An optical dipole trap for Erbiumwith tunable geometry
Simon BaierMASTERTHESIS
submitted to the
Faculty ofMathematics, Computer Science and Physics
of the University of Innsbruck

Supervisor:
Univ. Prof. Dr. Francesca Ferlaino

October 2012Abstract
In this thesis, we describe the realization of a novel optical dipole trap for
erbium atoms. The trap is based on time-averaged potentials and is thus tunable in geometry. We investigate both theoretically and experimentally the
dynamic polarizability of ground-state erbium atoms. The polarizability is avery important quantity for the understanding of the atomic properties of erbium and, prior to this thesis, its value was unknown. We measure a dynamic
polarizability of Re(α) = (84 ± 2 ± 18) a.u. for erbium atoms in a 1064-nm
laser field, which is 47 % lower than the one we calculated based on the best
knowledge of the erbium atomic spectrum. This discrepancy might points toa too rough knowledge of the atomic level structure or to novel unexpected
effects arising in sub-merged shell atoms. Further investigations both in theory
and experiments are highly needed.
For our novel optical dipole trap setup we use a scanning system consistingof an acousto-optical modulator, electronics and a customized optical setup.
The dipole trap beam is shifted perpendicular to its horizontal axis, creatingtime-averaged potentials when the scan over a range of positions is fast enoughcompared to the trap frequency. The aspect ratio of the dipole trap can betuned from 1.5 to 15. With the new dipole trap system we can load up to
35 % of MOT atoms to the dipole trap, which is related to mode-matchingarguments. Further the density at each evaporation step can be optimized,
leading to a large overall evaporation efficiency of 3.5. This improvements
result in up to three times larger numbers in the pure BEC compared to ourprevious experiments.
With the new tunable dipole trap we are confident that in future we caninvestigate geometry-dependent anisotropic quantum effects, unique to dipolargases.

iii

Contents
Abstractiii

1 Introduction

12 Erbium5
2.1 A glance at erbium . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Energy spectrum of erbium atoms . . . . . . . . . . . . . . . . . 7
2.3 Interaction between erbium atoms . . . . . . . . . . . . . . . . . 10
3 Optical dipole trap13
3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 The oscillator model . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Multilevel atoms . . . . . . . . . . . . . . . . . . . . . . 15
3.2 The case of erbium . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Modeling of the trapping potential . . . . . . . . . . . . . . . . 20
4 Atomic polarizability of erbium
29
4.1 Contribution of different energy levels . . . . . . . . . . . . . . . 29
4.2 Dependence on the laser wavelength . . . . . . . . . . . . . . . . 35
5 Setup of a flexible optical dipole trap for erbium
39
5.1 Scanning system . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.1 Requirements on the scanning system . . . . . . . . . . . 40
5.1.2 Technical realization . . . . . . . . . . . . . . . . . . . . 41
5.1.3 Implementation in the experiment . . . . . . . . . . . . . 44
5.1.4 Technical limitations . . . . . . . . . . . . . . . . . . . . 45
5.2 Offline characterization . . . . . . . . . . . . . . . . . . . . . . . 49
6 Measurements with atoms
53
6.1 Characterization of trap geometries . . . . . . . . . . . . . . . . 54
6.1.1 Trap frequency measurements . . . . . . . . . . . . . . . 54v

CONTENTS

6.2

6.1.2 Atomic polarizability . . . . . . . . . . . . . . . . . . . .
6.1.3 Comparison to offline characterization . . . . . . . . . .
6.1.4 Polarization dependence of the trapping geometry . . . .
A new Bose-Einstein condensate . . . . . . . . . . . . . . . . . .

56
60
61
63

7 Conclusion and Outlook

67

A Electronics

71

B Dipole trap model75

C PRL: Bose-Einstein Condensation of Erbium

83

Bibliography

89

Danksagung

95

viChapter 1
Introduction
Experiments on ultracold gases deal with atomic systems at temperatures near
the absolute zero, where the atoms start to reveal their quantum behavior.
Ultracold atomic gases provide an unprecedented level of control over many
external and internal degrees of freedom. Remarkably, the interparticle interaction and the confining geometry can be changed almost "on demand", giving
access to the investigation of various exciting phenomena belonging to a wide
range of physical researches. Quantum simulations [Fey82], high-temperature
superconductivity [Chi06], condensed matter [Gre08], few body physics [Fer10]
and quantum chemistry [Osp10] just name a few of these phenomena.
The first experimental realization of a Bose-Einstein condensate (BEC)
in 1995 [And95, Bra95, Dav95] marked the starting point of a new field inphysics. The attainment of quantum degenerate Fermi gases further pushed theinterest on this still rapidly growing field [DeM99, Sch01b, Tru01]. The levelof control was increased by the ability to modify the interparticle interaction
via Feshbach resonances [Chi10]. This controllability leaded to the observationof various fascinating phenomena, such as the superfluid to Mott insulatortransition [Gre02] or the creation of weakly bound dimer states [Win06, Joc03,
Fer09]. Even though the list of exciting results could be almost arbitrarily
extended, all of these physical pioneering works were performed with elementsfrom the alkali series, where the interaction is ruled by the surprisingly simple
isotropic and short-range contact interaction.
To make the story even more exciting, efforts were taken to explore novel
systems, in which the particles additionally interact via an anisotropic andlong range interaction, called dipole-dipole interaction (DDI). Polar moleculesand polar atoms carry an electric and a magnetic dipole moment, respectively,
and, therefore, give access to the DDI. Because of the DDI new fascinating
quantum phases are expected, which in particular depend on the trapping11 INTRODUCTION
geometry [Lah09, Bar08].
The first species used to study dipolar effects in experiments was chromium[Gri05, Bea08]. Among many fascinating phenomena the observation of the dwave collapse of a Bose gas stands out [Lah08]. In order to deepen the experimental possibilities on dipolar physics novel species with even higher magnetic
moments have become promising candidates in ultracold experiments. The recent realization of a BEC and a quantum degenerate Fermi gas of dysprosiumby the group of B. Lev at Stanford [Lu11, Lu12] and the BEC of erbium by
our group in Innsbruck [Aik12] promise to open a new chapter in the field ofdipolar gases.
Erbium is part of the lanthanide series and has a rich energy spectrum. Dueto its multivalence-electrons character complex scattering properties are expected. The investigation of optical transitions in erbium opened the possibility
to control this species by means of laser cooling techniques [Ban05, McC06b].
The first goal of our experiment was to reach a BEC. In march, 2012 we
could set an end to this over two years lasting journey. Worldwide we havebeen the first to cool down erbium to quantum degeneracy [Aik12]. In our
experiment we load erbium atoms from the MOT to an optical dipole trap
and evaporatively cool them down.
The purpose of my master thesis was to design an optical dipole trap withtunable geometry. This was realized by means of a scanning system, which cancreate time-averaged potentials. This improvement gave us the possibility todynamically optimize the efficiencies in the loading and the evaporation phaseof our experiment. Furthermore, the flexible geometry opens access to the investigation of various fascinating geometry-dependent quantum effects, unique
to dipolar gases [Lim11, San03, Bla12]. To gain a deeper understanding of thedipole trap for erbium I also focused on the development of a mathematicalmodel on dipole traps. This model calculates the dynamic polarizability of erbium taking all dipole allowed transitions into account. The difficulty of thiscalculation is connected to the complex and rather unknown energy spectrumof erbium [Ral11, Ban05]. Furthermore, with this model the trapping geometryof the dipole trap configuration in our experiment can be fully simulated.
This thesis is structured as follows. In Chap. 2 we present the properties of
erbium and focus on the energy spectrum of erbium. Furthermore, the up-todate known scattering properties of erbium are discussed. Chapter 3 reviews
the theory of optical dipole traps and outlines the influence of the dipole effecton erbium atoms. The mathematical model used to calculate dipole trap parameters for erbium at our experiment is explained. Chapter 4 focuses deeplyon the investigation of the atomic polarizability of erbium. In Chap. 5 the optical and electronic setup of the scanning system is discussed, as well as the21 INTRODUCTION
performance of the scanning system is characterized. Chapter 6 investigates
the trapping geometry of the new dipole trap for erbium atoms and shows themeasurement of the dynamic polarizability of erbium. The production of thenew, improved Er-BEC is discussed. Finally, Chap. 7 concludes and outlines
future experiments being accessible with the new dipole trap system.

3

Chapter 2
ErbiumUltracold gases are ideal systems to observe exciting many- and few-bodyquantum effects. In recent years there is an increasing interest on the investigation of non-alkali atoms. Such species have rich atomic spectra, opening
possibilities for novel laser cooling approaches and leading to fascinating interaction properties. Particularly interesting are multi-valence-electron atoms
with a non-S electronic ground state, such as lanthanides. The interaction ofalkali atoms can be described by a short-range and isotropic contact potential.
Lanthanide atoms, such as erbium, provide a large magnetic dipole moment,
leading to a long-range and anisotropic dipole-dipole interactions [Lah09]. This
interaction is expected to give access to unexplored physical scenarios [Gor00].
In this Chapter, we briefly discuss the main properties of erbium and willthen focus on the energy spectrum and the interaction of erbium atoms. For acloser insight into the properties of erbium see e.g. Ref. [Sch11].

2.1

A glance at erbium

General properties
Erbium (Er) is part of the lanthanide series and has an atomic number of 68.
It is a soft, silver-colored metal, which has a comparatively high melting pointof about 1530 ◦ C [Jam92]. Erbium has six stable isotopes with masses ranging
from 162 to 170 amu. Table 2.1 shows the natural abundance and nuclear spinof each isotope. There are three bosonic isotopes with a natural abundanceexceeding 14 %. Remarkably, the natural abundance of the fermionic isotope isextremely high compared to the typically used fermionic species 40 K [DeM99]
and 6 Li [Sch01a] belonging to the alkali class. This is a very favorable featurefor experiments on degenerate Fermi gases.
5

2 ERBIUM

Table 2.1: Natural abundance and nuclear spin quantum number I of all
staple erbium 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

4f -block atoms
Several species from the lanthanides, among which erbium, are also known asatoms of the 4f -block. They are named in this way because of an unfilled 4f
inner shell, shielded by a closed outer shell. This peculiarity plays a crucial rolein the atom behavior. In the case of erbium, the electronic configuration of theground state reads as [Xe]4f 12 6s2 . The [Xe]-like core indicates that all shells arefilled up to the 5p shell. From the 14 valence electrons only 12 are located in the
4f shell, leading to a two-electron vacancy. The two remaining electrons fill up
the 6s shell. This leads to the so-called submerged shell structure. The f -shellis characterized by a strong anisotropy in the electron density distribution.
One of the most relevant consequences is the large magnetic moment endowedby such atoms.
Another important feature is that the ground state of erbium has large
quantum numbers. The bosonic erbium isotopes have a nuclear spin quantum
number I of zero. Therefore the used quantum number to describe transitions
in erbium is the total angular momentum quantum number J. For the groundstate this quantum number yields J = 6. The fermionic isotope 167 Er has a
nuclear spin quantum number I of 7/2, which leads to a hyperfine structure.
The ground state splits up to eight hyperfine states from F = 19/2 to F = 5/2.
The magnetic quantum numbers mJ are the projection of the total angularmomentum quantum number and range between -6 and 6.
In erbium the valence electrons couple in a LS-scheme to the 3 H6 term of
the ground state; see Fig. 2.1, leading to a large orbital angular momentumof L = 5. Due to the non-S ground state an anisotropy in the interaction of
atoms is expected [Kre04].

62.2. Energy spectrum of erbium atoms

2.2

Energy spectrum of erbium atoms

The rich and complex energy spectrum of erbium is shown in Fig. 2.1. From
the NIST Atomic Spectra Database [Ral11] there are 672 excited states knownwith J values ranging from 1 to 12. The energy of the excited states in relation to the ground state ranges from 5035 to 46971 cm−1 and the ionization
limit is 49262 cm−1 . Black and red lines indicate even and odd parity states,
respectively. Only levels with J values ranging from 5 to 7 and even parity cancouple to the ground state via optical dipole transitions. These transitions areshown in Fig. 2.2. The natural linewidths ∆ν of these states are located in awide range from the MHz to the subHz regime.
To describe the quantum states of erbium two different angular momenta
coupling schemes are needed. In the spin-orbit (LS-) coupling the individual
electronic orbital angular momenta ~l and spins ~s add to the total orbital an~ and to the total spin S,~ respectively. Finally they couple
gular momentum L~~~ [Dem05]. In erbium the 4f 12 6s2
to the total angular momentum J = L + S
electrons of the outer shells follow these coupling scheme forming the 3 H6term of the ground state1 . Other levels of erbium have to be described by
J1 J2 coupling. In this case the 4f and the 6s electrons couple independently
in a LS-coupling scheme. The formed states J~1 and J~2 add then up to a total
angular momentum.
For laser cooling of erbium atoms at our experiment we use the strong
401-nm transition and the narrow 583-nm transition. These transitions were
proposed in Ref. [Ban05]. The 401-nm line is used for the Zeeman slower (ZS)
and has a linewidth of 29.7 MHz; see Fig. 4.2. For the operation of the magnetooptical trap (MOT) we use the 583-nm line, which has a linewidth of ∆ν = 190
kHz. At this transition one of the valence electrons is excited to the 6p shell,
thereby changing its spin. This kind of transitions are forbidden in LS-coupling
and are called intercombination lines. This leads to the observed narrower
linewidth. For evaporative cooling we use an optical dipole trap (ODT) operating at a wavelength of 1064 nm. The contribution of every dipole allowedtransition to the trapping potential is discussed in Sec. 4.1.

The term symbol for this quantum state is (2s+1) LJ where L = 5 is correlated to theletter H.
17

2 ERBIUM

energy H1000 cm-1L

40 00030 000

20 000

10 000

4f 126s2 3H6 ground state

0
1

2

3

4

5

6

7

8

9

10

11

12

J

Figure 2.1: Full energy spectrum of erbium with total angular momentum
quantum numbers J ranging from 1 to 12. The energy is given with respect
to the ground state [Xe]4f 12 6s2 (3 H6 ). Red lines have even parity where blacklines have odd parity. The data were taken from [Ral11].

8

2.2. Energy spectrum of erbium atoms

4f 12H3H6L6s6pH1P10LH6,1L70

25 000

5000

H583
ght
T li
MO

nmL
H401
ZS l
ight

10 000nmL

4f 12H3H6L6s6pH3P10LH6,1L70

15 000ODT light H1064nmL

energy H1000 cm-1L

20 000

4f 126s2 3H6 ground state

0
5

6
J

7

Figure 2.2: Zoom of the energy levels relevant for laser cooling. The excitedlevel with odd parity (black line) can couple to the ground state 4f 12 6s2 (3 H6 )via optical dipole transition. The blue and the yellow arrow indicate the levels,
which are used for the deceleration of atoms in the ZS and the capturing ofatoms in the MOT, respectively. The red arrow shows the energy of the usedODT laser. The data were taken from [Ral11].

9

2 ERBIUM

2.3

Interaction between erbium atoms

For the achievement of quantum degeneracy the scattering properties playa crucial role. An important quantity is the background scattering length a,which is an intrinsic property of a specific isotope and describes the strength ofthe contact interaction between the atoms. For erbium, a is so far not known.
During this thesis work we could Bose condense the 168 Er isotope by means ofevaporative cooling. This already indicates a favorable background scattering
length a for experimental work2 . In theory the interaction is described by theisotropic van der Waals C6 value and scales with the interatomic distance R
as 1/R6 . For non-S state atoms, such as erbium, additionally to the isotropic
contact interaction anisotropic interactions are expected [Kre04].

Anisotropy in the interaction
In the case of dysprosium the anisotropy in the interaction was discussed by
Kotochigova et al. [Kot11]. For erbium a similar behavior is expected. It isshown that the van der Waals potential also has a long-range part, whichstrongly depends on mJ . This is a consequence of an anisotropic coupling of theopen f -shell electrons of the two atoms. Different projection quantum numbersof the total angular momentum lead to a spread of the C6 value. This spreadis defined by the anisotropic dispersion potential ∆C6 /R6 . Additionally there
can be found a magnetic dipole-dipole interaction (DDI) and an electrostatic
quadropole-quadropole interaction. The DDI scales as C3 /R3 where C3 has anangle dependence and will be discussed later.
Due to these interactions higher partial waves can couple with each other,which can lead to inelastic losses. Figure 2.3 shows that, depending on theinteratomic distance, different interactions are dominant. At small separation
the anisotropic interactions lead to mixing of rotational levels, where at large
R the Zeeman splitting dominates.
The presence of anisotropy also changes the nature of Feshbach resonances.
Due to the anisotropy couplings to bound states with non-zero l can occur.
Therefore, for submerged-shell atoms multiple Feshbach resonances are predicted [Pet12]. In our experiment we already could confirm this theoreticalprediction by finding not less than six resonances in a narrow magnetic-field
range up to 3 G; see Appendix C.
2Preliminary cross-dimensional thermalization measurements give a168 Er = 180 ± 50 a0 .

10

2.3. Interaction between erbium atoms

Figure 2.3: Level splitting due to interactions between two Dy atoms as afunction of their interatomic seperation. For short distances R of the atoms
the energy shifts of the anisotropic interactions are dominant over the Zeeman
shifts, leading to mJ -changing collisions. The graph is taken from [Kot11].

Magnetic properties
One of the anisotropic interactions giving rise to novel scattering behavior isthe magnetic dipole-dipole interaction, which is related to the magnetic dipole
moment. The ground state of erbium has a total angular momentum quantum
number J of 6 and a magnetic quantum number mJ of 6. This leads to a Landég-factor gJ of 1.16. The magnetic moment µ is calculated byµ = gJ mJ µB .

(2.1)

For the ground state of erbium atoms this gives µ = 6.98 µB , which is about
seven times larger than the magnetic moment of alkali atoms. For two polarizedmagnetic dipoles the DDI is defined as
Udd =

C3
µ0 µ2 1 − 3cos2 (θ)
=
,
|~r|3
4π
|~r|3

(2.2)

where ~r is the relative coordinate between the dipoles, θ is the angle between
the quantization axis and the interatomic axis, defined by ~r [Lah09] and µ0
is the magnetic constant. In contrast to the short range contact interaction,
which scales with 1/|~r|6 , the DDI scales with 1/|~r|3 , and is thus a long-range
11

2 ERBIUM
interaction. The anisotropy of the DDI is indicated by the dependence on theangle θ. When the dipoles are placed side-to-side they repel each other, whilea head-to-tail configuration leads to attraction. The characteristic length scale
for the DDI of erbium ismµ0 µ= 201 a0 ,
(2.3)
add =
4π~
where m is the mass of erbium and a0 is the Bohr radius [Bar08]. The large
mass of erbium enhances the dipolar character.
As discussed earlier due to the anisotropic DDI novel collisional behavior is expected, since also higher partial waves can couple with each other.
This coupling could, for example, induce collisions between ultracold identicalfermions, which are usually forbidden by the Pauli exclusion principle [Bar08].
This opens up new possibilities for experiments on degenerate Fermi gases.
To visualize the DDI one can "turn off" the contact interaction by tuning
the scattering length via Feshbach resonances, leading to a d-wave collaps ofthe atomic cloud. For the collapse the relative strength of the dipolar and thecontact interaction has to be compared. The collapse occurs when [Lah08]
dd =

add
≥ 1.
3a

(2.4)

As a proof-of-principle experiment this collapse was also shown at our experiment; see Appendix C.

Scattering experiments
Scattering experiments on submerged-shell atoms are a new and challenging
field. In the case of erbium the scattering behavior is only weakly known. In
2004 it was shown that erbium can successfully be trapped in a He buffer
gas cooled magnetic trap at temperatures below 1 K [Han04]. This indicated,that in Er-He collisions the interaction anisotropy can be suppressed by thespherically symmetric 6s-shell, allowing magnetic trapping and evaporative
cooling. However, this conclusion could not hold for Er-Er collisions. In 2010
large spin relaxation rates of erbium in a magnetic trap at temperatures near
500 mK were measured, implying that evaporative cooling in magnetic traps
will be highly inefficient [Con10]. In our experiment an optical dipole trap isused to evaporative cool erbium atoms. Chap. 3 gives a theoretical discussion
on dipole traps for erbium.
The goal of our experiment is to further investigate the scattering properties
of erbium. The observations of the dipolar collapse and the first Feshbachresonances lead in the right direction.
12

Chapter 3
Optical dipole trap
In recent years new atomic species have been used in ultracold experiments tostudy novel physical phenomena. It has been discovered that for some species
magnetic traps are not appropriate for loading and evaporative cooling ofatoms because of large collisional relaxation rates [Söd98, Con10]. This has
led to the investigation of optical dipole traps, which are nowadays a crucial
tool in many ultracold experiments. In this Chapter we will give an overview
on the basic mechanism at the foundation of the interaction between light field
and atoms, which is at the origin of a trapping potential. Furthermore, we will
discuss this interaction in the case of erbium, which is correlated to the atomic
polarizability1 of erbium. Finally, we will introduce the theoretical model that
we use to simulate the trapping potential at our Er-experiment.

3.1

Theoretical backgroundAn intuitive picture of the dipole force acting on the atoms can be obtainedby considering the case of a classical oscillator [Phi92]. By placing a neutral
atom in an oscillating electric field, e.g. laser field, the centers of mass of boththe positive and negative charge are separated by the field. This separation
leads to an induced electric dipole moment in the atom. The atom can thenbe treated as a dipole that interacts with the driving field. When the dipole
is driven below resonance it oscillates in phase with the field and when it isdriven above resonance it oscillates out of phase. Below (above) resonance theinteraction energy is negative (positive) and therefore the atom is roped into
(out of) the region with the highest intensity of the electric field. For atoms the1

The atomic polarizability describes the coupling strength of an atom to an electromagnetic field and can, therefore, be used to calculate the potential depth of an optical dipole
trap.

13

3 OPTICAL DIPOLE TRAP
resonance frequency is defined by the energy of the excited state with respect
to the ground state. We call the driving field red-detuned when its frequency
is below resonance and blue-detuned when it is above resonance. Laser beams
can be described by Gaussian beams, where the intensity profile has a Gaussian
shape. This means that the intensity is the highest in the center of the beamand decreases with radial and axial distance. Due to these facts one can createa trap by sending red detuned laser light on atoms. All atoms will be pushed
to the center of the beam and thus can minimize their energy.
Of course the dipole force can also be seen as an scattering process of an
atom and a photon. Usually the photon is absorbed by the atom and thenspontaneously re-emitted, each time leading to a momentum transfer. In contrast to normal scattering for far-detuned light absorption and emission cannot
be seen as independent events anymore. This is the correlation, which leads tothe non dissipative nature of the dipole force.
The mathematical discussion in the next section follows the review article
of Grimm et. al [Gri00].

3.1.1

The oscillator model~ r) with frequency ω induces an electric dipole moment in
An electric field E(~
an atom given by~ r).
p~(~r) = αE(~
(3.1)
α is the complex polarizability of the atom and depends on the atomic species,
the light frequency ω and the polarization of the electric field. From this rela~tion one can calculate the potential rising from the interaction of p~ and E:
1~ r)i = − 1 Re(α)I(~r).
Udip (~r) = − h~p(~r)E(~
2
20 c

(3.2)

~ r)|2 is used, 0 being theHere the relation for the field intensity I(~r) = 20 c|E(~
dielectric constant and c the speed of light in vacuum. The real part of thepolarizability accounts for the in-phase oscillation.
In addition to the generated potential the driven oscillator absorbs energy,
which is then re-emitted as dipolar radiation. One can describe this absorption
by the out of phase component, thus the imaginary part of the polarizability:~ = ω Im(α)I(~r).
Pabs = hp~˙Ei
(3.3)
0 c
This absorption can be seen as a photon scattering process and, therefore, thescattering rate can be written as
Γsc (~r) =

Pabs
1
=
Im(α)I(~r),
~ω
~0 c
14

(3.4)

3.1. Theoretical background~ being the Planck constant divided by 2π.
In general it is very difficult to calculate the atomic polarizability α, butthere exist theoretical models, which make this calculation more easy. For
the calculation one can either use the classical Lorentz-oscillator model or asemiclassical approach. In the classical model an electron is bound elastically tothe atom core and oscillates with the eigenfrequency ω0 , which is the opticaltransition frequency of a two-level atom. This oscillation is damped due toradiative energy loss. The damping rate is proportional to the spontaneous
decay rate Γ of the excited level. The spontaneous decay rate Γ is correlated to
the natural linewidth ∆ν with Γ = 2π∆ν. The semiclassical approach considers
an atom as a two-level quantum system in a classical radiation field. The case ofa far-detuned electric field is of particular interest for us, because the scattering
rate becomes very low and, therefore, heating processes can be neglected. In
this case both models give the same result and thus the atomic polarizability
can be extracted. For large detunings the following expressions for the dipole
potential and the scattering rate can be derived [Gri00]:


Γ
Γ
3πc2
+
I(~r),(3.5)
Udip (~r) = − 32ω0 ω0 − ω ω0 + ω 3 
2
3πc2 ω
ΓΓ+I(~r).
(3.6)
Γsc (~r) =2~ω03 ω0
ω0 − ω ω0 + ω
These two equations are probably the most important relations to understand the physics of dipole traps using far-detuned laser fields. Equation (3.5)
indicates, that the sign of the dipole potential depends on the sign of thedetuning ∆ = ω − ω0 (∆ < 0: red detuning; ∆ > 0: blue detuning). Thisis the mathematical explanation of the above-discussed "intuitive picture". If
the laser frequency is smaller (larger) than the atomic resonance frequency, thedipole potential is negative (positive) and, therefore, atoms are located at theregion with maximum (minimum) intensity of the laser beam. The comparisonof Eq. (3.5) and Eq. (3.6) also shows why usually far-detuned laser light is used
for optical dipole traps. As Udip scales with ∆I and Γsc scales with ∆I2 it is clearthat the scattering rate can be suppressed for large detunings.

3.1.2

Multilevel atoms

The following model provides a picture on the interaction between a lightfield and an atom, which can be used for atoms with more than two internalenergy states. To extend the above discussed model to a multilevel quantum
system we have to introduce the influence of a far-detuned electric field on all
15

3 OPTICAL DIPOLE TRAP
the atomic levels. Mathematically this influence is treated as a second-ordertime-independent perturbation with interaction Hamiltonian H1 = µE, whereµ = −e~r represents the electric dipole operator.

Two-level system
For a two-level atom the coupling to the electric field leads to an energy shiftof the ground state:|he|µ|gi|2 2
|E| ,
(3.7)
∆E =
∆
which is nothing else than the ac-Stark shift. For the excited state the shift
is the same but with opposite sign. |he|µ|gi| is the dipole matrix element between the ground state |gi and the excited state |ei and corresponds to thespontaneous decay rate Γ of the excited state:Γ=ω03|he|µ|gi|2 .
3π0 ~c3(3.8)

Because of these relations the energy shift by the light, corresponding also to
the dipole potential seen by the atoms, only depends on the spontaneous decayrate Γ, the detuning ∆, and the intensity I of the laser light. Figure 3.1(b)shows the ac-Stark shift for a two-level atom due to an electric field. Applying
a Gaussian laser beam generates a trapping potential for atoms in their groundstate as depicted in 3.1(c).

Multi-level system
For a multilevel atom all allowed transitions from a ground state |gi i to an
excited state |ej i are involved. Dipole transitions are only allowed when the
total angular momentum quantum number J changes by 0, ±1 with respect tothe ground state and the parity of the excited state is different2 [Dra96]. For
the contribution of each excited state to the ac-Stark shift, furthermore, thereal transition coefficients cij have to be taken into account. These transitiondependent factors determine the specific dipole matrix elementsµij = cij ||µ||,(3.9)

where ||µ|| is the fully reduced matrix element, which is related to the spontaneous decay rate Γ according to Eq. (3.8). The transition coefficients cij , which
2

The parity gives information about how the electronic wave function ψ(~r) is inverted
through the origin. For ψ(−~r) = ψ(~r) the parity is even, where for ψ(−~r) = −ψ(~r) theparity is odd.

16

3.1. Theoretical background

(a)

(b)

(c)

𝐸

𝐼=0

𝐼≠0

𝐼 = 𝐼0 𝑒

2𝑟2

−𝑤

0

Δ𝐸𝑒

𝑒

ℏ𝜔

𝑔

Δ𝐸𝑔

Figure 3.1: Schematic of the ac-Stark shift of a two-level atom for red-detuned
light with intensity I (b). ∆Eg and ∆Ee are the energy shifts of the groundstate |gi and the excited state |ei, respectively. In (c) the potential seen by anatom in a Gaussian beam with peak intensity I0 and beam waist w0 is shown.

consider the coupling strength between specific sub-levels |gi i and |ej i, dependon the laser polarization and the electric angular momenta involved. They canbe calculated with the use of the 3J-symbol to2
Ji 1 Jj2
cij = (2Jj + 1),
(3.10)
mi p −mjwhere Ji and Jj are the total angular momenta of the states |gi i and |ej i,respectively. The parameters mi and mj are the corresponding magnetic quantum numbers and p denotes the laser polarization. Equation (3.10) is adaptedfrom Ref. [Gra07]. The 3J-symbol is only non zero when the relation
mi + p = mj

(3.11)

is fulfilled. For linear π and circular σ ± polarized light p is equal to 0 and±1, respectively. The relation indicates that for linear polarized light, e. g. , a
17

3 OPTICAL DIPOLE TRAP
ground state in the stretched state mi = −Ji will not couple to an excitedstate with Jj = Ji − 1, since in this case the magnetic quantum numbers
mi and mj cannot be equal. For different laser polarization different excitedstates contribute to the ac-Stark shift. The line strength factors are included
by defining a new variable Γ̃jΓ̃j = c2ij Γj = (2Jj + 1)

Ji 1 Jjmi p −mj2
Γj ,(3.12)

where Γj is the spontaneous decay rate of the corresponding excited state |ej i.
The overall energy shift of a state |gi i is given by the sum over the contributions of each coupled excited state |ej i according to the detuning (ω − ωij ),the spontaneous decay rate Γj and the line strength factors c2ij . This leads tothe following generalization of Eqs. (3.5) and (3.6):!
X 3πc2Γ̃jΓ̃j+I(~r),(3.13)
Udip (~r) = −2ωij3 ωij − ω ωij + ωjΓsc (~r) =X 3πc2  ω 3j2~ωij3

ωij

Γ̃j
Γ̃j
+
ωij − ω ωij + ω!2I(~r).

(3.14)

To calculate the trapping potential and the scattering rate the intensity
distribution of the electric field has to be known. The simplest realization ofan optical dipole trap is to focus a single red-detuned laser beam onto theatoms. In Gaussian optics [Sal91], the intensity of a laser beam propagating
along the z-axis with radial symmetry r is described by


r2
2P
exp −2 2(3.15)
I(r, z) =πw2 (z)w (z)with P being the total power3 . w(z) is the 1/e2 beam radius4 and it is definedby the minimal beam waist w0 , the axial coordinate z, and the Rayleigh range
zR :
s
z2
w(z) = w0 1 + 2 .
(3.16)zR
3

P can be calculated by integrating the intensity I(r) over the whole area in polar
coordinates.
4
This means that within a circle of radius r = w(z) approximately 86% of the power iscontained.

18

3.2. The case of erbium
2
The Rayleigh range zR = πw
√0 /λ describes the distance over which the waist of
the beam has increased to 2w0 . Equation (3.15) indicates that the intensity
is the highest in the focal point at the center of the beam.

I0 = I(0, 0) =

2P
πw02

(3.17)

This peak intensity I0 and the polarizability Ũ define the total trap depth!
X 3πc2Γ̃jΓ̃jÛ = −+
I(0, 0) = Ũ I0 .
(3.18)
3
2ω
ω
−
ω
ω
+
ω
ij
ij
ij
j
For convenience we express the polarizability in units of 20 c5 . In literature it
is often distinguished between the scalar and the tensorial part of the polarizability. The formula given in this thesis includes already both, the scalar andthe tensorial part.

3.2

The case of erbium

As discussed previously, the total trapping potential is determined by the contribution of every dipole allowed transition according to the detunings ∆ij ,the spontaneous decay rates Γj and the line strength factors c2ij . In the case of
erbium the calculation of the polarizability is difficult, since this rare-earth element has such a complicated level structure; see Sec. 2.2. To provide a complete
set of excited states and their corresponding linewidths precise spectroscopy
and complex calculations are necessary. Up-to-date still many lines in erbium
are not known.
Due to new data from J.-F. Wyart and the group of O. Dulieu [Law10,
Wya12], we can calculate the polarizability for the ground state of erbium
atoms, as well as for the 583-nm and 401-nm state. For these calculations we
use linear polarized light at 1064 nm and assume that the atomic sample isprepared in the stretched state mi = −Ji :Ũ|gi = −4.9384 · 10−37 Jm2 /W,
Ũ583 = −9.0164 · 10−39 Jm2 /W,
Ũ401 = −1.3633 · 10−38 Jm2 /W.
5

According to Eq. (3.2) Ũ is correlated to the complex polarizability α with Ũ =1
20 c Re(α).

193 OPTICAL DIPOLE TRAP
In atomic units the atomic polarizabilities are Re(α|gi ) = 159.01 a.u., Re(α583 ) =2.90 a.u. and Re(α401 ) = 4.39 a.u. The resulting potential depths for a Gaussianlaser beam with a 30 µm waist and a power of 10 W are:Û|gi = −253.01 µK,
Û583 = −4.62 µK,
Û401 = −6.98 µK.
In Chap. 4 we will focus deeply on the calculation of the polarizability and will
also discuss the influence of trapping laser light with a different wavelength.

3.3Modeling of the trapping potential

On the route to quantum degeneracy, the design of an optimal dipole trap
plays a crucial role. The most important parameters of an optical trap are thetrap depths and the trap frequencies, which give insight on the confinement ineach spatial direction. The confinement can be manipulated by changing thepower, the waist or the propagation direction of the laser beams. For a single,
circular beam the calculation of the relevant parameters is straightforward.
But a single beam cannot create a strong confinement in the axial direction.
In our experiment we cross two elliptic beams within a specific angle to create
the dipole trap. The angle is not avoidable, because the optical access to thechamber is limited. The ellipticity of the horizontal laser beam is tunable as
will be shown in Sec. 5.1.
A major part of my thesis work was about setting up a calculation6 todesign the optimal trapping potential for our experiment. This model can also
be adapted to any other atomic element and calculates trapping parameters
for arbitrary crossed elliptic beams.
In this Section, we will follow a step-by-step mathematical approach tomodel optical dipole traps and derive relevant parameters. We will start byconsidering a single circular laser beam. We will then allow the beam to be
elliptic. Further we will consider the case of two beams crossing at an angle of90◦ . Then we will describe the trapping geometry for a crossed beam dipole trap
intersecting at an arbitrary angle. Finally we will take gravity into account,which plays an important role for shallow trapping potentials.
In a last step, we will use this model to simulate the trapping potential of
the used dipole trap in our experiment.
6

The source code of this file can be found in Appendix B (Program version: Wolfram
Mathematica 8.0)

20

3.3. Modeling of the trapping potential

Single circular beam
The trapping potential of a dipole trap was already discussed in Sec. 3.1.2. It
is defined as:
Udip = Ũ I(~r),
where Ũ is the atomic polarizability and I(~r) is the laser intensity. For a circular
Gaussian beam it has the following form:


2P
r2
I(r, z) =exp −2 2.
(3.19)
πw2 (z)w (z)For small atomic clouds compared to the size of the laser beam it is possible
to use a harmonic approximation to extract the trap frequencies. This is areasonable assumption, since atoms populate only the center of the trap. The
harmonic approximation gives for the potential of a single circular beam:
1 ∂ 2 Udipr2z2
1 ∂ 2 Udip22(0)r +(0)z = Û 1 − 2 2 − 2 .
(3.20)
Ucircular ≈2 ∂r2
2 ∂z 2w0 zR
Here w0 represents the minimal beam waist at the focal point, zR is theRayleigh range and Û = Ũ I(0) is the total trap depth. By comparing Eq. (3.20)
to the classical harmonic potential 12 m(ωr2 r2 + ωz2 z 2 ) one finds for the trap frequencies in radial and axial direction:
s
s
4Û
2Û
ωr ≡
, ωz ≡.
(3.21)
2
mw0
mzR2
The axial confinement is much weaker than the radial confinement, becausezR √is much larger than w0 and the axial trap frequency is smaller by a factorof 2. Because Û is proportional to P/w02 and zR ∝ w02 , the dependence of thetrap frequencies can also be written as√√ωr ∝ P /w02 , ωa ∝ P /w02 .
(3.22)

Single elliptic beam
For the description of an elliptic beam one has to switch to Cartesian coordinates. The axes definition are the same as for the horizontal beam in Fig. 3.2.
As already seen in Eq. (3.19) the peak intensity is in principle nothing elsethan the total power divided by the area of the beam. For an elliptic beam thewaists in the radial directions x and y are different and defined as horizontal
21

3 OPTICAL DIPOLE TRAP
waist wx (z) and vertical waist wy (z). Therefore, the total cross section of thelaser beam changes, and leads to the following intensity profile:
  2

2P
x
y2
I(x, y, z) =exp −2+.
(3.23)
πwx (z)wy (z)wx2 (z) wy2 (z)The harmonic approximation of the trapping potential gives"!#
2x2 2y 2 1 2
1
1
Uelliptic ≈ Û 1 − 2 − 2 − z
+
,
wx
wy
2
zR2 x zR2 y(3.24)

where wx and wy are the minimal beam waists along the radial directions x
and y. zRx and zRy are the corresponding Rayleigh ranges along the axial z
direction and
2PÛ = Ũπwx wyis the total trap depth. For convenience we define a new Rayleigh range zRell :zRell = qzRx zRy.

1 2(z + zR2 y )2 Rx

Finally, the trap frequencies for a single elliptic beam read as
s
s
s
4Û
4Û
2Û
,
ω
≡
,
ω
≡
.
ωx ≡
y
z
mwx2
mwy2
mzR2 ell(3.25)(3.26)

Crossed beams at 90◦
The crossing of two beams in principle only adds another dipole trap potential
to the system. Here we define the propagation direction of the second beamalong the y-axis (axis of gravity), which is shown schematically in Fig. 3.2.
The first beam is defined as horizontal beam and the second beam as verticalbeam. The new trapping potential of the vertical beam is added to the trapping
potential of the horizontal beam and leads to the following total potential:
2x2
2y 2 1 z 2Ucrossed ≈ Û1 1 − 2 − 2 −) −w1x w1y 2 zR1,ell

2x2
2z 2 1 y 2− Û2 1 − 2 − 2 −) .
(3.27)
w2x w2z 2 zR2,ell
The definition w1i (w2i ) indicates the quantity for the minimal beam waist
along the axis i = x, y, z of the horizontal (vertical) beam. zR1,ell and zR2,ell
22

3.3. Modeling of the trapping potentialyvertical beamxz

horizontal beam

Figure 3.2: Schematic of two dipole trap beams crossed at an angle of 90◦ .
The horizontal beam propagates along the z-axis and the vertical beam along
the y-axis.

are the corresponding Rayleigh ranges of the horizontal and the vertical beam,
respectively. Û1 and Û2 are the total trap depths for these beams. The harmonic
approximation gives the following trap frequencies:
v
!
u
u4
Û
Û
2
1
+ 2 ,
ωx,crossed ≡ t
2
m w1x
w2x
v
!
u
u 1 4Û12
Û
2
ωy,crossed ≡ t+ 2,(3.28)
2m w1y
zR2,ellv!
u
u 1 4Û2
2
Û
1
ωz,crossed ≡ t
+ 2
.
2
m w2z
zR1,ell
These equations point out two things:• In the case of similar beam waists and powers of the two laser beams theconfinement in the x-direction is the highest, because the radial confinement of each beam is added up.
• The confinement in the axial direction of one beam is mainly given by
the second beam.
In the experiment the weakest confining axis plays the most significant role,
because atoms can be lost in this direction. Therefore, the effective potential23

3 OPTICAL DIPOLE TRAP
depth is always determined by the axis with the smallest trap frequency.

Crossed beams at an angle
When the vertical beam is rotated, the confinement is also changed. The rotation can be described by two angles α and β. Their definition is given in Fig. 3.3.
When both angles are 0◦ this model matches the discussion for crossed beams
at 90◦ .
Tilting one beam does not change the total trap depth, but the influence ofthe beam waists and the Rayleigh range on each axis on the trap frequencies isdifferent. To investigate this change we describe the trap geometry created by
2
2
the beam propagating along the y-axis as an ellipse xa2 + yb2 = 1 with boundaries
a = w2z and b = zR2,ell . When this ellipse is rotated to the used coordinate
system at an angle α, the new boundaries w̃0 and z̃R on the x-, and y-axis
have the following form:vu
1w̃2z (α) = u,t cos2 (α)
sin2 (α)
+ 2z
2
2
w2z
R2,ellv
u1(3.29)z̃R2,ell (α) = ut sin2 (α) cos2 (α) .
+
w2
2z 2
2z

R2,ellThe factor 2 at zR2,ell includes that the confinement in the axial direction is
weaker than the radial confinement, which was already mentioned in Eq. (3.21).
Figure 3.3(a) shows the tilt of the vertical beam to the axis of gravity, i.e. yz-plane, and Fig. 3.3(b) indicates the rotation of the vertical beam in theplane perpendicular to the axis of gravity, i.e. x-z-plane. The calculation of
the adapted boundaries for the angle β is done analogously. The only difference is that for this angle w2x and w2z have to be adapted. Note that forthe calculation for angle β the new boundary from the calculation with angle α, hence w̃2z (α), has to be taken into account. As a consequence the trap24

3.3. Modeling of the trapping potential

Figure 3.3: Definition of the angles embedded between the two beams. The
horizontal beam (beam 1) propagates along the z-axis. The vertical beam
(beam 2) in principle propagates along the y-axis. α defines the angle of thevertical beam to the axis of gravity in the plane spanned by the propagation
direction of the horizontal beam and the direction of gravity. If α would be
90◦ the vertical beam would overlap with the horizontal beam. β defines therotation of the vertical to the horizontal beam in the plane perpendicular to
the axis of gravity. The change of the effective waists and the Rayleigh rangeof the vertical beam in the x-, y- and z-axis due to these two angles cannot becalculated independently.

frequencies transform to:
ωxαβ

ωyαβ

ωzαβ

v
u
u4
≡ t
m
v
u
u4
≡ t
m
v
u
u1
≡ t
m

!
Û1
Û2
+ 2
,
2
w1x
w̃2x (α, β)
!
Û1
Û2
+ 2
,
2
w1y
z̃R2,ell (α)
4Û2
2Û1
+ 2
2
w̃2z (α, β) zR1,ell(3.30)!
.

How the beam waists and the Rayleigh range of the tilted vertical beam are
actually calculated can be found in Appendix B.

25

3 OPTICAL DIPOLE TRAP

(a)
60

40

20

20

Udip gravity µK

Udip + Ugravity (µK)(b)

dip +gravity
UUdip
UgravityµK(µK)

4060yy (µm)
µm

10

50

5

100
60

150

40

20

20

40

60

µm
yy (µm)

5

200

10

Ueffective

250

Figure 3.4: Influence of gravity on the trapping potential for different powersof the trapping laser beam along the plane x = z = 0. For this simulation we
use a single circular beam with a waist w0 of 30 µm and a power P of (a) 10 W
and (b) 0.3 W. The trap depths correspond to the polarizability of bosonicerbium atoms. This simulation shows that the effect of gravity only becomes
crucial for shallow trapping potentials.

Gravity
For low trap depths the gravitational force (Fgrav = mg) can push the atoms
out of the trap. m is the mass of the atom and g is the gravitational acceleration. It is important to know at which confinement this effect becomes relevant.
Only the trap frequency ωy has to be adapted.
Adding the gravitational potential (Ugrav = mgy) to the dipole potential(Udip ) makes the effect visible. In Fig. 3.4 the tilt of the potential is shown for
two different trapping configuration of a single beam. Figure (a) is for a 10 W
circular laser beam with a waist w0 of 30 µm and Fig. (b) shows the same beam
for a power of 0.3 W. The shown trap depths are calculated for erbium atoms.
The change of the trap frequency along the axis of gravity is described by thefollowing equation:v!
uu 4 Û1, gravity Û2, gravity+ 2,(3.31)
ωyαβ, gravity ≡ t2mw1yz̃R2,ell (α)
where Û1, gravity (Û2, gravity ) describes the reduced effective trap depth of thehorizontal (vertical) dipole trap beam due to the gravitational potential. The
reduced trap depths are found numerically. Therefore, we compare the total
trap depth with the maximum value of the combined potential in the directionof gravity. If the trapping potential is less steep than the potential of gravity,
all atoms are lost out of the trap.
26

3.3. Modeling of the trapping potential

Figure 3.5: Scheme of the cooling and trapping setup in our experiment. For
the Zeeman slower we use the strong 401-nm transition. The MOT operates on
the narrow 583-nm transition. For the dipole trap we use 1064-nm laser lightin a crossed configuration.

Real trap
In our experiment we use two crossed laser beams to create an optical dipole
trap. The vertical beam has an angle of α = 15◦ as defined in Fig. 3.3. This
angle is because of the limited space around the viewport, which is used forthe vertical beam due to the MOT operation. Additionally the vertical beamis rotated by the angle β = 14◦ in the plane perpendicular to the axis ofgravity. This angle is because of the angle of the viewport, which is used forthe horizontal dipole trap beam. The scheme of our cooling and trapping setup
can be found in Fig. 3.5. The experimental implementation of the dipole trap
beams will be discussed in detail in Sec. 5.1.3.
Figure 3.6 shows a simulation of the real trapping potential in our experiment. For this simulation we use a power of 1 W for the horizontal beam, anda power of 10 W for the vertical beam. The waists of the beams are specified
in the caption of Fig. 3.6.
27

3 OPTICAL DIPOLE TRAP

Figure 3.6: Trap potential of the crossed dipole trap in our experiment in all three coordinates. In this simulation
the horizontal beam has a power of 1 W, a horizontal waist (x-axis) of 27 µm, and a vertical waist (y-axis) of 16 µm.
The vertical beam crosses the horizontal beam under an angle α = 15◦ and β = 14◦ because of the limited opticalaccess in our experiment. The vertical beam has a power of 10 W, a horizontal waist (close to x-axis) of 110 µm, anda vertical waist (close to z-axis) of 55 µm. The direction of gravity coincides with the y-axis.

28

Chapter 4
Atomic polarizability of erbium
In this Chapter, we focus on the atomic polarizability of bosonic erbium atoms.
For the bosonic isotopes the nuclear quantum number is zero and, therefore, nohyperfine splitting is observed. We briefly introduce the up-to-date knowledgeof the energy level structure of erbium. Further, we give an estimate for thedepth of the dipole trap potential for the ground state of erbium atoms. Additionally, we discuss the polarizability of the narrow-line transition at 583 nm
and the strong 401-nm transition, which we use for the operation of the MOT
and the Zeeman slower, respectively. We show the dependence of the trapping
potential on the laser frequency ω and how the scattering rate depends on
ω. Finally, we calculate the relevant parameters of an optical dipole trap for
erbium atoms for a selection of technically feasible laser wavelengths.

4.1

Contribution of different energy levels

Erbium has a very rich energy spectrum; see Sec. 2.2. The electronic groundstate of erbium is in a [Xe]4f 12 6s2 (3 H6 ) configuration. This state has a total
angular momentum quantum number J of 6 and an even parity. From the NIST
Atomic Spectra Database [Ral11] there are 672 excited states known with J
values ranging from 1 to 12. The energy of the excited states in relation to
the ground state ranges from 5035 to 46971 cm−1 . From these levels only 144
can couple to the ground state via optical dipole transition; see Sec. 3.1.2. In
the case of erbium electrons can only be excited to levels with odd parity andJ values ranging from 5 to 7. These dipole allowed transitions are shown in
Fig. 4.1.
We have an intense collaboration with J.-F. Wyart and the group of O. Dulieu
at the Laboratoire Aimé Cotton in Orsay, France, which have derived transi29

4 ATOMIC POLARIZABILITY OF ERBIUM

40 000energy H1000 cm-1L

30 000

20 000ODT light H1064nmL

10 000

4f 126s2 3H6 ground state

0
5

6
J

7

Figure 4.1: Energy of all levels which can couple to the ground state
4f 12 6s2 (3 H6 ) due to the selection rules for electric dipole interaction. The
shown energy spectrum is incomplete, since to date not all energy levels in
erbium are known. The red arrow shows the energy of the used dipole trap
laser. For a low scattering rate of dipole trap photons it is important, thatthe detuning to the excited states is large. In particular, the detuning to thestrongest transition should be as large as possible. In our experiment a 1064nm laser source was chosen for the creation of the dipole trap. The data weretaken from [Ral11].

tion probabilities for many levels of erbium [Law10]. The data are the resultof combined spectroscopic measurements and calculations. Experimental data
are derived by combining branching fractions measured with a Fourier transform spectrometer with radiative lifetimes from time-resolved-laser-inducedfluorescence measurements. The calculations use parametric fits based on the
Cowan code [Cow81] and can also predict unobserved levels. The calculations
are than rescaled by using the observed levels as a reference value. The theoretical energies can be extracted from fitted parameters. Those values minimize
the root mean square of the difference between the experimentally observed30

4.1. Contribution of different energy levels
1 6 6

E r 2 9 .1 8 (2 1 ) M H z
E r 3 0 .2 9 (4 1 ) M H z
1 7 0
E r 2 9 .7 3 (2 2 ) M H z

2 .5 x 1 0 7

1 6 8

a to m n u m b e r

2 ,0

c o m b in e d∆ν4 0 1 = 2 9 . 7 4 ( 1 7 ) M H z

1 ,5

1 ,0

0 ,5

0 ,0
-4 0

-2 0

0

2 0

4 0

6 0

d e t u n in g ( M H z )Figure 4.2: Natural linewidth ∆ν401 of the strong 401-nm level. The quantity∆ν401 = 29.74(2) MHz was measured in our experiment by using an absorption
imaging technique. The atom number was measure versus the detuning of theimaging light for three different isotopes (166 Er, 168 Er, 170 Er). The data werefitted by a Lorentzian function and the arithmetic mean of the linewidths istaken to extract the actual linewidth. The temperature for the measurements
was in the range between 11 and 15 µK.

and the theoretically calculated energies of the excited states. This is done forall considered experimental energies.
Newest calculations predict 1176 excited levels that can couple to theground state [Wya12]. This new data set provides indeed more transitions,however, compared to [Ral11] there are also levels missing1 . The new set ofdata also includes the experimental values for the linewidths ∆ν = Γ/2π ofthe 401-, and the 583-nm state. Because of the complexity of the atomic spectrum, the knowledge of the transition linewidths is not very precise. For instance, even one of the strongest erbium transitions, located at 401 nm, wasoverestimated by 20 %. The group of McClelland et al. [McC06a] has measured
a linewidth of Γ401 /2π = 35.6 MHz. This measurement was done via fluorescence spectroscopy on atomic beams. Lawler et al. [Law10] found a value ofΓ401 /2π = 29.5 MHz from lifetime measurements. We measure a linewidth of
29.74(2) MHz by using an absorption imaging technique on magneto-opticallytrapped atoms; see Fig. 4.2. This little historical excursus shows exemplary
how difficult it is to determine all the levels in erbium and their corresponding
linewidths precisely. The linewidth of the 583-nm state is Γ583 /2π = 190 kHz.
1
In the new set of data transitions with linewidths below 100 Hz are excluded, which
explains the missing levels.

31

4 ATOMIC POLARIZABILITY OF ERBIUM

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401-nm level

æ

æ

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æ

200

æ

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400

600

800

Energy (nm)Figure 4.3: Natural linewidth ∆ν of all relevant excited states versus thecorresponding energy difference between the ground state |gi and each excitedstate |ei [Wya12].

Figure 4.3 shows the energies of the 1176 states plotted to their corresponding natural linewidths ∆ν. The strong 401-nm transition has the broadest
linewidth. The detuning of the laser trapping light is always given with respect to this level because it is the strongest transition in erbium. Dipole traps
with lower (higher) laser energies relative to this level are called red-detuned(blue-detuned) traps. Most of the excited levels in erbium are located in thehigh energy2 region below 300 nm. From these data one can already realize thata laser emitting at a wavelength above 900 nm will give a low scattering rate,
because the detuning to most of the levels is large. A more detailed discussion
on the dependence of the dipole force on the used laser wavelength is given inSec. 4.2.
As previously discussed, all the dipole allowed transitions contribute to theoverall trapping potential according to their line strength factors. However,the polarizability strongly depends on the linewidth ∆ν, making the weightof narrow transitions less an less important; see discussion below. Based on
the new data set, we derive the atomic polarizability following Eq. (3.18). For
linear polarized laser light at 1064 nm, we findŨ|gi = −4.9384 · 10−37 Jm2 /W.
2The energy E is correlated to the wavelength λ with E = ~c
λ .

32

4.1. Contribution of different energy levels
In atomic units3 this value converts to:Re(α|gi ) = 159.01 a.u.
The derived value already takes the scalar and tensorial part of the polarizability into account. For the ground state of erbium atoms the tensorial partdoes not have a big influence on the polarizability. We can now calculate theresulting dipole potential and potential depth. For a Gaussian laser beam of
30 µm waist and a power of 10 W, we find a potential depth for atoms in the
ground state ofÛ|gi = −253.01 µK4 .
Figure 4.4 shows the contribution of all dipole allowed transitions to the overall
trapping potential of the ground state. In Sec.6.1.2 we use trap frequency
measurements to extract the dynamic polarizability of erbium experimentallyand compare it to the derived theoretical value.
There are 4 (34) excited states with a natural linewidth ∆ν > 10 (1) MHz.
The coupling to these states already gives a contribution to the trap depth of
62.7 (84.1) %. The contribution of the strongest line (401-nm) is 11.1 %. Thisshows that mainly the strong lines contribute to the dipole trap potential.
For a deeper understanding of the trapping behavior of erbium atoms, it isnecessary to know the polarizability of specific excited states. From particularinterest are the 583-nm and the 401-nm state. These are the transitions at
which the MOT and the Zeeman slower are operated and hence atoms canpopulate these levels. For these calculations the excited state is treated as aground state, where all allowed transitions of this state can contribute to theenergy shift. Energy levels located below this excited state will increase theenergy, while energy levels located above this state will decrease the energy.
Fig. 4.5 shows the resulting shift of the 583-nm state due to the laser light.
The shift of the 401-nm state is shown in Fig. 4.6. The dynamic polarizabilities
for a linear polarized 1064-nm laser light are found to be:Ũ583 = −9.0164 · 10−39 Jm2 /W,
Ũ401 = −1.3633 · 10−38 Jm2 /W.
In atomic units this corresponds toRe(α583 ) = 2.90 a.u.,
The conversion factor to atomic units is Re(α) = −Ũ 20 c · /(1.6488 · 10−41 ); see Eq. (3.2)
and [Sch06].
4
To convert the dipole potential to the temperature scale, one has to divide the effective
potential depth by the Boltzmann constant kB .
333

4 ATOMIC POLARIZABILITY OF ERBIUM
0-50
Udip HµKL

-100
-150
-200
-250
-60

-40

-20

0
Radius HµmL

20

40

60

Figure 4.4: Dipole trap depth for a spherical 1064 nm laser beam with waist
w = 30 µm and power P = 10 W for the ground state of erbium atoms. The blue
curves are the contribution to the trap depth of each dipole allowed transition,
which is related to their linewidths and line strength factors. The purple curveis the total trapping potential.

Re(α401 ) = 4.39 a.u..
For the modeled laser beam the polarizabilities correspond to total trap depthsofÛ583 = −4.62 µK,
Û401 = −6.98 µK.
Unfortunately for these states only eight and three transitions are known,
respectively [Law10]. Therefore, the calculated polarizabilities might not becorrect. For the excited states it is expected that the tensorial part of thepolarizabilities plays a more significant role. The scalar part only depends on
the intensity of the electric field, whereas the tensorial part also depends onthe direction of the electric field compared to the direction of the quantization
axis of the atoms [Ang68]. The influence of the tensorial part on the total
polarizability can be huge, as shown recently for 40 K87 Rb molecules [Ney12].
To determine a complete set of dipole allowed transitions of the discussed
excited states will be part of future theoretical investigations.

34

4.2. Dependence on the laser wavelength0

Udip HµKL

-1

-2

-3

-4

-60

-40

-20

0

Radius HµmL

20

40

60

Figure 4.5: Dipole trap depth for a spherical 1064 nm laser with waist w =30 µm and power P = 10 W for the 583-nm state of erbium atoms. The blue
curves are the contribution to the trap depth of each dipole allowed transition
to the 583-nm state. The purple curve is the total trapping potential.
0-1

Udip HµKL

-2
-3
-4
-5
-6
-7
-60

-40

-20

0

Radius HµmL

20

40

60

Figure 4.6: Dipole trap depth for a spherical 1064 nm laser with waist w =30 µm and power P = 10 W for the 401-nm state of erbium atoms. The blue
curves are the contribution to the trap depth of each dipole allowed transition
to the 401-nm state. The purple curve is the total trapping potential.

4.2

Dependence on the laser wavelength

To calculate the dependence of the total trap depth Ûdip on the used laser wavelength λlaser we use Eq. (3.13) and change λlaser = 2πcfrom 200 to 1100 nm.
ωThe result is shown in Fig. 4.7. In the red-detuned region (700-1100 nm) the35

4 ATOMIC POLARIZABILITY OF ERBIUM
0-100

Udip HµKL

-200
-300
-400
-500
-600
-700
200

400

600
Λlaser HnmL

800

1000

Figure 4.7: Dependence of the total trapdepth Ûdip on the used laser frequency λlaser ranging from 200 to 1100 nm. For this calculation the model ofa single beam ODT with 30 µm waist and 10 W is used. This example gives
an estimate about the created trap depth in µK. The vertical lines show theasymptotic behavior of Ûdip when the wavelength is resonant to an atomic
transition.

total trap depth increases for smaller detunings as expected from Eq. (3.18).
Every time the wavelength coincides with one of the levels in erbium the contribution of this level to the total trap depth approaches infinity. This leads toan asymptotic behavior when the energy of the laser is close to an excited state.
For wavelengths between 200-700 nm the behavior of the total trap depth gets
more complicated. In this region the laser is to some lines red-detuned andto the other lines blue-detuned, leading to a deepening or steepening of thetrapping potential. Below the strong 401-nm line the trap depth is positive,
leading to the repulsive trapping behavior, as discussed in Sec. 3.1.
Furthermore, we simulate the dependence of the scattering rate Γsc on thelaser wavelength using Eq. (3.14). Figure 4.8 shows the strong 1/∆2 correlationof the scattering rate on the detuning. In the region close to the 401-nm line
the scattering rate gets extremely large, making it impossible to trap atoms
by means of optical dipole traps.
For experimental reasons the heating rate of the laser, which stems fromthe scattering of photons generating the dipole trap, is from interest. In the
far-detuned case this scattering is an elastic process. Therefore, the energy
transfer is determined only by the energy ~klaser of the scattered photon, whichis related to the wavelength of the laser. The overall heating increases the total
thermal energy by 2Erec per scattering event, where Erec = (~klaser )2 /2m is the36

4.2. Dependence on the laser wavelength
0.30

0.25

Gsc H1sL

0.20

0.15

0.10

0.05

0.00
400

500

600

700
800
Λlaser HnmL

900

1000

1100

Figure 4.8: Dependence of the scattering rate Γsc on the used laser wavelength
λlaser ranging from 400 to 1100 nm. For this calculation the model of a single
beam ODT with 30 µm waist and 10 W is used. The smaller the detuning to
the main transition (401-nm line) is, the larger becomes the scattering rate. For
wavelengths below this line (blue detuning) the scattering rate is not relevant.
In this blue-detuned case atoms are pushed out of the laser beam, as discussed
in Sec. 3.1. Therefore, scattering processes hardly occur.

recoil energy and is related to the recoil temperature by Trec = 2Erec /kB . For
a red-detuned 3D harmonic trap the heating rate can be expressed as [Gri00]:
1
Ṫred = Trec Γsc .
(4.1)
3
In the blue-detuned case atoms are located in the intensity minimum and thepotential depth is determined by the height of the surrounding walls. Such aconfinement can be created for example by a blue-detuned optical lattice. The
heating rate Ṫblue is given by [Gri00]
Ṫblue =3 kB TṪred ,
2 Û

(4.2)

where Û is the total trap depth and T is the temperature of the trapped atomic
sample. In Tab. 4.1 we simulate different dipole trap parameters for a single
spherical beam with 30 µm waist, 10 W power, and an atom cloud temperatureof 1 µK for different laser wavelengths. The considered wavelengths are notresonant with one of the excited states in erbium and are technically feasible.
For each laser wavelength we calculate the recoil temperature, the dynamic
polarizability, the effective potential depth, the scattering rate and the heating
rate for the red- and the blue-detuned case.
37

4 ATOMIC POLARIZABILITY OF ERBIUM

potential
depth Udip(µK)

scattering
rate Γsc (1/s)

heating rate
red detunedṪred (nK/s)

heating rate
blue detuned
Ṫblue (nK/s)

Table 4.1: Calculation of dipole trap parameters for a selection of technically relevant laser wavelengths. The values
are derived for a single beam with 30 µm waist, 10 W power, and an atom cloud temperature of 1 µK.

dynamic
polarizability
Re(α) (a.u.)

16.55

0.47

recoiltemperature
Trec (K)0.12

265.31

2.02

laser
wavelength
λlaser (nm)-447.32

0.87

50.34

0.38

1.20
281.12842.18

0.12

64.74

0.04

407.6

-529.28

37.47

0.12

-253.01

532
915.4-23.55252.97159.01355

1281.8-158.99

101.9

300

1630.4

1064

266

38

Chapter 5
Setup of a flexible optical dipole
trap for erbium
In the previous chapters we reported on the polarizability of erbium atoms.
From this knowledge we can estimate the trap depth and trap frequencies forvarious geometries of a dipole trap. Based on this knowledge, we are now able
to design an optimal dipole trap for erbium.
The main challenge is to find a configuration able to assure both, an optimalloading of atoms from the MOT to the dipole trap and a high efficiency during
the evaporative cooling. To match these requirements, we have chosen to build
an optical dipole trap based on a time averaged potential. With this technique
we can dynamically change the effective beam waist of the trapping beamand thus realize a tunable and flexible confinement. In addition, a numberof interesting effects in the quantum regime are expected to strongly dependon the trap geometry and on the trap aspect ratio AR. These effects include,
for instance, the roton instability [San03, Bla12] and the quantum-fluctuationdriven expansion [Lim11]. With our tunable dipole trap, we can have accessto these interesting phenomena.
The total number of atoms, which can be loaded from the magneto-opticaltrap (MOT) to the dipole trap, is significantly determined by the spatial overlap of these two traps. Based on mode-matching arguments, the larger theoverlap is, the more atoms will be loaded to the dipole trap. For this processa large beam size of the trapping laser beam is needed.
After atoms are loaded to the dipole trap, forced evaporation begins. The
trap depth is lowered by decreasing the intensity of the dipole trap beam. Thus,
the hottest atoms are lost out of the trap and the remaining atoms thermalize
to a lower final temperature via elastic collisions. For this process the collisional
rate between atoms is crucial. This rate is in principal determined by three39

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM
parameters. One parameter is the scattering length a, which is an intrinsic
property of an atomic isotope. Further, the mean relative velocity of the atoms
changes the collisional rate. Finally, the density determines how often atoms
collide per time. The density can be tuned by changing the geometry of the
trap. This gives a versatile tool to adjust the collisional rate, thus leading toefficient evaporative cooling. By changing the shape of the trapping potential
we can address both conditions, a large overlap of the MOT and the dipole
trap and a tunable density. This leads to an increased controllability of theatoms.
To realize this tunable dipole trap we build a scanning system, which allows
to change the waist of one dipole trap beam in one axis. The idea of thisscanning system was already realized in different teams of our group [Sch07,
Koh07].

5.1

Scanning system

For the scanning system we use an acousto-optic-modulator (AOM)1 . The
AOM consists of a crystal and diffracts a laser beam using sound waves. The
frequency of the sound waves fAOM defines the angle of deflection. When a lensis positioned after the AOM in the distance of the focal length, the diffraction
is translated to a parallel displacement of the beam. This is only true when the
beam is passing the AOM collimated. fAOM can be varied within the bandwidth
BWAOM of the AOM, leading to a different displacement of the beam. When
fAOM is tuned fast enough, compared to the trap frequencies, the scan of thebeam leads to a time-averaged potential for the atoms. Figure 5.1 shows theschematic of the scanning system.

5.1.1

Requirements on the scanning system

From previous measurements at our experiment we know that a beam waist of
about 30 µm leads to efficient evaporative cooling of the 168 Er isotope [Aik12].
The loading efficiency at this beam waist from the MOT to the dipole trap isabout 2 %.
With the new dipole trap we aim to increase the loading efficiency. Herewe consider a single beam configuration for the dipole trap. We use an elliptic
beam with waist in horizontal direction x of around 25 µm and waist in vertical1

We use the model 3110-197 from Crystal Technology, Inc, which has a center frequency
fcenter of 110 MHz and a bandwidth BWAOM of ±15 MHz. The numerical aperture of thismodel is 3 mm.

40

5.1. Scanning system
direction y of around 15 µm. The waist in horizontal direction can be increasedby the scanning system. The ellipticity of the beam is defined as aspect ratio
AR =

wx
ωy
=
,
wy
ωx

(5.1)

where wx and wy are the horizontal and vertical beam waists2 , respectively.
The quantities ωx and ωy are the corresponding trap frequencies along thehorizontal and vertical direction, as defined by Eq. (3.26).
The AR will then be changed by our scanning system. The maximal possible AR is limited by the resolution R = wDa of the AOM. D is the distanceby which the center of the beam can be deflected within the bandwidth of the
AOM and wa is defined as the beam waist after the lense; see Fig. 5.1. The
resolution only depends on the initial waist wi of the beam in the AOM andon intrinsic properties of the AOM [Koh07]:
R=

2π
wi BWAOM .
4cs

(5.2)

For our AOM the speed of sound cs is 4200 m/s and BWAOM is 30 MHz. The
resolution does not depend on the focal length of the used lens and cannot bechanged by implementing a telescope after the AOM. The largest beam waist
wi , which can pass through the AOM is limited by the numerical aperture of
the AOM and is about 700 µm. This configuration would give a resolution R
of about 8. With this resolution the horizontal waist wx at the chamber could
be tuned from 253 to 200 µm, leading to aspect ratios in the range of 1.5 to 13.
An important requirement on the scanning system is the modulation frequency ωmod . This frequency defines how fast the frequency of the AOM fAOM
is tuned. Only if the modulation is fast enough, atoms will not follow thechange of the potential. This so called time-averaged potential is typically created for modulation frequencies ωmod 100 times larger than the trap frequency
ωx . In this case no heating of the atomic cloud is expected.

5.1.2

Technical realization

To realize the required scanning system the driver of the AOM is adapted.
The scheme of the electronic setup is shown in Fig. 5.2. For the creation of anelliptic Gaussian beam we control the frequency of the AOM as follows:2
In the following discussion wx (wy ) is always used for the horizontal (vertical) beam
waist at the focal point in the chamber, at the position where atoms are trapped.
3
The lenses used to achieve this beam waist are shown in Fig. 5.3.

41

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM

wi

wa
D
f

f

Figure 5.1: Shown is the basic principle of the scanning system. An AOM
diffracts a collimated beam depending on the frequency of sound waves fAOMin the AOM. fAOM can be tuned around the center frequency fcenter andwithin the bandwidth BWAOM of the AOM. A lense having a distance of its
focal length f to the AOM translates the deflection to a parallel displacementof the beam. The bright and the flat beams show exemplary the diffraction fordifferent frequencies of the sound waves. A fast sweep over different frequencies
leads to an elliptic beam. The ellipticity depends on the range of the sweep.

• The angle of deflection is determined by the frequency of sound waves
fAOM in the AOM. This frequency is given by a voltage controlled oscillator (VCO). Depending on the input voltage of the VCO the output
frequency can be changed.
• The voltage Umod , which controls the VCO, is created by a mixer4 . Umod
is determined by three different parameters:
Umod = Uof f set +Usignal Ugain
.
10 V

(5.3)

• Uof f set is used to adjust the central working frequency of the VCO to thecenter frequency fcenter = 110 MHz of the AOM. The used VCO has aworking range of 75-150 MHz. 110 MHz correspond to an input voltage of
7.6 V. With this voltage one can also displace the atom cloud dynamicallyfor, e.g., trap frequency measurements.
• Usignal is created by an external frequency generator. This signal defines
the shape of the modulation signal Umod and, therefore, determines theshape of the scanned beam. In previous experiments an arcus-cosinefunction5 has proved to be suited for the creation of a Gaussian shape4

We used the model AD 734 from Analog Devices; see Appendix A.
As frequency generator we use the model Agilent 33522A. We create the desired function
with Mathematica, scale it to a 16 bit amplitude resolution and write the values in a text
file. This file is loaded to the Agilent via a USB-stick. The Agilent 33522A has a maximum
sampling rate of 250 MSa/s. The number of samples written to the text file determines the
maximum frequency. In our case we choose 252 samples and obtain a maximum frequency
maxof ωmod= 2π · 0.992 MHz.
542

5.1. Scanning system

Figure 5.2: Scheme of the electronic setup of the scanning system. To control
the deflection and the power of the dipole trap beam, we use the AOM-driver.
The signal Umod , which determines the potential form of the dipole trap iscreated by a mixer. This signal is sent to the frequency input of the AOMdriver and controls the radio frequency fAOM created by a voltage controlled
oscillator. The deflection of the beam depends on this radio frequency. The
power of the beam is controlled by a PID controller, which compares the set
voltage Uset with the actual voltage UP D monitored with a photo diode. Ucontrol
regulates the power of the AOM.

[Koh07]. The voltage of this signal is set to 10 Vpp . By using a different function one can create an arbitrary trapping geometry, e.g. a boxpotential or a double-well-potential.
• Ugain defines the amplitude of Umod . This parameter gives the knob totune the AR of the trap during the experimental cycle online. A voltageof 1 V corresponds to a sweep of 6 MHz.
The power of the dipole trap beam is monitored by a photodiode andcompared to the set value Uset by a PID-controller. The set value is given via
the experiment control system. The PID-controller regulates the power of thesignal, which is sent to the AOM. Less power corresponds to a less efficientdiffraction, thus more power is lost to the 0th order of the AOM. The power ofthe 1th order is adjusted until it reaches the set value. The photo diode only
monitors the fraction of the power of the beam, which is lost through a mirror.
43

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM
The voltage of the photo diode UP D has to be calibrated to the actual powerof the beam in the experimental chamber.

5.1.3

Implementation in the experiment

To find the optimal configuration of lenses and beam waists to create therequired dipole trap beam we built up a test setup. A copy of the optimized
setup was then built at the experiment table.
We use a 1064-nm light with a power of 42 W, generated from a single
frequency diode-pumped solid-state laser6 . Figure 5.3 shows the final setup
for the horizontal dipole trap beam. We put the laser system directly on theexperiment table. The advantage is that we can avoid the use of optical fibers,
which limit the final power, while the disadvantage is that the spatial mode ofthe laser light is not "cleaned" by a fiber.
To avoid back reflection into the laser from optical elements, we placea Faraday-isolator right after the laser. The transmission was optimized to90.2 %, which is equal to 38 W. The beam is diverging after the laser and canbe collimated by inserting a lens7 with a focal length of 1000 mm at around109 cm after the output of the laser.
For the designing of the setup after the AOM, we have to consider thefollowing technical facts:• Due to the vacuum chamber the focal length of the last lens cannot be
smaller than f = 150 mm.
• The beam at the AOM has to be collimated and as large as possible toachieve a large aspect ratio.
• Each distance between the lenses after the AOM has to be the sum of
their own focal lengths to maintain the parallel displacement of the beam.
This leads to the setup shown in Fig. 5.3. The lenses f2 and f3 are used as atelescope to adjust the beam size. The cylindrical lenses fcyl,1 and fcyl,2 onlychange the beam size of the vertical beam axis and do not affect the waist of
the scanned axis. Since we produce an elliptic beam, the focal points of the xand y-axis after the last lens do not coincide. This can be adjusted by changing
the distance between the two cylindric lenses.
6This laser is a Mephisto MOPA from InnoLight. The spatial mode of the laser lightstrongly depends on the injection current of the laser amplifier. Therefore, the calibration of
the beam and its beam waist should only be done, when the laser is working on maximum
power.
7
Due to the high power we use Fused Silica lenses with a coating for 1064-nm light. The
lenses are from LENS-Optics, GmbH and LASER COMPONENTS, GmbH.

44

5.1. Scanning system
The optimized overall optimized efficiency through the AOM is 75 %. This
relative small percentage is related to the beam size, which is too large for thenumerical aperture of the AOM8 . The diffraction efficiency between the 0th
and the 1th order is 90 %. Because the space after the last lense is limited, weplace a flip-mirror after the cylindrical lenses and copy the setup to measure
the beam size and the power "in the chamber". At the window of the vacuum
chamber 13 % of the power is lost due to reflections. The maximum power at thechamber is 18 W. For the beam waists at the chamber we obtain wx = 24 µm
and wy = 17 µm, respectively.
The technical limitations of our scanning system will be discussed in thefollowing Section. To obtain the desired performance some of the electronicparts had to be exchanged.

5.1.4

Technical limitations

The performance of the scanning system is determined by the following parameters:
Scanning range: The range of the scanning system sets the maximum achievable aspect ratio of the scanned beam, as already discussed in Sec. 5.1.2.
The range is technically limited by BWAOM and depends thus on the
AOM performance as specified by the manufacturer.
Modulation frequency: A modulation frequency ωmod of about 100 times
ωx is needed to create a time-averaged-potential. The modulation frequency is limited by the speed of the VCO that creates the radiofrequency signal. The used AOM-driver is designed for a usage of the POS
150 from Mini-Circuits; see Appendix A. For a scanning range of 30 MHz
the 3dB-modulation bandwidth was measured to be roughly 2π·100 kHz.
For higher modulation frequencies the range of the scanned frequency
fAOM decreases. From the calculations discussed in Sec. 3.3 we expect
trap frequencies ωx of up to 2π·2 kHz, for the used laser beam; see Sec. 5.2.
Therefore, we exchange the POS 150 by the model VCO190-112T from
Varil. With this model we can achieve a modulation bandwidth of up to
1 MHz. In Sec. 5.2 the performance of the new VCO will be discussed.
Shape of scanned radio frequency signal: By testing the finished electronic box we discovered, that the power shape, monitored by a spectrum
analyzer, strongly depends on Ucontrol . This voltage controls the power of
8

In a further optimization step we implement a 2:1 telescope before the AOM. With thenew, smaller beam waist wi the efficiency through the AOM increases to 90 %, leading toa higher laser power at the chamber. The quantity for beam waists and power discussed inthis Chapter refer to the first setup.

45

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM

Lift

wi
1th order

f2L=L100Lmm
wa

wx

PD

AtomcloudatLchamber

f4L=L150Lmm
λ-2
fcyl=2L=L100Lmm cylindricLlenses
fcyl=1L=LµL50Lmm verticalLbeamLaxis

AOM Driver
Box

PID

Faraday

Distances

358Lmm

426Lmm

149LmmBeamLwaistsL

wiL=LLL825Lµm

waL=LLLLL40Lµm

wbL=L2060Lµm

λ-2

wb

PBS

MephistoLMOPA
1064nm=L42LWL

wxL=LLLLL24Lµm

Dump

Isolator

f3L=L250Lmm

f1L=L1000Lmm

Figure 5.3: Optical setup for the dipole trap scanning system. The setup of the lenses is designed to fulfill our
requirements on the scanning system. The distances between the crucial lenses are depicted by the dashed, dasheddotted, and dotted line. The initial beam waist at the AOM is 825 µm along both, x- and y-direction. The final beam
waists at the chamber are wx = 24 µm and wy = 17 µm, respectively.

46

0th orderDump

5.1. Scanning system
the radio frequency signal. For the maximum power of 2 W, which corresponds to 33 dBm, the scanned signal had roughly the same power at
each frequency. However, for a lower Ucontrol the power distribution overthe scanned frequency range showed a strange shape. Some frequencies
maintained the set value of the power, where other frequencies had by
far a lower power. The problem could be traced to components of theAOM-driver. To fix this problem we bridged the TTL-switch OPA693,
the preamplifier ERA4-SM and the amplifier MHW9267. This components were replaced externally (outside the box) by coaxial components
from Mini-Circuits. As TTL-switch the model ZASWA-2-50DR is used.
For the preamplifier we use the model ZFL-500LN+ and for the amplifier
the model ZHL-1-2W is implemented. The overall gain is 34 dBm. With
the new components a sufficient shape of the power to frequency profilein scanning mode for different Ucontrol and Ugain could be achieved. In
Fig. 5.4(a) the shape of the scanned frequency monitored by a spectrum
analyzer is shown for a selection of Ugain . This Figure also shows thetranslation of the scanning range on the actual beam size; see Fig. 5.4(b)and (c). Deeper investigation on the overall performance of the scanningsystem follow in Sec. 5.2.

47

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM
(a )
5 0

5 0

U

p o w e r (d B m )3 0

g a inU= 0 V

g a in= 5 V
Ug a in= 1 0 V

4 0
3 0

2 0

2 0

1 0

1 0
0

0

-1 0
-2 0
-3 0
-4 0

-1 0
-2 0
-3 0
-4 0

7 0

8 0

9 0

1 0 0

1 1 0

fA O M (M H z )

1 2 0

1 3 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

p o w e r (d B m )4 0

fA O M (M H z )fA O M (M H z )(b )Ug a in= 0 V
wx= 2 4 .5 µ m

Ug a in

w

= 5 V

x

= 1 7 5 .0 µ m
Ug a inw= 1 0 Vx

= 2 2 2 .0 µ m

1 ,0

0 ,8

0 ,8

0 ,6

0 ,6

0 ,4

0 ,4

0 ,2

0 ,2

0 ,0

0 ,0

-4 0 0 -3 0 0 -2 0 0 -1 0 0

0

1 0 0 2 0 0 3 0 0 4 0 0 -4 0 0 -3 0 0 -2 0 0 -1 0 0

x - a x is ( µ m )

(c)
y

0V

3V

0

1 0 0 2 0 0 3 0 0 4 0 0 -4 0 0 -3 0 0 -2 0 0 -1 0 0

x - a x is ( µ m )5V0

1 0 0 2 0 0 3 0 0 4 0 0

x - a x is ( µ m )

10 V

x

Figure 5.4: Scanned radio signal that is sent to the AOM monitored by aspectrum analyzer (a) and overall performance of the scanning system (b) and(c) for a selection of scanning gains Ugain . For this measurement the modulation
frequency ωmod was fixed to 2π · 200 kHz. In (a) one can see the shape of thescanned radio frequency in terms of power (dBm) for Ugain = 0 V, 5 V, and
10 V. Due to the arcus-cosine-scanning-function the scanned range does nothave sharp edges. The scanned radio signal is translated by the AOM to thebeam size; see Figures (b) and (c). The profiles of the scanned beam fit nicely
to a Gaussian beam function. At the edges of the bandwidth BWAOM of the
AOM the diffraction efficiency decreases, what finally limits the maximum
beam size. The angle between the scanning axis and the x-axis is due to aslight tilt of the AOM.

48

in te n s ity ( a r b . u .)in te n s ity ( a r b . u .)1 ,0

5.2. Offline characterization

5.2

Offline characterization

Beam profile measurements
For the characterization of the laser beam one of the most crucial points isthe measurement of the beam profile. This can be particularly demanding forthe small beam waists we are using for our dipole trap. We compare threedifferent technical applications to measure the waist of a focused beam. For all
methods the beam profile at various points along the propagation direction isdetermined. From a fit with a Gaussian beam function the waist at the focal
point can be extracted:
CCD-camera9 : The total beam profile can be monitored with the provided
software. A Gaussian beam function is fitted online to the beam in xand y-direction and the two times 1/e2 -radius can be directly read by theuser. A disadvantage is, that the camera cannot be used at high power.
This can be circumvented by deflecting the high power beam with amirror and measuring the beam profile of the transmitted part.
Waist meter10 : This model uses a scanning knife-edge technique. The beam
waist size 1/e2 is calculated by measuring the rise-time of the integrated
waveform as the knife-edge passes through the input beam. The poweris monitored by a photodetector. The minimal beam waist that can bemeasured is provided to be 10 µm, with a resolution of 2 µm.
Knife edge: For this method one uses a calibrated micrometer-stage, where arazor blade is mounted to cut the beam. The power of the beam is monitored by a power meter. The total power of a beam can be calculated byintegrating the intensity of the beam over the whole area. By cutting thebeam, part of this area is blocked, leading to a lower value at the power
meter. From calculations using the Error-function it can be derived, thatthe distance from the point where 86 % of the total power passes the
knife edge to the point giving 14 % of the total power corresponds to the
1/e2 beam waist. This method can also be used at high power.
To determine the most reliable method, we use a test setup to measure a beamprofile with the different applications. In Fig. 5.5 our results are summarized.
The fitted waists for the CCD-camera and the knife edge method give similar
results, while the waist meter measurement results in a smaller waist. Aroundthe focal point the waist meter and the knife edge method give larger numbers
by a factor of two in comparison to the fitted quantity. This indicates that
9

Model BC106-VIS-CCD Camera Beam Profiler from Thorlabs. The pixel size of thiscamera is 6.45 µm, which allows waist measurements down to 30 µm.
10Model WM100 Omega Meter from Thorlabs.

49

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM
1 8 0 0

1 4 0
1 2 0

1 6 0 0w a is t ( µ m )1 4 0 0w a is t ( µ m )

1 0 0
8 0
6 0
4 0

1 2 0 02 0
0

1 0 0 0

8 ,0

8 ,5

9 ,0

9 ,5

1 0 ,01 0 ,5

1 1 ,01 1 ,5

1 2 ,0d is ta n c e ( c m )8 0 0

C C D -c a m e ra
W a is t m e te r
K n ife e d g e
w 0= 4 4 .5 7 µ m

6 0 0
4 0 0

w
0

= 3 9 .8 2 µ mw
0= 4 3 .5 6 µ m

2 0 0
0
0

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 0

2 2

2 4

2 6

2 8

3 0

3 2

3 4

d is ta n c e ( c m )Figure 5.5: Comparison between different methods for measurements of thebeam waist in a test setup. The beam profile is measured along its propagation
direction and fitted by a Gaussian beam function. The inset shows a zoom of
the measurement around the focal point.

these methods cannot give reliable results for small beam waists. The obtained
quantities with the CCD-camera, however, match nicely to the fitted curve.

Horizontal dipole trap
To investigate the dependence of the beam waist wx on the scanning voltage
Ugain we use the CCD-camera method. In Fig. 5.6 the measurements of thehorizontal and vertical waists are shown. The scanning voltage is varied from
1 to 5 V. One can clearly see the large effect on the beam waist. The waists forthe scanned beam are fitted by a Gaussian function and not by the formula for
a Gaussian beam, due to the fact that the scanning system does not actuallyenlarge the beam waist, but just shifts the beam very fast. The slope of thescanned beam waist in axial direction is, therefore, much to steep to be fittedwith the theory of a Gaussian beam. It seems that the scanning has also an
influence on the vertical beam axis. The slight enlargement of wy is just a relict
coming from the fact that the scanning axis has a slight angle to the x-axis;
see Fig. 5.4(c). The small shift of the focal position of about 300 µm can also
be related to this fact.
During the experimental cycle, Ugain is tuned to adjust the aspect ratioof the beam, to obtain the highest phase-space density of the atomic sample.
Therefore, it is crucial to explore the performance of the scanning system. For
50

5.2. Offline characterizationw x 0 Vw y 0 Vw x 1 Vw y 1 Vw x 2 Vw y 2 Vw x 3 Vw y 3 Vw x 4 Vw y 4 Vw x 5 Vw y 5 Vw = w 0 * s q r t(1 + ((z -z 0 )* 1 0 ^ -3 * la m b d a * 1 0 ^ (9 )/(p i* (w 0 * 1 0 ^ -6 )^ 2 ))^ 2 )V a lu e
E rro rU _ g a in

2 1 0
0 V

fo c u s y - a x is

1 9 5

2 4 ,7
6 ,3

1 ,8
0 ,2

w y (µ m )1 6 ,80 ,5z 0 (m m )1 8 0

U _ g a in

1 6 5

1 V

1 5 0

2 V

1 3 5

w a is t ( µ m )w x (µ m )z 0 (m m )

3 V

1 2 0
1 0 5

4 V

9 0

5 V

7 5
1 V

6 0
2 V

4 5
3 0

3 V

1 5
3

4 V

4

5

6

7

8

9

d is ta n c e ( m m )

1 0

1 1
5 V

7 ,1

0 ,1

M o d e l: G a u s s
w x (µ m )4 7 ,1 7

z 0 (m m )6 ,3 3w x (µ m )8 5 ,5 3z 0 (m m )6 ,3 7w x (µ m )1 2 1 ,8 7

z 0 (m m )6 ,1 9w x (µ m )1 5 2 ,8 3

z 0 (m m )6 ,0 2w x (µ m )1 7 9

z 0 (m m )5 ,9 3w y (µ m )z 0 (m m )1 8 ,9
7 ,2 8w y (µ m )1 9 ,3z 0 (m m )7 ,3 7w y (µ m )1 8 ,7z 0 (m m )7 ,4 2w y (µ m )1 9 ,6 1z 0 (m m )7 ,4 5w y (µ m )1 9 ,4 1

z 0 (m m )7 ,4 7

Figure 5.6: Measured beam size of the horizontal dipole trap beam in horizontal and vertical direction without and with scanning the beam. The scanningvoltage Ugain is tuned from 0 to 5 V with a step size of 1 V. In axial direction
the scanned beam does not show a Gaussian behavior anymore. The dashedline shows the position of the focal point of the vertical beam axis. Atoms are
sitting most likely at this position. For this set of measurements the modulation
frequency was fixed to fmod = 200 kHz.

this set of measurements we monitor the waists at the focal point of the ydirection. Figure 5.7(a) shows the dependence of the waist in x- and y-direction
on the voltage Ugain . Again the strong increase of wx can be seen. At around
5 V the curve levels out. This is due to the lower diffraction efficiency of the
AOM at the edge of its bandwidth, as discussed at Fig. 5.4.
In Fig. 5.7(b), the dependence of the scanned beam waist on the modulation
frequency fmod = ωmod /2π is shown. The maximum modulation frequency
strongly depends on the scanning voltage Ugain . The more the beam is scanned
the earlier the beam size breaks down for higher modulation frequencies. Formax
a scanning voltage of up to 2 V the maximum modulation frequency ωmod≈2π · 1 MHz can be used. In our experiment we choose the modulation frequency
ωmod = 2π · 200 kHz as working point. At this frequency the beam shape athigh scanning voltages is still Gaussian and the horizontal waist can be tuned
up to 222 µm. This corresponds to an aspect ratio of 13 and matches the
expectation of Sec. 5.1.1. The chosen frequency further fulfills the requirementωmod  ωx since we do not expect larger frequencies than ωx = 2π · 2 kHz at
our experimental cycle; see Sec. 6.1.
Summarizing, we find as minimal beam waists wx = 24 µm and wy = 17 µm.
51

5 SETUP OF A FLEXIBLE OPTICAL DIPOLE TRAP FOR ERBIUM
(a )
a

2 0 0
b

c

3 ,6 7 2 4 9

2 5 4 ,7 5 2 8

3 ,5 5 2 0 3

0 ,7 6 9 7 90 ,0 0 9 8 6

2 0 0 k H z

2 5 0

4 5

U g a in = 1 0 V

4 0
2 0 0

3 5

1 5 0

3 0

1 2 5

2 5

1 0 0

2 0

7 5

1 5

1 5 0

U g a in = 7 V

w

5 0

w

2 5

w
0
0

1

2

3

4

U

5

6

7

8

9

w
w

x

y

x

(µ m )

1 7 5

2 4 0 ,0 0 2 1

(b )

5 0

fm o d = 2 0 0 k H z(µ m )

2 2 5

A s y m p to tic : w x = a - b * c ^ U g a in
V a lu e
E rro r(µ m )

2 5 0

1 0
x

U g a in = 3 V

5 0

U g a in = 0 .5 V

5
y

1 0

1 0 0

0
0

0

2 0 0

g a in ( V )

4 0 0

6 0 0

8 0 0

1 0 0 0

fm o d (k H z )Figure 5.7: Dependence of the scanned beam waist on the voltage Ugain (a)and on the modulation frequency (b). The enlargement of wy in (a) is only due
to the slight angle of the scanning axis to the x-axis. In (b) the dependence on
fmod was measured for different aspect ratios. Higher scanning gains lead to alower maximum modulation frequency. For this measurements the new voltage
controlled oscillator VCO190-112T is used. In our experiment the modulation
frequency fmod = 200 kHz is chosen as working point.

By scanning the horizontal direction, wx can be enlarged up to 222 µm, leading
to an AR of 13. The maximum power at the chamber is 18 W. At the loading
phase a large beam area and a high power are crucial to increase the loading
efficiency. For the maximum scanning and the highest power we expect thefollowing trap frequencies according to Eqs. (3.26) and (3.31):
ωx = 2π · 106 Hz,

ωygravity = 2π · 1.34 kHz,ωz = 2π · 14 Hz.

In this configuration we want to load atoms from the MOT to the dipole trap.

Vertical dipole trap
The vertical dipole trap beam is produced by a 10 W broadband Yb fiber laser
source at 1064 nm and has an initial power of about 8 W in the chamber. The
beam profile is elliptic with waists w2z = 55 µm and w2x = 110 µm. Due tothe limited access to the view ports of the chamber the vertical beam has
an angle α of 15◦ to the y-axis. The angle to the propagation direction z of
the horizontal beam amounts to α = 14◦ . The definitions of these angles areshown in Fig. 3.3. The vertical trapping beam is crucial in the later phase of
the evaporation process, to "close the door" such that atoms cannot escapealong the propagation axis of the horizontal beam. The evaporation process
will be discussed in detail in the next Chapter.
52

Chapter 6
Measurements with atoms
In the previous Chapter we described our novel dipole trap setup based on thescanning system. With this improvement we are now able to produce BECs
with larger atom numbers, as compared to our previous setup [Aik12]. The
parameters, which are crucial to reach this goal, are the spatial overlap of MOT
and dipole trap as well as the evaporation efficiency. The optimal evaporation
ramp for a high evaporation efficiency is part of the discussion in Sec. 6.2.
Before we start our search on the best evaporation ramp, it is necessary tounderstand the geometry of the dipole trap. For this purpose we measuretemperature and trap frequencies in a single beam configuration for various
scanning voltages. Section 6.1 is dedicated to the investigation of the trapping
geometry. From these measurements we also can extract the experimental value
for the dynamic polarizability of erbium. We find Re(α) = (84 ± 2 ± 18) a.u.
for a dipole trap laser light operating at 1064 nm.
For a loading time of 5 s we prepare typically 2 · 107 atoms in the MOT at
a temperature of about 10 µK. Thanks to the scanning system we can load up
to 6 · 106 atoms to the optical dipole trap at about 20 µK for a loading time of
500 ms. This corresponds to a loading efficiency of 30 %, which is exceptionallyhigh. For a 10 s loading time we gain 3.5·107 atoms in the MOT and can load up
to 1.2 · 107 atoms to the dipole trap, corresponding to 34 %. The huge numbersin the dipole trap are a great improvement compared to the values obtainedin our earlier experiments [Aik12], where the maximum number was limited
to 3 · 106 atoms even for large numbers in the MOT. The results presented inthis chapter are for a loading time of 10 s of the MOT. These are the starting
conditions for the evaporative cooling process.
If not labeled differently the results shown in this Chapter have been measured with the further improved dipole trap setup; see Sec. 5.1.3. The values
for the beam waist and the power for this setup are:53

6 MEASUREMENTS WITH ATOMS• The vertical beam waist wy at the focal point is 16.1 µm.
• The horizontal beam waist wx at the focal point can be tuned from
26.9 µm to 190 µm. This corresponds to a maximal possible AR of about
12.
• The maximum power at the chamber of the beam depends on the scanning voltage. Without scanning a power of 27 W can be reached. For ascanning voltage Ugain of 10 V we measure a maximum power of 17 W.
This is due to the lower diffraction efficiency of the AOM at the edge ofits bandwidth.

6.1

Characterization of trap geometries

A precise knowledge of the trap frequencies is crucial to determine the properties of the trapped sample, e.g. phase-space density and peak number density.
The trap frequencies are related to the depth and dimension (beam width) of
the trap, as indicated by Eq. (3.26). Different methods can be used to measure
the trap frequencies. From comparison between the trap frequency, the dipole
beam profile, and the laser intensity one can extract the dynamic polarizability of the trapped sample. The determination of this quantity is from particular importance for theorists, since it gives them the possibility to test theirmodels for various parameters for the particular species. Before our studies,
experimental data on the dynamic polarizability of erbium was not available.
Additionally, we find a huge dependence of the lifetime of the atoms in the
dipole trap depending on the laser polarization.

6.1.1

Trap frequency measurements

The measurements of the trap frequencies give insight into the confinement of
the dipole trap laser beam. The basic principle of trap frequency measurements
is as follows. A thermalized sample is prepared in a specific trap. The cloudis then excited by compressing, decompressing, or shifting the sample for ashort period of time. When the cloud is released to the original trap it starts
to oscillate. The trap is switched off after a variable holding time te and anabsorption image is taken. Additional free expansion, which is named time-offlight (TOF), increases the signal-to-noise ratio. Different methods are probed
to obtain reliable trap frequency measurements:
Power modulation: The strength of the trapping potential is changed byeither increasing the intensity or switching off the trap for a short period of time. Therefore, the atomic cloud is compressed or released and54

6.1. Characterization of trap geometries
switching on the original trap leads to excitations of the breathing mode.
The typical modulation time for the switching-off method is one fourthof the trap frequency. The breathing mode is observed by monitoring thewidth of the atomic cloud for various holding times te in the final trap
after modulation. For an ideal thermal gas the breathing mode is equal totwice of the trap frequency. In principle, with this method the frequencies
of all axes could be observed. However, the oscillation on the strongest
confining axis is the most dominant one. Therefore, the oscillations on
the weaker confining axes are hardly visible. An example measurement
for this method is shown in Fig. 6.1(b).
Magnetic gradient field: The gradient field can be used to tilt the dipole
trap along the vertical y-axis. Additionally the gradient field has also
an influence on the horizontal x- and z-axis1 . This leads to a couplingof the individual oscillations, thus making the distinction between theindividual trap frequencies less reliable.
Individual modulation of each axis: Most precise measurements can bedone when the sample is only modulated along the monitored trap axis.
In our case the strongest confining axis is the y-axis (axis of gravity).
Therefore, the corresponding trap frequency can be obtained by switching off and on again the horizontal dipole trap beam, as discussed above.
For the measurement of νx we use the scanning system itself. By slowly
changing the voltage Uof f set the atomic sample is shifted adiabaticallyalong the x-axis and starts oscillating, when it is released to its original
position. The center-of-mass motion around the equilibrium position isobserved to extract the trap frequency. This measurement can be doneeven during wx is enlarged by the scanning system. An example of thismeasurement is shown in Fig. 6.1(a). For the z-axis we can use the vertical beam, which provides a strong confinement in the axial axis of
the horizontal beam to excite oscillations. The vertical beam is typicallyswitched on for a few milliseconds.
In conclusion, best results are obtained by modulating each axis individually.
An additional reason for this is that the axis of our imaging system has anangle of 28◦ to the z-axis in the x-z-plane; see Fig. 3.5. As a consequence,
simultaneous oscillation along the x- and the z-axis cannot be distinguished ifthey have similar frequencies. Too strong excitations of the trapped cloud canlead to large atom loss and heating. An adequate modulation manifests itself inmany observable oscillations of the trapped atomic cloud. A damping of these1

The vertical gradient is produced by two coils in Anti-Helmholtz configuration. Due to~ = 0 the creation of a vertical gradient also leads to horizontal gradients.
divB

55

6 MEASUREMENTS WITH ATOMS
(b )

(a )

0 ,2 5fx = 4 5 4 H z

fy =0 ,2 4f σy / 2 = 1 0 8 5 H z

0 ,2 3
2 ,2 8σy ( m m )x c e n te r ( m m )2 ,2 90 ,2 2

2 ,2 7
0 ,2 1

0 ,2 02 ,2 6
0 ,1 9
0

1

2

3

4

5

6

7

0 ,0

te (m s )

0 ,5

1 ,0

1 ,5

2 ,0

2 ,5

te (m s )Figure 6.1: Typical trap frequency measurements in a single beam dipole trap
configuration. In Figure (a) the sample was prepared at a laser power of 2 W,
and an AR of 2. To excite the center-of-mass oscillation along the x-axis thetrapped cloud was shifted by changing the value of Uof f set for 60 µs. In Figure(b) a breathing mode oscillation is shown. Switching off the beam for 50 µs
leads to an excitation of this mode, especially along the strong confining yaxis. For this measurement the sample was prepared at 2 W with the scanningsystem being off. We obtain trap frequencies ωx = 2π·454 Hz and ωy = 2π·1085
Hz, respectively.

oscillations indicates that the trapping potential is not perfectly harmonic.

6.1.2

Atomic polarizability

The knowledge of the atomic polarizability of erbium is of great importancefor experiments and theories. For instance, theorists are able to extract the C6
and ∆C6 values of this element if a precise knowledge of the polarizability isavailable; see Sec. 2.3. The atomic polarizability Re(α) can be obtained fromtrap frequency measurements. Here a single beam dipole trap is used. As discussed in Chap. 3, there is a very convenient relation between Re(α) and theoptical dipole trap frequencies ν:s1
8A · Re(α)
νx =
P· 3
,
2π
wx wy mπs
18A · Re(α)
νy =
P· 3
,
(6.1)
2π
wy wx mπ56

6.1. Characterization of trap geometries
where P is the power of the dipole trap beam, wx and wy are the beam waists in
the horizontal and the vertical axis, respectively, and A = 1.65 · 10−41 /(20 c)
is the conversion factor to extract the polarizability in atomic units. These
formula can be derived from Eqs. (3.2), (3.18), and (3.26), where
s
s
1
4Û
4Û
1
νx ≡
, νy ≡,
2
2π mwx2π mwy2
are the trap frequencies obtained from the harmonic approximation of thedipole trap potential andÛ = Ũ I0 = −

2P
1
Re(α)
,
20 c
πwx wyis the total trap depth. In the experiment we measure independently the trap
frequencies as well as the beam waists wx and wy . The latter are obtainedby beam profile measurements with a CCD-camera, see Sec. 5.2, and have in
general the highest uncertainty. For the measurements of the trap frequencies
we use the methods discussed above.
In a single beam configuration the trap frequencies are measured at different
dipole trap conditions. To change the trapping potential we either vary thetrapping power or change the trap aspect ratio with the scanning system.
Power variation: For this measurement we prepare a cold and thermalizedatomic sample. Therefore, the laser power is first decreased to 0.7 W,
leading to temperatures of about 2 µK. Then the power is increased
to the desired value, thus recompressing the sample. After one secondof thermalization the atomic cloud is modulated for a few micro seconds to excite oscillations and to extract the trap frequencies νx and νy .
The TOF time is typically 10 ms. Here the scanning voltage was fixed
to 0 V. This measurement was done with our first dipole trap setup,
discussed in Sec. 5.1.3, with beam waists wx = (24.0 ± 1.8) µm andwx = (17.0 ± 0.5) µm. Equation (6.1) is used as fit model to extract
the atomic polarizabilities. In Fig. 6.2 we summarize our measurementsof the trap frequencies as a function of the laser power. The solid lines
are the fits to the set of data.
Aspect ratio variation: Similar to the above discussed procedure a cold andthermalized atomic sample is prepared by evaporating down to about
2 µK and recompressing the sample with a power of 2 W. During therecompression the trap aspect ratio is adjusted by changing Ugain to thedesired value. We measure trap frequencies νx and νy for various scanningvoltages Ugain of up to 3.5 V. Our current dipole trap setup was used to57

6 MEASUREMENTS WITH ATOMS
(a )

(b )
1 5 0 02 4 0 0

1 4 0 0α= 8 4 (4 ) a .u .

1 3 0 0α= 8 2 (6 ) a .u .

2 2 0 02 0 0 01 2 0 0
1 8 0 0νy ( H z )1 1 0 0νx ( H z )1 0 0 0
9 0 0

1 6 0 0
1 4 0 01 2 0 0

8 0 0
7 0 0

1 0 0 0

6 0 0
5 0 0

w x = 2 4 µ m
w y = 1 7 µ m

8 0 0

w x = 2 4 µ m
w y = 1 7 µ m

6 0 0

4 0 0
0

1

2

3

4

5

P o w e r (W

6

7

8

9

1 0

0

1

2

3

4

5

P o w e r (W
)

6

7

8

9

1 0

)

Figure 6.2: In a single beam configuration we have measured trap frequencies
for different powers of the horizontal dipole trap beam in x- (a) and y-direction(b). The solid line is a fit to the data according to Eq. (6.1). The fit yields forthe atomic polarizabilities Re(α) = (84±4) a.u and Re(α) = (82±6) a.u for thedata of Fig. (a) and Fig. (b), respectively. The obtained errors are statistical
only.

perform this set of measurements. The beam waists without scanningare found to be wx = (27.0 ± 2.0) µm and wy = (16.1 ± 0.5) µm. The
dependence of the horizontal waist on Ugain in the region up to 3.5 V isfound from offline measurements similar to Fig. 5.7:
wx = [−345.8 + 372.7 · exp(0.078 · Ugain )] µm.

(6.2)

This dependence is used to refine the fit model of Eq. (6.1). Figure 6.3
shows the fit to the measured trap frequencies as a function of Ugain .
In Fig. 6.4 we summarize the extracted atomic polarizabilities of each set ofmeasurements. The data points are fitted with a constant to find the best valueof Re(α):Re(α) = (84 ± 2) a.u.
(6.3)
As discussed above the error of this value is statistical only. In the fits theuncertainties of the trap frequencies and the powers are included2 . Up to now
the uncertainties of the beam waists have not been considered. In the following
discussion we want to give an upper and a lower bound of the uncertainty ofthe measured atomic polarizabilities.
2

In our experiment the trap frequency measurements typically have an uncertainty of
3 %. The uncertainty of the power is about 1 % and is related to the used photo-diode andto the whole control circuit.

58

6.1. Characterization of trap geometries
1 2 0 0w

1 1 0 0tra p fre q u e n c y ν(H z )1 0 0 0x ,U

= 2 7 µm

= 0

g a in

w

9 0 0

y

= 1 6 µm

8 0 0
7 0 0

α= 8 7 (4 ) a .u .

6 0 0
5 0 0
4 0 0
3 0 0
2 0 0

α= 7 7 (1 0 ) a .u .

νx

1 0 0

νy
0
0 ,0

0 ,5

1 ,0

1 ,5

U

2 ,0
g a in

2 ,5

3 ,0

3 ,5

(V )

Figure 6.3: The trap frequencies were measured for different scanning voltagesof the horizontal dipole trap beam in y- and x-direction. The power of the beam
was set to 2 W. In the fitting function the dependence of the waist wx on thevoltage Ugain is included; see Eqs. (6.1) and (6.2). From the fit we can extract
the atomic polarizabilities Re(α) = (87 ± 4) a.u and Re(α) = (77 ± 10) a.u forthe set of measurements in the y- and x-direction, respectively.
1 0 0
9 5

p o la r iz a b ility ( a .u .)

9 0
8 5

α= 8 4 (2 ) a .u .

8 0
7 5
7 0
6 5
1

2

3

4

d a ta p o in ts

Figure 6.4: All obtained values are plotted in one graph to extract the bestvalue and the statistical error of the polarizability. The solid line is a constant
fit to the data that gives Re(α) = (84 ± 2) a.u.

Lower bound: To determine the lower bound value we use the theory of error
propagation. Equation (6.1) is solved to Re(α) and for each data point
we calculate the corresponding error of the polarizability according to
the above discussed uncertainties of ν, P , wx and wy . We then use all
the errors to calculate the total error and find ±3 a.u.
59

6 MEASUREMENTS WITH ATOMS
Upper bound: Trap frequencies and beam powers can be measured with ahigh precision, where for the beam waist measurements systematic errors
hardly can be avoided. For the offline beam characterization the last partof the dipole trap setup is copied since it is not possible to measure thebeam profile in the vacuum chamber; see Sec. 5.1.3. Imperfect alignmentof the optical elements in the copied path can lead to a deviation between
the measured and the actual beam waists. To obtain the beam waists wetake multiple data points along the propagation direction of the laser
beam and fit the set of data with the theory of a Gaussian beam. Since
the beam profile of the used laser is not perfectly Gaussian, additional
deviations are expected. To take the systematical error into account weassume the beam waists to be wrong by ±2 µm. This assumption is used
to calculate the polarizability again by varying the beam waists in thefit functions of Figs. 6.2 and 6.3. The observed differences to each above
discussed polarizability are then used to determine the overall systematic
error. The error propagation yields ±18 a.u.
In conclusion we find for the dynamic polarizability of erbium at 1064-nm:
Re(α) = (84±18±3 ± 2) a.u.

(6.4)

including lower and upper bound of the systematical errors as well as thestatistical error.
This quantity is in striking contrast to the theoretical predicted polarizability of 159 a.u. ; see Sec. 4.1. This seems to be typically for lanthanides,
since the dynamic polarizability of dysprosium has also been overestimated incomparison to the measured quantity [Dzu11, Lu11].
To cross-check the obtained quantity for the dynamic polarizability we use
the mathematical model to calculate the expected trap depth; see Appendix
B. The depth is compared to the corresponding, measured temperature of
the atomic sample. We find well agreement to the typical correlation kB T ≈1/10 · Udip .
In future we plan to add lithium to our experiment. For this element the
dynamic polarizabilities are known [Tan10]. This knowledge can be used to
extract the polarizability of erbium. A proposal by Cronin et al. [Cro09] showsthat the ratios of polarizabilities of different species can be determined accurately.

6.1.3

Comparison to offline characterization

From the results shown in Fig. 6.3 we can also extract the trap aspect ratio ARof the scanned beam for different scanning voltages Ugain . In Fig. 6.5 we plot the60

6.1. Characterization of trap geometries
1 1
1 0

A R
9

A R

fro m
fro m

tr a p fr e q u e n c ie sb e a m p r o file8

a s p e c t- r a tio

7
6
5
4
3
2
1
0
0 ,0

0 ,5

1 ,0

1 ,5

2 ,0

U

(V )
g a in

2 ,5

3 ,0

3 ,5

Figure 6.5: Comparison between the aspect ratio of the horizontal dipole
trap obtained from trap frequency measurements to the expected AR from theoffline beam characterization.

measured AR and the AR expected from our offline characterization. For smallvalues of Ugain the measured and the expected AR nicely agree. At a higher
scanning voltage the two values disagree by up to 30 %. An explanation for thiscan be an introduced anharmonicity in the x-axis lowering the trap frequencies
νx , thus increasing the AR. Suspects are the slight angle of the scanning axis
to the x-axis, and a non perfectly Gaussian shape of the time-averaged beam.
Systematic errors in the measurement of the beam profile could also contributeto the observed differences. One problem is, that the measurement of the beamprofile can only be done at a "copied chamber", and thus could differ from theactual profile. However, for the experimental process we are only interested in
the actual trap geometry, determined by trap frequency measurements. From
the trap frequency measurements we find a maximal possible AR of 15, which
gives us a huge flexibility for the modification of the trapping potential.

6.1.4

Polarization dependence of the trapping geometry

In a recent paper on the polarizability of polar molecules, it was shown thatthe polarizability strongly depends on the angle of the linear polarization of
the dipole trap laser beam [Ney12]. Tuning the polarization thus changes thetrap geometry. This is related to the tensorial part, which accounts for the
dependence of the polarizability on the angle between the electric field andthe quantization axis. The observation also could be valid for dipolar nonS state atoms, such as erbium. As a preliminary investigation we measured
61

6 MEASUREMENTS WITH ATOMS
3 ,5
P o la r iz a tio n 2 3 2 °P o la r iz a tio n 1 8 8 °P o la r iz a tio n 2 1 2 °3n u m b e r (1 0 6)

2 ,5
2
1 ,5

t life = 7 s

a to m1

t life = 3 st life = 0 . 6 s

0 ,5

0

1

2

3

4

5

6

7

8

9

1 0

1 1

h o ld in g tim e ( s )Figure 6.6: We load atoms to the dipole trap and simply let them evolve inthe same trap. Thereby we monitor the atom number right at the end of theholding time. The linear polarization of the trapping laser light is adjusted
manually, rotating a λ-half wave-plate, where 232◦ and 188◦ correspond to pand s-polarization, respectively. For s-polarization the field of the laser light isaligned along the quantization axis of the atoms. We find a strong dependenceof the lifetime on the laser polarization, indicating loss processes for a specificpolarization angle. The lifetimes are extracted by fitting the data with an
exponential decay function.

the trap geometries for different angles of laser polarization, but could notobserve a measurable difference in trap frequencies and temperature. This
matches our theoretical calculations, where we found that the tensorial partof the polarizability for the ground state of erbium does not play a significant
role; see Sec. 4.1. However, we found a dramatic change in the stability of thetrapped atoms. For this measurement we prepare 3.5 · 106 atoms in a single
beam configuration with a power of 18 W and an AR of 10 after a loading timeof 5 s of the MOT. For a specific angle of the laser polarization we observe hugeand very fast losses of atoms. In Fig. 6.6 we report the number of atoms for avariable holding time directly after loading to the optical dipole trap. The loglin scale reveals the non-exponential behavior of the loss dynamics. At shorter
holding time we generally see faster losses compared to the one observed atlonger time. This losses can be related to plain evaporation. For longer holding
times a change of 20◦ in the angle of the linear polarization gives a change of
one order of magnitude in tlif e . The investigation of the reason for these losses
will be part of future experiments.

626.2. A new Bose-Einstein condensate

6.2

A new Bose-Einstein condensate

In our recent papers, see Ref. [Fri12] and Appendix C, we demonstrated asimple and efficient approach to trap erbium atoms and cool them to quantum
degeneracy. During this thesis, we could further improve our system and reacheven higher atom numbers in the Bose-Einstein condensate. In this Section,
we will discuss the optimized evaporation process used with the new trap. Fora high evaporation efficiency the ratio between elastic and inelastic collisions
in the trap is crucial. This ratio is determined by the density, which can bechanged by modifying the trap geometry. In our previous setup simply thepower of the horizontal and vertical dipole trap were changed to reach a good
evaporation efficiency. The scanning system gives us an additional, optional"knob" to optimize the collisional rate. Especially at the end of the evaporation
process high densities can lead to large three-body losses. With the scanningsystem we have a versatile tool to reduce the losses. To control the scanningsystem Ugain is calibrated to the trap AR. Therefore, we fit the data of Fig. 6.5
and implement the obtained function in the experiment control system. The
AR can then be controlled for every evaporation step via this system.
The evaporation efficiency is determined by the ratio of the change in phasespace-density (D) to the change in atom number in a double-logarithmic scale:dDD = − dN,
N

where the phase-space-density D is defined as
 3
~
N
D=
ωx ωy ωz 3 .
kB
T

(6.5)

(6.6)

This equation outlines that high atom numbers and low temperatures are
crucial for a high phase-space-density. When the phase-space-density is equal toabout 1.2 quantum degeneracy is reached. An additional parameter to describe
the atomic sample is the peak-number density. It gives the maximum numberof particles per area and can be derived from the density distribution in aharmonic trap: 23mn0 = Nωx ωy ωz .
(6.7)
2πkB T

The new BEC
We load 1.2 · 107 atoms to the optical dipole trap with a power of 20 W and ascanning AR of 10. The thermalized temperature of the atoms directly after63

6 MEASUREMENTS WITH ATOMS
loading is about 20 µK. These are the starting conditions for the evaporation
process. Figure 6.8 shows the whole evaporation ramp for the creation of the
new BEC. At every evaporation step we adjust the power of the horizontal andthe vertical dipole trap beam, as well as the AR. We monitor the atom numbers, the temperatures, and the trap geometries by frequency measurements fora selection of evaporation steps. These data are fed to our mathematical model
to determine the trap depths, the phase-space-densities and the peak-number
densities. For these calculations we use the measured atomic polarizability of
erbium. The calculation of the trap depths already includes the contributionof both the horizontal and the vertical dipole trap beam. For the beam waistsof the horizontal beam we use wy = 16.1 µm and wx = (AR · wy ) µm where
for the vertical beam wx = 112 µm and wz = 70 µm are the expected beam
waists3 . The dashed lines in Fig. 6.8 show the expected trap frequencies foreach power and AR of the crossed dipole trap according to our mathematical model, discussed in Sec. 3.3. In most cases we find well agreement withthe actual measured trap frequencies. The maximal discrepancy between themeasured and the calculated trap frequencies is about 20 %. One explanation
for this is the difficulty to measure the beam waists precisely, as well as thesystematic errors discussed in Sec. 6.1.2. Another effect that could justify thisdiscrepancy is the movement of the atomic sample when the vertical dipole
trap beam is switched on. We see a shift of about 200 µm of the center of thecloud in the axial direction of the horizontal dipole trap beam. This is related
to the non perfect crossing of the beams. The atoms are shifted out of the focal point of the horizontal beam, which results in a change of the confinement.
Thus, different trap frequencies than expected can occur.
After about 5 s of evaporation an onset of a BEC is achieved, which corresponds to a phase-space-density of 1.2. A very interesting feature of the applied
evaporation process can be seen in the plot of the peak-number density. Whenthe vertical trapping beam is turned on a huge increase of n0 is observed, where
it is then slowly decreased to reach a high evaporation efficiency. Just at thevery last evaporation step we gain a pure BEC.
The whole evaporation process takes 6.9 s and leads to a final number inthe pure BEC of up to 2 · 105 atoms. In comparison to our previous setup
this number is larger by a factor of 3 ; see Appendix C. At the end of theevaporation process the increase of the AR by the scanning system is ideal todecrease the density, and thus three-body losses. Best performance is found for3

The used beam waists of the vertical dipole trap beam differ from the measured waistsin the focal point. For the alignment of the crossing of the two dipole trap beams only theatom number in the crossed region is monitored. Therefore, it is very likely, that the focal
points of the beams are not perfectly overlapped.

646.2. A new Bose-Einstein condensate

p h a s e - s p a c e - d e n s ity D

1

0 ,1

0 ,0 1

1 E -3

1

A to mn u m b e r x 1 0 6

1 0

Figure 6.7: Double-logarithmic plot of the phase-space-density during theevaporation ramp as a function of the atom number. The overall evaporation
efficiency  until quantum degeneracy is reached is 3.4. When the vertical beamis switched on a huge evaporation efficiency is observed.

an AR of 7 at the end of the evaporation process. Due to the large AR at theloading phase we start with low trap frequencies, then compress the sample
and finally evaporate it down. In the pure BEC we find peak-number densities
of up to 8 · 1013 cm−3 .
Figure 6.7 shows the phase-space-density as a function of the atom number
during the evaporation process. The overall evaporation efficiency until D =1.2 is reached is  = 3.4, which is very high. When the vertical dipole trap
beam is switched on we find an evaporation efficiency of 16. In the latter phaseof the evaporation process the efficiency decreases to  = 2.

65

6 MEASUREMENTS WITH ATOMS

2 0
1 8
1 6
1 4
1 2

0

2

4

6

8

1 0

0

2

4

6

8

1 0

1 0 0

1 0

1

0

0

2 0 W

0 W

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

E v a p o r a tio n tim e ( s )

6

7

7

3 0 0 m W
8 0 0 m W

66

7

1 6 0 0
1 4 0 01 2 0 08 0 0

1 0 0 0

6 0 0
4 0 0
2 0 0
0

5
4
3
2
1
0

w
w
w
w

x
y
x
z

= A R *w y µ m= 1 6 .1 µ m= 1 1 2 µ m= 7 0 µ m

f cx a l c u l a t e d

7

f mx e a s u r e d67

f cy a l c u l a t e d2

6

f cz a l c u l a t e d55f my e a s u r e d

4

4

E v a p o r a tio n tim e ( s )2f mz e a s u r e d33

h o r iz o n ta l b e a m

1

1v e r tic a l b e a m

0

0

1 0

1

0 ,1

1 0

1

0 ,1

1

0 ,1

0 ,0 1

1 E -3

1 E -4
0

0

0

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

E v a p o r a tio n tim e ( s )

6

6

6

7

7

7

66

n u m b e r x 1 0 6
A to mT e m p e ra tu re (µ K )p h a s e - s p a c e - d e n s ity D

Figure 6.8: This figure shows the evolution of different parameters during the forced evaporation. The whole evaporation process takes about 7 seconds. For each evaporation step we measure the atom number, the temperature, as
well as selected trap frequencies. From these quantities we can calculate the phase space density and the peak number
density at each step. Our theoretical model is used to calculate the potential depths of the crossed dipole trap andthe expected trap frequencies. The latter are indicated by the dashed lines. We find well agreement with the measured
data with a maximum deviation of 20 %. We obtain up to 2 · 105 atoms in the pure BEC which is an improve in numberof a factor of 3 compared to our previous dipole trap setup.

0

)
-3

(c m
1 4

x 1 0

tr a p fr e q u e n c ie s ( H z )p e a k - n u m b e r d e n s ity n 0

P o w e r (W )A R
T ra p d e p th (µ K )Chapter 7
Conclusion and Outlook
The goal of my master thesis was to implement an optical dipole trap withtunable geometry to our erbium experiment. Optical dipole traps are used toload atoms from the MOT and evaporatively cool them down to quantum
degeneracy. A flexible geometry can help to optimize the loading of the optical
dipole trap and the evaporative cooling efficiencies. This improvement can beused to reach large atom numbers in the trap.
The potential created by a laser beam depends on the intensity and profileof the laser. The depth of the created dipole trap is related to the atomic
polarizability of the trapped atomic species. The polarizability depends on theatomic spectrum of the particular species, as well as on the wavelength ofthe laser beam. Erbium is a multi-valence electron system and shows thus acomplex and rather unknown energy spectrum. Theoretical and experimental
efforts are taken to further improve the knowledge of the energy level structure.
During my thesis work I developed a mathematical model to calculate theatomic polarizability of erbium. For the calculation every known dipole allowedtransition is taken into account according to its line strength factor. For the
ground state of erbium atoms we find a dynamic polarizability of 159 a.u. for a
dipole trap operated at 1064-nm. This value accounts for both the scalar andthe tensorial part of the atomic polarizability. Furthermore, the mathematicalmodel allows to fully simulate the trapping geometry of the crossed dipole trap
used in our experiment.
In a first crossed dipole trap setup we managed to evaporatively cool down
erbium atoms to quantum degeneracy – for the first time. The corresponding
publication is given in Appendix C. This knowledge was used to design the
optical setup for the horizontal dipole trap laser beam with tunable ellipticity.
To dynamically tune the trapping geometry we use a scanning system consisting of an AOM. An AOM deflects a laser beam depending on the frequency of67

7 CONCLUSION AND OUTLOOK(a)y Ugain = 0 V
x

(c)

(b)

5V

Ugain = 3 V

Ugain = 3 V

arcus-cosine

squareFigure 7.1: Scanning system gallery for a scanning voltage of 0 V (a) and 3
V; (b) and (c). The shape of the beam depends on the function used to scan
the beam. Here we show two examples. An arcus-cosine function leads to an
elliptic Gaussian beam shape (b), whereas a square function creates two beams(c). The pictures were taken with the CCD-camera beam profiler.

sound waves in the crystal. A change of this frequency thus changes the angleof deflection and a scan over a range of frequencies leads to a time-averaged
beam profile. A lens positioned one focal length after the AOM translates thedeflection into a horizontal displacement of the beam. If the frequency of the
AOM is scanned fast enough compared to the trap frequency of the dipole
trap a time-averaged potential is created. The ellipticity of this time-averaged
potential can be tuned by the scanning system. The optical setup of the horizontal beam was optimized several times and we finally find ideal working
conditions with beam waists at the position of the atoms of wx = 27 µm andwy = 16 µm along the horizontal and vertical axis, respectively. The horizontal
waist can be tuned up to 190 µm, leading to a theoretical maximal possible
aspect ratio of 12. The vertical dipole trap beam has a waist of 55 µm along
the axial direction of the horizontal beam and a waist of 110 µm perpendicular
to this direction in the horizontal plane. In Fig. 7.1 the profile of the horizontal
dipole trap beam for a selection of scanning functions is shown.
With the new flexible dipole trap we can load up to 35 % of the atoms from
the MOT to the optical dipole trap at a temperature of about 20 µK. For aloading time of 10 s of the MOT this number typically corresponds to 1.2 · 107
atoms. For the optimized evaporation process we reach up to 2 · 105 atoms
in the pure BEC. The evaporation efficiency of this process is 3.5, which isvery high. In comparison to our first setup we could triple the atom numberin the BEC. Especially at the end of the evaporation process the scanningsystem is used to weaken the confinement in the trap thus suppressing threebody losses. In Fig. 7.2 a 3D density profiles of absorption images for different
evaporation steps are shown. For lower, and lower temperatures the matterwaves of the atoms start to overlap finally forming a Bose-Einstein condensate,a macroscopic occupation of the lowest quantum state.
68

7 CONCLUSION AND OUTLOOK

Figure 7.2: 3D density profile showing the phase-transition of the erbium
atomic cloud to a Bose-Einstein condensate.

Additionally the scanning system can be used for trap frequency measurements. Here the atomic sample is first adiabatically shifted in x-direction. Thena sudden switch back to the original trap leads to center-of-mass oscillations.
From trap frequency measurements we find that we can reach even higher
aspect-ratios than expected from the offline beam configuration. This behavior
could be due to anisotropical effects, and further increases the tunability ofthe dipole trap. The maximal possible aspect-ratio is measured to be 15.
First results based on my work are measurements of the atomic polarizability of erbium. Here we compare trap frequency measurements with ourmathematical model to extract the dynamic polarizability. In atomic units wefind a value of Re(α) = (84±18±3 ± 2) a.u. where ±3 and ±18 indicate the lower
and the upper bound of the systematical error of the measured polarizability
and ±2 is the statistical error. This value is smaller by 47 % compared to theone calculated by our mathematical model. A similar discrepancy also has been
observed for dysprosium [Lu11], which is part of the lanthanide series as well.
The obtained value will help to theorists to further improve their theoreticalmodels for erbium, including investigations of the isotropic and anisotropic partof the van der Waals interaction as well as theoretical researches on Feshbachresonances.
In future experiments we plan to use the scanning system to investigate
the geometry dependence of anisotropic quantum effects in dipolar gases. The
influence of the geometry on the behavior of a dipolar gas was already demonstrated by the group of T. Pfau [Koc08]. We are confident that with our tunable
dipole trap we can precisely map the behavior of the system when the geometry
is changed in a continuous way. Due to the high magnetism of erbium quantum fluctuation are expected to be sizable and to manifest themselves in the
TOF dynamics of a Bose gas as well as in the collective modes [Lim11]. Since69

7 CONCLUSION AND OUTLOOKquantum fluctuations are expected to depend on the aspect-ratio of the trap,
our new flexible dipole trap setup will be an ideal environment for this investigation. Other proposals show that in a dipolar BEC confined in a quasi-2D
trap a rotonlike spectrum can emerge [San03, Bla12]. This fascinating phenomena is unique to dipolar BECs and may also can be accessed by means of
the scanning system.

70

Appendix A
Electronics
Figure A.1 shows the electronic circuit for the creation of the frequency signal,
which is sent to the AOM-driver. The modulation signal Umod consists of thethree signals Uof f set , Ugain and Usignal and is generated by a mixer:
Umod = Uof f set +Usignal Ugain
10 V

(A.1)

The supply voltage is ±18 V. The maximum values for Uof f set and Ugain are+10 V, and for Usignal the voltage should not exceed ±10 V. The full control
on the signal Umod is then used to change the radio-frequency signal created bya VCO in the AOM-driver. The signal is then amplified and sent to an AOM.
In Fig. A.2 the electronic circuit of the used AOM-driver is shown. As
discussed in Sec. 5.1.4 some electronic parts had to be exchanged to achieve anutilizable radio frequency signal.
In Fig. A.3 the box of the AOM-driver is shown. The electronic circuit toproduce Umod is implemented in this box on the left hand side. The right handside was changed, as discussed at Fig. A.2.

71

A ELECTRONICS

Figure A.1: Electronic circuit used to generate the signal to scan the frequency of the AOM; see Sec. 5.1. The mixer AD 734 from Analog Devices
creates the output signal Umod from the input signals Uof f set , Usignal and
Ugain .

72

P4

VCC

L3

Supply

Vo

3 S2

P3

P0J1 *02

R6
5k1

D4

2

R28
5k1

1

P0L101

L1

Vpos

P0S302

GND

S3

D1

Vtune

1N4148 1N4148

GND ON / OFF

1

3

P0S301

P0S303

GND

C13
1u

J1 *
P0J1 *01

GND

GND

Vamp U10 LM78L15ACZVpos
P0U1003
1P0U1001
3
IN
OUT
C25
GND
1u

GND

P0P301

Select Frequency

P0S201

2

P0S202

100n

R3
540

P0J2 *02

Int / Ext Frequency

GND

P0L301 P0C2201 P0C2202

C22

P0J2 *01

Int / Ext Frequency

GND

P0S2031Out

C7 10u2

P0U302

P0C701 P0C702

Tune

P0U308

J2 *

4 GND

P0U1704* J2 = Intern Frequency

U3 POS-150 / POS-300

1

10uH

P0L302R15
50k

P0R1503

P0P401

R18
4k7

GND

Frequency

Vpos

BNC
GND

Frequency

Vtune

6
P0U1706* J1 = Extern Frequency

H11L1

Vcc
5
GND P0U1705

1n

GND

R8
56

GND

1n

R16
1k5

3P0L103

R9
1k5

R7
1k5

2

GND

C18
220u

3

P0U503

2

P0U502

GND

C9
1n

GND

P0C601 P0C602

C6

R11
1k5

GND

P0C2601 P0C2602

GND

LCL-Filter

P0C1001 P0C1002

C10

DIS

C26
100n

330

Rg

C4
100n

GND

C16
100n

GND

3

P0U103

2

P0U102

330

Rg

C21
VCC
100n

GND

C3
1n

GND

3 P0U903
INU9U8
1 P0U801
IN

GND

240
R20
4k7

P0R1901

4u7

330

Rf

GND

DIS
C5

C14
1n

GND

GND

Number: 1

3

Date: 12.01.2006
Time: 10:10:00
File: F:\AOM\Protel\AOM.SchDoc

A4

AOM Driver

GND

C2
1u

R1
50k

GND

CON

CON

GND

R14
40

4P0U604

3

P0U403

4

P0U404

3

4

P0U203

P0U204

R17

RF IN

P0U701

GND

10k

P0U705

VCC

CON

9

5

P6
Output

GND

GND

GND

P0P601

C12 10u

P0C1201 P0C1202

100n

P0C2301 P0C2302

C23

L2
10uH / 0.8A

Vamp

P0R501

4

Drawn by: Kluibenschädl Manuel

RF OUT

P0U709

GND

C11
100n

VCC

MHW9267

2 P0U702
GND
3 P0U703
GND
7 P0U707
GND
8 P0U708
GND

1

U7

40

R5

4

P0R402
P0R502

P0R1701

GND

GND

VCC

GND

GND

Exphystronix
University of InnsbruckTechnikerstraße 25
6020 Innsbruck
Austria1n

P0C1501 P0C1502

C15

L4
10uH

40

P0R1201

R4
Res Adj2

P0R1202 P0R1702

GND

R12

Output

2 P0U402
Control
TFAS-2SMU4
1 P0U401
Input

TFAS-2SM

Output5k1

P0R2201

2 P0U202
Control

Sheet 1 of 1

Revision: 2.0

GND

R22

5k1
P0D302
P0R2202

2

P0S102P0R401

P0R2101

P0R2703R21
P0D201
P0R210250k

3

1

P0S103

P0S101

S1
Attenuation Select

1 P0U201
Input

U2

P0D301D3

P0D202

D2

P0R2701

R27

ERA4-SMU6

P0D502D5
1N4148

P0P201P0D501

DS2
LED0

R24
2600

Vamp

1

P0U60150

P0R103

P0R1002

GND

R10
P0R1001

DS1
LED0

R23
1200

VCC

GND
P2
Attenuation

P0C1401 P0C1402

1n

VCC

Attenuation

3

P0C501 P0C502

GND
Monitor

Size:

Title

Vamp
C20
6u8

GND

C17
4u7

P5

6P0U106

8P0U108
DIS

U1
OPA693GND

P0P501
P0R1301

VCC

50

P0R1302

R13

1n

6P0U506 P0C2401 P0C2402

C24

U5
OPA693

8P0U508
DIS

P0R1902

P0U902

R19

OUT
ADJ

GND
LM317T2

LM7812CT
P0U803
3

330

Rf

4u7

C1
P0C101 P0C102

GND

GND

P0C801 P0C802

OUT
GND

GND

C19
100n VCC

C8

P0C1902
P0C1901

GND

P0R2702

D

C

B

A

Figure A.2: Electronic circuit of the analog AOM-driver. Umod is sent to input P4 and controls Vtune . Instead of themodel POS-150 we use the model VCO190-112T from Varil. The TTL-switch OPA693, the preamplifier ERA4-SM
and the amplifier MHW9267 are bridged and replaced coaxially by components from Mini-Circuits (ZASWA-2-50DR,
l ZFL-500LN+, ZHL-1-2W). C13 had to be removed, because of influences on the signal at high frequencies.

D

C

B

8

K

A

3P0U1703
NC

2

VCC

1P0U301A

P0P102

1
P0U1702

P0U303
P0U307
P0U306
P0U305
P0U304

P0R302
P0R301

P0U1701

P0C1302 P0C1301

P0D102
P0R602P0D101
P0R601

P0R802
P0R801

P0R702
P0R701
P0R902
P0R901

P0R202

P0C901 P0C902

R2

P0C1802 P0C1801

1K

P0C301 P0C302

2

P0C2101 P0C2102

P0P101P0R201

2

P0P402

P0R1501

P0R1802
P0R1502

P0R1801

5
4

1
7
5
4

DISABLE

P0P302

P0U505

P0U101

P0U105VCC

P0L102

P0R1102
P0R1101
P0R1602
P0R1601

1P0U501
7P0U507
P0U504

P0U107

P0U104

P0R101
P0R102
P0P202U17

2

P0P502

P0D401
P0R2802P0D402

P0R2801
P0C2502 P0C2501

2

P0C1102 P0C1101

P0C202 P0C201

3P0U603
P0U602

P0R1402
P0L402
P0R1401
P0L401

P0L202
P0L201

1

P0U1002

2
1

P0C401 P0C402
P0C1601 P0C1602

P0U802

P0U901
P0R2002

P0R2001

P0C1702 P0C1701
P0C2002 P0C2001

P0R2302
P0DS101
P0R2301
P0DS102

P0R2402
P0DS201
P0R240173
P0DS202

P0P602P1A ELECTRONICS

A ELECTRONICS

Figure A.3: Realized electronic box for the control of the AOM.

74

Appendix B
Dipole trap modelOn the following pages the source code of the MATHEMATICA file to model
the trapping potential of our dipole trap for the ground state of bosonic erbium atoms is shown. The definitions of the various formulas can be found in
Cha. 3.3.
The file is subdivided into five main parts:
Constants and data import: In this part the constants used in the file aredefined. Also the data of all dipole allowed transitions contributing to
the dipole trap are imported, hence the wavenumber of the allowed transitions from the ground to the excited states, as well as the total angular momentum quantum number J and the linewidth of the excitedstates [Wya12]. Furthermore, each linewidth is adapted according to theline strength factors related to the 3J-symbol. The total dipole potential and the scattering rate are calculated from Eq. (3.13) and Eq. (3.14)(Here the intensity I(~r) is set to 1).
Dipoletrap formula: In this section the functions to calculate the total intensity profile for crossed elliptic beams under an angle are defined. With
this intensity the real total dipole potential and the scattering rate – foreach beam – can be calculated.
Parameter: In this part all known parameters as beam waists, powers, atom
number and temperature are added.
Output: Before the trapping parameters are calculated, the trapping geometry has to be recalculated each time the input parameter are changed.
Therefore we calculate the trap depth of the horizontal and the vertical
dipole trap, the effective beam waists and Rayleigh ranges due to theangles α and β between the beams, the trap frequencies, the scatteringand heating rates, the phase space density and the peak number density,
the influence of the gravity on the vertical confinement as well as the75

B DIPOLE TRAP MODEL
magnetic field gradient to levitate the trap against gravity.
Figures: The calculated data are used to plot the change of the dipole trap
potential because of the gravity, as well as the 3D trapping profile alongeach axis.
This model also can be used in any other ultracold experiment. Here is a list of
the values, which have to be changed to adapt this model to any other element
and trapping geometry:• the wavelength λ of the used dipole trap laser – here given as λ1064.
• the trapping potential Udip for an intensity I of one – here given as U 1064.
• the scattering rate Γsc for an intensity I of one – here given as Γ1064.
• In this file the two values U 1064 and Γ1064 are calculated from implemented data, which includes the contribution of all the dipole allowedtransitions from the ground to the excited states for bosonic erbium
atoms according to their line strength factors [Wya12]. U 1064 is nothingelse than the atomic polarizability and already includes the scalar andthe tensorial part.
• the horizontal and the vertical waist of each laser beam – here given
as waist1y, waist1x for the horizontal beam and waist2z , waist2x for thevertical beam.
• the power of each laser beam – here given as power1 and power2.
• the angles between the two laser beams α and β; the definition of thesevalues can be found in Cha. 3.3.
• As an additional feature the model also calculates the phase-space-density(psd ) and the peak number density (npeak ). To get the right results, one
also needs to change the values for the atom number (Natom) and thetemperature (Tatom) in the trap.

76

Clear@"Global`*"Dmm
c = 299 792 458H* *L;
g = 9.804864698H**L;ss2
hbar = 1.054571726 * 10-34 H*Js*L;h = hbar * H2 PiLH*Js*L;JJ
kB = 1.3806488 * 10-23 H* *L;µB = 9.27400968 * 10-24 H* *L;K
TAs
ermass = 166 * 1.660538921 * 10-27 H*kg*L; Ε0 = 8.854187817 * 10-12 H**L;
Vm
2 Pi * cΛ1064 = 1064 * 10-9 H*m*L;Ω1064 =
Λ1064
H*s-1 *L;

Import data

H*Import data from .dat file,including all allowed transitions from the ground to excited states*L
H*wzgr imports all the wavenumbers from the transition of the ground to the excited states*L
H*Jex imports all the total angular momentum quantum numbers of the excited states*L
H*ggr1 imports all the linewidths of the excited states times the statistical weightof these levels H15L*L
AppendTo@$Path, NotebookDirectory@DD;
file = ImportA"Er_groundtransition.dat", "Table"E;ground = file@@19 ;; 1194DD;
wzgr = ground@@All, 7DD;
Jex = Round@ground@@All, 3DDD;
ggr1 = ground@@All, 10DD;
H*Calculate linewidth G and substitute by known measured G values for the 401 nm and 583 nm line;
The multiplication with the WignerJ accounts for the line strength factor of each transition*L
H*Calculate the optical transition frequencies*L
ggr1
ggr =
;
15
WignerJ = Table@H2 J + 1L * ThreeJSymbol@86, -6<, 81, 0<, 8J, 6<D ^ 2, 8J, Jex<D;
Ggr = ReplacePart@ggr, 8836< ® 1.8686 * 10 ^ 8, 88< ® 1.17 * 10 ^ 6<D * WignerJ;Ωgr = 2 * Pi * c 

10-2
wzgr * 103

H*s-1 *L;H*Dipoletrap formula*L
H*Calculates the potential depth of a dipole trap depending on the transitionfrequencyΩa and the linewidth Gamma of the excited state Hwith incoming light Λ=1064nmL*L
Gamma
1
1
UAGamma_, Ωa_E := - 3 * Pi * c2 **+3Ωa - Ω1064 Ωa + Ω1064
2 * Ωa2
1Ω1064 3Gamma
Gamma
GammascatAGamma_, Ωa_E := 3 * Pi * c2 *
*
*
+
3
Ωa
Ωa - Ω1064 Ωa + Ω1064
2 * hbar * Ωa
H*Potentialdepth Hatomic polarizabilityL and scattering rate of the dipole trap;
just intensity has to be added*L
H*Consider all known transitions from groundstate to other states*L
m2
U1064= Sum@U@Ggr, ΩgrD@@iDD, 8i, Length@ΩgrD<DH*J*LWReΑ = -U1064 * 2 * Ε0 * c  H0.16487776 * 10 ^ -40L a.u
m2
Gsc1064 = Sum@HGammascat@Ggr, ΩgrDL@@iDD, 8i, Length@ΩgrD<DH*s-1
*L
W
-37

-4.93837 ´ 10
159.009 a.u

4.98914 ´ 10-12

Dipole trap formula77

Dipole trap formula
H*Gaussian beam waist*LzwAw0_, z_E := w0
H*Rayleigh length*L

2

1+
zR@w0D

w02
zRAw0_E = Pi *;
Λ1064
H*Rayleigh length elliptic beam*L

zRellAwx_, wy_E = HzR@wxD zR@wyDL .

* IzR@wxD2 + zR@wyD2 M ;2
H*Calculates waist and Railey length of an elliptic beam having an angle alpha tothe coordinate system in the y-z planeΗ parameter used for =1: determining only the new beamwaist; =2: for including into trap frequency
Hcontribution of axial confinment is half of radial confinementL*LwΑangle Awz_, wx_, Α_, Η_E :=1

Cos@Α * Pi  180D2

zRellangle Awz_, wx_, Α_, Η_E :=w@wz, 0D2

+ Sin@Α * Pi  180D2  IΗ * zRell@wz, wxD2 M

Sin@Α * Pi  180D2
w@wz, 0D2-

1
2

+ Cos@Α * Pi  180D2  IΗ * zRell@wz, wxD2 M

;

-

1
2

;

H*calculates waist change of an elliptic beam having an angle beta to the coordinate
system in the x-z plane*Lw2zΒangle Awz_, wx_, Β_, Α_, Η_E := Cos@Β * Pi  180D2  wΑangle @wz, wx, Α , ΗD2 +w2xΒangle Awz_, wx_, Β_, Α_, Η_E := Sin@Β * Pi  180D2  wΑangle @wz, wx, Α , ΗD2 +H*Intensity of the light beam depending on waists w0x and w0y,Power P,
radial compontents xHhorizontalL and yHverticalL and axial distance z*LintensityAP_, w0x_, w0y_, x_, y_, z_E :=H2 * PL  HPi * w@w0x, zD * w@w0y, zDL * ExpA-2 *x2
w@w0x, zD2y2+

w@w0y, zD2

1
Sin@Β * Pi  180D2 - 2
w@wx, 0D2
1
Cos@Β * Pi  180D2 - 2
w@wx, 0D2

E

H*Horizontal beam: Potential depth in J and µK
Scattering rate*L
Upotgr1Ax_, y_, z_E := U1064 * intensity@power1, waist1x, waist1y, x, y, zD;
1

106
kB
Gscat1Ax_, y_, z_E := Gsc1064 * intensity@power1, waist1x, waist1y, x, y, zD;
Tpotgr1Ax_, y_, z_E := Upotgr1@x, y, zD

H*Vertical beam: Potential depth in J and µK
Scattering rate*L
Upotgr2Ax_, y_, z_E := U1064 * intensity@power2, waist2x, waist2z, x, z, yD;
1

106
kB
Gscat2Ax_, y_, z_E := Gsc1064 * intensity@power2, waist2x, waist2z, x, y, zD;
Tpotgr2Ax_, y_, z_E := Upotgr2@x, y, zD

H*Euler rotation on the x and z angle*L
xangleAx_, z_E := x * Cos@beta * Pi  180D + z * Sin@beta * Pi  180D;

yangleAx_, y_, z_E := x * Sin@alpha * Pi  180D Sin@beta * Pi  180D + y * Cos@alpha * Pi  180D ;78

z * Sin@alpha * Pi  180D Cos@beta * Pi  180D;
zangleAx_, y_, z_E := -x * Cos@alpha * Pi  180D Sin@beta * Pi  180D + y * Sin@alpha * Pi  180D +z * Cos@alpha * Pi  180D Cos@beta * Pi  180D;
H*Spatial potential of vertical beam due to angles Α and Β*L
Upotgr2angleAx_, y_, z_E :=U1064 * intensity@power2, waist2x, waist2z, xangle@x, zD, zangle@x, y, zD, yangle@x, y, zDD;
1
Tpotgr2angleAx_, y_, z_E := Upotgr2angle@x, y, zD
106 ;
kB

Parametery .. direction of gravity Haxial direction of vertical beamL
x .. horizontal, perpendicular to horizontal beamz .. axial direction of horizontal beam
H*Horizontal beam*L

waist1y = 16.1 * 10-6 ;H*Vertical beam*L

waist2z = 55 * 10-6 ;waist1x = 27 * 10-6 ;power1 = 1;

waist2x = 110 * 10-6 ;power2 = 10;

H*Angles between vertical and horizontal beam; Α..rotation in y-z-plane;
Β..rotation in x-z-plane;
Corresponds in our experiment to the angle of the vertical beam to the axisof gravity*Lalpha = 15H*°*L;

H*Corresponds in our experiment to the angle of the horizontal beam in thehorizontal planeHhalf angle between viewports = 14°L*Lbeta = 14H*°*L;

H*Measured data*L

Natom = 2.5 * 106 ;Tatom = 3.6 * 10-6 ;OUTPUT

H*Effective potential depth of the dipole trap in µK depending on the radial andaxial parameters of the horizontal beam*L
U1gr = -Upotgr1@0, 0, 0D;
T1gr = -Tpotgr1@0, 0, 0D;
H*Effective potential depth of the dipole trap in µK depending on the radial andaxial parameters of the vertical beam*L
U2gr = -Upotgr2@0, 0, 0D;
T2gr = -Tpotgr2@0, 0, 0D;
H*WaistRayleigh length change of vertical beam due to angle between beams*L
w2znew = NAw2zΒangle @waist2z, waist2x, beta, alpha, 2DE;
w2xnew = NAw2xΒangle @waist2z, waist2x, beta, alpha, 2DE;

zRell2new = NAzRellangle @waist2z, waist2x, alpha, 2DE;
H*Gravity deforms potential in vertical direction;
horizontal and vertical beam calculated independently
Hz-y-: particles with higher "altitude" have higher potential energy and could
therefore escape from the DT;

79*L

Hz-y-: particles with higher "altitude" have higher potential energy and could
therefore escape from the DT; with a magnetic field gradient Blev this can be compensated*L
Ugrav Ay_E = ermass * g * y;Blev =

Ugrav @1D
7 * µB* 102 H*

G

*L;

cm

T1gravity Ay_E = IUgrav Ay * 10-6 E + Upotgr1A0, y * 10-6 , 0EM

106

H*µK*L;kB
Min1 = FindMinimumA9T1gravity @yD, -waist1y * 106 £ y £ waist1y * 106 =, 8y, 0<EH*µK*L;

Max1 = FindMaximumA9T1gravity @yD, -10 * waist1y * 106 £ y £ Hy . Last@Min1DL=, 8y, 0<EH*µK*L;
T1real = -HFirst@Min1D - First@Max1DLH*µK*L;
T1real,y = If@power1 <= 0, 0, If@ T1real < 0, 0, T1real DDH*µK*L;
kB
U1real,y = T1real,y *106

H*J*L;

T2gravity Ay_E = IUgrav Ay * 10-6 E + Upotgr2angleA0, y * 10-6 , 0EM

106

H*µK*L;kB
Min2 = FindMinimumA9T2gravity @yD, -zRell2new * 106 £ y £ zRell2new * 106 =, 8y, 0<EH*µK*L;

Max2 = FindMaximumA9T2gravity @yD, -10 * zRell2new * 106 £ y £ Hy . Last@Min2DL=, 8y, 0<EH*µK*L;
T2real = -HFirst@Min2D - First@Max2DLH*µK*L;
T2real,y = If@power2 <= 0, 0, If@ T2real < 0, 0, T2real DDH*µK*L;

H*J*L;
106
H*Trapfrequencies due to angle Α and Β between the horizontal and the vertical beam;same coefficient for zR and waist contribution because Η parameterHsee aboveL takes
care of this*L
U1gr
U2gr
4
Ωx
Ωx = .
+; Νx =;
22
ermass waist1x
2 * Piw2xnewkB

U2real,y = T2real,y *Ωy = .

ermassΩz = .

U1gr

4

1

waist1y2+

U2gr
zRell2new 2; Νy =Ωy;
2 * PiI2 U1gr M  zRell@waist1x, waist1yD2 +4 * U2gr
2; Νz =Ωz;
2 * Pi

H*Trapfrequencies due to angle Α and Β between the horizontal and the vertical beamincluding also gravity in y directionHlowering of potential depthL*LΩygravity
U1real,yU2real,y4Ωygravity = .
+; Νygravity =;ermass waist1y2
2 * PizRell2new 2
E
1
H*Heating rate in three dimension Theat = *Trec *Gscmean ; Trec =2* rec =101nK;
3kB
Erec =Hhbar*klaser L2 2m*L
ermassw2znew

H*Phase-space-density and peak number density according to temperature,atomnumber and trap frequencies*L21h1Theat = *** HGscat1@0, 0, 0D + Gscat2@0, 0, 0DL;
3
Λ1064
ermass * kB
hbar
psd =

3
* Natom * Ωx * Ωy * Ωz;kB * Tatom
ermassnpeak = Natom *3
2 * Ωx * Ωy * ΩzH*m -3 *L;

2 * Pi * kB * Tatom

80

StyleFormA
NumberFormA98"Er DT Groundstate", "beamsangle", unit, "with gravity"<, 8<,

TableFormA

9"U0,beam horizontal ", T1gr , µK, T1real,y =, 9"U0,beam vertical ", T2gr , µK, T2real,y =,9"Νx,axis horizontal ", Νx, s-1 =, 9"Νy,axis vertical ", Νy, s-1 , Νygravity =,9"Νz,axis axial ", Νz, s-1 =, 8phspden, psd, 1<, 9n0, npeak * 10-6 , cm-3 =,8"Total scattering rate", Gscat1@0, 0, 0D + Gscat2@0, 0, 0D, 1  s<,
9Heating rate, Theat * 109 , nK  s=, 8zRhorizontal , N@zRell@waist1x, waist1yDD * 1000, mm<,
8zRvertical , N@zRell@waist2x, waist2zDD * 1000, mm<,8"mag. field gradient to levitate", Blev , G  cm<=,

TableSpacing ® 81, 1<E,
5E,

Background ® LightBlue, FontFamily ® "Times", FontWeight ® "Bold", FontSize ® 16E

Er DT Groundstate

beamsangle unit with gravity

U0,beam horizontal
U0,beam vertical

52.38337.638µKµK

47.1010.81345

Νx,axis horizontal

618.85Νy,axis vertical

1014.8960.33

Νz,axis axial

237.73

1
s
1
s
1
s

phspden

0.88431n02.4278 ´ 1015

Total scattering rate

0.012556

Heating rate

0.42652

1
cm3
1
s
nK
s

zRhorizontal
zRvertical

1.019812.254mag. field gradient to levitate 4.1633

Figures

81

mm
mm
G
cm

Figures

H*Deformation of the horizontal beam potential Hy-directionL due to gravity*L"Gravity on horizontal beam \Gravity on vertical beam"
GraphicsRowA9

PlotAT1gravity @yD, 8y, -2 * waist1y * 10 ^ 6, 2 * waist1y * 10 ^ 6<, PlotRange ® Full,
AxesLabel ® 8"vertical position y@µmD", "total potential@µKD"<E,

H*Deformation of the vertical beam potential Hy-directionL due togravity: angles Α and Β are included*L
PlotAT2gravity @yD, 8y, -2 * zRell2new * 10 ^ 6, 2 * zRell2new * 10 ^ 6<, PlotRange ® Full,=E

AxesLabel ® 8"vertical position y@µmD", "total potential@µKD"<E"VerticalHyL potential  HorizontalHxL Potential  AxialHzL Potential "GraphicsRowA9

Plot3DATpotgr1@x * 10 ^ -6, 0, z * 10 ^ -6D + Tpotgr2angle@x * 10 ^ -6, 0, z * 10 ^ -6D, 8z, -100, 100<,
8x, -100, 100<, PlotRange ® Full, AxesLabel ® 9"z HµmL", "x HµmL", "Udip HµKL"=,Mesh ® None, MaxRecursion ® 0, PlotPoints ® 815, 15<E,

Plot3DATpotgr1@0, y * 10 ^ -6, z * 10 ^ -6D + Tpotgr2angle@0, y * 10 ^ -6, z * 10 ^ -6D, 8z, -100, 100<,
8y, -100, 100<, PlotRange ® Full, AxesLabel ® 9"z HµmL", "y HµmL", "Udip HµKL"=,Mesh ® None, MaxRecursion ® 0, PlotPoints ® 815, 15<E,

Plot3DATpotgr1@x * 10 ^ -6, y * 10 ^ -6, 0D + Tpotgr2angle@x * 10 ^ -6, y * 10 ^ -6, 0D, 8x, -100, 100<,
8y, -100, 100<, PlotRange ® Full, AxesLabel ® 9"x HµmL", "y HµmL", "Udip HµKL"=,=E

Mesh ® None, MaxRecursion ® 0, PlotPoints ® 815, 15<E

Gravity on horizontal beam

Gravity on vertical beam

total potential@µKD

total potential@µKD
vertical position y@µmD

-30

-20

10

-10

20

30

50

-10
-20

vertical position y@µmD
-400

200

-200

400

-30
-40

-50

-50

VerticalHyL potential  HorizontalHxL Potential  AxialHzL Potential

82Appendix C
PRL: Bose-Einstein Condensationof Erbium83

PRL 108, 210401 (2012)

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

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25 MAY 2012

Bose-Einstein Condensation of Erbium
K. Aikawa,1 A. Frisch,1 M. Mark,1 S. Baier,1 A. Rietzler,1 R. Grimm,1,2 and F. Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Received 6 April 2012; published 21 May 2012)
We report on the achievement of Bose-Einstein condensation of erbium atoms and on the observation ofmagnetic Feshbach resonances at low magnetic fields. By means of evaporative cooling in an optical dipole
trap, we produce pure condensates of 168 Er, containing up to 7  104 atoms. Feshbach spectroscopy reveals
an extraordinary rich loss spectrum with six loss resonances already in a narrow magnetic-field range up to
3 G. Finally, we demonstrate the application of a low-field Feshbach resonance to produce a tunable dipolar
Bose-Einstein condensate and we observe its characteristic d-wave collapse.
DOI: 10.1103/PhysRevLett.108.210401

PACS numbers: 03.75.Nt, 37.10.De, 51.60.+a, 67.85.Hj

quantum gas experiments, Feshbach resonances and scattering lengths are a priori unknown. Magnetic lanthanides
such as erbium with their large magnetic moments andtheir non-S electronic ground states present a completely
unexplored terrain in ultracold scattering physics. Here theanisotropic interaction is expected to give rise to novel
scattering scenarios, which are not accessible with alkali
atoms [18,19].
In this Letter, we report on the attainment of
Bose-Einstein condensation of erbium atoms and on theobservation of Feshbach resonances in the region of low
magnetic fields. We obtain pure optically trapped BECs of
168
Er containing 7  104 atoms. The remarkably highefficiency of evaporative cooling in a standard optical
dipole trap indicates favorable scattering properties of the
168 Er isotope. In addition, the magnetic Feshbach spectroscopy provides first valuable information on the scattering
behavior of submerged-shell atoms at ultralow temperatures. Moreover, we demonstrate low-field Feshbach tuning of the contact interaction in our strongly dipolar BEC.
Our experimental procedure to create a BEC of Er
follows a simple and straightforward scheme, inspired bywork on Yb atoms [20,21]. Our starting point is thenarrow-line yellow magneto-optical trap (MOT) describedin our very recent work [22]; it operates on the 583 nm line
(natural linewidth 190 kHz). We choose this approach
because narrow-line MOTs permit us to obtain samples
with a large number of atoms at temperatures in the lower
microkelvin region. This allows a direct and efficienttransfer of atoms into optical dipole traps without theneed for additional cooling stages [20,21]. Our MOT gives
about 108 atoms at a temperature of 15 K [23].
An additional very advantageous feature of our approachis that the MOT light automatically pumps the atoms into
the lowest Zeeman sublevel mJ ¼ 6, where mJ is the
projection quantum number of the total electronic angularmomentum J ¼ 6. This effect results from the interplay
between gravity and weak radiation pressure, which leads
to a spatial down shift with respect to the zero of the

Ultracold quantum gases have proven to be ideal
systems for observing spectacular many- and few-bodyquantum effects. The large majority of these phenomena
rely on the high degree of control over the interparticle
interaction achieved with ultracold atoms. In the widelyused alkalis, ultracold atoms interact isotropically via ashort-range contact potential. A novel exciting frontier inquantum gas experiments is to access unexplored physicalscenarios based on the anisotropic and long-range dipoledipole interaction (DDI) [1,2]. A dipolar quantum gas is
expected to exhibit fascinating phenomena, including
novel many-body quantum phases [3–6]. The DDI actsin systems having sizable electric or magnetic dipole
moments [1].
In the context of ultracold atomic quantum gases,
pioneering experimental work on strong DDI has been
carried out with chromium atoms [7–9]. Magnetic lanthanides offer new possibilities for dipolar physics. In such
systems, the combination of a large magnetic moment anda large atomic mass leads to a particularly strong dipolar
character. The demonstration of the first magneto-opticaltrap of erbium atoms [10] stimulated growing interest insuch species for quantum gas experiments. Very recently, a
Bose-Einstein condensate (BEC) and a degenerate Fermi
gas of dysprosium have been produced [11,12]. We choose
erbium as a promising candidate for experiments on dipolar quantum gases. This species has a number of very
appealing features, including a large magnetic moment of 7 times the Bohr magneton, several stable isotopes, arich energy level scheme [13] with a non-S electronic
ground state [14], and interesting cold collisional
phenomena [15,16].
In strongly magnetic atoms, the competition between the
DDI and the contact interaction is very important and gives
rise to many intriguing phenomena. The contact interaction
is determined by the s-wave scattering length a and can beoften tuned with external magnetic fields via Feshbachresonances [17]. Tuning of a also controls the balance of
these two interactions. In the case of a novel species in
0031-9007=12=108(21)=210401(5)210401-1

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Ó 2012 American Physical Society

PRL 108, 210401 (2012)

week ending
25 MAY 2012PHYSICAL REVIEW LETTERS

magnetic quadrupole field [24] and thus to a preferential
absorption of the vertical MOT beam with  polarization[22]. The polarization of the sample is confirmed by
Stern-Gerlach-type measurements.
Our optical dipole trap geometry follows the concepts
originally applied in experiments with Yb BEC [20]. The
trap is created by crossing a tightly confining horizontal
beam (y axis) and a less focused vertical beam (z axis). The
basic idea is that initially the atoms are predominantly
trapped by the horizontal beam, whereas the verticalbeam provides confinement relevant in the final stage of
evaporation. The horizontal beam is derived from a 100 W
broadband Yb fiber laser operating at 1075 nm and has an
initial power of 10 W. The beam has an elliptic crosssection with a waist of 30ð40Þ m along the vertical(horizontal) direction. The vertical beam is produced by
a 10 W Yb fiber laser source at 1064 nm and has an initial
power of 8 W. The beam profile is elliptic with a waist of55ð110Þ m along (perpendicular to) the axis of the horizontal beam.
We load the dipole trap during the MOT compression
phase. We observe that the time period in which the compressed MOT and dipole trap coexist is crucial for efficientloading. The number of atoms in the optical dipole trap
exponentially approaches its maximum value with a time
constant of about 150 ms. After 600 ms of loading, we turnoff the MOT beams and the quadrupole magnetic field, andwe switch on a weak homogeneous magnetic field of about
300 mG along the vertical direction to preserve the spinpolarization of the sample. We obtain 2:6  106 atoms at atemperature of 42 K in the optical dipole trap. At thispoint, the atoms are mainly trapped by the horizontal
beam. We measure oscillation frequencies ðx ; y ; z Þ ¼
ð1:3; 0:016; 1:95Þ kHz; the potential depth is estimated to
be 560 K. The peak density and the peak phase-spacedensity are 1:7  1013 cm3 and 1:6  104 , respectively.
These are our starting conditions for the evaporative
cooling process.
Forced evaporative cooling is performed by reducing thepower of the trapping beams in a nearly exponential ramp.
The overall evaporation sequence has a duration of 5.5 s[25]. We then turn off the trapping beams and let the atomic
cloud expand before applying standard absorption imaging. For imaging, we illuminate the atomic cloud with a
50-s probe beam pulse [26]. The probe beam propagateshorizontally at an angle of 14 with respect to the propagation axis (y axis) of the horizontal trapping beam.
The phase transition from a thermal cloud to BEC manifests itself in a textbooklike bimodal distribution in the timeof-flight absorption images. Figure 1 shows the absorption
images and the corresponding linear density profiles fordifferent final temperatures, i.e., for different stages of theevaporation. At higher temperatures the atomic distribution
is thermal with the expected Gaussian profile resulting
from the Maxwell-Boltzmann distribution; see Fig. 1(a).

(a)

T=1100 nK(b)linear density (108 m-1)

408 nK0.4(c)

222 nK

(d)

0.3
0.2
0.1
0

1mm

position (µm)

FIG. 1 (color online). Absorption images and integrated
density profiles showing the BEC phase transition for different
evaporation times. The absorption images are an average of four
images taken after 24 ms of expansion. The color bar shows the
optical density. The solid lines are fits to the data using
Gaussian (a), bimodal (b) and (c), and Thomas-Fermi (d)distribution. The dotted lines represent the Gaussian part ofthe bimodal fit, describing the thermal atoms. From the fit
we extract: N ¼ 3:9  105 , T ¼ 1100 nK (a), N ¼ 2:1  105 ,T ¼ 408 nK (b), N ¼ 1:6  105 , T ¼ 222 nK (c), N ¼
6:8  104 (d), where N is the total atom number. For (b) and(c), we extract a condensate fraction of 5% and 20%, respectively.

By cooling the atomic sample below the critical temperature, we clearly observe that the atomic density distribution
has a bimodal profile with a narrower and denser peak at thecenter, which represents the BEC (b). By further evaporating
the sample, the BEC fraction continuously increases(c) until the thermal component is not anymore discernible
and an essentially pure BEC is formed with 7  104
atoms (d).
To analyze our data, we fit a bimodal distribution to theintegrated time-of-flight absorption images. This distribution consists of a Gaussian function, which accounts for thethermal atoms, and an inverted integrated parabolic function for the BEC component in the Thomas-Fermi limit.

210401-2

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Just after the onset of quantum degeneracy (BEC fraction5%), we measure trap frequencies of ðx ; y ; z Þ ¼
ð208; 70; 299Þ Hz, atom number of N ¼ 2:1  105 , and atemperature of T ¼ 408 nK. The critical temperature of
417 nK as calculated from standard BEC theory (without
interaction shift) is consistent with this observation.
The evaporation efficiency is found to be remarkably
high as 3.5 orders of magnitude in phase-space density are
gained by losing a factor of 10 in atom number. Thisobservation already points to favorable scattering parameters of the 168 Er isotope. First evaporative cooling experiments on the most abundant 168 Er isotope reveal a lower
efficiency in the final stage of evaporation, suggesting thata different strategy might be needed to reach BEC.
To gain insight into the ultracold collisional properties
of erbium we perform Feshbach spectroscopy [17] at low
magnetic fields. This measurement is done in a way thatallows us to identify both the poles and zero crossings ofFeshbach resonances [27,28]. The basic idea here is toprepare the system at a variable target value of the magnetic field and then to rapidly (50 ms) decrease the depth ofthe optical dipole trap by almost a factor of 2. The sample
stays near thermal equilibrium with an effective temperature of 2:2 K but features a truncated energy distribution.
We then let the system evolve at a constant trap depth
for 250 ms, during which plain evaporative cooling andinelastic losses can occur depending on the scattering
length. We finally switch off the trap and take time-offlight images to determine the temperature and number of
the atoms. The measurement is then repeated for variablemagnetic-field values. Such a Feshbach scan shows resonance poles as loss features and zero crossings as temperature maxima.
Figure 2 shows the loss spectrum and the corresponding
temperatures in the low magnetic-field range up to 3.2 G[29]. Already in this narrow magnetic-field range, the loss
spectrum is very rich. We identify six pronounced resonant
minima in the atom number that we interpret as beingcaused by Feshbach resonances. For convenience, we determine the resonance positions with Gaussian fits, yielding 0.72, 0.91, 1.51, 2.16, 2.48, and 2.85 G. The lossfeatures show different strengths and widths. For the threebroader resonances at 0.91, 2.16, and 2.48 G, we also
observe the appearance of temperature maxima to the rightof the loss minima (arrows in Fig. 2). These temperature
maxima mark the zero crossings of the scattering length.
The other loss features are too narrow to provide clear
signatures of the zero crossing. From the difference inpositions between the minima in the atom number andthe maxima in temperature we estimate the widths  ofthe resonances. We find  ¼ 65, 60, and 180 mG for the
resonances at 0.91, 2.16, and 2.48 G, respectively.
In a strongly dipolar atomic gas, universal dipolar scattering is present [30–32], so that the total cross section forelastic scattering does not vanish at the zero crossings ofweek ending
25 MAY 2012

FIG. 2 (color online). Observation of Feshbach resonances in
Er-Er collisions. The measured temperature (a) and atom
number (b) are plotted as a function of the magnetic field. The
minima in the atom number indicate the Feshbach resonancepoles, marked by the thin vertical lines. The maxima in thetemperatures to the right of the three stronger loss features(arrows) are attributed to the respective zero crossings of thescattering length. The varying background in the atom number ispresumably due to ramping effects caused by the sweep of themagnetic field across the resonances.

the scattering length. For Er, a minimum cross section
dip ¼ 8ð30a0 Þ2 results from universal dipolar scattering, where a0 is the Bohr radius. The fact that we observe
temperature maxima near the zero crossings suggests a
dominant role of s-wave scattering and not of dipolar
scattering. Preliminary cross-dimensional thermalization
measurements indeed point to a scattering length between
150 and 200a0 .
The existence of Feshbach resonances at low magnetic
fields makes the manipulation of the contact interaction in
the Er BEC very convenient and straightforward. As aproof-of-principle experiment, we explore the controlled
d-wave collapse of the BEC, following the procedure
successfully applied by the Stuttgart group [7]. We first
produce a pure BEC by evaporative cooling at 1.2 G, which
is above the position of the first broad Feshbach resonance(0.91 G). Here we obtain 3  104 atoms in the BEC,
indicating that forced evaporation at this magnetic field isslightly less efficient. We then ramp down the magnetic
field within 2 ms to a variable target value and let the
sample evolve for 2 ms before switching off the trap. The
magnetic field is kept constant at its target value during thefirst stage of the expansion (15 ms), where the main
dynamics happens. We then set the magnetic field along
the y axis and we image the atomic cloud after an additional 11 ms of expansion. Our results are summarized inFig. 3, where we show time-of-flight absorption images fordifferent values of the target magnetic field. We observe a
dramatic change in the shape of the condensate when the210401-3

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PHYSICAL REVIEW LETTERS

PRL 108, 210401 (2012)
(a)

(b)

(c)

[4] B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S.
Kotochigova, P. Julienne, and P. Zoller, Phys. Rev. Lett.
90, 110401 (2003).
[5] G. Pupillo, A. Micheli, H. P. Büchler, and P. Zoller, in
Cold Molecules: Theory, Experiment, Applications, editedby R. V. Krems, W. C. Stwalley, and B. Friedrich (CRC
Press, Boca Raton, 2009), Chap. 12.
[6] D. S. Jin and J. Ye, Phys. Today 64, 27 (2011).
[7] T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A.
Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda,
Phys. Rev. Lett. 101, 080401 (2008).
[8] T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A.
Griesmaier, S. Giovanazzi, and T. Pfau, Nature (London)
448, 672 (2007).
[9] B. Pasquiou, G. Bismut, E. Maréchal, P. Pedri, L. Vernac,
O. Gorceix, and B. Laburthe-Tolra, Phys. Rev. Lett. 106,015301 (2011).
[10] J. J. McClelland and J. L. Hanssen, Phys. Rev. Lett. 96,
143005 (2006).
[11] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys.
Rev. Lett. 107, 190401 (2011).
[12] M. Lu, N. Q. Burdick, and B. L. Lev, arXiv:1202.4444[Phys. Rev. Lett. (to be published)].
[13] H. Y. Ban, M. Jacka, J. L. Hanssen, J. Readerand, and J. J.
McClelland, Opt. Express 13, 3185 (2005).
[14] The Er electronic configuration is [Xe] 4f12 6s2 . The
electronic ground state is a 3 H6 state, meaning L ¼ 5(H state), S ¼ 1, and J ¼ 6. L, S, J denote the quantum
numbers for the orbital angular momentum, for the spin,and for the total electronic angular momentum, respectively. 168 Er has zero nuclear spin (I ¼ 0) and consequently does not exhibit a hyperfine structure.
[15] R. V. Krems, G. C. Groenenboom, and A. Dalgarno, J.
Phys. Chem. A 108, 8941 (2004).
[16] C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle, andJ. M. Doyle, Phys. Rev. A 81, 010702 (2010).
[17] C. Chin, R. Grimm, P S. Julienne, and E. Tiesinga, Rev.
Mod. Phys. 82, 1225 (2010).
[18] S. Kotochigova and A. Petrov, Phys. Chem. Chem. Phys.
13, 19165 (2011).
[19] A. Petrov, E. Tiesinga, and S. Kotochigova, arXiv:1203.4172.
[20] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M.
Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev.
Lett. 91, 040404 (2003).
[21] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A
76, 051604(R) (2007).
[22] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,
E. Zupanic, R. Grimm, and F. Ferlaino, arXiv:1203.1460[Phys. Rev. A (to be published)].
[23] The MOT stage includes a 15 s loading phase, in which thelight is red detuned by 9:5 MHz, and a compression
phase, in which we reduce the detuning to 0:8 MHz and
the MOT beam intensity by about a factor of 400. The
compression phase is used to obtain colder and denser
samples and it is optimized for loading efficiency into thedipole trap.
[24] H. Katori, T. Ido, Y. Isoya, and M. Kuwata-Gonokami,
Phys. Rev. Lett. 82, 1116 (1999).
[25] Initially only the power of the horizontal beam is reduced
to its final value of 230 mW, while the vertical beam power
is reduced during the last 600 ms of evaporation to its final

0.4

0.3

(d)

(e)

(f)

week ending
25 MAY 2012

0.2

0.1
0

FIG. 3 (color online). Absorption images showing the d-wave
collapse of the BEC. The field of view is 290 m  290 m.
The images are an average of eight pictures. The color bar shows
the optical density. The images are taken for different target
values of the magnetic field: 1.208 G (a), 0.963 G (b), 0.947 (c),0.942 G (d), 0.939 G (e), and 0.934 (f). We note that the actual
magnetic-field value might be slightly higher (10 mG) becauseof ramping issues and eddy currents [7].

magnetic field is reduced towards the zero crossing of
the scattering length. At the magnetic field of evaporation,
the aspect-ratio of the cloud is close to the one observed atzero magnetic field; see Fig. 3(a). By changing the magnetic field to lower target values, the BEC shows a more
and more anisotropic shape (b) and (c). Below a criticalmagnetic-field value, the BEC develops a complicatedcloverleaf pattern (d)–(f) which is the striking signatureof the d-wave collapse in a dipolar BEC [7].
In conclusion, we have demonstrated the first BEC of
Er atoms and the tunability of its interparticle interaction
via Feshbach resonances. Our scattering data provide firstsensitive input to understand the complex collisional
behavior of submerged-shell atoms. The observation of
the d-wave collapse is very encouraging in view of future
experiments dedicated to the rich phenomena expected instrongly dipolar quantum gases.
We are grateful to F. Schreck, S. Stellmer, and
B. Pasquiou (Innsbruck Sr team) for their continuous support. We thank the Yb teams in Kyoto and Tokyo for theiradvice, M. Springer for technical support, and E. Zupanič
for contributions in an earlier stage of the experiments.
This work is supported by the Austrian Ministry of Scienceand Research (BMWF) and the Austrian Science Fund(FWF) through a START grant under Project No. Y479N20 and by the European Research Council under ProjectNo. 259435.

[1] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T.
Pfau, Rep. Prog. Phys. 72, 126401 (2009).
[2] M. A. Baranov, Phys. Rep. 464, 71 (2008).
[3] K. Goral, K. Rzazewski, and T. Pfau, Phys. Rev. A 61,
051601 (2000).

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value of 1.5 W.
[26] We use the strong blue transition at 401 nm for imaging.
[27] S. Jochim, M. Bartenstein, G. Hendl, J. H. Denschlag, R.
Grimm, A. Mosk, and W. Weidemüller, Phys. Rev. Lett.
89, 273202 (2002).
[28] M. Zaccanti, C. D’Errico, F. Ferlaino, G. Roati, M.
Inguscio, and G. Modugno, Phys. Rev. A 74, 041605(2006).
[29] The magnetic field is calibrated via rf-spectroscopy
between Zeeman sublevels. From the calibration weweek ending
25 MAY 2012

estimate a short-term stability of 1 mG and a long-term
stability of about 10 mG.
[30] V. Roudnev and M. Cavagnero, Phys. Rev. A 79, 014701(2009).
[31] J. L. Bohn, M. Cavagnero, and C. Ticknor, New J. Phys.
11, 055039 (2009).
[32] K. K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B.
Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and
D. S. Jin, Nature (London) 464, 1324 (2010).

210401-5

88

Bibliography[Aik12]K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm,
and F. Ferlaino, Bose-Einstein Condensation of Erbium, Phys. Rev.
Lett. 108, 210401 (2012).

[And95]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, andE. A. Cornell, Observation of Bose-Einstein condensation in diluteatomic vapor, Science 269, 198 (1995).

[Ang68]J. R. P. Angel and P. G. H. Sandars, The Hyperfine Structure Stark
Effect. I. Theory, Proc. R. Soc. Lond. A 305, 125 (1968).

[Ban05]

H. Ban, M. Jacka, J. Hanssen, J. Reader, and J. McClelland, Lasercooling transitions in atomic erbium, Opt. Express 13, 3185 (2005).

[Bar08]M. A. Baranov, Theoretical progress in many-body physics with ultracold dipolar gases, Physics Reports 464, 71 (2008).

[Bea08]Q. Beaufils, R. Chicireanu, T. Zanon, B. Laburthe-Tolra,
E. Maréchal, L. Vernac, J.-C. Keller, and O. Gorceix, All-opticalproduction of chromium Bose-Einstein condensates, Phys. Rev. A
77, 061601 (2008).

[Bla12]P. B. Blakie, D. Baillie, and R. N. Bisset, Roton spectroscopy in aharmonically trapped dipolar Bose-Einstein condensate, Phys. Rev.
A 86, 021604 (2012).

[Bra95]C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidenceof Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75, 1687 (1995).

[Chi06]J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu, and W. Ketterle, Evidence for superfluidity of ultracold
fermions in an optical lattice, Nature 443, 961 (2006).
89BIBLIOGRAPHY[Chi10]C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).

[Con10]C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle, and J. M. Doyle,
Large spin relaxation rates in trapped submerged-shell atoms, Phys.
Rev. A 81, 010702 (2010).

[Cow81]R. D. Cowan, The Theory of Atomic Structure and Spectra, Berkeley, CA: University of California Press (1981).

[Cro09]A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Optics andinterferometry with atoms and molecules, Rev. Mod. Phys. 81, 1051
(2009).

[Dav95]K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S.
Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation
in a gas of sodium atoms, Phys. Rev. Lett. 75, 3969 (1995).

[DeM99] B. DeMarco and D. S. Jin, Onset of Fermi Degeneracy in a Trapped
Atomic Gas, Science 285, 1703 (1999).
[Dem05]

W. Demtröder, Experimentalphysik 3, Atome, Moleküle und Festkörper, Springer Verlag (2005).

[Dra96]

G. Drake, Atomic, Molecular and Optical Physics Handbook, American Institute of Physics (1996).

[Dzu11]V. A. Dzuba, V. V. Flambaum, and B. L. Lev, Dynamic polarizabilities and magic wavelengths for dysprosium, Phys. Rev. A 83,
032502 (2011).

[Fer09]F. Ferlaino, S. Knoop, and R. Grimm, Cold Molecules: Theory, Experiment, Applications, chapter Ultracold Feshbach molecules, Taylor & Francis (2009).

[Fer10]F. Ferlaino and R. Grimm, Forty years of Efimov physics: How abizarre prediction turned into a hot topic, Physics 3, 9 (2010).

[Fey82]R. P. Feynman, Simulating Physics with Computers, Int. J. Th.
Phys. 21, 467 (1982).

[Fri12]

A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler, E. Zupanič,
R. Grimm, and F. Ferlaino, Narrow-line magneto-optical trap for
erbium, Phys. Rev. A 85, 051401 (2012).
90

BIBLIOGRAPHY[Gor00]K. Goral, K. Rzazewski, and T. Pfau, Bose-Einstein condensation
with magnetic dipole-dipole forces, Phys. Rev. A 61, 051601 (2000).

[Gra07]

C. Grain, T. Nazarova, C. Degenhardt, F. Vogt, C. Lisdat, E. Tiemann, U. Sterr, and F. Riehle, Feasibility of narrow-line cooling in
optical dipole traps, Eur. Phys. J. D 42, 317 (2007).

[Gre02]

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch,
Quantum phase transition from a superfluid to a Mott insulator ina gas of ultracold atoms, Nature 415, 39 (2002).

[Gre08]

M. Greiner and S. Föllig, Condensed-matter physics: Optical lattices, Nature 453, 736 (2008).

[Gri00]R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Optical dipole
traps for neutral atoms, Adv. At. Mol. Opt. Phys. 42, 95 (2000).

[Gri05]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, BoseEinstein condensation of chromium, Phys. Rev. Lett. 94, 160401(2005).

[Han04]

C. I. Hancox, S. C. Doret, M. T. Hummon, L. Luo, and J. M. Doyle,
Magnetic trapping of rare-earth atoms at millikelvin temperatures,
Nature 431, 281 (2004).

[Jam92]A. M. James and M. P. Lord, Macmillan’s chemical and physicaldata, Macmillan (1992).

[Joc03]

S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl,
C. Chin, J. Hecker Denschlag, and R. Grimm, Bose-Einstein Condensation of Molecules, Science 302, 2101 (2003).

[Koc08]T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, andT. Pfau, Stabilization of a purely dipolar quantum gas against collapse, Nature Phys. 4, 218 (2008).

[Koh07]

C. Kohstall, A New Toolbox for Experiments with Ultracold 6 Li,
Diploma thesis, Universität Innsbruck (2007).

[Kot11]

S. Kotochigova and A. Petrov, Anisotropy in the interaction of ultracold dysprosium, Phys. Chem. Chem. Phys. 13, 19165 (2011).

[Kre04]R. V. Krems, G. C. Groenenboom, and A. Dalgarno, Electronic Interaction Anisotropy between Atoms in Arbitrary Angular Momentum States, J. Phys. Chem. A 108, 8941 (2004).
91

BIBLIOGRAPHY[Lah08]

T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier,T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, d-wave collapse andexplosion of a dipolar Bose-Einstein condensate, Phys. Rev. Lett.
101, 080401 (2008).

[Lah09]T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau,
The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72,
126401 (2009).

[Law10]J. E. Lawler, J.-F. Wyart, and E. A. D. Hartog, Atomic transition
probabilities of Er I, J. Phys. B: At. Mol. Opt. Phys. 43, 235001(2010).

[Lim11]A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose
gases, Phys. Rev. A 84, 041604 (2011).

[Lu11]

M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Strongly Dipolar
Bose-Einstein Condensate of Dysprosium, Phys. Rev. Lett. 107,
190401 (2011).

[Lu12]

M. Lu, N. Q. Burdick, and B. L. Lev, Quantum Degenerate Dipolar
Fermi Gas, Phys. Rev. Lett. 108, 215301 (2012).

[McC06a] J. J. McClelland, Natural linewidth of the 401-nm laser-cooling transition in Er I, Phys. Rev. A 73, 064502 (2006).
[McC06b] J. J. McClelland and J. L. Hanssen, Laser cooling without repumping: A magneto-optical trap for erbium atoms, Phys. Rev. Lett. 96,
143005 (2006).
[Ney12]

B. Neyenhuis, B. Yan, S. A. Moses, J. P. Covey, A. Chotia,
A. Petrov, S. Kotochigova, J. Ye, and D. S. Jin, Anisotropic Polarizability of Ultracold Polar 40 K87 Rb Molecules, arXiv:1209.2226(2012).

[Osp10]S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, D. S. Jin, andJ. Ye, Quantum-State Controlled Chemical Reactions of Ultracold
Potassium-Rubidium Molecules, Science 327, no. 5967, 853 (2010).

[Pet12]

A. Petrov, E. Tiesinga, and S. Kotochigova, Anisotropy-Induced Feshbach Resonances in a Quantum Dipolar Gas of Highly Magnetic
Atoms, Phys. Rev. Lett. 109, 103002 (2012).
92

BIBLIOGRAPHY[Phi92]

W. Phillips, Laser cooling and trapping of neutral atoms, International School of Physics "Enrico Fermi" (1992).

[Ral11]

Y. Ralchenko, A. Kramida, J. Reader, and N. A. Team, NIST
Atomic Spectra Database (ver. 4.1.0) (2011).

[Sal91]B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, John
Wiley & Sons, Inc. (1991).

[San03]

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Roton-Maxon
Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates, Phys. Rev. Lett. 90, 250403 (2003).

[Sch01a]F. Schreck, G. Ferrari, K. L. Corwin, J. Cubizolles, L. Khaykovich,
M.-O. Mewes, and C. Salomon, Sympathetic cooling of bosonic andfermionic lithium gases towards quantum degeneracy, Phys. Rev. A
64, 011402 (2001).

[Sch01b]

F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel,
J. Cubizolles, and C. Salomon, Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea, Phys. Rev. Lett. 87, 080403 (2001).

[Sch06]

P. Schwerdtfeger, Atomic Static Dipole Polarizabilities, ed. G.
Maroulis, IOS Press, Amsterdam (2006).

[Sch07]

H. Schöbel, Ultrakalte Cs2 -Moleküle in einer optischen Dipolfalle
mit kontrollierbarer Elliptizität, Diploma thesis, Universität Innsbruck (2007).

[Sch11]

J. Schindler, Characterization of an Erbium Atomic Beam, Master’s
thesis, Universität Innsbruck (2011).

[Söd98]J. Söding, D. Guéry-Odelin, P. Desbiolles, G. Ferrari, and J. Dalibard, Giant spin relaxation of an ultracold cesium gas, Phys. Rev.
Lett. 80, 1869 (1998).

[Tan10]L.-Y. Tang, Z.-C. Yan, T.-Y. Shi, and J. Mitroy, Dynamic dipole
polarizabilities of the Li atom and the Be+ ion, Phys. Rev. A 81,
042521 (2010).

[Tru01]A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge,
and R. G. Hulet, Observation of Fermi Pressure in a Gas of Trapped
Atoms, Science 291, 2570 (2001).

93

BIBLIOGRAPHY[Win06]

K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. Büchler, and P. Zoller, Repulsively bound atom pairs in an optical lattice, Nature 441, 853(2006).

[Wya12]J.-F. Wyart, M. Lepers, and O. Dulieu, Private communication(2012).

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DanksagungFirst of all I want to thank Francesca for letting me be part of the erbium
team. She supported me in every possible way – in preparing my talks, in thedecision concerning my scientific future and finally in the writing phase of mythesis. I want to thank her for giving me the possibility to work independently
on such a crucial part of our experiment. I had the luck to join the erbium
team in a very exciting phase and really look forward to my time as a Ph.D
student.
Des Weiteren möchte ich mich bei Rudi Grimm bedanken – für die Aufnahme in eine so professionell arbeitende wissenschaftliche Gruppe mit einemäußerst angenehmen Arbeitsklima. Besonders viel Freude hat mir die Abstimmung unserer Vorträge auf der YAO Konferenz gemacht, die es mir ermöglichte
unsere sensationellen experimentellen Ergebnisse als Erster präsentieren zu
dürfen – "The correct answer is . . . 13".
Vielen herzlichen Dank allen Gruppenmitgliedern, die mich schnell als einender Ihren aufgenommen haben. Christoph für seine nützlichen Tipps zum Scanning System, Simon für sein Know-how im Bereich von High-Power Linsen und
all jenen die mich bereitwillig in ihren Utensilien stöbern haben lassen.
Mein besonderer Dank gilt dem Erbium Team, für die unglaublich angenehme Atmosphäre auch bei Problemen im Labor, sowie bei unseren "Breakfast & Besprechung" Meetings, die gerne auch mal über Mittags hinaus gedauert
haben. Von Alberts technischen Wissen, seiner Fähigkeit sich von Problemen
nicht aus der Bahn werfen zu lassen und seiner Liebe für Details konnte ich
mir hoffentlich vieles abschauen. I enjoyed the talks with Kiyotaka concerningscience, attitude and lifestyle and want to thank him for always encouraging
me to push the limits. Ein großer Anteil dieser Masterarbeit ist Michael zu verdanken. Er hat alle wichtigen Entscheidungen bezüglich dieser Arbeit mit mir
gemeinsam getroffen und mich in meinem Kampf mit Elektronik, Laserprofilen und MATHEMATICA-Code stets unterstützt. Besonders hervorzuheben
ist die intensive Zeit die wir zur Fertigstellung des Laseraufbaus gemeinsam
investiert haben – Vielen Dank! Michael S. und Ben danke ich für die Diskussionen auf Augenhöhe im Labor.
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DANKSAGUNG
Alle Institutsmitarbeiter waren nie um eine Hilfe verlegen. Im organisatorischen Bereich haben sich vor allem Christine und Nicole um alle meine
Anfragen und Rechungen gekümmert. Michael und Wolfgang waren stets dierichtigen Ansprechpersonen für Fragen rund um IT und nicht zu funktionierend scheinende Elektronik. Helmut hat mit seinem unverwechselbaren Stilmeine anfänglich noch bescheidenen Kenntnisse im Bohren, Fräsen und Drehen
maßgeblich verbessert. Armin und Toni möchte ich für die Fertigung der maßgeschneiderten Breadboards danken.
Furthermore I want to thank Maxcence Lepers from the Université of Lille
and Jean-François from the Laboratoire Aimé Cotton in Orsay for providing
me always with the newest set of data for my calculations.
Meine Studienkollegen Florian, Christian und Michael haben viel ihrerseits
beigetragen, sodass mein Physikstudium vieles, aber sicher nicht langweilig
war. Legendäre Physikerfeste, intensive Lerneinheiten im 8. Stock und aufgelockerte Vorlesungen und Proseminare sind nur einige Anekdoten aus dieser
Zeit. Besonders Michael danke ich für die langjährige Zusammenarbeit und die
zahlreichen Diskussionen.
Der wahrscheinlich größte Dank gilt meinen Eltern, die mir durch ihre Unterstützung in allen Lebenslagen eine vollkommen freie Entscheidung bezüglichmeines Werdeganges ermöglichen. Meiner Mutter danke ich für all die Diskussionen über Gott und die Welt, meinem Vater für all die horizonterweiternden
Touren in den Bergen dieser Welt und meinem Bruder Stefan danke ich für das
so angenehme Zusammenleben. Bei meinen Großeltern möchte ich mich für ihr
immerwährendes Interesse bedanken und für die traditionellen wöchentlichen
Telefonate. Mein Onkel Gottfried hat schon früh den Wissenschaftler in mir
erkannt und fördert mich auch weiter auf diesem Weg. Meine Freundin Katrin
ist immer für mich da und gibt mir stets Kraft, sodass ich daran glaube alles
schaffen zu können – dafür schätze ich sie besonders. Vielen Dank euch allen,
dass ihr mich immer an die wirklich wichtigen Dinge im Leben erinnert.

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