Università di Pisa
Facoltà di Scienze Matematiche Fisiche e Naturali
Dipartimento di Fisica “E. Fermi”

Tesi di Laurea Magistrale in Fisica della Materia

Optical dipole trap for an erbium anddysprosium mixture

Relatore esterno:
Candidato:

Prof.ssa Francesca Ferlaino(University of Innsbruck)

Claudia Politi
Matr. 527904

Relatore interno:
Prof.ssa Donatella Ciampini

Anno Accademico 2016/2017

Alla mia famiglia, la mia forza.

Contents
Introduction1

1

Properties of erbium and dysprosium
1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Electronic configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
6
7
7
11

2

Optical dipole trap: theory and simulations
2.1 Optical trapping: basic concepts . . . . . . . . . . . . . . . . . . . . . . .
2.2 Optical trap with Gaussian beams . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Single beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Crossed beams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Gravity and astigmatism . . . . . . . . . . . . . . . . . . . . . . .
2.3 A trap for Er and Dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
17
21
23
24
26
27

3

Optical dipole trap: experimental realization
3.1 Why an optical dipole trap with tunable geometry? . . . . . . . . . . . . .
3.1.1 Loading efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Role of the tunability in evaporative cooling . . . . . . . . . . . . .
3.1.3 Tool to probe the anisotropy of DDI . . . . . . . . . . . . . . . . .
3.2 How to get a tunable dipole trap? . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 System with cylindrical lenses . . . . . . . . . . . . . . . . . . . .
3.2.2 Time-averaged potential . . . . . . . . . . . . . . . . . . . . . . .
3.3 Time-averaged potential with acousto-optic devices . . . . . . . . . . . . .
3.3.1 Optical setup: scanning system . . . . . . . . . . . . . . . . . . . .
3.3.2 Driving RF signal . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Measurement of the waist . . . . . . . . . . . . . . . . . . . . . .

35
35
36
37
37
38
38
40
42
46
50
54

vi

Contents
3.3.4

4

Implementation in the experiment . . . . . . . . . . . . . . . . . .

57

Measurements with atoms
4.1 RARE experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 ODT loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Trap frequencies measurements . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Towards evaportive cooling . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
64
66
71

Conclusion

77

Bibliography

81

A Scanning circuit

89

B Optical transport

91

Acknowledgements

97

Introduction
At extremely low temperatures, on the order of a billionth of degree above the absolute zero,nine orders of magnitude lower than the coldest temperature of the universe, atomic gases
reveal their quantum nature. In 1925 Satyendra Nath Bose and Albert Einstein predicted
the appearance of a new state of matter, the Bose-Einstein condensate (BEC) [15, 27]. In
this state, atoms occupy the same lowest energy level and display their wave-like properties,
forming one macroscopic quantum mechanical wave. For decades scientists thought that the
low temperatures needed to reach the BEC regime could never be achieved in laboratories.
After the invention of the laser and the development of new laser cooling and trapping
techniques in 1995, 70 years after the theoretical prediction, the first BEC was achieved
at JILA [5] and shortly after at MIT [24]. Among these new techniques, the fundamental
tool that enabled scientists to reach such low temperatures was the evaporative cooling. For
fermions, that behave differently with respect to bosons, it took four years more to achieve thefirst quantum degenerate gas in 1999 [26]. In this case, due to the Pauli exclusion principle,
atoms do not “condense” in the same state, rather they fill the lowest energy levels with at
most one particle per state.
Ultracold atoms, confined in traps formed of magnetic fields and laser beams, provide
ideal systems in which few- and many-body physics can be investigated with high control
of internal and external degrees of freedom. In experiment, this is done through the useof Feshbach resonances as a “knob” to tune the interparticle interactions [19], or actively
controlling the traps to change the dimensionality of the system. In the last decades, theinterest in using ultracold gases for several applications grew enormously. One of the most
exciting one, suggested by Feynman in 1981 [29], regards the use of these highly controllable
quantum systems to simulate quantum phenomena in nature that are too complex to fully
understand or simulate with standard computer, e. g. high temperature superconductivity[72]. Furthermore, laser beams can be combined in several ways to create artificial lattices,in which ultracold atoms can be easily manipulated to study solid-state problems. Other
applications are precision measurements [75] and the development of atomic clocks [14],which are involved in our daily life as standard reference for GPS.

2

Introduction

Since 1995 the field of ultracold gases keeps expanding with the addition of new ingredients, e. g. dipole-dipole interactions [9], atoms in highly excited states (Rydberg atoms)[56] or systems in which the spin becomes a new degree of freedom (spinor gases) [61].
The first experiments with ultracold gases used alkali atoms, whose interactions, at low
temperatures, have a short-range and isotropic character. On the contrary, dipolar gases
exhibit an additional interaction, the dipole-dipole interaction (DDI), which is long-range
and anisotropic. The geometry-dependence provides a new “knob” to manipulate the interactions. The experimental studies on dipolar gases started in 2005 with the Bose-Einsteincondensation of chromium [34]. The exciting results determined by the presence of the DDI
encouraged scientists in further investigation of dipolar systems. Since the dipolar strength ishigher for large mass and high magnetic moment, several groups have switched to heavy andstrongly magnetic lanthanides to study systems with dominating dipolar interactions. The
first lanthanide to be “condensed” was dysprosium [58] in 2011 by the group of B. Lev in
Stanford and a year later erbium [3] was brought to the quantum degeneracy in Innsbruck by
the group of F. Ferlaino.
The RARE experiment (Rydberg Atoms with Rare Earths) in Innsbruck, where thisthesis work has been performed, is the first mixture experiment that combines two strongly
magnetic atomic species, erbium and dysprosium. Both have several bosonic and fermionicisotopes with high abundances, which allow the study of Bose-Bose, Bose-Fermi and FermiFermi mixtures. The energy spectra with similar cooling transitions and the similar atomic
properties, such as mass and melting temperatures, enable a comparatively easy doublespecies implementation. Three science chambers, connected to each other by optical transport,
will be implemented: main chamber, Rydberg chamber and single-site imaging chamber.
This will allow us for the investigation of dipolar behaviour of rare earth atoms not only inthe lowest energy state (main chamber) but also in highly excited states (Rydberg chamber).
Furthermore, in the single-site imaging chamber, an high resolution imaging system will
enable us to manipulate and control the two species on the single-atom level.
When I joined the group in October 2016, the main chamber was just implemented and,soon after, the experiment was at the stage of MOT lights implementation and achievementof our first Dy MOT. Thus, it was the time to develop and implement trapping scheme tosimultaneously confine Er and Dy and to then further cool the mixture to quantum degeneracy.
The design and the realization of such optical dipole trap is the subject of my thesis. The
structure of the thesis is the following:Introduction

3

Chapter 1 introduces the basic properties of erbium and dysprosium as the particularelectronic configurations and energy spectra, which lead to their strongly magnetic nature.
Furthermore, it describes the main features of dipole-dipole interactions.
Chapter 2 gives the theoretical description of atoms trapping by means of far detuned laser
beams and the introduction of the tools necessary to characterize it. Particular emphasis isgiven to one of the key quantities involved in atom-light interaction, the atomic polarizability.
Chapter 3 describes the experimental realization of the optical dipole trap and its tunablegeometry. The tunability is obtained using a time-averaged potential created by an acoustooptic deflector. This feature is a fundamental tool to investigate phenomena triggered by theanisotropic character of the DDI. It is used to increase the loading efficiency of the ODTreaching the “mode-matching” condition with the magneto-optical trap. Additionally, it
can be used to tune the elastic collision rate and maximize the efficiency of the evaporative
cooling necessary to reach the degenerate regime.
Chapter 4 gives an overview of the RARE apparatus and describes the implementation of
the optical setup in the experiment and a preliminary characterization of the optical dipole
trap, e. g. loading efficiency of atoms and trap frequency measurements.

Chapter 1
Properties of erbium and dysprosium
Erbium and dysprosium are both rare-earth metals and they belong to the group of lanthanides1 . Despite from their name, rare-earth elements have quite high abundance, e. g. ceriumis the 25th most abundant element in nature, while dysprosium (Dy) and erbium (Er) are the43rd and 45th , respectively. In nature, they are found together in form of alloy and it is quitedifficult to isolate one from the other, indeed the name dysprosium has Greek origins and itmeans “hard to get”.
These elements got their name for historical reasons, in fact in the 19th century only one
deposit of rare earths, located in Sweden, was known. This explains the adjective “rare”, but
why “earth”? Rare-earth elements were mainly extracted as oxide that in French was known
as “terre” while in German “erde” which means “earth” [85]. Erbium and dysprosium areboth extensively used for technological applications.
Er has attracted particular attention as a dopant for YAG crystals and phosphate glasses
to produce lasers with an output wavelength of 2.9 µm and 1.54 µm, respectively. Bothwavelengths are absorbed by water, leading to interest in medical applications and eye
safe military range-finders2 [11]. In the lately years erbium played also an important roleenhancing the energy conversion efficiency in solar cells. Indeed Er3+ ions convert thephotons with an energy that exceeds the bandgap of the absorber, which typically would betransformed in heat, into infrared photons [76].
Dy is usually added to neodymium to produce the strongest permanent magnets and it isalso used as a dopant in different hosts for laser applications.
In this chapter the first section introduces the basic properties of Er and Dy as lanthanides.
The second and third sections treat the electronic configuration and the energy spectrum,
1 Not all the rare-earth elements are lanthanides, i. e. scandium and yttrium belong to this group just becauseof the similar chemical properties.
2 The water present in the eyes absorbs the light before it reaches the retina.

6

Properties of erbium and dysprosium

respectively. Finally, the last section is devoted to the interaction between highly magnetic
atoms.

1.1

Basic properties

Dysprosium has an atomic number Z = 66 and an atomic mass of 162.50 u3 , while erbium
has Z = 68 and an atomic mass of 167.26 u, see Fig. 1.1.
Samarium
62

Europium
63

Gadolinum64

Terbium
65

Dysprosium
66

Holmium
67

Erbium68

Thulium
69

Yitterbium
70

Lutetium71

Pr

Nd Pm Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

140.91144.24151.96157.25158.93162.50164.93167.26168.93173.05174.97Lanthanum
57

Cerium58

Praseodymium
59

La

Ce

138.91140.12

Neodymium
60

Promethium
61

145

150.36

Figure 1.1 Periodic table’s row of lanthanides. Dy has an atomic number Z = 66 and an
atomic mass of 162.50 u, while Er has Z = 68 and an atomic mass of 167.26 u.
Erbium and dysprosium have several bosonic and fermionic stable isotopes4 . In Table 1.1
all the stable isotopes and their abundances are listed.
Table 1.1 Relative abundances and type of quantum statistics for all stable isotopes of Er and
Dy. B and F stand for bosonic and fermionic statistics, respectively .
IsotopeAbundance(%)

Statistics

162

Er
Er
166
Er
167
Er
168
Er
170
Er

0.14
1.61
33.6
23.0
26.8
15.0

B
B
B
F
B
B

156

< 0.1
< 0.1
2.3
18.9
25.5
24.9
28.3

B
B
B
F
B
F
B

164

Erbium

Dy
Dy
160
Dy
161
Dy
162
Dy
163
Dy
164
Dy
158

Dysprosium

3 1 u = 1.660 540 2 × 10−27 kg.
4 Among the rare-earth elements only promethium has no stable isotopes.

7

1.2 Electronic configuration

Another important property of erbium and dysprosium is their large magnetic momentof 7 µB and 10 µB , respectively, where µB is the Bohr magneton5 . Among the rare-earth
elements also europium (7 µB ), terbium (10 µB ) and holmium (9 µB ) present this feature.
For comparison, alkali atoms, which are extensively used as quantum gas, have magnetic
moments of the order of 1 µB in the ground state. The main reason for this is that theelectronic ground state of Er and Dy has a large total angular momentum, as will be shownin the next sections.

1.2

Electronic configuration

The ground-state electronic configuration of the rare-earth atoms consists of a core identicalto xenon plus additional electrons in orbits with higher energy, as shown below:[1s2 2s2 2p6 3s2 3p6 3d 10 4s2 4p6 4d 10 5s2 5p6 ]Xe 4 f m 6s2
where m is the index representing the number of electrons occupying the 4 f shell.
Starting from the electronic configuration of xenon, the rare-earth atoms fill the innervacant 4 f orbits, e. g. the first is cerium with m = 1, erbium and dysprosium instead havem = 12 and m = 10, respectively. Due to the Madelung rule, electrons occupy the 6s orbital(n + l = 6) before the 4 f (n + l = 7) that remains an open shell. This type of electronic
configuration is called submerged-shell.
Figure 1.2 displays the angular dependence of the 4f electronic wavefunction, which ishighly anisotropic in comparison to the spherically symmetric cloud of alkali and alkalineearth atoms. This is the origin of a strongly anisotropic van der Waals potential [43] and it
plays an important role on the interaction between light and atoms, as it will be mentioned in
Chapter 2.

1.3

Energy spectrum

The bosonic isotopes of Er and Dy have nuclei with an even number of protons and neutrons,
this causes a zero nuclear spin I. On the other hand, fermionic isotopes have an even numberof protons and an odd number of neutrons leading to a nuclear spin I = 5/2 and I = 7/2 for
Dy and Er, respectively. Due to a non zero nuclear spin I, the total angular momentum J
couples with I giving rise to the hyperfine structure. This chapter will focus mainly on the
5 The Bohr magneton is defined as µ

mass.

eh̄
B = 2me , where e and me are the elementary charge and the electron

8Properties of erbium and dysprosium(a) ml = 0(b) ml = 1(c) ml = 2(d) ml = 3(e) 1sFigure 1.2 Angular dependence of the 4 f wavefunction in the four different ml states. The
electronic clouds manifest a high anisotropy, which clearly affects the properties of rare-earth
elements [36].
properties of bosonic isotopes since all the measurements of this thesis work are performed
with bosons.
According to Hund’s rules, in the state with the lowest energy, electrons have to fill the
4 f orbital6 in the following way:
• the multiplicity, defined as 2S + 1, and so the total spin S, has to be maximized;• for fixed S, the angular momentum L has to be maximized;• finally, for Er and Dy, which have the outermost subshell more than half filled, thetotal angular momentum J has to be maximized;
This leads to a ground state with
L=5, S=1, J=6
L=6, S=2, J=8

for Er.
for Dy.

In the Russel-Saunders coupling scheme (2S+1 LJ ) the ground states are written as 3 H6 and5 I for Er and Dy, respectively. The angular momentum is large compared to the ground8
state with L = 0 of the alkali and alkaline-earth atoms, as listed in Table 1.2.
6 Only electrons in the unfilled shell (“open shell”) have to be considered.

9

1.3 Energy spectrum

Erbium and dysprosium present two and four vacancies in the 4 f orbital, respectively.
According to the first Hund’s rule7 , these unpaired electrons are aligned in the states ml = 2, 3
for Er and ml = 0, 1, 2, 3 for Dy with the spin in the same direction and this gives rise to
their large magnetic moment. The presence of unpaired electrons is also the reason why
the spectrum of rare-earths is so complex compared to the case of alkali and alkali-earth
elements. Indeed, in the case of lanthanides, several transitions can take place involving the
4 f electrons together with other valence electrons.
Table 1.2 Ground state of different elements belonging to the group of alkali, alkaline-earth
and rare-earths, which have been “Bose-Einstein-condensed”. Dy and Er have a large angularmomentum L in the ground state with respect to other species.
Element
Lithium
SodiumPotassium
Rubidium
Caesium
Calcium
Strontium
Dysprosium
ErbiumElectronic configuration

Ground state[He]2s1[Ne]3s1[Ar]4s1[Kr]5s1[Xe]6s1[Ar]4s2[Kr]5s2[Xe]4 f 10 6s2[Xe]4 f 12 6s2

2S
2S
2S
2S
2S
1S

1
2
1
2
1
2
1
2
1
2

0

1S

0
5I
8
3H
6

Figure 1.3 shows the energy spectrum of Er and Dy up to an energy of E = h × 749 THz,which corresponds to the wavenumber k = 25 000 cm−1 . The levels shown in red (black)present an even (odd) parity. According to the selection rules, optical dipole transitions canonly couple the ground state J = 6 with J = [5, 7] and the state J = 8 with J = [7, 9] for
erbium and dysprosium, respectively.
In the Er-Dy experiment atoms are laser cooled in a first stage with resonant light. This
stage includes the transversal cooling (TC), in which the atomic beam is optically collimatedand the Zeeman slower (ZS) which reduces the velocity of the atoms in such a way to be
captured by the magneto optical trap (MOT). In the picture the broad transitions chosen
for ZS and TC (401 nm, 421 nm)8 are shown in light blue, while the narrow lines chosen
7 The first Hund’s rule is also known as the “bus seat rule” because electrons, like the bus passengers, firsttend to singly occupy all the double seats.
8 A broad transition, which is characterized by short lifetime of the excited state, is more suited for the ZS
because it guarantees a large number of scattering events with the light. On the other hand, a narrow line ismore suited for the MOT because it leads to lower temperature achievable. The request is that the velocity of
the atoms coming from the ZS has to be close to the capture velocity of the MOT.

Properties of erbium and dysprosium
10

25

20

15

10

5

3

4

7

8

ERBIUM6

J value5

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

11

4f126s2 3H6

10

12

1

583 nm (MOT)

64f12(3H ) 6s6p(3P )401 nm(slowing transition)

9

30
25

8

J value

79421 nm (slowing transition)
4f10(5I8)6s6p(1Po1)(8,1)o9

DYSPROSIUM

6

333

400

10

1000

10

11

12

13

4f106s2 5I8 ground state

2000
0

5

626 nm (MOT)

500

5

20

4

15

3

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

2

Wavelength (nm)02Figure 1.3 Energy spectrum of erbium and dysprosium [8, 87]. The levels shown in red (black) have an even (odd) parity. The groundstate, which has an even parity, according to the selection rules, can only be coupled to an energy level with a total angular momentum
within the interval J = [5, 7] for erbium and J = [7, 9] for dysprosium and with an odd parity. The broad cooling transitions used for
the Zeeman slower and transversal cooling (401 nm, 421 nm) are displayed in light blue, while in yellow and orange are shown the
narrow lines used for the magneto optical trap (583 nm, 626 nm) for erbium and dysprosium, respectively. To trap both species andreach the degeneracy, an optical dipole trap running at a wavelength of 1064 nm was chosen.

Wavenumber (1000 cm-1 )11

1.4 Dipole-dipole interaction

for the MOT (583 nm, 626 nm) are shown in yellow and orange for Er and Dy, respectively.
The energy spectrum of Er and Dy shows similar cooling transitions. The informations
regarding the lines used for cooling are listed in Table 1.3. After the laser-cooling stage,
the mixture is loaded into an optical dipole trap (ODT). This is a conservative potential andthe wavelength (λ ) of the ODT should be far off resonance. Thus, an ODT running at awavelength of 1064 nm was chosen for the optical trapping and the evaporative cooling toreach the degenerate phase of both species.
Table 1.3 Wavenumber, wavelength, linewidth of the cooling transition and the Doppler
temperature used for the Zeeman slower and the magneto optical trap of Er and Dy [18].

1.4

Wavenumber(cm−1 )

Wavelength(nm)

Linewidth TDoppler
(MHz)
(µK)

Erbium

24943.272
17157.307

401
583

29.7
0.19

714
4.6

Dysprosium

23736.61
17727.15

421
626

32.2
0.136

773
3.3

Dipole-dipole interaction

The interaction potential between two atoms, in a first order approximation, is given by thevan der Waals potential
C6
UvdW (r) = − 6 ,(1.1)
rwhere r is the internuclear distance and C6 is the van der Waals coefficient.
For a potential that scales as r−6 the total scattering cross section becomes negligible forpartial waves with l ̸= 0 as the wavenumber k approaches zero, so in the limit of vanishingcollision energy. Therefore, in the case of ultracold quantum gases usually only s-waves
scattering play a role [69].
The real interatomic potential can then be replaced by a pseudo-potential which is fully
described by a single parameter: the s-wave scattering length a [22]. These potentials areshort-range and isotropic and the pseudo-potential is usually referred as contact interaction
Ucontact (r) =where m is the atomic mass.

4π h̄2 aδ (r) ,
m

(1.2)

12

Properties of erbium and dysprosium

In the case of highly magnetic atoms, like Er and Dy, it has been shown [9] that theinteratomic potential has also a second contribution which is long-range and anisotropic: thedipole-dipole interaction (DDI).
The interaction potential between two dipoles with magnetic moment µ1 and µ2 pointing
in direction e1 and e2 , respectively, is expressed by the following relation
UDDI (r) =Cdd (e1 · e2 )r2 − 3(e1 · r)(e2 · r)
,
4π
r5

(1.3)

where Cdd is the dipolar coupling constant, which in case of magnetic dipoles9 is given byµ0 µ1 µ2 , with µ0 the vacuum permeability and r the distance between the dipoles.
Applying an external magnetic field it is possible to polarize the sample in such a waythat all dipoles point in the same direction, in this case Eq. (1.3) takes the simple form
UDDI (r) =µ0 µ1 µ2 1 − 3cos2 θ
,
4π
r3

(1.4)

where θ is the angle between the direction of the magnetic field and the relative position of
the dipoles.
Relation (1.4) shows the anisotropic and the long-range7 character of the DDI. In particular, tuning the angle θ from 0 (head-to-tail configuration) to π/2 (side-by-side configuration),
the interaction potential switches from attractive to repulsive. Furthermore, since 1 − 3cos2 θgoes from −2 to 1, the maximum repulsion is half the maximum attraction and there is a
critical angle equal to θm = 54.7° at which the dipole-dipole interaction vanishes. A sketchof the DDI in the not polarized and polarized case is shown in Fig. 1.4a and in Fig. 1.4b,
respectively.
Since DDI scales as r−3 , the total scattering cross section contains the contributions given
by all the partial waves, not only the s-wave10 . Furthermore, it does not vanish for identicalfermions at low temperature.
The magnetic moment’s projection along the direction of the magnetic field can be
expressed asµi = mJi gJi µB
with i = 1, 2
(1.5)
9 This relation can easily be applied to atoms with an electric dipole moment d writing C = d 2 /ε .
0
dd
7 The DDI shows a long-range character in three dimensions, in one and two dimensions is short-range.
10 δ ∼ k for all partial waves, thus for small k the phase shift does not vanish [69].
k13

1.4 Dipole-dipole interaction

θ
r
B

e2
r
e1
(a) Not polarized sample.

(b) Polarized sample.

Figure 1.4 Dipole-dipole interaction between two particles for a not polarized sample (a) anda sample polarized by an external magnetic field (b). The head-to-tail configuration shows
an attractive interaction while when the dipoles sit side-by-side the interaction is repulsive.
where mJ is the projection of the total angular momentum along the direction of the magnetic
field and gJ is the Landé g-factor given by
gJ = 1 + (gS − 1)

J(J + 1) − L(L + 1) + S(S + 1),
2J(J + 1)(1.6)

with gS ≈ 2.002 the gyromagnetic ratio of the electron spin. In the lowest energy level Er occupies the state |J, mJ ⟩ = |6, −6⟩ while Dy is in the state |J, mJ ⟩ = |8, −8⟩, by approximatinggS = 2 in Eq. (1.6), the magnetic moments read:µEr = 6.98 µB ,µDy = 9.93 µB .

(1.7)

Our experiment, called from now on “RARE”, combines for the first time two dipolar species,
thus for an Er and Dy mixture one has to consider the dipole-dipole intra-species interactions:µ0 µEr µEr 1 − 3cos2 θ
intra,
UDDI (r) =34πr
and the inter-species interactions:

inter
UDDI
(r) =

µ0 µDy µDy 1 − 3cos2 θ
intra
UDDI (r) =34πµ0 µEr µDy 1 − 3cos2 θ
.
4π
r3

r

(1.8)

(1.9)

14

Properties of erbium and dysprosium

Table 1.4 Comparison of dipole moments and relative dipolar strengths between different
atoms and heteronuclear molecules. The relative dipolar strength can be tuned changing thes-wave scattering length through Feshbach resonance [51].

AlkalisElement

Dipole moment

εdd

Rb
Cr
Dy
Er
KRb
HCN

1 µB
6 µB
10 µB
7 µB
1.5 D
3.0 D

0.007
0.16
0.1 – 1
0.1 – 120240

Chromium

Erbium Dysprosium

Heteronuclear
molecules

10−2

10−1

1

102

Rydberg
atoms
> 106

Figure 1.5 Sketch of the relative dipolar strengths for different species. The hard sphere
represents the contact interaction. In the alkalis the dipolar contribution is almost negligible
while it becomes relevant with dipolar atoms like Er and Dy.
To quantify the absolute strength of the dipolar interactions usually the dipolar length isdefined:
Cdd m
add ≡
.
(1.10)
12π h̄2
DDI is also often expressed relatively to the strength of the contact interactions, given by thescattering length a, by the quantity
εdd ≡

add
Cdd m
=
.
a
12π h̄2 a(1.11)

Figure 1.5 shows the relative dipolar strengths of different species. In particular, the valuesof the dipole moment and εdd for different atoms (Rb, Cr, Er and Dy) are compared with thehuge values of the heteronuclear molecules in Table 1.4. Tuning the contact interactions
through Feshbach resonances [19, 49] allows the system to get into the strongly dipolaredd

1.4 Dipole-dipole interaction

15

regime in which εdd > 1. In this regime new exotic many-body phases are raising particularinterest [20, 21, 28].

Chapter 2
Optical dipole trap: theory andsimulations
One of the crucial steps to create a quantum-degenerate gas, such as a Bose-Einstein condensate for bosons or a degenerate Fermi gas for fermions, is to use evaporative cooling to getphase-space density that could not be reached in a magneto optical trap.
The first Bose-Einstein condensate was obtained in 1995, at JILA [5], with laser precooled,magnetically trapped atoms of 87Rb using RF evaporative cooling. Afterwards, several groups
tried to switch to optical schemes. Compared to magnetic traps, that can only confine atoms
in “low-field seeking states1 ”, optical traps can trap all Zeeman sublevels. Furthermore,they allow the use of Feshbach resonances to tune the interaction strength without changing
the parameters of the trapping potential. Only in 2001, in the group of Chapman [10], a
Bose-Einstein condensate of 87Rb was achieved by evaporative cooling in a crossed optical
dipole trap generated by a CO2 laser (λ = 10.6 µm).
In this chapter the basic principles of optical trapping will be introduced to understand andcharacterize the optical dipole trap implemented in the Er-Dy-experiment. Special attention
will be devoted to the important role of the polarizability in trapping efficiency. Finally, thelast section will be dedicated to the simulations of the trapping potential implemented for thetwo species.

2.1

Optical trapping: basic concepts

When an atom is exposed to an electric field, the displacement of the electrons with respect
to the nucleus will induce an atomic dipole moment given by ⃗p = ∑i −e⃗ri , where ⃗ri is the
1 From Wing’s theorem only local minima of the magnetic field modulus can be achieved in free space.

18

Optical dipole trap: theory and simulations

position vector of the i-th electron. For laser light of a certain frequency ω, the induced
dipole moment oscillates at the driving frequency and his amplitude is related to the electricfield amplitude by the following relation [35],p(r) = α(ω)E(r),(2.1)

where α(ω) is the complex dynamical polarizability which determines the interaction energyof the induced dipole moment with the driving field.
The real part of α, which corresponds to the in-phase dipole oscillation, is important forcalculating the dipole potential:
1
1
Udip (r) = − ⟨⃗p · ⃗E⟩ = −Re{α}I(r) ,
2
2ε0 c

(2.2)

where ⟨·⟩ denotes the time average, ε0 is the vacuum permittivity, c is the speed of light andI = 2ε0 c|E|2 is the intensity of the electric field.
Atoms can also absorb energy and re-emit it, this can be seen as a scattering process with
rate Γsc , which represents the out-of-phase oscillation and it is given by the imaginary part ofthe polarizability,
1
Pabs
Γsc (r) ==Im{α}I(r),
(2.3)
h̄ω
h̄ε0 cwhere Pabs = ⟨⃗ṗ · ⃗E⟩ is the power absorbed by the atom.
To get an expression for the polarizability, it is possible to start considering the atom asa classical oscillator like in the Lorentz’s model, where an electron is elastically bound to
the nucleus with an eigenfrequency equivalent to the frequency of the optical transition ω0 .
Denoting Γ as the damping rate corresponding to radiative energy loss, me as the electron’smass and integrating the equation of motion ẍ + Γẋ + ωo2 x = −eE(t)/me , the polarizability
takes the form:Γ/ωo2α(ω) = 6πε0 c3 2.
(2.4)
ωo − ω 2 − i(ω 3 /ωo3 )Γ
The dynamical polarizability can be replaced by the scalar polarizability α if the conditionω ≪ ω0 is fulfilled. In the semiclassical approach, for a two-level quantum system, Γ corresponds to the spontaneous decay rate for an allowed transition in the dipole approximationfrom the excited state |e⟩ to the ground state |g⟩ and it is proportional to the dipole matrix
element
ω0 3Γ=|⟨e|µ|g⟩|2 ,(2.5)
3πε0 h̄c3

19

2.1 Optical trapping: basic concepts

where µ̂ = ⃗p · ê is the electric dipole moment operator and ê is the direction of the electricfield.
In the case of optical dipole traps, which use light far-detuned from the optical transition
ω0 , since the saturation can be really small and the population of the excited state is basicallynegligible, the condition Γsc ≪ Γ is fulfilled and the classical model yields the same results
given by the semiclassical approach. This leads to the following expressions for the dipole
potential and the scattering rate:
3πc2
Udip (r) = −
2ωo 3



Γ
Γ
+
ω0 − ω ω0 + ω


I(r),

  
2
3πc2 ω 3ΓΓ
Γsc (r) = −+I(r) .
2h̄ωo 3 ω0
ω0 − ω ω0 + ω

(2.6)

(2.7)

If the detuning from the resonance ∆ = ω − ω0 is not too large (|∆| ≪ ω0 ) , the rotating-wave
approximation can be applied, by neglecting the term that involves ω0 + ω, to get the two
simple expressions: 
3πc2 Γ
I
Udip (r) =I(r) ≃ ,
(2.8)
3
2ωo
∆
∆
 
I
3πc2 Γ 2
I(r) ≃ 2 .
Γsc (r) =32h̄ωo ∆∆

(2.9)

Equations (2.8) and (2.9) provide us the first tool to underline two important features of
optical dipole traps:• if the frequency of the laser light is below the frequency of the atomic transition (∆ < 0),
the dipole potential assumes negative values and the atoms, trying to minimize theinteraction energy, will be attracted towards regions of higher intensity (red detunedtrap). Whereas, if the detuning is positive (∆ > 0) the atoms will be pushed away fromthe high intensity regions (blue detuned trap).
• by increasing the intensity of the light and the detuning, the same trap depth can bereached with a reduced scattering rate. This shows the advantage of using optical
dipole trap at high power and large detunings, since less scattering means less heatingof the atomic sample.
Up to now the polarizability has been considered independent on the orientation of the electricfield. This would be right in case of alkali or alkali-earth atoms which are characterized by aground state with zero angular momentum. In the contrary, as described in Chapter 1, dipolargases like erbium and dysprosium feature an electronic configuration with an incompletely

20

Optical dipole trap: theory and simulations

filled 4 f shell which leads to a ground state with angular momentum L ̸= 0. Therefore, theelectronic distribution around the core is not spherically symmetric, resulting in an highly
anisotropic response to the light.
This anisotropy manifests a tensorial form of the polarizability and the trapping potential
accordingly contains also terms dependent on the light’s polarization and on the different MJ
sublevels. Equation (2.2) has to be modified ton1
MJ
UMJ (r, ω, θ p , θk , A) = −I(r) Re[αs (ω)] + A cos θkRe[αv (ω)]+
2ε0 c
2J
o
3MJ2 − J(J + 1) 3 cos2 θ p − 1+
Re[αt (ω)] ,J(2J + 1)
2(2.10)

where αs (ω) contains the diagonal elements of the polarizability tensor, αv (ω) is the antisymmetric part and αt (ω) is the symmetric part of the off-diagonal terms. θk is defined by
the angle between the quantization axis, given by the magnetic field direction, and light’s
polarization. θ p is defined by the angle between the quantization axis and wave vector.
Finally, A is the ellipticity coefficient of the polarization [53].
The polarization of an optical dipole trap beam is usually linear, in this case the expression
above simplifies because A = 0. Thus, the term due to the vectorial part plays no role andthe dipole potential will depend only on the polarization angle θ p and on the given state|J, MJ ⟩. Up to now both theoretical calculations [53, 54] and experimental results [12] havedemonstrated that even though for their electronic configuration erbium and dysprosium
should respond to the light in an anisotropic way, the trapping potential created by an infraredlaser beam is almost isotropic. On the other hand, this is not the case for the scattering rate
which results extremely anisotropic. Indeed, as can be seen from the experiment, the lifetimeof the atoms in the dipole trap depends strongly on the light’s polarization.
So far, we have reported only about the dipole potential and the scattering rate for a
two-level system. However, in reality atoms have several electronic transitions between
different energy levels. Furthermore, the spectrum is very rich in the case of Er and Dy.
The effect of the electric field on the atom can be seen as a perturbation given, in the dipole
approximation, by the term H (1) = −µ̂ · ⃗E. Since atomic states have a definite parity andthe perturbation is an odd operator, the first order’s contribution to the perturbation vanishes.
Thus, the energy shift of the ground state |gi ⟩ is given by the second order term and it isrelated to the dipole matrix elements of all the excited states |e j ⟩ as below:|⟨e j |µ|gi ⟩|2 I,∆i j
2πε0 cj∆Ei = ∑

(2.11)

21

2.2 Optical trap with Gaussian beams

where ∆i j is the detuning between |gi ⟩ and |e j ⟩. The dipole matrix elements |⟨ei |µ|gi ⟩| andthe scattering rate can be expressed in terms of the transition coefficients (ci j ) which take
into account the coupling strength between the specific sublevels and can be obtained by theuse of 3J-symbol2 :Γ̃ j = c2i j Γ j ,µi j = ci j ||µ|| ,with ||µ|| the reduced matrix element given by relation (2.5). Equations (2.6) and (2.7) canthen be generalized as below:"

3πc2
Udip (r) = − ∑3j 2ωi j"Γsc (r) =



Γ̃ j
Γ̃ j
+
ωi j − ω ωi j + ω  
3πc2 ω 3∑ 2h̄ω 3
j

ij

ωi j

#
I(r) = ŨI(r),

Γ̃ j
Γ̃ j
+
ωi j − ω ωi j + ω(2.12)

2 #I(r),(2.13)

where the quantity in square brackets in Eq. (2.12) represents the polarizability Ũ. In Fig. 2.1
the total polarizability, including the scalar and tensorial parts, is plotted as a function of
the wavelength. The strongest transition is at 401 nm for Er in contrast to 421 nm for Dy,
with the effect that everyline above (below) this wavelength is red (blue) detuned. The stars
represent the experimental data listed in Table 2.1.
Table 2.1 List of experimental values of the polarizability expressed in atomic units andmeasured by other groups [12, 58, 60]. The huge systematic uncertainties derive from themeasurement of the beam waist, as explained in section 2.3.
Erbium2.2

Dysprosium

Wavelength
(nm)

Re{α}
a.u.

Wavelength
(nm)

Re{α}
a.u.

532
1064
1570

430 ± 8st ± 80sys [12]166 ± 3st ± 61sys [12]163 ± 9st ± 36sys [12]1064

136 ± 15 [60], 116 [58]Optical trap with Gaussian beams

Optical trapping of neutral atoms relies on two types of radiative forces:
2 The Wigner 3J-symbols are related to the Clebsch-Gordan coefficients, which quantify the wavefunction

overlap.

Dynamical polarizability (a.u.)

22

Optical dipole trap: theory and simulations

1000
Erbium

500

0

-500

-1000
200

300

400

500

1000

2000

Dynamical polarizability (a.u.)

Wavelength (nm)1000
Dysprosium

500

0

-500

-1000
200

300

400

500

1000

2000

Wavelength (nm)Figure 2.1 Dynamic polarizability as a function of the wavelength for Er (top) and Dy(bottom). In this simulation the contribution of the tensorial part has been taken into account
[12].

23

2.2 Optical trap with Gaussian beams• the scattering force due to the absorption of the radiation followed by the spontaneousemission, which is classified as a scattering process with the rate Γsc ,• the dipole force resulting from the spatial gradient of the dipole potential Udip and thus
the gradient of the light’s intensity.
With lasers it is possible to achieve high intensity and high spatial intensity gradient thatallow to modify the motion of the atoms and stably trap them. Thus, the simplest form of an
optical dipole trap is the focus of a red-detuned Gaussian laser beam.

2.2.1

Single beam

Let’s consider an elliptic Gaussian beam3 propagating along the y direction, with z direction
coincident with the axis of gravity, with beam waist wx,z and power P. The intensity profile
of an elliptic beam for a fixed power can be expressed by the following relation:
  2

2P
x
z2
I(x, y, z) =exp −2+.
πwx (y)wz (y)w2x (y) w2z (y)(2.14)The beam radius, in terms of the beam waist wx,z and the Rayleigh length zRx,z , can be writtenassπw2x,zy 2
wx,z (y) = wx,z 1 +.
(2.15)
with zRx,z =zRx,zλFor a given laser power P, the intensity reaches its maximum in (x, y, z) = (0, 0, 0) where it
assumes the expression
2P
I0 ≡ I(0, 0, 0) =.
(2.16)
πwx wzOne of the important parameters to be defined for an optical dipole trap is the total trap depthÛ, which is given by Eq. (2.12) replacing I(r) with I0 :Û = ŨI0 = Ũ2P.
πwx wz(2.17)

If the thermal energy kB T is much less than the trap depth Û, the atomic cloud will occupy
mainly the central region of the trap where the dipole potential can be approximated as" #2x2 2z2 1 2 1
1 2
Udip ≃ Û 1 − 2 − 2 − y
+
.
wx
wz 2
zRx zRz(2.18)

3 The choice of an elliptic Gaussian beam is made because the shape of the beam is modified during theexperimental cycle as it will be described deeply in Chapter 3.

24

Optical dipole trap: theory and simulations

Equation (2.18) can be rewritten as an harmonic potential. The trapping frequencies can thenbe obtained writing the harmonic dipole potential in terms of the oscillation frequencies:" #2x2 2z2 1 2 1
1 2
Udip ≃ Û 1 − 2 − 2 − y
+
≃
wx
wz 2
zRx zRz


1
1
1
2 2
2 2
2 2
≃ Û 1 − mωx x − mωy y − mωz z
222sωx,z =

4Û
,mw2x,z

s
ωy =

2Û
,
mz2Rh

(2.19)(2.20)

where zRh is the effective Rayleigh length for the elliptic beam defined as:
1
=
z2Rh

1
1
+
zRhx zRhz!2.

(2.21)

As we can see from Eq. (2.17) and (2.20), the choice of the beam waist is highly critical. The
trap depth and the trapping frequencies, which are the most important parameters to control
during the forced evaporative cooling, rely exactly on the dimension of the waist.
The basic idea of the forced evaporative cooling is to allow the hotter atoms to escapelowering the trap depth while the colder atoms remain trapped and rethermalize. In optical
dipole traps this is usually done reducing the power of the laser beam. The desired condition
would be to reduce the depth keeping the curvature of the trap, i. e. the trapping frequencies,
constant but, as it is written in the equations above, together with the trap depth which scales√
as P also the trapping frequencies scale down as P. This situation is depicted in the simple
cartoon representing the evaporative cooling in Fig. 2.2. Since evaporative cooling depends
on the elastic collisions rate which decreases with decreasing frequencies, the cooling process
can be stuck. This is the reason why trap depth, trapping frequencies and evaporation’stime steps have to be chosen carefully to get an efficient evaporative cooling. Indeed if theevaporation rate is too low, atoms can be lost due to collisions with background gas or photon
scattering processes. In contrary if the evaporation rate is too high, with almost constant
trapping frequencies, three-body losses can become important.

2.2.2

Crossed beams

From relations (2.20) it comes out that a single focused beam creates an anisotropic trap witha longitudinal confinement that is weak compared to the radial one. Instead, crossing two

25

2.2 Optical trap with Gaussian beams(a) Ideal case.

(b) Real case.

Figure 2.2 Sketch representing the basic idea of evaporative cooling. (a) Ideal case in which
the trap depth is lowered without influencing the trapping frequencies. What happens inreality during evaporative cooling in an optical dipole trap is depicted in (b). Lowering thepower of the laser not only reduces the trap depth but also the trapping frequencies.
focused laser beams, it is possible to create a trapping potential with a tight confinement
along all the three directions.
As a second beam let’s consider another elliptic Gaussian beam propagating along thedirection of the gravity and crossing the first beam at an angle of 90°. The resulting trapping
potential is just given by the sum of the two dipole potentials:
"


2 #
2
2
2x
2z
1
1
1
+
Ucross ≃ Uˆh 1 − 2 − 2 − y2
+
wx
wz 2
zRx zRz" #1 2
2x2 2y2 1 2 1+ Ûv 1 − 2 − 2 − z
+
,
wx
wy 2
zRx zRy(2.22)

where Uˆh and Ûv are the total trap depth of the beam propagating along y and z direction,
respectively. In the following pages the first will be referred as horizontal beam while thesecond one vertical beam. Applying the harmonic approximation and defining the effective

26

Optical dipole trap: theory and simulations

Rayleigh length also for the vertical beam:
1
=
z2Rv

1
1
+
zRvx zRvy!2,(2.23)

the trapping frequencies can be expressed as below:
s


Ûv
Uˆh
ωx =
+
,
w2hx w2vxv!
u
u 1 4Ûv 2Uˆh
ωy =t
+ 2
,
m w2vyzRhs 
1 4Uˆh 2Ûv
.
+ 2
ωz =
m w2vzzRv4m(2.24)The Rayleigh lengths are much smaller than the waist thus, from relations (2.24), it is clearthat each beam contributes to the trap frequency orthogonal to the direction of propagation,
while the trap frequency along the direction of propagation is mostly given by the other beam.

2.2.3

Gravity and astigmatism

When dealing with real life, the parameters of the trap, like the depth and the frequencies,
may be influenced due to several reasons. The most obvious one is the presence of the gravity
which adds to the trapping potential the quantity Ug = mgz. Thus, along the z direction, thetrapping potential is tilted, the trap depth is reduced and the atoms can be lost. This effectis mainly visible in shallow trap rather than in the tight ones. Below a certain minimum
power the trap is not able anymore to hold the atoms. Considering the conservative forcedarising from the dipole potential Fdip (r) = − drUdip (r) it is possible to get the minimum
power necessary to beat the gravity, equating the dipole force to the force due to gravity,
which for our trap parameters4 are P ≃ 70 mW and P ≃ 60 mW for Er and Dy, respectively.
In this case again the polarizability is crucial, the lanthanides are characterized by quite
small values of the polarizability with respect to other species. For example in comparison
with Cs or Rb is more than a factor of three lower5 . This means that, for a fixed waist, more
power is needed to trap the atoms. The effect of gravity is taken into account substituting thetrap depths, in the expression (2.24) relative to the z direction, with the reduced ones.
4 w = 20.5 µm, ReDy {α} = 136 a.u. and ReEr {α} = 166 a.u.
x,z
5 Re{α}Cs = 1153 a.u., Re{α}Rb = 677 a.u.

27

2.3 A trap for Er and Dy

Another source of reduction of the trap depth is the astigmatism. A finite astigmatismhas to be taken into account, indeed the position of the foci along the two different axes may
not coincide∆y = yfocus,x − yfocus,z ̸= 0 .
When the astigmatism is on the same size or larger than the Rayleigh lengths, the relation(2.15) has to be modified considering that the waists along the different axes reach theminimum in two different points along the y-direction:vuu
wx (y) = wx t1 +

y

!2

zRhx

v
u
u
wz (y) = wz t1 +y − ∆yzRhz!2(2.25)

The astigmatism modifies the dipole potential with the presence of two minima, inducing a
reduction of the effective trap depth and trapping frequencies.

2.3

A trap for Er and Dy

To simulate the trap implemented in the Er-Dy experiment, let’s first consider a single laser
beam propagating along the y direction with a wavelength of 1064 nm, a power of 30 W
and radial waists equal to wx,z = 20.5 µm. The trapping potential for the two species arecalculated taking as values for the polarizability the experimental values measured in othergroups, Re{α} = 136 a.u. [60] and Re{α} = 166 a.u. [12] for Dy and Er, respectively. The
parameters chosen to reproduce the trapping potential are listed in Table 2.2.
Table 2.2 Parameters of the horizontal and vertical beam chosen to simulate the optical
dipole trap implemented in the Er-Dy experiment. The values of the polarizability have been
experimentally measured by other groups.
Power
(W)
Horizontal beam

30

Vertical beam

8

Beam waist
(µm)
wx
20.5
wx
103.5

Polarizability(a.u.)wz
Er20.5
wy 166
95.0

Dy
136

The polarizabilities’ values play a fundamental role in the trapping efficiency and, although the measurements of the polarizability may seem straightforward, the latter still dohave a huge uncertainty. This is due to the strong dependence on the waist, indeed, from

28

Optical dipole trap: theory and simulations

Eq. (2.17) and (2.20), writing the polarizability in atomic units6 , one gets:
Re{α} =2 πε cm
ωx,z
0

4P × 1.6488 × 10−41

w3x wz ∝ w4 .

(2.26)

Thus, even a small error in the waist leads to a large error on the polarizability.
Furthermore, also from a theoretical point of view estimate the value of the polarizability
is quite tricky. In fact unlike alkali atoms, which have a simple electronic structure with anisotropic s-shell and a not crowded energy spectrum, the lanthanides are characterized by acomplex electronic structure and the knowledge of their spectroscopy is still not complete. On
the left side of Fig. 2.3 the trapping potential for dysprosium atoms is sketched considering
different values of the polarizabilities and on the right side is shown the trapping potential
resulting from the experimental values of the polarizability for Er and Dy7 .
0Er
Dy116 a.u
136 a.u
164 a.u

-500

-500

-1000

-1000

-1500

-1500

-2000
-100

-50

0

Z direction (µm)

50

100 -100

-50

0

50

Trapping potential (µK)

Trapping potential (µK)

0

-2000
100

Z direction (µm)

Figure 2.3 Trapping potential for Er and Dy given by a single elliptic Gaussian beam
propagating along the y direction. The curve on the left side of the figure represents thetrapping potential for dysprosium atoms choosing as values for the polarizability the two
experimentally measured values 116a.u. [58] and 136a.u. [60] and the theoretical value
164a.u. [54]. On the right side, instead, the trapping potential for Er and Dy is shownchoosing the measured values of the polarizability166a.u. and 136a.u..
As mentioned before, the trapping frequencies are crucial for further evaporative steps
to reach the degenerate regime. Table 2.3 shows the values of the trap depths and trapping
frequencies. These values point out the weak confinement of the trap along the propagation
axis, which means that the atoms may spread along that direction even more during the
6Ũ =

1
−41 .
2ε0 c Re{α} × 1.6488 × 10

7 In ultracold groups, the trapping potential is usually in units of k

B and expressed in Kelvin.

29

2.3 A trap for Er and Dy

Table 2.3 Trap depth and trapping frequencies of the horizontal beam. The frequency along
the y direction, i. e. the direction of propagation, shows a weak confinement with respect tothe radial directions.
Trap depth
(µK)

νx
(Hz)

νy
(Hz)

νz
(Hz)

Erbium

1698

4502

53

4502

Dysprosium

1391

4124

48

4124

forced evaporative cooling when the trapping frequencies are reduced due to the decreasingof the power. Therefore, another laser beam may be needed to keep a strong longitudinal
confinement and “close the doors” to the atoms during the evaporation and reach the criticalphase-space density for the quantum degeneracy.
-1300

-200

-1400

-600
-800

-1500

-1000
-1600

-1200
-1400

-1700

Trapping potential (µK)-400

Trapping potential (µK)

Er
Dy

Er
Dy

-1600
-1800
-2000

-1000

0

1000

Z direction (µm)

2000-500

-250

0

250

-1800
500

Z direction (µm)

Figure 2.4 Trapping potential for Er and Dy given by the two crossed elliptic Gaussian beams.
The vertical beam adds another dipole potential which is represented by the small dip at theorigin of the x-axis and it is enlarged in the picture on the right.
A second elliptic Gaussian beam, operating at the same wavelength and propagating
along the z direction, crosses the horizontal beam at an angle of 90°. Considering a powerof 8 W, the same polarizabilities as before and the waists wx = 101.3 µm and wy = 95.0 µm,
the resulting trapping potential of the crossed optical dipole trap is depicted in Fig. 2.4 while
in Table 2.4 the values for the horizontal (Uˆh ) and vertical (Ûv ) trap depths and the trapping
frequencies are listed. The contribution of the vertical beam to the total dipole potential
results in a small dip in the central region. Its depth is just around 20 µK deep, this meansthat as long as the atoms have a much higher thermal energy, they will not see the effect of

30

Optical dipole trap: theory and simulations

Table 2.4 Trap depth of the horizontal and vertical beam and trapping frequencies of thepotential resultant from crossing the two beams.
Uˆh
(µK)

Ûv
(µK)

νx
(Hz)

νy
(Hz)

νz
(Hz)

Erbium

1698

20

4503

117

4502

Dysprosium

1391

16

4125

107

4124

the vertical beam. Thus, the second beam becomes useful only when a first stage of plain
evaporation and then forced evaporation cooling is applied.
The RARE experiment is the first mixture experiment with two highly magnetic atoms.
One of the interesting features of this mixture, in comparison to other mixtures studied up to
now in the ultracold groups, is the small mass imbalance and the quite similar values of thepolarizability for wavelengths in the infrared region. Erbium and dysprosium offer a wide
choice of isotopes and both present isotopes with a mass of 162 u and 164 u. Thus, it would
be also possible to have a mixture of two species with the same mass. Table 2.5 shows a listof several mixtures that have been involved up to now in ultracold groups.
Table 2.5 List of mixtures adopted by ultracold groups. Almost all the mixtures are characterized by large mass imbalances except for RbSr and ErDy.
Mixtures87

Rb133Cs [71], 6Li41K [55], 23Na6Li [40] Alkali + Alkali
87
Rb84Sr [68]Alkali + Alkali earth metal
87
171
133 174
Rb Yb [83], Cs Yb [44]Alkali + Rare earth metal
168 162
Er Dy
Rare earth metal + Rare earth metal

The mass plays an important role in the efficiency of trapping both species. Indeed,strong mass imbalances result in a discrepancy between the confinement seen by atoms ofdifferent species. Combining Eq. (2.17) and (2.20), the trap frequencies can be expressed insuch a way to underline the mass and the polarizability dependence:sωx,z =

4 Ũ2P,
mw2x,z πwx wz(2.27)

31

2.3 A trap for Er and Dy

from which, considering two species 1 and 2 and calling Ũ1,2 and m1,2 the polarizability andthe mass of each specie, the trap frequencies ratio is given by
1
ωx,z
=
2
ωx,z

s

m2Ũ1
.
m1Ũ2

(2.28)

This feature can be used as a tool to reduce the discrepancy of the polarizability, changing
among the available isotopes the ones with the right mass ratio. Looking at Table 2.5 it ispossible to see that almost all the mixtures are characterized by large mass imbalances except
for RbSr and ErDy. In particular, by choosing the fermionic isotope 87Sr, a Bose-Fermi
mixture with atoms with the same mass can be realized. However, rubidium and strontium
present a larger difference in polarizability values with respect to Er-Dy.
As said in the previous paragraph, the trapping potential is tilted by the gravity. Figure 2.5
displays the trapping potential modified by the gravity for a tight trap with a power of 30 W
and a very shallow trap with a power of 0.1 W. In both cases the waists are the same as in theprevious simulations. From these pictures it is visible that the effect of the gravity is almost
negligible for strong trap as also shown by the values of trap depth and trapping frequencies
including gravity, in Table 2.6. Gravity covers an important role in the evaporation process.
Indeed, when the power of the laser beam is reduced it becomes more and more important its
contribution on the reduction of the trap depth. The crucial point is that gravity reduces thedeepness of the trap without influencing the trapping frequencies in the central region.
20

Trapping potential (µK)

Er
Dy

Er
Dy

15

0
10
5

-500

0
-1000

-5
-10

-1500

Trapping potential (µK)

500

-15
-2000
-100

-50

0

Z direction (µm)

50

100 -100

-50

0

50

-20
100

Z direction (µm)

Figure 2.5 Trapping potential for Er and Dy given by the horizontal beam and modified by
the gravity. On the left side, the laser beam has a power of 30 W and the effect of the gravity
is hardly visible. Instead on the right side, where the power is 0.1 W, the reduction of thetrap depth is clear.

32

Optical dipole trap: theory and simulations

Table 2.6 Trap depth of the horizontal and vertical beam and trapping frequencies of cODT
modified by the gravity.
Uˆh
(µK)

Ûv
(µK)

νx
(Hz)

νy
(Hz)

νz
(Hz)

Erbium

1690

0

4503

117

4491

Dysprosium

1383

0

4125

107

4112

Finally, the trapping potential may be modified by the astigmatism, whose effect has to beconsidered if the difference between the foci is on the same order of the Rayleigh length. The
Rayleigh lengths of the horizontal and vertical beam are shown in Table 2.7. Furthermore,

Figure 2.6 Trapping potential for dysprosium atoms along y and z directions for different
values of the astigmatism. The reduction of the trap depth increases when the distance of thetwo foci raises. Furthermore, when it becomes larger than the Rayleigh length, two minima
start to appear along the y direction.
for an high astigmatic beam this has to be corrected. As it is shown in Fig. 2.6 for dysprosium,along the z direction the depth of the trapping potential is more and more reduced for an
increasing astigmatism, whereas along the y direction it also assumes a two-well shape. The
trap depth for different values of astigmatism is listed in Table 2.8.

33

2.3 A trap for Er and Dy

Table 2.7 Rayleigh lengths of the horizontal and vertical beams.
Rayleigh length
Horizontal (zRh )(mm)

Vertical (zRv )
(mm)

1.2

28.3

Table 2.8 Trap depth for dysprosium atoms calculated for different values of astigmatism.
The deepness of the trap is strongly reduced when the astigmatism is on the same order ofthe Rayleigh length.
Dysprosium
Astigmatism (∆y)(mm)

Trap depth (Uˆh )
(µK)

0
0.6
1.2
2.4
3.0

1391
1253
1000
639
532

Chapter 3
Optical dipole trap: experimental
realization
Optical dipole traps are a fundamental tool to produce and manipulate ultracold atoms. Theinterest in building systems to actively control the parameters and the shape of the trap grewenormously. This, indeed, allows to increase the available number of degrees of freedom tomanipulate atoms. The first section of this chapter points out the motivations that drive thechoice of an optical dipole trap with tunable geometry. The second section introduces thedifferent methods that can be implemented to get a tunable trap and the final section treats
the experimental setup necessary to realize a time-averaged potential with an acousto-optic
device.

3.1

Why an optical dipole trap with tunable geometry?

The optical dipole trap implemented in the RARE experiment presents a tunable geometry.
Indeed, both the shape and the beam size can be changed dynamically during the experimental
cycle. This section points out why a tunable dipole trap is so crucial for the experiment. A
variable beam waist plays an essential role during the loading of the atoms into the dipole
trap, providing a nice “mode matching” between the MOT volume and the ODT volume.
Furthermore, the evaporation cooling strongly depends on the particles density, which can be
modified changing the trap’s volume. Finally, a lot of interesting phenomena arise from thegeometry dependence of the dipole-dipole interaction.

36

3.1.1

Optical dipole trap: experimental realization

Loading efficiency

Let’s assume that the atoms in the magneto-optical trap (MOT) follow the Maxwell-Boltzmannenergy distribution. When the optical dipole trap is switched on, considering that the atoms
in the trap reach the thermal equilibrium with the MOT atoms, which represent the reservoir,
the population of the ODT will be given by [1]:Z

NODT =n f (r, p)g(ε)drdp,(3.1)

where n is the MOT’s particle density, f (r, p) is the distribution function dependent on theposition and momentum vector (r, p) and finally, g(ε) is the ratio of the density of states forthe potential created by the ODT to the density of states of the MOT potential, with energy ε.
This critical parameter varies from 0 to 1. The first case, g(ε) = 0, is for a δ -like potential,
while g(ε) = 1, is for a very shallow trap.
To simplify the expression, let’s assume g(ε) to be constant and equal to g0 over theoverlapping volume, relation (3.1) assumes the form:
NODT = n0V0 ηloading ,(3.2)

where n0 and V0 are the MOT peak density and the overlapped volume, respectively, whileηloading :Z
4g0 q 2 −u2ηloading = √u e du,(3.3)
π 0with u and q defined as below:
|p|
,
u= √
KB T

q=

U0
.
KB T

An increase of the laser power makes the trap deeper and the parameter q as well, butηloading will not rise forever, on the contrary, it saturates and the loading efficiency becomes
independent on the laser’s power. Relation (3.2) points out that the atom number, loaded into
the optical dipole trap, increases linearly with the overlapped volume, as long as the trap
depth is above the saturation regime. The overlapped volume scales as wx wz zRh π 3/2 , thus tomake it larger the beam waist has to be increased, leading to a reduction of the trap depth
and consequently a less efficient trap for the evaporation cooling stage.
The ideal situation would be the one in which the initial waist of the laser beam is quite
large, in order to get a nice overlap between the ODT and the MOT volume and an high3.1 Why an optical dipole trap with tunable geometry?

37

loading efficiency as well. Then, it is reduced in such a way to get a tight trap useful forstarting the evaporation process.

3.1.2

Role of the tunability in evaporative cooling

The possibility of tuning the geometry of the trap plays an important role also in the evaporation process. The latter, indeed, relies on the elastic collision rate between the particles andthis depends on the density. The goal at the end of the evaporation steps is to get as many
atoms as possible in the degenerate phase. Thus, the elastic collision rate has to be muchhigher than the rate with which the atoms are lost due to three-body collisions or scatteringwith background gas.
Considering n0 the particle density at the center of the trap, the elastic collision ratedepends linearly on the density and it can be written as [63]:n0 σ v̄Γel = √ ,
2 2

(3.4)

where σ = 8πa2 is the total elastic cross section, a is the s-wave scattering length and v̄ isthe mean thermal velocity. In contrary, the three-body losses rate depends stronger on theparticles density:
L3B n2(3.5)
Γ3B = √ 0 ,3 3
where L3B is the three-body losses coefficient and can be determined measuring the atoms
number N after a certain time t from the rate equation [17]dN
= −L3B n3 (x,t)d 3 x ,
dt
V
Z

(3.6)

where n is the particles density over the volume V .
Tuning the geometry of the trap, by changing in a dynamical way the beam waist of
the laser beam along one axis, allows the density to be changed in order to improve theevaporation efficiency. Therefore, besides the light power, the tunability of the trap geometry
represents an additional experimental parameter to fine tune the evaporation steps.

3.1.3

Tool to probe the anisotropy of DDI

In Chapter 1 the anisotropic character of the dipole-dipole interaction was introduced throughthe Eq. (1.4). Thus, thanks to this feature, it is possible to tune the interactions from attractive
to repulsive, by rotating the magnetic field. Furthermore, it depends on the geometry, so for a

38

Optical dipole trap: experimental realization

given orientation of the magnetic field, atoms can organize in the head-to-tail configurationin which the potential is predominantly attractive or in the side-by-side configuration inwhich they tend to repel each other. To induce the atoms in choosing one or the other, thegeometry can be tuned by changing the aspect ratio of the laser beam and its power.
Let’s consider the trapping potential, taken in the harmonic approximation, as in Eq. (2.19),
with oscillation frequencies (ωx , ωy , ωz ). If the magnetic field points in the z direction, theaspect ratio of the trap (AR) is defined as
AR =

ωz wx
= ,
ωx wz

for AR > 1 the trap will be oblate (pancake trap) while for AR < 1 will be prolate (cigartrap). In the first case the atoms will sit side by side giving an overall repulsive interaction,
while for a cigar shaped trap the potential will be mostly attractive.
Lot of interesting phenomena came out in the last years taking advantage of this featureof the dipole-dipole interaction as the discovery of new exotic phase of matter [20, 21, 47].

3.2

How to get a tunable dipole trap?

Different methods can be used to change dynamically the beam waist, either with active
devices like electro-optic (EOD) or acousto-optic (AOD) deflectors, both based on thecreation of time-averaged potential, or with mechanical solutions which involve the use of
several optical elements. During the course of this thesis different systems had been tested:first, a system consisting of cylindrical lenses and then, a system built with an acousto-optic
modulator (AOM) which was afterwards changed with an AOD. The possibility of using theEOD, which is adopted by other groups to create tailored time-averaged potential [86], wasrejected from the beginning because this device may be not suited for high power1 .

3.2.1

System with cylindrical lenses

Following the description in the paper [73], a similar system, formed by five cylindricallenses, was built to test the possibility of tuning the aspect ratio of our optical dipole trap.
Among these lenses, three are convex and the other two are concave. The basic idea is to keep
the convex lenses fixed and move the other two jointly along a distance of 25 mm. Figure 3.1ashows a simulation with the software Zemax OpticStudio of the optical setup, in which a
1 In our setup a 50 W single mode fiber laser, from the company Azur Light System (ALS-IR-1064-50W-SF),

with a mean beam waist of 500 µm at 180 cm from the laser head, is implemented to create the optical dipole
trap.

39

3.2 How to get a tunable dipole trap?

magnification of the beam, along the vertical direction, can be seen. The lenses are planned
to get a system with a tunable magnification from 1 to a maximum value of 6. Table 3.1
shows a list of distances to observe between the lenses to get the predicted magnifications.
Starting from the configuration that leads to a magnification of 1, the concave lenses are

C1

C2 C 3C4C5(a) Cylindircal telescope.
6

Aspect ratio

5

4

3

2

1
0

5

10

15

20

25

30

Displacement (mm)(b) Measured aspect ratio.

Figure 3.1 Sketch of the operation of the cylindrical telescope. (a) Working principle of thecylindrical telescope, simulated with the software Zemax OpticStudio, by fixing the lensdistances to the values in Table 3.1. In the upper and lower part of the figure, the configuration
leading to magnification of 1 and 6 are shown, respectively. (b) Aspect ratio measured bychanging the travelling distances of the C2 and C4 lenses.

40

Optical dipole trap: experimental realization

Table 3.1 List of the distances, between the cylindrical lenses, necessary to reach a magnification of 1 and 6. The concave lenses, C2 and C4, are moved back and forth together, while
the convex lenses, C1, C3 and C5, are fixed. These values are taken from [73] and verified
with simulations using the software Zemax OpticStudio.
Magnification
×1
×6
LensesDistance(mm) (mm)C1 - C2
C2 - C3
C3 - C4
C4 - C5
C2 - C4

48.54
10.73
26.54
58.562
37.27

23.63
35.64
1.63
83.47
37.27

moved on a motorized linear stage2 in such a way to get the lens C4 closer to C3. Thus, thebeam size along the vertical direction (z) increases with respect to the horizontal one (x).
When the laser beam reaches the last lens, used to focus the light on the atoms in the chamber,the vertical dimension, which is now larger, will be focused more, resulting in a smaller
beam waist along the z direction. Figure 3.1b shows the AR reached moving the two concave
lenses on the linear stage. On the x-axis the distance travelled by the two concave lenses isplotted starting from the position correspondent to the theoretical unitary magnification. The
initial position leads to an AR of 1.26 ± 0.04, due to the fact that the laser beam is slightly
elliptical. The maximum achievable AR is 5.3 ± 0.2, which do not follow the predicted value,
probably due to technical problems involving the test mount, which did not allow to get thelens C4 exactly in the desired position. It is worth to underline that this method is completely
different from the one involving time-averaged potential. Indeed, the lenses change the real
beam waist and it is not an effect related to a fast modulation. As a consequence, the beam
behaves as a Gaussian beam also along the y direction. The big issue of this technique is theimplication of the motion of mechanical parts. Furthermore, it requires a linear stage with amaximum travelling velocity compatible with the time scale of experimental cycle (∼ 10 µs).

3.2.2Time-averaged potential

In the last years, time-averaged potentials have been used to investigate the physics ofultracold atoms in optical dipole traps with particular geometry, like ring traps [13], boxshaped trap [33] or traps with several numbers of wells [81]. Indeed, the trap can be shaped
2 Thorlabs KMTS50E/M.

41

3.2 How to get a tunable dipole trap?

in different ways making use of the principle of the time-averaged potential, in which thebeam is sweeped in one or two dimensions at a frequency much higher than the oscillation
frequency of the atoms in the trap. In this way, atoms do not follow the beam but just see theresultant averaged potential. Thus, the request for this type of potential is summarized in thefollowing relation:
Utrap ∝ Itrap = ⟨I(r,t)⟩Tmod

with ωmod ≫ ω̄ ,

(3.7)

mod
where Tmod is the modulation period, fmod = ω2π
is the frequency at which the beam is
1
sweeped and ω̄ = (ωx ωy ωz ) 3 is the average trapping frequency. Time-averaged potential canbe created in different ways using electro-optic or acousto-optic modulation. The principles
of operation is briefly presented below.
Electro-optic deflectors are based on the creation of a gradient in the refractive index
of the active material as a result of an applied electric field. Usually, these devices areused to realize time-averaged potentials with complex shape because, since the deflection
angle of the beam is directly proportional to the applied voltage across the medium, is mucheasier to spatially control the beam and thus the geometry of the resultant potential [74, 86].
Furthermore, EODs are preferred to AODs for high resolution applications because of few
reasons: the transmittance does not vary as much as in AODs with the deflection, the beam isentirely deflected and AODs usually suffer from systematic angular deviations from linear
response to the acoustic driving frequency [84]. On the other hand, EODs are usually quite
expensive and the commercial ones are not suited for high power.
Acousto-optic deflectors, as well as acousto-optic modulators (AOMs), consist of an
optically transparent material to which a piezo-electric element launches acoustic waves. In
this way a periodic change in refractive index, which acts as an optical grating, is created anddeflects the laser beam travelling through the crystal. The main difference between AOD and
AOM is that the latter has just one radio frequency (RF) signal which drives the acoustic wave.
In contrary, in the former there are more than one radio frequency signal which are delayed inphase in such a way to create a resultant signal that makes the response to the acoustic wave
uniform over the broadest possible range of deflection angles. The transmittance from an
acousto-optic device depends, indeed, on the driving acoustic frequency and it is maximum
at the Bragg angle, defined by the following relation:

θB =

λ fc
,
2va

(3.8)

where λ is the wavelength of the laser beam diffracted by the crystal, fc is the central
frequency of the total bandwidth, defined as the frequency at which the diffraction efficiency

42

Optical dipole trap: experimental realization

reaches its maximum value and va is the velocity of the acoustic wave in the material. Whenthe driving frequency is changed to deflect the beam in different points of the space, thediffraction efficiency decreases. With the presence of several electrodes with RF signals
progressively delayed in phase, the Bragg condition (3.8) is reached over a larger acoustic
frequency range [42].

3.3Time-averaged potential with acousto-optic devices

When acousto-optic devices are used for standard applications, like adjusting the intensity of
the laser beam or shifting the frequency in the RF range in a controlled way, a single driving
radio frequency is chosen, to which corresponds a fixed deflection of the beam. The basic
idea of creating time-averaged potential, either with AOMs or with AODs, is to send to thepiezo-electric element a RF signal modulated with a certain shape. In this way, the beam
is deflected horizontally back and forth according to the selected shape. The key feature ofthis method is the possibility of creating time-averaged potential with different geometry
just changing the shape of the driving signal, few simple example are in Fig. 3.2. The figure
shows how the beam profile looks like for different shapes of the modulation function, thesepictures were taken during a first test of the scanning system with an AOM.
The resultant potential is given by the time-averaged sum of the overlapping Gaussian
beams. In our case the beam will be scanned only in the horizontal direction x, this leads tothe following expression:Û
Utot (x, 0, 0) =Tmod

Z Tmod02(xS (t) − x)2
dt exp −
,
w2x

(3.9)

where Û is the total trap depth defined in Eq. (2.17), Tmod is the modulation period, which
has to follow the request (3.7), xS (t) is the function that defines the shape of the potential andwx is the beam waist along the x direction. The principal use of the scanning system will be
to increase the size of wx , while keeping a Gaussian shape. From earlier experiments [6, 48]
an arcus-cosine wave was proved to be suited for this aim.

RF amplitude

RF amplitude RF amplitude RF amplitude

3.3 Time-averaged potential with acousto-optic devices

43

1

0
1

Time

0
1

Time

0

Time
1

0

Time
Figure 3.2 Different trap geometry, along the horizontal direction, given by different shapesof the modulated RF signal. Starting from the top, a square wave results in two separated
angle deflections and thus two traps for the atoms. A sine and a triangular wave lead to adouble well and a box potential, respectively. Finally, a modulation with an arcus-cosine
shape, as proved in previous experiments [6, 48], leads to a Gaussian potential. The picturesof the laser beam, shown on the right, were taken with a CCD-camera beam profile during
the test of the scanning system with an AOM.

44

Optical dipole trap: experimental realization

Other functions like sine wave or triangular wave result in a final double well trapping
potential and a box potential, respectively. The tools necessary to create an arbitrary timeaveraged potential and control his shape during the experimental cycle are combined inFig. 3.3 and listed below:• Arbitrary Function Generator (AFG)3 , to create the desired function which has to besent to the acousto-optic device. The upper limit of the modulation frequency is fixedby the maximum sample rate of the AFG and the number of points of the arbitrary
function, directly loaded through USB-stick. In our case, since the maximum sample
rate is 250 Msample/s, we choose 1002 points to get a maximum modulation frequencyof ωmod = 2π × 250 kHz4 ;• Mixer5 , needed to combine the signal coming from the AFG (Vsignal ) with two other
input voltages. One, Vgain , is needed to control the amplitude of the RF wave, which
is directly related to the width of the resultant time-averaged beam. A second input
voltage, Voffset , is used as an offset to shift the central frequency of the signal. The
mixer combines all the three inputs and gives as output the following signal [4]:Vmod = Voffset +VsignalVgain.
10V(3.10)

Both Voffset and Vgain can be changed in the range 0 V – 10 V. The voltage of the former
is chosen accordingly to the voltage-frequency conversion of the VCO. Looking at
Eq. (3.10), Vgain = 0 V means no modulation, while Vgain = 10 V corresponds to amodulation with the maximum amplitude.
• Voltage Controlled Oscillator6 (VCO), which receives as input a voltage from themixer and gives as output a frequency which is directly related to the applied voltage.
The VCO has a 3 dB modulation bandwidth of 180 kHz and a working range of
50 MHz – 100 MHz. A voltage of 8 V is needed to get an output frequency of 75 MHz.
Since the mixer is specified with a maximum output of 10 V, an operational amplifier
was implemented in the electronic circuit after the mixer, to reach the voltage range
3 V – 13 V. See Appendix A for the scheme of the electronic circuit;• Radio frequency amplifier (RF-AMP), which receives the RF signal from the VCO andsupplies the acousto-optic device with the RF power needed to achieve a diffraction
3 Model Keysight 33521B.
4 An higher modulation frequency is not needed at the moment in the experiment. Furthermore, the VCO

implemented in the setup would not respond linearly to frequency higher than 2π × 180 kHz.
5 Model Analog Devices AD734.
6 Model Mini-Circuits ZX95-100.

45

3.3 Time-averaged potential with acousto-optic devices

efficiency as higher as possible. It is also used, combined with a Voltage Controlled Attenuator (VCA), to actively control the power of the laser beam during the experimental
cycle.
In the first part of the thesis a test setup with an acousto-optic modulator7 was implemented following the electronic scheme shown in Fig. 3.3a. The AOM has a central frequency
fc = 110 MHz, a RF bandwidth BW = 15 MHz and an active aperture of 1.25 mm8 . Afterwards, another test setup with an AOD9 was built in order to compare the performanceof these two devices for our purpose. The AOD has a central frequency fc = 75 MHz, abandwidth BW = 32 MHz and an active aperture of 2.5 mm. Among the measurements, an
VoffsetAFG

VsignalVmod

VCO

RF-AMPAOMVgain(a) AOM.
VoffsetAFG

VsignalVmod

VCO

RF-AMP

PS

AOD

RF-AMP
Phase delay

Vgain(b) AOD.

Figure 3.3 Sketch of the electronic scheme used for AOM (a) and AOD (b). A mixer combines
the signal coming from the AFG (Vsignal ) with Vgain and Voffset . The resultant Vmod is sent
to the VCO, which converts it in a radio frequency output. In the case of the AOM, this
output is amplified by the RF-AMP and sent to the AOM. For the AOD, a power splitter (PS)generates two signals, which feed two different RF-AMPs. SMA cables, with a well defined
length difference, are used to create a delay in phase between the two signals.
interesting one for our purpose examines the dependence of the diffraction efficiency on
the frequency of the RF signal and thus, on the angle of deflection. In Fig. 3.4 the large
bandwidth of the AOD is clearly visible compared to the restricted one of the AOM. The
diffraction efficiency of the AOM decreases dramatically when the frequency moves awayfrom the central one. After 10 MHz already a reduction of almost 30% is visible. In thecontrary, the efficiency of the AOD is kept almost constant up to 15 MHz from fc . The
AOD minimizes the decreasing of the diffraction efficiency when the frequency is shifted far
7 Model 3110-197 Gooch & Housego.
8 The active aperture is 2.5 mm × 1.25 mm but the smaller dimension has to be considered as an effective

aperture to not cut the beam along that direction.
9 Model 4075-2 from Crystal Technology, Inc.

46

Optical dipole trap: experimental realization

away from the central one, thus it is more suited for the purpose of creating a time-averaged
potential. Furthermore, the active aperture of the AOM is smaller than the one of the AOD,
this makes the latter even more suited for our system, which uses a high power laser. These
motivations drove us to the choice of a setup built with the AOD. A sketch of the electronic
setup implemented for this device, is shown in Fig. 3.3b. The deflector has two radio frequency inputs to compensate for the deviation from the Bragg condition (3.8). A power
splitter10 receives the output of the VCO and creates two RF signals. Each of them feeds an
RF-AMP, travelling along two SMA cables with different lengths11 , in such a way to create adelay in phase.

Diffraction efficiency

1

AOD
AOM

0.8

0.6

0.4

0.2

0
60

80

100

120

Frequency (MHz)Figure 3.4 Diffraction efficiency measured changing the driving frequency of the AOD(blue points) and AOM (orange points). Starting from the central frequency of the AOM( fc = 110 MHz), already at 10 MHz, the efficiency is reduced by almost 30%, while for the
AOD is kept nearly constant up to 15 MHz from fc = 75 MHz.

3.3.1

Optical setup: scanning system

In addition to the electronics needed to control the AOD, from the point of view of opticalelements, the setup necessary to create a time-averaged potential requires few parts: theAOD and a lens placed at a distance of one focal length, whose task is to transform angle
deflections in horizontal shifts. To adjust the beam waist to the desired value on the atoms at
the center of the chamber, a telescope can be realized using as a first lens the one in front of
10 Model MiniCircuits ZFSC-2-1-S+.
11 The right length difference (∆l) is provided by the company. In our case ∆l = 2.461 m.

47

3.3 Time-averaged potential with acousto-optic devices

the AOD. A simple sketch of the setup is shown in Fig. 3.5. The range, in which the laser

f

f

AOD
D

w0
wf

RF + delay

RF

Figure 3.5 Working principle of the scanning system. The AOD has two electrodes for two
RF inputs. The electrode on the left receives the RF signal with a delay in phase with respect
to the one on the right. A lens, placed at a distance of one focal length from the AOD,
converts the angle deflections in horizontal shifts. The main quantities that play a role in the
maximum aspect ratio achievable are depicted (w0 , w f , f and D).
beam can be deflected, fixes the maximum width of the resultant potential and the AR ofthe trap. This quantity is strictly related to two parameters: the resolution (R), which is thenumber of resolvable spots, and the bandwidth (BW) of the AOD. In our case the dimensionof one spot is given by the diameter of the beam in the focus of the lens (2wf ). The maximum
distance by which the beam can be deflected (D) is given by the RF bandwidth, which isfixed by the frequencies at which the diffraction efficiency decreases by 3 dB. Thus, theresolution can be written as:D.
(3.11)
R=
2wf
It is useful to express R in terms of the AOD’s parameters and the laser beam waist. To dothis, Eq. (3.8) and the following relations can be used:
wf =

λf
,
w0 π

D=

BWD0
,
fc

D0 = θB f ,where w0 is the waist of the incoming beam, f is the focal length of the lens and D0 is theshift of the beam relative to the zero order. Equation (3.11) can then be expressed as below:R=

2π
w0 BW.
4va(3.12)

48

Optical dipole trap: experimental realization

The resolution and thus the final aspect ratio achievable, depends only on the intrinsic
characteristics of the AOD and on the initial beam waist of the laser beam. The telescopebuilt afterwards will not influence this value at all. For fixed BW and va , which depend on
the device, by increasing the initial beam waist w0 it is possible to achieve larger values ofresolution. The limit is, of course, imposed by the active aperture of the AOD.
The crystal of the AOD used in the setup is made of TeO2, to which corresponds asound velocity va = 4200 m s−1 . Considering our values of va and BW and taking as w0
the maximum waist of the incoming laser beam, fixed by the AOD’s active aperture w0 =1.25 mm, Eq. (3.12) leads to a maximum theoretical value of AR = 15.
As a first test, instead of the system made by VCO, power splitter and cables with different
lengths, a function generator12 able to create two signals at a frequency of 75 MHz with adelay in phase, was used. The signal from the AFG, with an amplitude expressed in Vpp ,can be superimposed on the RF signals as external modulation. The radio frequency ranges,
covered during the scanning, are related to the voltage amplitude of the modulation, thecalibration is reported in Table 3.2.
Table 3.2 Radio frequency ranges correspondent to different values of voltage amplitude of
the modulation.
Modulation amplitude
Voltage(Vpp )

Frequency
(MHz)

0
1
2
3
4

75
70 – 80
65 – 85
60 – 90
55 – 95

The scanning is applied on the laser beam used for the horizontal ODT. This beam
propagates along the y direction and the modulation involves the x direction (z corresponds
to the axis of gravity). To verify the performance of the scanning, after selecting an arcuscosine function as modulation, the beam waist was measured using a CCD-camera beamprofiler13 along both directions (x and z). In the following measurements, the size of theincoming beam was adjusted with a telescope in order to get w0 = (1.15 ± 0.01) mm, thisleads to a predicted resolution of R = 14 according to Eq. (3.12). The plot in Fig. 3.6 shows,
12 Model RSDG5162 from RS Pro.
13 Model BC106-VIS CCD-camera beam profiler from Thorlabs. This camera has a pixel size of 6.45 µm ×6.45 µm, an aperture size of 8.8 mm × 6.6 mm and can measure beam diameter down to 30 µm.

49

3.3 Time-averaged potential with acousto-optic devices
15

350

beam waistX
beam waistY
AR

300

f(x)=a*x2+b*x+c
10250

Aspect ratio

Waist (µm)

400

200
150

5

100
50
0

0
0

1

2

3

4

Voltage amplitude (Vpp )Figure 3.6 Measurement of the waist as a function of the modulation amplitude, expressed in
Vpp . The axis on the left shows the waist along both directions x and z, whereas the axis on
the right shows the resultant AR. The orange curve shows the fit function f (x) = ax2 + bx + c.
−2 , b = 129.0V −1 and c = 30.2 µm.
Where a = −12.4Vppppon the left axis, the beam waist measured along the x (circles) and z (squares) directions
while, on the right axis, the resultant AR. The data shown in the plot are obtained taking
with the CCD the x and z profile of the beam for each chosen voltage amplitude and fitting
the curve with a Gaussian function. In a first test of the dipole trap the beam waist at
the center of the chamber was planned to be around w0 = 35 µm. Thus, starting from abeam waist of wx = (34.4 ± 0.5) µm, along x, and wz = (30.9 ± 0.8) µm, along z, the final
beam waist achieved with the maximum modulation amplitude reads wx = (344 ± 1) µm andwz = (31 ± 1) µm, respectively, leading to the aspect ratio AR = 11.1 ± 0.4. In Table 3.3 all
the measured waists for different modulation amplitudes and the relative AR are listed.
The next step is to check that the shape of the beam modulated with an arcus-cosine
wave follows a Gaussian function. The profile along the x direction, taken again using the
CCD-camera beam profiler, is shown in Fig. 3.7. This profile displays a single pixel rowof the received image and on the curve is shown the relative intensity (100% represents the
maximum on that row). On the left side of the figure the Gaussian profile of the beam isshown without any modulation on. Whereas, on the right side, the profile resultant from amaximum amplitude of the modulation (4 Vpp ) is displayed. The Gaussian fit follows in anice way the experimental points.

50

Optical dipole trap: experimental realization

Table 3.3 Beam waist measured along the x and z directions for different modulation amplitudes. The beam itself is slightly elliptic, thus the AR starts from 1.11 ± 0.03, with nomodulation, and reaches the maximum value of 11.1 ± 0.4.
Amplitude(Vpp )
0
1
2
3
4

wx
(µm)

wz
(µm)

AR

34.4 ± 0.5
137 ± 2242 ± 1310 ± 1344 ± 130.9 ± 0.8
31 ± 131 ± 131 ± 131 ± 1

1.11 ± 0.03
4.4 ± 0.2
7.8 ± 0.3
10.0 ± 0.3
11.1 ± 0.4

A second test was made choosing a triangular wave as modulation function in order tocheck the maximum range in which the time-averaged potential assumes the shape of a box
along the direction of the scanning. The measurements, relative to the triangular function, areshown similarly to the arcus-cosine case in Fig. 3.8. The beam profile looks quite uniform up
to a frequency range of 70 MHz – 80 MHz. Afterwards, it starts to assume a different shapequite close to a Gaussian function. This is probably due to the high harmonics induced by the
sharp corners of the triangular wave, which will need further investigation. The maximum
beam size, measured for the two different modulation shapes are shown in Table 3.4 with thecorrespondent aspect ratio.
Table 3.4 Maximum beam size and relative aspect ratio, choosing an arcus-cosine and atriangular function as modulation for the driving frequency.
Modulation function
Arcus-cosine
Triangular

3.3.2

Maximum beam size
(µm)

AR

344 ± 1388 ± 2

11.1 ± 0.4
12.5 ± 0.4

Driving RF signal

To fully test the performance of the scanning system, in the previous measurements, theinitial waist of the incoming beam was chosen close to the limit of the active aperture of the
AOD and the modulation frequency was fixed in a region in which the effective bandwidth
was not limited by the electronics (ωmod = 2π × 1 kHz). When shining the time-averaged
beam on the atoms, the modulation frequency has to be much higher than the oscillation
frequency of the atoms in the trap. In particular, it has been proved that fmod has to be at

51

3.3 Time-averaged potential with acousto-optic devices
Amplitude = 4 V pp

No scanning
Gaussian fit

w0 =34.4 0.5 m

w0 =344 1 m

100

80

80

60

60

40

40

20

20

0
-100

-50

0

X position ( m)

50

100

Power (a.u.)

Power (a.u.)

100

Gaussian fit

0
-500

0

500

X position ( m)Figure 3.7 Beam profile along the direction of the scanning (x) for a not modulated driving
frequency (left) and for an arcus-cosine shape of the modulation with the maximum amplitude
4 Vpp (right).
least 100 times larger than ωx to not induce heating [64]. Thus, from the simulation shownin the Chapter 2, the modulation frequency should be around ωmod = 2π × 400 kHz to satisfy
this rule. However, find a VCO suitable for a center frequency of fc = 75 MHz with suchhigh RF modulation bandwidth, is not straightforward. The VCO implemented, up to now,
has a modulation bandwidth of ωmod = 2π × 180 kHz.
Already at ωmod = 2π × 50 kHz, the VCO starts to cut the range of radio frequencies sent
to the AOD and this leads to a smaller aspect ratio achievable. In the experiment the AOD iscontrolled with the electronic scheme shown in Fig. 3.3b and the modulation frequency isfixed to ωmod = 2π × 100 kHz. In Fig. 3.9 the RF bandwidth, measured looking the signal
coming out of the RF-AMPs and sent to the AOD on a spectrum analyzer, is plotted forthree different modulation frequencies ( fmod = 10 kHz, 50 kHz and 100 kHz) changing thevoltage Vgain responsible for the modulation amplitude. For the modulation frequency of
100 kHz, the VCO levels out the maximum RF bandwidth at around 8 V. The RF signal sent
to the channels of the AOD, for three different values of Vgain (0 V, 4 V and 8 V), is shown in
Fig. 3.10 for a triangular wave as modulation function.
The influence of the VCO on the final achievable size of the beam is clearly seenin Fig. 3.11. For this test the initial beam waist was (46.4 ± 0.7) µm and the modulation
amplitude was fixed at Vgain = 5 V. Increasing the modulation frequency the width of thebeam decreases. For an arcus-cosine function, at the end of the bandwidth, the beam size is52Power (a.u.)

No scanning
Gaussian fit

Amplitude = 1 V pp

Amplitude = 3 V pp

Box shape

Gaussian fit

100

100

80

80

60

60

40

40

20

20

0
-100

0

X position ( m)

100-200

Power (a.u.)

Optical dipole trap: experimental realization00

X position ( m)

200 -500

0500

X position ( m)Figure 3.8 Beam profile along the direction of the scanning (x) for a not modulated driving frequency (left) and for a triangular shape of the modulation with an amplitude of 1 Vpp (center)and 3 Vpp (right). The beam profile looks quite uniform up to a range of 70 MHz – 80 MHz,afterwards it starts to assume a nearly Gaussian shape.
reduced by 15% with respect to the one measured at 60 kHz. The reduction is stronger forthe triangular wave where, at the modulation frequency limit, the reduction is of almost 25%.
To summarize the motivations that drove our choice for the acousto-optic deflector,
advantages and disadvantages of the three tested systems are listed as follows:• Cylindrical lenses: the setup built with cylindrical lenses has the advantage to change
the real aspect ratio of the laser beam. This leads to an optical dipole trap with
Gaussian behaviour in all the three directions. On the other hand, the lenses need
to be moved on a linear stage. This means dealing with the drawback of movementof mechanical parts, which could affect the beam pointing stability necessary for an
efficient optical trap. Thus, to be implemented, the system requires good translational
stages in terms of deviations from the true direction of travel and reproducibility.
Furthermore, during an experimental cycle, the beam waist has to be tuned on timescales down to 10 µs, demanding high maximum velocity. All these conditions require
an expensive translational stage, whose price would be comparable to the cost of two
acousto-optic deflectors.
• Acousto-optic modulators: these devices have been used since decades now andhave the advantage to be well known systems. However, for time-averaged potential
applications, diffraction efficiency curves as uniform as possible are required. Thus,
the short bandwidth of the AOM is often corrected rescaling the diffraction curve tothe minimum value, with a dramatic impact on the output power.

53

3.3 Time-averaged potential with acousto-optic devices
60
fmod = 10kHz
fmod = 50kHz

50

RF bandwidth (MHz)fmod = 100kHz
40

30

20

10

0
0

1

2

3

4

5

6

7

8

9

10

Voltage gain (V)Figure 3.9 Measurement of the radio frequency bandwidth as a function of Vgain for different
values of modulation frequencies (10 kHz, 50 kHz and 100 kHz). For a modulation frequencyof 100 kHz, the VCO levels out the maximum RF banwidth at around 8 V.
V gain = 8V

V gain = 4V

40

40

20

20

0

0

-20

-20

-40

-40

-60
60

80

Frequency (MHz)

100

60

80

Frequency (MHz)

100

60

80

Power (dBm)

Power (dBm)

V gain = 0V

-60
100

Frequency (MHz)Figure 3.10 Radio frequency signal, coming out from the RF-AMPs and sent to the AOD,
measured with a spectrum analyzer. The three plots show the signal for different modulation
amplitude (0 V, 4 V and 8 V), choosing as modulation function a triangular wave. Eachchannel of the AOD receives a maximum RF power of 33 dBm (2 W).
• Acousto-optic deflectors: these devices are made for applications in which the laser
beam has to be deflected back and forth maintaining the diffraction efficiency curvealmost constant. They, as the modulators, are easily controllable in experimental
cycles tuning the driving RF frequency and changing the RF-AMP parameters. Both
AOMs and AODs exhibit the drawback of the creation of a resultant potential which

54

Optical dipole trap: experimental realization
350
arcus-cosine
triangular

Beam width ( m)

300

250

200

150
60

80

100

120

140

160

180

Modulation frequency (kHz)

Figure 3.11 Reduction of the beam size for increasing modulation frequencies, induced by
the limited bandwidth of the VCO. In this test the initial beam waist was (46.4 ± 0.7) µm andthe modulation amplitude was fixed at Vgain = 5 V.
does not follow the behaviour of a Gaussian beam along the direction of propagation.
Furthermore, they require high modulation frequencies, which are quite often difficult
to reach for central radio-frequencies in the range 0 MHz – 200 MHz. Even if AOD
results in a more complex system with respect to the AOM because it requires several
RF inputs, it is the most suited for creating time-averaged potentials, among the threetested systems.

3.3.3

Measurement of the waist

The setup was built to reach a waist of around wx = wz = 20 µm. The measurement of thebeam waist is crucial to fully characterize the dipole trap and for further measurements of thepolarizability. To estimate the value of the beam waist, the CCD-camera beam profile wasmounted on a linear stage and moved in such a way to scan the laser beam along the directionof propagation (y). The data, taken for the horizontal beam of the dipole trap, are plotted inFig. 3.12. In each point, the 1/e2 radius of the beam along the x and z axis is measured. The
right side of the figure shows a zoom of the region around the foci. The points are fitted withthe expression for the radius of a Gaussian beam, given by Eq. (2.15). From the fit the beam
waists of the horizontal beam, along the x and z directions, are wx = wz = (20.5 ± 0.2) µm.
Where the uncertainty are given by the statistical errors of the fit. Around the focal points, thefit for both directions does not follow nicely all the data. This can be related to the difficulty

55

3.3 Time-averaged potential with acousto-optic devices
wx

wx

Gaussian beam fit
wz

Gaussian beam fit
wz

Gaussian beam fit

Gaussian beam fit

100

300

80

200
60
150
40
100

Waist ( m)Waist ( m)

250

20

50
0
0

0.9

1.8

Position ( m)

2.7 0.6
10 4

1

1.4

Position ( m)

0
1.8
10 4

Figure 3.12 Measurement of the beam waist along the x (blue) and z (red) axis of thehorizontal beam. The data points are the 1/e2 radius of the beam measured with the CCDcamera beam profiler. The plot on the right is a zoom of the region around the foci.
of measuring such small beam waists with the CCD-camera beam profiler. A beam waist of
20.5 µm is, indeed, quite close to the resolution limit of the CCD, which can measure beam
diameters down to 30 µm.
In the RARE experiment the vertical dipole trap, which crosses the horizontal one andpropagates along the z axis, is obtained splitting the initial beam coming from the laser headwith a polarizing beam splitter (PBS) and a waveplate (λ /2). In this way, with a motorized
waveplate, the power can be redistributed arbitrarily during the experimental cycle between
the horizontal and the vertical beam. The beam waist of the vertical beam is measured
similarly to the horizontal one. The results are shown in Fig. 3.13. As for the hODT, thedata are fitted with the relation of the Gaussian beam radius. The fit gives as results thebeam waists of the vertical beam, along the x and y directions, wx = (101.3 ± 0.2) µm andwy = (95.0 ± 0.2) µm. The uncertainties are given by the statistical errors of the fit.
From the measurement relative to the horizontal optical dipole trap, there is a clear
difference in the position of the foci along the two different axes. This value is comparable to
the Rayleigh length and leads to an astigmatic beam. On the contrary, the astigmatism does
not affect the vertical beam in which the distance of the foci is much less than the Rayleighlength. All the result of the beam waist measurements are summarized in Table 3.5.

56

Optical dipole trap: experimental realization
120
wx
Gaussian beam fit
wy115

Gaussian beam fit

Waist ( m)

110

105

100

95

90
0

0.9

1.8

2.6
10 4

Position ( m)Figure 3.13 Measurement of the beam waist along the x (blue) and y (red) axis of the verticalbeam. The data points are the 1/e2 radius of the beam measured with the CCD-camera beamprofiler.
Table 3.5 Measured beam waist of the horizontal and vertical optical dipole trap. ∆y and ∆z
represent the shift between the foci of the horizontal and vertical beam, respectively. zRh (zRv )is the Rayleigh length of the horizontal (vertical) beam.
Beam waist(µm)

Astigmatism(mm)Rayleigh length(mm)

Horizontal beamwx
20.5 ± 0.2

wz
20.5 ± 0.2∆y0.6 ± 0.1

zRh
1.2

Vertical beamwx
101.3 ± 0.2

wy
95.0 ± 0.2∆z1.2 ± 0.4

zRv
28.3

In Fig. 3.14 is shown how the beam waist, starting from wx = 20.5 µm increases as afunction of the modulation amplitude (Vgain ). The rise of the waist is levelled out due to thelimitation in the bandwidth of the VCO, as discussed previously. This effect is enhancedby the fact that to make the electronic circuit more compact the voltage range sent to the
VCO was reduced. At the moment in the experiment it seems we are not limited by thiseffect for the loading of the ODT, for both species of erbium and dysprosium, if necessary
the electronics can be easily modified as shown in Appendix A.

57

3.3 Time-averaged potential with acousto-optic devices
180
fmod = 100 kHz

160

Horizontal waist (µm)

140
120
100
80
60
40
wx

20

f(x)=a*x2+b*x+c

0
0

1

2

3

4

5

6

7

8

9

10

Vgain (V)Figure 3.14 Dependence of the beam waist on the modulation amplitude (Vgain ) choosing
the arcus-cosine function as modulation with a frequency of fmod = 100 kHz. The limited
bandwidth of the VCO levels out the curve around 7 V. The orange curve shows the fit
function f (x) = ax2 + bx + c. Where a = −1.3 V−2 , b = 25.9 V−1 and c = 18.5 µm.

3.3.4

Implementation in the experiment

After the test, the setup was built on a small breadboard to be implemented in the experiment.
The final setup of the horizontal and vertical dipole trap is shown in Fig. 3.15. The 1064-nm
light comes from a 50 W Yb3+ fiber laser. In the laser head is already included an high power
infrared (IR) isolator to prevent damage coming from intense back reflections. This reduces
the available power to 45 W. After the laser head the beam path is folded several times due toa bad shape of the profile in the near field. Two systems, PBS plus waveplate (λ /2), control
the distribution of the power. In particular, the first controls the total power14 and the second
controls the power distribution between the horizontal and vertical ODT. In the picture, thepath of the horizontal beam is depicted in red while the vertical one in yellow. On the hODT
path, after the AOD, a series of lenses are installed to get a final waist of wx = wz = 20.5 µm
on the atoms. All the lenses and mirrors are made by Fused Silica and coated for 1064 nm15 .
Indeed, due to the high power, it is preferable to use Fused Silica instead of the standard
N-BK7, which has a thermal expansion coefficient 14 times higher, to prevent thermal lensing
effect. As discussed in Chapter 2, the lifetime of the atomic sample strongly depends on the
14 This was installed for preliminary test, since the minimum power coming out of the laser is 400 mW, and it

will be removed afterwards.
15 The lenses and the high power mirrors are from the company LENS-Optics. High power mirrors: M1064/1"2"/45. Fused Silica lenses: LPX1"Q, LPV1"Q.

58

Optical dipole trap: experimental realization

light polarization due to the tensorial nature of the atomic polarizability. Thus, a waveplate isplaced behind the last lens to rotate the linear polarization. The AOM in the vertical path is
not involved in any scanning but plays the role of controlling the power of the laser beamduring the experimental cycle. All the shutters implemented in the setup of the optical dipole
trap are built with high power mirrors and fan-cooled16 beam dumps.

16 This system will be exchanged with water-cooled beam dumps.

500

λ/2
-50

PERISCOPEPD

POWER
STABILIZATION

200

AOD
75MHz

-1

250

POWER
STABILIZATION

150

-100

50

-50

λ/2

110MHz

AOM

+1

DUMP

λ/2

BREADBOARD

ALS-IR-1064-50-SF

Figure 3.15 Scheme of the optical dipole trap setup. After the laser head two systems, PBS plus waveplate (λ /2), control the
distribution of the power. The path of the horizontal beam is depicted in red while the vertical one in yellow. On the hODT path,
after the AOD, a series of lenses are installed to get a final waist of whx = whz = 20.5 µm on the atoms. The AOM on the vODT pathcontrols the power of the vertical beam during the experimental cycle. The beam waist is adjusted to the final size wvx = 101.3 µm andwvy = 95.0 µm. The grey circle in the main chamber represents the mirror used to reflect the MOT and vODT beams from the bottom
along the axis of gravity.

300

1000

PD

1064 nm
45W

MAIN
CHAMBER

3.3 Time-averaged potential with acousto-optic devices
59Chapter 4
Measurements with atoms
The previous chapters introduced the tools necessary to characterize an optical dipole trap, as
trap depth, trap frequencies, and polarizability. We also discussed the possibility to change
dynamically the trap geometry as an additional degree of freedom and how to implement itin the experimental cycle. This chapter is devoted to test the ODT on atoms. In particular,
the first section introduces the RARE experimental apparatus. The second section focuses on
the effect of the scanning system on the atom loaded into the ODT. The third section treats
the trap frequency measurements as a tool to characterize the trap geometry and to determine
the polarizability, and the final section is devoted to the planning of further evaporative steps.

4.1

RARE experimental setup

RARE experiment combines for the first time Er and Dy in a dual species machine which,with a three-chamber concept, will investigate several aspects of dipolar physics. The
apparatus is shown in Fig. 4.1. The atom flux of Er and Dy is created in a commercial oven1 ,in which solid pieces of Er and Dy in form of alloy are evaporated at a temperature of about1100 ◦C. The oven includes two sections: the effusion cell, which contains the material,
and the hot lip2 . The two species have close melting temperature (1414 ◦C and 1529 ◦C
for Dy and Er, respectively [43]) resulting in a similar atom flux density. To compensate
the difference in melting point, and thus atomic flux3 , Er was added recently in the hot lip
section, which operates at 1150 ◦C. The atomic beam is collimated first, through a set ofapertures in the oven section and then through two-colour transversal cooling, using the
1 Model DFC-40-10-WK-2B from CreaTec Fischer & Co. GmbH.
2 The effusion cell is usually operating at a temperature of 1100 ◦C, while the hot lip is set to a temperatureof 1150 ◦C.
3 Erbium has a lower vapour pressure, which leads to a lower evaporation rate.

Measurements with atoms
62

Oven

Main chamber
for Er-Dy degenerate
mixtureZeemanslowerGlass cellfor future quantum
gas-microscopeexperimentTransversal
cooling

Rydberg chamber

Figure 4.1 RARE experimental apparatus. The oven generates the atomic flux of Er and Dy, which is then collimated by a two-colour
transversal cooling (401-nm light and 421-nm for Er and Dy, respectively). The Zeeman slower slows the atoms down to the capture
velocity of the two-colour MOT (583-nm light and 626-nm for Er and Dy, respectively). The figure shows the three chambers: mainchamber, Rydberg chamber and the future glass cell for quantum gas-microscope experiment. Up to now, only the main chamber isfully implemented and operative.

4.1 RARE experimental setup

63

401-nm light for Er and 421-nm for Dy [31]. Atoms are then slowed by the Zeeman slower
down to the respective capture velocity of the MOT (few m s−1 ). The RARE experimental
setup plans to include three chambers. Atoms will be moved from one chamber to the others
by implementing an optical transport.
In the main chamber atoms of Er and Dy are trapped first in the MOT and then they
are transferred into the ODT for evaporative cooling to the degenerate phase. The MOT
provides a confining force which, in our case, is comparable to gravity. Thus, the shape of
the atomic cloud is deformed and atoms accumulate to the lower boundary of the MOT, fromwhich they predominantly absorb light from the bottom and get spin-polarized in the Zeeman
sublevel mJ = −6 and mJ = −8 for erbium and dysprosium, respectively. This particularfeature allows a MOT configuration, which employs only five beams instead of the standard
six beams. With only five beams, the laser beam coming from the bottom does not need tobe retro-reflected, leading to free optical access from the top side of the chamber. The MOTparameters as power, detuning, magnetic field gradient and compression time are optimized
to load both species simultaneously into the ODT. The ODT setup consists of an horizontal
beam (hODT) travelling along the y direction and a vertical beam (vODT) travelling along
the axis of gravity z. Figure 4.2 shows a top view of the main chamber with the laser beamsof MOT, ODT and ZS.
The second chamber is dedicated to Rydberg physics, in which atoms are excited instates with a principle quantum number n ≫ 1. The high value of n, usually in the range of
20 – 100, leads to typical sizes of the electronic wave function on the order of ≈ nm and to
extraordinarily large electric dipole moment [16, 32].
In the third chamber a single site resolution imaging [7, 80] is planned to detect single
atoms in a optical lattice. Among the three only the main chamber is, up to now, fully
implemented and operative.
The RARE experiment plans to investigate, in the main chamber, the behaviour of dipolar
atoms in presence of a magnetic field B, which polarizes the sample. In particular, theinfluence on the interactions of the dipolar imbalance induced by the different magnetic
moment can be studied together with the interspecies scattering physics. Furthermore, inthe single-site imaging chamber, dipolar atoms will be studied in an ideally zero magnetic
field (B ≈ 0). In this way, atoms are not forced by the external field to orient the spin in a
fixed direction and the magnetization is free. To study this particular behaviour, the B field
has to be controlled at the 100 µ G level [67]. Due to this reason, to compensate the earth’s
magnetic field and other external magnetic fields, compensation cages will surround all thethree chambers and the optical table. Furthermore, mostly all the equipment on the optical

64

Measurements with atoms

table devoted to the experimental apparatus (screws included) is made by non magnetic
materials.

ZS LIGHT

MOT LIGHT
yhODT LIGHT

x

Figure 4.2 Top view of the main chamber. Atoms, travelling along the x-axis, are slowed
down by the Zeeman slower with light at 401 nm (violet arrow) and 421 nm (blue arrow).
The yellow and red arrows indicate the direction of the MOT beams for erbium (583 nm) anddysprosium (626 nm), respectively. The dark red arrow, along the y-axis, shows the directionof the horizontal beam of the ODT (1064 nm). In the center of the chamber the other two
beams of the MOT and the vertical beam of the ODT are visible and propagate along thedirection of the gravity z.

4.2

ODT loading

Once atoms are slowed down by the Zeeman slower, they are captured by a two-colour
narrow-line MOT, which allows to reach samples with a number of atoms around 107 forboth erbium and dysprosium4 at a temperature of 10 µK. This temperature fixes the minimum
trap depth necessary to transfer the atoms from the MOT to the ODT. The ratio between
the trap depth and the thermal energy (Û/kB T ) has to be higher than 10 as a general ruleto get an efficient loading. In our case the maximum power available for the ODT beam is
4 Recently, the MOT atom number was improved up to 108 . On the other hand, this value does not concern

the measurements presented in this thesis.

4.2 ODT loading

65

around 38 W. For a single beam with P = 38 W (single hODT) and waist of about 20 µm,
we estimate a trap depth of about 1700 µK and 1400 µK5 for Er and Dy, respectively, which
are fulfilling the general rule. When Û/kB T ≫ 1 the integral in Eq. (3.3) saturates and thenumber of atoms loaded becomes independent from the laser power. Thus, assuming to bein the saturation regime, an efficient loading of the ODT relies on the overlapped volume
between MOT and ODT. This requires to increase the laser beam waist through the scanningsystem described in Chapter 3. Changing the trap geometry during the loading allows toreach the “mode matching” condition. “Mode matching” is referred to the ideal conditionin which, after the transfer, the atomic cloud maintains his volume and temperature. The
departure from the ideal case is given by a too tight trap, which heats up the atoms, or by a
too loose trap, which leads to a non-adiabatic expansion of the atomic cloud [46].
In our experiment, during the MOT loading, the ODT is kept on since the atomic clouds
are displaced with respect to the center of magnetic field gradient. The atoms are loaded
into the hODT during the MOT compression for a time of 100 ms. During the loading anhomogeneous magnetic field of about 1 G is switched on to preserve the spin polarization of
the sample.
To test the effect of the scanning system on the ODT loading, the number of atoms
loaded from the MOT to the ODT was measured as a function of the scanning parameter
Vgain . The measurement is shown in Fig. 4.3. The rise of the atom number shows the clear
effect of the scanning system, which leads to a number of atoms that is almost twice thevalue measured with the scanning off6 . The curve reveals the limit of the current electronics,
already discussed in section 3.3.2, indeed after Vgain = 7 V the atom number levels out. Thismeasurement was taken during the test of the ODT with a waist of 35 µm with only Dy atoms.
When the waist of the hODT is increased along the x-axis, the average power of the laser
beam at the atoms decreases due to the reduced diffraction efficiency of the AOD for highmodulation amplitude. Thus, when the volume of the ODT increases to get the maximum
overlapping with the MOT volume, the power can become the limiting factor of the ODTloading.
A 35-µm waist of the hODT gives a too low axial confinement to efficiently trap theatoms. We thus rebuilt the setup to get a waist of 20.5 µm. This allows us to transfer ≈ 106
atoms from the MOT to the ODT and hold them for a time sufficiently long to go on withfurther trap frequency measurements. In Fig. 4.4 the absorption images of the ODT shows
the simultaneous loading of both species Dy (left) and Er (right).
5 The details on trap depths and trap frequencies are listed in Table 2.5.
6 The atom number is plotted in arbitrary units (a.u.) because the measurement was taken in a stage in which

the imaging was not calibrated yet.

66

Measurements with atoms
12000

Number of atoms (a.u.)

10000

8000

6000

4000

2000

0
0

2

4

6

8

10

V gain (V)Figure 4.3 Number of atoms loaded into the optical dipole trap as a function of the scanningparameter Vgain . This measurement was done with the first setup of the ODT in which thebeam waist was 35 µm. The points show the clear influence of the scanning system on theenhancement of the atom loading. At around Vgain = 7 V the atom number levels out due totechnical limits induced by the electronics, discussed in Chapter 3.

4.3

Trap frequencies measurements

An important step to characterize the geometry of the ODT is the measurement of trap
frequencies. For this aim, two types of collective modes are usually excited in the atomic
cloud: the monopole mode and the dipole mode. The former is referred as “breathingmode” because involves the compression and de-compression of the atomic cloud and forthermal gas has a frequency of twice the trapping frequency [22]. The latter corresponds to atranslation of the total cloud and has the same frequency as the confinement.
We now briefly review how to get, using the classical Boltzmann equation, the frequencyof the “breathing mode”. Let’s consider the trapping potential in the harmonic approximation as defined in Eq. (2.18). At low temperature, below the critical temperature TC for
Bose-Einstein condensation, the system can be described by the hydrodynamic equationof superfluidity [70]. On the other hand, when the temperature is much higher than TC the
dynamic of a dilute gas is well described by the Boltzmann equation [41]:∂f
Fext+ v · ∇r f +· ∇v f = Icoll ( f ) ,
∂t
m

(4.1)

where f (r, v, t) is the phase space distribution function, r and v are the position and velocityof the particles, Fext is an external force acting on the particles and Icoll is the collisional

67

4.3 Trap frequencies measurements
ErbiumDysprosium

1495

1500

1495

1500

1505
0

500

1000 0

500

1505
1000

Figure 4.4 Simultaneous ODT loading of Er and Dy. During the MOT loading the ODT is
kept on since the atomic clouds are displaced with respect to the center of magnetic field
gradient. The atoms are loaded in the hODT during the MOT compression for a time of
100 ms. During the loading an homogeneous magnetic field of about 1 G is switched on topreserve the spin polarization of the sample.
integral describing the effect of collisions between particles. In our case Fext = Fdip = −∇Udip .
Two regimes can be identified: the hydrodynamic (collisional) and the collisionless regime(Icoll = 0).
The behaviour of the monopole mode can be obtained applying the Boltzmann equation
to the square radius of the atomic cloud [37]. Considering that Icoll (⟨r2 ⟩) = 0 because⟨r2 ⟩ is conserved during elastic collisions and assuming an isotropic harmonic potential
ωx = ωy = ωz ≡ ωtrap , the equation assumes the form:d⟨r2 ⟩= 2⟨r · v⟩ .
dt

(4.2)

Equation (4.2), together with the following two equations:d⟨r · v⟩2
= ⟨v2 ⟩ − ωtrap⟨r2 ⟩,
dt

d⟨v2 ⟩
2= ωtrap⟨r2 ⟩ ,
dt

lead to the solution ωm = 2ωtrap . In the case of a Bose-Einstein condensate at T = 0 also the√mean field interaction has to be included, thus the solution reads ωm = 5ωtrap .
There are several methods to determine the trap frequencies, which are based either onparametric heating or on direct observation of the atomic cloud oscillations in the trap. The
oscillations are induced by a change of the trap frequencies, e. g. obtained by power variationof the laser beam, a change of the beam waist or applying a magnetic field gradient.

68

Measurements with atoms

In the parametric heating method, the intensity of the dipole trap beam is modulated
forcing the atoms to oscillate. If the modulation is twice the frequency of the confinement,
atoms are parametrically excited and are lost. Thus, looking at the number of atoms asa function of the modulation frequency, the position of the resonance corresponds to themeasured trap frequency.
Experimental procedure In our experiment, we measured the trap frequencies by looking
at the atoms oscillations. In particular the monopole mode can be excited by suddenly
changing the confinement, for example reducing or increasing the power of the laser beam orchanging the aspect ratio of the ODT through the scanning system.
The dipole mode, which involves the center of mass oscillation and it does not dependon the temperature of the gas, can be excited by a displacement, in a controlled way, of thecloud from the central position of the trap. The scanning system provides a tool to probe thismode along the horizontal direction (x) changing the radio-frequency that controls the AOD,
which corresponds to the scanning parameter Voffset .
In the measurement presented in this section, we look at the monopole mode measuring
the width of the atomic cloud along the gravity axis z. The isotopes involved in this measurement are 168Er and 162Dy. We load approximately 106 atoms of both species from the MOT
into the single hODT beam in 100 ms, as explained in the previous section7 . The hODT laser
beam has a power of P = 37 W8 and waist wx = wz = 20.5 µm. After the loading, we switch
off the optical dipole trap for 100 µs9 and switch it on again. The absence of the trap leads
to a ballistic expansion of the atomic cloud. When the trapping is restored the atoms arerecaptured and they gain potential energy, which is converted into kinetic energy through theoscillation of the radial size of the cloud. The atoms are then held to register the oscillations.
The imaging beam hits the atomic cloud at an angle of 45° with respect to the propagation
axis of the laser beam (y), in the x − y plane. Due to this reason an oscillation along the x or y
direction would not be easily distinguished and the stronger frequency along the x-axis would
dominate on the other. This influenced our choice to look at the z-axis. The implementationof an high resolution imaging is planned to image the atomic clouds from the top of the mainchamber, along the z direction. Figure 4.5 shows the trap frequency measurements for Er(top) and Dy (bottom). The data points are the measured widths along z, while the orange
curve is the fit.

7 The scanning parameter V
gain was set to Vgain = 2 V.
8 Power measured with the thermal power sensor Thorlabs S322C, which has a measurements uncertainty of

3% at 1064 nm.
9 This time is usually chosen equal to a fraction of the oscillation period of the atoms in the trap.

69

4.3 Trap frequencies measurements
168

110

Er

Atomic cloud width ( m)

100
90
80
70
60
50
40
30
0

1

2

3

4

3

4

Evolution time (ms)

162

160

Dy

Atomic cloud width ( m)

140

120

100

80

60
0

1

2

Evolution time (ms)Figure 4.5 Trap frequency measurements for Er (top) and Dy (bottom). In this measurement
the monopole mode is excited. After the loading, the optical dipole trap is switched off
for 100 µs and switched on again to excite the monopole mode. The atoms are then held toregister the oscillations, measuring the width of the atomic cloud along the gravity axis z. The
widths are fitted by the function wfit (t) = Ae−γt sin(2πνmt) + B. The trapping frequencies
are estimated by νz = νm /2.

70

Measurements with atoms
The widths are fitted by an exponentially dumped sine function
wfit (t) = Ae−γt sin(2πνmt) + B ,(4.3)

where A is the initial amplitude, γ is the decay rate, νm is the frequency of the mode, t is theevolution time of the oscillation and B is an offset constant.
Table 4.1 Results from the fitting of the width of the atomic cloud with the function wfit (t) =Ae−γt sin(2πνmt) + B. A is the initial amplitude, γ is the decay rate, νm is the frequency ofthe mode, t is the evolution time of the oscillation and B is an offset constant. The final trap
frequency is given by νz = νm /2.
Fit parameters
A (µm)γ (ms−1 )νm (kHz)B (µm)

168Er−36 ± 3
1.3 ± 0.2
3.56 ± 0.02
75.0 ± 0.6

162

Dy

−55 ± 41.4 ± 0.2
3.72 ± 0.02
110.9 ± 0.8

Table 4.1 shows the results of the fit. The trap frequency along the z axis is obtained byνz = νm /2. This leads to the following values:νz = (1.78 ± 0.01) kHz
νz = (1.86 ± 0.01) kHz

for Erfor DyConsidering for the polarizability the theoretical values of Re{α} = 173 a.u. [12] andRe{α} = 164 a.u. [54] for Er and Dy, respectively, and a pure Gaussian beam, we expect
trap frequencies of νz ≈ 5.1 kHz. Our results are about 65% (Er) and 63% (Dy) smaller thanthe predicted values. The strong deviation could be induced by several reasons. A wrong
estimation of the beam waist and astigmatism are two possible reasons. On the other hand,to explain such discrepancy, the beam waist would have to be 70% larger than the measured
one10 . Whereas, the same discrepancy would be explained by an astigmatism of ≈ 10 mm.
Reflecting out the beam before the last mirror used to send the hODT light on the atoms, weinvestigated the beam shape using the CCD-camera beam profiler. The profile exhibited alarger astigmatism with respect to the one mentioned in Chapter 3, which is comparable tothe theoretical value needed to get the strong reduction of trap frequencies. The damping of
the oscillation is caused by the anharmonicity of the confining potential.
The knowledge of trap frequencies is crucial not only to define the geometry of the trap
seen by the atoms, but also to extract the experimental value of the polarizability. Applying
10 Section 3.3.3 shows the measurement of the beam waist for the hODT and vODT.

71

4.4 Towards evaportive cooling

Eq. (2.27) to the z-axis, the polarizability can be obtained for each specie measuring waist
and trap frequencies. The expression, written in atomic units, reads:
Re{α} =mπε0 c 3wz wx ωz2 ,
4PK

(4.4)

where K = 1.6488 × 10−41 and ωz = 2πνz .
In our case, the absolute values are not reliable because affected by large systematic
errors resulting from the astigmatism of the laser beam. On the other hand, this effect concern
both species. Thus, the polarizability ratio can be estimated.
In the evaluation of the polarizability ratio the only quantities that survive are the mass
ratio and the measured trap frequencies:
DyR≡

Re{α}Dy mDy (νz )2= Er Er 2 ,Re{α}Erm (νz )

(4.5)

from which the experimental result reads RExp. = 1.05 ± 0.02. Since all the measurements
are taken in the same apparatus under the same conditions of the dipole beam, e. g. power,
waist and astigmatism, the systematic errors cancel out and the uncertainty is dominated bystatistical errors. The predicted polarizability ratio, calculated considering the theoreticalvalues mentioned before, results RTheory = 0.95. In contrast to the theoretical prediction wemeasure a higher polarizability for Dy with respect to Er. So far, the measurements on theatomic polarizability of Dy manifest large discrepancy from each other (Re{α} = 116 a.u. ,Re{α} = 136 a.u.) and with the theoretical value (Re{α} = 164 a.u.). On the other hand,the experimental value measured by the Erbium lab in Innsbruck (Re{α}Er = 166 ± 3st ±61sys a.u. [12]), results in agreement with the predicted one. Taking this value as a reference,from Eq. (4.5) the polarizability of Dy reads Re{α}Dy = 175 ± 4st ± 64sys a.u. . Thus, we
expect for the atomic polarizability of Dy, once correcting the astigmatism, a measured value
close to this estimation.

4.4

Towards evaportive cooling

The evaporative cooling is the last step to produce a degenerate quantum gas. Above thecritical temperature for the degenerate regime, atoms in the phase space follow the MaxwellBoltzmann (MB) energy distribution:−En(E) = n0 e kB T1 ,

(4.6)

72

Measurements with atoms

where n0 is the peak atom density and T1 is the temperature of the atomic cloud. The conceptof evaporation is already well know in everyday life and relies on the leaving of moreenergetic particles from a system with a binding energy. Since the exponential distribution
extends to infinity, there is always a finite probability to find particles with high energy thatleave the system. Thus, this process happens spontaneously in nature and becomes more
and more slow as the energy is carried away and the system cools down, unless further
evaporation is forced modifying the system from the outside, in such a way that also theparticles with lower energy escape from the system.
Defining Et = kB T1 η, where η usually assumes values in the range 3 – 6 [30], let’s cut
the MB distribution at the energy Et . The truncated distribution has in total less energy per
atom thus, after the rethermalization, the new one is characterized by a lower temperature T2 .
Without any energy exchange between atoms the resulting distribution would be just cut andthe process would be a selection of the coldest atoms rather than a cooling. The same would
happen for a too fast evaporation, in which atoms do not have the time to collide. Thus,
the efficiency of the evaporative cooling relies on the elastic collision rate Γel given, for a
3D harmonic trap, by Eq. (3.4). Considering the expression of the mean thermal velocitypv̄ = 4 kB T /πm and the total elastic cross section σ = 8πa2 , Γel reads
n0 8πa2Γel = √ 4
2 2

r

kB T
.
πm

(4.7)

For a thermal cloud, once the atom number, trap frequencies and temperature are known, theparticle density n0 can be written asn0 = N ω̄

3m
2πkB T

3
2

,

(4.8)

where ω̄ = (ωx ωy ωz )1/3 is the average trap frequency. The s-wave scattering length, which
determines Γel and thus fixes the time scale of the evaporation, assumes different values foreach Er and Dy isotope. Table 4.2 shows a list of s-wave scattering lengths for two bosonicisotopes of Er and Dy.
In a gas mixture, during the rethermalization, the average energy transferred per eachcollision is [25]∆E1−2 = kB ∆T ζ ,1 m2
where ∆T = T2 − T1 is the temperature difference between the components and ζ = (m4m+m2.
1
2)
For an Er-Dy mixture, in which atoms have almost the same mass, approximately 3 elastic

73

4.4 Towards evaportive cooling

Table 4.2 List of s-wave scattering lengths for two bosonic isotopes of Er and Dy. a is usually
expressed in terms of the Bohr radius a0 .
ErbiumDysprosium

IsotopeaIsotopea

166 Er72(13) a0 [31]200(23) a0 [31]162 Dy112(10) a0 [82]92(8) a0 [82], 91(15) a0 [59]168 Er

164 Dycollisions per atom are needed for the thermalization and the stabilization of the new MB
distribution [62].
In optical dipole traps the evaporation consists of lowering the trapping potential by
reducing the power of the laser beam in several steps. The process can be iterated up to thedegenerate regime. At each point all the atoms with an energy greater than the cutoff energy
will escape from the trap and the remaining ones rethermalize leading to an overall lower
temperature of the sample. The cutoff energy of each step has to be chosen with high care toreach the degenerate regime in an efficient way.
The quantity to keep track of, during the evaporation for the achievement of a BEC, isthe phase space density (PSD). Atoms in a gas can be seen as wavepackets with an extension
given by the deBroglie wavelengthsλdB =2π h̄2
,
mkB T

(4.9)

where m and T are the mass and the temperature of the atoms, respectively. For lower
temperatures λdB increases. When the temperature is such that the extension of the particles−1/3
is comparable to the mean particle distance defined by n0 , a phase transition to the BoseEinstein condensation takes place. The quantity that relates the mean particle distance to λdB
is the PSD, defined as
3
PSD = n0 λdB,(4.10)
which, at the transition point, assumes the value PSD ≈ 2.6 [70].
During the experiment, the evaporation is usually interrupted at several points to checkatom numbers, temperatures and trap frequencies in order to evaluate the PSD. The overall
efficiency of the evaporative process is summarized in the quantity [45]γ =−d(ln PSD),d(ln N)(4.11)

74

Measurements with atoms

which is a measure of the orders of magnitude gained in the PSD at the cost of atom number.
Find the right parameters, as the rate of each step at which the trap depth is reduced, is notstraightforward. As a starting point we can take advantage of the knowledge on the optimized
evaporation scheme used for erbium [31]. Our trap geometry is given by the waists of thehODT and vODT, whose values are whx = whz = 20.5 µm and wvx = 101.3 µm, wvy = 95.0 µm,
respectively. The maximum available power is ≈ 38 W, which can be distributed between
hODT and vODT through a waveplate, as shown in Fig. 3.15. The polarizability values
considered for the simulation of the evaporative steps are the experimental ones Re{α} =136 a.u. [60] and Re{α} = 166 a.u. [12] for Dy and Er, respectively.
Figure 4.6 shows the simulations of the evaporation ramps for Er and Dy. The powerof the hODT and vODT together with the AR is changed in order to reproduce the trap
frequency settings used in the Erbium lab (Innsbruck) to get the degenerate phase.
Particular attention was dedicated to the frequency along the y-axis, where the confinement of the trap is weaker and the atoms can spread during the evaporation. This frequency,
which in the Erbium setting for the crossed ODT configuration was around 100 Hz, is mainly
given by the vertical beam of the ODT. The latter plays an important role during the evaporation, leading to a sudden increase of the PSD when switched on [6]. The process should
start with a large AR, to match the ODT volume with the MOT volume, and high power, toprovide a trap sufficiently deep to get the saturation regime. After the loading, the power
is quickly (≈ 50 ms) reduced to start the “exponentially-like” ramp and the AR is reduced
as well, in such a way to increase the strength of the confinement and compress the atomic
cloud. In the central steps of the evaporation the vODT is switched on to confine the atoms
that try to spread along the propagation axis of the hODT beam (y-axis). The last steps
are crucial and the power of the hODT and vODT and the AR as well are chosen in sucha way to optimize the ratio between elastic and inelastic collisions. The AR, at this stageof the evaporation, is usually increased to reduce three-body recombinations. As discussed
in the previous section, the atomic polarizability of Dy shows great discrepancy among thevalues measured by other groups and the theoretical one. This, of course, strongly affectthe resulting trap depths and trap frequencies. Figure 4.7 shows trap depths and average
trap frequency during the evaporation ramps, calculated choosing different values of thepolarizability Re{α} = (136, 116, 164) a.u., already mentioned in Chapter 2.
An interesting feature of dipolar gases concerns the evaporative cooling of fermions. The
key factors for the evaporation are the elastic collisions needed for the thermalization. Dueto this reason, forced evaporative cooling to achieve the quantum degeneracy of identicalfermions has been a challenge for many years with respect to the BEC, since the Fermi-Diracstatistics forbid s-wave collisions between spin-polarized identical fermions. Indeed, the75

4.4 Towards evaportive cooling

30

40

hODT
vODT

30

20

20

10

10

0

104

1

2

3

4

5

6

7

0

1

2

3

4

5

fx
fy
fz

103

0
7
104

6
fx
fy
fz

103

102
101

102

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Trap depth (µK)

600

600
hODT
vODT

400

hODT
vODT

400

200

200

0

0
0

10

Aspect ratio

101

Trap depth (µK)

Trap frequency (Hz)

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

hODT
w x = w z*ARvODT
w x = 101.3µm

hODT
w x = w z*ARvODT
w x = 101.3µm

w z = 20.5µm

w y = 95.0µm

w z = 20.5µm

w y = 95.0µm

7
10

5

5

0

Aspect ratio

Power (W)Re( )=166 a.u
exp. value

hODT
vODT

Power (W)Re( )=136 a.u
exp. value

40

ErbiumTrap frequency (Hz)Dysprosium

0
0

1

2

3

4

5

6

Evaporation time (s)

7

0

1

2

3

4

5

6

7

Evaporation time (s)Figure 4.6 Evaporation ramp simulated for Er and Dy. The power of the hODT and vODT
together with the AR is changed in order to reproduce the trap frequency settings used in the
Erbium lab (Innsbruck) to get the degenerate phase.

76

Measurements with atoms
1000
Re( )= 136 a.uRe( )= 116 a.uRe( )= 164 a.u

Trap depth (µK)

500
400

Re( )= 136 a.uRe( )= 116 a.uRe( )= 164 a.u

800
600

300
400
200
200

100
0

Average trap frequency (Hz)

600

0
0

1

2

3

4

5

Evaporation time (s)

6

7

0

1

2

3

4

5

6

7

Evaporation time (s)Figure 4.7 Simulations of trap depths and averaged trap frequency for Dy atoms, considering
the same settings of power and AR of Fig. 4.6.
total wave function for two identical fermions has to be antisymmetric for particle exchange.
At the low temperatures of interest for ultracold physics to reach the degenerate Fermi gas
the collisions are primarily s-wave. Several techniques have been adopted to overcome thislimit, by sympathetic cooling between fermionic and bosonic species [57, 62] or trapping
two different spin states [26]. Provided that these mixtures have to be stable against inelastic
collisions.
At temperature lower than 10 µK – 100 µK when, in the only presence of contact interaction, the total elastic cross section for identical fermions would be zero because containing
only s-wave contributions, the scaling as r−3 of the DDI prevents the cross section fromvanishing. Thus, identical dipolar fermions can collide in the limit of low temperature,making possible the production of a single component degenerate Fermi gas with direct
evaporate cooling [2].

Conclusion and outlook
The aim of my master thesis was to design and realize an optical dipole trap for the RARE
experiment in Innsbruck, enabling us to simultaneously confine ultracold Er and Dy atoms.
The working principle of the dipole trap relies on the atom-light interaction properties.
Erbium and dysprosium are both rare-earth metals with a multi-valence electrons character.
This gives rise to a complex and rich energy spectrum. The first task was to simulate thetrapping potential for the atoms with a determination of the most important parameters for an
efficient confinement of both species, e. g. trap depth and trap frequencies. A fundamentalrole is played by the atomic polarizability which, due to the anisotropic character of theelectronic distribution around the core, for Er and Dy manifest a tensorial nature. Estimating
the atomic polarizability is a non-trivial task with lanthanides. Indeed, the experimental
values are usually affected by systematic errors induced by the uncertainty on the waist.
Moreover, the complex spectrum makes the theoretical determination challenging.
The second part of my thesis was dedicated to the experimental realization of the optical
dipole trap with a tunable geometry. The tunability is a fundamental tool to increase theloading efficiency of the optical dipole trap, reaching the “mode-matching” condition with themagneto-optical trap. In this condition the atomic cloud maintains its volume and temperatureduring the transfer. Additionally, it can be used to optimize the efficiency of the evaporative
cooling, which is determined by the elastic collision rate, by tuning the cloud density. Finally,a tunable trap behaves as an additional experimental parameter to manipulate inter-atomic
interactions due to the anisotropic character of the dipole-dipole interactions. Several systems
were tested to find the most suited for our experiment. These include also a mechanicalsystem consisting of five cylindrical lenses and a translational stage. The tests led us to asetup based on time-averaged potentials created by an acousto-optic deflector. This device
deflects the laser beam horizontally according to a certain shape of the driving RF signal.
In contrast to an acousto-optic modulator, two RF inputs, with a given delay in phase withrespect to each other, overlap in such a way that the resultant signal compensates for thedeviation from the Bragg angle condition, allowing a diffraction efficiency almost uniform
over a broad range of deflection angles. If the beam deflection is modulated on a timescale

78

Conclusionmuch faster than the atomic oscillation, the atoms behave as in a smooth time-averaged
potential. Several trap geometries can be realized by modifying the shape and parametersof the driving RF signal. The ultimate limitations of the setup are mostly due to the VCOthat controls the deflector, in particular its limited frequency bandwidth. Indeed, in a regionin which the limited bandwidth does not play a role, the maximum aspect ratio (AR) was
measured to be AR ≈ 11. In contrast, by increasing the modulation to enter the time-averaged
potential regime, the AR is reduced to AR ≈ 7.
Finally, we describe the integration of the optical setup into the main experiment, and a
preliminary characterization of the optical dipole trap. If we denote by x the direction of theatomic beam and by z the axis of gravity, the ODT has a 90°-crossed configuration, created by
splitting in two paths the laser beam coming from a 50 W Yb3+ fiber laser with a wavelengthof 1064 nm. One beam, referred in the text as horizontal beam (hODT), is involved in thecreation of the time-averaged potential and propagates along the y direction. The verticalbeam (vODT) crosses the hODT and propagates along the z direction. A first test of thehODT was performed with a beam waist of 35 µm on the atoms. In this case, the scanningplayed an important role in the ODT loading, increasing the atom number by almost a factorof two. However, the weak y-confinement provided by the hODT let the atoms spread along
the direction of propagation, preventing accurate trap frequency measurements and further
evaporative steps. Reducing the radial waist to wx = wz = 20.5 µm allowed us to proceed withtrap frequency measurements, from which we deduced an efficient loading of ≈ 106 atoms
for both species, enabling us to extract the value of the atomic polarizabilities. Probing thebreathing mode along the gravity axis, we measured trap frequencies of Er νz = 1.78(1) kHz
and Dy νz = 1.86(1) kHz, for Er and Dy respectively. The results show great discrepancy fromthe predicted values, which is probably due to an astigmatism of the laser beam. So far, there
have been large discrepancy between polarizability measurements of different groups withdysprosium. For our latest measurements, the absolute values of the polarizablities are notreliable since affected by systematic errors induced by the astigmatism. This request to becorrected in future and leaves room for improvement of our laser source. Independent of
this, our experiment enables us to determine the polarizability ratio of Er and Dy, for which
the systematic errors cancel out, offering a better understanding of their atomic properties.
The beam waist of the vertical beam was chosen to be wx = 101.3 µm and wy = 95.0 µm. The
astigmatism does not affect the vODT, for which the distance of the foci is much smaller thanthe Rayleigh length. The vertical beam plays an important role in the evaporative cooling“closing the door” to the atoms along the weak longitudinal confinement of the hODT.
The RARE experiment plans to combine Er and Dy in a three-chamber apparatus. Atoms
will be moved from one chamber to the others by optical transport, which needs to be able toConclusion79

cover a total distance of around 70 cm. During my thesis I was also involved in the planningof the transport setup. A first test was conducted with focus-tunable lenses which revealed to
be not suited for our case. Therefore, the setup will involve the use of a translational stage.
Appendix B shows some preliminary consideration of the transport.

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82

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Appendix A
Scanning circuit
This appendix describes the electronic circuit built to combine the signals used to control the
acousto-optic deflector. Figure A.1 shows the scheme of the electronic circuit. The centralcomponent is the mixer (Analog Devices AD734). This device takes as inputs the signal
coming from the arbitrary function generator (AFG) Vsignal , which gives the modulation
function, and the two signals Voffset and Vgain , used to tune the central RF frequency ( fc ) andthe modulation amplitude, respectively. A similar electronic circuit was already implemented
in a previous experiment [6]. The output of the mixer is given by the following relation [4]Vmod = Voffset +VsignalVgain
.
10V

(A.1)

Voffset and Vgain can be tuned in the range 0 V – 10 V. In the circuit the switch enables tochoose the central frequency externally or internally rotating the 10-kΩ potentiometer. The
VCO1 receives as input a voltage from the mixer and sends to the AOD a RF signal. The
voltage range, needed to achieve an output RF signal wide enough to cover the AOD’s
bandwidth is 3 V – 13 V. However, the mixer is specified with a maximum output of 10 V,
thus an operational amplifier (op-amp) is implemented after the mixer. The full range wasreached supplying the op-amp with ±21 V. On the other hand, to make the circuit compact
the op-amp was supplied with the same voltages as the other electronic elements in the circuit.
This reduces the voltage range sent to the AOD resulting in a saturation of the maximum
beam waist achievable, as shown in Chapter 3. The VCO has a modulation bandwidth of
180 kHz. This quantity plays a fundamental role in the resulting time-averaged potential andcan be a limiting factor for the trapping efficiency, as explained in the main text. Thus, infuture the VCO will be probably changed with one with higher modulation bandwidth.

1 Model Mini-Circuits ZX95-100.

90

Scanning circuit

+18V

REF01

7815
0.1µF

0.33µF

0.1µF

0.1µF
10kΩ

-18V

7915
0.1µF

0.33µF

0.1µF

Voffset
INA114

external

internal

0.1µF
0.1µF

Vsignal

Vgain

1

14

2

13

3

12

4

11

5

10

6

9

7

8
AD734

NC

Vmod

+15V

Voutput
-15V

NC

0.1µF

2kΩ

1kΩ

Figure A.1 Electronic circuit implemented to generate the RF signal to control the AOD.
The central frequency can be tuned through Voffset or through the 10-kΩ potentiometer. The
modulation is sent to the circuit from the AFG through Vsignal and its modulation amplitude
is tuned through Vgain . The mixer AD734 creates Vmod by combining Voffset , Vsignal and Vgain .
Then, the op-amp OP27 receives Vmod and sends Voutput to the VCO.

Appendix B
Optical transport

Figure B.1 Side-view of the three chambers of the RARE experimental apparatus. A: Rydberg
chamber, B: main chamber and C: single-site imaging chamber.
Figure B.1 shows a side-view of the three chambers that will constitute the RARE
experiment. After precooling in the magneto-optical trap and loading into the optical dipole
trap, atoms will be optically transported from the main chamber (B) to either the Rydberg
chamber (A) or the single-site imaging chamber (C). Even if the latter is not designed yet,the distance from the center of the Rydberg chamber will be approximately ≈ 700 mm. Thus,
the optical transport will need a system able to cover the full distance, moving the atomsof ≈ 350 mm from B in both directions. Optical transport can be performed in several
ways. One method relies on the displacement of the focus of a red-detuned Gaussian laser

92

Optical transportbeam and involves the movement of the focusing lens on a linear stage [38, 60]. Anothermethod involves a one-dimensional optical lattice. In this case the frequencies of the twocounter-propagating beams are detuned with respect to each other, resulting in a moving
standing wave. The latter is usually used for short displacement [50, 77, 78].
As a possible solution for the optical transport setup we tested the focus-tunable lenses
from the company Optotune1 [65]. These lenses are plano-convex with a focal length thatcan be tuned in the range 100 mm – 200 mm and with an anti-reflection coating which covers
the range 700 nm – 1100 nm. A setup with two focus-tunable lenses and an offset lens can bebuilt to reach the requirements of our transport [52]. On the other hand, the tests revealed
drifts of the beam position after the change of the focal length. Furthermore, although thedamage threshold of the lenses is 10 kW cm−2 for 1064-nm light in continuous-wave, thelaser beam profile could be affected at a power of ≈ 50 W [66]. For these reasons we aredesigning a system which involves the use of a translational stage2 and a retro-reflector,
e. g. two-mirror or a corner-cube retro-reflector [79]. In this way, the laser beam path isfolded to save half of the space with respect to the simple scheme in which the last lens,
used to focus down the laser beam onto the atoms, is moved on the stage. It is worth noting
that, even if the corner-cube is less sensitive to beam fluctuations on both horizontal andvertical planes, small misalignments can strongly affect the polarization of the output beam[39]. This could be a drawback in our case in which, due to the tensorial nature of the atomic
polarizability, the polarization of the ODT has to be controlled with high care.
Thus, the optical transport consists in the shift of the focal point of the ODT from position
B either to A or to C. This appendix shows simulations performed considering erbium atoms.
In the moving frame, during the transport, atoms are confined by the trapping potential
modified by the presence of a new term determined by the acceleration with which the focus
is shifted. In particular the effective potential depth is reduced according to the chosen
function of the acceleration. The transport axis corresponds to the direction of propagationof the horizontal beam of the ODT, as shown in Fig. 4.2. Thus, from the intensity profile ofa Gaussian beam given in Eq. (2.14), for a beam propagating along the y axis, the trapping
potential during the transport is modified as follows
  2 Ũ2P
r
Utransport =exp −2+ may ,22πwh (y)wh (y)
1 Model EL-10-30-CiNIR-LD
2 Aerotech ABL15040-M-10-NC.

(B.1)

93
0

Trapping potential ( K)

-200

-400

Û

-600

-800
a = 2 m/s2
a = 0 m/s2
-1000
-15

-10

-5

0

5

10

15

Transport axis (mm)Figure B.2 Trapping potential along the transport axis for a 1064-nm laser beam withpower P = 35 W, waist wx = wz = 35 µm. The orange curve shows the potential without
any acceleration, whereas the blue curve shows the trapping potential in the case of anacceleration of a = 2 m s−2 . The black bar defines the reduced trap depth Û.
where r2 = x2 + z2 , w2h (y) = w2x (y) = w2z (y) and a is the acceleration with which the focus
is shifted along the y axis. Figure B.2 shows the trapping potential3 for a wavelength of
λ = 1064 nm, total power P = 35 W and a waist of 35 µm, without acceleration and with anacceleration of a = 2 m s−2 . The black bar in the figure defines the reduced trap depth Û.
In order to not lose atoms during the transport the acceleration function has to be chosencarefully. Figure B.3 presents two of the possible choices, i. e. a trapezoidal and a triangularfunction, and the corresponding velocity (v) and position (s) of the focus, for a total transport
time of 1 s and a total distance of 375 mm. The rising time of the trapezoidal function takes
into account that the translational stage will always have a finite response time. The transport
time Tt is one of the critical parameters for an optical transport. Usually, Tt is chosen muchlarger than the oscillation period Tosc , given by the trap frequency along the transport axis ωy .
This situation is referred as adiabatic condition
Tt ≫ Tosc =2π
ωy(≃ 50 msfor νy = 20 Hz) .

(B.2)

In the non-adiabatic regime Tt ≃ Tosc energy is transferred to the cloud, leading to heating
and oscillations [23]. The choice of the transport time is a trade off between the adiabatic
3 For these simulations, the polarizability values were chosen equal to 1064 Re{α} = 166 a.u., 532 Re{α} =430 a.u. and 1570 Re{α} = 163 a.u., as reported in Table 2.1 [12].

94

Optical transporta(m/s 2)v(m/s)
s(m)

3
2

2

1

1

0

0

-1

-1

-2

-2

-3

-3
0

0.25

0.5

0.75

a(m/s 2)v(m/s)
s(m)

3

1

0

0.25

Transport time (s)

0.5

0.75

1

Transport time (s)(a) Trapezoidal.

(b) Triangular.

Figure B.3 Acceleration, velocity and position curves in the case of a trapezoidal (a) andtriangular (b) function of the acceleration.
1950
TriangularTrapezoidal

1900

Trap depth (µK)

1850
1800
1750
1700
1650
1600
1550
1500
0

0.25

0.5

0.75

1

Transport time (s)Figure B.4 Trap depth as a function of the transport time for two possible choices: trapezoidal(blue) and triangular (orange) function.
regime and a reasonable time compared with the lifetime of the atomic cloud.
Figure B.4 shows how the trap depth is reduced during the transport for the two configurations previously discussed. In this case the wavelength and the power are the sameas in Fig. B.2, while the waist is chosen equal to wx = wz = 20.5 µm. Since the distance isfixed by the position of the chambers and the time is approximately given by the adiabatic

95
condition, an efficient transport relies on the acceleration function. For a triangular shape,
higher accelerations are needed to cover the same distance for a given time. Whereas, thetrapezoidal function keeps a high value for longer time. Thus, the potential depth is reduced
more when the acceleration reaches the maximum in the triangular case. Figure B.5 shows
3000
532
1064
1570

Trap depth (µK)

2500

2000

1500

1000

500

0
0

0.25

0.5

0.75

1

Transport time (s)Figure B.5 Trap depth as a function of the transport time, in case of a trapezoidal function of
the acceleration, for the wavelengths: 532 nm, 1064 nm and 1570 nm .
the reduction of the potential depth for three different wavelengths for a trapezoidal functionof the acceleration. In this simulation the power was set to 10 W, 35 W and 15 W for 1570 nm,
1064 nm and 532 nm, respectively. These are wavelengths suitable for red-detuned optical
dipole trap for Er and Dy. The trap depth determined by the available power of the 1570-nm
light could not be enough with a waist of 20.5 µm4 to confine the atoms, if additional heatingis transferred to the atomic cloud during the transport. On the other hand, the 532-nm light
could be an alternative to the 1064 nm.

4 Due to constraints imposed by the experimental apparatus, however, such small focus would not be

achievable with the 1570-nm light.

Acknowledgements
I would like to thank my supervisor in Innsbruck, Francesca Ferlaino for giving me thepossibility to work in this fantastic experiment. The enthusiasm she puts every day in thisproject is amazing. I want to thank her for the trust she gave me and for reading and correcting
my thesis.
I would also like to thank my supervisor in Pisa, Donatella Ciampini for transferring to me
the interest for the physics of ultracold atoms. She suggested to participate at the summer
school on ultracold atoms in Innsbruck, which was decisive for my future choice. I want tothank her for all the suggestions she gave me before and during the course of my thesis.
Many thanks go to all the RARE group. Thanks to Manfred J. Mark for all the discussions
and the patience he had, especially in the first period of my thesis.
Thanks to Philipp Ilzhöfer, Gianmaria Durastante, Arno Trautmann and Alexander
Patscheider for the great support in the lab.
Thanks to Maximilian Sohmen and again Arno for answering to my questions and forcorrecting part of my thesis. Additionally, I want to thank them for making always “great”
jokes in the office.
Thanks to all the ERBIUM team for sharing their knowledge and experience.
I would like to thank the secretaries from Innsbruck, Gaby Holzer and Nicole Jorda,
and from Pisa, Roberta Giusti, for helping me in coordinate everything concerning theadministrative work.
Thanks to Giulia, Dani, Elisa, Lorenzo and Deborah for all the great moments spent
together, making the time running super fast.
I want to thank Gabriele for standing always on my side and for being my team-matein life. A big thanks goes to Fabio for the time he spent helping me with my back injury,

98

Acknowledgementsallowing me to recovery before starting my thesis in Innsbruck. Thanks to my sister Giulia
for the encouragements and for being my official advisor on everything. Finally, I would liketo thank my parents for giving me the possibility to study away from home and for supporting
me in every choice I made.