atoms becomes possible. Like in the preceding chapters, this is usually achievedby means of electromagnetic radiation of some sort. Regardless, the tool, that I
am going to introduce in this subsection has some distinct differences to the lightused to address greater bulks of atoms. It was discovered in 1986 [50] by Arthur
Ashkin, which led to him being awarded the 2018 Nobel Prize in physics. Althoughback then these tweezers were used for the manipulation of single organic dielectric
particles like tobacco mosaic viruses and Escherichia coli bacteria [51], it turned outthat they can also be used for single atoms. Since the working principle and thecorresponding treatment in terms of ray optics of tweezers used for trapping larger
dielectric particles like viruses or bacteria is different from the one for atoms I will

3.2. OPTICAL TWEEZERS AND POLARIZABILITY OF ERBIUM

40

limit the discussion to the latter kind. Here, for successful trapping some prerequisites
have to be fulfilled, but first, let us briefly discuss the physics of which the gradient
forces, required for the successful operation of an optical tweezer, originate.
The core of optical tweezers, and optical dipole traps in general, is formed by the
Stark effect, more specifically, the dynamic or AC Stark effect. Without going into
detail, it represents the counterpart to the Zeeman effect, but now for electric fields.
~ for an atom with an electric dipole moment p~el its total angular
In an electric field E,
momentum J splits, analogously to the Zeeman effect, into 2J + 1 substates. Atoms
do not have a permanent dipole moment; if exposed to an electric field, however, thecharges in an atom get shifted, which ultimately results in an induced electric dipole
moment [16]~p~el,ind = αE,(3.16)
with α denoting a scalar quantity called the polarizability.
The sub-effect presented above is called the linear Stark effect, as the energetic
magnitude of the shift increases linear with the electric field strength. A second one
is called the quadratic Stark effect which originates in the induced dipole moment
not necessarily being parallel to the electric field leading to an energy shift [16]~ = αE~E~ = −|α|E 2 cos β,∆E = −~pel,ind E(3.17)

where its magnitude is dependent on the square of the electric field and the angle β
between the dipole moment vector and the electric field vector. The AC Stark effectthen emerges when an atom is exposed to a time-varying electric field like the oneof an electromagnetic wave, creating an intensity dependent energy potential andthereby a gradient force. Since an atom is usually much smaller than the wavelength
it is irradiated with, a polarized atom can be treated as a point dipole. The force
acting on such an atom can then be calculated by the Lorentz force [52]



1
d
2
~ r, t) +~ r, t) × B(~~ r, t) ,F~ (~r, t)grad = α∇E(~E(~(3.18)
2dt~ r, t) denotes the magnetic field of the electromagnetic wave. Further the part
where B(~
~ × B)/µ~in brackets in the second term is proportional to the pointing vector (E[53]with the Poynting vector being a measure for the transported energy per time andarea unit in the direction perpendicular to the electric and magnetic fields – in otherwords in the propagation direction of a laser. Since the laser’s power ought notchange, at least not on a short time scale, the time derivative in the second term
vanishes, leaving us with only [50]1 ~F~ (~r) = α∇E(~r)2 .
2(3.19)

Ultimately, this gradient force draws an atom to the point of greatest intensity,
which is the focal point in a focused beam. This is also the reason why, besides ofpolarizable particles, red-detuned, tightly focused beams with corresponding small
waist sizes (waist radius ≈ w1/e2 = 1 µm [54]) are required for such an optical tweezer
to work. If one would use, however, a blue-detuned laser beam such a particle would
be pushed away from the region of maximum intensity, that is the focal point. In
that sense, one could say that optical tweezers are a specialized case of the optical41

3.2. OPTICAL TWEEZERS AND POLARIZABILITY OF ERBIUM

dipole trap (ODT) that operates on the same principle and is widely used in theultracold atoms community. For the successful operation of such an ODT, just as forthe optical tweezers, a red-detuned, focused laser beam is needed. However, in thiscase larger waist sizes are chosen (waist radius ≈ w1/e2 ≈ 10 µm [55]) resulting in theability to trap larger collections of atoms (103 to 104 atoms [55]) at temperatures in
the mK-regime [55].
These tweezers can then be used to hold single atoms in three dimensions and evenmove them around. Spectacular work has been presented in [54] and [56], where in
the former they use optical tweezers created by a reflective spatial light modulator(SLM) to form a two-dimensional Rb87 array of unity-filled microtraps of arbitrary
shape. Due to light assisted collisions [57] within the tweezers, when loading fromthe used magneto-optical trap, initially about one half of the traps are filled with
one atom while the other half remains empty, creating a random atom pattern inthe process. With a single tweezer created by an acousto-optic deflector the trapped
atoms are subsequently rearranged into the desired shape. In the latter of the two
papers, the same approach is extended to three dimensions yielding the possibility offorming impressive three-dimensional trap geometries.
However, these experiments have not been conducted with Rydberg atoms whichpose a serious obstacle for the use of optical tweezers. The reason for this is thenearly free electron in a Rydberg atom and the electron’s negative polarizability
given by [58]e2α=−,(3.20)me ωp2where e denotes the electron’s charge, me its mass and ωp the frequency of the laser
used for trapping (for an exemplary wavelength of 532 nm: αe ≈ 5200 au). This
polarizability prevents the electron from being trappable in an intensity maximum
and repels it from the focal point of a focused beam. To give an example, let us
have a look at Rubidium in the ground state which, as an alkali metal, features justone valence electron. In the ground state it has a polarizability value of ≈ 317 au [9],whereas when excited to a Rydberg state the remaining ionic core features a value ofa mere ≈ 9 au [9]. Together with the repulsion of the excited, nearly free electron,such an atom can not be trapped in an optical tweezer. There are ways to trap
such atoms, like for example presented in [59], where so-called bottle beams havebeen used to trap Rb8 7 Rydberg atoms in an intensity minimum. However, sincefor this technique the Rydberg electron is used for trapping it must stay within theintensity minimum of the beam. Thus, as the atom’s radius increases quickly withthe principal quantum number, trapping becomes less efficient for higher principal
quantum numbers, which limits these traps in terms of the number of reachable
Rydberg states.
Regardless, there comes hope for conventional intensity-maximum tweezers withmulti-valence-electron-atoms. It has been shown that the remaining ionic coresof the Rydberg-excited alkaline earth metal Strontium and lanthanide Ytterbium
feature significant ground state polarizabilities of αSr+ ≈ 90 au and αYb+ ≈ 62 au
respectively, with ground state polarizabilities of the atoms as a whole of αSr ≈ 195 au

3.2. OPTICAL TWEEZERS AND POLARIZABILITY OF ERBIUM

42

and αYb ≈ 140 au [9]. The reason for these high values is the still optically active
ionic core, due to the multiple valence electrons. Now, if the waist of an opticaltweezer is smaller than the classical orbit of the Rydberg electron around the ioniccore, these polarizabilities can be large enough to successfully trap Rydberg atoms
in optical tweezers, as demonstrated with Ytterbium in [60]. Since Erbium features
14 valence electrons it is very likely, that the polarizability of the ionic core of an
Erbium-Rydberg atom is large enough to trap it in an optical tweezer or a wholearray of optical tweezers. However, the sheer amount of electronic transitions in
Erbium (see Fig. 3.7) makes finding suitable trapping wavelengths a non-trivial task.
Regardless, a promising wavelength candidate usable for optical tweezing of groundstate Erbium would be at 532 nm as it would induce a high polarizability of 430 au.

Er polarizability / au

wavelength / nm
600

1500

1000

667

10000

15000
20000
25000
-1
wavenumber / cm

500

400

350

320

400
200
0
−200
30000

Figure 3.7: Polarizability of ground state Erbium in dependence of the wavenumber
and wavelength, where steep slopes indicate resonances. The blue marks witherror bars are measured values of the polarizability with the one at 532 nm being
a promising candidate for optical tweezing, as it features a high polarizability of
430 au. Code from [38]; data from [61].

Chapter 4
Design and building of a Zeeman
slower for Erbium
In the previous chapters we learned about the fundamentals of Rydberg atoms, its
properties, ways of interaction and the excitation schemes, necessary to reach a
Rydberg state. General properties of Erbium, our chosen atomic species, have beenpresented, as well as details about its electronic configuration, magnetic properties
and its use for Rydberg physics. Furthermore, we had a look on atoms in electric,
magnetic and electro-magnetic external fields and the utilization thereof for coolingand trapping. With most of the necessary concepts covered it is time to present
and explain the hardware that will be used in the experiment. Special emphasis ison the parts I worked on, including a potential upgrade of the transversal coolingchamber to a magneto-optical lens by the introduction of a magnetic quadrupole
field and most importantly the Zeeman slower. For the Zeeman slower, the task was
to improve upon an existing design, that is currently used in the RARE experiment,
in a way that allows for a higher MOT loading rate, while additionally improving
the water cooling of the slower itself and ensuring the design to be easily buildable.
To achieve this, simulations of the needed magnetic field and the propagation of theatomic beam has been conducted, as well as a fluid dynamical calculation of thebuilt-in water cooling.

4.1

The vacuum apparatus

First of all, the most important parts of the whole vacuum apparatus (see Fig. 4.1)
shall be explained, starting with the oven which is our atom source. It consists of aneffusion cell which is loaded with a sample of solid Erbium and heated close to themelting temperature, yielding an appropriate Erbium vapor pressure. Consequently,
the oven then ejects an effusive atomic beam. Proceeding along the apparatus, thenext major part is the transversal cooling chamber, which consists of a vacuum chamber in which a two-dimensional optical molasses is formed in order to transversallycool and collimate the atomic beam coming from the oven. The next crucial part in
the setup (even more so than the transversal cooling chamber) is the Zeeman slower.
Its purpose is to significantly slow down the atoms in the beam to a velocity smallenough for the optical molasses in the MOT chamber (which is the last major partin the apparatus) to capture the atoms. In the MOT chamber our reliable five-beamopen-top narrow-line MOT will be deployed, providing us with a practical capture
43

Zeeman slower

Transversal cooling

Figure 4.1: The vacuum apparatus with the oven containing the effusion cell, that provides us with a continuous flux of Erbium atoms.
Thereafter is the transversal cooling, which collimates the atomic beam coming from the oven and thereby increases the flux through the
Zeeman slower that is situated after the transversal cooling. The Zeeman slower decelerates the atoms contained in the atomic beam down
to a velocity at which they can be captured within the capture volume of the magneto optical trap inside the MOT chamber. From [62]MOT chamberOven

4.1. THE VACUUM APPARATUS
44

45

4.1. THE VACUUM APPARATUS

velocity as well as low Doppler temperatures and free optical access from the top.
This easy access from the top gives us the possibility to implement the planned arrayof optical tweezers from the top, while not interfering with other parts of the opticalsetup around the MOT chamber. Now, with a rough picture of the whole setup in
mind, let us discuss the individual parts presented above, starting with the oven.

4.1.1

Oven

For the oven, a dual filament effusion cell made by CreaTec Fischer & Co. GmbHis used in which a solid Erbium sample will be heated to temperatures around
1200 ◦C yielding vapor pressures of around 0.006 mbar (see Fig. 2.1). Pressures ofthis magnitude are needed to form an atomic beam with a flux sufficiently large forfast loading of the magneto-optical trap in the MOT chamber and correspondingly
low cycle times during experiments. Details on the composition, the inner workings
and the control system of the oven can be found in [63]. The two sections of theoven important for the scope of this thesis are the actual effusion cell and anotherpart which is called “hot lip”. Within these parts the Erbium sample is heated (seeFig. 4.2). Thereby, the Erbium sample is situated in the effusion cell and the hot
lip remains empty, albeit being able to act as a secondary reservoir if experimentswith a second atomic species shall be conducted [64]. Furthermore, the hot lip is
operated at a higher temperature than the effusion cell to prevent Erbium vapor
from condensing in the hot lip, which would eventually result in the hot lip aperture
being overgrown with material [63]. The Erbium vapor generated in the effusion cell
escapes into the hot lip through an aperture made from a 30 mm long tube with an
inner diameter of 3 mm. The cylindrical aperture in the effusion cell ensures a better
collimated atomic beam as it only allows shallow escaping angles.

Effusion cell

d1 = 30 mm

Hot lip

d2 = 50 mm

d3 = 65 mm

Figure 4.2: Schematic of the crucible in the oven consisting of the actual effusion celland the hot lip. The effusion cell features a tube with a length of 30 mm and innerdiameter 3 mm which acts as the aperture from which the atomic vapor escapes into
the hot lip. Thereby, the use of a tube instead of a circular hole ensures a better
collimation of the atomic beam. The hot lip, which is operated at temperatureshigher then the ones the effusion cell is operated at, features a normal circular
aperture. As an additional measure to prevent atoms with too steep exiting angles
from getting into the vacuum apparatus, an additional aperture is be installed. Witha diameter of 8 mm it does not obstruct the appropriately collimated part of thebeam; Adapted from [63].

4.1. THE VACUUM APPARATUS

46

Together with the hot lip, which features a normal circular aperture, and an
additional aperture outside of the crucible, which prevents atoms with steeper
escaping angles from getting into the vacuum apparatus, this oven provides a highquality atomic beam for our experiment.

4.1.2

Transversal cooling chamber

In the previous subsection we said, that the apertures in the oven collimate theatomic beam emerging from the crucible. However, unsurprisingly it is not possible
to completely collimate a beam just with apertures, as the moving atoms passing
the apertures always have some maximal angle at which they can pass them. Unfortunately, over the length of the apparatus this angle allowed by the apertures would
result in a significant transversal drift due to which a large proportion of the atoms
escaping the oven would never reach their final destination in the MOT chamber. To
prevent this from happening, we will use a technique called transversal cooling. For
this purpose we use a vacuum chamber (see Fig. 4.3) specifically designed in-houseby Philipp Ilzhöfer.

Figure 4.3: The vacuum chamber for the transversal cooling, designed in-house by
Philipp Ilzhöfer. It features four CF40 viewports (the smaller ones) and four CF63
viewports. The larger ones are used to irradiate the atomic beam passing throughthe viewports on the back and front side with 401 nm laser light. The waists of thesefour beams exhibit an elliptic shape at the center of the chamber to ensure a large
interaction volume of the passing atomic beam and the laser beams and thereby
effective collimation. From [62]It features four CF40 and four CF63 viewports of which the four larger ones
(CF63) are used to resonantly irradiate the atomic beam passing through the chamberconcentrically with laser beams at a wavelength of 401 nm. This configuration
resembles a two-dimensional optical molasses and as a result cools the passing beamtransversally, i.e. perpendicular to its direction of propagation (see Fig. 4.4). To
increase the interaction time, the waists of the laser beams exhibit an elliptic shape47

4.1. THE VACUUM APPARATUS

with the dimensions of the semi-major axis and the semi-minor axis being 15 mm and
3 mm respectively, where the semi-major axis is parallel to the propagation directionof the atomic beam.
Ideally, the transversal temperature of the beam after passing through the twodimensional molasses corresponds to the Doppler temperature. Hence, the atomic
beam is again not fully collimated even though this measure has improved the
spread of the beam dramatically – based on previous experiments in our groupan enhancement of the MOT loading rate by a factor between 7 to 10 can be
expected [64], [65]. Furthermore, there are plans to upgrade the transversal cooling
section to a magneto-optical lens, which would then be able to truly collimate andeven focus the atomic beam. An appropriate coil configuration has already been
designed and is discussed in Appendix B.1.

h̄kphtnOvenatomic beamh̄kphtn
Figure 4.4: One dimensional schematic of the transversal cooling process. The atomic
beam escapes the oven at an angle given by the apertures in the crucible. In thetransversal cooling chamber this beam is irradiated perpendicular to its propagation
direction and cooled along the laser beams. Ideally, during passing the laser beams,
the atoms are cooled down to the Doppler temperature, effectively collimating theatomic beam in the process.
Now that we have discussed a way to limit the spread of the atomic beam while
traveling through the vacuum apparatus, let us address the MOT chamber, as it has
a direct impact on the design of the Zeeman slower.

4.1.3

MOT chamber

In our MOT chamber, we use a five-beam open-top narrow-line MOT, which has
already proven reliable in our RARE experiment. It follows the basic principles
explained in subsection 3.1.3 with the difference, that we do not use a top laser
beam. This grants us two advantages – firstly, we have free optical access fromthe top, which is important for the implementation of optical tweezer arrays. And
secondly, due to the absence of a σ − polarized laser beam, the atoms absorb, only σ −polarized photons, preparing them in the lowest Zeeman substate in the process [64].
In the five-beam design gravity acts as the force on the atoms counteracting the one
introduced by the bottom laser beam. To cool and trap the atoms in the chamber,
we make use of a narrow intercombination line transition at a wavelength of 583 nmwith a line width of just 186 kHz. This transition allows low Doppler temperatures,
important for directly loading the optical tweezers from the MOT.

4.1. THE VACUUM APPARATUS

48

The vacuum chamber in which our MOT will be used is a commercial sphericalvacuum octagon with eight CF40 flanges and two CF100 flanges on the top andbottom (see Fig. 4.1). Originally it was meant as a dedicated chamber for Rydberg
physics in our RARE experiment but was never used. Though, due to its special
features it will be now used instead in our new experiment. As discussed in chapter 1,
Rydberg atoms are extremely sensitive to electric fields. Hence, for the undisturbed
operation of experiments it is important to take care of possible stray electric fields.
Fortunately, the chosen vacuum chamber features a pair of grooves around the insideon either side of the chamber. This grooves provide excellent mounting points for
an in vacuum electric field control, capable of compensating for external electricfields. Additionally, the large number of viewports around the chamber allow theuse of multichannel plate detectors in order to detect Erbium ions created within
the vacuum chamber.
Unfortunately though, the vacuum chamber does not only have advantages (seeFig. 4.5). Compared to the vacuum chamber used in our RARE experiment, it has
significantly smaller lateral flanges and corresponding viewports, while additionally,
the distance between two opposing flanges is increased (150.72 mm in the RARE
versus 177.29 mm in the T-REQS). In terms of maximally allowed diameters of the
MOT laser beams the difference between the T-REQS and the RARE is then 29 mm
versus 38 mm respectively. Additionally, the electric field control imposes anotherconstraint on the MOT beams with a maximal allowed beam diameter of 25.4 mm.

(a)

(b)

Figure 4.5: a) The vacuum chamber used in the RARE experiment. The distance
between two opposing flanges is 150.72 mm and the diameters of the lateral andtop/bottom viewports are 63 mm and 64 mm,respectively, Which makes large MOT
beam diameters possible. b) The vacuum chamber used in the T-REQS experiment.
Here, the distance between two opposing flanges is 177.29 mm and the lateral viewports and the in vacuo electric field control impose maximal MOT beam diametersof 29 mm for the horizontal beams and 25.4 mm for the vertical one. The blue arrowsignifies the atomic beam in both images.
These differences have severe consequences for the loading rate of the MOT in the
T-REQS in comparison to the one in the RARE. The reason for this is for the most49

4.1. THE VACUUM APPARATUS

part due to the spread of the atomic beam after being slowed down to a velocity
MOT
which is slower than the capture velocity vcaptof the MOT. Consequently, theimpact of this beam divergence on the MOT loading rate has been simulated during
the design process. This spread arrises from the fact that the atoms in the beam
travel with a velocity of over 300 m s−1 , where an additional small transversal velocity(guaranteed by the transversal cooling) does not have a big impact on an atom’stransversal position. However, after being slowed down in the Zeeman slower, whichby itself increases the transversal velocity of the atoms in the beam through heating,a significant spread of the beam is introduced with highly detrimental effects on the
MOT loading rate. While a more thorough explanation will follow in subsection 4.2.3,
the key takeaway points are listed in the following (also see Fig. 4.6):• The larger the distance between the end of the Zeeman slower and the center ofthe MOT, the more time and space for atoms with greater transversal velocities
to drift sideways and thereby never reach the capture volume generated by the
MOT laser beams. The result is a lower loading rate compared to a scenario
with a shorter distance between the end of the Zeeman slower and the MOT
center (or in other words a shorter distance between two opposing flanges ofthe vacuum chamber).
• The larger the diameter of the MOT laser beams, the larger the generated
capture volume. Hence, the loading rate increases with increasing laser beam
diameters.
Capture volumes
Zeeman slower
Atomic beamdRARE
dT−REQS
Figure 4.6: Schematic of the MOT loading geometry. The atomic beam is slowed to avelocity that the magneto optical traps are able to capture in their capture volumes,
depicted by dotted and dashed circles for the RARE and T-REQS, respectively.
Due to transversal heating during the slowing process, the atomic beam becomes
divergent after the Zeeman slower resulting in an incomplete capture within the
MOT. Due to the larger distance between the end of the Zeeman slower and thecenter of the capture volume in the T-REQS, a smaller proportion of the atom flux
can be captured per second, compared to the RARE. Additionally, due to the MOT
laser beams being smaller in diameter in the T-REQS the loading rate is further
reduced.
However, a high MOT loading rate is desirable as it decreases the experimental cycle
time and thereby increases the number of individual experiments possible per hour.
Hence, with all possibilities regarding the MOT and its geometry being exhausted,
the next step is to optimize the Zeeman slower.

4.2. ZEEMAN SLOWER

4.2

50

Zeeman slower

In many experiments conducted in the ultracold atoms community an initially createdcontinuous atomic beam has to be decelerated in order to make it possible to trap
and cool the contained atoms in a MOT. A tool which has proven to be reliable is
the Zeeman slower, named after the Dutch physicist and discoverer of the Zeeman
effect Pieter Zeeman. Its working principle is somewhat similar to the one of the
MOT, as both exploit the Zeeman effect. But before explaining how it works, let us
start with answering the question why there even is the need for a separate tool for
atom deceleration and why a laser beam on its own does not suffice for this task.
Imagine an atom moving in the positive direction along the x-axis. A photon
travelling towards the atom along the x-axis can excite this atom if, also allowing forthe Doppler shift, it is resonant to an electronic transition in the atom. As describedin section 3.1, in the event of an excitation the atom transfers its momentum to theatom which results in a deceleration. The number of momentum transfers from thephoton to the atom at an unchanged photon frequency is dependent on the linewidthof the transition one is driving. However, due to the changing Doppler shift, at somepoint the photon will be out of resonance and can no longer decelerate the atom.
One of the techniques that where proposed to solve this problem was to chirp thelaser used to decelerate the atom. This means the lasers frequency is swept across acontinuous range of frequencies containing all the resonances needed for the slowing
process. This technique was first successfully demonstrated in [66] by slowing a beamof neutral sodium atoms, but soon it was taken over by another technique whichtackles the problem from the other way round and shall be presented in the following.

4.2.1

Working principle

Instead of continuously adjusting the light field’s frequency driving the transition,
the idea of this new strategy was to harness the Zeeman effect to continuously tune
the frequency of the used transition to fit a constant laser frequency. To achieve this,an atomic beam is sent through a tapered solenoid (see Fig. 4.7) from one side witha counter propagating circularly polarized laser beam coming from the other side. A
larger magnetic field at the beginning of the Zeeman slower shifts the transitions offaster atoms back to the laser frequency, while from there on decreasing magnetic
fields shift back the transitions of slower atoms. This way, the atom velocities are
mapped to a position along the Zeeman slower. This means, atoms that are initiallyZS
faster than some maximal velocity vcapt, the Zeeman slower is able to decelerate cannever be in resonance. Atoms that are slower than this maximal velocity will bein resonance at some position along the Zeeman slower. Such a design, like it waspresented in [67], is called decreasing field Zeeman slower. While a reverse design
with an increasing field is possible, it is associated with difficulties. In this case, theregion with the largest magnetic field is near the magneto-optical trap, a part of theapparatus where the influence of external fields should be as small as possible. A
good middle ground represents the so-called spin-flip Zeeman slower which will be
used in our experiment. It utilizes a decreasing field in the first part of the slower
and an increasing one in the second part (see Fig. 4.6).

51

4.2. ZEEMAN SLOWER
Zeeman slower
Atomic beam

ZS beam, σ +Figure 4.7: Schematic of a decreasing field Zeeman slower. It consists of a taperedsolenoid composed of several individual coils generating a decreasing magnetic
field along the atomic beam path. The atomic beam is irradiated by a counter
propagating σ + -polarized laser beam responsible for decelerating the individual
atoms. The decreasing magnetic field shifts the Doppler shifted frequency of thetransition used for the slowing process back to the frequency of the ZS beam. Thisway, the atoms velocities are mapped to a position along the Zeeman slower and it
ZS
is ensured that all atoms, up to a maximal velocity vcapt
supported by the Zeeman
slower, are slowed to the same velocity at the end of the slower.
While with this design the MOT can operate without interference, anotheradvantage is that there is less need for high currents in the coils as they are smaller
overall. Additionally, it is also possible to build a Zeeman slower using permanent
magnets [68], [69]. Due to the resulting transverse direction of the magnetic field
and the corresponding polarization splitting, however, a superposition of σ + - andσ − -polarized light, i.e. linearly polarized light has to be used with the consequencethat only half of the laser power is used for the slowing process [70]. The reason
is that for an increasing field or decreasing field Zeeman slower, where transverse
magnetic fields are used, circularly polarized light is still needed. σ + for a decreasing
field slower and σ − for a slower with an increasing field [71]. The laser beam used
for the excitation, can be tuned to resonance for only one of the emerging σ-states.
The second state that is not in resonance and an additional π-state are detrimental
for the excitation process of which the effect of the π-state can be eliminated byusing light that is polarized orthogonal to the magnetic field [68]. Hence, the needfor linearly polarized light.
Now, with a rough idea about the working principle in mind, let us discuss themathematics behind the slowing process. Some of the most important characteristics
of a Zeeman slower are the shape of the magnetic field needed to reliably slow down
an atomic beam as well as the total distance z0 required to achieve the desired amountof deceleration. To calculate this distance one has to first determine the decelerationaZS the atoms are experiencing and the total velocity change, which is given by the
ZS
maximum velocity vcapt
the Zeeman slower supports and the desired end velocity
ZS
vend. While the maximal deceleration amax , on the other hand, is calculated byequation 3.8, one does in general not use this maximal value for designing a Zeeman
slower but enhances it by a safety factor of usually η ≈ 2/3 to account for possible
imperfections in the field or too steep field changes. This way, the deceleration in
the Zeeman slower is given by aZS = ηamax which, using basic equations of motion,
results in a Zeeman slower length of

ZS
ZS 2
vcapt
− vend
z0 =
.
(4.1)
2aZS

4.2. ZEEMAN SLOWER

52

Assuming a constant deceleration within the Zeeman slower, the time-dependent
position reads
q

ZS
ZS 2
vcapt ±
vcapt
− 2aZS zaZS
ZSz(t) = vcapt·t−· t2 =⇒ t(z) =
.
2
aZS
ZS
The time-dependent velocity, in turn, is given by v(t) = vcapt− aZS · t, which can bemapped to a position-dependent velocity via the previous relation, which then yields
q



s
ZS
ZS 2
vcapt
±
vcapt
− 2aZS z
2aZS zZS
ZS
 = vcaptv(z) = vcapt− aZS · 
· 1−
 . (4.2)
ZS 2
aZS
vcapt

In general, the final velocity after the Zeeman
 slower is much smallerZSthan2 its capture
ZS 2
velocity, due to which we can replace vcapt
in equation 4.2 by vcapt = 2z0 aZS inorder to end up with
r
z
ZS
(4.3)
v(z) = vcapt · 1 − .
z0Further, because the magnetic field generated by the Zeeman slower should compensate for the Doppler shift in order for the ZS laser light to be in resonance, theresonance condition
ωphtn − kv(z) +!
ωphtn =ωexckh̄v(z)µ0 B(z)= ωexc ⇐=====⇒ B(z) =
h̄
µ0

(4.4)

has to be obeyed. Here µ0 = (me ge − mg gg ) µB denotes the difference of the magnetic
moments of the ground and excited state of the driven transition. ωphtn equals ωexc if
the Zeeman slower compensates for the Doppler shift. In this case we get a relation
between the position-dependent velocity and magnetic field, which eventually allows
us to formulate an equation for the position-dependent magnetic fieldrz
B(z) = Boffset + B0 · 1 − ,(4.5)
z0where B0 denotes the initial magnetic field and Boffset an offset-field to shift thewhole magnetic field by a given amount. This formula gives us the basis for thedesign/optimization of our Zeeman slower following in the next subsection.

4.2.2

Design and simulation

The design off the Zeeman slower for the new T-REQS experiment is heavily basedon the one used in the RARE experiment (designed by Albert Frisch) as it has provento be quite reliable. In this sense the design of the new slower is more an optimisationof the existing one than a completely new design. However, due to the limitations
imposed by the new vacuum chamber, these optimisations are necessary to ensurea fast and efficient MOT loading rate. The optimisations take place in three areas
with the first and arguably the most important being the increase of the atomic flux
through the capture volume of the MOT. The second one affects the manufacturing
in terms of an optimized placement of weld seams in order to prevent magnetizationin these sections. The last area of optimization concerns the enhancement of thecooling performance, as the Zeeman slower in the RARE experiment indeed operates
nicely in terms of the cooling of atoms – the Zeeman slower itself, however, runs
hotter than it should.

53

4.2. ZEEMAN SLOWER

To slow down the atoms in the Zeeman slower we will use the same 401 nm
transition, that will also be used for the transversal cooling. Its broad linewidth ofΓ = 2π × 27 MHz (see table 2.2) guarantees many scattering events per second andtherefore a large deceleration force, making this transition highly suitable for ouruse case.
As the Zeeman slower presented in this thesis is based on the one used in the RARE
experiment (see Fig. 4.8) let us first cover some basic information about it. It is built
ZS
in the spin-flip configuration with a capture velocity of vcapt= 370 m s−1 and consistsof a total of 14 coils of which the first and last one can be controlled individually.
Additionally, there is a bias/offset coil located underneath the remaining 11 coils
controlling the maximal strength of the magnetic field and the position of its zerocrossing along the slower. The used wire features an insulating coating and has thedimensions 1 mm × 2.6 mm, where the rectangular shape helps with the dissipation ofheat. All but the first and last coil (which are wound separately) are wound coaxiallyaround each other on a stainless steel tube of type 1.4404 with a length of 435 mm,
outer diameter of 26.9 mm and inner diameter of 23.7 mm, forming a coil setup witha total length of 367 mm. Within this outer tube, there is a differential pumping
tube made of 1.4429 type stainless steel with outer an diameter of 12 mm and an
inner diameter of 8 mm. On both ends CF40 flanges are welded onto these two tubes
in a way that the atomic beam passing through the differential pumping tube canpass through the flanges. On the side that is attached to the MOT chamber thedifferential pumping tube features a tapered widening over a distance of 100 mm inorder to not prevent parts of the divergent atomic beam from entering the MOT
chamber. The volume created in between the two tubes is used as a duct for thewater cooling that is necessary to transfer the heat dissipated by the coils out of thesystem.

(a)
Atomic beam(b)

Figure 4.8: The Zeeman slower designed by Albert Frisch, used in the RARE
experiment, which the Zeeman slower in the T-REQS experiment is based on. a)
It consists of 14 coils including a bias/offset coil hidden underneath all other coils
except the first and last one, which allows to control the maximal strength of themagnetic field as well as to shift the zero-crossing of the magnetic field along the
Zeeman slower. The two vertical tubes along with the connectors are the intakes forthe water cooling. b) A cut through the whole slower revealing the tapered widening
at the right end and the cooling water duct formed by the space between the innerand outer stainless steel tube. Thereby, the atomic beam propagates within the innertube (direction signified by the black arrow).

4.2. ZEEMAN SLOWER

54

For the optimization and the design of the new Zeeman slower, let us recall from
subsection 4.1.3 that the limitations are firstly posed by the larger distance between
the end of the Zeeman slower and the MOT center, due to which the divergence of
the atomic beam after the Zeeman slower is more detrimental. Secondly the smaller
diameters of the lateral viewports of the new MOT chamber limit the diameterof the MOT laser beams and thereby the size of the capture volume. But how
big is the impact of these limitations on the loading rate in the MOT chamber of
the T-REQS? To answer this and other questions, we used and enhanced a set ofsimulation scripts, originally written by Albert Frisch to design the Zeeman slower
for the RARE experiment. Without going into details here, the script to study theloading rate, or to be precise the atomic flux through the capture volume, sums
up all atoms in a Maxwell-Boltzmann distribution up the slower’s capture velocity
ZS
vcapt= 370 m s−1 . Considering the heating induced spread of the atomic beam within
the Zeeman slower, it calculates the proportion of the atoms, which travel throughthe capture volume.
·108

7

RARE chamber
T-REQS chamber
ZS
vcapt

Atomic flux / s−1

6
5
4
3
2
1
0

0

100

200
300
400
500
Atom velocity / m s−1

600

700

Figure 4.9: Comparison between the atomic flux passing the capture volume generated
by the magneto optical trap in dependence of the atom velocity. Thereby, the basefor both flux calculations is the Zeeman slower design used in the RARE experiment.
The black curve shows the calculated atomic flux through the capture volume in
the vacuum chamber used in the RARE experiment, while the red one shows thecalculated atomic flux through the capture volume in the vacuum chamber that will
be used in the T-REQS experiment. The dotted line indicates a Zeeman slower
ZS
capture velocity of vcapt= 370 m s−1 for which the calculations are carried out.
With an atomic flux of 4.8 × 108 s−1 in the case of the RARE MOT chamber and2.17 × 108 s−1 when using the T-REQS MOT chamber, we face a loading rate decreaseof 55 %. This means, that due to the vastly different designs of the RARE and
T-REQS MOT chambers, the Zeeman slower design used in the RARE experimentyields very different atomic fluxes at the MOT center for these vacuum chambers.
To reduce this difference, we aim to optimize the existing Zeeman slower accordingly.
Based on scripts by Albert Frisch.

55

4.2. ZEEMAN SLOWER

The results for the velocity dependent atomic flux (for this and all following
graphs, the oven temperature is 1200 ◦C) for the unoptimized (RARE) Zeeman slower
in both cases – the MOT chamber used in the RARE experiment and the T-REQS
experiment are depicted in Fig. 4.9. With an atomic flux of 4.8 × 108 s−1 using the
RARE MOT chamber and a flux of 2.17 × 108 s−1 in the case of the T-REQS MOTchamber, we would experience a massive loading rate reduction of 55 %, as shown in
Fig. 4.9.
Thinking of ways to improve the loading rate one comes to the conclusion that,naturally, the whole diameter of the lateral viewports will be used to their full extent
anyway. This means, unfortunately, that the latter of the two previously mentioned
limitations cannot be circumvented by increasing the capture volume any further.
For this reason, bringing the Zeeman slower, or to be precise its last coil, closer tothe MOT center is the name of the game.

Atomic flux / s−1

·109

·109
RARE
T-REQSopt.
T-REQSunopt.
ZS
vcapt

1

0.5

0

RARE
T-REQSopt.
Erbium
ZS
vcapt

1

0.5

0

200
400
600
−1
Atom velocity / m s
(a)

0

0

200
400
600
−1
Atom velocity / m s(b)

Figure 4.10: a) Comparison of the calculated atomic flux through the capture volume
generated inside the MOT chamber of the T-REQS experiment for the unoptimized(red) and the optimized Zeeman slower (teal), as well as the one used in the RARE
experiment. Thereby, the unoptimized Zeeman slower denotes the design currently
used in the RARE experiment; the optimized Zeeman slower the final design for the
T-REQS experiment. We see an increase of 20 % from 2.17 × 108 s−1 to 2.61 × 108 s−1b) Comparison of the calculated atomic flux trough the capture volume in the RARE
(black), T-REQS (red) and Erbium (blue) assuming their appropriate Zeeman slowers.
Based on that, the loading rate in the T-REQS with its optimized Zeeman slower sits
right in between the ones of our other two experiments. In both cases, the dotted
ZS
line represents the Zeeman slower capture velocity of vcapt= 370 m s−1 . Both basedon scripts by Albert Frisch.
Fortunately, we found a rather straight forward way to do this by shifting theentire active part of the Zeeman slower closer to the MOT, making use of a gap

4.2. ZEEMAN SLOWER

56

present in the previous Zeeman slower design. One of the reasons this 10 mm gap wasneeded, was to allow for sufficient space for the nuts that attach the Zeeman slower tothe MOT chamber. We found out, however, that this space can be saved, if the outer
diameter of the outer tube, the coils are wound around, is reduced appropriately.
Reducing the outer diameter from initially 26.9 mm to 23 mm allows to reduce thecoil diameters as well and in consequence shift the whole coil setup all the way tothe flange, while leaving enough space for the nuts to be tightened (see Fig. 4.11).
This measure leads to the distance between the last coil and the MOT center
being reduced from 111 mm to 101 mm. Comparing this distance to the one in the
RARE experiment of 98 mm reveals, that the larger size of the MOT chamber in
the T-REQS experiment is almost compensated distance-wise. We then simulated
the atomic flux through the T-REQS MOT center considering the new geometry.
Fig. 4.10 a) summarizes the the result, where we observe at a Zeeman slower capture
ZS
velocity of vcapt= 370 m s−1 that the optimized Zeeman slower yields a calculatedloading rate of 2.61 × 108 s−1 , which increases the loading rate in comparison to
the unoptimized case by 20 %! In our third experiment called Erbium the distance
between the last Zeeman slower coil and the MOT center is even larger than in
the T-REQS with smaller MOT beam diameters. There, despite the lower atomic
flux, we could efficiently load the MOT. An additional comparison of the calculated
MOT loading rates between all our experiments (see Fig. 4.10 b)) shows, that the
optimized loading rate in the T-REQS experiment now sits right in between the onesof our other two experiments.

(a)

(b)

Figure 4.11: Cut through the Zeeman slower depicting the flange attaching the
Zeeman slower to the MOT chamber. a) Unoptimized Zeeman slower consisting ofan outer tube with an outer diameter of 26.9 mm which results in the coils partiallycovering the nuts used for attaching the slower to the MOT chamber. b) Optimized
Zeeman slower with an outer tube of reduced outer diameter of 23 mm. In this case,even with the last Zeeman slower coil shifted all the way to the flange there is enoughspace to tighten the nuts.
Additionally, the active region of the Zeeman slower, i.e. the coil assembly
has been slightly extended by leaving 3 mm gaps between the main coils and thestart/end coils (resulting in a total extension of 6 mm) [72]. Firstly, this is necessary
for winding the coils and secondly, by separating the first coil from the rest of theassembly, the generated magnetic field reflects the theoretical, ideal field better (seeFig. 4.12 top and middle). This leads to the slower’s capture velocity matching thetheoretical one better as well. While this measure does not affect the actual capture

57

4.2. ZEEMAN SLOWER

velocity of the design, it should indeed increase it practically.

Magnetic field / G

400
200

Field single coil
Total field
Theoretical field
MOT position

0−200−100 −50 0

50 100 150 200 250 300 350 400 450 500 550 600

Magnetic field / G

400
200

Field single coil
Total field
Theoretical field
MOT position

0
−200

η

−100 −50 0
0.8
0.6
0.4
0.2
0
−100 −50 0

50 100 150 200 250 300 350 400 450 500 550 600
η parameter

50 100 150 200 250 300 350 400 450 500 550 600
Position / mmFigure 4.12: Top and middle calculated graphs: black illustrates the magnetic fields
generated by the single coils of which the Zeeman slowers are built, red the total
resulting magnetic field, blue the theoretically required field and the black dashedline the MOT position. The top graph shows the magnetic field of the RARE Zeeman
slower, where it is apparent that, while the total magnetic field approximates thetheoretical field in most places, it does not reach the maximal field at the zero
ZS
position, resulting in a practically lower vcapt. The calculated total field of the
T-REQS Zeeman slower depicted in the middle graph corrects this flaw. Additionally,
due to an improvement in the script code for the calculation, the exact geometryof every single coil (see subsection 4.2.4) is considered. The bottom graph depicts
the η parameter (safety parameter) along the Zeeman slower. It is crucial, that thisparameter stays below a threshold of η = 2/3 to prevent atoms from falling out of
resonance and being unavailable for slowing. Based on scripts by Albert Frisch.

4.2. ZEEMAN SLOWER

58

During the designing process it is crucial to take care that the η parameter along
the Zeeman slower coils stays within a safe range of 0 and 2/3 (see Fig. 4.12 bottom).
Otherwise, if η becomes to large at some position, it is possible that the atoms fallout of resonance and are lost for the slowing process. The magnetic fields as wellas the η parameter shown in Fig. 4.12 have been calculated by using scripts also
written by Albert Frisch. These scripts rely on the utilization of the Biot-Savart
law, which presents a way to calculate the magnetic field at a given point in space
provoked by a constant electrical current. Thereby, the magnitude of the electricalcurrent also determines the magnitude of the magnetic field. An improvement thathas been applied to these scripts, though, concerns the way how the magnetic fieldof each individual coil is being calculated. In the old version the individual fields
were calculated as if the coils were stacked consecutively along the Zeeman slower,
just as illustrated in Fig. 4.6 and 4.7. In reality, however, due to practical reasons(that will be explained in subsection 4.2.4) every coil is wound around another one
lying underneath. This way, with the exception of the first and last coil, the whole
coil setup forms a radial layer structure as opposed to a linear one.

(a)

Atomic beam(b)

Figure 4.13: Optimized Zeeman slower design for the T-REQS experiment. a) It
consists of 14 coils including a bias/offset coil hidden underneath all other coils except
the first and last one, which allows to control the maximum strength of the magnetic
field as well as to shift the zero-crossing of the magnetic field along the Zeeman
slower. The 3 mm gap separating the first/last coil and the main coil assembly inthe middle allows to better approximate the desired magnetic field and is necessary
for the building process. The two vertical tubes along with the connectors are theupgraded intakes for the water cooling, which now feature a larger cross section. b)
A cut through the whole slower revealing that the tapered widening at the right end(MOT chamber side) of the inner tube has been removed. Thereby the atomic beam
propagates within the inner tube (direction signified by the black arrow).
To facilitate the designing process and assisting in the current tuning process of59

4.2. ZEEMAN SLOWER

the finished Zeeman slower, the script has been altered in a way that accounts forthe actual geometry of the coil assembly (compare single coil fields in Fig. 4.12 top
and middle).
To wrap up the final design of the T-REQS Zeeman slower let us go throughits key data. It is designed in the spin-flip configuration with a capture velocity
ZS
MOT
vcapt
= 370 m s−1 and an end velocity vcapt= 5 m s−1 that is just below the capture
MOTvelocity of the MOT of vrecap= 5.6 m s−1 [34]. The vacuum part (see Fig. 4.13) has
a total length of 435 mm with a fixed CF40 flange on the end that is attached to
the MOT chamber and a rotatable one on the other end (the one where the atoms
enter the Zeeman slower). The cooling water intakes with an inner diameter of 8 mm
and outer diameter of 12 mm along with the connectors are also situated on this end.
The tubes used for the intakes are made from the same tube (1.4429 type stainlesssteel) as the differential pumping tube within the Zeeman slower. The tapered
widening of this tube that was present in the unoptimized design to allow as large
an amount as possible to enter the MOT chamber has been removed with the tube
now extending through the whole slower. This measure has been taken to prevent
possible magnetization of the material at the welding spots in the active region of
the Zeeman slower and to increase the cooling performance of the water cooling. The
outer tube (1.440 type stainless steel), which together with the aforementioned innertube and a pair of separators (see Fig. 4.14) forms the ducts for the water cooling,has decreased in outer and inner diameter to 23 mm and 20 mm, respectively. The
coil assembly has a total length of 373 mm and consists of 14 individual coils. The
start and end coils will be wound and controlled separately, while the remaining coils
are wound around each other starting with a long bias/offset coil, forming a radial
layer structure. Between the start/end coil and the main coil assembly there is a gapof 3 mm each, which allows better approximation of the theoretically desired field by
the generated one and is necessary for the building process. The used wire features
an insulating coating and has the dimensions 1 mm × 2.6 mm.
Table 4.1: Key data about the Zeeman slower that will be used in the T-REQS
experiment at a glance.
Quantity

Value

total length of vacuum part
435 mmdiameter of cooling water intake
8 mm
outer diameter of outer tube
23 mm
inner diameter of outer tube
20 mm
outer diameter of inner tube
12 mm
inner diameter of inner tube
8 mm
total length of coil assembly
373 mmnumber of coils
14
wire dimensions
1 mm × 2.6 mm
ZS
capture velocity vcapt
370 m s−1
MOT
end velocity vcapt
5 m s−1

4.2. ZEEMAN SLOWER

60

By reducing the outer diameter of the slower’s outer tube while leaving its wall
thickness and the inner differential pumping tube the same, one could be worriedby the cooling performance of the water cooling inside the slower. Due to thismeasure the flow cross section inside one channel of the water cooling is reduced by≈ 41 % from 164 mm2 (RARE) to 96 mm2 . This cross-sectional change raises twoquestions – does it have an impact on the cooling performance or the pressure needed
to operate the water cooling? To answer these questions it is necessary to have alook at the possible flow behaviour within the annular duct formed by the two tubes.
Broadly speaking, flow of a liquid can be divided into two regimes called laminarand turbulent flow. In the case of laminar flow this means that, if one would take asample volume that spans across the whole cross section of an annular duct, all atoms
contained in this volume would move in the same direction with the same velocity.
In a turbulent flow, on the other hand, the movement is completely unorganized.
Now, from a standpoint of water cooling, a laminar flow would mean that the layerof water adjacent to the (hot) outer tube would heat up until it becomes unavailable
for further cooling, as its heat capacity is depleted. A turbulent flow, however, would
mean that there is no single layer of water adjacent to the outer tube, as the water
is in constant exchange, distributing the heat over the whole volume in the process.
This means a situation with turbulent flow is clearly preferable.
Whether a flow is laminar or turbulent can be determined by a single dimensionlessparameter called Reynolds number [73]
Re =

DV
4A
cD
=
with D =
,
ν
νA
P

(4.6)

where c denotes the mean velocity, ν the kinematic velocity, V the volumetricflow rate, A the cross section of the geometry bounding the flow, P the wetted
perimeter and D the hydraulic parameter. Further, the Reynolds number at whicha laminar flow transitions to a turbulent flow is called critical Reynolds number
Recrit. . Assuming the volumetric flow rate to be similar to the one in the RARE’s
Zeeman slower with V = 3 L min−1 = 0.5 × 10−3 m3 s−1 [74] and a kinematic viscosityν = 0.8007 × 10−6 m2 s−1 [75] of water, the Reynolds number for our geometry (seeFig. 4.14) reads Re ≈ 4344. Further, the critical Reynolds number for an annular
duct is dependent on the ratio r2 /r1 of the outer radius of the inner tube and the inner
radius of the outer tube [76]. In the case of the T-REQS Zeeman slower, this ratio
reads ≈ 0.61, which in turn gives us a critical Reynolds number of Recrit. ≈ 3200 [76].
With our calculated Reynolds number being significantly above this threshold, we can
expect thorough mixing of the cooling water and therefore good cooling performance.

61

4.2. ZEEMAN SLOWER

r1
r2

Figure 4.14: Cross section of the (semi) annular duct, forming the water cooling ofthe Zeeman slower. The gray parts illustrate the outer/inner tubes and the walls
separating the two ducts, blue the water. r1 is the inner radius of the outer tube
and r2 is the outer radius of the inner tube. ⊗ signifies the flow direction away fromthe viewer (towards the MOT chamber) and the flow direction towards the viewer(away from the MOT chamber).
Now, to answer the question, if the diameter reduction of the outer tube has
an impact on the pumping pressure, let us have a look at the cross sectional areasof the ducts as well as the water intakes. As a precaution, the diameters of thewater intakes have already been increased from 6 mm (RARE design) to 8 mm, which
increases the cross sectional area by about 44 %. That being said, however, as theresulting cross section of the water intakes measures ≈ 50 mm2 and the one of thesemi annular duct ≈ 96 mm2 the necessary pumping pressure will still not increase.
As discussed above, the design of the T-REQS Zeeman slower does not exhibit atapered widening at the end attached to the MOT chamber anymore. What still has
to be addressed is whether this has an impact on the MOT loading rate and if thisis the case, how big the impact is. This question is being discussed in the following
section.

4.2.3

Impact of beam divergence

Whether the omission of the tapered widening has an impact on the MOT loading
rate essentially boils down to the extent of the divergence of the atomic beam after
cooling and the distance the atoms still have to travel within the inner tube of the
Zeeman slower. If the remaining travel distance inside the tube is still very long asignificant proportion of the beam can overlap with the tube and cannot reach the
MOT (see Fig. 4.15). Furthermore, if the divergence of the atomic beam after being
slowed down is high the overlap of the beam with the tube can still be significant,even if the remaining distance inside the tube is short. The divergence of the atomic
beam is determined by the heating introduced by the scattering of photons during
the slowing process. The amount of heating, indicated by the induced root meansquare transversal velocity, is then calculated by [77]rvrec ZS
rms
MOT
vheat = 3vcapt − vcapt
,
(4.7)
10

4.2. ZEEMAN SLOWER

62

where vrec = h̄k/matom denotes the recoil velocity. The fact that the amount ofheating is dependent on the number of scattering events, means that it does notreally come into effect until the far end of the Zeeman slower. The reason for thisis that at the beginning of the slower the atom’s velocity is still very high, which
results in fewer scattering events per centimeter than at the end, where the atoms
are slow and many photons can scatter. Additionally, a heating induced transversal
velocity does not have a big impact as long the atoms’ longitudinal velocity is highas opposed to slow atoms leaving the Zeeman slower.

Atomic beam

Figure 4.15: Schematic of a cut through spin-flip Zeeman slower exposing thedivergent atomic beam traveling within its inner tube and the water cooling (blue).
The divergence emerges from the scattering of photons at the atoms that are being
decelerated, resulting in a random walk of said atoms. Thereby, the translucent conewith black, dashed margins illustrates this divergent beam of which a proportion is
not able to freely leave the Zeeman slower as the atoms contained in this proportion
hit the inner tube of the Zeeman slower. The black opaque cone with attached arrowillustrates the proportion that is able to leave the Zeeman slower unhindered. A
measure that would allow the entire beam to leave the slower would be to widen theend of the tube which, however, also brings its own potential drawbacks.
If the amount of atoms traveling through the capture volume is limited by theremaining tube that catches the part of the beam that is too divergent, a possible
counter measure is to widen the end of the tube conically like it was done in the
RARE Zeeman slower. This way, the atoms can exit the tube freely and enter the
MOT chamber. However, introducing such a widening is not always advantageous
in terms of the MOT loading rate and can even introduce problems like discussed
above. Whether a tapered widening is useful in the T-REQS experiment or theomission of said widening has an impact on the flux through the capture volume
shall be resolved in the following. For this purpose, a one-dimensional simulationof the atomic flux propagation (see Appendix A for source code) along the Zeeman
slower has been written and conducted. First of all, in our case the flux through thecapture volume is in general limited by different loss factors:
η0 Geometrical constraints: For the simulation every part of the atomic beam
emerging from the oven that does not cross the optical molasses of the transversal cooling is considered lost.
η1 Due to different abundances in a piece of solid Erbium, in general the achievable
flux is dependent on the atomic isotope that is being decelerated. However,63

4.2. ZEEMAN SLOWER
as we seek for a general answer to a geometrical problem which is not isotope
dependent, isotopes are not taken into account for the simulation.
η2 The deceleration efficiency, i.e. the ratio of atoms that can be slowed to the ones
that enter the Zeeman slower. Again, this problem is not geometry dependent,
which is why we assume for the simulation that every atom that enters theslower is fully decelerated.
η3 The divergence of the atomic beam after being slowed down. This is the subjectof this discussion.

Additionally, for the simulation we assume the atoms to have a homogeneous radial distribution when leaving the oven and that the transversal cooling does notchange this distribution. Moreover, as indicated above for the loss factor η0 , weassume a region with a diameter of 6 mm where the transversal cooling takes place.
Furthermore, after the transversal cooling we assume the atoms to exhibit a radial
velocity distribution following a two-dimensional Maxwell-Boltzmann distribution (asopposed to a one-dimensional one which would describe the situation only accurately,
if we would neglect one axis of movement) with a temperature corresponding to
the Doppler temperature of the 401 nm transition (see Fig. 4.16). Fortunately, thesimulated propagation through the apparatus has shown, that the transversal velocity
distribution after the transversal cooling is narrow enough for the atomic beam tonot be constrained by the Zeeman slower’s differential pumping tube. From theend of the last coil on another two-dimensional Maxwell-Boltzmann distribution
ZS
with a calculated heating velocity vheat= 0.81 m s−1 given by equation 4.7 takes
over. The propagation of the atomic beam to the MOT center following this velocity
distribution forms the cone which is the subject of this study. During the propagation
the flux is redistributed radially symmetric around the propagation axis, eventuallyresembling the two-dimensional Maxwell-Boltzmann distribution (see Fig. 4.17) thatis the reason for the redistribution in the first place.
10,000

Atomic flux / s−1

8,000
6,000
4,000
2,000
0
−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Radial velocity / m s−1

Figure 4.16: The radial velocity of the atomic beam after passing the transversal
cooling. It follows a two-dimensional Maxwell-Boltzmann distribution where thetemperature is given by the Doppler temperature of the 401 nm transition used in
the transversal cooling.

4.2. ZEEMAN SLOWER

64

Now to find out whether the tube the atomic beam propagates within constrains
its path, one has to consider the remaining length of the tube, the remaining distancethe atoms have to travel (end of last coil to MOT center) and the diameter of the
MOT capture volume. This means that if the remaining tube after the last coil islong and the the capture volume is large a large proportion of the atomic flux is
cut away and a tapered widening can be advantageous. If, however, the capture
volume is small, the length of the remaining tube is not so important because of
the anyway smaller flux through the capture volume. However, to find a definitive
answer, we conducted simulations for the geometries of the T-REQS the RARE andthe Erbium experiment with and without the consideration of a tapered widening.
For the the T-REQS the considered widening has a radius of 13 mm with a remaining
tube length of 12 mm, which is the maximum length without being constrained by
the geometry of the MOT chamber itself. Due to the absence of a widening in the
Erbium experiment the same cone length of 21 mm as in the RARE experiment has
been considered. Furthermore, in the Erbium experiment the length of the remaining
tube and the distance between the last slower coil and the MOT center is significantly
larger than in the case of the T-REQS and RARE experiment with the capture
volume being additionally smaller.

Atomic flux / s−1

10,000
8,000
6,000
4,000
2,000
0
−6

−4

−2

0

2

Radial position / m

4

6
·10−2

Figure 4.17: The radial position distribution at the MOT center given by the
propagation of the radial position distribution at the end of the Zeeman slower by theradial velocity distribution by a two-dimensional Maxwell-Boltzmann distribution
ZS
where the temperature is given by heating velocity vheat= 0.81 m s−1 resulting fromthe deceleration process.
The results of these simulations are that with a relative performance of 0.706(with widening vs. without widening) a tapered widening would have a positive
impact in the Erbium experiment, which kind of contradicts the statement above,that the length of the remaining inner tube is not overly important with small
capture volumes. However, with a length of 52 mm it is more than twice as long as
the one in the RARE experiment and even four times as long the one in the T-REQS
experiment. Further, with a relative performance of 0.806 the results indicate thatthe tapered widening in the RARE experiment is a reasonable measure to maximize

65

4.2. ZEEMAN SLOWER

the atomic flux, despite introducing drawbacks. The most pleasant result, though, isthat, with a relative performance of 0.983 in the case of the T-REQS, the omission ofsuch a widening has only a very small impact on the atomic flux through the MOT
and thereby on the MOT loading rate.
Table 4.2: Proportions reaching the MOT capture volumes in our three experiments
and the corresponding relative performances considering their Zeeman slowers withand without tapered widening at the end. Due to the long remaining tube after
the atoms are slowed down such a widening would have been a worthwhile additionin the Erbium experiment. The results show that in the RARE experiment thewidening has a great impact too. However, regarding possible disadvantages, atapered widening would not make sense in the case of the T-REQS experiment.

4.2.4

Configuration

T-REQSRARE

ERBIUM

with widening
without widening
rel. performance

0.543
0.533
0.983

0.613
0.761
0.806

0.218
0.309
0.706

Planned building process

At the time of writing this thesis the whole world is in the midst of fighting the
global COVID-2019 pandemic. For this reason it is not possible to include the actual
building process of the Zeeman slower in this chapter. However, the current status
and the plan for the manufacturing shall be presented.
So far the vacuum part with the included water cooling has been made by theengineers of the IQOQI (Institute for quantum optics and quantum information)
workshop. The water cooling has been tested successfully for leakproofness as wellas the whole apparatus down to a minimal pressure of 1 × 10−7 mbar. The pumping
duration in advance of the actual leakage test was two hours for which we covered thewhole part as well as critical areas like the flanges and the water cooling with Heliumgas. Due to the small size of Helium atoms they are able to creep even into thesmallest cracks, which makes Helium perfect for finding leakages in vacuum criticalparts. Because large parts of the vacuum part are covered with coils and thereby
exposed to magnetic fields it is important that it shows only little magnetizability.
The finished part showed magnetizabilities between 1.004(1) and 1.059(1) in regions
far from weld seams while weld seams showed high magnetizabilities starting from1.07(1) and maxing out at 1.33(1). Such high values made it necessary to glow to
the whole part and thereby get rid of these high magnetizabilities.
The winding process of the coil assembly is planned to follow a protocol based on
the one already used for the Zeeman slower in the RARE experiment. Thereby wewould start with the last coil that touches the flange on the MOT side of the Zeeman
slower. After this one we would proceed with the bias/offset coil – the inner most of
the coils of the middle coil assembly – and wind all the other coils around each other4.2. ZEEMAN SLOWER

66(see Fig. 4.18). Because every coil consists of two layers it is possible to start thewinding in the middle of the coil assembly, wind until the end of the underlying coil
is reached and wind the same way back. This way, we need only very little space tolead the wire out and attach it to a current source. At last, the start coil (the sidewith the water cooling connectors) is wound.

Atomic beam

Figure 4.18: Schematic of a cut through Zeeman slower revealing the radial layer
structure of the middle coil assembly. The inner most coil is the bias/offset coilaround which all the other coils are wound. For the outer coils we start in the middle,
wind until we reach one of the ends and wind our way back to where we started.
To stabilize the coils during winding we use two winding aids, specifically designed
for this purpose (see Fig. 4.19), which have already been 3D printed. With a widthof 3 mm, we place them within the gaps of the coil assembly, where they act as
separators and additionally clamp down and fix the inner most wire into position on
the outer tube of the Zeeman slower. To lead the inner most wire out of the assembly
without interfering with other coils, a channel is embedded into the winding aid. To
fix all the wires in place we would use a special glue “Duralco® NM25 Adhesive epoxy”
by Final Advanced Materials. It is it thermally conductive (thermal conductivity:
1.87 W mK−1 ), yet electrically insulating (electrical resistivity: 1015 Ω cm), which isanother measure to increase the cooling performance. Furthermore, it is amagnetic
and specified up to a temperature of 260 ◦C, which means that the whole Zeeman
slower can be baked with the coils already in place.

Figure 4.19: Winding aid necessary for the winding process of the Zeeman slower.
It is clamped onto the outer tube by screws and thereby also fixes the inner mostwire of a coil into place. A channel additionally allows this wire to be fed out of
the assembly. With a width of 3 mm these aids also act as coil separators, necessary
to form the gaps specified in the design while also stabilizing the coils during thewinding process.

Chapter 5
Light for the Zeeman slower
It is crucial for the deceleration of the atoms contained in the atomic beam, thatthe frequency of the used laser is resonant with the Doppler shifted energy of thetransition used for the deceleration process. As we use a Zeeman slower to maintainthe resonance condition between the slowing-laser light and the atomic frequency,
this means that the stability of the laser’s frequency is of utmost importance toprevent the atoms from falling out of resonance. There are several techniques tomaintain a stable frequency, like for example the use of a high finesse cavity [78].
Our approach involves the Doppler-free spectroscopy of the transition utilized forthe slowing process. For this purpose, I built an optical setup in order to perform aso-called modulation-transfer spectroscopy within an Erbium hollow-cathode lamp
and generated an error signal , which we can lock the laser used for the Zeeman
slower to. The details of this approach shall be explained in the following.

5.1

Concept

The idea to use a spectroscopic signal to stabilize a laser for the subsequent use
in other parts of the experiment is not new ([79], [80]) and currently successfully
used in many labs. The laser in question, which shall be stabilized is an 401 nm
frequency-doubled diode laser TA-SHG by Toptica. Apart from the Zeeman slower,
its light will be used for other parts of the experiment, such as the transversal coolingor the imaging of the atoms stored within the MOT, as well. Naturally, all theseapplications require frequency-stabilized laser light.
The basic idea is to use a sample of Erbium and measure the absorption of the lightfield incorporated by the laser. A possible way of doing this could be to guide a laser
through a cell of Erbium gas. While doing this, one would need to sweep the lasers
frequency over a range around the 401 nm mark. The signal intensity after the cell in
dependence of the laser’s frequency could be picked up by a photo diode and madevisible with an oscilloscope. If the intensity measured by the photo diode matches
the initial one of the laser none of the atoms in the Erbium gas are being excited,
thus, the medium is transparent for the corresponding wavelength. If, however, thesignal intensity after the cell is lower, the laser is on resonance at the corresponding
frequency. This method, known as the laser absorption spectroscopy, has one major
disadvantage though - its inaccuracy, which arrises from line broadening mechanisms
presented in the following subsection.
675.1. CONCEPT

5.1.1

68

Spectral lines and line broadening mechanisms

Before talking about line broadening mechanisms, we should first recall the natural
shape of spectral lines and their emergence. One important figure when speaking ofspectral lines is their natural linewidth, i.e. the width of the frequency distribution
around some center frequency νc . This linewidth gives the minimal width of atransition that cannot be undercut and originates from the finite relaxation time ofan excited atom. Thereby, the distribution of its line profile follows a Lorentzianfunction and can be derived by describing the excited electron by means of thedampened classical harmonic oscillator [16] and forming the Fourier transformationof the resulting equation of motion. Due to the amplitude decreasing gradually withtime and the associated decrease of the corresponding frequency, this transformation
gives us a Lorentzian intensity distribution in dependence of the emitted frequency [16]P (ω) = P0

Γ/2π.
(ω − ωc )2 + (Γ/2)2(5.1)

Based on this equation and after performing some basic transformations we thenget a natural linewidth at FWHM of 2 (ω − ω0 )2 = 2∆2 = Γ = 1/τ . Although theidea of the derivation of the natural shape of a spectral line and its corresponding
linewidth is based on the relaxation of an excited atom, everything stated above also
holds true for the reverse process, i.e. the excitation of an atom.
On top of the natural linewidth there are different mechanisms that contributeto a broadening of the natural linewidth, like for example the so-called collisional
broadening and pressure broadening. These are the direct consequence of the atoms
contained in a gaseous sample sensing each other. For the collisional broadening,
imagine two atoms A and B in a gas with atom A having a ground state |gi andexcited state |ei and corresponding energies Eg , Ee . The presence of atom B in
the vicinity leads to a shift of these energy levels in dependence of the distance
between these atoms. In a gas in thermal equilibrium, these distances follow a
distribution around some average value. This, in turn, means that every resonance isshifted by a certain amount which is determined by the type of interaction between
the two atoms in question. Thereby, the frequency shift is positive for a repulsive
interaction and negative for an attractive interaction. Additionally, due to thestatistical distribution of the atom distances and the associated absorption/emission
frequencies, a temperature and pressure dependent broadening of the spectral lines
emerges. In this case, the energy for the broadening as well as for the frequency
shift is provided completely by the elastic collision of the atoms A and B. Apartfrom elastic collisional, inelastic collisions can happen as well. In this case – thecase of pressure broadening – an atom A in the excited state |ei can be relaxed by
the inelastic collision with an atom B, which reduces the overall occupation numberof the excited state |ei in the medium. This leads to an effective reduction of the
lifetime of the excited state and thereby to an increased linewidth – in other words
to a broadening of the spectral line. Both of the above explained mechanisms follow
a Lorentzian shape.
The most important broadening mechanism for us and this part of the experiment,
however, is the Doppler broadening that originates from the thermal velocity of the69

5.1. CONCEPT

atoms. The importance of this mechanism for us arises from the fact that we need
to be able to select the right laser wavelength for each Erbium isotope, where eachisotope’s resonance frequency is shifted. Due to the Doppler broadening, however,
all the resonances overlap, which makes it impossible to select one of them.
The origin of this effect is the velocity distribution of the atoms within a gas, which,in thermal equilibrium, follows a Maxwell-Boltzmann distribution, where its mostprobable velocity is given by [16]
r
2kB T
vpr =
.
(5.2)
matom
This means for a laser beam, that for every atom moving along its direction of
propagation its resonance is either blue or red detuned. For a laser of which thefrequency is swept across a certain range, containing the resonances one is interestedin, this leads to multiple velocity classes and multiple transitions being resonant at
once. The number of atoms contained in a certain velocity class [vz , dvz ] along thez-direction corresponding to a frequency range [ω, dω] and available for excitation
can be obtained by [16]2c)
N · c − c(ω−ωωc vpr√n(ω)dω =edω,
vpr ωc π





(5.3)

where N denotes the overall number of atoms available for excitation. The resulting
intensity distribution for absorption/emission is then proportional to equation 5.3,
yielding [16]−I(ω) = I(ωc )ec(ω−ωc )ωc vpr2(5.4)

with a peak intensity I(ωc ) at the center of thep distribution.
The FWHM Doppler linewidth δωD = (ωc /c) (8kB T ln 2) /matom [16] of this gaussian distribution yields then, for example, for Er170 a width of δωD ≈ 711 MHz atroom temperature (T = 300 K). As this is almost as much as the isotope shifts
presented in subsection 2.1.1, the importance of a technique to circumvent Doppler
broadening is apparent. The technique that we use shall be explained in the following
subsection.

5.1.2

Modulation transfer spectroscopy

In general, single beam spectroscopy methods, like the one briefly discussed at the
beginning of this chapter, suffer from serious drawbacks regarding the resolvable
linewidths of spectral lines. As shown in the preceding subsection, the Doppler
linewidth can be almost as large as, for instance, the isotope shifts of Erbium
isotopes, which would prevent us from obtaining a clear signal for laser locking.
To resolve this issue, we use a spectroscopic technique called modulation-transfer
spectroscopy [81], which uses two counter propagating laser beams to achieve subDoppler resolution (see Fig. 5.1). The lower intensity beam, called probe beam,
ultimately carries the spectroscopic information that is picked up by some sort ofdetector like a photo diode. The other one, called pump beam, exhibits a higher
intensity and is guided through an electro-optic modulator (EOM) before being
superimposed with the probe beam within a cell that contains the atom species the
spectroscopy is conducted upon. This EOM is fed with an alternating voltage of5.2. BUILDING AND CHARACTERIZATION

70

some frequency ωm that, by means of phase modulation, imprints sidebands of thisexact spacing from the carrier frequency ωc onto the pump beam. Here, the carrier
frequency is just the normal unmodulated frequency of the laser before entering theEOM. The modulated light can then be expressed by its time-varying electric fieldin terms of Bessel functions [82]
!
∞
∞
X
X
E(t) = E0
Jn (δ) sin(ωc + nωm )t +(−1)n Jn (δ) sin(ωc − nωm )t . (5.5)
n=0

n=1

E0 denotes the maximal amplitude of the electric field and Jn (δ) the Bessel functionof order n in dependence of the modulation index δ. Although, the two terms inequation 5.5 are infinite sums the whole expression can be simplified, as normallyδ < 1. In this case only Bessel functions of the zeroth and first order contribute,yielding
E(t) = E0 (J0 (δ) sin(ωc )t + J1 (δ)t (sin(ωc + ωm ) − sin(ωc − ωm ))) .

(5.6)

This means, that the pump beam exhibits in good approximation just the carrier
frequency and two sidebands. Within a non-linear medium (in our case Erbium
vapor) sidebands are imprinted onto the probe beam as well. This happens via
four-wave mixing [83], where two bands of the pump beam create one sideband in theprobe beam, resulting in two additional sidebands. The intensity signal eventually
resulting from sweeping the lasers frequency over a continuous range of frequencies ispicked up by a photodiode that is attached to a mixer. There the intensity signal is
combined with the signal from the frequency generator that is also driving the EOM.
The resulting error signal can then be used to gain knowledge about the resonancesof electronic transitions or, in our case, be fed back to the laser system to lock it toa specific frequency.

5.2

Building and characterization

The laser light for the spectroscopy is provided by a 401 nm frequency doubled diodelaser TA-SHG from Toptica. The light is guided through a λ/2-waveplate WP1 andinto a polarizing beam splitter cube PBS1 (see Fig. 5.1). By means of the waveplate
the optical intensity of either of the two beams exiting the polarizing beam splittercan be controlled. Thereby in the sketch, the horizontal one is the probe beam andthe vertical one is the pump beam with corresponding intensities of 1.23 mW and7.45 mW. After leaving the first beam splitter, the pump beam crosses an additional
λ/2-waveplate WP2 responsible to turn the polarization by 90° which is necessary
for properly entering the EOM later down the path. After this waveplate a telescope,consisting of two lenses L1 and L2 with corresponding focal lengths of 100 mm and−50 mm, reduces the diameter of the pump beam in order two fit through the crystal
in the EOM. A mirror M1 after the telescope deflects the pump beam in a way thatit propagates parallel to the probe beam and eventually reaches the EOM.
The EOM consists of a crystal with bevelled entrance and exit surfaces, which makes
it necessary to twist the crystal (and consequently the whole EOM) so that theincident angle of the probe beam equals the Brewster’s angle. This condition is also
the reason for the use of waveplate WP2 because, in this case, for the light to enter

71

5.2. BUILDING AND CHARACTERIZATION

the EOM it has to be polarized parallel to the plane of incidence – otherwise all or atleast part of the light would be reflected [84]. The used EOM exhibits a resonance at
28.7 MHz, however, it is operated at 26.35 MHz coming from a frequency generator.
After being modulated within the EOM, the pump beam is deflected at another mirror
M2 and guided through a third λ/2-waveplate WP3, which turns the polarization
back, in order to match the one of the probe beam. A second polarizing beam splittercube superimposes the now modulated pump beam with the unmodulated probe
beam within an Erbium hollow-cathode lamp HL (Model: 3QQAYEr by Heraeus
Noblelight). This lamp consists of a glass cell in the center of which a cylindricalcathode is situated that additionally acts as the Erbium source. This cathode features
a hole along its rotational axis and is accompanied by an anode on either side. By
applying a voltage to these electrodes an electric field is generated, which results in aplasma discharge of the Argon gas the glass cell is filled with. The cations generated
in that process are accelerated to the cathode and – if their kinetic energies allow it– knock neutral Erbium atoms out of the cathode that can be used for spectroscopy.
Thereby, although the lamp is rated for a maximum current of 15 mA it is operated
at a current of 7.4 mA and a voltage of 109.5 V.
Mixer
Error signal
to laser
Frequency generator

PD

EO

M2

M

M1

L3

L2
L1

WP3
WP2
from laser
M3

PBS2

HL

PBS1 WP1

Figure 5.1: The setup of the modulation-transfer spectroscopy used to lock a 401 nm
laser to a desired frequency/wavelength. The laser is divided into a high intensity
pump (dark blue) and a low intensity probe beam (light blue). The pump beam’s
diameter is reduced in order to fit through the EOM were its frequency is modulated.
By superimposing the two beams within an Erbium hollow-cathode lamp an intensity
modulation of the probe beam is achieved, which is picked up by a fast photo diode.
By means of a mixer and the intensity signal in conjunction with the frequency
signal that also drives the EOM, an error signal is obtained to which the laser canbe locked.
The probe beam of which the intensity is modulated during the spectroscopy is
passed through the polarizing beam splitter PBS2 and is deflected by a third mirror

5.2. BUILDING AND CHARACTERIZATION

72

M3. A third lens L3 with a focal length of 50 mm eventually focuses the probe beam
onto a fast photo diode which picks up the signal and sends it to a home-built unit
containing a mixer and the aforementioned frequency generator. There, the photo
diode signal and the signal driving the EOM are mixed, generating an error signal in
the process. Ultimately, this error signal is fed back to the laser, where it is used tolock it to the desired frequency/wavelength.
The whole setup is built onto 30 cm×45 cm breadboard which allows for convenient
positioning on the experimental table. Furthermore, all components are positioned
inside a home-built aluminium housing that sits flush with the outer edges of thebreadboard. It protects the setup from dust and prevents stray light from entering
the setup and interfering with the measurement. Additionally, it also prevents laser
light from leaving the setup and consequently the experimental table and doing
harm to people present in the laboratory. Thereby, the in- and outlets for cables are
obvious points, where dust and light can easily pass, which is why flexible rubber
curtains have been manufactured and attached to these areas.

5.2.1

Transmission signals

Before discussing the error signal let us first have a look on the single beam signalthat we obtain by irradiating the hollow-cathode lamp from only one side. Despite
the signal being Doppler broadened, however, we can guess where the Gaussian
shaped spectral lines are. It is safe to assume that the only Erbium isotopes thatcan be recovered are the three most abundant bosonic ones – Er166 , Er168 and Er170 ,which is why we, to get a somewhat exact picture, fit a triple Gaussian function to
the data. These three isotopes should be positioned equidistant in terms of theirresonant frequencies, due to which we can eliminate a degree of freedom during thefitting process and also fix Er166 at 0 GHz.Strictly speaking, the Doppler linewidth√
should be different for every spectral line. However, it scales with 1/ matom , whichleads to deviations of only about one percent. For this reason, we assume the same
Doppler linewidth for all three spectral lines in order to eliminate another degree of
freedom. The resulting formula for fitting then has the form [16]V (x) = a − b1 · e


−

c/

√x

4 ln(2)2

− b2 · e

−

√x−d

c/

4 ln(2)

2

− b3 · e

−

x−2d
√

c/

4 ln(2)

2(5.7)

with a accounting for an offset (in our case the base intensity of the signal), b1 , b2 ,b3 denoting the amplitude, c the Doppler linewidth and d the displacements of thecenter positions of the spectral lines. The obtained value of d = 0.840 GHz gives
estimations for frequency offsets of δν168 = 0.840 GHz and δν170 = 1.680 GHz for theisotopes Er168 and Er170 .
Despite having acquired these isotope shifts from a Doppler broadened spectrum,
they are somewhat consistent with the ones (δν168 = 0.81(1) GHz and δν170 =
1.66(2) GHz) reported in [85]. Additionally, another estimation can be undertaken inwhich we use the amplitudes of the fitted Gaussian functions to find the abundancesof the isotopes contained in the Erbium gas present in the hollow-cathode lamp.
Thereby, by forming ratios of the obtained amplitudes and correcting the result
by factoring in the abundance of the fermionic Er167 [31] that cannot be recovered

73

5.2. BUILDING AND CHARACTERIZATION

from the spectrum, we end up with abundances of 0.345, and 0.296 and 0.1305 for
Er166 , Er168 and Er170 respectively. While these numbers do not align perfectly withthe ones (abundances: 0.335 03(36), 0.269 78(18) and 0.149 10(36) for Er166 , Er168
and Er170 respectively) presented before in table 2.1 it is still surprising how muchinformation can be pulled from a broadened spectrum like the presented one.

Signal strength / V

0.6

0.5
Measured data
Triple-Gaussian
Er166
Er168
Er170

0.4

0.3
−3

−2

−1

0

1

2

3

4

5

Detuning from Er166 resonance / GHz

Figure 5.2: Measured single beam transmission signal (black) with fitted triple
gaussian (orange) exposing the underlying Doppler broadened spectral lines. Thereby
the frequency positions and abundances of the three most abundant bosonic isotopesof Erbium – Er166 (blue), Er168 (red) and Er170 (green) – can be estimated.
If we know deploy the pump beam with the EOM still turned off, we see thatthe overall signal becomes weaker. However, the important feature in Fig. 5.3 isthe emergence of so-called Lamb dips (the three sudden spikes in the signal) at
the positions of the actual resonances. While their linewidths are approximately
two orders of magnitude smaller than the corresponding Doppler linewidth [16], it
still does not allow the Doppler free spectroscopy that we need. However, theiremergence signifies a well aligned setup that allows us to turn on the EOM andapply a frequency modulation to the pump beam. It is not entirely clear, though,why in this case the Lamb dips are rather weak, but a probable reason is that thethree most pronounced dips are surrounded by weaker resonances from the hyperfine
levels of the fermionic isotope Er167 and are in that way slightly obstructed.

5.2. BUILDING AND CHARACTERIZATION

74

Signal strength / V

0.8
0.6
0.4
0.2
0
-0.08

Single beam
Two beams
-0.04

0

0.04
0.08
Time / s

0.12

0.16

Figure 5.3: Comparison of the Doppler broadened signal obtained with a single
beam spectroscopy setup (blue) and the signal obtained with an unmodulated pumpbeam superimposed with the probe beam (black). The emerging Lamb dips (thethree sudden spikes in the signal) in the latter exhibit narrower linewidths thanthe Doppler broadened spectral lines. However, they still do not allow Doppler free
spectroscopy. This figure is only used for illustrative purposes, hence, the values on
the x-axis were not converted into frequencies.

5.2.2

Error signal and locking sensitivity

The resulting error signal (see Fig. 5.5; 16 averages) has been optimized for maximum
steepness and thereby manages to resolve the four bosonic isotopes Er164 , Er166 ,Er168 and Er170 as well as the hyperfine structure of the fermionic isotope Er167 (seealso [36]). This opens the possibility to lock the laser not only to the resonances ofthe bosonic isotopes, but locking to some of the hyperfine levels of the fermionicisotope, visible in-between the four pronounced resonances in Fig. 5.5, might bepossible as well. A simple linear fit of the form V (x) = a + b · x, where a denotes
an offset and b the slope of the curve, reveals the resonances and slopes of the error
signal at V (x) = 0. When locking a laser, these slopes give the response speed, orin the other words the sensitivity, when the lasers frequency deviates from the set
lock point. The resonance of the Er166 isotope has been fixed at 0 GHz, while theother bosonic isotopes are located at detunings of −1.68(3) GHz, −0.839(8) GHz and0.86(4) GHz for Er170 , Er168 and Er164 , respectively (for the expected values see also
Fig. 5.4), where the margins of error correspond to the fitting errors.

5.2. BUILDING AND CHARACTERIZATION

Isotope mass / au

75

170
168
166
164
−2

−1.5

−1

−0.5

Detuning from Er

166

0

0.5

1

resonance / GHz

Figure 5.4: Expected isotope shifts of the four most abundant bosonic isotopes of
Erbium – Er170 , Er168 , Er166 and Er164 – for the broad resonance at 401 nm. Here,the resonance of Er166 is fixed at 0 GHz. Isotope shift values from [85].
Considering the margins of error, these values are in good coincidence with theones stated in [85]. The accompanying values for the resonance slopes for Er170 ,Er168 , Er166 and Er164 read 10.60(13) V GHz−1 , 13.72(9) V GHz−1 , 15.96(8) V GHz−1
and 1.28(5) V GHz−1 , which is sufficient for reliably locking the laser to the desired
frequency/wavelength [86]. It has to be noted that the overall offset of the error
signal (≈ 0.01 V) is very likely caused by a floating ground during the measurement.
This offset has been considered for the obtained values above.
Meas. data
Er170
Er168
Er166
Er164
Centerline

Signal strength / V

0.4

0.2

0

−0.2
−2

−1.5

−1

−0.5

166

Detuning from Er

0

0.5

1

resonance / GHz

Figure 5.5: Error signal obtained (16 averages) by means of modulation transfer
spectroscopy. The clear signal shows the four most abundant bosonic isotopes of
Erbium – Er170 , Er168 , Er166 and Er164 – as well as the resonances emerging from thehyperfine structure in the fermionic Er167 [36]. Situated in-between the resonancesof the bosonic isotopes, some of these hyperfine resonances could be used for laser
locking as well. The resonance positions and slopes have been determined by fittingof a linear function to the corresponding slope (blue, red, violett, orange). The
small of offset (cyan) of ≈ 0.01 V is very likely caused by a floating ground during
measurement and has been considered during the fitting process.

Chapter 6
Conclusion and outlook
The aim of this thesis was to produce a design for a spin-flip Zeeman slower for our
new experiment called T-REQS that is under construction at the time of writing.
Thereby, the focus lied on the maximization of the loading rate of the open-top
narrow-line magneto-optical trap (MOT), that we will use in the experiment. Thisis a natural requirement for any experiment involving the use of a MOT as a higher
MOT loading rate also means shorter experimental cycle times and, in consequence,a higher experimental cycle rate. However, in the RARE experiment, one of our othertwo experiments, a MOT chamber that is highly optimized for fast MOT loading
rates is used, which added additionally to the desire for fast loading rates in the newexperiment. Apart from this, other aspects like the divergence of the atomic beam
resulting from the slowing process within the Zeeman slower and the demand for a
sufficient water cooling have been factored in to the design. Additionally, I built an
optical setup used to conduct modulation-transfer spectroscopy on a gaseous Erbium
sample and thereby produce an error signal that is subsequently used to lock thelaser needed for the Zeeman slower.
The long-term goal of the new T-REQS experiment is to establish a novel platform
for quantum simulations using single Erbium-Rydberg atoms confined in opticaltweezers. Therefore, we started by summarizing the discovery of Rydberg atoms
and Rydberg physics by Johann J. Balmer and Johannes R. Rydberg. Balmer
came up with the first formula describing the spectral lines in hydrogen, while
Rydberg presented a more versatile equation that was not restricted to just hydrogen.
Although both equations required the use of integer factors, neither Balmer nor
Rydberg realized that these numbers represented an important notion in quantum
physics, called the principal quantum number n. After all quantum physics were notborn yet.
We proceeded with introducing some extreme properties of Rydberg atoms like,for example, their size (radius) which increases quadratic with n, their high sensitivity
for electric fields as well as the suppressed relaxation back to the ground state. The
strong interactions among Rydberg atoms, which are governed by dipole-dipole andVan der Waals interactions cause a phenomenon called Rydberg blockade where theprobability of Rydberg excitation of an atom is dependent on the distance to anotheratom that is already in the same Rydberg state. Therefore, this mechanism could
become an ingredient for quantum computing [87], [88]. Additionally, we presented
excitation schemes, necessary to achieve Rydberg states.
76

77
In the next chapter, Erbium, the atomic species we have chosen for our experiment,
was the star of the show. General properties regarding its stable isotopes, its
temperature dependent vapor pressure as well as other use cases outside of ultracold
atoms have been covered. We discussed some of its electronic transitions in detail
as well as its ground state electronic configuration and the corresponding magnetic
properties, including its high magnetic moment of µ = 7 µB . Eventually, we shedlight on Erbium as a Rydberg atom which, as a multi-valence electron atom, needs
to be described in terms of a multi channel defect theory. Additionally, it offers
multiple excitation paths which is also one of the reasons for choosing Erbium forthe experiment in the first place.
We discussed atoms in external fields (electric, magnetic and electro-magnetic),
the working principle of an optical molasses and magneto-optical trapping. The
Zeeman effect covered therein forms the foundation for the operation of a magnetooptical trap as well as for the Zeeman slower which was presented in this thesis.
Further, as they form a key ingredient for our experiment, the working principle of
optical tweezers was discussed as well as the polarizability of Erbium. Due to its
multiple valence electrons it makes for a promising candidate for optical tweezing
even while being in a Rydberg state.
Finally, we explained the parts of the vacuum apparatus that are most important
for the design of the Zeeman slower, including the oven that includes an effusion
cell capable of producing a directional atomic beam that is collimated within atransversal cooling section, decelerated along the Zeeman slower and captured aswell as cooled in our open-top narrow-line MOT. We stated the difficulties I had to
overcome during the design process in terms of achieving a high MOT loading rate
due to the, in comparison with the MOT chamber used in the RARE experiment,
unfavorable dimensions of the new commercially available MOT chamber. There,
the large chamber diameter as well as the smaller diameter of the lateral viewports
resulting in a smaller capture volume of the MOT, make the divergence of the atomic
beam after being slowed down more detrimental. This thesis presents an optimized
design of the Zeeman slower that features a maximal velocity of 370 m s−1 , which inpart overrides the above stated limitations and increases the expected MOT loading
rate by about 20 % in comparison to the unoptimized Zeeman slower. Additionally,a simple fluid dynamical calculation that attests unchanged or maybe even increased
cooling performance of the built-in water cooling is presented. Finally, this thesis
additionally reported on the necessity of a tapered widening of the tube, in which
the atomic beam propagates, by writing a simple one-dimensional simulation which
propagates the lateral extent of the atomic beam throughout the apparatus. The
result is that the new design no longer needs such a widening, which insures easier
manufacturing of the vacuum part of the Zeeman slower as well as unimpeded
operation due to the lack of welding seams inside the part. These welding seems
could have otherwise been areas of facilitated magnetization of the material which
in turn can interfere with the magnetic field generated by the coils of the Zeeman
slower. Additionally, this thesis gives a plan for the winding process of the Zeeman
slower.
In the last part of this thesis a method and an optical setup to lock the laser,that is necessary for the operation of the Zeeman slower, is presented. Thereby,
the technique of modulation transfer spectroscopy was introduced which is used torecord a Doppler-free spectrum of the spectral lines around 401 nm – the wavelength

78of the broad transition we use for the Zeeman slower. The atomic sample used
for the spectroscopy was generated within an Erbium hollow-cathode lamp thatis irradiated from two directions with a probe and a pump beam. The former ofthese two eventually contains the spectroscopic data needed to generate an error
signal to which the laser can be locked reliably. Following this method this thesis
demonstrates a clear signal, containing the resonances of the four most abundant
stable bosonic isotopes of Erbium – Er164 , Er166 , Er168 and Er170 – as well as thehyperfine structure of the fermionic isotope Er167 .
In the future the coils of the Zeeman slower have to be wound following the protocolpresented in subsection 4.2.4. However, as the Zeeman slower is only a small part of
the whole setup, which is currently in construction, many other parts have to be built
as well. For example a setup to distribute 401 nm laser light for the spectroscopy
and the Zeeman slower covered in this thesis, as well as other parts of the experiment
like the transversal cooling or the imaging of the MOT. On the physics side and once
the apparatus is complete, intense spectroscopic measurements have to be conductedin order to find intermediate states for the two-photon excitation schemes explainedin subsection 2.3.2. Additionally, as long-lived Rydberg states are favorable forexperiments due to their potentially longer interaction time, the discovery of suchstates would also be a step closer to a working quantum simulator.

Appendix A
Matlab code
I have to thank Manfred Mark at this point for the help, he provided me with during
writing parts of the code presented below.

A.1

Beam propagation% This script calculates the atomic beam propagation% of Erbium along the Zeeman slower
clear all ;close all ;h = 6.62606896 e -34;
kB = 1.3806503 e -23;
h_bar = h /(2* pi ) ;amu = 1.66053886 e -27;
m_Er = 168* amu ;l a m b d a _ z e e m a n_ E r = 401 e -9;
k_zeeman_Er = 2* pi / l a m b d a _ z e e m a n _ E r ;v_recoil_Er = h_bar * k_zeeman_Er / m_Er ;d e l t a _ n u _ z e e m a n _ E r = 29.7 e6 ;p r o p a g a t i o n _ d i s t a n c e _ Z S _ s t a r t = 0.492;
p r o p a g a t i o n _ d i s t a n c e _ Z S _ e n d = 0.870;
v_capt_ZS = 370;
d o p p l e r _ t e m p _ E r _ 4 0 1 = ( h_bar * 2 * pi * d e l t a _ n u _ z e e m a n _ E r ) / (2 * kB ) ;d o p p l e r _ v e l o ci t y = sqrt (( kB * d o p p l e r _ t e m p _ E r _ 4 0 1 ) /(167 * amu ) )N = 1000000;
tic
rpos = ( rand (N ,1) *6 -3) *1 e -3; % rpos in m / random distribution of atoms right after
transversal coolingvtrans = boltzdist (N , d o p p l e r _ v e l o c i t y ) ; % double sided Boltzmann distribution oftransversal velocity right after transversal coolingalpha = atan ( vtrans ./ v_capt_ZS ) ;and transversal velocity% trajectory angle given by allowed longitudinal

dartboard = zeros (3 ,2001) ;dartboard_x = linspace ( -4 ,4 ,2001) ;dartboard (1 ,:) = dartboard_x ;d a r t b o a r d _ c o u n t e r = 1;

79

A.1. BEAM PROPAGATION80

p o s i t i o n s _ e n d _ Z S = ( rpos + tan ( alpha ) .* p r o p a g a t i o n _ d i s t a n c e _ Z S _ e n d ) ;for i = -4e -3:0.0040 e -3:3.996 e -3
treffer = (( rpos + tan ( alpha ) .* p r o p a g a t i o n _ d i s t a n c e _ Z S _ e n d ) > i ) & (( rpos +tan ( alpha ) .* p r o p a g a t i o n _ d i s t a n c e _ Z S _ e n d ) < ( i + 0.0040 e -3) ) ;disp ( sum ( treffer ) ) ;dartboard (2 , d a r t b o a r d _ c o u n t e r ) = sum ( treffer ) ;da r t b o a r d _ c o u n t e r = d a r t b o a r d _ c o u n t e r + 1;
end
d a r t b o a r d _ c o u n t e r = 1;
for i = -4e -3:0.0040 e -3:3.996 e -3
treffer = (( rpos + tan ( alpha ) .* p r o p a g a t i o n _ d i s t a n c e _ Z S _ s t a r t ) > i ) & (( rpos +tan ( alpha ) .* p r o p a g a t i o n _ d i s t a n c e _ Z S _ s t a r t ) < ( i + 0.0040 e -3) ) ;disp ( sum ( treffer ) ) ;dartboard (3 , d a r t b o a r d _ c o u n t e r ) = sum ( treffer ) ;d a r t b o a r d _ c o u n t e r = d a r t b o a r d _ c o u n t e r + 1;
end
save ( '~ p l a c e _ h o l d e r _ p a t h \ dartboard . mat ' , ' dartboard ') ;save ( '~ p l a c e _ h o l d e r _ p a t h \ p o s i t i o n s _ e n d _Z S . mat ' , ' p o s i t i o n s _ e n d _ Z S ') ;dlmwrite ("~ p l a c e _ h o l d e r _ p a t h \ rpos_ after_t cc . txt " , rpos ,"\ t ") ;figure (1)plot ( dartboard (1 ,:) , dartboard (2 ,:) , 'b ') ;hold onplot ( dartboard (1 ,:) , dartboard (3 ,:) , 'r ') ;xlabel ( ' Radial distance / mm ') ;ylabel ( ' Atomic flux ') ;title ( ' Atom distribution in ZS ') ;legend ( ' End of ZS ' , ' Start of ZS ') ;figure (2)histogram ( rpos )xlabel ( ' Radial distance / m ') ;ylabel ( ' Number of atoms ') ;title ( ' Radial atom distribution at end of TCC ') ;toc

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Function definition :function out = boltzdist (n , v_heating )out = zeros (n ,1) ;vdist = [ -2.5:0.001:2.5];
dist = abs ((( vdist .*2) /( v_heating ^2) ) .* exp ( -( vdist .^2) /( v_heating ^2) ) ) ;dist = dist / sum ( dist ) ;distcum = cumsum ( dist ) ;for ind = 1: nx = rand ;out ( ind ) = vdist ( find ( distcum > x ,1) ) ;end
end% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

81

A.2

A.2. IMPACT OF BEAM DIVERGENCE

Impact of beam divergence% This script calculates the impact of a tapered widening% at the end of the ZS tube
clear all ;close all ;t e m p _ p o s i t i o n s _ e n d _ Z S = load ( '~ p l a c e _h o l d e r _ p a t h \ p o s i t i o n s _ e n d_ Z S . mat ') ;h = 6.62606896 e -34;
h_bar = h /(2* pi ) ;amu = 1.66053886 e -27;
m_Er = 168* amu ;l a m b d a _ z e e m a n_ E r = 401 e -9;
k_zeeman_Er = 2* pi / l a m b d a _ z e e m a n _ E r ;v_recoil_Er = h_bar * k_zeeman_Er / m_Er ;c a p t u r e r a d i u s _ t r e q s = 0.0145;
c a p t u r e r a d i u s _ r a r e = 0.038 / 2;c a p t u r e r a d i u s _ e r b i u m = 0.025 / 2;d i s t a n c e _ t r e q s _ Z S = 0.10135;
d i s t a n c e _ r a r e _ t r e q s _ Z S = 0.08806;
d i s t a n c e _ r a r e _ r a r e _ Z S = 0.09786;d is ta nc e _e rb i um = 0.12816;
b l o c k i n g r a d i u s _ c o n u s _ t r e q s = 0.005;
b l o c k i n g r a d i u s _ c o n u s _ r a r e = 0.01195;b l o c k i n g r a d i u s _ s t r a i g h t = 0.004;
l e n g t h _ b l o c k i n g _ t r e q s = 0.01153;
l e n g t h _ b l o c k i n g _ r a r e = 0.02133;
l e n g t h _ b l o c k i n g _ e r b i u m = 0.05163;
v_capt_mot = 5;
v_capt_ZS = 370;
v_heating = sqrt (3/10* v_recoil_Er *( v_capt_ZS - v_capt_mot ) )N = 1000000;
tic
rpos = t e m p _ p o s i t i o n s _ e n d _ Z S . p o s i t i o n s _e n d _ Z S ; % r pos in m / calculated
distribution by propagation from transversal coolingvtrans = boltzdist (N , v_heating ) ;alpha = atan ( vtrans ./ v_capt_mot ) ; % trajectory angle given by allowed longitudinal
and transversal velocity
pos_end_mot = rpos + tan ( alpha ) .* d i s t a n c e _ t r e q s _ Z S ;treffer0 = (( rpos + tan ( alpha ) .* d i s t a n ce _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ t r e q s ) & (( rpos +tan ( alpha ) .* d i s t a n c e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ t r e q s ) ;treffer1 = (( rpos + tan ( alpha ) .* d i s t a n ce _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ t r e q s ) & (( rpos +sin ( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) < b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + sin (alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) > - b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + tan ( alpha ).* d i s t a n c e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ t r e q s ) ;treffer2 = (( rpos + tan ( alpha ) .* d i s t a n ce _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ t r e q s ) & (( rpos +sin ( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) < b l o c k i n g r a d i u s _ c o n u s _ t r e q s ) & (( rpos + sin (alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) > - b l o c k i n g r a d i u s _ c o n u s _ t r e q s ) & (( rpos + tan (alpha ) .* d i s t a n c e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ t r e q s ) ;fprintf ( '\ n ')message = ' Ratios for T - REQS experiment : ';

A.2. IMPACT OF BEAM DIVERGENCE

82

disp ( message ) ;abs_capture = sum ( treffer0 ) / N ; % ratio of captured atoms to total number of atoms (without constraint )message_abs = [ ' Proportion of atoms through capture volume without constraints : ' ,num2str ( abs_capture ) ];disp ( message_abs ) ;s t r a ig h t _ c a p t u r e = sum ( treffer1 ) / N ; % ratio of captured atoms to total number ofatoms ( tube at end - no cone )m e s s ag e _ s t r a i g h t = [ ' Proportion of atoms through capture volume without cone :' , num2str ( s tr a i g h t _ c a p t u r e ) ];disp ( m e s s a g e _ s t r a i g h t ) ;conus_capture = sum ( treffer2 ) / N ; % ratio of captured atoms to total number of atoms( cone at end )message_cone = [ ' Proportion of atoms through capture volume with cone :',num2str ( conus_capture ) ];disp ( message_cone ) ;rel_change = sum ( treffer1 ) / sum ( treffer2 ) ; % relative difference of captured atoms tube and conem e s s a g e _ r e l _ c h a n g e = [ ' Relative performance of setup with cone and without cone :' , num2str ( rel_change ) ];disp ( m e s s a g e _ r e l _ c h a n g e ) ;fprintf ( '\ n ')treffer0 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & ((rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ r a r e ) ;treffer1 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & ((rpos + sin ( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) < b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + sin( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) > - b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + tan ( alpha ).* d i s t a n c e _ r a r e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ r a r e ) ;treffer2 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ t r e q s _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & ((rpos + sin ( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) < b l o c k i n g r a d i u s _ c o n u s _ t r e q s ) & (( rpos +sin ( alpha ) .* l e n g t h _ b l o c k i n g _ t r e q s ) > - b l o c k i n g r a d i u s _ c o n u s _ t r e q s ) & (( rpos + tan (alpha ) .* d i s t a n c e _ r a r e _ t r e q s _ Z S ) > - c a p t u r e r a d i u s _ r a r e ) ;treffer3 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ r a r e _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & (( rpos
+ tan ( alpha ) .* d i s t a n c e _ r a r e _ r a r e _ Z S ) > - c a p t u r e r a d i u s _ r a r e ) ;treffer4 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ r a r e _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & (( rpos
+ sin ( alpha ) .* l e n g t h _ b l o c k i n g _ r a r e ) < b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + sin (alpha ) .* l e n g t h _ b l o c k i n g _ r a r e ) > - b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + tan ( alpha ) .*distance_rare_rare_ZS ) > - captureradius_rare );treffer5 = (( rpos + tan ( alpha ) .* d i s t a n c e _ r a r e _ r a r e _ Z S ) < c a p t u r e r a d i u s _ r a r e ) & (( rpos
+ sin ( alpha ) .* l e n g t h _ b l o c k i n g _ r a r e ) < b l o c k i n g r a d i u s _ c o n u s _ r a r e ) & (( rpos + sin (alpha ) .* l e n g t h _ b l o c k i n g _ r a r e ) > - b l o c k i n g r a d i u s _ c o n u s _ r a r e ) & (( rpos + tan ( alpha ).* d i s t a n c e _ r a r e _ r a r e _ Z S ) > - c a p t u r e r a d i u s _ r a r e ) ;fprintf ( '\ n ')message = ' Ratios for RARE experiment : ';disp ( message ) ;abs_capture = sum ( treffer0 ) / N ; % ratio of captured atoms to total number of atoms (without constraint )message_abs = [ ' Proportion of atoms through capture volume without constraints (T REQS ZS ) : ' , num2str ( abs_capture ) ];disp ( message_abs ) ;s t r a ig h t _ c a p t u r e = sum ( treffer1 ) / N ; % ratio of captured atoms to total number ofatoms ( tube at end - no cone )m e s s ag e _ s t r a i g h t = [ ' Proportion of atoms through capture volume without cone (T REQS ZS ) :' , num2str ( s t r ai g h t _ c a p t u r e ) ];disp ( m e s s a g e _ s t r a i g h t ) ;conus_capture = sum ( treffer2 ) / N ; % ratio of captured atoms to total number of atoms( cone at end )message_cone = [ ' Proportion of atoms through capture volume with cone (T - REQS ZS ) :' , num2str ( conus_capture ) ];disp ( message_cone ) ;rel_change = sum ( treffer1 ) / sum ( treffer2 ) ; % relative difference of captured atoms -83

A.2. IMPACT OF BEAM DIVERGENCE

tube and conem e s s a g e _ r e l _ c h a n g e = [ ' Relative performance of setup with cone and without cone (T REQS ZS ) :' , num2str ( rel_change ) ];disp ( m e s s a g e _ r e l _ c h a n g e ) ;fprintf ( '\ n ')abs_capture_2 = sum ( treffer3 ) / N ; % ratio of captured atoms to total number of atoms( without constraint )message_abs = [ ' Proportion of atoms through capture volume without constraints (RARE ZS ) :' , num2str ( abs_capture_2 ) ];disp ( message_abs ) ;s t r a i g h t _ c a p t u r e _ 2 = sum ( treffer4 ) / N ; % ratio of captured atoms to total number ofatoms ( tube at end - no cone )m e s s a g e _ s t r a i g h t _ 2 = [ ' Proportion of atoms through capture volume without cone (RARE ZS ) :' , num2str ( s t ra i g h t _ c a p t u r e _ 2 ) ];disp ( m e s s a g e _ s t r a i g h t _ 2 ) ;c on us _c a pt ur e _2 = sum ( treffer5 ) / N ; % ratio of captured atoms to total number ofatoms ( cone at end )mess age_cone _2 = [ ' Proportion of atoms through capture volume with cone ( RARE ZS ) :' , num2str ( c on u s_ ca pt u re _2 ) ];disp ( mes sage_co ne_2 ) ;rel_change_2 = sum ( treffer4 ) / sum ( treffer5 ) ; % relative difference of captured atoms- tube and conem e s s a g e _ r e l _ c h a n g e _ 2 = [ ' Relative performance of setup with cone and without cone (RARE ZS ) :' , num2str ( rel_change_2 ) ];disp ( m e s s a g e _ r e l _ c h a n g e _ 2 ) ;fprintf ( '\ n ')treffer0 = (( rpos + tan ( alpha ) .* d is t an ce _e r bi um ) < c a p t u r e r a d i u s _ e r b i u m ) & (( rpos + tan( alpha ) .* d i st an ce _ er bi um ) > - c a p t u r e r a d i u s _ e r b i u m ) ;treffer1 = (( rpos + tan ( alpha ) .* d is t an ce _e r bi um ) < c a p t u r e r a d i u s _ e r b i u m ) & (( rpos + sin( alpha ) .* l e n g t h _ b l o c k i n g _ e r b i u m ) < b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + sin ( alpha ).* l e n g t h _ b l o c k i n g _ e r b i u m ) > - b l o c k i n g r a d i u s _ s t r a i g h t ) & (( rpos + tan ( alpha ) .*d is ta nc e _e rb iu m ) > - c a p t u r e r a d i u s _ e r b i u m ) ;treffer2 = (( rpos + tan ( alpha ) .* d is t an ce _e r bi um ) < c a p t u r e r a d i u s _ e r b i u m ) & (( rpos + sin( alpha ) .* l e n g t h _ b l o c k i n g _ e r b i u m ) < b l o c k i n g r a d i u s _ c o n u s _ r a r e ) & (( rpos + sin (alpha ) .* l e n g t h _ b l o c k i n g _ e r b i u m ) > - b l o c k i n g r a d i u s _ c o n u s _ r a r e ) & (( rpos + tan (alpha ) .* di s ta nc e_ e rb iu m ) > - c a p t u r e r a d i u s _ e r b i u m ) ;fprintf ( '\ n ')message = ' Ratios for ERBIUM experiment : ';disp ( message ) ;abs_capture = sum ( treffer0 ) / N ; % ratio of captured atoms to total number of atoms (without constraint )message_abs = [ ' Proportion of atoms through capture volume without constraints (Erbium ZS ) :' , num2str ( abs_capture ) ];disp ( message_abs ) ;s t r a i g h t _ c a p t u r e = sum ( treffer1 ) / N ; % ratio of captured atoms to total number ofatoms ( tube at end - no cone )m e s s a g e _ s t r a i g h t = [ ' Proportion of atoms through capture volume without cone (Erbium ZS ) :' , num2str ( s tr a i g h t _ c a p t u r e ) ];disp ( m e s s a g e _ s t r a i g h t ) ;conus_capture = sum ( treffer2 ) / N ; % ratio of captured atoms to total number of atoms( cone at end )message_cone = [ ' Proportion of atoms through capture volume with cone ( Cone openingof RARE ZS ) : ' , num2str ( conus_capture ) ];disp ( message_cone ) ;rel_change = sum ( treffer1 ) / sum ( treffer2 ) ; % relative difference of captured atoms tube and conem e s s a g e _ r e l _ c h a n g e = [ ' Relative performance of setup with cone and without cone :' , num2str ( rel_change ) ];disp ( m e s s a g e _ r e l _ c h a n g e ) ;fprintf ( '\ n ')

A.2. IMPACT OF BEAM DIVERGENCE

toc
figure (1)histogram ( rpos )xlabel ( ' Radial distance / m ') ;ylabel ( ' Number of atoms ') ;title ( ' Radial atom distribution at end of last ZS coil ') ;figure (2)histogram ( vtrans ,400)
xlabel ( ' Radial velocity / m / s ') ;ylabel ( ' Number of atoms ') ;title ( ' Velocity distribution after last ZS coil ') ;figure (3)histogram ( pos_end_mot )xlabel ( ' Radial distance / m ') ;ylabel ( ' Atomic flux ') ;title ( ' Radial atom distribution at MOT center ') ;% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Function definition :function out = boltzdist (n , v_heating )out = zeros (n ,1) ;vdist = [ -2.5:0.001:2.5];
dist = abs ((( vdist .*2) /( v_heating ^2) ) .* exp ( -( vdist .^2) /( v_heating ^2) ) ) ;dist = dist / sum ( dist ) ;distcum = cumsum ( dist ) ;for ind = 1: nx = rand ;out ( ind ) = vdist ( find ( distcum > x ,1) ) ;end
end% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

84

Appendix B
Supplementary
B.1

Magneto-optical lens

The application of transversal cooling by forming a two-dimensional optical molasses
is an efficient measure to increase the atom flux through, for example, a Zeeman
slower. This works due to the collimation that is taking place within the 2D molasses.
However, even though we speak about collimation, in reality the beam that is leaving
the transversal cooling is not fully collimated, but rather the angle of its divergenceis reduced. Thereby, in the simulation of the beam propagation along the Zeeman
slower, we estimated that the divergence after the transversal cooling is given by the
Doppler temperature associated with the used transition. Nonetheless, there is atechnique that has been successfully applied by some groups (see [48], [49]), which is
not only able to collimate a beam, or reduce its divergence for that matter, but also
to focus it. In our case, this could lead to an additional increase of the MOT loadingrate. Such a magneto-optical lens (MOL) would require a magnetic quadrupole field
like the one use for a magneto-optical trap, hence the similar names. The only realdifference is that the atoms, instead of being trapped, travel through the magnetic
field and four crossed laser beams.
The vacuum chamber we use for the transversal cooling exhibits an indentationon either side (see Fig. 4.3), where the coils could be positioned. These indentations
impose restrictions on the coil dimensions with a minimal radius of rmin = 37 mm,
a maximal radius of rmax = 80 mm and a minimal coil distance of dmin = 89 mm.
Additionally, as the coils should protrude as little as possible from the indentationin order to not interfere with other parts of the apparatus, each coil should have amaximal length of lmax = 13 mm. A coil configurationthat fits these restrictions is√
the so-called Maxwell coil pair, where d = 3R with d denoting the coil distanceand R the coil radius [89]. Based on that, we conducted several simulations of thegenerated magnetic field using scripts written by Manfred Mark. Eventually we ended
up with a configuration that makes use of the complete depth of the indentations,where the coils are separated by dmin = 89 mm. When choosing the same wire for thecoils as for the Zeeman slower (dimensions: 1 mm × 2.6 mm) the maximal length oflmax = 13 mm then allows for 5 windings. Combined with 10 layers and an operating
current of 20 A this yields a gradient at the MOL center of Bz /dz = 1.182 G cm−1
along the direction of beam propagation and Bx,y /dx,y = −0.591 G cm−1 in theperpendicular direction.
85

B.2. HOUSING OF SPECTROSCOPY

B.2

86

Housing of spectroscopy

Figure B.1: Home-built housing of the spectroscopy to protect the inside from dust
and stray light as well as to prevent laser light from leaving the setup.

Figure B.2: All in- and outlets for fibers and cables are held shut by home-built
rubber curtains.

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Eidesstattliche Erklärung
Ich erkläre hiermit an Eides statt durch meine eigenhändige Unterschrift, dass ich
die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen
Quellen und Hilfsmittel verwendet habe. Alle Stellen, die wörtlich oder inhaltlichden angegebenen Quellen entnommen wurden, sind als solche kenntlich gemacht.
Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als
Magister- /Master-/Diplomarbeit/Dissertation eingereicht.

Innsbruck, on June 26, 2020

Benedict Franz Hochreiter95