Narrow-Line Cooling Light for a Magneto-Optical
Trap of Erbium Atoms
master thesisbyAlexander Rietzler
submitted to the Faculty of Mathematics, Computer
Science and Physics of the University of Innsbruckin partial fulfillment of the requirements
for the degree of master of scienceadvisor:Prof. Dr. Francesca Ferlaino,
Institute of Experimental Physics, University of InnsbruckInnsbruck, September 2012

ii

SummaryIn this thesis, we describe the main steps for the production and stabilization of the laser
light used to cool erbium atoms in a magneto-optical trap (MOT). The line that we use for
the MOT has a narrow-line character with a 190-kHz linewidth and has a wavelength of
583 nm.
To generate the light at 583 nm, we use a dye laser, operated with Rhodamine 6G. The
laser light from the dye laser has a linewidth of about 1 MHz and a frequency drift of about
100 MHz/h. For MOT operation, we need to reduce the laser linewidth and to stabilize thelaser on the resonance frequency for the different Er isotopes. For this purpose, we use apre-locking cavity, which narrows the linewidth down to 45 kHz and we build a ultra low
expansion (ULE) cavity for long-term frequency stability. A special and convenient feature of
our stabilization scheme is that it allows locking of the cooling light on the atomic transitionof each Er isotope in a very straightforward manner - it can in fact bridge the isotope shifts,
corresponding to several GHz.
The part of our locking setup that is responsible for long-term frequency stability is basedon the Pound-Drever-Hall locking technique in combination with a home-build ULE cavity.
This cavity is made of ultra low expansion glass, which we cool down to the zero-expansion
temperature. At this temperature, the thermal expansion of the cavity is minimized. By
stabilizing the laser to the ULE cavity, we reduce the long-term drift to 0.16 Hz/s.
Using this light, we cooled down the five most abundant erbium isotopes to 15 µK in the
MOT, and recently we achieved the first BEC of erbium atoms.

iii

Zusammenfassung
In dieser Arbeit werden die entscheidenen Schritte für die Produktion und Stabilisierungvon Laserlicht beschrieben, welche für das Kühlen von Erbium Atomen in einer magnetooptischen Falle (MOT) benötigt werden. Der schmalbandige MOT-Übergang hat eine Linienbreite von 190 kHz und eine Wellenlänge von 583 nm.
Für die Erzeugung dieses Lichts verwenden wir einen mit Rhodamin 6G betriebenen
Farbstofflaser. Dieser Laser emittiert Licht mit einer Linienbreite von ca. 1 MHz, die Frequenz des Lichts driftet mit einer Rate von ca. 100 MHz/h. Für den Betrieb der MOT ist
es notwendig die Linienbreite zu reduzieren und den Laser auf die Resonanzfrequenzen derverschiedenen Erbium Isotope zu stabilisieren. Für diesen Zweck verwenden wir einen optischen Resonator zur Reduktion der Linienbreite auf ca. 45 kHz und einen zweiten, selbst
gebauten optischen Resonator zur Erreichung der Langzeitstabilität des Laserlichts. Eine
spezielle Eigenschaft unseres Aufbaus zur Frequenzstabilisierung ist, dass es sehr einfach ist,
die Frequenz in Resonanz mit den Übergängen der verschiedenen Isotope zu stabilisieren eine Verschiebung des Stabilisierungsbereichs entsprechend der Isotopenverschiebungen vonmehreren GHz ist möglich.
Der Teil unseres Aufbaus, der für die Langzeitstabilisierung des Laserlichts verantwortlich
ist, basiert im Wesentlichen auf dem Pound-Drever-Hall-Verfahren in Verbindung mit einem
optischen Resonator aus “ultra low expansion” Glas (ULE), welches einen sehr geringen
thermischen Ausdehnungskoeffizienten besitzt. Wir regeln die Temperatur des optischen
Resonators auf eine bestimmte Temperatur (“zero-expansion temperature”), bei welcher
die Wärmeausdehnung des optisches Resonators vernachlässigt werden kann. Nach Stabilisierung des Lasers auf den ULE Resonator erzielen wir eine Frequenzdrift von 0.16 Hz/s,
gemessen über den Zeitraum von ca. einem Monat.
Wir verwenden das Licht zum Kühlen der fünf meist vorkommenden Erbium Isotope ineiner MOT und erzielen dabei eine Temperatur von 15 µK.
Erst kürzlich gelang es uns, das erste BEC von Erbium Atomen zu erzeugen.

iv

DanksagungFirst of all I want to thank Francesca for giving me the opportunity to do my master thesisin the erbium team. I want to thank her for her support in preparing my seminar talk andfor the assistence in writing this thesis, especially during the time of my community service.
I am also thankful for allowing me to work independently on such an important part of our
project.
Furthermore I want to express my gratitude to Rudi Grimm - the research group he built
up is very professional and has been a supporting and pleasant environment for me.
Besonderer Dank gilt auch den Institutsmitarbeitern. Zu nennen ist hier Nicole, diesehr viele meiner Bestellungen entgegen nehmen musste, Helmut für seine energiereichen
Unterhaltungen und seine Unterweisung im Drehen. Mathias und Toni haben einen großen
Beitrag zur Realisierung der Cavity geleistet.
Ich möchte mich herzlich bei allen Gruppenmitgliedern - Diplomstudenten, Doktoranden
und Post-Doktoranden - bedanken. Besonders viel verdanke ich Hans, von ihm konnte
ich einiges vom Design der Cavity übernehmen. Raffaels Know-How über das optische
Kontaktieren ersparte mir viel Zeit im Reinraum. Dank an Benjamin, der mich auf viele
Probleme aufmerksam gemacht hat, die beim Zusammenbau einer Cavity auftreten können.
Vielen Dank an das Erbium Team. Interessante und lehrreiche Diskussionen bei Almdudler, Prinzenrollen und Thorlabs-Süßigkeiten haben mir sozusagen die Arbeitszeit versüßt.
Albert lehrte mich wie man Probleme am besten anpackt und ohne sein technisches und elektronisches Wissen wäre vieles nicht möglich gewesen. Kiyotaka half mir sehr bei den vielen
Messungen und ohne seine magischen Hände wäre der Farbstofflaser nicht so schnell angesprungen - Thanks a lot! Die Unterhaltungen mit Johannes und die lockeren Sprüche vonMichi wären auch nicht wegzudenken gewesen.
Spezieller Dank gilt Stefan. Danke für die vielen Korrekturen und Verbesserungsvorschlägefür die schriftliche Arbeit und für die langjährige Freundschaft und Zusammenarbeit während
des Studiums.
Es gebührt natürlich auch vielen Menschen Dank, die nicht direkt an meiner Arbeit
beteiligt waren, mich aber in allen Belangen großzügig unterstützt und geprägt haben undsomit dieses Studium erst möglich gemacht haben. An erster Stelle steht meine Familie.
Ich danke meiner Mutter für all die philosophischen Diskussionen rund um die Physik
und ihr grenzenloses Vertrauen in mich. Ich danke meinem Vater für die Entscheidungshilfekurz vor Beginn meines Physikstudiums. Danke Martin und und Laura, ohne euch wäre das
Leben nur halb so schön. Vielen Dank an meinen Onkel Stefan, der es immer wieder schafft,
dank des Einflusses von Raumschiff Enterprise, mich daran zu erinnern, dass das scheinbar
Unmögliche vielleicht doch möglich sein könnte. Dank an meinen Großvater - ich glaube zwarv

vi
immer noch nicht an UFOs, aber unsere Diskussionen haben mich dem wissenschaftlichen
Denken wesentlich näher gebracht. Danke auch an alle in meiner Familie, dass ich immer
auf euch zählen kann.

Contents
Summaryiii

Zusammenfassung

iv

Danksagungv

Introduction

1

1 Erbium
1.1 Laser cooling transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Narrow-line transition for magneto-optical trapping . . . . . . . . . . . . . .

3
3
6
7
8

2 Dye laser for MOT operation
2.1 Dye-laser active medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Optical setup of the dye laser . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dye-laser locking scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
11
14

3 ULE cavity
3.1 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Cavity design considerations . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Mechanical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Temperature control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
22
24
27

4 Experimental setup
4.1 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Optical setup on the dye laser board . . . . . . . . . . . . . . . . . .
4.1.2 Frequency-shifting setup . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Optical setup for the ULE cavity . . . . . . . . . . . . . . . . . . . .
4.1.4 Spectroscopy setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Pound-Drever-Hall locking . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 PDH error signal of the ULE cavity . . . . . . . . . . . . . . . . . . .

29
29
31
32
34
36
38
38

vii

viii

Contents

5 Measurements
5.1 Free spectral range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Zero-expansion temperature of the ULE cavity assembly . . . . . . . . . . .
5.4 Long-term stability of the ULE cavity . . . . . . . . . . . . . . . . . . . . . .
5.5 Linewidth of the locked laser . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
44
45
52
53

6 Conclusion and Outlook

57

A Cavity
A.1 Theoretical description of the PDH locking technique . . . . . . . . . . . . .
A.2 Assembling of the ULE cavity . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Cavity temperature monitoring . . . . . . . . . . . . . . . . . . . . . . . . .

59
59
61
64

B Electronics

69

C Hyperfine structure of the fermionic isotope

73

D PRL: Narrow-line magneto-optical trap for erbium

75Introduction
At temperatures near absolute zero, atomic gases begin to reveal their quantum behavior
and fascinating effects can be observed. A key feature is the high level of control over theinterparticle interaction, which allows us to study phenomena that are strongly connected
to other branches of research like high-temperature superconductivity [Chi06], quantum
chemistry [Osp10], condensed matter [Gre08] and few body physics [Fer10].
The achievement of the first Bose-Einstein condensation in 1995 marked the beginningof a rapid growth in the field of ultracold quantum gases [And95, Bra95, Dav95]. A further
important development has been the cooling of atomic Fermi gases well below the degeneracy
temperature [DeM99, Tru01, Sch01]. The large majority of experiments have been performed
with elements from the alkali series, as these atoms show isotropic contact-like interaction
and provide suitable optical transitions for laser cooling.
The progresses made in cooling and trapping alkali atoms have stimulated studies on more
exotic species for ultracold gas experiments [Ste09, Esc09, Kra09]. Of particular interest
are the atomic species, which interact via dipole-dipole interaction (DDI). The anisotropic
and long-range character of the DDI leads to totally new phenomena in quantum systems.
Predictions show spectacular quantum phases and new kind of experiments exploiting thegeometry-dependent tunability in these systems [Lah09, Bar08]. Systems with DDI are
atoms with a permanent magnetic dipole moment. Another example are polar molecules,
which carry an electric dipole moment.
Up to the moment three species of magnetic atoms have been Bose condensed. Amongthem are a condensate of chromium [Gri05], dysprosium [Lu11] and erbium1 [Aik12]. Chromium
was the first dipolar atom to be Bose condensed by the group of T. Pfau in Stuttgart [Gri05]
and later by the group of O. Gorcieux in Paris [Bea08]. With Cr, many fascinating effects
have been observed. Among all, the most spectacular is the d-wave collapse of the Bose gas[Lah08].
In order to increase the dipolar strength, novel species with higher magnetic moments
have to be considered. With the recent BECs of dysprosium by the Group of B. Lev at
Stanford [Lu11] and of erbium by our group in Innsbruck [Aik12], strongly dipolar gases are
now available in the laboratories.
As a rare earth element with multivalence-electrons character, Er has a very rich atomic
spectrum, which opens up many opportunities for laser cooling. Narrow, intermediate andbroad optical transitions are thus available [Han04, McC06b, Ber08].
Our main experimental goal is the creation of the first BEC of erbium atoms. With an
1

During writing of this thesis, our group achieved a Bose-Einstein condensate of erbium.

1

2

Contents

ultracold sample, we plan to investigate the unknown scattering properties of Er, to produce
Feshbach molecules, and to explore few-body physics with DDI.
The first fundamental step towards quantum degeneracy requires the application of laser
cooling - laser cooling has proven to be a very powerful technique to cool the atoms by ordersof magnitude down to the µK regime [Met99]. In this context, erbium is a rather unknown
species. The first magneto-optical trap of Er has been demostrated at NIST by the groupof J.J McClelland, using a broad J → J + 12 transition at 401 nm. Although the 401-nm
line has the advantage to produce a strong radiation force, this transition has the drawbackof not being optically closed and to have a high Doppler temperature3 . Inspired by works
on Ca [Kra09], Sr [Ste09], Yb[Py09], we choose a narrow-line transition at 583 nm for MOT
operation.
The purpose of my master thesis was to create narrow-line cooling light, which we use
for magneto-optical trapping (MOT) of the erbium atoms. For creation of the light, we use
a dye laser, which is locked to a self-made ultra low expansion (ULE) cavity. The setting upof the locking setup including the construction of the ULE cavity has been the main part of
my work.
This thesis is structured as follows. In Chap. 1 we discuss properties of erbium, which
are relevant for the MOT. The requirements for the cooling light are outlined.
Chapter 2 explains the setup of the dye laser. Chapter 3 decribes the ULE cavity including construction details and temperature stabilization. In Chap. 4 the optical setup needed
for creation of the cooling light is explained as well as details of our locking technique.
Chapter 5 shows the results of the measurements done on the cavity performance and thelaser linewidth. Finally, Chap. 6 concludes and gives a short outlook on the future of ourexperiment.

2
3

J is the total angular momentum quantum number
Cooling limit for Doppler coolingChapter 1
Erbium
Erbium is a chemical element with an atomic number of 68. It belongs to the lanthanide
series and it is part of a set of 17 elements with similar geochemical properties, the so-called
rare earth metals. In the field of ultracold quantum gas experiments, there is an increasing
interest on those elements, since they have exotic level schemes with several transitions suitedfor laser cooling. Further, most of the rare earth elements have a high magnetic moment,
which allows the investigation of dipolar effects.
In this Chapter, we discuss properties of erbium relevant to our experiment, in particular
the energy level scheme of Er and the laser cooling transitions we use and may want to use
in the future. Finally, we discuss the requirements for the light which we use for magnetooptical trapping of Er.

1.1

Laser cooling transitions

The ground state configuration for Er is [Xe]4f 12 6s2 , where [Xe] means that the electronic
shells are filled with electrons up to the 5p shell like in [Xe]. In addition, two of the remaining
valence electrons fill up the 6s shell, whereas the subjacent 4f shell is filled with 12 electrons
having a two-electron vacancy. This kind of structure is called submerged shell structure,
which results in a rich, complex energy level scheme [Ban05, Mar78]; see Fig. 1.1.
Erbium possesses six stable isotopes. Five of them are bosons, which have no hyperfine
structure because the nuclear spin quantum number I is zero. This means that the relevant
quantum number describing the different transitions is the total angular momentum quantum
number J.
In Fig. 1.1, 110 levels of erbium with J ranging from 2 to 12 are shown. Black marked
horizontal lines indicate even parity states, whereas odd parity states are shown in red. The
arrows identify 5 different transitions, which have been proposed to be suitable for laser
cooling applications [McC06b]. These dipole-allowed transitions couple the J = 6 groundstate to an excited state with ∆J = +1 and opposite parity [Dra96].
For describing the quantum states of erbium, we need two different angular momenta
coupling schemes. If the coulomb interaction is dominant over the spin-orbit (LS-) coupling,P~=
the spin ~si of each electron adds up to the total electron spin S
si . The same is true for
i~
3

4

Chapter 1. Erbium

Figure 1.1: Level scheme of atomic erbium. Black (red) horizontal lines indicate even (odd)parity states. Five transition marked by arrows have been identified to be suited for coolingand trapping of Er [Ban05]. So far we use the 401-nm and the 583-nm transition in our
experiment. Grafic taken from [Ban05].
P~~ =the total angular momentum, Li li . The total spin and total orbital angular momentum~ +S~ [Dem05]. The resulting quantum
then couple to the total angular momentum, J~ = L
2s+1
state is denoted by
LJ . The L states are labeled with the following letters [Dra96].
Lsymbol0
S

1
P

2 3 4 5D F G H

6 ...
I ...

This coupling scheme is often used to describe light elements.
In the case of heavier atoms, which have a large nuclear charge, the spin-orbit interaction
becomes stronger than the interaction between individual spins or orbital angular momenta.
Here, the spin and orbital angular momentum of each electron couples individually to form
~i + S~i . These momenta then add up to the total
an individual angular momentum,
J~i = LP
angular momentum J~ = i J~i . This coupling scheme is referred to as the jj-coupling.
For the lanthanides, a special case of this scheme, the J1 J2 -coupling, is applied to describe
many of the excited states [Wyb07]. Here, the 4f electrons and the remaining outer electrons
couple independently via LS-coupling to form two states with J~1 and J~2 , which then add up
to a total angular momentum. These states are labeled with (J1 , J2 )J .

1.1. Laser cooling transitions

5

The 4f 12 6s2 electrons in the outer shells of the Er ground state follow a LS-coupling
scheme, which gives a 3 H6 term. Other states of erbium, e.g. the 583-nm transition, which
we use for the MOT, is described by J1 J2 coupling.
This state is denoted as [Xe]4f 12 (3 H6 )6s6p(3 P01 )(6, 1)07 , where one of the electrons in the 6s
shell is excited to the 6p shell. In this expression, the 4f electrons couple to a 3 H6 term,
whereas the two 6s6p electrons couple to a 3 P01 state. The superscript ’0 ’ indicates odd parityof the state. Here, the 6p electron changes its spin state, which leads to a narrower transition
compared to the 401-nm transition. These kind of transitions are called intercombinationlines.
Table 1.1 lists the main relevant parameters for laser cooling of the five transitions marked
in Fig. 1.1.
parameterΓ (s−1 )τ∆νIsat
TDopplervDoppler
Trecoilvrecoil

400.91 nm
1.7 × 108
5.8 ns27.5 MHz55.7 mW/cm2
659 µK
0.26 m/s
717 nK
6.0 mm/s

582.84 nm
1.2 × 1060.857 µs
190 kHz0.13 mW/cm2
4.8 µK
15 mm/s
170 nK
4.1 mm/stransition
631.04 nm 841.22 nm 1299.21 nm
1.8 × 105 5.0 × 104
13
5.6 µs
20 µs
75 ms
28 kHz8.0 kHz
2.1 Hz
15 µW/cm2 1.8 µW/cm2 0.13 nW/cm2
680 nK
190 nK
51 pKmm
mm
5.8 /s3.1 /s
50 µm/s
140 nK
81 nK
34 nK
mm
mm
3.8 /s
2.8 /s
1.8 mm/s

Table 1.1: Laser cooling parameters for five transitions in Er: spontaneous decay rate Γ,
lifetime τ , linewidth full width at half maximum (FWHM) ∆ν, saturation intensity Isat ,Doppler temperature TDoppler , Doppler velocity vDoppler , recoil temperature Trecoil and recoil
velocity vrecoil . Parameters are taken from Ref. [Ban05] with corrections from [McC06a,
Den10, Fri12].
The scattering force acting on an atom resonant to a laser is proportional to the linewidthof the transition used. At the same time, the linewidth limits the Doppler temperature, whichis the fundamental limit of the Doppler cooling mechanism1 . Therefore, it is beneficial to use
combinations of strong lines and narrow lines to cool atoms down to very low temperatures.
In our experiment, we use the strong 401-nm transition with a linewidth of 27.5 MHz for
the Zeeman slower (ZS) and the narrow 583-nm transition with a linewidth of 190 kHz for
the MOT. The 583-nm transition provides a low Doppler temperature of 4 µK and a low
recoil temperature2 of 170 nK due to the high mass of erbium. In general, a transition used
for laser cooling needs to be optically closed, which means that no atoms decay to metastablestates and thus, get dark to the cooling light. The branching ratio, which is the probabilityof a decay to a metastable state per cooling cycle, limits the atomic population reaching thedesired temperature.
~∆ν[Foo05]
TDoppler = 4πkB
2 22The recoil temperature Trecoil = ~ mk is the fundamental limit for sub-Doppler cooling [Foo05].
16

Chapter 1. Erbium

Figure 1.1 shows that the 401-nm as well as the 583-nm transition are optically open.
Decay can happen via electric dipole allowed transitions to several states with even parity
and J = 6, 7, 8. According to Ref. [Ban05], the branching ratios can be estimated to about
10−4 for the 401-nm transition and about 2 × 10−8 for the 583-nm transition.
The branching ratio for the 401-nm transition suggests that operating a MOT on this line
shows bad performance. However, the group around J.J McClelland has demonstrated a
MOT with an atom number of more than 106 atoms using the 401-nm transition [Ber08].
They explain this result with a recycling mechanism, where the atoms decaying to metastablestates stay in the vicinity of the MOT due to trapping by the quadrupole magnetic field,
until they decay back to the ground state and are recaptured by the MOT. Trapping is only
possible due to the high magnetic moment of 7µB of erbium.

1.2

Isotopes

Er possesses six stable isotopes with masses ranging from 162 amu to 170 amu. Table 1.2
shows the abundance and the nuclear spin quantum number I for each of the isotopes.

Table 1.2: Abundance and spin quantum number for the different Er isotopes.
mass m (amu)
162
164
166
167
168
170

abundance (%)
0.1
1.6
33.5
22.9
27.0
14.9

nuclear spin I
0
0
0
7
2

0
0

statistic
boson
boson
boson
fermion
boson
boson

None of the bosonic isotopes shows hyperfine structure due to I = 0, whereas thefermionic isotope with I = 27 has eight hyperfine states from F = 19
to F = 25 . The abun2
dance of 167 Er of more than 20% is exceptionally high compared to other fermionic alkaline
atomics species such as 40 K [DeM99] and 6 Li [Sch01], which are often used for cold atoms
experiments. The exact transition frequency between two atomic states depends slightly on
the mass of the atom core and therefore, changes with different isotopes. This effect is calledisotope shift [Foo05]. Figure 1.2 shows the inter-isotope shifts for three different transitions(401 nm, 583 nm, 841 nm) of the bosonic isotopes of erbium. These transition couple tostates with J = 7 and therefore, 167 Er has 21 different transitions between the hyperfine
states, which are not shown in this plot.

1.3. Magnetic properties

7

Figure 1.2: Isotope shift of three laser cooling transitions of Er vs. the natural abundance.
The transitions for the fermionic isotope is not shown, as it involves 21 different transitions
due to hyperfine splitting. Data taken from [Con10, Lip93, Jin90].
Typically, the isotope shift for Er is in the order of GHz, so one can reach the transition
frequency of every isotope with a single laser system.

1.3

Magnetic properties

Compared to alkali atoms, Er has a high magnetic moment in its ground state. The absolute
ground state has a total angular momentum quantum number of J = 6 and magnetic
quantum number of MJ = 6. The magnetic moment µ is then calculated byµ = gJ MJ µB ,(1.1)

where gJ is the Landé g-factor. For the Er ground state gJ = 1.16, which gives a magnetic
moment of 6.98 µB . The high magnetic moment has a huge impact on the quantum behavior
due to the dipole-dipole interaction (DDI) energy. For polarized dipoles, the DDI reads asUDD =µ0 µ2 1 − 3 cos2 (θ).
4π| ~r |3(1.2)

Here, ~r is the relative coordinate between the two dipoles and θ is the angle between ~r andthe dipole axis [Lah09]. Interesting properties of the DDI is its anisotropy and long-range
character, which could introduce novel collisional behavior. For example, the prohibition ofultracold collisions between identical fermions could be lifted due to the anisotropy of the
DDI [Bar08].

8

Chapter 1. Erbium

1.4

Narrow-line transition for magneto-optical trapping

In summary, the 583-nm laser used for magneto-optical trapping of Er has to fulfill thefollowing specifications.
• The linewidth of the laser light has to be in the order of or below the linewidth ∆ν
of the 583-nm transition, which is 190 kHz, to get enough power into the transition
during MOT operation.
• To get a large capture volume for our 3D-MOT, the laser beams are expanded tothe diameter of the viewports of the MOT chamber, which is about 3 cm. For agood cooling performance, the beam intensity should be several times the saturation
intensity of the transition; e.g. for the ytterbium MOT a intensity of 45 × Isat is used
[Yam08]. For the 583-nm transition the saturation intensity is 0.13 mW/cm2 . For
example, if we use 45 × Isat , a power of about 40 mW per beam is needed.
In order to have enough power including losses from optical elements, the laser should
provide an output power of several 100 mW.
• The laser-frequency has to be stable during the typical timescale of an experiment,
before the laser drops out of the resonance. An acceptable change in frequency of one∆ν = 190 kHz in one hour implies that the frequency drift of the laser has to be lessthan about 50 Hz/s.
• The isotope shift for the different Er isotopes has to be bridged and thus, the laser
and the locking setup has to be capable of changing its working-frequency continuously
over several GHz.

Chapter 2
Dye laser for MOT operation
As a laser source for MOT operation, we use a continuous-wave ring dye laser1 . The laser
emits light at a wavelength of 583 nm. The active medium of the dye laser is pumped by
a 10 W solid-state laser2 at 532 nm. Our dye laser, as provided by the manufacturer, isspecified to an output power of about 1 W, a linewidth of around 1 MHz and a frequencydrift of 100 MHz/h. An output power of 1 W is sufficient to operate the MOT, but we need tonarrow down the linewidth below 190 kHz, which is the linewidth of the 583-nm transition of
Er. In addition, we need to stabilize the laser frequency to reduce the long-term drift below
several kHz/h. This requirement is necessary to keep the laser light resonant to the 583-nm
transition during the data acquisition time, which is typically several hours. Therefore, thelaser has been modified and locked to an external ULE cavity; see Chap. 3. Here, we present
the basic concepts of the operation principle of our dye laser and the modified locking scheme.

2.1

Dye-laser active medium

One of the important parts of a laser is the active medium or gain medium. External energy
is used to create a population inversion in a metastable upper state of the active medium.
The population inversion is needed to make the stimulated emission to be the dominatingprocess.
In a dye laser, the active medium is a liquid, which consists of organic dye-molecules mixed
with a solvent. These molecules have a complex energy level diagram with several bands
composed of a large number of rovibrational states. Absorption and emission of light canoccur between bands and additional relaxation processes occur between rovibrational states
within the band. This gives a large tuning range of several tens of nanometers, which is oneof the advantages of a dye laser over other laser sources like diode lasers. Another advantage
is, that there are dyes for nearly every wavelength in the visible spectrum available; see
Fig. 2.1. We use Rhodamine 6G dissolved in ethylene glycol, as it has the largest conversion
efficiency at 583 nm and further, it has the advantage of not being toxic.
1
2

Radiant Dyes CW ring dye laser
Coherent Verdi V109Chapter 2. Dye laser for MOT operationeﬃciency (%)

10wavelength (nm)Figure 2.1: Tuning curves for different dyes in the visible spectrum. Dashed lines are fordyes that can only be used for pulsed lasers. The curve with the black filled backgroundis the one for Rhodamine 6G, which is one of the interest for the present work. Graphic
adapted from [Rad11].

(b)

(a)
singletstatesS2
non-radiative
relaxation~10-12sT2absorption

T1

emission
excitation

50100nmfree-airdye jetwatercooled
reservoirabsorption

S1

S0

catcher tube

tripletstates

532nm
pump
beam
dye
nozzlepump

non-radiative
decay
10-7 - 10-3s

Figure 2.2: The active medium. (a) Level structure of a typical dye. The structure on whichlasing occurs is a four-level system between the zeroth and first singlet state, which are verybroad due to rotational and vibrational splitting. Vibrational states are indicated by thicklines, rotational states by thin lines. (b) Flow diagram of the dye circulator.
The lasing mechanism is based on absorption of pump light at 532 nm and emission of abroad spectrum of 560 − 585 nm, from which one can choose the desired wavelength using
optical filters; see Fig. 2.2(a). Dye molecules are pumped from the S0 ground state into ahigher rovibrational state of the excited singlet state S1 , followed by a non-radiative decay tothe lowest rovibronic state of S1 . Lasing can occur on any transition going from the lowest
S1 state to the S0 state. Loss processes are mainly absorption into the singlet state S2 andtriplet absorption between the states T1 and T2 . Here, the problem is that the emission band,on which lasing occurs, and the absorption band may overlap. Triplet absorption is only
possible, if there is a population in the T1 state. The probability of S1 ⇒ T1 is low, while
the transition rate of T1 ⇒ S0 can be as high as 103 s−1 . As a result, the population can get2.2. Optical setup of the dye laser

11

piled up fast in the triplet states. This would stop the laser emission after a time of about
0.1 ms. For this reason and due to heating of the dye liquid one has to constantly remove thedye solution from the interaction region with the pump laser, using a dye circulator1 . The
dye liquid is pumped with a pressure of about 7 bar into a nozzle, creating a homogeneous
free-air dye jet; see Fig. 2.2(b). After interacting with the pump laser, the dye liquid iscatched by a tube, in where the dye liquid flows back into a water-cooled reservoir.

2.2

Optical setup of the dye laser

Another important part of a laser is the laser cavity, which amplifies and frequency-selects
the emission from the active medium. The simplest laser cavity is set up by two or more
highly reflective mirrors acting as a resonator for light. The ideal geometrical alignment andthe mode matching of the cavity mirrors sets the spatial mode of the output beam, whichin our case is a Gaussian mode (also called TEM00 mode [Dem03]). For an overview oncavity behavior, see Sec. 3.1. Transmission through such a resonator is only possible if the
optical length of the resonator is a multiple of the laser wavelength. This creates a comb-like
transmission curve. In the case of a dye laser, the tuning curve of the active medium is verybroad. Thus, the laser will run on multiple longitudinal modes (multiple frequencies) which
are separated by the frequency distance between the transmission peaks of the laser cavity
called free spectral range (FSR). To get a dye laser to operate on a single mode, additional
optics are needed inside the laser cavity.
The optical setup for our dye laser is illustrated in Fig. 2.3. We use a ring cavity madeof the four mirrors M1 to M4. The optical path in the cavity is about 1.5 m long. An
astigmatism-correction rhomb is placed inside the cavity to increase the quality of the spatial
mode of the laser beam. The rotate-able Brewster plate as well as the piezo-attached mirror
M2 (“tweeter”) are able to change the optical length of the ring cavity for fine adjustment
and control of the output frequency. For selecting a single, longitudinal mode, a birefringent
filter (BF), thin etalon (TNE) and thick etalon (TKE) are introduced into the beam path.
An optical diode, which consists of a Faraday rotator followed by a thin λ/2 plate, is used
for unidirectional operation. Without the optical diode the laser-beam would run in both
directions around the ring cavity. This would lead to beam interference and spatial holeburning in the active medium. The frequency of the laser is stabilized to a reference cavity,
the photodiodes PD 1 and PD 2 are used to generate the error signal for feedback.

Frequency selective elements
The transmission curves for birefringent filter, thin etalon and thick etalon are schematicallyillustrated in Fig. 2.4(a). They are all periodic in frequency space. The periodicity is given by
the free spectral range. Another important quantity is the width of one transmission peak.
Often a small width is important to suppress the oscillation of all modes except one. If theseelements are combined in the right way, which means a multiplication of the transmission
curves, a single longitudinal mode can be selected from the ring cavity; see Fig. 2.4(b).
1Radiant Dyes RD1000CW dye circulator

12

Chapter 2. Dye laser for MOT operation
M3
intracavity
EOM
opticaldiodeastigmaticrhomb

brewsterplate

M2"tweeter"
PZT-mirror

beam
dump

TNE

TKE

beam
pick off

583nm
laser
outputM4
output coupler

BF
PD 1

dye jetPD 2532nm pump lightpump mirror

M1

reference cavity
with galvo plate

Figure 2.3: Optical setup of the dye laser. The dye jet is pumped by 532-nm light. The ringcavity is set up by the four mirrors M1 to M4. The elements inside the cavity are for single
mode operation and to tune the laser frequency. The intracavity EOM has been added to theoriginal configuration of the laser in order to narrow the linewidth of the laser below 1 MHz.
Additionally, the reference cavity is drawn, which has been used to frequency-stabilize thelaser in its original configuration.
Birefringent filter
Our three-stage birefringent filter is a combination of three birefringent plates in series.
Three stages are used to amplify the filter effect. In the following, I explain the filter effectof one birefringent plate.
A birefringent plate has different refraction indices for different propagation directions inside
the crystal. These directions are defined by the optical axis of the crystal, which is set by
its structure. For a uniaxial crystal, this would be the direction along the speed of light islowest or highest compared to the rest of the crystal. If polarized light is transmitted througha birefringent plate, the different velocities for ordinary and extraordinary rays result in apolarization rotation of
2πd∆n
∆Φ ≈
sin2 (β).
(2.1)
λ sin(α)
Here, λ is the optical wavelength, d is the distance covered by the ray inside the plate and∆n the difference of the refraction indices for extraordinary and ordinary ray [Bon86]. The
angle β is between the optical axis of the birefringent crystal and the wave vector directionof the beam inside the crystal; see Fig. 2.5. If the plate is placed at Brewster’s angle α into
the beam path, a vertical polarized beam will have almost no losses in transmission, whereas
other polarizations will exceed substantial losses. As the polarization rotation angle ∆Φ
depends on the wavelength, filtering is achieved. Tuning of the transmission peaks is doneby turning the plate around the rotation axis as depicted in Fig. 2.5. This changes the angle

2.2. Optical setup of the dye laser

13

(a)

(b)

thin etalontransmission profile

thick etalon

lasing threshold

transmission

transmission

birefringent filter

laser-cavity modes

frequency

frequency

Figure 2.4: Transmission peaks of a dye laser resonator. (a) Schematic illustration of thetransmission curves of birefringent filter, thick etalon and thin etalon. The curves are drawn
out of scale to show the selection of a single frequency (dashed line). (b) Combined transmission curve. To get the combined transmission curve, the transmission curves of all thethree elements have to be multiplied. Only one laser cavity mode is above lasing threshold
due to the losses generated by the combined transmission curve.
β without changing the angle of incidence.
Using three birefringent plates with integer thickness ratios in series, the transmission peakwidth of the filter can be chosen independently of the FSR. Our three-stage filter has a FSR
in the order of THz and has a broad transmission peak width compared to TNE and TKE.
thin etalon

birefringent
plate

thick etalon

tiltingplaterotation axis

laser beam

crystaloptical axis

laser beamlaser beamddFigure 2.5: Schematics on the intracavity elements. The physical quantity, which is used
to tune the transmission frequency, is the angle β for the birefringent plate (and filter), theangle θ for the thin etalon and the distance d for the thick etalon.
Thin and thick etalon
Thin and thick etalon are illustrated schematically in Fig. 2.5. These etalons work in exact
the same way as the laser cavity or any other optical resonator; see Sec. 3.1. If the wavelengthof the light matches the optical distance (nd) inside the plate, a standing wave builds up in

14

Chapter 2. Dye laser for MOT operation

the optical element and the plate gets transmissive. The transmission peaks of an etalon areat wavelengths
1(2.2)
λm = 2nd cos(θ),
mwhere m can be any integer number greater than zero and n is the refraction index of
the glass. The angle of incidence is θ and d is the thickness of the plate. Tuning of thetransmission curve is done in two ways. The thin etalon changes the angle of incidence θ byusing a galvo motor, while the thick etalon, which is made up of two Littrow prisms, canchange its distance d with a piezoelectric transducer (PZT) mounted to one of the prisms.
The thin etalon has a FSR of about 200 GHz and is used for rough filtering in comparison
to the thick etalon with a FSR of 10 GHz and narrower transmission peak width.

2.3

Dye-laser locking scheme

Thermal and mechanical noise introduce fluctuations in the length of the ring cavity. The
frequency drift of an intracavity element results in a mode hop in the laser output, whereas
length fluctuations of the ring cavity broadens the linewidth of the laser light. The birefringent filter and the thin etalon have a large enough transmission peak width to be manually
tuned, but the distance of the thick etalon prisms and the length of the laser cavity have to
be stabilized by feedback from a frequency reference. Generally speaking, a feedback-loop
is used to keep constant a physical quantity, which in our case is the frequency of the laser
light. For this purpose, the frequency of the light is compared to a stable frequency reference, which then gives the so-called error signal. The error signal is fed to a servo thattransforms it in such a way that the frequency fluctuations are minimized, if the output of
the servo is applied to an frequency-tuning element inside the laser cavity. In our setup
we use analog Proportional-Integral-Derivative (PID) controllers. I will not go into detail
explaining feedback theory and refer to literature [Tie99].
The locking scheme of the dye laser both before and after our modification is illustrated
schematically in Fig. 2.6. We now discuss the different parts of the locking scheme in detail.
Thick etalon lockAs already mentioned, the distance of the two etalon prisms can be changed with a PZT. A
small-amplitude sinusoidal voltage of 2 kHz frequency modulates the distance of the prisms,
creating a slight amplitude modulation on the light at 2 kHz, which is detected by PD 1.
The original 2 kHz signal is compared with the signal from PD 1 in a phase-sensitive detector (PSD). If the transmission maximum of the thick etalon is below the target frequency(transmission maxima of BF and TNE) in frequency space, the PZT-voltage increase leads
to an increase of output power, resulting in a phase relation of 0◦ between the two signals in
the PSD. Correspondingly, a transmission maximum above the desired wavelength ends up
in a phase relation of 180◦. Thus, for locking the TKE to the transmission maximum, theinternal phase shifter of the PSD is set to 90◦ , creating an error signal which is integrated
and fed back to the piezo.

2.3. Dye-laser locking scheme

15
dye laserthin etalon
offsetscan
generator

galvo
driver2 kHz
sinewavethick etalon

∫

PSD
PD 1

referencecavityPD 2+

reference cavity
error signal

thin etalon

HV
TKE
feedback signal

servo LF feedbackreferencecavitygalvo
driverbrewster plate"TOPTY"cavity"tweeter"galvo plate
MF feedback"TOPTY" PDH
error signal

mixer

ULE
cavityHVLO"TOPTY" cavity

LF feedback

DYE
servo

HF feedback

PZT

intracavity
EOM

MF feedback

HV

ULE cavity
HVconnections inactive formodified locking scheme
connections active only when
scanning laser frequency

LO
mixer

ULE cavity PDH
error signal

servo

connections active formodified locking scheme

Figure 2.6: Feedback scheme of the dye laser. Both, the old locking setup and the modified
locking setup are shown. The setup consists of 3 main parts. The thick etalon lock, which gets
its error signal from PD 1, the original lock using the reference cavity and the modified lock,which gets the error signal from the “TOPTY” cavity and the ULE cavity. For narrowing thelinewidth to lower values, the intracavity EOM has been added. Additionally, the mechanism
to scan the laser frequency is shown.
Stabilization of the laser cavity length - linewidth reduction
Before the modification, the laser has been locked only to the reference cavity by a side of
the fringe lock; see Fig. 2.6 and Fig. 2.7. In this configuration, the error signal is connected
to a servo and split up into a mid frequency part (MF feedback) and a low frequency part
(LF feedback). The low frequency component of the feedback signal is applied to the galvo
driver of the Brewster plate, which has a frequency response of several 100 Hz and is able
to correct the laser frequency by 15 GHz; see Tab. 2.1. The high frequency part is applied
to the piezo of the “tweeter” mirror. This piezo has a frequency response of several kHz,and has a frequency correction range of 1 GHz. Here, the piezo is the element with thelargest frequency response, which limits the servo bandwidth to several kHz and thus, is
not fast enough to reduce the linewidth below 1 MHz. Further, the long-term drift of about
100 MHz/h of the reference cavity is not sufficient for operating the MOT.

16

Chapter 2. Dye laser for MOT operation
Total Power - PD 1
Virtual Zerovoltage

Cavity transmission - PD 2

Lockpoint

frequency

Figure 2.7: Error signal for the side-fringe locking technique. The reference cavity transmission is detected with PD 2. The error signal is generated by subtracting this signal by
the signal on PD 1, which measures the total laser power. This creates a virtual zero at theside of a fringe at approximately 2/3 below the maximum . The advantage of subtracting thetotal laser power is that the error signal gets insensitive to intensity fluctuations.
Therefore, we changed the locking scheme. To cancel out the faster frequency fluctuations, an intracavity electro-optical modulator3 (EOM) with a frequency response up to
100 MHz, is introduced into the laser cavity. This EOM has a half-wave voltage of 1000 V. In
combination with a laser cavity FSR of about 150 MHz this results in a frequency correction
range of 75 kHz/V. In the new feedback scheme, the laser is pre-locked to a scanning FabryPerot cavity4 (“TOPTY”) by using the Pound-Drever-Hall locking technique; see Sec. 4.2.
The PDH error signal from the “TOPTY” cavity is fed to the DYE servo, which splits up
the feedback signal into a low frequency part (LF feedback), a mid frequency part (MF feedback) and a high frequency part (HF feedback). The parts are applied to the galvo driverof the Brewster plate, the “tweeter” piezo and the electrodes of the EOM, respectively. The
electronic circuit for the DYE servo is found in App. B.5. For the feedback onto the EOM,a home-made high-voltage (HV) amplifier is used. This amplifier has a 3 dB bandwidth ofabout 800 kHz and a large-signal gain of about 25.
Table 2.1: Properties of the three elements which tune the laser frequency. The correction
range is the maximum frequency change an element is able to correct.
element
Brewster plate“tweeter“ piezo
intracavity EOM

frequency response
several 100 Hz
several kHz
100 MHzcorrection range
15 GHz
1 GHz
75 kHz/V

The “TOPTY” cavity itself is not stable for a long term. For this purpose, I built the
ULE cavity, which has a long-term stability in the order of Hz/s; see Chap. 3. The error
3
4Linos PM25
Toptica FPI-100-0500-1

2.3. Dye-laser locking scheme

17signal from the ULE cavity is also generated with the PDH locking technique. This error
signal is applied to a simple PID controller followed by an HV amplifier and fed to the piezo
controlling the length of the “TOPTY” cavity. In this way, the long-term stability of the
ULE cavity is transferred to the “TOPTY” cavity and the frequency of the laser light.
Scanning of the laser frequency
The scan generator changes the laser frequency by scanning the Brewster plate inside thereference cavity. By scanning slowly and feeding the scanning voltage forward to most of theintracavity elements, the laser stays locked during the process. A scanning range of 30 GHz
is achieved.

18

Chapter 2. Dye laser for MOT operation

Chapter 3
ULE cavity
In ultracold atom experiments, sophisticated lasers systems are used, for instance to trap andcool atomic clouds. Often these kind of experimental techniques require a high frequency
stability of the lasers. Therefore, they have to be stabilized using frequency references.
A typical stabilization scheme is to lock the laser to an atomic reference for long-term stability
and to stabilize it to a mechanically stable cavity for suppression of frequency fluctuations
on short timescale. Such a scheme is not feasible in cases where atomic references are notavailable at the desired wavelength or it is too involved to implement. In addition, for narrowline atomic transitions, with linewidths below several kHz, weak absorption and broadening
effects in atomic references may be a limiting factor.
Since we want to address narrow-line transitions in our experiment, I have built an ultra low
expansion (ULE) cavity, serving as a long- and short-term frequency reference. In the next
Sections, the theoretical description of our cavity is discussed, followed by a description of
the cavity design and the vacuum chamber enclosing the cavity.

3.1

Theoretical description

We here consider a lossless cavity consisting of two identical plane mirrors. The steady-statesolution of cavity modes is known in literature [Dem05] and I here briefly summarize themain features of a cavity.
An overview of the working principle of the cavity is illustrated in Fig. 3.1. In deriving
the steady state solution of the cavity-problem, it is sufficient to describe the incident lightby a monochromatic wave Ein = E0 eiωt . The light is sent through two cavity mirrors at
distance L, having reflection coefficient r and transmission coefficient t. In the case ofnegligible absorption processes in the mirrors, the transmission and reflection coefficientsare connected by the following simple relationr 2 + t2 = 1.

(3.1)

All the partially reflected waves have to be summed up, to get an expression for the reflected
light field. Each reflection shifts the phase of the light field by π. In addition one has totake the phase shift Φ into account, which is accumulated due to propagation of the wave
19

20

Chapter 3. ULE cavity
r,t

r,t

Ein
rEin
t2rEint2r3Ein...
L

Figure 3.1: Scheme of a symmetric lossless cavity. The cavity consists of two mirrors withreflection coefficient r and transmission coefficient t. The transmitted and reflected lightfield are generated by multi-beam interference. For a better illustration, the angle of theincident beam is drawn 6= 90 ◦.
during every round-trip,

2L
,
(3.2)
c
where ω is the frequency of the light field and c the speed of light in vacuum. The totalreflected light field Eref is then given in the limit of infinite partially reflected waves.

Eref = Ein −r + t2 reiΦ + t2 r 3 ei2φ + ...
Φ=ω

= Ein reiΦ − 1.
1 − r 2 eiΦ

(3.3)

From expression (3.3), the total reflected light intensity Ir is derived by taking the absolutesquare. We get the transmitted intensity It with the same approach, by adding up thetransmitted partial waves and taking the absolute square. Finally, one obtains
4R sin2 (Φ/2),
T 2 + 4R sin2 (Φ/2)
T2= Iin 2,
T + 4R sin2 (Φ/2)

Ir = Iin

(3.4)

It

(3.5)

2
2
, R = r 2 and T = t2 are the ingoing intensity, the reflectance andwhere Iin
= cǫ20 Ein
transmission of the mirrors, respectively. The transmission function It (ω) has two main
properties as illustrated in Fig. 3.2. First, the function is periodic. The periodicity is givenby the frequency distance between two adjacent transmission peaks and is called free spectral
range
c
.
(3.6)
FSR =
2L
For instance, a cavity length of L = 15 cm, results in a FSR of about 1 GHz.
The second important parameter is the full width at half maximum FWHM or also called
linewidth ∆ν of the transmission peaks. We are only interested in cases, where the mirrors arehighly reflective (R > 0.5) and thus, the transmission peaks are very sharp. The linewidth is
an important parameter, because it is directly related to performance of the Pound-DreverHall lock; see Sec. 4.2. A simple expression for the FWHM of the cavity is found by defining3.1. Theoretical description

It(ω)/Iin

21

FSR

Δν,
FWHM
ν
Figure 3.2: Transmission of a cavity as a function of the frequency. The periodicity of thetransmission function is given by the FSR. The measure for the quality of the resonator, the
FWHM linewidth ∆ν of a cavity mode is marked.
another parameter, the finesse F . The finesse is a measure for the reflectance of the mirrors,
the equation
√
π R
F≈
(3.7)
1−R
is used for R > 0.5. Examples for various finesses specified for our cavity can be found in
Tab. 3.1. With the finesse defined, a simple expression for the linewidth of the cavity can befound,
FSR
.
(3.8)
∆ν =
F
The linewidth of the cavity, can be measured indirectly via the average photon storage timeτ . This is the characteristic time for the exponential decay of the transmitted light intensity,
when the light incident on the cavity is suddenly switched off.
τ=

1
.
2π∆ν

(3.9)

In our derivation of the cavity transmission function, the spatial profile of the intensity
distribution has not be taken into account. This profile can be described by a superpositionof so called Hermite functions called TEMnm modes [Dem03], where n and m are positive
integers. Depending on the geometry of the cavity, length and mirror curvatures, the TEM
modes have different transmission frequencies. The locking performance degrades, if other
TEM modes lie close to the TEM00 modes in frequency space.
We use a cavity configuration, where one mirror is plane and one mirror has curvature ρ.
The transmission frequencies of the different TEMnm modes in the resonator are then given
by
FSR
.
q
δν = (n + m + 1)
(3.10)π cos
1 − Lρ

For example, with a cavity length of 15 cm and a mirror curvature of ρ = 400 mm, evaluationof Eq. (3.10) gives a frequency distance of 210 MHz between adjacent cavity modes. In thiscase, only cavity modes with n + m > 4 could lie near the TEM00 mode. These modes

22

Chapter 3. ULE cavity

can easily be suppressed by a good cavity coupling. For a good coupling the laser beam
propagation axis has to match exactly the cavity axis. Further the radius of curvature of
the phase front has to be matched with the radius of curvature of the mirrors.
This is achieved by focusing a laser beam, which already has a good Gaussian shape, into
the cavity. For our cavity geometry, the mode matching condition corresponds to a beam
waist ofv!
ruu Lλ r − L
w=t.
(3.11)πL
Here, λ is the wavelength of the light that is resonant to the cavity.

3.2

Cavity design considerations

Our cavity mirrors have been designed for several wavelengths of interest for the erbium
project. The full chart of the reflectance of the mirrors depending on the wavelength isshown in Fig. 3.3. The cavity has different finesses for the different wavelengths optimized
for a good locking performance, and because of Eq. (3.11) also the mode matching condition
is wavelength dependent. Specifications of these quantities can be found in Tab. 3.1.
Within the scope of this thesis, we want to lock only the 583-nm light to the cavity. The
locking setup has to be capable of changing immediately the lock between different isotopes.
For tuning the transmission frequency we use two acousto optical modulators with limited
tuning capabilities. Therefore, I chose the cavity to have a length of 15 cm, which gives afree spectral range of 1 GHz, which roughly matches the isotope shift of the erbium bosonicisotopes.

par.
L
FSR
δν
ρ

value
15 cm
1 GHz
210 MHz
400 mmλ (nm)
583 (spec.)
583 (meas.)
631
841

R (%)
99.91
99.83
99.95
99.96

F
∆ν (kHz)
3500
286
1865
536
6500
154
8000
125

(w µm)
190
190
200
230

Table 3.1: (left): Cavity parameters resulting from the geometry. For evaluation of δν Eq.
(3.10) is used. The FSR has been measured; see Sec. 5.1. (right): Specifications for different
wavelengths resonant to erbium. For calculation of the finesse F , the linewidth ∆ν and themode-matching waist w, Equations (3.8, 3.11) are used. The finesse at 583 nm has a valueof 1865 (see Sec. 5.2
for details).

3.2. Cavity design considerations

23

Figure 3.3: Chart of the reflectivity versus wavelength for our mirror coating. Below theupper reflectivity curve a zoom into the region between 99.90 and 100 % is shown. The
mirrors are highly reflective in a range of 560 − 650 nm and 750 − 890 nm. The red points
mark the wavelength of the Er transitions.
Frequency stability requirements
Absolute frequency stability of a cavity is directly related to the stability of its length andthe fluctuations of the refraction index of the medium between the cavity mirrors. Therefore,
one has to minimize temperature changes that influence the length of the spacer as well aschanges in the refraction index, pressure changes and vibrations. The narrowest transition
in Er lockable to our cavity is the 841-nm transition, with a linewidth of 8 kHz.
This means, that the cavity resonance should not drift by more than 8 kHz in a typicalexperimental timescale of several hours. The frequency change ∆f of the cavity is directly
connected to the changes of the optical length nL of the cavity. Therefore, the following
criterion has to be fulfilled.
∆f < 8 kHz →
∆f
∆L ∆n
=
+
< O(10−11).
fLn(3.12)

Here, ∆n is the change in the refraction index of the air between the cavity mirrors.
Using the equation for an ideal gas, relative changes in the refraction index can be related
to the temperature T and to the pressure p of the gas via the Clausius-Mosotti equation[Jac06], which gives the relationp(n − 1) ∝
⇒
T
s
2 
2
∆p
∆T
n−1
∆n
+
.
(3.13)=nnpT
At standard lab conditions, the refraction index is about (1 + 10−4 ). This means that thetemperature T and the pressure p of a gas should have relative fluctuations below 10−7 .

24Chapter 3. ULE cavity

By placing the cavity in a vacuum chamber (see Fig. 3.4), and pumping the chamber off toa pressure of about 5 × 10−8 mbar, pressure fluctuations and temperature fluctuations aresuppressed to an acceptable amount.
Further, the length of the cavity has to be insensitive to temperature fluctuations. This isachieved by optically contacting the mirrors to a spacer made of ultra low expansion glass1(ULE). The optical contacting is a glass bonding technique. Two highly cleaned glass surfaces with a flatness in the order of tens of nanometer are connected by overlapping andputting small pressure on them. The two surfaces stick together due to the Van der Vaals
force. We did the optically contacting by ourselves in a clean room; see App. A.2. The
material ULE is suited best for applications where low thermal drifts are required, because
it has a low coefficient of thermal expansion (CTE) in the order of 10−8 /K around room
temperature. A characteristic feature of ULE is that it is designed such, that the CTE has
a zero-crossing near room temperature. This temperature is called zero-expansion temperature TC , which is specific of every cavity assembly and has to be experimentally determined.
If the cavity temperature is stabilized near TC , only small frequency drifts of the cavity ofaround 0.1 Hz/s remain due to the degeneration of the crystal structure of the ULE [Leg10].
Another noise source that contributes to length changes of the cavity is mechanical deformations of the spacer due to vibrations. Especially, vibrations in the frequency range of1 − 10 Hz, which are present in every building, can not be damped effectively by an optical
table. An estimation for our cavity can be made using [Che06] and [CVI09]. For our spacer
geometry (see Fig. 3.5), we estimate the vibrational sensitivity to be about 200 kHz/m s−2along the vertical axis. Further, we estimate the maximum acceleration acting on our cavity
vertically to about 3 mm/s2 . This estimation gives a frequency change of 600 Hz due tovibrations, which is not a limiting factor for us.

3.3

Mechanical setup

Figure 3.4 shows a 3D drawing of the complete assembly. A cylindrical vacuum chamber witha diameter of 250 mm and a height of about 220 mm houses the cavity. The chamber has two
CF402 viewports3 for optical access. An additional flange is used to connect a 25 pin Sub-D
electric feedthrough, which is needed to supply temperature sensors and Peltier elements
to cool the inside of the vacuum chamber. To evacuate and pre-evacuate the chamber, an
ion-getter pump and an edge-valve to connect a turbopump are mounted, respectively. To
protect the spacer, the vacuum chamber is not baked. The ion-getter pump has a pumping
speed of 20 L/s, we reached a pressure of 5×10−8 mbar several weeks after closing the vacuum
chamber. This pressure is low enough for our purposes and also guarantees a long lifetimeof the ion pump, as it is continuously powered. The vacuum chamber is mounted on analuminum baseplate to guarantee a good heat release from the bottom CF250 flange of the
1

ULE, Corning Code 7972
CF (ConFlat) flanges have knife-edges cutting into a gasket made of copper (or other soft metals), when
two flanges are pressed together. This kind of sealing is used for ultra high vacuum (UHV). For example,
CF40 (more precisely CF DN40) means that inner diameter of the flange is 40 mm.
3
VACOM - silica viewports with a broadband anti-reflection coating at 560-780 nm.
2

3.3. Mechanical setup

valve forturbopump

25

ion-getter pump 20l/s

CF250 flanges
CF63 flange
electric feedthroughCF40 flange
viewport

CF40 flange
unusedaluminium baseplate
heat release

Figure 3.4: 3D drawing of the vacuum chamber. All the parts for the vacuum chamber
are made of steel and connected by CF flanges. Pumping is done by a 20 L/s ion-getterpump after pre-pumping the chamber with a turbopump connected to the edge-valve. The
CF63 flange is connected to a 25 pin Sub-D electric feedthrough to connect Peltier elements
and temperature sensors. The steel CF250 flange on the bottom is used as a heat sink for
the Peltier units. For heat release from this flange and for mounting the chamber onto theoptical table, the flange is screwed to a baseplate made of aluminum. For optical access tothe cavity two CF40 viewports are used (one is not seen).
chamber the Peltiers are glued to.
Figure 3.5 shows a sectional view through the vacuum chamber. Our cylindrical cavity sitson a V-shaped block, where it is supported on four small Viton4 cylinders. These cylinders
are glued onto the V-block with vacuum epoxy5 . Along the cavity axis, these suspension
points are at the so called Airy distance AD = 0.578L. If the cavity is suspended atthese points, the end-facets of the spacer stay parallel during cavity deformation and at thesame time the vibrational sensitivity is minimized for this kind of 4-point suspension frombelow, to about 200 kHz/m s−2 . The central bore drilled through the center of the spacer
guarantees passage for the laser beam. For evacuation of the inside another small bore is
drilled vertically. For a good temperature isolation, the cavity is surrounded by two cuboidal
boxes made of aluminum, the inner and the outer shield. The inner shield is a passive heatshield working as a temperature low-pass filter, and is separated from the outer heat shieldby five Teflon6 spacers. The isolation between the shields is very high, I measured a time
constant of about 21 hours, and therefore, it takes several days until the temperature of
the inner shield has thermalized completely; see Sec. 5.3. The temperature of the outer4

Viton is similar to rubber and has low outgassing rate in vacuum.
EpoTek - 353ND optical adhesive
6
Teflon has a low thermal conductivity and low outgassing rate in vacuum.
526

Chapter 3. ULE cavity

heat shield is controlled by four Peltier elements, which support the outer shield. The coldsides of the Peltier elements are glued with silver loaded epoxy7 onto the bottom plate of
the outer shield, whereas the hot sides are glued to the bottom CF250 flange of the vacuum
chamber. Heat from this flange is released by the aluminum baseplate mounted to it. For thetemperature regulation and monitoring of the temperature inside the chamber at different
places, we use six temperature sensors glued with vacuum epoxy at different positions inside
the chamber; see App. A.2.
Further, two AR-coated glass plates prevent the thermal radiation to reach the cavity throughthe bores inside the walls of the heat shields along the cavity axis. These glass plates are glued
onto the outer shield slightly tilted, about 5 ◦ , from the vertical axis to prevent interference
effects. Due to the same reason, also the AR-coated viewports have a tilt from the verticalaxis. Inside the cavity, Capton-coated wires are used, which are all soldered onto a 25 pinin vacuum connector.

inner thermal shield
AR-coated glass
broadband 580-860nm
tiltedouter thermal shield

AR-coated viewport
broadband 560-870nm
tiltedV-block
cavity suspension
CF250 flange
heat sinkteflon posts

temperature
sensors

peltierelements

Figure 3.5: Sectional view through the center of the cavity. The cavity is suspended by four
small Viton cylinders glued onto the V-block (two of the cylinders are seen). The cavity
is surround by two thermal shields, which are separated by five Teflon spacers. The outershield is temperature controlled by four Peltier elements. For further shielding of thermalradiation, two tilted AR-coated glass plates are glued onto the outer shield. The viewports
for optical access are also tilted from the vertical axis by a small angle.

7

RS-Components - Silver loaded epoxy

3.4. Temperature control

3.4

27

Temperature control

We cool the cavity down to the measured zero-expansion temperature of −5.65 ◦ C; seeSec. 5.3. The cooling is done by using four Peltier elements8 placed inside the vacuum
chamber. For keeping the cavity at this temperature, the Peltier elements need a total electric power of about 13 W. The Peltier coolers control the temperature of the outer shield,
which surrounds the inner heat shield for suppression of fast temperature fluctuations.
For the measurement of the temperature, three negative temperature coefficient (NTC) sensors with a nominal resistance of 10 kΩ and three PT1000 (1000Ω) temperature sensors areused. The schematics of the complete setup to control and monitor the temperature of thecavity is depicted in Fig. 3.6. The electronics for the temperature control loop are placed
Picolog
datalogbox

Picolog
datalogger

25-pin Sub-D
feedthrough1xNTC 10kΩ
measurement
bridgePCtemperaturemonitoring

current
output stage

PI controller

6xtemperature
sensors4x
peltierelements

Figure 3.6: Scheme for controlling and logging the temperature of the cavity. The temperature control loop is split from the logging setup after the feedthrough. The cable-paircarrying the signal of one NTC is connected to the measurement bridge, whereas the othercables that are soldiered to the other five sensors are connected to the Picolog datalog box.
The orange boxes represent electronic boxes made by myself.
inside three different boxes. The measurement bridge creating the error signal, the PI controller generating the feedback signal and the current output stage, which amplifies the low
power feedback signal. The electronic circuits for these three boxes are found in App. B. The
resistances of the four of the other five temperature sensors are measured inside the Picolog
datalog box by four voltage dividers, which are soldered to a pre-build PCB board9 . The
electronic circuit for this box can also be found in App. B. The voltage signals proportional
to the resistance of the sensors are then transformed into digital signals inside the Picolog
data logger10 . The Picolog data logger is connected to a computer, where the digital signals
are converted into temperature and monitored11 ; see App. A.3.

8

Laird Technologies - ThermaTEC HT4,7,F2,3030
This pre-build PCB board is shipped together with the Pico data logger.
10
Picolog Datalogger ADC-24bit
11
Picolog Recorder9

28

Chapter 3. ULE cavity

Chapter 4
Experimental setup
The preparation of the MOT light includes control over the frequency, the linewidth and theintensity of the laser light. For frequency and linewidth control, the dye laser is stabilized
to a scanning Fabry-Perot cavity and to a ULE cavity via the Pound-Drever-Hall (PDH)
method.
In the first Section of this Chapter, I will describe the optical setup for the light at 583 nm.
In the second Section, I will present details about our realization of the PDH setup.

4.1

Optical setup

Figure 4.1 gives a global overview of the optical setup used to generate the magneto-opticalcooling light. The setup is divided into several main sections, which serve to different specificscopes. In general, we prepare the MOT light on the optical table and we transport the light
via optical fibers to the different sections, which are mainly placed on single breadboards(round-edged boxes). The sections include for instance the optical setups for locking, forspectroscopy, for wavelength detection (fiber to the wavemeter), and for distributing thelight into the different MOT beams.
Here, we list and briefly describe the main sections of the full optical setup:
Wavelength detection
To monitor the absolute frequency of the laser light with an accuracy of several MHz, wesend a part of the light to a wavemeter. The use of a wavemeter significantly facilitates thelocking of the laser to the correct ULE cavity mode and the adjustment of the dye laser
wavelength.
Pre-lock section
In the pre-lock section, we prepare the light with a PDH locking setup and send it to a
mid-finesse cavity for narrowing the linewidth of the dye laser.
MOT light preparation section
Here, we use an acousto-optical modulator to stabilize the intensity of the MOT light on the
MOT distribution board and to fine-tune the frequency of the light for the MOT.

29

30

Chapter 4. Experimental setup

Figure 4.1: General concept of the optical setup for the 583- nm light. The setup is divided
into several parts marked by colored boxes. Round-edged boxes mark parts of the opticalsetup, which are placed on their own breadboard. The different breadboards are connected
via fibers.
MOT distribution board
For generating the beams for the MOT, we split the light into three paths. The light is thencoupled into three fibers, which are connected to fiber couplers on the main experimental
table. We use a standard six-beam 3D-MOT with retro-reflected beams.
Spectroscopy and imaging preparation section
We use an AOM to shift the frequency of the light to the desired erbium atomic transition.
Using this light, we perform spectroscopic measurements either by using a hollow cathodelamp (HCL) or by imaging the atoms in the MOT chamber, which to referred as “yellow
imaging”.
HCL spectroscopy setup
We perform modulation transfer spectroscopy on erbium in a hollow cathode lamp. The
spectroscopic measurements are used to optimize the locking setup.
Frequency shifting setup
We use the frequency shifting setup to tune the frequency of the laser light to an arbitraryvalue while the laser is locked to the ULE cavity. For this, we use two AOMs to shift thelaser frequency by amounts smaller than the FSR of the ULE cavity (about 1 GHz).

4.1. Optical setup

31

ULE cavity setup
This setup consists of the ULE cavity, the optics for coupling to the cavity and for PDHlocking.

4.1.1

Optical setup on the dye laser board

Figure 4.2 shows the optical setup on the dye laser board. About 1 mW of light emitted

Figure 4.2: Scheme of the optical setup for the 583-nm light on the dye laser board. The
setup is divided into the pre-lock, MOT light preparation and spectroscopy and imaging
preparation section. For simplicity, lenses are not shown.
from the dye laser is coupled to a single-mode fiber1 which is connected to a wavemeter2 formonitoring the frequency of the light. Most of the laser-field intensity is coupled to a shortfiber3 . This fiber is used for alignment purposes since the angle and position of the beamemitted from the laser changes if the internal laser cavity is realigned. After this short fiber,
the light is split into several paths by using polarizing beamsplitters (PBS) and half-wave(λ/2) plates.
Pre-lock section
In the pre-lock section, the light is sent to a Fabry-Perot cavity4 (“TOPTY”), which has
1Thorlabs HP-460
HighFinesse WS/7
3
Thorlabs PM-460
4
Toptica FPI-100-0500-1
2

32

Chapter 4. Experimental setup

a finesse of about 1000 at 583 nm. One of the cavity mirrors is attached to a piezoelectric
actuator (PZT), which is capable to scan the cavity over more than one free spectral rangeof 1.5 GHz. This cavity is used for narrowing the linewidth of the dye laser. The error signal
is generated with the Pound-Drever-Hall locking technique5 ; see Sec. 4.2.
MOT light preparation section
To prepare the light for the MOT, the intensity is stabilized after the fiber connected to the
MOT distribution board. For this purpose, we use the acousto-optical modulator 1 (AOM1)6 . The feedback signal for the intensity stabilization is created by a photodiode placed on
the MOT distribution board and a proportional-integral-derivative (PID) controller. The
feedback signal is applied to an analog attenuator inside the RF-amplifier which amplifies
the signal driving the AOM. Further, we use the AOM to fine-tune the frequency of thelight, e.g. for optimizations and frequency scans in the range of several MHz. The AOM hasa center-frequency of 80 MHz and is set up in double-pass configuration. Currently we use
the +1st order of the light.
Spectroscopy and imaging section
In the spectroscopy and imaging preparation section, we shift the frequency of the light tothe desired value needed for spectroscopy or imaging of erbium atoms. For this, we use AOM
26 in double-pass configuration.

4.1.2

Frequency-shifting setup

To tune the dye laser to any desired frequency while it is locked to the ULE cavity, we put
optical elements into the beam path before the cavity. For this, we use a combination of two
AOMs7 . We use the AOMs either in single-pass or double-pass configuration [Don05]. As
shown in Fig. 4.3, the double-pass configuration includes a lens with f=75 mm to preserve
the alignment, a quarter-wave plate and a PBS to separate the incoming from the reflected
light. Further, AOM S1 is used to stabilize the light intensity for the ULE cavity; see Fig. 4.6.
After passing the two AOMs, the light is coupled to a fiber8 connected to the ULE cavity
breadboard.
To get an estimation of the frequency tuning range of the two AOMs, we measured
the transmission efficiencies of AOM S1 in double-pass configuration for several frequencies.
We changed the driving frequency of the AOM and re-optimized the transmission after the
double-pass for the first order diffraction, which gives a frequency shift of two times thedriving frequency. Figure 4.4 shows the double-pass efficiency as a function of the frequency
shift. We see an about 260-MHz broad profile with a peak efficiency of about 0.8 at 330 MHz.
As we have enough light for the frequency shifting setup available, both AOMs can be usedwith a low diffraction efficiency of about 10% in double-pass configuration. Therefore, we
can shift the frequency in a range of about 260 − 520 MHz with one AOM in double-pass
5
To create the sidebands on the light, an electro optical modulator (Linos PM25) with high transmission
is used.
6
AOM 1 and AOM 2 are Brimrose Model TEF80
7
Crystal Technology, Inc. - AOMO 3200-125 (200 MHz)
8
Thorlabs - PM-460

4.1. Optical setup
fiber fromlaser table

33
λ/2
λ/4

f=6.1mm

f=75mm(140-260)MHz+1

AOM S1intensity stabilization
for ULE cavity setup(PD1)

AOM S2(140-260)MHz+1
λ/2
f=75mmλ/4
f=4.6mm

λ/2

fiber to
ULE cavity

Figure 4.3: Optical setup for tuning the frequency of the laser relative to the transmission
frequency of the ULE cavity. Here, the setup is shown for the two AOMs in double-passconfiguration using the +1st order diffraction, also denoted as DP+DP+.
and about 130 − 260 MHz in single-pass.
To continuously tune the dye laser-frequency, while the laser is locked to the ULE cavity,
we have to bridge the FSR of the ULE cavity, which is about 1 GHz. For this, we combine
the two AOMs in three different configurations; see Tab. 4.1.
Table 4.1: Frequency-shifted ranges for three combinations of the two AOMs S1 and S2.
Double-pass is denoted as DP, single-pass as SP. Using the plus 1st order is denoted as ’+’,
minus 1st order as ’-’. Using these combinations together with the periodicity of our FSR of
1 GHz provides continuous tuning of the locked laser.
AOM combination
DP+DP+
DP-DPDP+SP+

shifted frequency range (MHz)+520 to +1040-1040 to -520
+390 to +780

If we want to change the lock to a different erbium isotope, we need to simply change
the driving frequencies for the two AOMs. To calculate these frequency changes, we have toconsider the following relevant frequencies:• The transition frequency of the target Er isotope, which is denoted as νatom .

34

Chapter 4. Experimental setup

double-pass efficiency

1.0

usable range H260-520 MHzL
double-pass, re-optimized angle

à

0.8

à

à

à
à

0.6

à
à

à

à

à

0.4

à
à

0.2
à

0.0

à

à

250

300

350

400

450

500

à

550

frequency shift HMHzL

Figure 4.4: Measurement of the transmission efficiency through the double-pass system fordifferent frequency shifts. For each datapoint, we re-optimized the adjustment of the doublepass setup. Further, the frequency range for efficiencies higher than 10% is shown.
• The transition frequency νcav of a TEM00 mode of the ULE cavity, which is a multipleof the free spectral range.
• All frequency-shifting elements inside the beam path between the ULE cavity andthe MOT chamber are important, which are three AOMs; AOM 1 (in Fig. 4.2), AOM
S1 and AOM S2. We denote the corresponding frequency shifts as ν1 , νS1 and νS2respectively.
The following condition has to be fulfilled for resonant locking
νatom = νcav + (ν1 − νS1 − νS2 ) .
|
{z
}

(4.1)

∆νshift = −592.72(1) MHz.

(4.2)

∆νshift

An overview of how the different frequencies are connected is shown in Fig. 4.5. We
measured νatom of 166 Er relative to νcav by spectroscopic means. The corresponding shift isThis value has been measured on the 21.11.2011 and changes about 2 MHz per month due
to a drift of the ULE cavity.

4.1.3

Optical setup for the ULE cavity

Figure 4.6 shows the optical setup for the ULE cavity. Light coming from the frequencyshifting setup is collimated with a f = 6.1 mm fiber coupler to a beam width of about 1 mm.

4.1. Optical setup

35

n

n+1

amplitude HarbitraryL

FSR

Νatom

Νcav

DΝshift

DP+SP+
DP+DP+

0200

DP-DP-

400

600

800

1000

1200

frequency HMHzL

Figure 4.5: Position of the different resonances in frequency space. The plot shows an atomic
resonance of Er, the nearby cavity resonances and the regions which are accessible due toa shift of the three AOMs 1, S1 and S2. In this example AOM 1 shifts the frequency by
162 MHz. The regions DP+DP+ and DP+SP+ are shifted relative to the mode ’n+1’,
whereas the region DP-DP- is shifted relative to the mode ’n’ to illustrate the capability ofcontinuous tuning.

The intensity of the light is stabilized after the reflection of the first PBS. The intensity
stabilization setup stabilizes the light entering into the cavity at a level of about 100 µW.
The error signal for locking the dye laser to the cavity is generated by a PDH setup; seeSec. 4.2. The light is sent through an EOM9 for sidebands generation. To match the polarization axis of the light with the crystal axis of the EOM, a half-wave plate before theEOM is used. A good matching is necessary to minimize polarization fluctuations producedby the EOM crystal. After the EOM, a lens with a focal length of 500 mm is used to focus
the collimated beam to a beam having a waist of about 200 µm at the position of the planecavity mirror. Two mirrors before the cavity are used to couple the light with the correct
angle and direction. The curved mirror is made of fused silica, and therefore, the substrate
has a refraction index of about n = 1.5. This mirror has a radius of curvature ρ of 400 mm
and thus, is working as a plano-concave lens with a focal length of f = −800 mm. We founda distance of about 35 mm between the 500 mm lens and the curved cavity mirror to give agood cavity coupling.

9

Linos PM-CBB

36

Chapter 4. Experimental setup
fiber fromfrequency shifting
board

λ/2

f=4.6mm

intensity PD1 f=35mm
stabilization
ULE
cavityLO - 31MHz
λ/2
EOM
PM-C-BB

PDH
error signal
PD2

ρ=400mm
f=-800mm

λ/4

f=500mmλ/2

Figure 4.6: Optical setup for coupling the light to the cavity and generating the PDH error
signal for locking. Further, a PD is used to stabilize the light after the first PBS.

4.1.4

Spectroscopy setup

To find the exact transition frequencies of the 583-nm line for the different isotopes of erbium,
we perform spectroscopic measurements on erbium. A vapor of Er atoms is generated via
LO 14.7MHz

spectroscopy
signalλ/2

f=-50mmEOM

f=100mm
f=35mm

PD

λ/2

HCL
λ/2

fiber fromlaser breadboard

Figure 4.7: Scheme of the optical setup for the modulation-transfer spectroscopy. The probeand phase-modulated pump beam are overlapped inside the hallow cathode lamp (HCL). By
phase-sensitive detection of the probe beam signal, a spectroscopy signal is obtained.
a discharge process in a hollow cathode lamp (HCL). Due to the high melting point of Er,
normal vapor cells are not available. The hollow cathode lamp is a glass-tube with an Er
coated cathode inside. The tube is filled with argon serving as a buffer gas at a pressureof about 4 mbar. By applying high voltage to the electrodes Ar atoms are ionized, and arethen accelerated towards the cathodes sputtering off Er atoms. To obtain a signal, we use
modulation transfer spectroscopy [Raj80, McC08]. This technique is based on pump-probe4.1. Optical setup

37

schemes to reach sub-Doppler resolution. The spectroscopic setup is schematically illustrated
in Fig. 4.7.
The light coming from the laser breadboard via a fiber is split up by a PBS into pumpand probe beam in transmission and reflection, respectively. The beam diameter of thepump beam is scaled down by a telescope, to match the aperture of the EOM. The EOM
is driven by a local oscillator (LO) at about 14.7 MHz imprinting sidebands onto the pump
beam. The pump beam is then overlapped with the probe beam inside the HCL. If thefrequency of the light is resonant to an atomic transition of Er, the sidebands are transferred
via four-wave mixing onto the probe beam. By mixing the signal of the photodiode detecting
the probe beam with the local oscillator, we obtain the spectroscopy signal. This signal isdispersive-like, as shown in Fig. 4.8. Three transitions of the bosonic and five transitions of
the fermionic isotope can be identified. The signal has a flat, nonzero baseline, and is very
sensitive to the alignment of the probe and pump beam due to interference effects. This
gives rise to frequency uncertainties in the order of MHz. Therefore, with the present setup
the signal can not be used for locking the dye laser.
1.5
spectroscopy signal
166

Er

1.0

168

Er

167

170

Er

8.5-9.5

9.5-10.5

6.5-7.5

0.0

7.5-8.5

0.5

5.5-6.5

signal HVL

Er - HFS

1500

2000

- 0.5
- 500

0

500

1000

detuning relative to 166Er HMHzL

Figure 4.8: Signal of the modulation transfer spectroscopy. Three transitions of the bosonicspecies and five hyperfine transitions of the fermionic isotope have been identified. The
hyperfine transitions are denoted by F → F ′ . A theoretical calculation of the hyperfine
states is found in App. C.

38

Chapter 4. Experimental setup

4.2

Pound-Drever-Hall locking

Stabilization of the laser frequency (locking) is performed with a servo loop, which involves
techniques to generate an error signal. The error signal is then transformed into a feedback
signal with a servo. The feedback signal is applied to an element capable of changing thefrequency of the laser. An introduction into the theory of a commonly used servo, theProportional-Integral-Device (PID) is given in Ref. [Tie99].
The PDH locking scheme is widely used to lock a laser to a cavity. Its advantage is thatthe locking range can be chosen independently of the steepness of the zero-crossing. Further,
the detection mechanism using the reflection from a cavity is sensitive to the phase of theinvolved light field providing locking-bandwidths in the order of several MHz.
A PDH error signal is created by phase modulating a laser beam, which imprints sidebands
at the modulation frequency, sending it to a cavity and detecting the reflection from thecavity. The reflection is separated by the incoming beam via a polarizing beamsplitter anda quarter-wave plate; see Fig. 4.9. The sidebands on the signal create a beat signal with thecarrier. The phase of these signals depends on the detuning of the carrier from resonanceand by detecting this phase with a LO, the error signal is obtained, which can be used for
feedback onto a laser. In the following the error signal for the ULE cavity is discussed. The
theoretical background for the PDH locking technique is found in App. A.

laser

EOM
resonant
circuit

λ/4cavityservo
LO

low-passﬁlter

error signal
mixer

Figure 4.9: PDH locking setup. The EOM creates sidebands on the light, which is thenreflected from the cavity and detected by the photodiode. The error signal is created byfiltering the detected signal with a low-pass filter, which has a cutoff frequency above thelocal oscillator frequency, and mixing it with the LO. A servo generates the feedback signal.
Often a resonant circuit is used to amplify the voltage over the EOM.

4.2.1

PDH error signal of the ULE cavity

An overview of the most important parameters for our PDH error signal can be found in
Tab. 4.2. For creating our error signal we use an EOM10 . The facets of this crystal are cut at
10

Linos - PM-C-BB

4.2. Pound-Drever-Hall locking

39

Brewster’s angle and therefore, residual amplitude modulation (RAM) is minimized, whereas
the transmission is maximized. RAM creates an offset of the error signal after the mixer
and therefore, has to be reduced to a low level for a good locking performance. Using a LO
running at 31 MHz and a resonant circuit to amplify the voltage drop over the EOM (seeFig. B.2), we get sidebands having about 25% of the carrier height (P s/P c ≈ 0.25), whichcorresponds to a modulation index of roughly 0.9. This value is only about 5% below the
optimum value achievable for the slope of the error signal. As a low-pass filter we use a filter
with a cutoff frequency at 60 MHz. The beam intensity incident on the cavity is 100 µW. We
get an error signal that has a peak-to-peak amplitude of about 1.8 V and a slope of about
4 mV/kHz; see Fig. 4.10.
Table 4.2: Parameters connected to the PDH error signal.
LO frequency
P s/P cmodulation index
peak-to-peak amplitude
slope
intensity incident on cavity

1.0

31 MHz
0.25
0.9
1.8 V
4 mV/kHz
100 µWerror signal

voltage HVL

0.5

0.0

- 0.5

- 1.0
- 100

- 50

0

50

100

frequency detuning, calibrated HMHzL

Figure 4.10: PDH error signal for the ULE cavity. The sidebands are at ±31 MHz, which is
the EOM modulation frequency. The distance between the two largest extrema in frequency
space is corresponding to the linewidth of the cavity. The baseline of the signal is not atzero, because of a standard offset of the photodiode electronics.

40

Chapter 4. Experimental setupChapter 5
Measurements
In this Chapter, we present measurements related to the ULE cavity and the linewidth of
the yellow light. We measure important parameters like the free spectral range, finesse,
and zero-expansion temperature of the ULE cavity. Finally, we investigate the long-term
stability of the ULE cavity and the linewidth of the locked laser.

5.1

Free spectral range

To determine the FSR of the ULE cavity, we make use of two adjacent cavity TEM00 modes
and the 166 Er atomic resonance at 583 nm. We lock the dye laser to each cavity mode
separately and measure the corresponding frequency shift to the 166 Er atomic resonance;
see Fig. 5.1(b). To resolve the atomic resonance, we image the atomic cloud in the MOT
chamber and we scan the frequency of the yellow light with AOM 2 (see Fig. 4.1 - “yellow
imaging”).
The cavity modes are separated by one FSR and their transmission frequencies are denoted as νcav,n = n · FSR and νcav,n+1 = (n + 1) · FSR. The frequency shifts ∆νs,1 and ∆νs,2
are a result of the combination of the frequency shifts performed by AOM 2, AOM S1 andAOM S2; see Figures 4.2 and 4.3. According to Eq. (4.1), the FSR is then given by∆νs,1 = νAOM2,1 − νAOMS1,1 − νAOMS2,1 ,∆νs,2 = νAOM2,2 − νAOMS1,2 − νAOMS2,2 ,FSR = ∆νs,1 − ∆νs,2 .

(5.1)

In order to get an accurate value for the FSR, we have to determine the peak position of
the atomic resonance very precisely; see Fig. 5.1(a). For this spectroscopic measurement
we already used our MOT, which we achieved during writing of this thesis. For a closer
discussion on our Er MOT at 583 nm, see Chap. 6 and Ref. [Fri12]. The atomic spectroscopy
measurement will be explained in more detail below.
The frequency shifts for AOM S1, AOM S2 and AOM 2 which are used to calculate the FSR
are shown in Tab. 5.1. From these values, we calculate the FSR and the length of the cavity
41

42

Chapter 5. Measurements
Table 5.1: Frequency shifts of the AOMs for both measurement settings.

νAOM2,iνAOMS1,iνAOMS2,i∆νs,ifrequencies for i = 2 (MHz) frequencies for i = 1 (MHz)162.618(10)162.640(12)400.010280.008355.328(10)-523.338(6)-592.720(14)405.970(14)by using Eq. (3.6),
FSR = 998.69(2) MHz,L = 15.0093(2) cm.

(5.2)
(5.3)

The length is within the specifications given by the manufacturer of the spacer of the cavity.
Atomic spectroscopy on 166 Er at 583 nm - “yellow imaging”
For the measurement of FSR of the cavity, we need first to accurately determine the atomic
transition frequency of 166 Er corresponding to the 583-nm line. We measure the atomic
transition by performing absorption imaging on an atomic cloud released from the MOTwith the yellow light. The probing light has an intensity of about I ≈ 36 µW/cm, which
is about 1/4 of the saturation intensity of the transition. The temperature of the atomic
cloud has been measured by time of flight experiments [Pet02]. We found a temperature ofT ≈ 20 µK. For each data point a new sample is used, we scan the laser frequency with
AOM 2 and image the cloud with a CCD camera.
The result of the measurement is shown in Fig. 5.1(a).
The scale of the y-axis is already converted to the number of atoms in the cloud.
We fit the data with a Voigt profile1 , because we expect both Gaussian and Lorentzian
contribution to the lineshape. In the following, we discuss the lineshape as we expect it fromtheoretical considerations. In our case, the line is broadened by• natural broadening, which has a Lorentzian shape with a FWHM ∆νnat = 190 kHz.
• power broadening resulting from the intensity of the probing light. In firstporder power
broadening changes the FWHM of the Lorentzian lineshape by a factor of 1 + I/Is ≈1.12. This changes the FWHM to about 210 kHz.
• thermal Doppler broadening, which results from the finite temperature of the sample.
Thermal Doppler
q broadening gives a Gaussian lineshape with a FWHM calculated

8kT ln(2)ν583 , where m is the mass of an erbium atom and ν583 theby ∆νDoppler =mc2transition frequency of the 583-nm transition. With a temperature of 20 µK we get
Doppler broadening of ∆νDoppler = 127 kHz.

1

A Voigt profile is the convolution of a Lorentzian with a Gaussian profile.

5.1. Free spectral range

43

(b)

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
- 1.0

data

amplitude HarbitraryL

number of atoms x107(a)

Voigt profile fit

- 0.5

0.0

0.5

1.0

atomic resonancecavity resonance
n
DΝ s1

0

n+1

FSR

200

detuning AOM 2 Hf+162.618L HMHzL

DΝ s2

400

600

800 1000

frequency HMHzL

Figure 5.1: Measurement of the free spectral range. (a) Spectroscopy of the atomic clouds
released by the MOT. The number of atoms detected by the 583-nm imaging light vs. thefrequency shift performed by AOM 2 is plotted. The center of the peak is determinedby fitting a Voigt profile. For the spectroscopy, we have a maximum expansion time of
tof = 25 ms, I/Is = 0.25, and T = 20 µK. (b) Schematics on the measurement of the FSR.
• the finite linewidth of the yellow light. The lineshape of a dye laser usually has
Lorentzian shape. In our case, we measured the linewidth of our laser over a timescaleof 10 ms and assume a Gaussian shape; see Sec. 5.5. The corresponding linewidth is∆νlaser = 45.0(1) kHz• other broadenings, e.g. caused by the inhomogeneous magnetic field in the MOT
chamber. During the spectroscopy measurement, the magnetic field produced by the
Zeeman slower is still on and shifts the transition frequency of the magnetic substates
dependent on the position in space. This can create an asymmetric type of broadeningof unknown strength.
Therefore, we expect a spectral line with a Gaussian contribution of about 140 kHz and a
Lorentzian contribution of about 210 kHz, with an additional unknown broadening due tothe inhomogeneous magnetic field.
From the fit we extract the following values:∆νGaussian = 310(20) kHz,∆νLorentzian = 140(20) kHz.

(5.4)
(5.5)

The Lorentzian contribution is smaller than we expected, whereas the Gaussian contribution
is about 100 % higher than we expected. This discrepancy could be to the inhomogeneous
magnetic field, but further investigations are needed.

44

5.2

Chapter 5. Measurements

FinesseIn general, the finesse of a cavity is measured by scanning a laser over a cavity resonanceand by that determining the linewidth of the cavity mode. This kind of measurement is only
possible if the linewidth of the laser is much smaller than the linewidth of the cavity mode.
In our case, the linewidth of the 583-nm laser is in the order of linewidth of the ULE cavity and therefore, we use another technique to measure the finesse, the so-called ring-down
measurement [Dem03]. Here, we measure the decay time of the light stored in the cavity.
For this purpose, we first tune the laser to the cavity resonance for maximum transmission,
which is done by locking the dye laser to the cavity. Then we use a radio-frequency (RF)switch connected to the RF-amplifier of AOM S2 (see Fig. 4.3) to suddenly switch off theintensity of the light coupled to the ULE cavity. The light transmitted through the cavity
follows an exponential-like decay, which is measured by a photodiode.
Another important point is, that the speed of switching the intensity has to be shorter than
the timescale corresponding to the bandwidth of the servo loop for the laser-frequency stabilization, because a frequency change during the measurement leads to interference effects.
Our ULE cavity is only used for the slow part of the frequency stabilization, which has abandwidth in the order of kHz. Therefore, we assume no influence on the stabilization.
The cavity ringdown is a process described by the following first order differential equation1(−y(t) + u(t)) .
(5.6)
τ
In this equation y(t) is the output intensity of the cavity, u(t) is the input intensity and τ isthe decay constant of the cavity. If the AOM switches off the intensity with infinite speed,u(t) can be described by an step function u(t) = σ(−(t − t0)), where t0 is the switch-off
time. This gives the well-known solution with an exponential decay with a characteristic
time τ . In our case we use a Gaussian error function for the input function, which models
our AOM switching process; see Fig. 5.2.
t − t0a−b
1 + erf −
+ b.
(5.7)
u(t) =
2a4
ẏ(t) =

Here, a and b are the values for the upper limit and the lower limit of the switching curve.
To be sure that the switching process has negligible influence on the exponential formof our decay curve, I compared the fit of a true exponential decay to a calculated solutionof the numerical differential equation for the same decay time τ . This means that in Eq.
(5.7), we have τ and a4 fixed, and adapt a, b and τ0 manually to our data; see Fig. 5.3. We
see, that there is a small influence of the switching process in the first 100 ns, but there isa negligible difference to a true exponential decay after the 100 ns. The decay time we getfrom the fit isτ = 297(1) ns.
(5.8)
We calculate the finesse and cavity linewidth by using the decay time, the measured FSR,
and Equations (3.8) and (3.9).
F = 1865(6).
∆ν = 536(2) kHz.

5.3. Zero-expansion temperature of the ULE cavity assembly

1.8

45

data
fit: a1+a2*erf @-Ht-a3La4D

1.6
PD voltage HVL

1.4
1.2
1.
0.8
0.6
0.4
0.2
0.
0

100

200

300

400

500

time HnsL

Figure 5.2: Switching curve of the AOM. For estimating the influence of the AOM responseon the ringdown, we fit an errorfunction to this data to model the switching process numerically by a single parameter, which is a4 = 55.3(1) ns. The other parameters a1, a2, a3 areused for adjusting the scaling, translation and offset of our data.
The finesse is not within specifications of the finesse calculated from the reflectivity specified
by the company2 , which is 3500. There may be unexpected absorption losses due to dust or
imperfect cavity coupling. However, the finesse is high enough for locking of the dye laser.

5.3

Zero-expansion temperature of the ULE cavity assembly

One of the advantages of using a spacer made of ultra low expansion glass is that the coefficient of the thermal expansion (CTE) has a zero-crossing around 20 ◦ C, called zero-expansion
temperature (TC ). If the temperature of the cavity is stabilized near this temperature, its
length becomes insensitive to temperature fluctuations, which results in a low thermal driftof the cavity mode in the order of Hz/s.
We use a 15 cm long ULE spacer with fused silica mirrors optically contacted to it. The CTEof ULE is specified to below 3 × 10−8 /K and fused silica has a CTE around 500 × 10−8/K.
Due to the mismatch of the CTEs, the zero expansion temperature of the whole cavity assembly shifts to lower temperatures with respect to the TC of the ULE itself. The rigidity ofthe optical contacted connections between mirrors and ULE spacer leads to a bending of themirrors along the cavity axis. This bending causes a shift of the zero-expansion temperatureof about −12◦ C for our cavity geometry, according to Ref. [Leg10].
2

LayerTec

46

Chapter 5. Measurements
0.2
data ringdown
fit: A+Exp@-Ht-t0LΤD
numerical solution for Τ=297ns

PD voltage HVL

0.15

0.1

0.05

0.

0

300

600

900

1200

1500

1800

time HnsL

Figure 5.3: Cavity ringdown. We fit an exponential decay to the data, which gives a time
constant of τ = 297(1). For validation of the assumption of a infinitely fast switching-off
process, we compare the calculated solution for τ = 297(1) to our data.
We determine the zero-expansion temperature by changing the temperature of the cavity
and measuring the resulting length change. The length change is connected to the thermal
expansion properties of the ULE cavity. Near the zero-expansion temperature, thermal expansion has a strongly linear behavior [Leg10]. Therefore, we use a simple model where we
describe the instantaneous CTE, α(T ), around TC byα(T ) = a(T − TC ),(5.9)

where a is the linear coefficient of thermal expansion. For ULE cavities, a has typically avalue around 2 × 10−9/K2 .
Small temperature shifts lead to a small length changes, which is described by the following equationdL = Lα(T )dT.
(5.10)The relative change in frequency is directly connected to the relative change of the lengthand can be expressed as
∆ν
∆L
a
=
= (T − TC )2 + C0 ,(5.11)
νL2−a 2) . To measure TC , wewhere C0 is an integration constant which contains the term ( 2Tcchanged the temperature of the cavity and we monitored the resulting frequency shift ofa cavity mode with respect to the modulation transfer spectroscopy signal of the 583-nm
transition in 166 Er; see Sec. 4.1.2.
Figure 5.4 shows the relative frequency shift as a function of the set temperature of the innershield. We first changed the set temperature from about 0 ◦ C to 6 ◦ C and then to about

5.3. Zero-expansion temperature of the ULE cavity assembly

47

150

relative frequency DΝΝ 10-9

ð

H4L

data
fit: a2HT-Tc L2+C0

ð
ð

ð
ð
ðð

100

ð

50

H5L

H3L

ððð
ðð
ð
ðð
ð
ðð H2L

ð
ð

H6L
H7L
H8L

0
- 15

- 10

ðð
ððð
ðððð

H1L
ð

ð ðð

ðð

-5

0

5

10

temperature inner shield H°C L

Figure 5.4: Plot of the relative frequency drift of the cavity versus the temperature of theinner shield. The measurement took 8 days, the time when the datapoints were taken aremarked by “#” (in days). One can clearly see a hysteresis behavior, due to the time-delayin thermalization of the cavity. We fit a curve only taking the linear coefficient of thermal
expansion into account, to get a rough estimation of the zero-expansion temperature of ourcavity assembly.

−8 ◦ C. The overall measurement took 8 days, as indicated in brackets. For the temperature
measurement we had to use a PT1000 sensor attached to the inner heat shield, since a direct
measurement of the cavity temperature is not possible. The data clearly shows hysteresis
behavior, which is caused by a fast change in temperature of the inner heat shield during
the measurement. The temperature of the cavity was not able to follow instantaneously thechange in temperature.
Equation (5.11) is not able to fully capture the observed behavior and leads to an estimation of TC with a 800 mK error, which is far above the required precision level; see solidline in Fig. 5.4. The reason for this mismatch can be attributed to the fact that the modelaccounts for the actual temperature of the cavity while we consider the temperature of theinner shield. To overcome this problem, we need to set a more developed model which accounts for this difference.
Estimation of the cavity temperature
Here, we introduce a model based on thermal radiation heat transfer processes between twogray diffuse surfaces to estimate the temperature of the cavity as a function of the tem-

48

Chapter 5. Measurements

perature of the inner shield, following Ref. [Sie02]. Here, the term “gray” indicates that theradiation emitted by the surface of the material is described by a parameter called emissivity
ǫ, which can have a value between one and zero. The model considers diffuse surfaces. Thismeans that the radiation incident on a spot of the surface is reflected into all directions,according to the Lambertian law.
Let us assume a radiation exchange between two concentric diffuse-gray spheres with uniform temperature, see Fig. 5.5(b). The relevant parameters describing the problem are theso-called net energy loss Qk from a surface k having area Ak . This net energy loss is composedof the outgoing thermal radiation per unit area (flux) qo,k and ingoing flux qi,k . Other important parameters describing the radiation process are the emissivity ǫk and the temperature
Tk of the surface. To solve the problem, we set up the following relations.
Q2, T2, ε2(a)parameter
AiS
Acav
ǫiS
ǫcav
mcav
ccav

enclosure

(b)
value
0.062 m2
0.034 m2
0.07
0.92
0.9 kg767 J kg−1 K−1

A2qO,2A2qI,2

A1qI,1

A1qO,1Q1
T 1 , ε1

enclosed
sphereFigure 5.5: (a) Parameters used in the model for estimating the temperature of the cavity.
The values for the area and volume of the spacer have been measured with a computer-aided
design (CAD) software. The values for the emissivity for aluminum for the inner shield androughened glass for the cavity are taken from Ref. [Ome03]. The value for the density of ULE
is taken from [Cor08]. (b) Radiation heat transfer between two concentric gray and diffuse
spheres. The parameters relevant are the temperatures Tk , the surfaces Ak , the emissivitiesof the involved materials and the radiative heat losses Qk . These losses are composed ofincoming and outgoing thermal radiation.
Qk = (qo,k − qi,k )Ak .
qo,k = ǫk σTk4 + ρk qi,k .
|{z}(5.12)(5.13)1−ǫk

The first equation is describing the energy balance. The second equation states, that theemitted flux is dependent on the temperature of the surface following the Stefan Boltzmann
law of radiation including the reflected ingoing flux modified by the reflectivity ρk . The
reflectivity is connected to the emissivity by ρk = 1 − ǫk . Another relation is needed to solve
the problem. Equation (5.14) is related to the geometrical arrangement of the two surfaces.
For example, the outgoing flux from surface 1 completely reaches surface 2, whereas a part

5.3. Zero-expansion temperature of the ULE cavity assembly49of the flux emitted from surface 2 is also reabsorbed by itself. This geometrical arrangementof two surfaces is described by the so-called configuration factor Fj−k , which is the portionof flux emitted from the surface j reaching the surface k. Therefore, the relation
qi,k =

2
X

Fj−k qo,k(5.14)

j=1

holds. In our case the configuration factors are F1−2 = 1, F2−1 = A1 /A2 and F2−2 =1 − A1 /A2 . If we combine the three Equations (5.12 - 5.14), we can reduce the problem toone equation, which connects the physical quantities relevant to our problem.
Q1 =A1 σ(T14 − T24 ).
1/ǫ1 + (A1 /A2 )(1/ǫ2 − 1)

(5.15)We extend the problem of the enclosed sphere to our more complicated geometry bytaking the following assumptions into account:• The inner shield is identified as the enclosure and the cavity is identified as the enclosed
sphere.
• The geometrical arrangement between the cavity and the inner shield is not the one oftwo spheres. To compensate the change in configuration factors and other inaccurate
assumptions of our model, we correct the whole equation by a free parameter α, which
accounts for the model corrections.
• The net change of heat Q1 = Q̇cav of the cavity spacer and mirrors is instantly transformed into a change in temperature according to the equation Q̇cav = ṪULE ccav mcav .
Here, ccav is the specific heat of ULE and mcav is the mass of our cavity.
Within these assumptions, we derive the differential equation (5.16), which gives a direct
relation between the measured temperature at the inner shield TiS (t) and the cavity temperature TULE (t).
4αAcav σ(TULE(t) − TiS4 (t))
ṪULE (t) =.
(5.16)
ccav mcav 1/ǫcav + (Acav /AiS )(1/ǫiS − 1)
The fixed parameters Acav , AiS , ccav , mcav , ǫcav , and ǫiS are calculated based on thespecifications of the spacer and the inner shield (geometry and material) and are listed inFig. 5.5(a).
For solving this differential equation, we measured the temperature of the inner shieldTiS (t) for the whole measurement period with a resolution of better than 1 data point every
5 s; see Fig. 5.6(a). At the beginning of the measurement, the set temperature of the outershield was ramped to about 6 ◦ C. We see an exponential relation between the temperature of
the outer and the inner shield. The time constant extracted from the exponential fit describes
the shielding quality and is about 21 hours. At the end of day five of the measurement period
we ramped the cavity set temperature to values around −8 ◦ C. We stopped the measurement
at the end of day 8.
The only free parameter of the model is α, which we determine using a fitting procedurethat is explained in the following:

50

Chapter 5. Measurements• First we solve differential equation (5.16) for several specific values of α, e.g. from 1.20
to 1.30 with a step size of 0.01; see Fig. 5.6(b). As a starting condition, we assume that
TULE (0) = TiS (0), because we let the cavity thermalize for more than one week before
we started the measurement.
• As a solution we get the temperature of the cavity TULE,α (t) depending on the model
correction parameter α. We further assume that for one specific value of α, the model
is valid.
• To validate our model, we create a dataset, where the y-values correspond to therelative frequency shift of the cavity at a certain time t (what we measured). The
x-values of the dataset correspond to the corrected values of the temperature, which
is now the temperature of the cavity at a certain time t. If we plot such a dataset,
it looks comparable to Fig. 5.4. For a specific value of α, the hysteresis behavior ofthe data disappears. We find the right value of α numerically by fitting the quadratic
curve given by Eq. (5.11) to the datasets. The hysteresis behavior is minimized if thequadratic fit matches best. We determine the matching of the fit by a parameter called
residual sum of squares (RSS)3 , which we minimize with respect to α; see Fig. 5.6(b).
We find that the model suits best for α = 1.25.

9

T iS
exp. fit
T ULE

6
3
0
-3
-6

RSS x 10-18(b)temperature H°C L

(a)

53
52 ì
51
50
49
48
47

ì

ì
ì
ì
ì
ì
ì
ì

0

1

2

3

4

5

time HdL

6

7

8

1.22

1.24

ì

1.26

1.28

1.30

model correction parameter ΑFigure 5.6: (a) Temperature of the inner shield TiS during the measurement of the zeroexpansion temperature. In this graph we also plot the temperature of the cavity, calculated
by using the differential equation (5.16) for α = 1.25. An exponential function is fit to thetemperature curve of the inner shield to determine the time constant for the heat-exchange
between inner and outer shield. (b) Plot of the model correction parameter versus theresidual sum of squares (RSS), which measures the matching-quality of a quadratic fit to
the corrected temperature data.

3

fit.

RSS =

P

2
i (yi − f (xi )) , where yi are the measured values and f (xi ) are the predicted values from the5.3. Zero-expansion temperature of the ULE cavity assembly

51

150

relative frequency drift DΝΝ 10-9

corrected data
fit: a2HT-Tc L2 + C0

100

50

0
- 15

- 10

-5

0

5

10

corrected temperature H°C L

Figure 5.7: Plot of the relative frequency drift of the cavity versus the corrected temperature
for the cavity. Now we clearly see a good agreement of the fit to our data within the error
bars. With the corrected data, we are able to measure the zero-expansion temperature andthe linear coefficient of thermal expansion accurately for our cavity assembly.

The data plotted as a function of the corrected temperature is shown in Fig. 5.7. We see
a good agreement between the model and the data. From the fit we extract the following
values for the zero-expansion temperature and the linear coefficient of thermal expansion.
a = 2.02(3) × 10−9 /K2 .
TC = −5.65(9) ◦ C.

(5.17)(5.18)The value for the linear coefficient of thermal expansion is similar to values found in literature[Leg10]. The zero-expansion temperature is significantly below room temperature. As wealready noted at the beginning of this section, we experience a shift of the zero-expansion
temperature of the cavity of about −12 ◦ C with respect to the zero-expansion temperature of
the ULE spacer due to the usage of fused silica mirrors. This means that the zero-expansionof the ULE spacer lies at about 6 ◦ C. Such a low value is not within the specifications givenby the manufacturer of the ULE spacer, which usually lies at around 20 ◦C. Anyhow, thisproblem is known in literature [Aln08] and we were prepared for this situation by using four
Peltier units.

52

Chapter 5. Measurements

5.4Long-term stability of the ULE cavity

Information about the long-term stability of our ULE cavity is important for our experiment.
The sources of instability are temperature drifts, material aging [Aln08], or imperfections in
the optical contacting as suggested in Ref. [Leg10].
Figure 5.8(a) shows the temperature of the outer and the inner shield over 10 days. We
(b)

(a)

AOM detuning HkHzL

temperature H°C L- 5.4
outer shield
inner shield- 5.45- 5.5- 5.55- 5.6- 5.65
- 5.7
4

5

6

7

8

2200
data
fit: A*x+B

2000
1800
1600

9 10 11 12 13

time HdL

0

10

20

30

40

time HdL

Figure 5.8: (a) Temperature of the inner shield and the outer shield measured over several
days with two PT1000 sensors. The temperature difference between both shields is probably
due to the error in manufacturing the sensors. (b) Temporal drift of the cavity over 40
days. The drift has been measured by comparing the locking setpoint relative to the 166 Er
resonance measured with the “yellow imaging” setup.
take the fluctuations of the inner shield as a reference for the temperature excursion ∆T ofthe cavity itself for estimating an upper bound for frequency excursions due to temperature.
We estimate ∆T to about 30 mK. By further assuming that we stabilized the temperature
to 0.1◦ C within the zero-expansion temperature, we get a temperature sensitivity of about
100 Hz/mK. This results in a frequency excursion range of∆νtemperature = 3 kHz.

(5.19)

This value is low enough for our experiment, and we expect no large drifts due to temperaturefluctuations.
We further measured the frequency drift of the cavity relative to the 166 Er atomic resonancevia spectroscopy4 on an erbium atomic cloud released from the MOT, over a timescale ofabout 40 days; see Fig. 5.8(b). We assume a linear drift and from the fit we extract adrift-rate of∆νdrift = 0.16(2) Hz/s.
(5.20)We see that the data are not fully matching the linear fit. We think that the problem ismagnetic field fluctuations over a longer timescale as well as the magnetic field fluctuations
4
same procedure as in Sec. 5.1. For determination of the peak position of the resonance Gaussian fits
have been used.

5.5. Linewidth of the locked laser

53

during spectroscopy due to Eddy currents produced by the induced currents when switching
off the MOT coils. Anyhow, it is improbable that the unknown magnetic field fluctuations
compensate the real drift of the cavity, so this value should hold as an upper bound for thereal cavity drift rate. Compared to literature values [Leg10], where the lowest temporal drift
rate of several ULE cavities examined is about 0.1 Hz/s, our value is in the right order ofmagnitude and low enough for our requirements for the stability of the cavity.

5.5

Linewidth of the locked laser

The usual way to measure the linewidth of a laser is to beat it with another laser of the same
wavelength and of similar or smaller linewidth. As we do not have another laser source at
583 nm available, we estimate the linewidth of the locked laser from the transmission signalof the ULE cavity.
Figure 5.9(a) shows the ULE cavity transmission signal over a timescale of 10 ms. The
intensity noise of the transmission signal has components from three different noise sources.
• Laser intensity noise• Laser frequency noise• Photodiode noise
To estimate the photodiode noise, we measured the background voltage of the photodiode,which is I0 = −0.011(1) V. This means that fluctuations are in the order of 1 mV. Thus, we
neglect the photodiode noise compared to the cavity transmission signal, which has about
10 times higher noise fluctuations.
We want to extract the laser frequency noise from the signal and convert it to a value infrequency space. If the light has zero intensity noise, the cavity transmission signal would
be completely flat. Further, no fluctuations to higher voltages could be seen. A histogram ofsuch a laser intensity noise-free signal would give a Gaussian-like distribution, which is cutat the center; see function “original f ” in Fig. 5.9(b). To extract the pure frequency noise,
we apply the following steps:• The slowly varying component of the transmission signal is removed by subtracting thesignal from its moving average having a averaging time of 0.4 ms; see Fig. 5.9. Here,
we remove a large amount of pure laser intensity noise.
• The dataset for the corrected signal is used to create a histogram with a bin-size of
2 mV; see Fig. 5.9(b).
• There is still noise left, as we observe no pure distribution that is cut at the center.
−( x )2
Therefore, we assume a function g(x) ∝ e 2σg , which models the non-frequency
noise source. This function modifies our original pure-frequency noise function f (x) ∝x−x−( 2σ 0 )2

σstep (x0 − x)efby convolution, f (x) ∗ g(x). Here, σstep is the step function.

54

Chapter 5. Measurements(b)

PD-voltage HVL

0.27
0.26

transmission signal
moving average 0.4ms

0.25
0.24
0.23
-8

-6

-4

-2

0

2

absolute frequency

(a)

1200
fit: f*g
1000
original f
800
600
400
200
0
2.35 2.40 2.45 2.50 2.55 2.60
PD-voltage HV10L

time HmsL

Figure 5.9: (a) Transmission signal of the cavity with the laser locked to it over a timescale of
10 ms. (b) Intensity noise spectrum of the transmission signal. A convolved curve f ∗g, where
f is a “half-Gaussian” function describing laser frequency noise and g, which is Gaussianfunction describing non-laser other noise sources is fit to the data. Thereby we extract theoriginal function f , which is also shown.
• We fit the convolved function f (x) ∗ g(x) to the data to extract the center x0 of thefunction f .
From the fit we get σg ≈ 7 mV, σf ≈ 36 mV and the parameter x0 = 0.2571 V. This meansthat the contribution from pure laser frequency noise is about five times higher than the one
given by the laser intensity noise and photodiode noise. The parameter x0 , which we willcall the maximum intensity Imax in the following, is especially important, because with thisparameter we are able to convert the voltage scale of the transmission signal into a frequency
scale by using the ULE cavity transmission function; see Eq. (3.5). Rewriting this equationfor our needs gives
I(ν) = I0 +Imax − I0, −80 kHz < ν ≤ 0 kHz.
1 + (F π2 )2 sin2 (2πνl/c)(5.21)

Here, F = 1865 is the finesse of our cavity, l = 0.15 m its length, and c the speed of light invacuum. By converting the voltage axis with the inverse of the function above we get thefollowing plot; see Fig. 5.10. We used the original data for the conversion except the last twobars from the right side of the histogram 5.9(b), which show a large discrepancy comparedto the function f (black marked “half-Gaussian” profile). By fitting a Gaussian profile witha fixed center at 0 kHz, we estimate the upper bound of the laser linewidth to
FWHM10 ms = 45.0(5) kHz.

(5.22)

This value is a factor of four smaller smaller than the natural linewidth of the 583-nm
transition and therefore, fulfills our needs for the short-term stability of the dye laser.

5.5. Linewidth of the locked laser

55

3000

absolute frequency

2500

gaussian fit
data

2000
1500
1000
500
0
- 80

- 60

- 40

- 20

0

frequency HkHzL

Figure 5.10: Frequency noise spectrum calculated by using the transmission function of thecavity. We fit a Gaussian profile to the data to estimate an upper bound for the laser
linewidth.

56

Chapter 5. Measurements

Chapter 6
Conclusion and Outlook
The goal of my master thesis was to stabilize a dye laser such that it emits light at 583 nmwith a linewidth below 190 kHz and that the laser has a stable locking performance forseveral hours meaning that the frequency drifts in the order of Hz/s.
For improving the long-term frequency stability, we locked the laser to an ultra low
expansion cavity. The construction of the ULE cavity has been the main part of my work.
The ULE cavity consists of a spacer made of ultra low expansion glass with mirrors
optically contacted to it. To gain a high degree of isolation from the environment, the cavity
is surrounded by two thermal heat shields, placed inside a vacuum chamber with a pressureof about 10−8 mbar and cooled to the zero-expansion temperature. I further implemented atemperature stabilization and temperature monitoring.
Finally, all the measurements describing the performance of the locking setup have been
carried out. We measured a FSR of 998.69(2) MHz, which is within specifications of the
manufacturer. This value is also needed for changing the lockpoint to another isotope.
The finesse has been measured by a cavity ringdown, the corresponding value is 1865(6).
The determination of the zero-expansion temperature has been challenging due to the highthermal isolation of the heat shields and no possibility to measure the temperature of thecavity directly. Therefore, I introduced a theoretical model to account for the effect of atime delay in temperature between inner shield and cavity. We got an very accurate valueof Tc = −5.65(9) ◦C. With knowledge of this value, we have been able to reduce the longterm drift of the cavity to an acceptable amount for our experiment. We observed a linear
frequency drift of 0.16(2) Hz/s over a timescale of about one month. This value assures ahigh degree of frequency stability and therefore, the ULE cavity serves as a stable frequency
reference.
The first result based on my work was the narrow-line MOT at 583 nm. The corresponding publication is given in App. D. In brief, the MOT is set up in a standard six-beamconfiguration. Our MOT beams have a diameter of about 30 mm to get a large capture
volume. Typically, we use a laser intensity of 12 times the saturation intensity. With theseparameters set, we achieve a MOT with a temperature as low as 15 µK and atom numbers
up to 2 × 108 . We further demonstrate a MOT for all the isotopes except the low abundant162
Er; see Fig. 6.1. For the fermionic MOT, we use the F = 19/2 → F ′ = 21/2 transition.
We observe about 3 × 107 atoms, which is substantially below the atom numbers for the57

58

Chapter 6. Conclusion and Outlookbosonic isotopes. We think that this is due to the statistical weight of about 20% concerning
the hyperfine states [Fri12].

Figure 6.1: MOT behavior of different Er isotopes. Plotted is the atom number as a functionof the natural abundance. Squares mark a MOT loading time of 20 s, circles a loading timeof 5 s. Figure adapted from [Fri12].
Recently, we managed to load about 107 Er atoms from the MOT directly into a crossed
dipole trap at 1064 nm. We then performed evaporative cooling and observed the first BEC
for the 168 Er isotope [Aik12]. After 5 seconds of evaporative cooling, we achieve a BEC with
N = 8 × 104 atoms; see Fig. 6.2.
The next steps of our experiment will be the investigation of scattering properties andthe production of a degenerate Fermi gas. Because of the more complex energy structureof the fermionic isotope, additional laser cooling stages can be needed. The ULE cavity
developed during my master work can be used with even narrower transitions (e.g 841 nm),which can be used for cooling of the fermionic isotope.

Figure 6.2: 3D density profiles of the Er cloud for different evaporation times show the BEC
phase transition.

Appendix A
CavityA.1

Theoretical description of the PDH locking techniqueIn the following the PDH error signal is derived. Our discussion follows Ref. [Bla98]. The
electric field of a monochromatic laser beam that is phase-modulated by an electro-opticalmodulator driven at frequency Ω, can be described by
E = E0 ei(ωt+β sin Ωt) ,(A.1)

where β is the modulation index. The upper equation can be approximated for β < 1E ≈ E0 J0 (β)eiωt + J1 (β)ei(ω+Ω)t − J1 (β)ei(ω−Ω)t ,(A.2)

where the Jn (β) are Bessel functions. In first order, two sidebands at the sum and thedifference frequency appear in addition to the carrier with their height depending on themodulation index. After phase modulation, the beam is sent to the cavity, where it ispartially reflected and partially transmitted. The reflected component is the relevant one for
the PDH locking scheme. The reflection coefficient F (ω) of the cavity depends on frequency
and is given by the reflected monochromatic wave Eref over the incident wave Ein ; see
Equations (3.2) and (3.3).
F (ω) =ErefeiΦ(ω) − 1=r,
Ein
1 − r 2 eiΦ(ω)(A.3)

where r is the reflectivity of the cavity mirrors and Φ(ω) the frequency dependent phase. To
get an expression for the reflected phase modulated beam, the reflection coefficient has tobe multiplied to the carrier and side-band signals. This gives
Er = E0 (F (ω)J0(β)eiωt+ F (ω + Ω)J1 (β)ei(ω+Ω)t − F (ω − Ω)J1 (β)ei(ω−Ω)t ).
59

(A.4)

60

Appendix A. Cavity

A photodiode detects the power Pr =| Er |2 of the reflected wave, which leads top
Pr = 2 Pc Psℜ[F (ω)F ∗(ω + Ω) − F ∗ (ω)F (ω − Ω)] cos Ωt+ ℑ[F (ω)F ∗(ω + Ω) − F ∗ (ω)F (ω − Ω)] sin Ωt+ (DC terms)+ (2Ω terms),(A.5)

where Pc = (J0 (β)E0 )2 is the power in the carrier and Ps = (J1 (β)E0 )2 is the power in the
sidebands. The terms oscillating at a frequency of 2Ω are filtered by low-pass filter. The DCpart of this signal is negligible because by mixing the signal with a local oscillator (LO) at afrequency of Ω, only the signal proportional to the cosine or sine of Ω is detected, dependingon the phase relation between LO and the signal. For a LO with 0◦ phase relative to thedetected signal, the slope of the error signal has a maximum. This leads topǫ = 2 Pc Ps ℑ[F (ω)F ∗(ω + Ω) − F ∗ (ω)F (ω − Ω)].
(A.6)
The error signal with an optimized LO phase is illustrated in Fig. A.1. Assuming highly
amplitude
error signal
-DΝ2

-W
+W

frequencyDΝ2

Figure A.1: Example of a calculated PDH error signal for a LO Phase of 0 ◦ . The locking
range is determined by the sideband-frequency Ω. The slope of the zero-crossing is dependent
on the cavity linewidth ∆ν, which is also the distance between the two largest extrema infrequency space. The slopes at the sidebands have a different sign than the slope at thecarrier frequency and thus, prevent locking to them.
reflective mirrors (r ≈ 1) and a modulation frequency much higher than the linewidth of thecavity (Ω >> ∆ν), the error signal near the zero crossing can be rewritten asǫ = −8

√

Pc Ps
δf.
∆ν

(A.7)

A.2. Assembling of the ULE cavity

61

Here δf is the frequency deviation of the laser from the cavity resonance.√For a good locking
performance, a suitable cavity linewidth has to be chosen and the part Pc Ps of the upper
expression should be optimized depending on the modulation index β. The expression above
is a product of the Bessel functions of first and second order and it is maximal for β = 1.08.
At this modulation index, the approximation (A.2) is not valid anymore. However, the independence of the slope of the error signal on the modulation index is discussed in reference[Kir06].

A.2

Assembling of the ULE cavity

In this Section, I describe the crucial steps I followed for assembling the ULE cavity. Some
problems and suggestions for improvement are also discussed. For visual support of mydescription, I will refer to a series of images, which have been made during the task ofassembling the cavity; see Figures A.2 to A.11.
Cleaning of the cavity parts
All the surfaces of the parts that are inside the vacuum chamber had to be cleaned properly
for high vacuum applications. For the cleaning of the cavity parts, I followed a procedure
which has been previously successfully used in our group, combined with cleaning hints from
Ref. [Bir05].
For parts contaminated with oil, mainly coming from our workshop, I used soap and distilledwater for pre-cleaning. The main cleaning has been done with technical grade1 acetone andisopropanol. For the aluminum parts, we additionally used “P3 Almeco 18”2 , which is asophisticated free degreaser. In the following, the main cleaning steps are described.
1. Pre-cleaning with distilled H2 O and standard soap to remove oil and dirt. (only ifnecessary)
2. 10 min in supersonic bath at 60 ◦ C. Solution: Almeco dissolved in distilled water witha concentration of about 30 g/L (only aluminum parts)
3. 10 min in supersonic bath at 40 ◦C. Solution: distilled water (only aluminum parts)
4. 15 min in supersonic bath at 40 ◦C. Solution: acetone
5. 15 min in supersonic bath at 40 ◦C. Solution: isopropanol
6. Cleaning surfaces with special wipes3 and blasts of nitrogen
Viton degrades if exposed to acetone. Therefore, cleaning step 4. is not used for the V-block
after the Viton has been glued to it. All the other parts, including the flanges, the Capton
1

About 99.5% purity
Henkel - P3 Almeco 18C
3
VWR - Spec wipes
2

62

Appendix A. Cavity

wires and the Peltier elements have been cleaned this way.
Viton cylinders
The spacer is supported on 4 Viton cylinders glued to the V-block. First, the cylinders havebeen cut from a large Viton ring. As a next step, they are glued into accurately fitting
cutouts; see image A.2. For this, we use high temperature epoxy. After that, the V-blockincluding the Viton cylinders has been machine-finished with a CNC-milling-machine while
taking care not to contaminate the pre-cleaned V-block. Finally, the finished V-block has
been cleaned with isopropanol only, as acetone degrades the Viton.
Temperature sensors
Inside the vacuum chamber we have placed six temperature sensors. Figure A.3 shows fiveof the sensors which are glued to the bottom plate of the outer shield with high-temperature
epoxy. The sensors are soldiered to shielded Capton wires with vacuum soldier. For positional stability of the sensors, the Capton wires are clamped to the aluminum plate. For
drying of the glue, we placed the whole aluminum plate on a heating plate.
There is an additional sensor (PT1000-3) glued to a side plate of the inner shield.
Peltier elements
For cooling of the cavity, we use four Peltier elements. In the beginning, we controlled all
four Peltier elements connected in series. Unfortunately, three months after successfully
controlling the cavity temperature in this way, one Peltier element broke and the resistanceof a second Peltier increased. We think, that this problem occurred due to several heating
and cooling cycles, which decreased the lifetime of the Peltier elements. Now we control only
one Peltier and use a fixed current for two Peltier units left. We are still able to cool thecavity down to the zero-expansion temperature.
Figure A.4 shows the Peltier coolers (labeled PE1 to PE4), which are glued to the bottom
plate of the outer shield with silver loaded epoxy4 .
When we first cooled down the cavity, the difference in thermal expansion coefficient of thePeltier, aluminum and steel put mechanical stress on the Peltier elements and the outershield. There may be better solutions for connecting the Peltier surfaces. For instance byusing a flexible graphite sheet and nylon screws for fixation.
Glass plates
Figure A.5 shows the two glass plates, which are glued to front- and back- plates of theouter shield. We use vacuum epoxy, which dries at room temperature. The usage of hightemperature epoxy would put mechanical stress onto the glass plates due to the temperature
gradient caused during the drying process. The glue is placed on the edges of the glass
plates.
Closing the chamber
Figure A.6 shows the vacuum chamber before we closed it and pumped it off. We maneuvered
the optically contacted spacer inside the heat shields by using wires.

4

RS Components - This epoxy has a good thermal conductance.

A.2. Assembling of the ULE cavity

63

Closed chamber
Figure A.7 shows the finished vacuum chamber, which is already placed on the ULE cavity
breadboard and surrounded by optics.
Optical contacting
We did the optical contacting ourselves in an clean room of ISO class 5. We recommend to
do the contacting in a more clean atmosphere, but with a precise cleaning of the optics it
was still possible.
First of all, we need to prepare the utensils we need for the process; see Fig. A.8. All theutensils are properly cleaned for usage in the clean room. We have a holder for the spacer
made of Teflon, a lens to investigate the surface of the spacer, a bright light source, lenscleaning tissues. For cleaning of the surfaces, we use isopropanol and ethanol.
After bringing all the utensils to the clean room, we started with cleaning of the spacer
by flushing it with isopropanol (Fig. A.9) and drying it with blasts of a clean nitrogen source.
As a next step, we put the spacer into the holder and cleaned its end-facets with isopropanolin the beginning, and ethanol at last. The surface is ready for contacting if no dust at all
can be seen anymore. After this task, the mirrors have been cleaned by using ethanol only,
as the surfaces have been very clean from the beginning.
Figure A.10 shows the microscope in the clean room, which we used to inspect the mirror surfaces. Also here, no impurities should be seen. If all the surfaces are clean, we put
the mirrors onto the spacer facets and put pressure on them. If the spacer is successfully
contacted, no interference stripes should be seen anymore and one is able to lift the wholeassembly by pulling the mirrors only. If the contacting is not successful, one should put
some methanol between the surfaces and slide the mirror to the side and rip it off at a place,where the surface damages do not disturb a further try of contacting. Figure A.11 shows
the successfully contacted spacer.
Electric wiring inside the vacuum chamber
Figure A.12 shows how the six temperature sensors and the four Peltier elements are connected to the D-Sub 25 pin connector on the vacuum side.

64

Appendix A. Cavity

A.3

Cavity temperature monitoring

For logging the temperature of the inside of the cavity, we use a data logger5 , which has a 24bit voltage resolution in a voltage range of 0 to 2.5 V which gives a temperature resolutionof about 1 mK if used with our PT1000 sensors. For transforming the resistance of thetemperature sensors into voltage, we use a measurement bridge; see Fig. A.13. Four precisionresistors measure the voltage drop over the sensors; we use the NTC-2 and the three PT1000
sensors for the monitoring. As a reference voltage, we use the internal 2.5 V voltage referenceof the Pico data logger, which we measured in the used configuration to VRef = 2.4973 V.
It is very important that all the four sensors are connected to the logger at the same time
because one disconnection changes the total current and has an influence on the reference
voltage in the millivolt-range, which can cause a change in the measured temperature of up
to 1 ◦C.
The NTC-10K6 sensors we use are linearized in the following way.
T (RNTC )/◦ C =B · Tn− 273.16.
B + ln(RNTC /Rn ) · Tn

(A.8)

Here, RNTC is the resistance of the used sensor, B = 3450 is a material constant, Tn =(273.16 + 25) K is the nominal temperature and Rn = 10 kΩ is the nominal resistance of thesensor.
For the PT1000 sensors7 , we transform the resistance into temperature in the following way.
T (RPT )/◦ C =RPT − 1000 −6
10 + δ.
T CR

(A.9)

Here, RPT is the resistance of the sensor, and T CR = 3.85 × 10−6 is the temperature
coefficient of the sensor. The offset δ = −1.076 is introduced by us. It is only used tocalibrate the temperature of the PT1000 sensors at the zero-expansion temperature. We get
the value by manually measuring the resistance of the PT1000-3 sensor and comparing it to
the resistance measured by the Pico data logger8 .
The voltage drop U over the temperature sensors is transformed into resistance by thefollowing function.
RINP RPR
R(U) = U.
(A.10)
VRef RINP + U(RPR − RINP )In this equation, RINP = 1 MΩ is the input impedance of the measurement channel and RPRis the resistance of the precision resistor (for values see Fig. A.13) used for the particularchannel. By combining the three above equations, the measured voltage U is converted into
temperature.

5

Pico Technology - ADC-24
EPKOS G560/10K/+
7INNOVATIVE SENSOR TECHNOLOGIES - Platinum 600 ◦C MiniSens
8
The resistance of the PT1000-3 sensor is 978.2 Ω at the zero-expansion temperature.
6A.3. Cavity temperature monitoring

65

Figure A.2: Gluing of Viton cylinders

Figure A.3: Gluing of temperature sensors

Figure A.4: Gluing of Peltier elements

Figure A.5: Gluing of glass plates

Figure A.6: Closing the vacuum chamber

Figure A.7: Finished chamber

66

Appendix A. Cavity

Figure A.8: Utensils used for optical contacting

Figure A.9: Cleaning of the spacer

Figure A.10: Checking of the mirror surface

Figure A.11: Finished optical contacting

Cavity ground andCable shielding connected

NTC-1

NTC-2

NTC-3

PT1000-3

PT1000-1

PT1000-2

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

GND

G2

G1
14
15
16
17
18
19
20
21
22
23
24
25

D-SUB-25PIN-VACUUMSIDE

GND

PE1
PE2
PE3
PE4

Figure A.12: Electric connections inside the vacuum chamber

A.3. Cavity temperature monitoring

67

reference voltage @0.9mA+2.4973V

each channel has an
imput impedance of 1M

R1
8.065K

R2
19.992K
CH5

PT1000-1
GND

R3
8.058K
CH6

NTC10K-2
GND

R6
8.061K
CH7

PT1000-2
GND

CH8

PT1000-3-I.S
only sensor on inner shield
GND

Figure A.13: The measurement bridge used for temperature monitoring with the Pico datalogger.

68

Appendix A. Cavity

Appendix B
Electronics
In this Section, diagrams of several electronic circuits used are shown.
+20V

Input feedback signal

linearisation oftransistor characteristic

BNC-INPELTIERS
~25

-15V

7

GND

R7
10K
3

D1
1N4148

IC2

4

R1
10K

100

2

OP07

R6
100K

R4
1K

3

R3
1

1

R8
1K

2

6

Q1
APT20M19JVR

R5

+15V
GND

GND
R2
power resistor
0.1

GND

Figure B.1: The current output stage.
Linos- L
EOM PM-C-BB
18-20pF
RF-source in
6.65uH
L1
2uH
L2

GND

Figure B.2: Resonant circuit for amplifying the voltage drop over the EOM. The inductances
are self-wound toroidal coils suited for HF-applications. The circuit is resonant at 31 MHz.

69

70

Appendix B. Electronics

+

+15V
C1

IC1
V+
OPT
TRM
4

+10V
GND

GND

REF01Z

VO

0.1uF

5

GND
+10V

6
+

3

2

C2
0.1uF

GND
NTC10K-1

R1
34K

C3
100uF

IC2
2

2

1

R5
1K

6

1

3

R6
100K 8
3

R3
10K

R4
10K

V+
V-

5
7

GND
4

INA114P

+15V
-15V

GND

Figure B.3: The measurement bridge used for the control of the Peltier elements.

4

Vorverstärker

10k

3

Interner EingangsoffsetVm+
321

VerrVinerr-

6

10k

7

2

5
GND
LM2904

1 J1

5

GND
Rd2*
10k

Vs-

Rd1*10k

1

Vm+

12

R7 10k

Vext+
GND
Vext-

3
2
1
V+ EXT/INTern

V+
Ve+
Vext+

3
2
1
V- EXT/INTern

VVeVext-

2
1
3
4
GND

IN
NC
NC
GND
REF01AG

6
8
7
5

5
GND

LM2904
7

2

8

3

2
R9 10k

3

P4
U4A

D2
1

1

1
2
3

out
Diode 1N4006

Vs-

GND

Gleichrichter

LM2904Vs-

10k

C- Integrator
VsP7

Ci*

Cd*
Rpot3
10k
Rsum510k

U4B

6
R5 10k

4

Vs+
8

OUT
NC
NC
TRIM

10
GND

D-part (unused):
Cd*=unused
Rd2*=unused
Rd1*=unused

Vs+

U3C
LM324AM

1
2
3
4
C- Integrator

Cd1
Cd2

1
2
3
4

1
2
3
4

Cap

Con

MonitorC- Integrator

MonoutIout

GND

R8 10kVm4

3
2
1

Externe Versorgung

U10

Vs+
9

U3D
LM324AM
14 Rsum4

13

Cd2

OffsetR6 10k

Vs+

10k

VsVs+

Cd1

GND

P10

U3B
LM324AM
Rsum3
CI

2
3
INV/NINV Ionly

P3
Ionly/PID

P9

7

2

8

6
Ri* 10k

U2B

D1

Diode 1N4006

10k
4

8

1

1

LM2904
R3

Vm-

P8

11

Con

Vs+

Vs-

Rpot1
10k

3

Vs+

U2A

I-part (switched oﬀ):
Ri*=58K
Ci*= 100uF

R1010kRsum2*
Vs+

3
2
1

Eingangsoffset

GND

2

P5

PID

Con

R4
10k

4

R2
10k
2

VmoninR1

Cap

P-part:Rp1=47K
Rp2=1K

Gleichrichter

4

1
Rpot2
100k

Vin

S1
INT OFF / ON

Invertierer/Nichtinvertierer

2

Vs- = -15V

11

Diff

PID-settings

Vs+ = +15V

Vs-

4

Vs-

1
2
3

11

4

1
GND

GND

4

Vin
Vbridge*plugable

10k

11

3

3

8

6

VerrVinerr-

U3A
LM324AM
Rsum1

1

1

Cap

4

1
2
3
4
Input

8

2

Rp2*10k

U1
INA114AG
P2

3

GND

P1

Vs+

7

5

2

VsP11
GND
out
Ionly

1
2
3
4

5
6
7
8

Iout (feedback signal)

Mode select
Rpot6
50k

GND

71

Figure B.4: The temperature PI controller used for the control of the cavity temperature.
This controller is made by our electronic workshop and modified by myself.

Vs+ Rp1* 10k

Differenz-Eingang

72

Appendix B. Electronics

EOM servo
Differential amplifier+2x Integrator

C5
2.53uF

C4
800pF

10

+

1M
1uF

R10

-15V
4

R3 5K
1

2

3
2

2

3

220pF
1M

OP07

2

6
3
7

IC1

-15V

R4
1K

+

C7

IC3

GND

Output Offset
+15V

R6

1

GND

2

3

C10
10uF
OP07

2

22uF

-15V

+

4

IC2

3

2

R14 5K

OP07

1

6
7

Monitor OUT

6
3

2

+15V
100K
R15

1

+15V

R13

IC4

IC6

GND

57

3
2

OP07

PIEZO OUT

GND

3

3

-15V

R17

IC7

7
4

+15V

4

120

6

-15V

GND

7

OP07

2

GND

2

1K
R16

3
7

C1

C11
10uF

+

6

+15V

R20
3.3K

Integrator

R5
50K
4

R1
51

+

EOM OUT
C8
10uF

-15V
4

1

7

C3

R33
1K

Error In

OP07

3

-15V

C9

10uF
6

Buffer

+

33uF

R9

R8

C6
33uF

C2

10

R18
1M

-15V
4

+15V
2

OP07
6

3

Inverter

1K
R12

R7
1K

3
2

-15V

1

7

R11
1K

GND
3

1

+15V

2
4
2

OP07
6

S1

3
7

IC8

1K

IC5

R31

R30
1K

GND

-15V
4

+15V
2

OP07
6

Output Offset

3

41K

7

-15V

R24

R21
8.2K

DYE CONTROL OUT

GND

1K

6

R25

IC9

SW1
1

R23

OP07

3
7

3

41K

2

4

+15V

2
1

100
R32

-15V
2

R2
10K

IC11

+15V
R22
8.2K

41K

1K

R28

R29

-15V
4

Ramp IN
2
2
1

R27
10K

GND

41K
R26

OP07
6

3
7

3

+15V

IC10

+15V

Figure B.5: Dye servo. The servo is used to apply feedback onto the EOM, the piezo mirror
and the galvo plate of the dye laser.

Appendix C
Hyperfine structure of the fermionicisotope
For calculation of the energy E of hyperfine states of the 167 Er isotope, we use the following
formula [Ari77]
K
3K(K + 1) − 4I(I + 1)J(J + 1)
+B,
2
8I(2I − 1)J(2J − 1)
K = F (F + 1) − J(J + 1) − I(I + 1).

E[J, F ] = A(C.1)

Here, A and B are the hyperfine constants, which are known for the ground state andthe excited 583 nm state. They are taken from Refs. [Smi65, Chi83], and have the values
AGS = −120.4864 MHz, BGS = −4552.959 MHz, A583 = −172.5 MHz, B583 = −4440 MHz.
The nuclear spin quantum number is I = 7/2. For the ground state, we have J = 6 and F
ranging from 5/2 to 19/2, whereas for the excited state we have J = 7 and F ′ ranging from
7/2 to 21/2. In first order, only transitions having F ′ − F = ∆F = 0, +1, −1 are allowed.
The corresponding frequency shifts are calculated by∆E = E[J = 7, F ′] − E[J = 6, F ] − δE,(C.2)

where δE = −335.2 MHz is the shift of the unperturbed 167 Er transition relative to the thetransition of 166 Er at 583 nm [Jin90]. Figure C.1 shows all the transitions F → F ′ for the
583-nm line.

73

74

Appendix C. Hyperfine structure of the fermionic isotope- 20009.5-10.5Er170

8.5-9.5- 1000Er168
7.5-8.5

6.5-7.50
5.5-6.5

DF=0

1000

detuning relative to 166Er HMHzL

Er166

4.5-5.5

Er164

3.5-4.5
2.5-3.5
DF=-1

2000
3000

Figure C.1: Erbium hyperfine structure at 583 nm.

Er162

Appendix D
PRL: Narrow-line magneto-opticaltrap for erbium

75

76

Appendix D. PRL: Narrow-line magneto-optical trap for erbium

Narrow-line magneto-optical trap for erbium
A. Frisch,1 K. Aikawa,1 M. Mark,1 A. Rietzler,1 J. Schindler,1 E. Zupanič,1,2 R. Grimm,1,3 and F. Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria2
Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
3
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, A-6020 Innsbruck, Austria(Received 7 March 2012; published 7 May 2012)
We report on the experimental realization of a robust and efficient magneto-optical trap for erbium atoms,
based on a narrow cooling transition at 583 nm. We observe up to N = 2 × 108 atoms at a temperature of about
T = 15 µK. This simple scheme provides better starting conditions for direct loading of dipole traps as compared
to approaches based on the strong cooling transition alone, or on a combination of a strong and a narrow kHz
transition. Our results on Er point to a general, simple, and efficient approach to laser cool samples of otherlanthanide atoms (Ho, Dy, and Tm) for the production of quantum-degenerate samples.
DOI: 10.1103/PhysRevA.85.051401

PACS number(s): 37.10.De, 67.85.−d, 37.10.Vz

Laser cooling of non-alkali-metal atoms has become avery active and challenging field of research. The great
appeal of unconventional atomic systems for experiments on
ultracold atomic quantum gases stems from the possibilityof engineering complex interactions and of accessing richatomic energy spectra. Both features are at the foundation of anumber of novel and fascinating phenomena. For instance,
the energy spectra of two-valence-electron species, suchas alkaline-earth-metal and alkaline-earth-metal-like atoms,
feature narrow and ultranarrow optical transitions, which are
key ingredients for ultraprecise atomic clocks [1], efficientquantum computation schemes [2], and novel laser coolingapproaches, as demonstrated in experiments with Sr, Yb, andCa [3–5].
As a next step in complexity, multivalence-electron atoms
with a non-S electronic ground state such as lanthanides are
currently attracting increasing experimental and theoreticalinterest. Among many, one of the special features of lanthanides is the exceptionally large magnetic dipole momentof atoms in the electronic ground state (e.g., 7µB for Er and10µB for both Dy and Tb), which provides a unique chance
to study strongly dipolar phenomena with atoms. Highly
magnetic atoms interact with each other not only via theusual contact interaction but also via an anisotropic and longrange interaction, known as the dipole-dipole interaction [6].
Chromium was the first atomic species used for experiments
on atomic dipolar quantum gases [7,8], and the even moremagnetic lanthanides are nowadays in the limelight thanks tolaser cooling experiments on Er and Tm [9,10] and to therecent realization of quantum-degenerate Dy gases [11,12].
Similarly to Yb and the alkaline-earth-metal atoms, theatomic energy spectra of magnetic lanthanides include broad,narrow, and ultranarrow optical transitions. This collectionof lines is reflected in a wide choice of possible schemes
for laser cooling experiments. However, all experiments on
Zeeman slowing and cooling in a magneto-optical trap (MOT)with magnetic lanthanides so far have relied on an approachthat is essentially based on the strongest cycling transition[9,10,13]. This broad transition typically lies in the blue
between 400 and 430 nm and has a linewidth on the order of
few tens of MHz. As a consequence, the Doppler temperature
is close to 1 mK. Such a high temperature makes direct loading
1050-2947/2012/85(5)/051401(5)from a MOT into a dipole trap difficult and inefficient, evenwhen sub-Doppler mechanisms take place [14,15]. To further
decrease the temperature of atoms prior to the dipole trap
loading, an additional MOT stage based on an ultranarrow
kHz-linewidth transition was applied in Refs. [11,13], making
the whole experimental procedure more involved.
Taking advantage of the rich atomic spectrum of lanthanides, we identify a different transition to be the mostsuitable one for MOT operation toward production of aquantum-degenerate gas. This transition, which for Er liesat 583 nm and has a linewidth of 190 kHz, represents anintermediate case between the broad (blue) and the ultranarrow(red) transitions available in the lanthanide spectra and hasvery similar properties to the Yb intercombination line [16,17].
Based on this narrow yellow line, we demonstrate a MOT of
2 × 108 Er atoms at a temperature as low as 15 µK. Our
approach, inspired by Yb experiments [4], provides better
starting conditions for direct loading into a dipole trap thanthe conditions achieved in other experiments with magnetic
lanthanides [11].
Erbium is a heavy rare-earth element of the lanthanide
series. It has six stable isotopes, among which the three bosonicisotopes 166 Er (34%), 168 Er (27%), and 170 Er (15%) and thefermionic isotope 167 Er (23%) have a high natural abundance.
The Er electronic configuration is characterized by a xenonlike
core, an inner open 4f shell, and an outer closed 6s shell,[Xe]4f 12 6s 2 . The electron vacancies in the inner f shell are acommon feature of all the lanthanides (with the exception ofYb) and are at the origin of the strong magnetism as well as
various interesting collisional effects [18–20].
The atomic level spectrum of Er is shown in Fig. 1.
In the ground state, Er has a highly anisotropic electronic
density distribution with a large orbital angular momentum
L = 5 (H state) and a total electronic angular momentum
J = 6. The bosonic isotopes have zero nuclear spin (I = 0)
and consequently do not exhibit a hyperfine structure. On
the contrary, the fermionic isotope 167 Er has a nuclear spin
I = 7/2, leading to eight hyperfine levels, from F = 5/2 to
F = 19/2, in the electronic ground state.
In Ref. [22], five different J → J + 1 laser coolingtransitions were identified with linewidths ranging from tensof MHz to a few Hz. Here we focus on the blue and051401-1©2012 American Physical Society

77energy (1000 cm -1 )25

20

4f12(3H6) 6s6p(1P1)

15

4f12(3H6) 6s6p(3P1)

10

ZS light(401 nm)5

MOT light(583 nm)
4f126s2 3H6

0
2

3

4

5

6

7

8

9

10

11

12

J
FIG. 1. (Color online) Energy levels of atomic erbium up to
25 000 cm−1 for different total electronic angular momentum quantum numbers J [21]. States with odd (even) parity are indicatedby black (red/gray) horizontal lines. The two relevant laser coolingtransitions at 401 and 583 nm are indicated by arrows [22].

the yellow transition at 401 and 583 nm, respectively; see
the arrows in Fig. 1. The corresponding excited levels are
the singlet 1P1 and triplet 3P1 states coming from the transitionof an s electron into a p shell. The strong blue transition
at 401 nm has a linewidth of Ŵ401 /2π = 27.5 MHz [23],corresponding to a Doppler temperature of h̄Ŵ401 /(2kB ) =660 µK. We use the blue light for both Zeeman slowing
and transversal cooling of the Er atomic beam. The MOT
operates on the narrow yellow transition at 583 nm, which has anatural linewidth of Ŵ583 /2π = 190 kHz and a corresponding
Doppler temperature of 4.6 µK. The line may exhibit weakleaks from the excited level into two intermediate levels
with very low calculated leakage rates (0.017 and 0.0049
s−1 ) [22]. Note that we find these losses irrelevant for allpractical purposes. For 167 Er the hyperfine structure in the401- and 583-nm excited states gives rise to eight levels
ranging from F ′ = 7/2 to F ′ = 21/2. For the 583-nm line,
the hyperfine constants are known [26]. On the contrary
the hyperfine constants for the 401-nm line are currently
unknown, making the operation of the fermionic MOT morechallenging.
We generate the blue light from two independent blue diodelasers injection locked to a master laser. The master laser
light is produced by frequency doubling methods based ona tapered amplified diode laser at 802 nm. The blue light fromthe master laser is locked to a hollow-cathode discharge lamp[4] via modulation transfer spectroscopy. With this setup wespectroscopically resolve the lines of the four most abundantbosonic isotopes as well as the hyperfine structure of thefermionic isotope 167 Er. Since the hyperfine constants of theexcited level at 401 nm are unknown, we could not assign
the absorption features to specific hyperfine transitions. We
derive the yellow light from a dye laser operating with

Rhodamin 6G. By using an intracavity electro-optic modulator(EOM) and an external reference cavity, we stabilize the laser
to a linewidth of about 50 kHz. By additionally locking thelaser to an ultralow-expansion cavity, we achieve a long-term
stability better than 30 kHz within a day [27].
Our experimental procedure is as follows. We load the Zeeman slowed atomic beam from an effusive, high-temperatureoven directly into the narrow-line MOT. Our commercial oventypically operates at a temperature of 1300 ◦ C, which is about
200 ◦ C below the Er melting point. Two 3-mm apertures,
separated by 50 mm, provide a geometrical collimation of theatomic beam. In addition, the atomic beam is further collimatedand transversally cooled by a two-dimensional (2D) opticalmolasses, working on the broad 401-nm transition with a total
power of about 100 mW. The beams are elliptically shaped
to increase the interaction time between the atoms and thelight. Thanks to the transversal cooling stage, we increase theloading flux by almost an order of magnitude.
The atomic beam then enters the Zeeman slower (ZS).
Because of the limited capture velocity (a few m/s) imposedby the narrow cooling transition used for the MOT, it is
crucial to design a ZS that provides enough atomic flux at
low velocities [4]. We build a 360-mm-long spin-flip Zeeman
slower, which can slow the atoms from 500 to about 5 m/s. The
ZS light is focused at the oven position and has a total powerof about 60 mW. At the MOT position, we estimate a beam
diameter of about 11 mm, corresponding to an intensity of
about one Is,401 , where Is,401 = 56 mW/cm2 is the saturation
intensity. Our ZS operates with light detuned by about −20Ŵ401(−540 MHz) from the unshifted resonance.
The narrow-line MOT is operated in a standard sixbeam configuration with retroreflected beams. To increase
the capture velocity of the MOT, we use large MOT beam
diameters of about 30 mm. Typical MOT loading parameters
include a magnetic field gradient B ′ (along the symmetryaxis) of 4 G/cm, a laser intensity of 12Is,583 per beam with
Is,583 = 0.13 mW/cm2 , and a detuning δ583 from the atomic
transition of −50Ŵ583 (−9.5 MHz). To measure the number ofatoms in the MOT after loading, we optically compress the
MOT by reducing δ583 to −0.5Ŵ583 and we apply standard
absorption imaging on the blue transition.
A special feature of our narrow-line MOT is the large
detuning of the MOT light (typically −50Ŵ583 ) needed for
optimal loading. At this detuning, we observe a very strong
effect of gravity on the position and shape of the atom cloud [3].
The atoms are located well below the center of the magnetic
quadrupole field, and the cloud takes the form of a large
semishell. To elucidate the reason for the large detuning, wemonitor the loading dynamics and the lifetime of the MOT;
see Fig. 2. Our measurements focus on the 166 Er isotope, but
we have observed the same qualitative behavior also for theother isotopes.
Figure 2(a) shows the atom number in the MOT as afunction of the loading time for different values of δ583 . We
fit our data by using a standard loading rate equation [28],which includes a capture rate R and a decay rate γ ; thelatter accounts for both collisions between trapped atoms
and collisions with the background gas. For a detuning of−50Ŵ583 the atom number approaches its steady state in about
10 s with Nss ≈ 2 × 108 . For a lower value of the detuning

051401-2

78

Appendix D. PRL: Narrow-line magneto-optical trap for erbium

FIG. 2. (Color online) Loading (a) and decay (b) of the narrowline MOT with 166 Er atoms for δ583 = −50Ŵ583 (circles) and δ583 =−32Ŵ583 (triangles). (a) Atom numbers are plotted as a function of
the MOT loading time. The solid lines are fits to the data using
N (t) = Nss (1 − e−γ t ) with Nss = R/γ . From the fits we obtain
γ = 0.137(2) s−1 , R = 2.6(4) × 107 s−1 , and γ = 0.200(4) s−1 ,R = 1.1(1) × 107 s−1 for −50Ŵ583 and −32Ŵ583 , respectively. The
inset shows the atom number after 10-s loading time as a function of
the MOT detuning. (b) Atom numbers are plotted as a function of theholding time in the MOT in presence (open symbols) and absence of
the ZS light (solid symbols). The solid lines are fits to the data using
a double-exponential function; see text.

to −32Ŵ583 we observe a substantial decrease of the atom
number to Nss ≈ 5 × 107 . This behavior is clearly shown in
the inset, where we monitor the number of atoms Nss for a fixedloading time of 10 s as a function of the MOT light detuning.
For detunings exceeding δ583 = −30Ŵ583 we observe a rapid
increase in the atom number. When further increasing thedetuning, Nss first stays constant and then rapidly decreases.
We believe that the large detuning of the MOT light serves
to minimize the detrimental effects of off-resonant pumping
processes driven by the ZS light. According to the Zeeman
shift in the quadrupole field of the MOT, we observe a
spatial displacement of the atomic cloud with δ583 of about
1.4 mm/MHz [29]. For large detunings, this shift becomes solarge that the atoms can no longer be kept in the MOT. Forintermediate detunings in the range from −40 to −55Ŵ583 ,the cloud displacement is advantageous for MOT operation
since the atoms become spatially separated from the region ofinteraction with the ZS light. For small detunings, the ZS lightleads to substantial losses of atoms.
The effect of the ZS light also shows up in lifetime
measurements, where we monitor the number of atoms in the
MOT as a function of the holding time with the ZS light being
present or absent; see Fig. 2(b). For these measurements weswitch off the ZS magnetic field and the atomic beam after
10 s of MOT loading. For large detuning (δ583 = −50Ŵ583 ) the
evolution of the atom number is not affected by the ZS light. In
both cases, i.e., with the ZS light on and off, we observe fasterlosses in the earlier stage of the decay, which we attribute toinelastic two-body collisional processes, and a slower decay at
a later time, which is finally limited by background collisions.
For simplicity we use a double-exponential fit function toestimate the time constants [30]. We extract time constants of
19(3) and 80(20) s for the fast and slow dynamics, respectively.
At lower MOT detuning (−32Ŵ583 ), the ZS light strongly
affects the decay. When the blue light is turned off, ourobservations are qualitatively similar to the ones at −50Ŵ583
while with the blue light turned on the atomic loss dramaticallyincreases. In this case the decay curve is well described by asingle exponential function with a time constant τ = 9.0(1)s. This decay time is consistent with a simple estimate ofpumping losses. By considering the absorption rate Ŵa of the
401-nm light at the actual intensity and detuning [31] andthe branching ratio b for decay from the excited state to all
the possible metastable states [9], we estimate a decay rate
1/τ = bŴa of the order of 0.1 s−1 .
We could also demonstrate trapping of all the other Er
isotopes, with the exception of the rare 162 Er (0.1% naturalabundance). For all the bosonic isotopes we used about thesame values for the detuning of the ZS and the MOT light.
Figure 3(a) shows the atom number in the MOT for thedifferent isotopes as a function of their natural abundance.
For a long loading time of 20 s we observe similar atom
numbers exceeding 108 for the three most abundant bosonicisotopes, indicating that saturation effects might apply. For ashort loading time of 5 s, the atom number increases with thenatural abundance. However, we observe a more complicated

FIG. 3. (Color online) Behavior of different Er isotopes. (a) MOT
atom number for the four most abundant bosonic isotopes as well
as for the 167 Er fermionic isotope is plotted as a function of thenatural abundance for 20-s (squares) and 5-s (circles) loading times.
(b) Spectroscopy signal of the blue transition shows the isotope shifts
for the bosonic isotopes and the HF structure of 167 Er. The arrow
indicates the locking point used for the ZS light to produce the 167 Er
MOT.

051401-3

79

behavior than the expected linear growth. This might be due
to slight differences in the optimal MOT parameters, but may
also point to differences in scattering and collisional properties
among the different isotopes.
For the fermionic isotope 167 Er we observe a MOT withatom number of about 3 × 107 , which is substantially lower
than the numbers measured for the bosonic isotopes. A simple
explanation of this behavior can be that we decelerate andcool only atoms in the F = 19/2 hyperfine state, which has astatistical weight of about 20%. Note that a similar behavior
has been observed with the fermionic 161 Dy MOT, which also
shows lower atom numbers than the ones of bosonic MOTs[13]. An additional complication stems from the unknown
hyperfine splitting of the 401-nm line. From the spectroscopic
signal we could not identify a priori where to lock the Zeeman
slower light to be resonant with the desired F = 19/2 → F ′ =21/2 hyperfine transition. To produce the 167 Er MOT, we had toproceed blindly by first locking the 583-nm MOT light on thecooling transition and then by trying different locking points
for the ZS light until the MOT was visible. We finally succeedin creating a MOT by locking the ZS light to the spectroscopic
line located 150 MHz below the blue transition frequency of
the 166 Er isotope; see the arrow in Fig. 3(b).
After loading the MOT we apply a stage of compression to
reduce the temperature of the atomic cloud and to achieve
good starting conditions for direct loading of an optical
dipole trap. With the optimized parameters δ583 = −0.5Ŵ583 ,I = 0.16Is,583 , and B ′ = 0.8 G/cm, we measure a temperatureof 15 µK via time-of-flight experiments. In addition, SternGerlach experiments indicate that atoms in the compressed
MOT phase are naturally pumped in the lowest Zeeman
sublevel mJ = −6 by the MOT light. This can be explained byconsidering a combined effect of the narrow cooling transition
used for the MOT and of gravity. The latter pushes the atoms

downward, creating an effective imbalance in the trap andleading to a preferential absorbtion of the σ − polarized lightfrom the lower vertical beam. For 2 × 108 atoms at T = 15 µK
we estimate a peak number density of the polarized sample
of 1.5 × 1011 cm−3 , corresponding to a phase-space densityof about 4 × 10−6 . These values are similar to the ones
observed in Yb experiments using the intercombination light[4]. Compared to other experiments on magnetic lanthanides,
such as Dy [11], our much simpler approach based on a single
cooling light for the MOT provides higher atom numbers andsimilar final temperatures. We suggest that this scheme canbe successfully implemented with Dy, Ho, and Tm, using
the 626-nm (Ŵ/2π = 135 kHz) [32], the 598-nm (Ŵ/2π =146 kHz) [33], and the 531-nm (Ŵ/2π = 370 kHz) [34]transitions, respectively.
In conclusion, we have demonstrated an efficient and simple
approach for an Er MOT based on a single narrow-line
transition. Our scheme works with all abundant Er isotopes
and allows for direct loading of an optical dipole trap. In
first experiments we were able to load up to 107 Er atoms
into the dipole trap in a single focused-beam configuration.
Optimization of the dipole trap loading and evaporative
cooling experiments are under way in our laboratory.

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We are grateful to J. J. McClelland and A. J. Berglund forsharing with us their knowledge on Er and to S. Kotochigova,
O. Dulieu, M. Lepers, and J. F. Wyart for fruitful discussions.
We also thank the Sr team in Innsbruck and the Yb team inTokyo for their support. This work is supported by the Austrian
Ministry of Science and Research (BMWF) and the Austrian
Science Fund (FWF) through a START grant under Project
No. Y479-N20 and by the European Research Council under
Project No. 259435.

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while in Ref. [25] a value of Ŵ401 /2π = 29.5 MHz has beenobtained from lifetime measurements. We measure a linewidthof 27.5(4) MHz by using an absorption imaging technique onmagneto-optically trapped atoms in the regime of low saturation
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moments between the ground and excited state.
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