University of Innsbruck
Master Thesis

Stable Reference Cavity
for Er and Dy MOT Light
Alexander Patscheider

Supervised by Prof. Dr. Francesca Ferlaino
Institute of Experimental Physics, University of Innsbruck

Submitted to the Faculty ofMathematics, Computer Science and Physics

February 10, 2017

2

Abstract
In this thesis we describe the setup and characterization of a reference cavity for thestabilization of laser light at several wavelengths. On the one hand, the cavity is used tostabilize light used for the magneto-optical trap of erbium and dysprosium atoms. On theother hand, we want to stabilize laser light for Rydberg excitation at a future date of theexperiment. Additionally, the thesis covers the generation of the light at 583 nm and 626 nm,
corresponding to the transitions in Er and Dy used for the narrow line MOT, respectively.
For the generation of the light at 583 nm we use a seed laser emitting at 1166 nm. This lightis then coupled into a Raman fiber amplifier and then frequency doubled within a second
harmonic generation (SHG) crystal. With a seed power of 25 mW, we obtain slightly lessthan 2 W of 583-nm light. For the 626-nm light we use two fiber lasers at 1550 nm and1050 nm together with two fiber amplifiers and mix the light in a sum frequency generation(SFG) crystal. With a maximal output power of 5 W for each fiber laser we obtain morethan 2 W of light at 626 nm. The light at 626 nm is stabilized with a stable reference cavity.
The two mirrors of the cavity and the spacer between them are made of ultra low expansion
glass (ULE), which has a measured minimal thermal expansion coefficient at 21.78(5) ◦C.
The locking of the laser light is done with a fiber-coupled electro-optical modulator thatmodulates sidebands at two different frequencies. While the laser is resonant to the atomic
transition, one of these sidebands is shifted to the cavity resonance. The second sidebands
are then used to generate the Pound-Drever-Hall error signal for the electronic locking
setup.

4

Zusammenfassung
In dieser Masterarbeit beschreiben wir den Aufbau und die Charakterisierung eines optischen Resonators, welcher zur Stabilisierung von Laserlicht verwendet wird. Zum Einen
wird damit das Licht, welches für die magneto-optische Falle von Erbium und Dysprosium
benötigt wird stabilisiert. Zum Anderen soll der Resonator, zu einem zukünftigen Zeitpunkt
des Experiments, für die Stabilisierung von Laserlicht für Rydberg Anregungen verwendet
werden können. Des Weiteren behandelt die Masterarbeit die Erzeugung von Laserlicht bei
einer Wellenlänge von 583 nm und 626 nm. Diese Wellenlängen entsprechen den optischenÜbergängen, welche für die magneto-optische Falle für Erbium und Dysprosium verwendet
werden. Für das Laserlicht bei 583 nm verwenden wir ein System aus einem Pump-Laser bei
einer Wellenlänge von 1166 nm, zusammen mit einem Raman Faserverstärker und einem
Kristall zur Frequenzverdopplung. Mit einer Pumpleistung von ungefähr 25 mW erhalten wir etwas weniger als 2 W Licht bei 583 nm. Für die Erzeugung von Licht bei 626 nm
werden zwei Faserlaser bei den Wellenlängen 1550 nm und 1050 nm mit Faserverstärkern
verstärkt und in einem nichtlinearen Kristall für Summenfrequenz addiert. Hier erhalten
wir bei maximaler Leistung der Faserlaser (5 W) eine Ausgangsleistung von etwas mehrals 2 W. Die Spiegel des Resonators und der Abstandshalter dazwischen sind aus einem
speziellen Glas (ultra low expansion glass), dessen thermischer Ausdehnungskoeffizient bei
gemessenen 21.78(5) ◦C minimal ist. Zur Stabilisierung des Laserlichts werden mit einem
fasergekoppeltem elektro-optischen Modulator Seitenbänder bei zwei unterschiedlichen
Frequenzen auf das Licht aufmoduliert. Während der Laser zum atomaren Übergang resonant ist, wird ein Seitenband zur Resonatorresonanz geschoben und das zweite Seitenband
zur Erzeugung des Pound-Drever-Hall Fehlersignals verwendet.

6

Contents
1. Introduction

9

2. Overview on the Main Properties of Erbium and Dysprosium
2.1. Erbium and Dysprosium at Glance . . . . . . . . . . . . . . . . . . . . . . .
2.2. Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Laser-Cooling Transitions in Erbium and Dysprosium . . . . . . . . . . . .
2.4. Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
14
15
17

3. Laser Systems for Er-Dy Double MOT
3.1. Dual Species Magneto Optical Trap . . . . . . . . . . . . . . . . . . . . . .
3.2. Laser System for 583-nm Light for Erbium . . . . . . . . . . . . . . . . . .
3.3. Laser System for 626-nm Light for Dysprosium . . . . . . . . . . . . . . . .

21
21
23
29

4. Design of the Vacuum Chamber and the ULE Cavity
4.1. Frequency-Stability Requirements of the Light . . . . . . . . . . . . . . . .
4.2. Basic Concepts of Cavity Physics . . . . . . . . . . . . . . . . . . . . . . .
4.3. Cavity Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Mechanical and Electrical Design of the Vacuum Chamber . . . . . . . . .

33
33
34
38
44

5. Experimental Part
5.1. Optical Setup for the Coupling of the Lasers to the Cavity . . . . . . . . . .
5.2. Modulation-Transfer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
5.3. Zero Crossing of the ULE Cavity . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Locking of the Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5. Performance of the Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51
53
56
58
60

6. Conclusion and Outlook
6.1. Er–Dy Quantum Mixture: Motivation and Vision . . . . . . . . . . . . . .
6.2. Rydberg Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
68
69

A. Appendix
A.1. Pound-Drever-Hall Error Signal . . . . . . . . . . . . . . . . . . . . . . . .
A.2. Modulation-Transfer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
A.3. Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4. Assembly of the Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . .

71
71
74
75
79

Bibliography

85

7

Contents8

1. Introduction
At temperatures close to the absolute zero the atoms begin to reveal their quantum behavior.
In 1911 Heike Kamerlingh Onnes observed that mercury looses suddenly its electric resistivity at a temperature of 4.2 K [43]. This discovery was related to the first realization of liquid4
He only three years before. The superfluidity effect of liquid 4 He could be explained in1938 by Kapitza, Allen, and Misener [3, 44]. In the following, in 1933 Walther Meißner andRobert Ochsenfeld discovered that superconducting materials repel an applied magnetic
field and are therefore perfectly diamagnetic [66]. It took until 1957, when John Bardeen,
Leon Neil Cooper, and John Robert Schrieffer found a theoretical microscopic approach
to explain the macroscopic effect of superconductivity [9]. However, the understandingof superconductivity changed in 1986, when Johannes Georg Bednorz and Karl Alexander
Mueller observed that the critical temperature, where the superconductivity appear, forcopper oxide is much higher than for the materials known up to then. While for othermaterials the critical temperature is below 23 K for copper oxide the critical temperature is
at 135 K [10]. Both, superfluidity and superconductivity, are macroscopic consequences ofthe bosonic and fermionic character or, more precisely, of the different statistical behaviorof the particles. The Bose-Einstein statistics describes particles of whole-integer spin, calledbosons. For zero temperature all identical particles occupy the lowest possible energy
state in the system. On the other hand the Fermi-Dirac statistic describes the behavior offermions (particles with half-integer spin) and at zero temperature each energy level up
to the Fermi-energy is occupied by one single particle. The resulting states formed by theparticles are called Bose-Einstein condensate for bosons and degenerated Fermi gas forfermions, respectively.
The complex nature of the superconductivity calls for the simulation of the systems
in a better-controllable environment. The development of laser cooling techniques tocool particles with laser light to the µK regime and even lower temperatures lead to newopportunities for the investigation of many-body quantum systems and superfluidity.
With the observation of the first Bose-Einstein condensates (BEC) in 1995 [5, 13, 22],the field of ultracold atomic gases experienced a fast expansion. A special property ofultracold gases is that they allow a high controllability of internal and external degrees of
freedom. Interparticle interactions can be controlled via Feshbach resonances [17], andoptical lattices can be used to create spatial structures. This gives opportunity to enter awide range of physical research fields connected to strongly correlated many-body systems
as Quantum Simulators [30] and high-temperature superconductivity [19].
Among others, important aspects reached by the tunability of the interaction strength
are the crossover from a BEC to a Bardeen-Cooper-Schrieffer superfluid state [39] and thesuperfluid to Mott transition with atoms in an optical lattice [37]. Early experiments withultracold atomic gases haven been realized with alkali atoms as rubidium, lithium, cesium,

91. Introduction
and sodium. With the condensation of chromium in 2005 [38] for the first time an atomic
species was brought to degeneracy that carries a higher magnetic moment than alkali
atoms, i. e. 6 µB instead of only 1 µB . While for alkali atoms the properties are dominatedby the isotropic and short range contact interaction for dipolar atoms the anisotropic andlong range dipole-dipole interaction (DDI) comes into play. This makes dipolar elementsof special interest for the investigation of new quantum phenomena. With chromium forexample the d-wave collapse of the BEC [48] and the spin relaxation in an optical lattice
have been investigated [68].
Since then, BECs and degenerated Fermi gases of erbium and dysprosium that carryeven higher magnetic moments of 7 µB and 10 µB , respectively, have been realized. The
BEC of dysprosium was realized for the first time in the group of B. Lev [55] at Stanford,
whereas the first realization of an erbium BEC was here in Innsbruck [1]. Dysprosium
shares together with terbium the largest magnetic moment of all elements. While forchromium the contact interaction has to be reduced with a Feshbach resonance in order tomake the DDI dominant, for erbium and dysprosium the DDI is dominant even without
reducing the contact interaction.
Beside the large magnetic moment, erbium and dysprosium share additional special
properties, which are the multi-valence electron character and the rich atomic electronic
spectra containing many broad and narrow optical transitions [11, 54, 64] that can be used
for laser cooling to very low temperatures. For both elements the 4 f shell is not completely
filled, which results in the high total angular momentum in the ground state. This highangular momentum, the large number of broad and narrow optical transitions, and themulti-valence electron characteristic give new possibilities for the growing field of Rydberg
atoms. These are atoms where one electron is excited to a high principal quantum number
n having a large spacial expansion, e. g. a rubidium atom excited to a state with n = 43 hasan orbital radius that is larger by a factor of 40 compared to the ground state [57]. The
combination of ultracold gases with Rydberg excitation has already lead to many exciting
aspects such as the Rydberg blockade[77, 78] or the spatially ordered structure in a Rydberg
gas [73]. Similar to atoms with two valence electrons, the core of the Rydberg atom remains
optically active allowing imaging [65] and manipulation by light scattering [27].
The goal of our experiment is to combine for the first time two different strongly magnetic
elements, erbium and dysprosium. The experiment will consist of three parts: the mainchamber where the mixture experiments will be done and two additional chambers that will
be attached at a later point. One of these additional chambers will be used to do Rydberg
excitation experiments and for the other one a quantum gas microscope is planed.
About this thesis:
The purpose of this master thesis was to work on the generation and stabilization of thecooling light for the narrow-line MOTs of erbium and dysprosium and on the stabilizationof future lasers for Rydberg excitation. For the cooling light for erbium, a seed laser in
combination with a Raman fiber amplifier and a second harmonic generation crystal isused. For dysprosium we use a sum frequency generation crystal to add up the light of
two fiber laser systems consisting of a fiber amplifier that is seeded by a fiber laser. For thestabilization we use a cavity with a spacer out of ultra low expansion glass (ULE) to lock10

the lasers to the cavity resonance using a Pound-Drever-Hall locking scheme [25]. The
assembling of the cavity and locking of the lasers was the main part of the presented thesis.
The thesis is structured as follows: In Chapter 2 the main properties of erbium anddysprosium such as atomic structure, energy spectra, and magnetic properties are discussed.
The topic of Chapter 3 is the generation of the laser light for the MOTs of erbium anddysprosium. This contains a comparison of the parameters used in the already running
experiments as well as a description of the laser systems that we will use in our experiment.
In Chapter 4 a theoretical overview about cavities is given. Further, the requirements forour cavity such as thermal stability are discussed and the final design of the cavity is shown.
The characterization and discussion of the cavity parameters is done in Chapter 5.

11

1. Introduction

12

2. Overview on the Main Properties of
Erbium and Dysprosium
In the last years the interest on strongly magnetic lanthanide atomic species increased
significantly. These species give access to the study of dipolar phenomena. Recently, erbium(Er) and dysprosium (Dy) have been brought to quantum degeneracy with the creation of
Bose-Einstein condensates and Fermi gases [1, 55]. In this chapter some general propertiesof Er and Dy such as level schemes, cooling transitions and magnetic moment are discussed.

2.1. Erbium and Dysprosium at Glance
Both, Er and Dy are rare-earth elements being part of the Lanthanide series of the periodic
table. Er has an atomic number of 68 leading to a mass of 167.27 amu and Dy has an atomic
number of 66 leading to a mass of 162.5 amu. The melting point of Er is 1529 ◦C, slightly
higher than for Dy (1412 ◦C). This similarity is in principle an important added value
when combining the two species for a quantum mixture experiment. Both, Er and Dy
have different fermionic and bosonic isotopes. Er has five stable bosonic isotopes (162 Er,
164
Er, 166 Er, 168 Er, and 170 Er) and only one fermionic isotopes (167 Er). On the other hand Dy
has also five stable bosonic isotopes (156 Dy, 158 Dy, 160 Dy, 162 Dy, and 164 Dy) but two stablefermionic isotopes (161 Dy, and 163 Dy) compared to Er. In Table 2.1 a list of all isotopes,
which have a natural abundance above 1 % are listed for Er and Dy.
Table 2.1.: Relative abundances of the stable isotopes of Er and Dy which have a minimum abundanceof 1 %. 167 Er, 161 Dy and 163 Dy are fermionic whereas the other isotopes are bosonic.

Isotopeabundance (%)

Isotopeabundance (%)

164

Er

166

Er

167

Er

168

Er

170

Er

1.61

33.6

23.0

26.8

15.0

boson

boson

fermion

boson

boson

160

Dy

161

162

163

164

2.3

18.9

25.5

24.9

28.3

boson

fermionboson

fermion

boson

Dy

Dy

Dy

Dy

A special property of Er and Dy is their large magnetic moment of 7 µB and 10 µB whereµB denotes the Bohr magneton, respectively. Together with terbium, Dy has the largest

13

2. Overview on the Main Properties of Erbium and Dysprosium
magnetic moment of all elements. Table 2.2 gives an overview about some properties of Er
and Dy.
Table 2.2.: Overview over some properties of Er and Dy. Z is the atomic number.
Note: 1 amu ≡ 1.600 539 040(20) × 10−27 kg [67].

Z

mass (amu)melting point (°C)

electron config.

mag. moment

Er68

167.271529[Xe] 4 f 12 6s2

7 µB

Dy

66

162.5

1412

[Xe] 4 f 10 6s2

10 µB

2.2. Atomic Structure
Following Mandelung’s rule, the electrons for Er and Dy fill up as

and[Xe] 4 f 12 6s2(Er)(2.1a)[Xe] 4 f 10 6s2(Dy).

(2.1b)The notation in square brackets represent the electron configuration of Xenon,(1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 ).
While the 4 f -orbital is only partially filled, the 6s-orbital is completely filled up. This
electron configuration is referred to as submerged-shell structure. The electrons in the
4 f -orbital have an angular momentum quantum number of l = 3. Considering the angularmomentum projection quantum numbers ml = −l, . . . , l and the spin projection numbers
m s = ±1/2 result in 7 states (ml = −3, −2, −1, 0, 1, 2, 3), which can be doubly occupied.
Therefore, in total up to 14 electrons can sit in the 4 f -orbital. The very similar electron
configuration of the two atomic species to be stressed whereas Dy has 2 electrons less inthe 4 f -shell compared to Er.
Depending on the spin-orbit coupling energy relative to the Coulomb interaction, different
coupling schemes are necessary to find the respective quantum numbers. If the energy
for the spin orbit coupling E s−o is lower than the residual electrostatic interaction Ere thespin-orbit coupling scheme (LS-coupling1 ) can be applied. Otherwise, jj-coupling become
relevant. For Er and Dy only for the ground state the LS-coupling scheme is sufficient,whereas for the excited states the jj-coupling has to be applied [80]. For the LS-couplingP
all spins and angular momenta add up to a total spin S~ = i ~si and a total orbital angularP
momentum ~L = i ~li . The resulting total spin and total orbital angular momentum couple
then to the total angular momentum, J~ = ~L + S~ . Finally, the quantum state is denoted
1

Russell-Saunders coupling scheme.

142.3. Laser-Cooling Transitions in Erbium and Dysprosiumby 2s+1 L J . For Er and Dy all electron shells, except the 4 f -shell are completely filled, which
results in a high total angular momentum in the ground state of J = 6 for Er and J = 8 for
Dy, respectively.
Following the LS-coupling for the ground state results in 3 H6 for Er and 5 I8 for Dy. On
the other hand, for the jj-coupling first the 4f electrons and the outer electrons couple
independently via the LS-coupling scheme. This leads to two different quantum numbers J1
and J2 which by adding up form the final state written as (J1 , J2 ) J . As an example one cantake the 583-nm MOT transition of Er, where an electron from the 6s-orbital is excited to a
6p-orbital ([Xe] 4 f 12 (3 H6 )6s6p(3 P01 )). First, the inner electrons couple to 3 H6 and the outerelectrons couple to 3 P1 by LS-coupling. Then, these two states couple to (6, 1)07 , where thesubscript 7 denotes the sum J~ = ~j1 + ~j2 and the subscript 0 refers to odd party.
The bosons of Er and Dy have no hyperfine structure because their nuclear spin I is zero.
In contrary, the fermionic isotope 167 Er has I = 7/2, which results in a hyperfine structure
quantum number F = 19/2, whereby the angular momentum J couples to the nuclear spin I.
For Dy the nuclear spin is I = 5/2, which leads to F = 21/2 for the electronic ground state.
Table 2.3 gives a summary over all quantum numbers in the ground state of Er and Dy.
Table 2.3.: Quantum numbers for the ground state of Er (3 H6 ) and Dy (5 I8 ).

total orbital angular momentum L
total spin S
total angular momentum J
nuclear spin I for bosons
nuclear spin I for fermions
total angular momentum F for fermionsErbiumDysprosium

5
1
6
0
7/2
19/2

6
2
8
0
5/2
21/2

Figure 2.1 shows the atomic spectra for Er and Dy. States, which are labeled in red, referto even parity and states in black have odd parity. This very rich atomic spectra containmany different possible cooling transitions.

2.3. Laser-Cooling Transitions in Erbium andDysprosium
The plan for the experiment is to have first a transversal cooling section (TC) to collimate
the divergent beam coming out of the oven. To decelerate the atoms before trapping them a
Zeeman slower is used (ZS). For the TC and the ZS the broad blue transitions (401 nm for Er
and 421 nm for Dy) are used. Since one atom has to scatter many photons over a short time
period transitions with a short lifetime are preferred. Additionally, the wavelength should
be small, leading to a higher recoil velocity2 . For Er (Dy) the transition at 401 nm (421 nm)
has a lifetime of 4.94 ns (5.35 ns) and a recoil velocity of 6 mm/s (5.8 mm/s). Assuming, that2vrec = ~k/m. Here, ~ is the reduced Plack’s constant, m the atomic mass, and kB the Boltzmann constant.

15

Melting Slowing transition
MOT transition
point
(°C)
[nm]
[MHz]
[nm]
[kHz]
Er
1529
401
30
583
190
2. Overview on the Main Properties of Erbium and Dysprosium
Dy
1421
421
32
626
136
ERBIUM

401 nm(slowing transition)

15

F
1
2

7
10

421 nm (slowing transition)
4f10(5I8)6s6p(1Po1)(8,1)o9

333

25

400

20

500

15

4f10(5I8)6s6p(3Po1)(8,1)o9 667

4f ( H6) 6s6p( P1)
12 3

3583 nm (MOT)

626 nm (MOT)

10

5

10

1000

5

2000

4f126s2 3H6

0
2

3

4

5

6

7

J value

8

9

10

11

12

0

Wavelength (nm)

Wavenumber (1000 cm-1 )4f12(3H6) 6s6p(1P1)

20

B
5
5

μ
(μ_B)

DYSPROSIUM

30

25

Isotope

4f106s2 5I8 ground state
2

3

4

5

6

7

8

J value

9

10

11

12

13

Energy le
and Dy (electronic
quantumodd (evenby black
The groua orbital
number Lof the bo
hyperfinefermionic
main lasare indic
pictures51].

Figure 2.1.: Atomic spectra and cooling transitions of Er and Dy. In our experiment we will use
the 583-nm and 626-nm transitions for the Er and Dy MOT whereas the transitions of 401 nm andtube. After the Zeeman slower, the atomic beam will be captured into the magneto
421 nm will be used for the transversal cooling and the Zeeman slower, respectively. Red lines
chamber.
theparity
Er andwill with
operateon theirrespective
narrow
represent
statesBoth
with evenandDyblackMOT
lines states
odd parity.
Graphics
taken from[8, 81]. inter-comb

nm for Er and at 626 nm for Dy); see Fig. 3. The stainless steal MOT chamber will hTable 2.4.: Electron configurations and quantum numbers of the excited states. First the electrons
for optimal optical access and will integrate a multi-channel plate and a appropriate eleof the inner and outer shell couple independently via the LS-coupling scheme and then they couple
In this chamber, we will perform evaporative cooling of the mixture in a threeviasetup.
the j j-coupling.

dipole trap and studies on multi-electron Rydberg atoms. Additionally, the MOT chambelectronconfigurationhigh-resolution
j j-coupling imaging. The glass cell wil
to a glass cell, where we plan
to implement
1 0
nm [Xe]4 f 12by(3 Ha
P1 ) box(6,made1)07 of mu-metal and aluminum.
6 )6s6p(
undesired stray 401
magnetic
fieldsshielding
12 3

3

0

0

583 nmf (useH6 )6s6p(
P1 )high-temperature(6, 1)7• Dual species oven:
We [Xe]plan4 toa singleeffusion cell to sublima
10 5
1 0
0
nm [Xe]4 f ( I8 )6s6p((8, 1)and Dy. At an 421
operatingtemperatureof P1100C,we
1)
9 checked that Er and Dy are n
10 5
3 0
low-temperature626
alloyswith[Xe] 4 f between
nm neither( I8 )6s6p(themP1 ) nor(8,1)09 tantalum, of which the crucible
different of the melting points between the two atomic species will lead to a difference i
plan
to overcome
this problem
by using
an Er-Dy alloywiththe appropriate
mixingtheWe
atoms
comingfrom the ovenhave a velocity
of approximately
500 m/s,roughly
80 000ACI ALLOYS
can produce
a sample.
photonshave to be scattered
to slowsuch
the atoms
down to 10 m/s, which is a adequate capture
velocity [33, 61]. The broadness of the transition is a useful characteristic because more

•photons
Dual canspeciesZeeman slower and MOT: The dual-species Zeeman slower and MOT
be scattered by the atoms. After the ZS the atoms reach the main chamber
tivelyon theline
intertransition
combination
line;thesee
Tabletransition
X. Given that bothwith
the MOT.
Forstrong
the MOT
weand
use onthe the583-nmfor Er and626-nm2
forvalues
Dy. Forof
both,and Dymeansparametersthat one electron
fromthe 6sstate isexcited to a they can be
theEr
mass
andthissimilarfor thelaser
coolingtransitions,
3
6pusingstate andwith the remaining
6s electron
to a triplet
P1 state.show
Table that
2.4 showsthe couple
same Zeeman
slower. Our
preliminary
calculations
a dual-species Z
the electron configuration of the excited states of the laser cooling transitions.
cm length will be sufficient to decelerate the thermal atomic beam from 400 m/s to do
The temperature limit of the Doppler cooling is given by the Doppler temperature3 ,small
final velocities
arelinewidth
necessaryto capture
theTheatoms
in ournarrow-line
whichis proportional
to theof thetransition.
smallerthe linewidth,
theMOT. The Z
includingthe magnetic
fieldThe
windings,
have a homemade
andareit will be bu
lower
the reachable
temperature.
Doppler willtemperatures
for the used designtransition
workshop.
3/2k . Here, ~ is the reduced Plack’s constant, ∆ν the natural linewidth /2π, and kB the Boltzmann
D =• TQuantumgas microscope: This is a powerful technology, which allows single-parti
constant.
solved detection in many-body quantum systems. It has been developed by the groupand the group of I. Bloch and S. Kuhr [56]. So far it has been achieved only with u
16 in deep pinning optical lattices. Combining this technology with atomic species polevel-structure, such as Er and Dy, is a very interesting goal by its own. The variety
transitions will offer many opportunities for the investigation of new high-resolution-i
particular interest is the strong transition in the blue (401 nm for Er and 421 nm for D
an order of magnitude stronger than the alkali one. The quantum gas microscope willvant in studying the dipolar spinor gas in optical lattices because of the possibility to o
~∆ν

B

Γ

2.4. Magnetic Properties
T D = 4.6 µK for Er and T D = 3.3 µK for Dy. In principle, also the 401-nm and 421-nm
transitions could be used for the MOT, but due to the relatively high Doppler temperatureof T D = 714 µK for Er and T D = 773 µK for Dy additional cooling steps would have to be
implemented. A low temperature is preferred because of a higher loading rate into a dipole
trap. For laser cooling it is very important to have a close cycling transition. This means,that most of the excited atoms do not get lost in a metastable state, but decay back into theground state. In a metastable state the atoms would became dark to the cooling light. A big
advantage of the MOT transitions for Er and Dy is that no repumper is needed because
the decay probability into intermediate states is very low. For Er there are two possible
intermediate states with transition probabilities of 0.017 and 0.0049 [8]. Table 2.5 gives, alist of relevant parameters of the cooling transitions, which will be used in our experiment.
Table 2.5.: Important parameters of the cooling transition of 401 nm, 421 nm, 583 nm, and 626 nm [8,56, 58].

ErbiumParameterstransition rate
lifetimenatural linewidth
saturation intensity
doppler temperaturedoppler velocity
recoil temperature
recoil velocityΓ (1/s)τ (ns)∆ν (MHz)IS (mW/cm2 )T D (µK)vD (mm/s)
T r (nK)
vr (mm/s)

Dysprosium

401 nm583 nm421 nm626 nm1.87 × 108
5.35
29.7
60.3
714
267
717
6.0

1.17 × 106
857
0.19
0.13
4.6
21
339
4.1

2.02 × 108
4.94
32.2
56.3
773
198
659
5.8

0.85 × 106
1170
0.136
0.072
3.3
12.9
298
3.9

The different isotopes of Er and Dy have different numbers of neutrons and therefore
different masses. This leads to a different distribution of the electrons, resulting in a changeof the charge distribution, causing a shift of the energy levels. Figure 2.2 shows this isotope
shift for the bosonic isotopes of Er and Dy, which have a natural abundance above 1 %.
The isotope shift for both elements is on a comparable order.

2.4. Magnetic Properties
A very special property of these two elements is their large magnetic moment of 7 µB for
Er and 10 µB for Dy, where µB denotes the Bohr magneton. This is in contrast to otherelements as alkali atoms, having a 1 S 0 state as ground state. The exact value of the magnetic
moment µ can be calculated usingµ = m J g J µB ,(2.2)

where mJ is the projection of the total angular moment in the direction of a magnetic fieldand g J is the Landé g-factor, also known as the gyromagnetic ratio. The Landé factor canbe calculated by [23]17

2. Overview on the Main Properties of Erbium and Dysprosium"8 (GHz)

2
(a)
1
0

583 nm626 nm

-1
160

162

164

166

168

170

isotope"8 (GHz)

2
(b)
1
0

401 nm421 nm

-1
160

162

164

166

168

170

isotopeFigure 2.2.: The isotope shifts of (a) the 583-nm (Er) and 626-nm (Dy) transitions and (b) the 401-nm(Er) and 421-nm (Dy) transitions [20, 41, 42, 50].

J(J + 1) − L(L + 1) + S (S + 1).
(2.3)
2J(J + 1)
However, corrections due to deviations from the LS-coupling and other effects, e. g. relativistic corrections, have to be considered. Inserting the Landé factor g J = 1.163801(1)[21] for Er and g J = 1.2415867(10) [29] for Dy and assuming that the atoms in the groundstate are sitting in the m J = 6 or m J = 8 level leads to a magnetic moment of 6.98 µB and9.93 µB , respectively. By applying an external magnetic field, the energy level with total
angular momentum quantum number J split up in 2J + 1 sub states with magnetic quantum
number mJ . In Figure 2.3 the Zeeman splitting for ground state of the bosonic isotopes of Er
and Dy is plotted for a magnetic field B up to 5 G. The energy shift can be calculated by
gJ = 1 +∆E(B) = m J g J µB B.

(2.4)

For the fermionic isotopes the hyperfine structure has to be taken into account and theunderlying values are the mF state and the Landé g-factor gF . For the lowest hyperfine state(F = 19/2 and mF = −19/2 for Er and F = 21/2 and mF = −21/2 for Dy) the magnetic moment
has the same value as for the bosonic ground state.
This large magnetic moment leads to completely different scattering properties since thedipole–dipole interaction is fundamentally different from the contact interaction. For two

18

2.4. Magnetic Properties
80

mJ mJ
+8
+7

60

+6
+5

40

energy/2: (MHz)

+4

20

0

-20

+4
+3

+2

+2

+1

+1

0

0

-1

-1

-2

-2

-3

-3

-5
-6

-60

+5

+3

-4

-40

+6

-4
-5
-6

-7
-8

-80

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

magnetic -eld (G)Figure 2.3.: Linear Zeeman energy splitting for the the ground state of the bosonic isotopes of Er(orange) and Dy (blue) with J = 6 and J = 8, respectively. On the right hand site of the graph the
projection quantum number m J is given.

particles with dipole moments along the unit vectors e1 and e2 the dipole–dipole interaction
reads as [49]µ0 µ2 (e1 · e2 )r2 − 3(e1 · r)(e2 · r)
Udd (r) =
,
(2.5)
4π
r5
where r is the relative position of the atoms to each other. For a polarized sample thissimplifies toµ0 µ2 1 − 3 cos2 (θ)
Udd (r) =
,
(2.6)
4π
r3
where θ is the angle between the direction of the polarization and the position vector r. The
two main properties of Udd are the long-range character, identifiable from the factor 1/r3 andthe anisotropy deriving from the symmetry of the Legendre polynomial of second order,
P2 (cos θ) = 12 (1 − 3 cos θ). Since θ can vary between 0 and π/2, the dipole-dipole interaction
can be either repulsive for atoms sitting side by side or attractive for atoms sitting in a19

2. Overview on the Main Properties of Erbium and Dysprosium
103

Udd ( 74:r
2)
07

0.5

-0.5
-1
-1.5
-2

0

10

20

30

40

50

60

70

80

90

relative angle between two dipoles 3 (:)
Figure 2.4.: Strength of the dipole-dipole interaction Udd in dependence of the relative angle
between the dipole orientation at a fixed distance r. The green part stands for attractive interaction
and the red line for repulsive interaction, respectively. The blue dot marks the so called ’magic
angle’, where the dipole-dipole interaction vanishes.

’head to tail’ configuration. Figure 2.4 shows the strength on the dipole–dipole interaction
in dependence of the relative angle between the polarization and the relative position of thedipoles at a fixed distance r. For the special angle θ = 54.7 °C the dipole–dipole interaction
vanishes.

20

3. Laser Systems for Er-Dy Double
MOT
In this Chapter, we will describe the two laser systems used for the narrow line magnetooptical traps (MOTs) of Er and Dy and the expected requirements for MOT production.

3.1. Dual Species Magneto Optical Trap
Among the different possible laser cooling transitions, Er and Dy feature an intercombinationtype line, which is broad enough to provide efficient laser cooling but narrow enough tohave conveniently low Doppler temperatures. This is the case of the transition at 583 nm for
Er and at 626 nm for Dy. For both species, single-species MOTs are currently successfully
operating on such transitions [24, 33, 61]. The relatively narrow linewidths of aroundΓ ≈ 2π · 150 kHz arise as a result of a transition from the ground state being a singlet state
to an excited state being a triplet state (intercombination line).
The experimental sequence in the experiments, where the narrow line MOTs are realized,
is as follows: First, the atoms are loaded from the Zeeman slower (SL) into a MOT withlargely detuned laser light compared to their natural linewidth, i. e. approximately −40 Γ583
for Er and −35 Γ626 for Dy, respectively. Due to the large detuning the atoms do not feela homogeneous light force over the volume of the MOT but are resonant with the lighton an ellipsoid of constant magnetic field [32]. Due to the gravitational force, the atoms
will accumulate in the lower region of this ellipsoid and form a characteristic bowl-shaped
MOT. This automatically prevents the Zeeman slower light to reduce the atom number inthe MOT by exerting a light force on the atoms. Since for the double species MOT both
atomic species feel the same magnetic field gradient, the position relative to each other canbe influenced by the detuning. In a second step the MOT is compressed (cMOT) to reach
lower temperatures and a higher density of the atomic cloud. During the compression themagnetic field gradient is changed as well as the detuning and the light intensity. Figure 3.1
shows a schematic representation of the passage from a MOT to a cMOT for the simple
case with a J = 0 → J 0 = 1 transition. Table 3.1 gives some typical parameters used in [61]and [33] for the realization of the narrow line MOTs.
In summary this narrow line transitions provide a exceptionally good starting point forthe subsequent direct loading of a dipole trap due to the already low final temperatures andlarge atom densities reached after the cMOT step. Both species, Er and Dy reach similar
final temperature, atom number, and atom density for comparable detunings and magnetic
field gradients. This leads to the promising chance of realizing a simultaneous double
species MOT using adequate trade-off parameters especially for the magnetic field gradient.

21

3. Laser Systems for Er-Dy Double MOT
E

mJ = -1
mJ = -1
small δlarge δmJ = +1
mJ = +1z

Figure 3.1.: Representation of a transition from a MOT to a cMOT. For the cMOT the magnetic field
gradient, the beam intensity and the detuning are reduced compared to the starting MOT.
Table 3.1.: Summary of the parameters used for the narrow line MOTs in [61] and [33].

transition
natural linewidth Γmagnetic field gradient MOT
detuning MOT

ErbiumDysprosium

583 nm0.19 MHz
2.6 G/cm−40 Γ583

626 nm0.136 MHz
3 G/cm−35 Γ626

magnetic field gradient cMOT
0.9 G/cm
detuning cMOT
−9.5 Γ583
final temperature
10 µK
final atom number
2 × 108
final atom density
1.5 × 1011 /cm3

1.5 G/cm−2.75 Γ626
6 µK
1.5 × 108
8.6 × 1010 /cm3

An issue that could have a negative influence on the double-species MOT are possible crosstalks. This means that the light at 583 nm for the Er MOT could have an effect on the Dy
atoms and the light at 626 nm could have an effect on the Er atoms. Table 3.2 and 3.3 givethe closest known transitions with respect to the MOT transition of each species.
Table 3.2.: Closest Er transitions near Dy MOT line.

J → J0λ (nm)∆ν = Γ/2π (kHz)

6→6
6→6

622.102
620.2

133.7
55

Table 3.3.: Closest Dy transitions near Er MOT line.

22

J → J0λ (nm)∆ν = Γ/2π (kHz)

8→7
8→8
8→7

565.36597.6
599.0271
67
90

voltage (1/40mV)

voltage (1/20mV)

3.2. Laser System for 583-nm Light for Erbium

time (1/50ms)

time (1/100ms)(a) Acoustic noise on the laser of 1166 nm.

(b) The acoustic noise is removed with the flexure laser design.

Figure 3.2.: Figure (a) shows the acoustic noise on the light of 1166 nm. The noise was removed (b)by changing the laser design to the flexure design. Blue is the voltage applied for the piezo scan,orange is the cavity transmission, and in (a) purple is the cavity reflection. The scan ramp is not
symmetric to the cavity transmission.

The final performance of the double-MOT will also crucially depend on interspecies
inelastic collisions, which have to be investigated experimentally [36].

3.2. Laser System for 583-nm Light for Erbium
To create the 583-nm light for the narrow line MOT of Er we use a commercial Raman
amplifier from MPB Communications, which is seeded from a homemade grating-stabilized
external cavity diode laser (ECDL) at 1166 nm.

External cavity diode laser at 1166 nm
As already mentioned above we plan to use a home built ECDL as a seed laser. There are
only a few laser diodes available covering the needed wavelength of 1166nm. We chosea high power gain chip1 based on an InAs Quantum Dot with a central wavelength at
1178 nm and a tuning range of 50 nm.
Initially, we used a laser design, which was frequently used in Innsbruck, e. g. in [76].
However, we observed a high sensitivity to acoustic noise. Figure 3.2(a) shows the cavity
transmission of the 583 nm while scanning the laser over the cavity resonance. Althoughhigher transversal modes are suppressed by the coupling, many other modes appeared
when clapping or knocking on the laser table. Changing to a flexure-based laser design [47]solved this issue. Figure 3.2(b) shows the cavity transmission of the 583 nm laser with theflexure-based laser as a seed laser. The laser design works in Littrow-configuration, where1

GC–1178–TO–200 from Innolume, 200 mW @ 1178 nm.

23

3. Laser Systems for Er-Dy Double MOT
the 1th diffraction order of the grating is reflected into the gain chip and the 0th order is used
as an output. In comparison to the first Innsbruck design the diffraction grating and theoutput coupling mirror of the new design are glued on a compact flexure instead of a mirror
mount. Figure 3.3 shows the design of the flexure laser. The gain chip is mounted in ascrews for verticaladjustment
diode holder
flexurePeltier elementout coupling
mirrorgratingscrews to turnthe flexure

piezo

Figure 3.3.: Drawing of the flexure type laser design. The gain chip is fixed in the diode holder,which is temperature stabilized by the Peltier element. Two screws are placed in the diode holder
for the vertical adjustment. For rough wavelength adjustment the flexure (blue) is turned with thebrown screws on the side. The piezo element is for the fine adjustment of the wavelength.

standard collimation package from Thorlabs2 which is used for collimation of the light. The
collimation package tube sits inside an aluminum holder which gets temperature stabilized
via a Peltier element3 , see Fig. 3.3. The heat from the Peltier elements is removed via a heatsink, which is glued on the Peltier element and connected to the housing. The temperaturestability is further improved by stabilizing the laser housing itself using additional Peltierelements sitting between the laser housing and the baseplate.
The wavelength of the laser is adjusted by turning the flexure, whereby the grating4 ischosen in such a way, that the angle between the 0th order and the 1th order is approximately
45 degree. This ensures that the light leaves the laser without hitting the housing. The
2

e. g. Thorlabs LT230P-B.
e. g. Peltier element with 5.1 W, 2.2 A, and 3.8 V.
4
For 1166 nm the grating GH13-12V from Thorlabs is chosen.

3243.2. Laser System for 583-nm Light for Erbium
rough wavelength adjustment is done by turning the flexure via two adjustment screws thatpress on the flexure. The fine adjustment as well as the scanning of the wavelength is doneusing a piezo element placed in the flexure and fixed with a screw that presses on the piezo.
The special structure in the flexure allows to turn the grating without turning the wholeflexure when applying a voltage to the piezo. The laser housing provides additional space
for a filter- or prestabilization cavity and the possibility to evacuate the whole housing foradditional temperature- and acoustic shielding. However, the linewidth and acoustic noise
shielding are already sufficient without those options.
Figure 3.4 shows the characterization of the achievable output power of the 1166 nm
laser for three different temperatures. For a better clearness the data points are shown
without error bars. The error can be assumed with approximately 5 %. The threshold of
140

output power (mW)120

T = 21/ C
T = 22/ C
T = 23/ C

100
80
60
40
20
0
50

100

150

200

250

300

350

400

450

current (mA)
Figure 3.4.: Output power of the 1166 nm laser as a function of the current for three different
temperatures. Error bars are not shown for clarity.

the gain chip was between 109 mA for 21 ◦C and 115 mA for 23 ◦C. Although the gain chipis specified up to 800 mA the output power shows a saturation after 400 mA. Since only
20 mW of 1166-nm light are needed for seeding the Raman fiber amplifier we restricted
the the maximal current used to below this value to avoid this problem. Unfortunately,
the Quantum Dot chip behaves in frequency domain differently than what expected (seediscussion below). Therefore, we finally decided to buy a commercial system as a seed laserat 1166 nm.

Raman fiber amplifier
The commercial system consists of three elements:• Ytterbium Fiber Laser

25

3. Laser Systems for Er-Dy Double MOT• Raman Fiber Amplifier• Second Harmonic Generator
The working principle of the amplifier is based on induced Stokes-Raman-scattering, where
both a pump photon and a photon of the desired frequency enter the optical material.
For the Raman fiber amplifier the optical material is based on a single-mode polarizationmaintaining fiber. The Stoke-Raman-scattering describes a photon at a certain frequency ν1interacting with the material and leaving either with lower frequency ν s = ν1 − νR (StokeRaman-scattering) or with higher frequency νA = ν1 + νR (anti-Stoke-Raman-scattering). νR
is the frequency of an oscillation- or ration mode of a molecule or a solid. The entering
photon either transfer energy to the medium or gets energy from the medium. The induced
Stoke-Raman-scattering occurs when a photon enters the medium together with a pump
photon of higher frequency (Stoke-Raman-scattering). The photon induces the emission of
a second photon, which has the same frequency as the first one. In our case, this leads toan amplification of the seed light by about a factor of ≈ 300.
After the amplification stage, the light gets frequency doubled in a Second Harmonic
Generator (SHG), which in this case uses a periodically-poled lithium niobate crystal(PPLN) in single-pass configuration. The special property of a non linear crystal is that thepolarization of the medium does not depend linearly on the electric field. The polarizationof the material can be written as Taylor expansion. For a non linear crystal, where higher
orders except the second can be neglected, the non-linear polarization of the medium independence of the electric field E = Re{E(ω) exp(iωt)} of the incident light can be written
as(3.1)
Pnl = χ(2) E 2 .
χ(2) is the second order electric susceptibility, describing the strength of the non linear
response. In general, χ(2) is a tensor of 3th rank. However, for simplicity we take Pnl and
E to be scalar quantities. Inserting the electric field E = Re{E(ω) exp(iωt)} gives for thepolarization
Pnl (t) = Pnl (0) + Re{Pnl (2ω) exp(i2ωt)},which consists of the linear term Pnl (0) and the frequency doubled term Re{Pnl (2ω)ei2ωt }.
PPLN is a material engineered being quasi-phase-matched. This means, that the orientation of the electric susceptibility of the crystal is inverted periodically. This avoids that thegenerated photons run out of phase to each other and interfere destructively. The needed
period of the alternating crystal is set by the desired wavelength to be generated. The
doubling efficiency, according to the company, is about 30 %.
Figure 3.5 shows the optical setup of the home made 1166 nm seed laser together withthe commercial Raman fiber amplifier and the SHG crystal. An optical isolator5 is placed toprevent that light is back reflected into the laser and disturbs the behavior of the gain chip.
The 1166-nm light is fed into the Raman fiber amplifier and transported to the SHG crystal
via optical fibers. Afterwards, the 583-nm light is spitted into three paths, which go to the5

Isowave Manufacturing: I-11B-4.

26

f = 500mm

626 nmPBS

HCL

fiber output
couplerdichroic mirror

fiber output
coupler

3.2. Laser Systemcavity
for 583-nm Light for Erbium
experiment, the cavity, and the wavemeter. The intensity of the 583-nm light, which goesto the experiment is stabilized with an acousto-optical modulator6 (AOM).

1166 nmλ/2

experimentfiber laser+
fiber amplifier

PBS

λ/2

583 nmwavemeter

experiment

ECDL

f = -30 mm f = -50 mm

BS
f = 50 mm
cavitycavitywavemeter

λ/2

optical isolator

f = 100 mm
SHG

λ/2

f = 200 mm

f = -50 mm

AOM

λ/2

PBS
PBS

Figure 3.5.: Drawing of the optical setup for the generation of the 583-nm light. The setup is divided
into the seed laser at 1166 nm and the SHG crystal with the 583 nm light.

1550 nm

Figure 3.6(a) shows the output power of the 583-nm light for different pump laser currents.
The current is representative for the pump power. The output power also depends on thetemperature of the crystal as the phase matching condition for the SHG crystal has tof = 50mm
be fulfilled.
As the temperature inside the crystal changes with the total power, the set
temperature of the SHG crystal has to be adjusted for each current to maximize the output
power. Figure 3.6(b) shows the output power as a function of the crystal temperature when
the pump power was kept constant (4.5A). To extract the temperature at which the output
power is maximal we fit P(T ) = A + B · sinc2 (C(T − D)) to the data points. The fit describes
thefcentral
part well and we obtain a maximum power at a temperature of 64.97(2) ◦C. The= 200mmidea for the locking of the laser is to generate the error signal (see Sec 5.4) with the 583-nm
light
feed the error signal
to the 1166-nm ECDL.
wavemeter
f =and50mm

f = 200mm
λ/2
λ/4

cavityf = 100mm
6

Gooch & Housego, 3080-215.

PBS

λ/2

BS
AOM

experiment27

3. Laser Systems for Er-Dy Double MOT

2.5

power 583 (W)

2
1.5
1
0.5
0

0

1000

2000

3000

4000

5000

6000

7000

8000

current (mA)(a) Power of the 583-nm light as a function of the current of the pump laser.

power 583 (mW)150
data points
A + B " sinc2 (C(T ! D))
100

50

0
60

61

62

63

64

65

66

67

(b) Power of the 583-nm light as a function of the temperature of the SHG crystal.

Figure 3.6.: Output power of the system for the generation of 583 nm light. (a) shows the output
power of 583-nm light vs. the current of the Ytterbium fiber laser. In (b) the output power of 583-nm
light vs. the temperature of the SHG crystal is plotted. The amplifier current was at 4.5 A. For bothmeasurements the seed power at 1166 nm was at 25 mW.

28

3.3. Laser System for 626-nm Light for Dysprosium

Issues with the Seed Laser at 1166 nm
After coupling the 583-nm light into our home built cavity (see Chap. 4) we observed thatthe laser was not always running single mode and sensitive to the current. Already a veryslight change of the current lead to a mode-hop or a change from single-mode to multi-mode.
Furthermore, it happened that after running single mode for some time the laser jumps
into multi mode. The minimum time between we observed being single mode and multimode was only a few minutes, whereas the maximum time observed was 1 to 2 hours. The
current plateau, where the laser was single mode, was determined to be 0.6 mA, whichmakes single-mode operation at the desired wavelength very challenging. Additionally, it
was observed that the long term stability of the wavelength was not satisfying. To test this,the laser was kept switched on for one night, whereby the next morning the laser was multimode and it was not possible to get the laser back to the desired wavelength by adjusting thecurrent and the temperature. A priori it was not clear if the laser diode itself, the electronics
for the diode current or the temperature stabilization are responsible for this behavior. After
careful testing of different electronics and checking the temperature stability, we concluded
that the laser diode itself was responsible for the frequency instabilities. A possible solution
for this issues is to use a different type of gain chip. For an ECDL, two types of gain chip are
available: type A and type B. While type A has a straight stripe normal to the facets type
B has a curved stripe. The tilted stripe has the advantage of reducing the self lasing andminimizing gain ripples. Another improvement would be the reduction of the cavity lengthof the ECDL since this would increase the current plateau where the laser runs single-mode.
From the experiences of previous experiments [33] an optimal intensity for the narrow
line MOT of Er is 12 × Isat , which leads to a total power requirement of
P583 = 12 · Isat · A · 3 ≈ 60 mW,(3.2)

where the factor 3 refers to the three MOT beams (we use them in retro reflecting configuration), A = π · 4 cm2 is the beam area with a beam waist of 2 cm and the saturation
intensity Isat = 0.13 mW/cm2 (see Table 2.5). This value for the power is just a reference
value estimated with the experience of the running experiment. The exact necessary power
has to be determined empirically.

3.3. Laser System for 626-nm Light for Dysprosium
The laser system we use for the creation of 626-nm has been implemented successfully
in different experiments, e. g. in the groups of D. J. Wineland at NIST [79] and T. Pfau in
Stuttgart [61]. The light is generated via sum frequency generation (SFG) of 1550-nm7 and1050-nm8 light in a periodically poled lithium niobate (PPLN) crystal9 . Fig. 3.7 shows the
optical setup for the generation of the light 626 nm. An AOM10 stabilizes the intensity,
which goes to the experiment. The two laser systems consist of a seed laser and a fiber
7

NKT Photonics: Koheras Boostik HPA Y10 System.
NKT Photonics: Koheras Boostik HPA E15 System.
9
Covesion: MSFG626-0.5-40.
10Gooch & Housego, 3080-215.
8

29

PBS

3. Laser Systems for Er-Dy Double MOT

f = 50mm

1550 nm1050 nmf = 100mm

f = 200mm
f = 150mm
f = 175mm

f = 50mm

λ/4
λ/2

wavemeterf = 200mm
λ/2
λ/4

cavityf = 100mm

PPNL

PBS

λ/2
PBS
AOM

λ/2

experiment

λ/2
Figure 3.7.: Optical setup for the SFG of the 626-nm light. The 626-nm light is splitted into threepaths: one is going to the experiment, one to the fiber coupled EOM/cavity and one goes to thewavemeter.

amplifier. The fiber amplifier for the 1550-nm system is a Er doped fiber laser, whereas
the fiber amplifier for 1050-nm is ytterbium doped. The laser systems provide an output
power of 5 W each and have a free running linewidth of < 1 kHz (1550 nm) and < 20 kHz(1050 nm). Both lasers can be tuned with a piezo element that has a modulation frequency
limit up to 20 kHz. To lock the 626-nm we couple it into the ULE cavity and generatea PDH error signal, which we feed back to the 1050-nm seed laser while the system of1550 nm remains free running (linewidth < 1 kHz). This is possible, because the lasers havea large mode-hop-free detuning range (10 GHz for 1050 nm and 8 GHz for 1550 nm) and a
sufficient long term stability. Since the linewidth of the laser systems are small enough,the locking only needs to account for long term drifts. A more detailed description of thelooking scheme is given in Sec. 5.4.
To generate the 626-nm light as efficient as possible the phase matching condition~k626 = ~k1550 + ~k1050 has to be fulfilled for the SFG crystal. The phase matching condition
is fulfilled via the PPLN crystal, as explained already for the SHG crystal in Sec. 3.2. For the
SFG crystal three different periodicities are available within one single crystal, 11.12 µm,
11.17 µm, and 11.22 µm, whereby we use the 11.22 µm. The quasi-phase-matching can be
influenced slightly by varying the temperature of the crystal. In Figure 3.8(a) the powerof the 626-nm light is plotted against the temperature of the PPLN crystal and fitted with30

3.3. Laser System for 626-nm Light for Dysprosium
P(T ) = A + B · sinc2 (C(T − D)). Although the data points are asymmetric, the sinc2
describes the central part sufficiently good to extract the temperature with maximal power
to 170.00(1) ◦C.
Additionally to the phase-matching, the intensity of the pump beams is important. Therefore, the 1550-nm and the 1050-nm light is focused into the crystal fulfilling approximately
the condition z0 = l/2, where z0 is the Rayleigh length and l the length of the crystal. This is arule of thumb recommended by Coevision. Our crystal has the dimensions (40×10×0.5) mm.
Similar to the SHF in Sec. 3.2, the SFG is based on the polarization P of the non-linear
crystal. Inserting Einc = E(ω1 )eiω1 t + E(ω2 )eiω2 t in Eq. 3.1 leads to
PNL (ω1 + ω2 ) = χ(2) · E(ω1 )E(ω2 )ei(ω1 +ω2 t) ,(3.3)

oscillating at ω1 + ω2 . Therefore, the amplitude of PNL (ω1 + ω2 ) depends on the product of
the amplitudes of the two incoming waves. In Figure 3.8(b) the product of the power of thepump lasers is plotted against the power of the 626-nm light. The conversion efficiency
can be determined from the slope of a linear fit (B · x), to the data points toηconv = 9.48(34) %As for the 583-nm light the optimal intensity for the MOT can be estimated from theexperience of a running experiment. In [24] a power of ≈ 50 · Isat is used for each beam,leading toP626 = 50 · Isat · A · 3 ≈ 135 mW,(3.4)
where Isat = 72 µW/cm2 is the saturation intensity and A = π · 4 cm2 is the beam areawith a beam waist of 2 cm. The factor 3 arises from the three beams in retro reflecting
configuration, which we will have for the MOT. This value is just a reference estimatedwith the experience of the running experiment [24]. The final necessary power has to be
determined by optimizing the MOT loading phase in the experiment.

31

3. Laser Systems for Er-Dy Double MOT

2.5
data points
A + B " sinc2 (C(T ! D))power (W)

2

1.5

1

0.5

0
165

166

167

168

169

170

171

temperature (/ C)

172

173

174

175

(a) Output power vs. Temperature of the SFG crystal at an amplifier current of I1550 = 8.3 A(I1050 = 6 A) and a seed laser output power of P1550 = 6.7 mW (P1050 = 6.6 mW).
2.5

P626 (W)

2

1.5

1

0.5

0

0

5

10

15

P1550 " P1050 (W2 )2025(b) The output power of 626-nm light is plotted versus the product of the power of the 1550 nm
and 1050 nm laser (circles). The data points are fitted with a linear function B · x.

Figure 3.8.: Output power of the system for the generation of 626 nm light. (a) shows the output
power of 626-nm light in dependence of the temperature of the PPLN crystal. (b) shows the output
power of 626-nm light in dependence of the product of the power of the 1550 nm and 1050 nm laser.

32

4. Design of the Vacuum Chamber
and the ULE Cavity
The present Chapter describes the core project of my master thesis. Indeed, the projecttargets to build a stable cavity to lock narrow-line laser beams. As discussed in Chap. 3, thecavity will be used for the Er and Dy MOT light. In addition, our experiment also aims at
studying Rydberg physics in Er. To excite ground-state atoms to a Rydberg level, we plan
to use a two photon excitation, both in the UV. For efficient transfer, the laser light for theexcitation needs to have small linewidth. Our cavity is indeed designed for light for the
MOTs as well as for the Rydberg excitation light.
In this chapter, first, we discuss the frequency stability of light and give a short overview of
the theoretical aspects regarding the cavity. Afterwards, we discuss the cavity requirements
and describe the setup of the vacuum chamber for the cavity.

4.1. Frequency-Stability Requirements of the LightThe narrow line magneto optical traps (MOTs) of Er and Dy [33, 61] as well as Rydberg
excitation schemes require lasers with a high frequency stability. The stability of a laser
is described by the frequency jitter, which can be divided into fast and slow jitter. The
division between slow and fast jitter depends on the time scale. The fast jitter is practicallynot resolvable on the respective timescale and corresponds therefore to the linewidth ofthe laser light. For the short term stability of the laser the fast jitter is relevant and thewidth in frequency should be smaller than the linewidth of the transition. For Example, if
the linewidth of the laser light for the MOT is broader than the transition linewidth of theatom, it is possible, that the necessary detuning is not guaranteed anymore. In [16] it isshown, that trapping of atoms with a natural linewidth of ≈ 6 MHz with laser light with alinewidth of 10 MHz leads to a 40 % lower atom number compared to trapping with laser
light with a linewidth of 1 MHz.
The linewidths of the cooling transitions for Er and Dy are on the order of 100 kHz
and 2 kHz–10 kHz (see Table 2.5 for transition parameters), whereby the linewidths for thelasers for Rydberg excitation are on the low kHz level [57]. For laser cooling the time scale
for the fast jitter is in the range of 1 µs to 1 ms, because it is the time scale where the atoms
scatter photons.
The slow jitter can in practice be resolved and is therefore referred to as frequency drift
and related to the long term stability. In our case, the frequency drift over one day should besuch low that the laser remains resonant to the atomic transition. This means, the laser issatisfying stable if the frequency drift over one day is on the order of a few kHz. If the long33

4. Design of the Vacuum Chamber and the ULE Cavity
term stability of the lasers is unsatisfying, the lasers would have to be readjusted regularly.
One possibility to stabilize the lasers to the desired wavelength is to use a lock to astable reference cavity. For example the frequency drift of a similar cavity [46] over 7 h
was estimated to 2.77 Hz/s leading to a satisfying long term stability. Cavities can also be
used to improve the short term stability, as it has been shown in [71], where for a dye laser
at 583 nm a linewidth of 45 kHz was obtained and in [53] the linewidth of 200 kHz of a
741 nm external cavity diode laser was reduced to 1 kHz.

4.2. Basic Concepts of Cavity Physics
The aim of this section is to give a brief overview of the theoretical basics of a Fabry-Perotcavity formed by two flat mirrors as illustrated in Fig 4.1. A more detailed description canbe found in many textbooks, e. g. [72]. For the derivation of the most important properties
for this thesis it is sufficient to concentrate on ray optics describing the incoming wave as
Einc = E0 eiωt ,(4.1)

with ω being the laser frequency and E0 the complex amplitude. The electric field picks upt,rt,rt3 r2 Einct2 r2 Einct2 r Einct r Einct EincEincFigure 4.1.: Fabry-Perot cavity and electric field of the incoming beam. t and r label the transmission
and reflection coefficient, respectively.

an additional phase while traveling between the mirrors which can be expressed as∆φ = ω

(n + 1)L
,
c

(4.2)

where L is the distance between the mirrors of the Fabry-Perot cavity, c is the speed oflight and n labels the number of reflections before leaving the cavity. To obtain the totaltransmitted light field, the partially transmitted fields have to be summed up as
Etrans = Einc tei∆φ + r2 tei3∆φ + r4 tei5∆φ + · · · ,(4.3)
where r is the reflection coefficient and t the transmission coefficient of the mirrors. For
the derivation we assume perfect mirrors neglecting any losses. Both coefficients are thenconnected via
r2 + t2 = 1.
(4.4)

34

4.2. Basic Concepts of Cavity Physics
Using the geometric series and taking the square of the transmitted electric field leads tothe transmitted intensity
T2
.
(4.5)
Itrans = Iinc
T 2 + 4R sin2 (∆φ)
In the same way, the reflected intensity can be written as
4R sin2 (∆φ)
Iref = Iinc − Itrans = Iinc,
T 2 + 4R sin2 (∆φ)

(4.6)

where T = t2 is the transmission and R = r2 is the reflectivity. The transmission signal of acavity is plotted in Fig. 4.2 for two different reflectivity values.
Cavity Transmission

relative intensity1

FSR

r = 0.9
r = 0.99

0.8
0.6
"8
0.4
0.2
0

n-2

n-1

n

n+1

n+2

frequency/FSR

Figure 4.2.: Theoretical plot for the transmission of a cavity. The signal in blue is for a reflectivityof 0.9, whereas the red signal is for a reflectivity of 0.99. The spacing between two cavity-modesccan be expressed as νn − νn−1 = 2L
and is equal to the FSR.

Important parameters of a cavity are the free spectral range (FSR) and the full width at
half maximium (FWHM). The FSR
c
(4.7)
FSR =
2L
describes the distance in frequency between two neighboring longitudinal modes. The
full width at half maximum (FWHM) corresponds to the linewidth ∆ν of the transmission
peaks. Assuming low losses in the cavity it can be calculated as"#
FS R(1 − R)2∆ν =arccos 1 −
.
(4.8)
π
2R
An additional important parameter to characterize a cavity is the finesse F , which simply
is the ratio between the FSR and the FWHM:
F =

FSR
.
FWHM

(4.9)

35

4. Design of the Vacuum Chamber and the ULE Cavity
For the special case of low losses and high reflectivity as in our case, the finesse can be
expressed as
√
π R
F =
.
(4.10)
1−R
The linewidth of the resonance can be interpreted as a decay of optical energy due toresonator losses and is therefore related to the average photon lifetime, τP , in the resonator:
τP =

1
,
2π∆ν

(4.11)

where ∆ν is the linewidth of the cavity. If we consider wave optics instead of beam optics,
beside the longitudinal modes, only differing in frequency, also higher transversal modes
exists, which differ in their intensity pattern and can be different in frequency. In wave
optics the solution of a resonator is described by the Hermite-Gauss functions, written as
TMEnm with the intensity distribution  √ ! 2   √ !2  2 −x2    2 −y2  Hm  , exp expI(x, y, z) = I0 Hn 
w(z)
w(z)2
w(z)
w(z)2
where Hk is the kth Hermite polynomial, w(z) is the waist (see below), and n and m arepositive integers. In Figure 4.3 the intensity distribution in the x-y plane for different TME
modes is plotted. The TME00 is called the Gaussian mode and is the only mode which has
the same resonance conditions as obtained with ray optics, whereas higher TME modes
have different resonance conditions depending on the cavity geometry. Those higher
cavity modes can have a negative effect on the locking if they are to close in frequency
to the TME00 mode and therefore influence the PDH-error signal (see App. A.1). For thegeometrical case in which one cavity mirror is flat whereas the other mirror is spherical,
the frequency difference between two TMEnm modes is given by [28]
s



c (n + m + 1)
L
arccos  1 −  ,δν =(4.12)
2Lπρ
where L is the length of the cavity and ρ is the radius of the spherical mirror. For ourcavity parameters of L = 0.15 m and ρ = −500 mm this leads to δν ≈ 210 MHz. Since wewill lock the laser to the TME00 mode it will be important, that the adjacent cavity modes
are far enough away from the sidebands for the generation of the PDH error signal (seeSec A.1). Otherwise, the latter could be disturbed by the close lying cavity modes. Since
our FSR equals approximately 1 GHz, cavity modes with m + n ≥ 5 have to be sufficiently
suppressed. This is easily possible with a good cavity coupling.
For the description of the coupling to a cavity the laser beam has to be treated as a
Gaussian beam, which can be characterized by the q-parameter
1
1
iλ
=
−
,
q R(z) πW(z)236(4.13)

-20

-20

-10

-10

y (nm)y (nm)

4.2. Basic Concepts of Cavity Physics

0
10
20

0
10
20

-20

0

20

-20

x (nm)(b) TME01

-20

-20

-10

-10

y (nm)y (nm)20

x (nm)(a) TME00

0
10
20

0
10
20

-20

0

20

-20

x (nm)

020x (nm)(c) TME11(d) TME22

-20

-20

-10

-10

y (nm)y (nm)

0

0
10
20

0
10
20

-20

0

x (nm)(e) TME04

20

-20

0

20

x (nm)(f) TME33

Figure 4.3.: The intensity distribution of different transversal cavity modes is plotted.

37

4. Design of the Vacuum Chamber and the ULE Cavity
containing the curvature"R(z) = z 1 +and the waist of the beamsw(z) = w0

 z 2 #
0

z

!2
z
,
1+
z0

(4.14)(4.15)

where λ is the wavelength, z is the expansion direction of the beam, w0 is the minimum
waist in the focus of the beam and z0 is the position where the curvature of the beam ismaximal, respectively. To couple a laser beam to a cavity, it is important that the beam
travels exactly along the axis of the mirrors and that the curvature of the wavefront of
the beam equals the radius of the mirrors. Since we use one planar mirror and one mirror
with a radius of ρ = −500 mm the beam has to be focused on the planar mirror and bycombining Eq. 4.14 and Eq 4.15 one obtains that the beam waist in the focus W0 has to ber
Lλ ρ − Lw0 =.
(4.16)πL
By inserting our cavity parameters, L = 15 cm and ρ = −500 mm, we obtain for example abeam waist of W0 = 214 µm for λ = 626 nm. In Sec. 5.1 the optical setup for the coupling to
the cavity is explained.

4.3. Cavity Requirements
Reflectivity Requirements
An important requirement for our cavity is the possibility to frequency stabilize light atseveral wavelengths. On the one hand, we want to stabilize the light for the MOT of Er and
Dy and, on the other hand, we want to lock the light for Rydberg excitation. Table 4.1 gives
a list of all required wavelengths or wavelength ranges. The 583-nm transition for Er andthe 626-nm transition for Dy are used for the MOTs, respectively. The 631-nm transition
and the 741-nm transition are even narrower transitions, that can be used for additionalcooling [56] or state preparation. The exact wavelengths for the Rydberg excitations are
not known yet, as many different excitation schemes are possible. For this reason a broad
wavelength range was chosen. Furthermore, the largest isotope shift for Er and Dy is about
1 GHz (see Fig. 2.2), that suggests to use a cavity with L = 15 cm, which leads to a FSRof 1 GHz. This makes the switching between the isotopes easier because the laser can belocked to the next cavity resonance (see Sec. 5.4).
The multiwavelenght operation requires to employ cavity mirrors with high reflectivity
over a wide frequency range. The reflectivity of a mirror is related to the Bragg diffraction
condition. For high reflection coatings a periodic layer structure composed of different
materials enhances the reflectivity of the mirror because they lead the reflected beams tointerfere constructively. By varying the thickness and the composition of the layers thecharacteristics of the mirror can be manipulated and high reflective and anti-reflective38

4.3. Cavity Requirements
Table 4.1.: List of all wavelengths we want to stabilize.

ErbiumDysprosium

Wavelength (nm)Linewidth (kHz)Wavelength (nm)Linewidth (kHz)583
631
380–450

190
28
few kHz

626
741
–

136
1.78
–

regions can be created. Due to technical limitations, it is hard to produce one highlyreflective (> 99.96 %) mirror coating for all our required wavelengths. Therefore, we use
two sets of cavity mirrors of different mirror coatings. The mirrors for one cavity are coated1
for the 583 nm, 626 nm, 631 nm and 741 nm transitions, while the mirrors for the othercavity are coated for all wavelengths in the range from 380 nm–450 nm. Figure 4.4 shows
the theoretical transmission curves for the coatings, provided by Lens-Optics. Table 4.2
gives the theoretical reflectivity and the related values for F . For each wavelength thefinesse for the theoretically best and worst case are calculated by Eq. 4.10. In comparison toother experiments, e. g. [4, 74] our cavities will have a relatively low finesse. Nevertheless,this is sufficient since the light of the lasers does not have to be narrower than 1 kHz.
Table 4.2.: Reflectivity and rough estimation of the expected finesse for the different wavelengths.
For the calculations the theoretical curves from Fig. 4.4 and Eq. 4.10 are used. The 410 nm line is one
possible transition for the Rydberg excitation.

FR(%)
wavelength (nm)583
626 & 631
741
410

min

max

min

max

99.91 99.96 4500 7800
99.95 99.98 6300 20 000
99.95 99.98 6300 15 700
99.92 99.98 4000 15 700

Cavity Stability
To improve the stability of the length of the cavity and therefore of the resonance frequency a spacer out of ultra low expansion glass2 (ULE) between the cavity mirrors isplaced. Although we have two cavities we use only one ULE spacer. This design is already
implemented successfully in a running experiment [74]. The special property of ULE isthat the linear coefficient of thermal expansion (CTE) has a zero-crossing point near room
temperature providing a CTE3 below 2 × 10−9 K1 within one degree around this point and1

The mirrors are manufactured and coated by Lens-Optics GmbH.
From Hellma-Optics, ULE premium grade.
3
Corning, manufacturer for glass and ceramic.
239

4. Design of the Vacuum Chamber and the ULE Cavity

(a)

(b)

Figure 4.4.: The different coatings, used for the two sets of cavity mirrors are shown. (a) Coating for the stabilization of the 583 nm, 626 nm,
631 nm and 741 nm lasers. (b) Coating for lasers in the range of 380 nm-450 nm used for Rydberg excitation. Graphs are provided by Lens-Optics.

40

4.3. Cavity Requirements
therefore high stability against length fluctuations. Unfortunately the exact temperature
for the zero-crossing can be different for each production series and is therefore specified
only to be between 0 ◦C and 30 ◦C.
The cavity mirrors are optically contacted to the ULE spacer, which is manufactured
by Hellma-Optics. Also the optical contacting of the mirrors to the ULE spacer was doneby Hellma-Optics. Before choosing the mirror substrate we considered several aspects:
Combining different production series of ULE or even different materials as ULE and Fused
Silica (when using mirrors out of Fused Silica) could shift the zero-crossing temperatureout of the specified region. Fused Silica has a CTE of about 5 × 10−7 K1 being roughly
independent of temperature. Due to the fact, that the CTE of Fused Silica is positive andthe CTE of ULE gets negative for temperatures below the zero-crossing, the minimum CTEof the ULE spacer together with the Fused Silica mirrors gets shifted to lower temperatures.
This makes the temperature regulation more elaborate. Fused Silica mirrors have theadvantage that they can reduce the fractional instability from thermal noise compared
to ULE mirrors [51]. Nevertheless, for our purpose of a relatively low finesse cavity thestability of ULE mirrors does not limit us.
Beside the reflectivity of the mirrors, it is important that the mirror substrate does notabsorb to much light. The transmission of ULE at our desired wavelengths is sufficiently
high (see gray area in Fig. 4.5) and therefore, we decided to use cavity mirrors made out of
ULE glass.

Figure 4.5.: Transmission curve of ULE glass. The gray area indicates approximately our wavelength
range. Graphic taken from Präzisions Glas & Optik GmbH. December, 2016.

Regarding the long term frequency stability, we aim to stabilize the lasers in such a
way that they do not drift more than the linewidth of the transition over a long enoughtime period, ideally one day. In our case the smallest linewidth is 1.78 kHz for the 741 nm
transition of Dy. Thus, the cavity should have a long term stability on the order of ∆ν .
1 kHz. The resonance frequency of a cavity depends on the optical length nL (see Eq. 4.7)
and therefore the relative frequency deviation ∆ν of the cavity resonance can be expressed41

4. Design of the Vacuum Chamber and the ULE Cavity
by the relative length change ∆L/L and the relative change of the refraction index ∆n/n:
s
!2
!2
∆n
∆ν
∆L
=
+
.
(4.17)
νLn
For our requirement of ∆ν . 1 kHz we need to reach a relative stability of∆ν≤ O(10−12 )ν(4.18)First, we look at the influence of change in the refraction index: by using the ClausiusMossotti Equation [52]n2 − 1 Nα=(4.19)
n2 + 2
3
we can make a connection between the refraction index n and both, the temperature T
and the pressure p. Since the refraction index for air is on the order of 1 + O 10−4 andfor vacuum even much lower, we can use the taylor expansion for Eq. 4.19, expanding n2
around 1, leading to
n2 − 1 n2 − 1=.
n2 + 2
3
Furthermore, inserting the molar refractivity A, which is a measure for the total polarizabilityof a mole, one obtains for the refractive index
r
3Ap
n= 1+
,
(4.20)
RT
where R is the universal gas constant. For the relative change of the refractive index one
obtainss!2!2∆p∆n n − 1∆T=+.
(4.21)nnp
T2
Calculating the refractive index for a temperature and pressure comparable with our system,
i. e. 20 °C and 10−8 mbar, leads to a relative change of refraction index on the order of 10−15 .
Obviously, the change in temperature and pressure should be moderate for this estimation.
Therefore, in the following we can neglect the influence of the refraction index on thestability of the cavity resonance and focus on the length stability.
The length change of the the ULE spacer can be described as∆L = α(T ) · L · ∆T,(4.22)

where α is the coefficient of thermal expansion (CTE) and L the length of the cavity. A special
property of ULE is that the CTE has a so called zero-crossing around room temperature. In
Figure 4.6 the CTE is plotted between 0 ◦C and 44 ◦C with the zero-crossing temperature
T C at 20 ◦C.
This shows, that having a temperature stability of 10 ◦C around T C gives a CTE on theorder of 1.5 × 10−8 /K. Assuming the CTE to be linear around T C gives a maximal CTE

42

-150

-100

-50

0

50

100

150

200

250

300

-140 -120 -100

Temperature [˚C]

4.3. Cavity Requirements• CTE verification is ach• Stability: Excellent lon
No residual figure cha
quench.
• Delayed elastic effect:effect in Corning 7973
strain is present durin
present, such as durin• No measurable hyster

Instantaneous CTE
30

Expansivity [ppb/C]
Expansivity(ppb/˚C)

30.0

20

20.0

10

10.0

0

0.0

-10

-10.0

-20

-20.0

-30

-30.0

10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
00 22 44 66 88 10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Temperature [˚C]

Temperature (˚C)

Figure 4.6.: This Figure shows the zero-crossing of the instantaneous CTE for ULE at an example
3
temperature of 20 ◦C. Figure is taken from Corning, Product Information Sheet 7973 July, 2015

on the order of 1.5 × 10−11 /K for a temperature stability of 10 mK. Therefore, having atemperature stability on the order of 10 mK gives a relative length stability on the order of∆L= O(10−11 ) · O(10−2 ) = O(10−13 ).
L(4.23)

This is easily sufficient for our application (see Eq. 4.17).
Another effect to take into account is the length change due to vibrations. The effect ofvibrations to relative length changes can be calculated by [15]∆L ρgh=σ,L
2E(4.24)

where ρ = 2.21 g/cm3 is the mass density, g is the acceleration, h the hight of the ULE
spacer, E = 67.6 GPa the elastic modulus and σ = 0.17 the Poisson’s ratio4 . This results ina sensitivity of the relative frequency stability to vertical accelerations of 2.8 × 10−10 /ms−2 .
Vibrations in the horizontal direction should have a much lower influence because of thesymmetric mounting [45]. For the lab typical values for the vertical accelerations are on
the order of 3 mms−2 , which leads to a minimum stability on the order of 10−13 . Therefore,
we can assume that the vibrational stability of the cavity is sufficient for this purpose.
Another point to be noted is the drift of the frequency caused by relaxation effects ofthe cavity due to aging of the ULE spacer. In [26] the long term drifts of the cavity is fitted
by two exponential decay functions superimposed on a linear function with a slope of
13(2) mHz/s. The two time constants of the decay functions describing the long term drifts
are 32(7) days and 260(80) days. The positive sign of the linear slope indicates that thecavity length decreases and is therefore an indication for aging of the material of the spacer.

4

Values for the material specific parameters can be found into data sheet of ULE® Corning Code 7972.

43

4. Design of the Vacuum Chamber and the ULE Cavity

4.4. Mechanical and Electrical Design of the Vacuum
Chamber
The design for the housing of the cavity was inspired by the master theses of Alexander
Ritzler and Maximilian Segl [71, 74].

Vacuum Chamber Design

20 l/s ion-getter pumpedge-valve

CF63 viewport
CF63 viewport

Sub-D25 feedthrough CF63

aluminum baseplate

Figure 4.7.: Drawing of the vacuum chamber which houses the ULE cavity. The baseplate has a sizeof 400 mm × 400 mm. The height of the chamber is about 300 mm. The vacuum chamber is made ofstainless steel, whereas all interior parts as well as the baseplate are made of aluminum. All vacuum
parts are of the CF type. For optical access two CF63 viewports are implemented. A feedthroughwith a Sub-D25 connector is used for the connection of the NTCs and the Peltier elements and a 20
l/s ion-getter pump is installed for pumping.

Fig. 4.7 shows the complete design. The vacuum chamber has a cylindrical shape and isclosed by two CF2505 flanges. The vacuum chamber has two CF63 viewports for opticalaccess and an electrical feedthrough with a Sub-D25 connector for electrical connections.
The viewports are attached at an angle of 3° to avoid disturbing reflections. We use an
5

ConFlat (CF) is a special flange design which uses a knife edge in combination with a copper gasket as asealing technique. The number stands for the DN type (DN250 in this case) and indicates the diameter ofthe system.

44

4.4. Mechanical and Electrical Design of the Vacuum Chamber
all-metal angled valve for the initial pump down with a turbo pump and a 20 l/s ion-pump6
to hold the pressure afterwards at the low 10−8 mbar level.
The inner part of the vacuum chamber is formed by two consecutive shieldings as shownin Fig. 4.8 which shows a cut through the vacuum chamber. The ULE spacer forming thecavity together with the mirrors is sitting on a V-shaped block supported by 4 rings madeof Viton, intended to reduce vibrational distortions. The location of the Viton rings isdetermined by the so called Airy points, a/L ≈ 0.2113, at which the bending of a held bar
is minimized and the facets of the end of the bar are parallel to each other. L relates to thelength of the ULE spacer and a is the distance between the Airy point and the nearer endof the bar.
We temperature stabilize the outer shielding via six Peltier elements7 , whereas the innershielding is sitting on Teflon spacers to ensure heat is transported mainly by radiation fromouter to inner shielding. The reduced heat transport results in a long thermalization time
constant, which corresponds to the principle of a low-pass, i. e. fast temperature changes
below the characteristic time constant are suppressed providing a higher stability against
temperature fluctuations. In Figure 4.9 the temperature of the aluminum block that holds
the cavity is plotted against time. The temperature of the outer shielding was set from
32.5 ◦C to 30.5 ◦C. With an exponential fit the thermalization constant was determined
to 9.6(7) h. The dip around 25 h was probably caused by a temperature failure of the air
conditioning system of the lab and is an artifact deriving from the measurement method.
The temperature in the lab increased by a few degree and this influenced the values of theresistors used for the voltage dividers to read out the temperature. The inner part of thecavity should not have experienced the temperature change.
Due to possible flatness imperfections of the Peltier elements, we use tin sheets with athickness of 100 µm to compensate unevenness and therefore to increase the heat conductivity. We glue coated windows into the holes of both shieldings to have optical access tothe cavities. Further, this should give additional shielding against thermal noise. All metallic
inner parts of the vacuum chamber are made of aluminum for a better heat conductivity.
To remove the heat produced by the Peltier elements, the vacuum chamber sits on a large
heat sink. To reduce the temperature difference for the Pelter elements that stabilize theouter shielding six Peltier elements are used to cool the lower flange of the cavity. At thesame time this Peltier elements temperature stabilize the vacuum chamber. At the time
the housing for the cavity was built the CF250 flanges and the tube were available onlyout of stainless steal and therefore the vacuum chamber itself has a 5 times less thermal
conductance than aluminum. Therefore, a copper plate sitting on the Peltier elements was
included to increase the contact area from the Peltier elements on the baseplate to thelower flange of the vacuum. For future cavity projects aluminum CF vacuum components
are recommended. Figure 4.10 shows temperature simulations8 for both cases, with andwithout the additional copper plate. If we have to go to low temperatures (CTE of ULE
is specified between 0 °C and 30 °C) the heat sink is not big enough to remove the heat6

Agilent Vaclon Plus 20.
Multicomb MCPE-241-10-13 Peltier Cooler, 71.8 W.
8SolidWorks 2014.
745

4. Design of the Vacuum Chamber and the ULE Cavity

inner shielding

outer shielding

ULE cavity

3˚

Teflon spacer
Peltier element

NTC

aluminum heat sinkcopper block for better thermal connection(a) Cut through the vacuum chamber. The cavity spacer is seen from the side.

583nm, 626nm,
631nm, 741nm
380nm - 450nm(b) Cut through the vacuum chamber. The cavity spacer is seen from the front.

Figure 4.8.: Vertical cuts through the vacuum chamber. In (a) the cavity spacer is seen from theside. The Peltier elements are illustrated in green and the Teflon spacer in red. The position of the
NTCs which are used for temperature regulation are marked by a black dot. Many other NTCs are
distributed over the vacuum chamber. In (b) the cavity spacer is seen from the front and the twocavities with the different coatings are labeled.

46

4.4. Mechanical and Electrical Design of the Vacuum Chamber

33
32.5
32
31.5
31
30.5
30

0

5

10

15

20

25

30

35

40

45

50

time (h)Figure 4.9.: Record of the temperature of the aluminum holder of the cavity. Note that the dip at
25 h was caused by a failure of the air conditioning system of the lab. The data points are fittedwith an exponential function y = A + B exp(t/τ), where t denotes the time. The time constant τ wasdetermined to 9.6(7) h.

produced by the Peltier elements. The heat transport of the vacuum chamber to the outershielding by thermal radiation can be calculated by44Q̇ = σ Achamber T chamber− Ashielding T shielding(4.25)
where  is the emissivity of the material, σ is the Stefan-Boltzmann constant, A the surface
area and T is the temperature. By assuming 0.2 for  and a temperature difference of 10 ◦Cleads to a thermal radiation of 15 W from the vacuum chamber to the shielding. This meansthat 15 W of the cooling power of the Peltier elements are used to make up the heatingfrom the vacuum chamber walls.
The temperature difference between the shielding and the vacuum chamber should be
as low as possible and therefore a water cooling system is implemented in the baseplate.
The water cooling system consists of a copper tube that was pressed into a channel on thebottom of the baseplate. Testing the reachable temperature with the water cooling showedthat this simple solution is satisfying for our purpose.

Temperature Control
Thermistors9 (NTCs) for the temperature regulation and monitoring are allocated at different
positions in the vacuum chamber. The monitoring of the NTCs is done with an Arduinosystem10 . The two NTCs planned for the temperature regulation are sitting on the copper
plate below the lower flange of the vacuum chamber and on the outer shielding (see Fig. 4.8).
The temperatures are stabilized with a simple analog PID controller. In the Appendix A.3,
910

Epcos B57541G1, 10 kΩ at 25 ◦C.
Arduino Due.

47

4. Design of the Vacuum Chamber and the ULE Cavity
the measurement bridge, which generates the error signal, and the PID drawing are shown,respectively. For the connection of the electronic components inside the vacuum chamber
we use wires insulated with Kapton™11 . The advantage of using Kapton™is that it has alow outgassing rate compared to standard isolation materials. The wires are soldered to
the Sub-D connector at the feedthrough and the electronic components and to isolate thesolder joints we use tape out of Kapton™.

11

Twisted pair wire, 311-KAPM-060-PAIR1-20M.

48

4.4. Mechanical and Electrical Design of the Vacuum Chamber(a) Heat sink and lower flange without the additional copper plate.

(b) Heat sink and lower flange with the additional copper plate.

Figure 4.10.: Temperature simulations, done with SolidWorks 2014, for the two types of base plates
we tested. It is assumed, that six Peltier elements are sitting on the lower flange, which are heatingwith a total power of 100 W. Below the lower flange 6 Peltier elements should remove the heat andtransport it to the aluminium heat sink. In (a) no additional copper plate was placed and the heatcan not be removed satisfyingly. In (b) the copper plate solves the issue of the heat transport.

49

4. Design of the Vacuum Chamber and the ULE Cavity

50

5. Experimental Part
In this Chapter, we present the main experimental results concerning the characterizationof the cavity performance, i. e. the zero-crossing temperature of the CTE of the ULE glass,the finesse, and the free spectral range of the cavity.

5.1. Optical Setup for the Coupling of the Lasers to theCavityFor a good mode matching between the light and the cavity, it is important to fulfill thecondition from Eq. 4.16r
Lλ ρ − L
,
(5.1)
w0 =
π
L
where λ is the wavelength, L is the length of the cavity and ρ is the radius of the curved
mirror. Table 5.1 gives the theoretically calculated values for the necessary beam waistson the planar mirror wplanar of the cavity. These beam waists can be achieved, by putting a
Table 5.1.: Calculated beam waists on the planar mirror of the cavity wplanar for the light of 583 nm,
626 nm, 631 nm, and 741 nm. The cavity length is L = 15 cm and the radius of the curved mirror isρ = 500 mm.

λ (nm)

wplanar (mm)

583
626
631
741

0.206
0.21370.21450.2325lens in front of the cavity to focus the beam. Figure 5.1 shows the lens in front of the cavity,
forming a lens system with the curved mirror of the cavity. We assume the curved mirrorρwith ρ = −500 mm being a lens with f = n−1= −1000 mm, because the refractive index n
1of ULE is approximately 1.5 .
We calculate the beam waist on the planar mirror of the cavity in dependence of thewaist of the incoming collimated beam via ray transfer matrix analysis. For a lens with a
focal length of 500 mm the incoming beam should have a beam waist of ≈ 0.5 mm. Ourfiber output coupler2 is chosen such that the beam diameter is already on the right order
1
2

Corning, manufacturer for glass and ceramic.
Thorlabs, SM1Z in combination with CP11/M.

51

5. Experimental Part
35cm

15cm

incoming
beam

MHz

λ/2

M

ρ = 500 mm

f = 500 mm
windows

PBS

HCL

outer shielding

λ/2
Figure 5.1.: Schematic presentation for the coupling to the cavity. The laser beam gets focusedthrough the 500 mm lens into the cavity, whereby the curved mirror can be assumed as a lens withfocal length −1000 mm.
fiber output
couplerand for fine adjustment the position of the lens and therefore the beam size can be tuned.
Figure 5.2 shows the optical setup for coupling of the 626-nm light and the 583-nm lightinto the cavity.
fiber output
coupler
M1

M2

M1

lens
f = 50mm
EOM

experiment583 nm

M2

BS

lens
f = 50mmPD
626 nmfiber output
coupler

BS

BS

EOM

S

50 nm

inner shielding

PD

lens
f = 500mm
dichroic mirror
cavitycavityFigure 5.2.: Optical setup to couple the light of 583 nm and 626 nm into the cavity. After the fiber
output coupler the light gets coupled into the cavity with the mirrors M1 and M2. A dichroic mirror
overlaps the 583-nm light (yellow) with the 626-nm light (orange). The light is then focused onto
the planar mirror. The reflected light is picked up with a beam splitter (BS) and detected with aphotodiode (PD).

52

lens
f = 50mm

5.2. Modulation-Transfer Spectroscopy
We guide the laser beams via the fiber-coupled EOM to a breadboard, where the setup to
couple the light into the cavity is placed. A dichroic mirror3 overlaps the 583-nm and the626-nm light and we couple the light with the mirrors labeled by M1 and M2 into the cavity.
We focus both beams into the cavity through the same lens. After the output coupler a50-50 beam splitter splits the reflected signal of the cavity for the generation of the PDH
error signal. One reason to use a 50-50 beam splitter instead of a polarizing beam splitter in
combination with a λ/4 in front of the cavity, as in [71], is that we want to add the 631-nm
light and the 741-nm light in the future and there is no standard dichroic mirror available tooverlap the 626-nm and the 631-nm light. Therefore, the light has to be overlapped with a
PBS and with a λ/4 in front of the cavity the separation of the reflected signal is not possible.

5.2. Modulation-Transfer Spectroscopy
To achieve a stable operation of the cavity, one first needs to determine the value of thetemperature at which the cavity, i. e. ULE mirrors and ULE spacer, exhibits a minimal
thermal expansion. To determine this temperature, we measure the resonance frequency ofthe cavity because a change of the cavity length results in a change of the free spectral range(FSR) (see Sec. 5.3). We measure the relative length of the cavity by measuring changes in theresonance frequency with respect to the frequency of a reference signal. At the time when
this measurement was done we had only the laser of 626 nm for Dy available and therefore
we used this laser for the measurement. Since Dy has a high melting point (see Sec. 2.1),
one would need a high temperature oven to create neutral hot Dy vapor. Instead, we use ahollow cathode lamp (HCL) to produce the atomic sample. The HCL is a glass cell with acathode and an anode inside. The cathode is coated with the desired element, in our casewith dysprosium. The glass cell is filled with argon, which serves as a buffer gas and gets
ionized in an electric discharge process when a high voltage is applied between cathode andanode. The applied voltage during the spectroscopy was 117.1 V with a current of 14.9 mA.
After the ionization the Ar+ ions are accelerated and impact into the cathode. In this way
the Ar+ ions sputter the dysprosium away in form of atomic vapor by depositing theirkinetic energy. The hot atomic vapor exhibits a too large Doppler broadening to be used asa reference. In order to avoid this we use a Doppler-free spectroscopy technique, namely
the modulation-transfer spectroscopy. This method employ two counter-propagating laser
beams to interrogate the atoms. One beam, known as pump beam, has a higher intensitywith respect to the weak probe beam. Figure 5.3 shows the setup for the generation of the
MTS signal and Appendix A.2 outlines the generation of the spectroscopy signal.
We guide the laser light with a fiber from the setup of the 626-nm laser to the spectroscopy
setup. After the fiber output coupler the light is divided into pump and probe beam with aλ-waveplate and a polarizing beam splitter (PBS). The best signal-to-noise ratio has been
2
obtained using a relative power ratio of approximately Ipump/Iprobe ≈ 4. The pump beam goes
through an EOM, which imprints sidebands onto the beam at a modulation frequency of
5 MHz. The frequency is generated by a direct-digital synthesizer (DDS) and enhancedby an LC-circuit, formed by the EOM crystal that acts as a capacitor and an inductance3

Thorlabs, DMLP605.

53

5. Experimental Part
spectroscopy signal
LO 5 MHz
mixer

λ/2
EOM
PD

λ/2

lens
f = 50mm

λ/2

PBS

HCL

PBS

fiber output
coupler

Figure 5.3.: The setup for the MTS. The ratio between the pump beam and the probe beam is
Ipump/Iprobe ≈ 4. The modulation frequency applied to the EOM is 5 MHz and generated by a local
oscillator (LO), which consists of a frequency generator in combination with a LC-circuit. The probe
beam is detected with a photodiode. To obtain the final spectroscopy signal, the photodiode signal
is mixed with the modulation frequency of the EOM. PD: photodiode, PBS: polarizing beam splitter,
HCL: hollow cathode lamp.
M1

EOM

lens
λ/2 gets PBSof L = 56 µH. Afterwards, the pump beam
overlapped with the probe beam in the HCL.
f = 50mm
Figure 5.4 shows an example of the MTS signal for Dy.experiment
Two transitions can be assigned
164
162
BS
Laser
to the bosonic isotopes Dy and Dy, whereas three transitions can be assigned tofermionic isotopes. The amplitude of the resonances for the bosonic isotopes comparedf higher+ f because the additional hyperfine structure ofto the fermionic isotopes is much
mod
shift
P
the fermions leads to morestates,i.
e.
9for Dy, over which the fermions are distributed.
f
Therefore, the number ofmod
fermionsPDresonant to the respective light is only slightly more 626 nm
than 10 % of all fermions. The resonance of the bosonic 160 Dy and the resonances of thefiber output
PID
other fermionictransitions are out of the scan range. BS
coupler
The frequency difference between the 164 Dy and the 162 Dy resonance can be extracted
by fitting the theoretical spectroscopy signal, Eq. 5.9, to the data and is determined to becavity960.2(1) MHz. This agrees within the error bars with thevalue of 962(2) MHz obtainedin [41]. The transition frequencies of the hyperfine states relative to the bosonic isotope
162
Dy can be calculated by
EOM∆νHFS = ∆162 + Ee (Fe , Je , I) − Eg (Fg , Jg , I),(5.2)

where ∆162 is the isotope shift with respect to the bosonic 162 Dy and Ei are the hyperfine
energies
1
1 3Ci (Ci + 1) − 4I (I + 1) Ji (Ji + 1)
Ei = AiCi + Bi
(5.3)
2
2
2I (2I − 1) 2Ji (2Ji − 1)with
Ci = Fi (Fi + 1) − Ji (Ji + 1) − I (I + 1) .
(5.4)

550 nmlens
f = 100mm

050 nm54

lens
f = 50mm

5.2. Modulation-Transfer Spectroscopy

MTS signal (mV)

60
40

163

Dy
F = 15! F = 17
2
2

20

164

162

Dy

Dy

0
-20
-40
-60
-1.7

161

Dy
F = 21! F = 23
2
2
-1.5

-1.3

163

Dy
F = 17! F = 19
2
2
-1.1

-0.9

-0.7

-0.5

-0.3

-0.1

frequency relative to the 162 Dy transition (GHz)

0.1

0.3

Figure 5.4.: Modulation-transfer spectroscopy signal for dysprosium. Two transitions can be
assigned to the bosonic isotopes, whereas three transition are of fermionic isotopes. The hyperfine
transitions are labeled by F → F 0 . The signal was averaged over ten recording cycles.

Table 5.2 gives the hyperfine coefficients A and B and Table 5.3 gives the calculated andmeasured frequency differences of the hyperfine transitions relative to the 162 Dy bosonicisotope that corresponds to Fig. 5.4. The measured frequency differences agree within theerror bars with the theoretically calculated ones.
Table 5.2.: Hyperfine A and B coefficients for the ground state (g. s.) and the excited state (e. s.) of
the 626-nm transition [41], [29] (as cited in [58]).

g. s.

A (MHz)

161

Dy
Dy
e. s.

−116.2322 1091.574
162.7543 1153.8684

161−187.0(1)261.7(2)

163

163

Dy
Dy

B (MHz)−317(4)−334(6)

In principle, already the spectroscopy signal could be used to lock the laser. However, forthe 583-nm transition of Er and for the 626-nm transition of Dy broadening effects suchas power broadening and pressure broadening affect the spectroscopy signal. The power
broadening can be calculated by [31]∆ωFWHM = 2π · ∆ν 1 +

I
Isat

!1/2
,

(5.5)

where ∆ν is the natural linewidth and Isat is the saturation intensity of the transition. For
Dy a power of 4 mW leads to a power broadening of ∆ωFWHM ≥ 2π · 1 MHz. Additionally,55

5. Experimental Part
Table 5.3.: Calculated hyperfine transitions relative to the bosonic 162 Dy. For the calculations Eq. 5.3
is used and Table 5.2 gives the hyperfine coefficients A and B.

transition∆ν (MHz)∆νmeasured (MHz)Dy: F = 17/2 → F 0 = 19/2
163
Dy: F = 15/2 → F 0 = 17/2
161
Dy: F = 21/2 → F 0 = 23/2

212(2)−1014(2)−1470(2)

210(8)−1011.0(1)−1473.0(1)163

the high background pressure in the HCL of 4 mbar, leads to pressure broadening of thetransitions. The pressure broadening can be as large as 8.2 MHz [32]. Due to this two
broadening effects the signal gets very large and therefore, this method is not optimal
to lock the light of 583 nm and 626 nm, because the linewidths of these transitions areΓ = 2π · 190 kHz and Γ = 2π · 136 kHz, respectively. By fitting the data points of ourspectroscopy signal we extract a linewidth of Γ = 2π · ∆ν = 12.051(5) MHz for a total
power of the two beams of approximately 10 mW. In Figure 5.5 one clearly sees, that thetransmission peaks of the cavity are much narrower that the MTS spectroscopy signal andthus better for the locking purpose.
The spectroscopy still gives an absolute frequency reference to calibrate the frequencyof the cavity modes by comparing the spectroscopy signal with the cavity transmission.
Furthermore, it gives a stable reference, which can be used to determine the zero-crossingof the CTE of the cavity as discussed in the following section.

5.3. Zero Crossing of the ULE Cavity
One of the advantages of using mirrors and a cavity spacer out of ULE is that the coefficientof thermal expansion (CTE), α of ULE has a zero-crossing around room temperature (seeSec. 4.3 and Fig. 4.6). If the cavity is temperature stabilized at this special zero-crossing
temperature T C the length of the cavity becomes more stable against temperature fluctuations. For cavities stabilized at T C frequency drifts on the order of Hz/s due to temperaturefluctuations can be achieved [46, 71].
The cavity mirrors are optically contacted to the spacer and since both elements are
made of ULE we expect T C to be between 0 ◦C and 30 ◦C.
Around T C the CTE of ULE can be approximated byα(T ) = a (T − T C ) + b (T − T C )2 .

(5.6)

The linear expansion coefficient a is on the order of 10−9 /K2 and the quadratic expansion
coefficient b is on the order of 10−12 /K2 [51]. Since we are interested only in a small region
around T C , keeping (T − T C ) small, and b is three orders smaller than a, the linear term isdominant (compare to Fig. 4.6) and we focus only on the linear approximation, which leads
to a length change of
dL = Lα (T ) dT = La (T − T C ) dT.
(5.7)

56

5.3. Zero Crossing of the ULE Cavity
The relative change in frequency is directly connected to the change in length by∆ν ∆L a== (T − T C )2
ν
L
2

(5.8)

To determine T C , we measure the frequency drift of the cavity resonance while changing thetemperature of the cavity. The frequency drift is quantified by measuring the position of thecavity transmission relative to the modulation-transfer spectroscopy signal of the 626-nm
transition for 162 Dy. For this the laser is scanned over more than one FSR and the respective
traces are recorded with an oscilloscope4 . Figure 5.5 shows the cavity transmission signal
and the spectroscopy signal. The cavity transmission signal is fitted with a double Lorentzianfunction, whereas the spectroscopy signal is fitted with the theoretical spectroscopy signal
derived in [63]C
S (ωm ) = p
2
Γ + ω2m
× L−1 − L−1/2 + L1/2 − L1 cos φ+ D1 − D1/2 − D−1/2 + D−1 sin φ ,
where

(5.9)

Ln =

Γ2
Γ2 + (∆ − nωm )2(5.10)Dn =Γ(∆ − nωm ).
2
Γ + (∆ − nωm )2(5.11)

andΓ is the linewidth of the transition, ωm the modulation frequency, ∆ the detuning fromthe atomic resonance and φ is the phase difference between the detected signal and themodulation field applied to the EOM to modulate the pump beam. The center position of
the transmission peak of the cavity labeled with T in Fig. 5.5 is taken as frequency reference.
Additionally, we record the voltage at an NTC sitting on the aluminum block that holds
the cavity (see Sec. 4.4) and use the associated temperature as actual cavity temperature.
The NTC is part of a voltage divider and the temperature can be read out by calculating theresistance of the NTC. We changed the temperature from 27.0 ◦C to 17 ◦C in steps of 2 ◦C.
Since the cavity has no thermal connection to the aluminum block, where the temperatureis measured, the time between setting the temperature and measuring the relative positionof the resonance was minimum 2 days. This ensures that the cavity has thermalized andreached the same temperature as the aluminum block. After reaching 17 ◦C we changed
the temperature back to higher values to check if the time we waited for thermalization
was enough or if we observe a hysteresis behavior as in [71]. Figure 5.6 shows the positionof the 162 Dy transition relative to the cavity transmission as a function of the temperatureof the aluminum block that holds the cavity. We do not observe a hysteresis behavior,
which means, that the time we waited for thermalization was enough. For each data point
4

LeCroy: SDA 760Zi-A.

57

PD / MTS signal (mV)

5. Experimental Part

60
162

40

Dy

T
"8

-0.2

0

20
0
-20
-40
-60

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

0.2

frequency relative to the cavity transmission (GHz)
Figure 5.5.: Example of the spectroscopic trace recorded with the oscilloscope for the determinationof the zero-crossing. The aluminum block that holds the cavity had a temperature of 16.58(6) ◦Capproximately 50 traces were recorded and the position was averaged. The error bars in
Fig. 5.6 are the standard deviation, whereby the error bars for the temperature are not visible,
because they are on the order of 5 × 10−2 ◦C. The data points are fitted with a quadratic
function ∆ν = c + νa(T − T C )2 , where c is a constant containing the minimal difference
between the atomic resonance and the cavity transmission. From the fit we extract T C to be
T C = 21.78(5) ◦C.
and the linear expansion coefficient a in Eq. 5.6 to bea = 1.65(5) × 10−9 /K.
The error is the statistical error from the fit. We determine the temperature of the zerocrossing of the CTE with a precision of 50 mK, which satisfies the requirements given
in Sec. 4.3. The result of 21.78(5) ◦C for T C is very convenient because the temperaturestabilization is very easy, i. e. no water cooling is required. Additionally, the difference tothe room temperature is small and therefore the Peltier elements do not have to work athigh power. In the following the cavity is stabilized at 21.78(5) ◦C and the locking of the
626 nm lasers is done.

5.4. Locking of the Laser Systems
The Pound-Drever-Hall (PDH) technique (see App. A.1) is used to lock the lasers for thelight of 583 nm and of 626 nm. The idea is to modulate the laser light via a fiber-coupled
EOM5 with two different modulation frequencies, which results in sidebands at frequencies
fshift and fmod . The frequencies are generated by two direct-digital synthesizer (DDS) andsuperimposed with a power combiner6 . While the laser frequency is tuned to the atomic
5
6

Jenoptik, PM635 (PM585) for the light of 626 nm (583 nm).
Minicircuits, ZSC-2-4+.

58

5.4. Locking of the Laser Systems

position "8 (MHz)

-115

-125

-135

-145
15

20

25

30

Figure 5.6.: The frequency difference of the 162 Dy spectroscopy transition and the cavity resonanceis plotted as a function of the temperature of the aluminum block that hold the cavity. The data
points are fitted with a quadratic function ∆ν = c + νa(T − T C )2 . Each point is averaged over about
50 data points. The error bars are given by the standard deviation. The zero-crossing temperature is
at 21.78(5) °C and a = 1.65(5) × 10−9 /K.

resonance, the locking is done with the additional sidebands by shifting fshift to the cavity
resonance and using this sideband together with the sidebands at fmod to generate the error
signal. In Figure 5.7 the locking scheme is illustrated schematically. fexp labels the atomic
resonance frequency and fn labels the cavity modes. Figure 5.8 shows the schematic setup
fmod

fshift

FSR
EOM
fn

fexp

fn+1

fn

fexp

fn+1

Figure 5.7.: Illustration of the locking scheme for the laser of light of 626 nm and 583 nm. fn labels
the cavity modes and fexp labels the experimentally desired wavelength. fshift and fmod are thesidebands that are modulated by the EOM.

for the laser locking. The light of the laser gets divided by a polarizing beam splitter (PBS).
Most of the power goes to the experiment and only approximately 5 mW are used for thelaser locking. A fiber-coupled EOM modulates the sidebands at frequencies fshift and fmod .
The reflected beam of the cavity is intercepted with a beam splitter and recorded with aphotodiode. The detected signal is mixed with the modulation frequency fmod to extract the
PDH error signal. The laser is then stabilized with a PID controller. This locking method
has several advantages compared to other systems realized with acousto-optic modulators(AOM) as in [58, 71]: The fiber-coupled EOM has a large bandwidth of 5 GHz, which gives59

5. Experimental Part
λ/2

PBS
experiment

Laser

f

mod

mod

+f

shift

PD

EOM

f

583 nmexperimen

BS

PID

cavitycavityw

λ/2
AOM

Figure 5.8.: Schematic drawing of the setup of the laser locking for the 626 nm light.

1050 nm

1550 nma large tuning range (large fshift is possible) without the need to optimize the couplings to
the AOMs. This facilitates the switching between the different isotopes. The optical setup
gets much more easier compared to standard realizations of such systems [32, 58] because
usually at least two AOMs in double pass configuration are necessary to obtain a sufficiently
large frequency scan range for the laser light. The fiber-coupled EOM also provides a spatial
mode cleaning, which makes the coupling into the cavity and the suppression of higher
transversal modes easier. In total only one fiber-coupled EOM and one AOM are requiredwith our system (see Fig. 3.7 and 5.8) for tuning, locking and power stabilization of the laser
light. Figure 5.9
shows
the PDH error signal that we obtain with a modulationf frequency= 50mmf=
100mm
fmod of 20 MHz. In Section 5.5 we extract the linewidth and therefore the finesse out of theerror signal. Figure 5.10 shows the cavity transmission signal and the signal of the MTS,
while the laser was scanned over more than one FSR. The sidebands are at a frequency of
200 MHz, whereas the sidebands at 20 MHz used for locking are not visible due to the lowamplitude. This plot can be used as a reference to check how large the frequency fshift mustbe, to shift the sideband to the cavity resonance to lock the laser light of 626 nm.
f = 200mm

f = 150mm

f = 50mm

5.5. Performance of theλ/4
Cavityw

λ/2Beside the zero-crossingof the CTE, the free spectral range (FSR), the finesse,thef = 175mm
f =and200mm
linewidth of the cavity are important parameters. In the following section we determine
this three properties with different measurement methods. For allλ/2
three measurements we
f = 100mm
λ/4
use the light of 626 nm.

PPNL

PBS

λ/2
PBS

60

AOM

λ/2
λ/2

experiment5.5. Performance of the Cavity

600

voltage (mV)

400
200
0
-200
-400
-600
-800
-30

-20

-10

0

10

20

30

frequency (MHz)Figure 5.9.: PDH error signal of the 626 nm light obtained with the cavity.

Free Spectral Range and Linewidth out of the Transmission Signal
In this subsection we extract the FSR and the linewidth out of the transmission signal of thecavity. For this, we modulate sidebands on the laser light with the fiber-coupled EOM andscan the frequency of the laser over more than one FSR (1 GHz). We record the transmissionof the cavity with an oscilloscope7 and use the sidebands at known frequencies to convert
the time axis into a frequency axis. Figure 5.10 shows the transmission of the cavity withsidebands modulated at 200 MHz. To measure the FSR we take the frequency difference
between the center positions of the carrier of the transmission peaks (labled with C in
Fig. 5.10), which are fitted with a Lorentzian function
A
1.
(5.12)
2
π s + (x − t)2
Figure 5.11 shows a zoom into one cavity transmission peak and the Lorentzian fit. We
did this with three different modulation frequencies, 100 MHz, 120 MHz, and 200 MHz andaverage the FSR to
F = 989(8) MHz.
f (x) =The slightly lower value than the expected FSR of 1 GHz can be explained by the opticallycontacted mirrors. Due to the mirrors, especially the curved mirror, the cavity distance isslightly increased, leading to a smaller free spectral range.
Since the linewidth of our 626 nm laser light is on the order of a few 10 kHz and therefore
lower than the linewidth of the cavity for the theoretically best coating case of the mirrors(≈ 50 kHz), it is possible to measure the linewidth by scanning the laser over the cavity
resonance. We fit each transmission peak in Fig. 5.10 with the Lorentzian Function 5.12 andextract the linewidth. All 6 linewidths are then averaged to ∆ν = 630(9) kHz.
7

LeCroy: SDA 760Zi-A.

61

5. Experimental Part

50
C

PD voltage (mV)

40

C

30
20
10
0
-10
-20

161

Dy
F = 21! F 0 = 23
2
2

164

162

Dy

163

Dy
F = 15! F 0 = 17
2
2
-1.6

-1.4

-1.2

Dy

163

Dy
F = 17! F 0 = 19
2
2
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

frequency relative to the cavity transmission (GHz)
Figure 5.10.: Cavity transmission and the MTS signal. The laser is scanned over more than one
free spectral range with a modulation frequency of 200 MHz switched on. The 20 MHz sidebands,
that will be used for the generation of the PDH-signal, are not visible here because there amplitude
is too small.

By combining the results obtained for the FSR and the linewidth of the cavity we calculate
the finesse with Eq. 4.9 to
F = 1569(23).
(5.13)

Finesse and Linewidth out of the Cavity Ring-Down Measurement
Another method to determine the linewidth of the cavity and to calculate the finesse with
Eq. 4.9 is the cavity ring-down measurement. The working principle of the cavity ring-down
measurement is based on the decay time of the light stored in the cavity. If the laser islocked to the cavity light enters the cavity and due to the reflections of the mirrors a part ofthe light remains trapped in the cavity modes. The equilibrium between the leaving Iout (t)and entering intensity Iin (t) is described by1
I˙out (t) = (Iin (t) − Iout (t)) .
ττ is the decay constant and defined as(5.14)

1,(5.15)
αc
where α is the total lost per length unit and c the speed of light. If at t = ξ, the incoming
intensity is suddenly switch of, Iin (ξ) = 0, then Eq. 5.14 reduces toτ=1
I˙out (t > ξ) = − Iout (t).
τ62

(5.16)

5.5. Performance of the Cavity

40

PD voltage (mV)

35

30
FWHM
25

20

15
-1000

-800

-600

-400

-200

0

200

400

600

800

1000

frequency (kHz)
Figure 5.11.: Zoom into one cavity transmission peak. The cavity resonance is fitted with theLorentz function from Eq. 5.12. The frequency axis is set to be zero at the resonance.

This differential equation has the simple solution
Iout = I0 e− /τ ,(5.17)twhere I0 is the intensity stored in the cavity. For this measurement we lock the 626-nm
laser onto the cavity resonance as explained in Sec. 5.4 and change suddenly the frequency
fshift of the sideband, which is shifted to the cavity resonance. Since the sideband is notresonant to the cavity anymore, the input intensity is turned off. The intensity transmitted
through the cavity is recorded with a photodiode. Figure 5.12 shows the exponential decayof the measured intensity. It is important that the switching occurs much faster than the
decay time. Otherwise, the assumption of Iin (t > ξ) = 0 is not valid anymore and the decay
is determined by the switching time. The EOM has a specified switching time of 0.2 ns andis therefore fast enough for this purpose, since we expect a decay time τ > 100 ns.
To extract the decay time, the exponential functiontf (t) = A + B · e− τ

(5.18)

is fitted to the data. The constant A describes the intensity offsets, including stray light andthe offset from the photodiode and B is the amplitude of the decay function. The photon
lifetime can be determined toτ = 260.4(7) ns,
which results, in a linewidth of the cavity of∆ν = 611(2) kHz,63

5. Experimental Part

35

PD voltage (mV)

30
25
20
15
10
5
0
-5
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (7s)Figure 5.12.: Cavity transmission signal as a function of time. The intensity decays after shifting
the sideband, on which the laser is locked, out of resonance.

by using Eq. 4.11. Combining the FSR of 989(8) MHz calculated with the cavity transmission
and the obtained linewidth, the finesse of the cavity can be calculated using Eq. 4.9 to
F = 1636(4).

Linewidth out of the PDH Error Signal
Beside the determination of the linewidth out of the cavity transmission signal and thecavity ring-down as a third method the linewidth is extracted out of the PDH error signal.
This is done, by fitting [62]





Γ∆Ω sin(ϕ)Γ Γ2 + ∆2 + Ω2 + cos(ϕ)Ω Γ2 − ∆2 + Ω2S (∆) = A + B ·(5.19)
Γ2 + ∆2 Γ2 + (∆ + Ω)2 Γ2 + (∆ − Ω)2to the data points, where A is a general offset, B is the amplitude of the signal, Γ is the
linewidth of the cavity, ∆ is the laser detuning with respect to the cavity resonance, Ω is themodulation frequency and ϕ is the phase between the reflected signal and the modulation
signal.
In Figure 5.9 shows the fit to the recorded error signal. We extract a linewidth of
410(20) kHz. By using the FSR of 989(8) obtained before we can calculate the finessetoF = 2213(120).
(5.20)
Function 5.19 does not fit perfect to the data. This explains the deviation from the results
obtained with the cavity transmission and the ring-down measurement. Nevertheless, thelinewidth and therefore the finesse is of the same order and confirms the other results.

64

5.5. Performance of the Cavity

Discussion of the Cavity Parameters
Table 5.4 gives a summary of the results obtained via the three different measurement
methods discussed above. Theoretically, following the coating curves for the mirrors in
Fig. 4.4 provided by Lens-Optics8 and using Eq. 4.10, one obtains for the finesse for light at
626 nm a minimum value of 6300, which is approximately a factor of 4 higher than in ourcase.
Table 5.4.: Summary of the results obtained for the finesse F , the FSR, the linewidth ∆ν, and thereflection coefficient R with the different measurement methods.

min. theo. calculatedtrans. signal

cavity ring-down

PDH error signal

1000
6300
159
99.95

989(8)
1569(23)630(9)
99.800(3)–1636(4)611(2)
99.8082(5)–2213(120)410(20)99.86(1)

FSR (MHZ)F∆ν (kHZ)R (%)

One reason for the low finesse could be that the mirrors are not as good as specified.
To verify the quality of the mirrors we measured the transmission for both type of mirror
coatings with two mirrors, which we have in spare. We used light of 421 nm for the mirrors
coated from 380 nm – 450 nm and light of 626 nm for the other mirror coating type. Table 5.5
gives the reflection coefficients that are calculated with the measured transmission. To
take into account that some of the light is absorbed from the mirror substrate we assume
an absorption of 10 %, according to Fig. 4.5. Although the absorption is estimated quite
high, the finesse calculated for 626-nm with the measured transmission coefficient lies
within the specifications and is higher than the finesses measured with the cavity. As acheck we estimate the finesse of the cavity also for the light at 421 nm. For this we record
the transmission with the oscilloscope and estimated the linewidth to be > 1 MHz. The
corresponding finesse is lower than 1000. This leads to the conclusion that the mirror
coatings are not casing the low finesse.
Table 5.5.: Measured transmission for both type of mirror coatings. Additionally the resulting
finesse is calculated with this reflection coefficient and a FSR of 989(8) MHz.

wavelength

power (mW)trans. power (µW)

ref. coeff. (%)

finesse421 nm626 nm38.0(2)10.0(2)28.5(2)3.45(1)99.92(2)99.96(9)3800(38)8277(62)

Another possible explanation for the low finesse is that during the evacuation water
steamed up on the mirrors. This happens especially if the evacuation occurs too fast or thevacuum chamber gets reopened without floating the chamber with argon. Since we pumpwithout any special needle valve that controls the pumping speed and had also to reopen
8Lens-Optics GmbH65

5. Experimental Part
the chamber once, this is a good explanation. According to Lens-Optics9 the steaming effectshould get stronger for wavelengths in the UV. This agrees with the estimated finesse of
the cavity for the light of 421 nm. Therefore this effect could be a plausible explanation for
the unexpected low finesse.
In summary, for light of 626 nm the finesse of the cavity is approximately by a factorof 4 lower than for the worst theoretical value. Since the mirrors coated for light of 626 nm,
583 nm, 741 nm, and 631 nm have the highest reflection at 626-nm we assume that thefinesses for the other wavelengths are even worse. For the locking of the laser light thismeans, that the slope of the PDH errror signal is 4 times less than expected, because1
the slope is proportional to ∆ν. The transitions of 626 nm and 583 nm have linewidths of
136 kHz and 190 kHz. Since the lasers for these transitions have a small enough fee running
linewidth the main task of the cavity is the long term stabilization. The lock of this twolasers is therefore possible. However, for the lasers of 741 nm and 631 nm as well for thelasers for Rydberg excitation the transition linewidths are below 10 kHz and therefore thelocking is, if at all possible, very challenging.

9

Private communication.

66

6. Conclusion and Outlook
The main goal of this master thesis was to set up and characterize an optical cavity, which
can be used to stabilize different lasers at several wavelengths. In the main experimentthese lasers will be used for slowing, cooling, and trapping of Er and Dy atoms and for
Rydberg excitation. To reach a high stability against temperature fluctuations we chose
ultra low expansion glass (ULE) as material for the mirrors and the cavity spacer. Dueto technical limitations it is not possible to produce mirrors with high reflectivity at allnecessary wavelengths. Therefore, we decided to built a double cavity. This means weused one ULE spacer but two different pairs of mirrors. One pair of mirrors is coated forthe wavelengths at 583 nm, 626 nm, 631 nm, and 741 nm, which are cooling transitions in
erbium and dysprosium. The second pair of mirrors is coated for the range from 380 nm to
450 nm to be able to lock lasers for Rydberg excitation.
To shield the cavity as good as possible from the environment, i. e. from acoustic andthermal noise, the cavity is placed in a vacuum chamber with a pressure on the order of
2 × 10−8 mbar. For further stability, the cavity inside the vacuum chamber is surroundedby two additional heat shieldings. The temperature of the outer one of these two shieldings
is stabilized via Peltier elements. For monitoring of the temperature several thermistors are
distributed over the vacuum chamber and can be read out with an Arduino system.
The temperature where the ULE exhibits a minimal thermal expansion was determined
to be at 21.78(5) ◦C and therefore the cavity is stabilized to this value. To determine thefree spectral range (FSR), the linewidth, and finesse of the cavity we performed threedifferent measurements. First, we determined the FSR to 989(8) MHz by scanning the626-nm laser over two cavity resonances and extracted the frequency difference of thetransmission peaks by fitting a Lorentzian function. Additionally, we modulated sidebandson the laser beam with an electro-optical modulator (EOM) and used them to calibrate
the frequency axis. Further, the linewidth was extracted from the fit of the transmission
peaks and was determined to 630(9) kHz, which leads to a finesse of 1569(23). As a second
measurement we performed a cavity ring-down and the resulting finesse was 1636(4).
Finally, we extracted a linewidth of 410(20) kHz out of the PDH error signal, leading to afinesse of 2213(120).
The finesse obtained with these three methods deviate by a factor of ≈ 4 form thetheoretically expected value. We come to the conclusion, that due to too fast pumping andreopening of the cavity water steamed up on the mirrors. This could have degraded thereflection of the mirrors and caused therefore the lower finesse compared to the expectedvalue. Nevertheless, it was possible to lock the 626 nm laser light to the cavity.
The next step in the experiment will be the realization of the single- and double-species
magneto optical traps (MOTs) of Er and Dy.

67

6. Conclusion and Outlook

6.1. Er–Dy Quantum Mixture: Motivation and Vision
So far, quantum-mixture experiments with dipolar atoms has not yet been realized and evennever considered in theory. Our experiment would then be the first to go in this direction.
We indeed plan to combine two strongly magnetic atomic species for the first time. Fromthe experimental realization point of view the Er–Dy mixture presents several advantages.
One advantage for the combination of Er and Dy is that they share many similar properties,
e. g. strength of the laser cooling transitions, polarizability, anisotropic interaction character,and similar atomic masses and melting points. Both elements give a convenient choiceof bosonic and fermionic isotopes (see Table 2.1) that gives opportunities to have threedifferent types of mixtures: Bose-Bose, Fermi-Fermi and Bose-Fermi.
A first aspect to be studied is the interspecies interaction of Er and Dy having an imbalanceon the dipolar interaction, where µ = 7µB for Er and µ = 10µB for Dy and different
total angular momentum J. Both elements offer a rich spectrum of intraspecies Feshbachresonances [34, 59, 60]. The Feshbach spectroscopy of the mixture will add important
information for the understanding of scattering physics of dipolar gases.
For the Fermi-Fermi mixture the crossover from a Bose-Einstein Condensate (BEC) to
Bardeen-Cooper-Schrieffer (BCS) superfluid (BEC–BCS crossover) with dipole-dipole interaction (DDI) is of large interest. The BEC-BCS crossover is important for the understandingof superconductors and the associated superfluidity [82]. Depending on the interaction
strength between the atoms, given by the scattering length a, the fermions can either
form molecules or weakly bound pairs, so called Cooper-pairs. Both the BEC and the
Cooper-pairs give explanation to different types of superfluids. In many experiments the
BEC–BCS crossover has been shown, e. g. in 6 Li and 40 K [18, 70], and the whole range from
BEC to BCS can be investigated. These experiments have in common that they work withalkali atoms having isotropic interaction. Dipolar atomic species give the possibility to
study the crossover under the influence of long-range and anisotropic dipolar interaction.
Atoms forming a Cooper-pair have opposite momentum, because of the conservation of
momentum [40]. In momentum space the Fermi surface for alkali atoms is spherical, due
to the isotropic interaction, whereas for dipolar atomic species the Fermi surface can be
deformed [2], which may have an impact on the Cooper-pairing mechanism. For alkali
atoms such as 6 Li and 40 K the experimental technique is to prepare the atoms equally intheir lowest two spin states and tune the interaction by a Feshbach resonance. However,for dipolar atoms this type of preparation is complicated by dipolar relaxation effects [14]occurring as a consequence of the long-range dipolar interaction. In a dipolar relaxation
process the spin is flipped, since for dipol-dipol interaction the total spin is not preserved
and the atoms relax into the state with lowest energy. This issue can be circumvented byusing two different atomic species. The idea is to prepare both species in their lowest spinstates. Again a Feshbach resonance can then be used to tune the interaction between theatoms.
Another interesting case is to investigate the behavior of the atoms without the external
magnetic field, that polarizes the atoms. Without the polarization, the dipoles of the atoms
are not frozen anymore and will organize their orientation by them self. Spinor Bose gases
are gases with a spin degree of freedom that, similar to the BEC–BCS crossover, have been68

6.2. Rydberg Excitation
studied in detail with alkali atoms [75]. The dipolar characteristic of Er and Dy will add new
aspects. Since the dipole dipole interaction is spin dependent, this will lead to a competition
with the spin dependent contact interaction for the local spin ordering of the spinor gas.
For 87 Rb in the F = 1 state the ratio (dd ) between the dipole-dipole interaction strength(add = µ0 µ2 m/12π~2 ) and the contact interaction strength given by the scattering length a is
dd = 0.002. In comparison, for Dy, dd is on the order of 1 for a typical scattering length [75].
The large total angular momentum quantum number J of Er (J = 6) and Dy (J = 8) gives
rise to the realization of spin-N systems with large N, i. e. 2J + 1 different states (13 for Er
and 17 for Dy).

6.2. Rydberg Excitation
An atom excited to a Rydberg state is a highly excited atom with a large principal quantum
number n. Rydberg atoms exhibit many special properties, e. g. a large orbital radius scaling
with n2 , which leads to a high polarizability, and a long lifetime proportional to n3 (forRubidium the theoretical lifetime of the 110S 1/2 state is 1.7 ms) [7, 57]. However, for Rydberg
states with a high angular momentum quantum number l, the lifetime increases further
with a scaling τ ∝ n5 [35]. Due to the high polarizability a large electric dipole moment canbe induced, resulting in a strong long-range interaction. Similar to a Feshbach resonance,
the interaction strength is tunable [6].
Rydberg excitation using a single photon excitation scheme is technically challenging
from the aspect of laser technology, as usually laser light in the UV is required. In orderto overcome this problem, two photon excitation schemes are commonly used in the fieldof ultracold Rydberg atoms. Here, the excitation lasers are operated with a detuning withrespect to the intermediate state in order to minimize photon scattering, which eventually
results in a heating of the ultracold sample. However, this intermediate state detuning also
results in a reduction of the coupling strength between ground- and Rydberg state. The
rich level structure of Er, as shown in Chap. 2, allows for various two photon excitation
schemes via a narrow line intermediate state, which potentially can be utilized to increase
the coupling between ground and Rydberg state.
Further, the sub-merged shell electron configuration of Er (see Sec. 2.1) allows to excite
either an electron in the 6s or the 4 f state. Utilizing a two photon excitation scheme, theexcitation of an 4 f electron to states with a high angular momentum and therefore, Rydberg
states up to l = 5 are reachable in a straight forward manner.
In addition, similar to alkaline earth atoms, the core of an Er Rydberg atom remains
optically active and thus still can be addressed, e. g. for trapping, detecting or further
cooling.

69

6. Conclusion and Outlook70

A. Appendix
A.1. Pound-Drever-Hall Error Signal
The Pound-Drever-Hall (PDH) technique [25] is a very powerful technique to frequency
stabilize laser systems using a Fabry-Perot cavity as frequency reference. The PDH error
signal is created by modulating sidebands on the laser beam and by extracting the phasedifference between the sidebands and the carrier of the laser beam after it is reflected fromthe Fabry-Perot cavity. The following Appendix gives a brief summary of the derivation of
the PDH error signal.
First, we want to concentrate on the reflection of a monochromatic beam from a FabryPerot cavity. The incoming electrical field can be written as
Einc = E0 eiωtwhile the reflected beam is

Eref = E1 eiωt

E0 and E1 are the complex amplitudes, containing the relative phase. The ratio between
the incoming and the reflected beam leads to the reflection coefficient [12] iωFS R − 1reEincF(ω) =
=
,
(A.1)
iω
Eref
1 − r2 e FS R
where r is the reflection coefficient of the mirrors and FSR the free spectral range. If thefrequency of the laser light is exactly an integer multiple of the FSR, the phase difference
between the incoming and reflected beam is 180° and therefore this two beams interfere
destructively. If the laser frequency does not match the FSR, the phase difference is different
from 180° and the two electric field do not cancel out each other. The phase differenceindicates, on which side of the cavity resonance the laser frequency is on. To measure thephase of the reflected beam, the incoming beam gets phase modulated, which generates
sidebands on the laser beam. After the reflection of the cavity the sidebands interfere withthe carrier and the phase is extracted from the beat pattern.
For the phase-modulation of the laser light and therefore, for the creation of the sidebandsa Pockels-crystal is used. An important attribute of a Pockels crystal is that the refraction
index depends linearly on the applied electric field:n(E) = n + aE
For the PDH technique a sinusoidal electrical field is applied to the crystal and the electricalfield of the laser beam after the crystal can be described by
Einc = E0 ei(ωt+β sin(Ωt)) ,71

A. Appendix
where Ω is the frequency of the electrical field, which is applied to the Pockels crystal
and β is the parameter for the modulation depth. If β < 1, the expression of the incoming
electrical field can be expanded using the Bessel functions [12]Einc ≈ E0 J0 (β) + 2iJ1 (β) sin(Ωt) eiωtwhere terms of higher order are neglected, because J1 (β)/J0 (β) ≈ 0.2 for β = 1. For β < 1 thisix
−ix
ration becomes even smaller. Using sin(ix) = e −ethe electrical field becomes
2ihiEinc = E0 J0 (β)eiωt + J1 (β)ei(ω+Ω) − J1 (β)ei(ω−Ω)(A.2)
In Eq. A.2 terms with frequencies ω, ω − Ω and ω + Ω appear, where ω is the frequency ofthe carrier and ω ± Ω are the frequencies of the two sidebands. The power in the carrier
and in each first-order sideband is given by [12]for the carrier and

PC = J02 (β) P0(A.3)

PS = J12 (β) P0(A.4)

for the sidebands, where P0 is the total power of the incident beam. The distribution of thepower between the carrier and the sidebands depends on the modulation depth β. With
the reflection coefficient from Eq. A.1 and the modulated incoming beam in Eq. A.2 thereflected beam can be written as [12]hiEref = E0 F(ω)J0 (β)eiωt + F(ω + Ω)J1 (β)ei(ω+Ω) − F(ω − Ω)J1 (β)ei(ω−Ω)(A.5)
The power measured at the photodiode is Pref = |Eref |2 . In comparison to a beam at
single frequency without sidebands Eq. A.5 leads to additional terms appearing due to theinteraction of the carrier with the sidebands. The most interesting terms are the one which
oscillate at the modulation frequency Ω, because they sample the phase of the reflected
carrier:p
2 PC PS Re [F(ω)F ∗ (ω + Ω) − F ∗ (ω)F(ω − Ω)] cos(Ωt)
and
2 PC PS Im [F(ω)F ∗ (ω + Ω) − F ∗ (ω)F(ω − Ω)] sin(Ωt).
p

Usually, depending on the modulation frequency only one of these two terms will survive,i. e. for low modulation frequencies F(ω)F ∗ (ω + Ω) − F ∗ (ω)F(ω − Ω) is purely real, whereas
for high frequencies it is purely imaginary. Using a mixer, which forms the product of two
signals, the terms can be extracted. The product of two sine waves issin(Ωt) sin(Ω0 t) =1
{cos[(Ω − Ω0 )t] + cos[(Ω + Ω0 )t]}
2

(A.6)

For the case that Ω ≈ Ω0 , the cosine which contains the frequency difference becomes adc signal, which can be isolated using a low-pass filter. Thereby mixing the output signal of72

A.1. Pound-Drever-Hall Error Signal

normalized intensity

0.5
0
-0.5

normalized intensity1
r = 0.999, + = 3
r = 0.9995, + = 4
r = 0.9999, + = 51

r = 0.999
r = 0.9995
r = 0.9999

0.5

0

-0.5

-1
-6

-4

-2

0

2

4

6

frequency (1/+)(a) Overview with differend reflection coefficients andmodulation frequencies.

-1
-0.2

-0.1

0

0.1

0.2

frequency (1/+)(b) Zoom-in on the slope of the central peak.

Figure A.1.: (a) PDH error signal with 3 different modulation frequencies and 3 different reflection
coefficients. (b) Slope of the PDH error signal with 3 different reflection coefficients.

the photodiode with the modulation signal and applying a low-pass filter we can extract the
PDH error signal for high modulation frequencies. Since at low modulation frequencies theterm F(ω)F ∗ (ω + Ω) − F ∗ (ω)F(ω − Ω) is purely imaginary only the cosine term is relevant
and the signal need to be phase shifted to obtain the same mixing behavior. Afterwards
the error signal can be fed to a PID controller that stabilizes the laser, e. g. by changing thelaser current and the grating angle with a piezo element for an external cavity diode laser.
In Figure A.1(a) three different error signals as a function of the detuning are plotted.
For each signal the modulation frequency of the EOM and the reflection coefficient of themirrors, thus the finesse (Eq. 4.9), are changed. The change of the modulation frequency
results in a different position for the wings of the error signal. A higher modulation
frequency leads to a higher capture range, i. e. the region where the error signal is unequal
zero and has the correct sign for driving the laser back onto resonance. However, thedrawback of a higher modulation frequency is that the value of the error signal between
the wings is small and therefore the influence on the regulation decreases. On the other
hand, the change of the reflection coefficient results in a displacement of the minima andmaxima, respectively. A higher reflection coefficient shifts the minima and maxima towards
the center and therefore the slope around the central point increases. This central slope ofthe error signal can be expressed by [12]√PC PS
D = −8.
(A.7)∆ν
The slope is proportional to the power of the carrier and the sidebands and to the inverseof the linewidth of the cavity. The sensitivity of the laser stabilization with the PDH
technique to frequency fluctuations is directly related to the steepness of the slope. The
steeper the slope, the higher the inherent gain of the laser stabilization. A simple electronic
amplification also amplifies the present technical noise and would therefore not lead to anincreased signal-to-noise ratio. Figure A.1(b) shows the slope for three different reflection73

A. Appendix
coefficients for the same modulation frequency. For a given√ linewidth of the cavity, one canmaximize the slope D by finding an optimal value for PC PS . The maximum value for D
can be found if PS/PC = 0.42, which corresponds to a modulation depth of β = 1.08.

A.2. Modulation-Transfer Spectroscopy
The modulation-transfer spectroscopy (MTS) is a spectroscopy technique based on two
counter-propagating beams, called pump and probe beam. The pump beam passes throughan electro-optical modulator (EOM) and sidebands are imprinted on the beam. The modulation is already described in Appendix A.1. After the EOM the pump beam can be represented
by the carrier frequency ωC and sidebands separated by the modulation frequency ωm [63]"X∞
E = E0Jn (δ) sin (ωC + nωm ) t
n=0

+

∞
X

#
n
(−1) Jn (δ) sin (ωC + nωm ) t − J1 (δ) sin (ωC − ωm ) t .

(A.8)

n=1

Jn (δ) represents the Bessel function of order n and δ is the modulation index. If themodulation index δ < 1, only the first order sidebands have to be considered and Eq. A.8
simplifies to [63]E = E0 [J0 (δ) sin ωC t + J1 (δ) sin (ωC + ωm ) t − J1 (δ) sin (ωC − ωm ) t] .

(A.9)

Therefore, the pump beam consists of three frequency components after the EOM, i. e. ωC ,
ωC + ωm , and ωC − ωm . When the pump and the probe beam are overlapped in the spectroscopy cell four-wave mixing occurs, where a fourth wave is generate by the combinationof three other waves [69]. The four-wave mixing is related to the non-linear polarization of
the medium
PNL = χ(3) E 3 .
If E is the sum of three waves of three different frequencies ω1 , ω2 , and ω3 , E 3 leads todifferent frequency combinations and the generation of a fourth wave. In the case of the
MTS, two frequency components of the pump beam combine with the frequency of theprobe beam and generate a fourth frequency component that can be treated as a sidebandof the probe beam. The sidebands of the probe beam, modulated in the vapor of thespectroscopy cell, beat with the probe beam leading to [63]
UPD (∆, t) = p

CΓ2 + ω2m
× L−1 − L−1/2 + L1/2 − L1 cos ωm t + φ+ D1 − D1/2 − D−1/2 + D−1 sin ωm t + φ ,

where
Ln =

74

Γ2
Γ2 + (∆ − nωm )2(A.10)(A.11)

A.3. Electronic CircuitsandΓ(∆ − nωm ).
(A.12)
Γ2 + (∆ − nωm )2∆ denotes the detuning of the laser frequency to the atomic resonance frequency and Γ isthe linewidth of the transition. The beat signal S (ωm ) is detected with a photodiode and aphase detector mixes the signal of the photodiode with the modulation signal. Using thetrigonometric formula A.6 leads to a DC term and an oscillating term. With a low-passfilter the DC term can be extracted and the spectroscopy signal
Dn =CS (∆) = p
Γ2 + ω2m
× L−1 − L−1/2 + L1/2 − L1 cos φ+ D1 − D1/2 − D−1/2 + D−1 sin φ(A.13)

obtained. It is important to note, that the modulation occur only, if the condition for the
sub-Doppler resonance is fulfilled. Otherwise, the four-wave mixing does not transfer the
sidebands onto the probe beam. This means that only atoms of the velocity class zero
contribute to the spectroscopy signal and therefore the transition is not broadened by the
motion of the atoms. The resonance position of the MTS signal is always located at thecenter of the sub-Doppler resonance.

A.3. Electronic Circuits
All used electronic circuits are designed by Dr. Manfred Mark.

Measurement Bridge
There are two measurement bridges (MBs) on one circuit, which are framed with the black
dashed line in the electronic circuit below. The MB for the resistor two with negative temperature coefficient (NTC2) consists of has a potentiometer to set the desired temperature.
The aim of the temperature PID is to regulate the resulting temperature difference of the
MB to zero. Therefore, with the MB for NTC2 we can reach the following resistance values:
R8 + Rpoti
R12(A.14)

R8
R12 + Rpoti(A.15)RNTC = R11
for minimum temperature and
RNTC = R11

for maximum temperature. If instead of a NTC a PTC (positive temperature coefficient) isused the minimum an maximum temperature are reversed. The resistor R8 , which defines
the reachable temperature range is not predefined and has to be chosen appropriately.

75

A. Appendix
For the NTC1, additionally to R16 , dip switches are implemented, that can be used to
add additional resistors in series to R16 . Therefore, with this measurement bridge a higher
temperature range can be reached:
RNTC = R5

R16 + Rpoti + Rswitch
R6(A.16)

R16
R6 + Rpoti(A.17)

for the minimum temperature and
RNTC = R5

for the maximum temperature. The other elements on the circuit are either for the supply
voltage or not used. The connection from the MB to the PID is done via a SUB-D15 connector.

Temperature PID
The temperature PID is adapted to the measurement bridge and consists also of two independent PIDs. The circuits of both PIDs are the same, whereby for the resistors and capacitors
different values can be inserted to adjust the proportional, integral and differential part.
Besides the PID, there are are also other parts on the circuits such as voltage supply andmonitor output. Table A.1 gives a list of all resistors and capacitors, which we have chosen
properly.
Table A.1.: Values for the different components of the temperature PIDs. The designations are
consistent with the circuit below. The derivative term is not used in our case.

Component

Value

Rp1 1
Rp1 2
Rd1
Cd1
C_In 1
Rp2 1
Rp2 2
Rd2
Cd2
C_In 213 kΩ
100 kΩ––
470 µF
11 kΩ
150 kΩ––
100 µFFor the temperature stabilization of the cavity the derivative part of the PID is not
used, because he reacts on the rate of change, which is slow for temperature. With thepotentiometer Rpot 1 and Rpot 2 the output current of the PID can be limited. For this theconnections P11 and P5 have to be jumpered.

76

1

2

3

4
a15V

PIR1302

COR13
R13
1206
3k3

COP2
P2

PIR1301
PIR302

a15V
10V/K
8
COR3
R3
aIn
aVS PIU208 COR4
R4 1206
3
7
PIU203 RG
PIR402
1206
Vout PIU207PIR401
Vin1
COR1
R1
2
6
100
PIR102
PIR101 PIU202
PIU206
2k2
RG
Ref
GND
1
5
PIU2011206-In-VS PIU205
110R COR38
AD8421
PIR1501
PIR3802 R38
PIR202
-15V
COR15
COR2
R15
R2
3RPot
TAP PIR38031K
1206
0805
1K
1k
a15V
COR5
COR6
R5
R6
GND
0805
0805
COU4A
10k
10k
U4A
TL052
2
PIU402
COR31
R31
GND
1
PIU401
PIR3101
PIR3102
A
3
PIU403
1206
100k

COU2
U2

4
PIU204

11
12
13
14
15
16
17
18
19
20

R21 R20 R19 R18 R170805 0805 0805 0805 0805
PIR2101 1K PIR20 1 1K PIR19011K PIR1801 1K PIR17011K

PI601 PI6012 PI6013 PI6014 PI6015 PI601 PI6017 PI6018 PI 6019 PI 602

COP6
P6

PIR4502COR45 PIR2502COR25 PIR2402COR24 PIR2302COR23 PIR2 02COR22 PIR501
R45 R25 R24 R23 R220805 0805 0805 0805 0805
PIR4501 1K PIR25011K PIR2401 1K PIR23011K PIR2 01 1K

PIR502

PIR1502

PIR3801
PIR601

COR29
R29

PIR1401

PIR2901

1206
100k

PIR201

PIR3202

PIR3201
PIR3 02COR33

410k

Put resistor on socket

PIR902

2

PIR701

BOT
TOP

COR9
R9
3RPot
TAP PIR9031K
COR11
R11
0805

PIR1 02 10k
C

PIR901
PIR1201

1206

RG
RG
-In
AD8421

PIR702PIU303

2
PIU302
1
PIU301

PI 501 PI 502 PI 503

Switch

COP10
P10

GND-LD

LD

COR12
R12
0805

PID102 COD1

PID10

PIR1202 10k

D1
7V5

1
2
3

PI 801 PI 802 PI 803

COJ1
J1

D

PID201

COAX-F
GND-LDD2
Diode

1
2 PIP702PIJ201
1 PIP701

COP8
P8
Vset1
Vin1
Vin2

Polarity LD
PIP102
PIP101

2
1

a15Vina15V
L1 NFM41P
COL1
1
3
PIL101
PIL103

PIC1302 COC13COC1
C13 PIC101 C1

PIL102

2
PIP301 1
OUT2a OUT2

2 2

PIP302GND

PIL202

GND
1

a15Vin

PIL201-15VinD Connector 15

PIJ202 PIJ203 PIJ204 PIJ205

COAX-F

OUT1- COP1
P1

OUT1a OUT1
OUT2- COP3P3-15VinC

COP7
P7

NTC2-Control
8
15
PIJ1015
7
PIJ107
14
PIJ1014
6
PIJ106
13
PIJ1013
5
PIJ105
12
PIJ1012
4
PIJ104
11
PIJ1011
3
PIJ103
10
PIJ1010
2
PIJ102
9
PIJ109
1
PIJ101
PIJ108

17

PIJ702 PIJ703 PIJ704 PIJ705

PID20 COD2

COJ2
J2
GND-LD

PIJ1017

Change Diode direction depending on LD Polarity
COJ7
J7

GND-LD

1PIJ701

GND

16
PIJ1016GND

2 PIP1002
1 PIP1001-15V

1

PIR1 01

a15V
10V/K
8
aVS PIU308 COR10
R10 1206
7
PIR1002
Vout PIU307PIR1001
Vin2
6
100
Ref PIU306 GND
5-VS PIU305

COU3
U3
4
PIU304 aIn
COR7
R7 110R
3

B

2
3
4
5

PIR802

NTC2

COR8
R8

COP5
P5 Polarity LD

GND-LD

-15V

R33
1206
PIR3 0182k

2
3
4
5

2 PIP402
1 PIP401

3 PIP1403
2 PIP1402
1 PIP1401

COL2
L2
NFM41P

PIC201 C2
COC2

PIC701 COC7C7PIC601 COC6
C6

COU1
U1
2
IN
1
PIU101 NC
3
PIU103 NC
4
PIU104 GND
PIU102

0805
0805
0805
0805
PIC1301 1206
2.2uF PIC102 100nF PIC202 100nF PIC702 100nF PIC602 100nF
PIC1402 COC14
PIC401 C4
PIC10 1 COC10
COC3
COC4
C14 PIC301 C3
C10
GND
1206
0805
0805
0805
3 PIC1401 2.2uF PIC302 100nF PIC402 100nF PIC10 2 100nF

REF01CS

a10V
6
OUT PIU106
8
NC PIU108
7
NC PIU107
5
TRIM PIU105

PIC501COC5C5PIC502 0805

100nF
GND
D

PIL203

Title

-15V

Size:

TempPID V8a0 Messbrücke TA
1

Vset1
1V/10K

PIU40

COR32
R32
1206
12k

1
2
3

PIR801

U4B
TL052
7PIU407

B

5
PIU405

-15V

COP14
P14

a10V

COP4
P4

PIU408 COU4B

6
PIU406

PIU40

B

PIR3002

1206
12k
a15V

PIU408

PIR602

COR30
R30

PIR2902
PIR3001

1206
82k

8

10
9
8
7
6
5
4
3
2
1

PIR2102COR21 PIR20 2COR20 PIR1902COR19 PIR1802COR18 PIR1702COR17

COR14
R14

PIR1402

4

PI601 PI609 PI608 PI607 PI60 PI605 PI604 PI603 PI 602 PI 601

0805
5k6

8

PIR1601A

PIR301

COR16
R16

2

PIR1602

BOT
TOP

NTC1

a10V

1

1 PIP201
2 PIP202

10 x Switch

A

2

3

A3

Number:Date: 18.05.2015
Time: 17:58
File: Q:\CsIII\Documents\Drawing
4

1

2

3

4

5

6

7

8

Diode 1N4148

COD2 optional
D2

4

a15V

PIU108 COU1B

8

CORd1
Rd1

1206
100k

CCW

GND

U1B
COP3
OPA2277
P3
7 PIP301
PIU107B
1
PIP302 2
PIP303 3

6

PIR901
PIU106

5
PIU105

PIU104

GND

PIRd102

PIRp1 102

a15V

PIU208 COU2B

PIRp1 201

B

PIRp1 20

6

PIRp1 101

PIU206

5
PIU205

R48
INV 1 On/Off 1206
PIR4801 100k

-15V

Offset 1 On

SW DIP-4

COCd1
Cd1
PIRd101
PICd102 PICd101

Poly
CORp1 11
Rp1

PIR4802COR48

5PIC0In105

U2B
OPA2277
7
PIU207

PIU204

GND

0805

COR8
R8
1206
PIR802100k

CORp1
Rp1 22
COR11
R11

PIR1102

3
PIU203

OUT1a COP4
P4

PIR40 1

PIC25011206
1k

PIC2502 0805

COR40
R40

PIU304

PIR40 2R41206

PIP401
PIP402

COC25
C25

1
2

100nF

-Vend

PIRpot30CW

PIR1601

CORpot3
Rpot3
Imax1
10k

COR16
R16

W
PIRpot30W

W

PIRpot30C W

PIR1602

GND

0R1

GND

GND

PIJ10 7 PIJ10 6

Diode 1N4148
optional
COD1
D1
INT OFF / ON
COS3
S3

GND

Poly

CORp2
Rp2 11

PIR4902COR49

6

PIRp2 101

PIU506R491206

INV 2 On/Off

B

5

PIR2602 100k
COR29
R29

PIR2902

PIU504

-15V

1
9
2
10
3
11
4
12
5
13
6
14
7
15
8

PIU1 05

PIRpot60CW

3
PIU1103

PIR1201

WImax2W PIRpot60W 10k

GND

GND

6
PIU1106

PIR701

PIR3401

aVendIn-18V

A14 C14 PIP10C14

PIP10A16

A16 C16 PIP10C16

PIP10A18

A18 C18

PIP10C18

PIP10A20 A20 C20 PIP10C20-VendInD

PIP10A22

A22 C22 PIP10C22

PIP10A24

A24 C24 PIP10C24

PIP10A26

A26 C26 PIP10C26

PIP10A28

A28 C28 PIP10C28

PIP10A30

A30 C30 PIP10C30

PIP10A32

A32 C32 PIP10C32

a18V

NFM41P

COU6
U6

a15V

7815

PID302GND
1
OUT PIU1201

100nF
4
PIU1204 NC
GND
5
PIU1205 NC

2
GND PIU1202
3
GND PIU1203
6
GND PIU1206
7
GND PIU1207

VCC

8

U12
COU12
IN

PIC3401
COC34
C34
PIC3402 0805

PIU1208

PIC2301
COC23
C23
PIC2302 0805

PIC3101
COC31
C31
PIC3102 0805

LM78L05ACM

GND

Vin1

PIS601

Vin2

PIS603

100nF

100nF

100nF

100nF

100nF

PIC1901
PIC2901
PIC501 COC5
C5
C19
C29
COC19
COC29
Elko
PIC1902 0805 PIC2902 0805 PIC502 100uF
GND
100nF
100nF
PIC2101
PIC30 1
PIC2401 COC24
C24
C21
C30
COC21
COC30
Elko
PIC2102 0805 PIC30 2 0805 PIC2402 100uF
-Vend
100nF

2 PPIU805
5
PIS602
IU805

Mon 1/2

B

6

OPA2277
7

3

PIU807

PIU804

PIU803

2

A

1

OPA2277
R39
COR39

PIU801
PIR3901

1206
4k7

COP9B
P9B
21PIP9021

PIU80

PIP902

a15V

PIR3802

R38
COR38
1206
PIR38051k
1

PID801
COD8
PID802

PIR3902

D8
red

PID902
PID901

Vin2

COD9
D9
blue

R46
COR46
PIR4601

1206
100k

PID402

GND

PIU904

6

Diode 1N4148

5

B

D7
COD7

-15V

PID701

7

PIU907
PID601

PIU904

PID602

U10D
COU10D13
PIU10013

GND
Diode 1N4148
U9B
COU9B
OPA2277
D6
COD6

PIU905

GND

12

PIU10012
PID702

PIR4702

PIU908

PIR4602
PIU906

Logic High if
Temperature
is away from Set point

1111

MC74VHC132DPIR5201COR52

P9A
COP9A ErrorOut

PIU10011
PIP9011

PIP9012

GND

R52
COU10A
U10A
11206 PIU1001
COR43
R43
PIR52051k
2
PIR4302
PIR4301
2
1206

3

PIU1003

PIU1002

100k

PIU1007
MC74VHC132D
PIU10014

D

-15V

GND

PIR3702

COR37
R37
1206

PIR3701k
1

4

PIU901
PID401

1206
100k
a15V

COU8A
U8A

PIU802

GND

3

1

A

PIR4701

GND

2

3

-15V

Monitor

100nF

PIU908COR42
COU10B
R42
U10B
41206 PIU1004
OPA2277
6
COR41 PIR420251k
Diode 1N4148 PIR4102R41
PIU1006
PIR4101U9A
COU9A
5PIU1005
1206
D4
COD4
MC74VHC132D
100k

PIU903

GND

COU8B
U8B

PIU806

2x16

1

1206
100k

2
PIR4402
PIU902R47
COR47

PIU804

a15V

aVend

PIR4401

Display9932844

PIU80

100nF

Vin1

R44
COR44

-15V

COS6
1 S6
3

VDD

100nF
100nF
GND
GND

a15V

PIC10 1
PIC1 01
PIC1201
PIC901
PIC20 1
PIC3201
COC20
COC32
C20
C32
C10
C11
C12
C9
COC10
COC11
COC12
COC9
PIC10 2 0805 PIC1 02 0805 PIC1202 0805 PIC902 0805 PIC20 2 0805 PIC3202 0805
GND
100nF
100nF
100nF
100nF
100nF
PIC1401
PIC1501
PIC1601
PIC1301
PIC2 01
PIC3 01 100nF
C22
C33
C14
C15
C16
C13
COC14
COC15
COC16
COC13
COC22
COC33
PIC1402 0805 PIC1502 0805 PIC1602 0805 PIC1302 0805 PIC2 02 0805 PIC3 02 0805
-15V

1
PIP1402 2
PIP1403 3
PIP1401

PIR4201

PIR4502

1206
100k
a15V

COP14
P14

GND

GND

PID502

12

A12 C12 PIP10C12

PIP10A14

COL2
L2

COD3
D3

R45
COR45
PIR4501

GND

PIR5102

8

PIP10A12

PIL201

PID301

1206
100

4

C8 PIP10C8A10 C10 PIP10C10

PIC802

PIR5002
PIR5101

8

A8

PIP10A10

PIU602C8COC8GND
PIC801 Elko1100uF
3
PIU601 IN
OUT PIU603 33uF

1206
10k

PID501

4

PIP10A8

1aVendInC7

PIC701 Elko

COR50
R50
PIR5001

4

C6 PIP10C6PIC702 COC7

Vset1

8

A6

Diode 1N4148

22

PIP10A6

PIU701

4

C4 PIP10C4

3
NO1 PIS503
1
PIS501 COMMON1
2
NC1 PIS502
SW DPDT
3
PIL203

0805
100nF
COR51
R51

C

COD5
D5

GND

PIC2801 PIC2802

IN

8

PIP10A4 A4PIL102
NFM41P
PIL20

-15V
3
OUT PIU703
PIC1702
PIC1802
GND
COC17
COC18
C17
C18
PIC1701 Elko
PIC1801 Elko
100uF
33uF

Diode 1N4148

C28
COC28

COU7
U7 7915

2

PIU702

1

C2 PIP10C2

2 2

GND

-18V
S5
COS5
6
NO2 PIS506
3PIS504
4
PIL103
COMMON25
NC2 PIS505

L1
COL1
1
PIL101
A2

GND-VendIn-Vend

C

PIP10A2

PI 130 PI 1302 PI 130

2

aVend

COP1
P1

COR34
R34
0R1

1
2
3

1
2
3

PI 120 PI 120 PI 1203

OUT2- Out 2-Vend

INT/EXT
COP13
P13

PIR3402
a18V

PIP801 1
PIP802 2

PIC201

COR7
COC2
R7
C2
1206 PIC202 0805
100nF
PIR7021k

PIU1 04

COR12
R12
1206
PIR12010k
2

PIRpot60C W

B

OUT2a COP8
P81
PIU1101

CORpot6
Rpot6

OUT2-

COU11
U11
OPA548T-1

5

2
PIU1102

COU5A
U5A
OPA2277
COR32
R32
1
PIU501
PIR3201
PIR32021206 PIR3501COR35
R35
10k
1206
PIR3502 470
-15V

A

3
PIU503
GND

7
PIU1107

a15V

PIU508

2
PIU502

PIR2901

1206
100k

PIU504GND

GND

INT/EXT
COP12
P12

1206

PIRp2 02

U5B
OPA2277
7
PIU5071206
100k

PIR2601

COR26
R26

PIU505

PIR4901100k

-15V

W

a15V

PIU508 COU5B

PIRd201
PICd202 PICd201

PIRp2 102

Offset 2 On

CORp2
Rp2 2

OUT1-

OUT2a

aVend

PIR2001

CW

PIU40

PIRd202

PIRp2 01

SW DIP-4

Vin2 Vin1 Vset1

OUT1a

4

GND

COCd2
Cd2

CORd2
Rd2

U4B
COP7
OPA2277
P7
7 PIP701
PIU407B
1
PIP702 2
PIP703 3

COR20
R20

PIR2002

0805
1nF

-a

1uF
GND

-15V

PIU408 COU4B

a15V GND -15V

CCW

PIU40

CORpot5
Rpot5
COR27
R27
WAtt2
6
PIR2701
PIU406W PIRpot50W5k PIR2702
1206
PIRpot50C W
5100k PIU405

2 PIP502
1 PIP501

8

PIC601COC6C6PIC602 1206

PIRpot50CW

PIC0In208

4

8

1206
10k

A

8

1206
10k

3
PIR2401
PIU403

4

GND

COR24
R24

PIR2301
PIR2402

CW

COR23
R23

PIR2302-a

2
1 PIP601
In 2

2
PIU402

2.2uF

Vin2
PIP602

CCW

COP6
P6

U4A
OPA2277
1
PIU401

4

B

a15V

PIU408 COU4A

COP5
P5

8
7
PIC0In207
6
PIC0In206
5
PIC0In205

PIC0In201

8

PIC402COC4C4PIC401 1206

COC0In2
C_In2
12
PIC0In202
3
PIC0In203
4
PIC0In204

PIR1901

1206
100k
a15V

COC27
C27
PIC2701 PIC2702

4

COR19
R19

PIR1902

a15V
Rpot4 CCWCWPIRpot40CWCORpot4
a- PIRpot40CCW
Offset2-15V PIRpot40W
10k

PIS302COJ1J1D Connector 15

PIJ10 PIJ109 PIJ102 PIJ10 PIJ103 PIJ10 1 PIJ104 PIJ10 2 PIJ105 PIJ10 3 PIJ106 PIJ10 4 PIJ107 PIJ10 5 PIJ108

PID101 PID102

W

PIS301A

OUT1- Out 1

PIR40210k

OPA2277
COR14
R14
1
PIU201
PIR1401
PIR1402
PIR1701
1206
COR17
R17
10k
1206
PIR1702470-15V

PIU204GND

PIR401COR4

COU2A
U2A

A

-15V

GND

3
PIU303

a15V

PIU208

2
PIU202

PIR1101

1206
100k

PIR2011206
100k

PIR801

6

PIU306

1
PIU301COR2
R2 1nF
PIR202

COU3
U3
OPA548T-1

16-a

PIR902

PIRpot20C W-15V

6
PIC0In106

4
PIC0In104

8W
W PIRpot20W
5k

PIU104COR9
R9

Offset1
10k
2 PIP1102
1 PIP1101

4

1206
100k

Rpot2
PIRpot20CW CORpot2
Att1

3
PIC0In103

PIR101

4

2
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PIR501
PIR602
PIR601
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2 PIP202PIR502
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C
3
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1
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1 PIP201 1206
COC3
C3
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10k
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In 1
1uF
GND

COP2
P2

U1A
OPA2277
1
PIU101CW

PIC101 1206

PIR102

COP11
P11

7
PIC0In107

PIU305

2

PIU30217

8

PIC0In108

2
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W

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C_In1

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PID202

COC26
C26

5

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CWPIRpot10CWRpot1PIRpot10CCWCCW
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8

PIS101

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Exphystronix
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G:\Libraries\Exphyslogo Protel.jpg
University of InnsbruckTechnikerstraße 25
6020 Innsbruck
Date: 18.05.2015
Time: 17:57:35
SheetofAustria
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TempPID V8a0

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78

A.4. Assembly of the Vacuum Chamber

A.4. Assembly of the Vacuum Chamber
In this Appendix, we describe the main steps of the assembly of the vacuum chamber.
Figure A.2 shows the base plate of the cavity with the copper plate on the Peltier elements.
The Peltier elements are not glued and therefore we fix the positions slightly with Kapton™
tape. The copper plate is screwed to the base plate with plastic screws for better thermal
isolation and to reduce the force acting on the Peltier elements. Figure A.3 shows the insideof the vacuum chamber without the shieldings. The lower flange is placed on the copper
plate. We used tin foil between the Peltier elements and the lower flange to increase thethermal contact and fixed the position of the Peltier elements again with Kapton™ tape.
The NTCs are glued1 into small holders (see Fig. A.7), which are screwed onto the vacuum
chamber at different positions. Figure A.4 shows the outer shielding sitting on the Pelter
elements. On both sides of the shieldings windows are glued into holes (see Fig. A.5). On
the bottom of the outer shielding, see Fig. A.6, spacers out of Teflon and additional holders
for NTCs are placed. The outer shielding is screwed onto the lower flange with vacuum
compatible M2.5 screws2 . The Teflon spacers reduce the thermal contact from the innershielding to the outer one. Figure. A.8 shows the cavity sitting on the V-shaped aluminum
block inside the inner shielding. To monitor the temperature of the aluminum holder two
NTCs are glued into two holes on the side of the aluminum block. The inner shielding isclosed (Fig. A.9) and the wires for to the NTC are guided through a hole in the top of theshielding to the feedthrough.

Figure A.2.: Heat sink of the vacuum chamber with the additional copper plate. Between the copper
plate and the heat sink six Peltier elements are placed to temperature stabilize the copper plate.

1
2

The NTCs are glued with Torr Seal.
Vacom, VFC-2.506-25 and VFC-2.510-25.

79

A. Appendix

Figure A.3.: Framework of the vacuum chamber with Pelter elements inside. The positions of
the Peltier elements are fixed with Kapton™and holders for the NTC are screwed on at different
positions.

Figure A.4.: The outer shielding sits on the Pelter elements on the lower flange.

80A.4. Assembly of the Vacuum Chamber

Figure A.5.: Windows on the outer and inner shielding are glued in. They are coated for thenecessary linewidths, respectively.

Figure A.6.: Inside of the outer shielding. The inner one is sitting on Teflon spacer to reduce thethermal contact.

81

A. Appendix

Figure A.7.: NTCs are glued into holders which can be screwed onto the vacuum chamber.

Figure A.8.: Open chamber with ULE cavity inside.

82

A.4. Assembly of the Vacuum Chamber

Figure A.9.: Both shieldings are closed and an additional temperature sensor is screwed on the top.

83

A. Appendix

84

Bibliography[1] K. Aikawa et al. Bose-Einstein Condensation of Erbium. Phys. Rev. Lett. 108 (2012),
p. 210401.
[2] K. Aikawa et al. Observation of Fermi surface deformation in a dipolar quantum gas.
Science 345. (2014), pp. 1484–1487.
[3] J. F. Allen and A. D. Misener. Flow of Liquid Helium II . Nature 141. (1938), pp. 75–75.
[4] J. Alnis et al. Subhertz linewidth diode lasers by stabilization to vibrationally andthermally compensated ultralow-expansion glass Fabry-Pérot cavities. Phys. Rev. A 77
(2008), p. 053809.
[5] M. H. Anderson et al. Observation of Bose-Einstein Condensation in a Dilute Atomic
Vapor. Science 269. (1995), pp. 198–201.
[6] J. B. Balewski. A single electron in a Bose-Einstein condensate. PhD thesis. Universityof Stuttgart, 2014.
[7] Jonathan B. Balewski et al. Coupling a single electron to a Bose–Einstein condensate.
Nature 502. (2013), pp. 664–667.
[8] H. Y. Ban et al. Laser cooling transitions in atomic erbium. Opt. Express 13. (2005),
pp. 3185–3195.
[9] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Microscopic Theory of Superconductivity.
Phys. Rev. 106 (1957), pp. 162–164.
[10] J. G. Bednorz and K. A. Mueller. Possible high Tc Superconductivity in the Ba–La–Cu–O
System. Zeitschrift fuer Physik B Condensed Matter 64 (1986), pp. 189–193.
[11] A. J. Berglund, J. L. Hanssen, and J. J. McClelland. Narrow-Line Magneto-Optical
Cooling and Trapping of Strongly Magnetic Atoms. Phys. Rev. Lett. 100 (2008), p. 113002.
[12] E. D. Black. An introduction to Pound-Drever-Hall laser frequency stabilization. American Journal of Physics 69. (2001), pp. 79–87.
[13] C. C. Bradley et al. Evidence of Bose-Einstein Condensation in an Atomic Gas with
Attractive Interactions. Phys. Rev. Lett. 75 (1995), pp. 1687–1690.
[14] N. Q. Burdick et al. Fermionic Suppression of Dipolar Relaxation. Phys. Rev. Lett. 114(2015), p. 023201.
[15] Lisheng Chen et al. Vibration-induced elastic deformation of Fabry-Perot cavities. Phys.
Rev. A 74 (2006), p. 053801.
[16] Y.C. Chen et al. Effect of the trapping laser linewidth on the atom number in a magnetooptical trap. Chinese Journal of Physics 38. (2000), pp. 920–926.

85

Bibliography[17] C. Chin et al. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82 (2010),
pp. 1225–1286.
[18] C. Chin et al. Observation of the Pairing Gap in a Strongly Interacting Fermi Gas.
Science 305. (2004), pp. 1128–1130.
[19] J. K. Chin et al. Evidence for superfluidity of ultracold fermions in an optical lattice.
Nature 443 (Oct. 2006), pp. 961–964.
[20] C. B. Connolly et al. Large spin relaxation rates in trapped submerged-shell atoms.
Phys. Rev. A 81 (2010), p. 010702.
[21] J. G. Conway and B. G. Wybourne. Low-Lying Energy Levels of Lanthanide Atoms andIntermediate Coupling. Phys. Rev. 130 (1963), pp. 2325–2332.
[22] K. B. Davis et al. Bose-Einstein Condensation in a Gas of Sodium Atoms. Phys. Rev.
Lett. 75 (1995), pp. 3969–3973.
[23] W. Demtröder. Experimentalphysik 3. Springer Berlin Heidelberg, 2010.
[24] D. Dreon et al. Optical cooling and trapping highly magnetic atoms: The benefits of aspontaneous spin polarization. ArXiv e-prints (2016).
[25] R. W. P. Drever et al. Laser phase and frequency stabilization using an optical resonator.
App. Phys. B 31. (1983), pp. 97–105.
[26] P. Dubé et al. A narrow linewidth and frequency-stable probe laser source for the 88Sr+
single ion optical frequency standard. Applied Physics B 95. (2009), pp. 43–54.
[27] F. B. Dunning et al. Recent advances in Rydberg physics using alkaline-earth atoms.
Journal of Physics B: Atomic, Molecular and Optical Physics 49. (2016), p. 112003.
[28] J. Eichler and H.J. Eichler. Laser: Bauformen, Strahlführung, Anwendungen. Physics
and astronomy online library. Springer, 2003.
[29] J. Ferch, W. Dankwort, and H. Gebauer. Hyperfine structure investigations in DyI withthe atomic beam magnetic resonance method. Phys. Lett. A 49. (1974), pp. 287 –288.
[30] Richard P. Feynman. Simulating physics with computers. International Journal of
Theoretical Physics 21. (1982), pp. 467–488.
[31] C.J. Foot. Atomic physics. Oxford University Press, 2005.
[32] A. Frisch. Dipolar Quantum Gases of Erbium. PhD thesis. University of Innsbruck,
2014.
[33] A. Frisch et al. Narrow-line magneto-optical trap for erbium. Phys. Rev. A 85 (2012),
p. 051401.
[34] A. Frisch et al. Quantum chaos in ultracold collisions of gas-phase erbium atoms. Nature
507 (2014), pp. 475–479.
[35] T.F. Gallagher. Rydberg Atoms. Cambridge Monographs on Atomic. Cambridge University Press, 2005.
[36] J. Goldwin et al. Two-species magneto-optical trap with 40 K and 87 Rb. Phys. Rev. A 65
(2002), p. 021402.

86

Bibliography[37] M. Greiner et al. Quantum phase transition from a superfluid to a Mott insulator in agas of ultracold atoms. Nature 415. (2002), pp. 39–44.
[38] A. Griesmaier et al. Bose-Einstein Condensation of Chromium. Phys. Rev. Lett. 94(2005), p. 160401.
[39] R. Grimm. Ultracold Fermi gases in the BEC-BCS crossover: a review from the Innsbruckperspective. eprint arXiv:cond-mat/0703091 (Mar. 2007).
[40] R. Gross and A. Marx. Festkörperphysik. Oldenbourg Wissenschaftsverlag, 2012.
[41] W. Hogervorst et al. Isotope shifts and hyperfine structure of natural dysprosium.
Physics Letters A 65. (1978), pp. 220 –222.
[42] W. G. Jin et al. Hyperfine Structure and Isotope Shift in Er I by the Atomic-Beam Laser
Spectroscopy. Journal of the Physical Society of Japan 59. (1990), pp. 3148–3154.
[43] H. Kamerlingh Onnes. The Superconductivity of Mercury. Leiden Com. (1911).
[44] P. Kapitza. Viscosity of Liquid Helium below the λ-Point. Nature 141. (1938), pp. 74–74.
[45] J. Keller et al. Simple vibration-insensitive cavity for laser stabilization at the 10-16level. App. Phys. B 116. (2014), pp. 203–210.
[46] G. Kirchmair. Frequency stabilization of a Titanium-Sapphire laser for precision spectroscopy on Calcium ions. MA thesis. University of Innsbruck, 2006.
[47] E. Kirilov et al. Compact, robust, and spectrally pure diode-laser system with a filteredoutput and a tunable copy for absolute referencing. Applied Physics B 119. (2015),
pp. 233–240.
[48] T. Lahaye et al. d-Wave Collapse and Explosion of a Dipolar Bose-Einstein Condensate.
Phys. Rev. Lett. 101 (2008), p. 080401.
[49] T. Lahaye et al. The physics of dipolar bosonic quantum gases. Reports on Progress inPhysics 72. (2009), p. 126401.
[50] N. Leefer, A. Cingöz, and D. Budker. Measurement of hyperfine structure and isotope
shifts in the Dy 421 nm transition. Opt. Lett. 34. (2009), pp. 2548–2550.
[51] T. Legero, T. Kessler, and U. Sterr. Tuning the thermal expansion properties of opticalreference cavities with fused silica mirrors. J. Opt. Soc. Am. B 27. (2010), pp. 914–919.
[52] Y. Liu and P. H. Daum. Relationship of refractive index to mass density and selfconsistency of mixing rules for multicomponent mixtures like ambient aerosols. Journalof Aerosol Science 39. (2008), pp. 974 –986.
[53] M. Lu. Quantum Bose and Fermi Gases of Dysprosium: Production and Initial Study.
MA thesis. Stanford University, 2014.
[54] M. Lu, S. H. Youn, and B. L. Lev. Trapping Ultracold Dysprosium: A Highly Magnetic
Gas for Dipolar Physics. Phys. Rev. Lett. 104 (2010), p. 063001.
[55] M. Lu et al. Strongly Dipolar Bose-Einstein Condensate of Dysprosium. Phys. Rev. Lett.
107 (2011), p. 190401.

87Bibliography[56] Mingwu Lu, Seo Ho Youn, and Benjamin L. Lev. Spectroscopy of a narrow-line lasercooling transition in atomic dysprosium. Phys. Rev. A 83 (2011), p. 012510.
[57] R. Löw et al. An experimental and theoretical guide to strongly interacting Rydberg gases.
Journal of Physics B: Atomic, Molecular and Optical Physics 45. (2012), p. 113001.
[58] T Maier. Interactions in a Quantum Gas of Dysprosium Atoms. PhD thesis. Universityof Stuttgart, 2015.
[59] T. Maier et al. Broad universal Feshbach resonances in the chaotic spectrum of dysprosium atoms. Phys. Rev. A 92 (2015), p. 060702.
[60] T. Maier et al. Emergence of Chaotic Scattering in Ultracold Er and Dy. Phys. Rev. X 5(2015), p. 041029.
[61] T. Maier et al. Narrow-line magneto-optical trap for dysprosium atoms. Opt. Lett. 39.
(2014), pp. 3138–3141.
[62] M. J. Mark. Wechselwirkungseffekte eines Caesium-BEC in eindimensionalen Gittern.
MA thesis. University of Innsbruck, 2007.
[63] D. J. McCarron, S. A. King, and S. L. Cornish. Modulation transfer spectroscopy inatomic rubidium. Measurement Science and Technology 19. (2008), p. 105601.
[64] J. J. McClelland and J. L. Hanssen. Laser Cooling without Repumping: A MagnetoOptical Trap for Erbium Atoms. Phys. Rev. Lett. 96 (2006), p. 143005.
[65] P. McQuillen et al. Imaging the Evolution of an Ultracold Strontium Rydberg Gas. Phys.
Rev. A 87 (2013), p. 013407.
[66] W. Meissner and R. Ochsenfeld. Ein neuer Effekt bei Eintritt der Supraleitfähigkeit.
Naturwissenschaften 21. (1933), pp. 787–788.
[67] P. J. Mohr, D. B. Newell, and B. N. Taylor. CODATA recommended values of thefundamental physical constants: 2014*. Rev. Mod. Phys. 88 (2016), p. 035009.
[68] B. Pasquiou et al. Spin Relaxation and Band Excitation of a Dipolar Bose-Einstein
Condensate in 2D Optical Lattices. Phys. Rev. Lett. 106 (2011), p. 015301.
[69] R. K. Raj et al. High-Frequency Optically Heterodyned Saturation Spectroscopy Via
Resonant Degenerate Four-Wave Mixing. Phys. Rev. Lett. 44 (1980), pp. 1251–1254.
[70] C. A. Regal, M. Greiner, and D. S. Jin. Observation of Resonance Condensation of
Fermionic Atom Pairs. Phys. Rev. Lett. 92 (2004), p. 040403.
[71] A. Rietzler. Narrow-Line Cooling Light for a Magneto-Optical Trap of Erbium Atoms.
MA thesis. University of Innsbruck, 2012.
[72] B.E.A. Saleh and M.C. Teich. Fundamentals of photonics. Wiley series in pure andapplied optics. Wiley, 1991.
[73] P. Schauß et al. Observation of spatially ordered structures in a two-dimensional Rydberg
gas. Nature 491. (2012), pp. 87–91.
[74] M. Segel. Stable and narrow-line diode-laser system for STIRAP. MA thesis. Universityof Innsbruck, 2015.

88

Bibliography[75] D. M. Stamper-Kurn and M. Ueda. Spinor Bose gases: Symmetries, magnetism, andquantum dynamics. Rev. Mod. Phys. 85 (2013), pp. 1191–1244.
[76] P. Unterwaditzer. Aufbau eines vollstaendigen Diodenlasersystems zur Laserkuehlung
und Detektion von gespeicherten Cs-Atomen. MA thesis. University of Innsbruck, 2005.
[77] E. Urban et al. Observation of Rydberg blockade between two atoms. Nature Physics 5.
(2009), pp. 110–114.
[78] M. Weidemüller. Rydberg atoms: There can be only one. Nature Physics 5. (2009),
pp. 91–92.
[79] A. C. Wilson et al. A 750-mW, continuous-wave, solid-state laser source at 313 nm forcooling and manipulating trapped 9Be+ ions. Applied Physics B 105. (2011), pp. 741–
748.
[80] B.G. Wybourne and L. Smentek. Optical Spectroscopy of Lanthanides: Magnetic andHyperfine Interactions. CRC Press, 2007.
[81] S. H. Youn et al. Dysprosium magneto-optical traps. Phys. Rev. A 82 (2010), p. 043425.
[82] W. Zwerger. The BCS-BEC Crossover and the Unitary Fermi Gas. Lecture Notes inPhysics. Springer Berlin Heidelberg, 2011.

89

Bibliography90

Acknowledgement
First of all I want to thank Francesca for giving me the opportunity to do my master thesisin the RARE experiment. She supported me during the whole time, especially with the
preparation of my seminar talk and during the writing phase of my thesis. Further, I wantto thank her that she gave me the opportunity to participate at the "Enrico Fermi" Summer
School in Varenna, which was a unique experience.
Ein großer Dank geht an alle Mitglieder des Erbium- sowie des RARE-Teams für ihre
ständige Hilfsbereitschaft und Geduld bei offenen Fragen. Ein ganz besonderer Dank gilt andieser Stelle Philipp, der mich von Beginn an bei der Masterarbeit unterstützt und bei Fragen
immer ein offenes Ohr gehabt hat. Weiters danken möchte ich Manfred, der mir immer
mit gutem Rat beiseite stand und mich sehr beim Schreiben der Masterarbeit unterstützt
hat. Arno möchte ich für seine Hilfsbereitschaft mit LATEX und das Lesen von Teilen derArbeit danken. Ein Dank geht auch an Emil, der mir beim Bauen des Lasers im flexure
design geholfen hat und auch sonst immer für Fragen zur Verfügung stand. Ein großer
Dank geht auch an die Werkstatt des IQOQI, wo Stefan, Andreas und Bernhard immer
wieder große Hilfsbereitschaft gezeigt haben. Abseits der Universität möchte ich mich bei
der Fußballmannschaft des ASV Riffian-Kuens bedanken, wo ich während meines Studiums
trotz sehr bescheidener Trainingspräsenz spielen durfte. Ein Dank geht auch an meine
langjährigen Mitbewohner Paul und Hannes, mit denen ich einige tolle Jahre zusammen
wohnen durfte.
Abschließend geht der allergrößte Dank noch an meine Eltern, meine Großeltern undmeine Freundin, welche mich immer unterstützt und motiviert haben.

91

Leopold-Franzens-Universität Innsbruck

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 Quellenund Hilfsmittel verwendet habe. Alle Stellen, die wörtlich oder inhaltlich den angegebenen
Quellen entnommen wurden, sind als solche kenntlich gemacht.
Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als Magister/Master- /Diplomarbeit /Dissertation eingereicht.

Datum

Unterschrift