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

Tesi di Laurea Magistrale in Fisica della Materia

Spin physics in dipolar quantum gases of
erbium atoms

Relatore esterno:
Candidato:

Prof.ssa Francesca Ferlaino(University of Innsbruck)

Gabriele Natale
Matr. 527902

Relatore interno:
Prof.ssa Donatella Ciampini

Anno Accademico 2016/2017

Alla mia f amiglia . . .

Contents
Introduction1

1

Erbium properties
1.1 Electronic configuration and energy spectrum . . . . . . . . . . . . . . . .
1.1.1 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Interaction with magnetic fields . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Landé g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Interactions properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The contact interactions . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 The magnetic dipole-dipole interactions . . . . . . . . . . . . . . .

4
5
6
7
8
8
8
10
12
12
14

2

Production of spinor Er quantum gases
2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Reaching quantum degeneracy . . . . . . . . . . . . . . . . . . . .
2.2 Overview on spin preparation . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Direct radio-frequency excitation of the Zeeman sublevels . . . . .
2.2.2 Landau-Zener sweep . . . . . . . . . . . . . . . . . . . . . . . . .

16
16
19
20
20
27

3

Experiment on spin-1/2 system
3.1 The optical lattice: basic concepts . . . . . . . . . . . . . . . . . . . . . .
3.2 Fermi-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Measurement of the interaction properties . . . . . . . . . . . . . . . . . .
3.3.1 Experimental sequences . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 The onsite interaction . . . . . . . . . . . . . . . . . . . . . . . .

33
34
36
38
39
41

CONTENTS

iv

4

45
45
45
49
52
54
56
61
64
64
65
66
68

Advanced Spin-preparation with light
4.1 AC-Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Stark shift: theory . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Ac-Stark shift: calculation at 631 nm . . . . . . . . . . . . . . . .
4.2 Realization of a laser source at 631 nm . . . . . . . . . . . . . . . . . . . .
4.2.1 Design of the laser system . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Diode laser characterization . . . . . . . . . . . . . . . . . . . . .
4.2.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Overview on spin preparation/manipulation with light . . . . . . . . . . . .
4.3.1 Optical pumping scheme . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Raman scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Stimulated raman adiabatic passage . . . . . . . . . . . . . . . . .
4.3.4 Single spin state manipulation . . . . . . . . . . . . . . . . . . . .

Conclusion

70

A Lock of the laser system

79

Acknowledgements

83

Introduction

One of the fundamental principle of quantum mechanics relies on the revolutionary conceptof wave-particle duality [De 25]. In every-day life, ordinary objects at room temperatures
have an extension of the corresponding wavepacket so small that their wave nature is notapparent. On the other hand, at extremely low temperatures, atomic gases reveal their wavelike nature. Albert Einstein, following Satyandra Bose’s work on photons, predicted that, inthese extreme conditions, atoms occupy a single macroscopic quantum wave function. This
new state of matter is known as Bose-Einstein condensate (BEC) [Bos24; Ein25].
Einstein predictions remained for decades a mere theoretical concept. The main limiting
factor to realize such an exotic state of matter was the requirement of extremely low temperatures. For a million atoms, it would be indeed necessary to reach temperatures on the orderof tens of nano Kelvin. Only after the development of new laser cooling and atomic trapping
techniques in the 1980’s and ultimately the implementation of evaporative cooling, the first
Bose-Einstein condensate was achieved with Rb [And95] and Na [Dav95] .
In this temperature regime, where thermal wavelength and inter-particle distance becomeof the same order, quantum statistics start to play a role.
Whereas bosons undergo Bose-Einstein condensation, fermions begin to occupy one-byone all the energy levels up to the Fermi energy, forming the so called degenerate Fermi gas(dFg). The first dFg was reached only later in the 1999 [DeM99]. The delay in reaching a
dFg is due to the Pauli principle, which makes the collision properties different with respect
to bosons and the implementation of evaporative cooling not straightforward.
Over the last two decades, quantum gases have proven to be ideal systems to study novel
few- and many-body quantum phenomena, e.g [Kno08] [Cho16]. These atomic gases provide
a clean (impurities are essentially absent) and dynamically tunable system. The versatilityof ultracold atomic gases relies on the possibility of tuning the strength of the interactions
between atoms, e.g. via a magnetic Feshbach resonance [Chi10], or by designing the externalpotential landscape seen by the atoms thanks to light and magnetic fields engineering.

1

Introduction

2

The level of control was even improved when ultracold bosonic and fermionic atoms
were stored in artificial periodic potentials, created by standing waves of light, namely opticallattices. The use of optical lattices opens the doors for studying the strongly-interacting
regime (via the introduction of a large effective mass to the atoms, mitigating their kinetic
energy) and probing fundamental questions of interdisciplinary fields of research by its
similarities to electrons in solid crystals. A first pioneering experiment with optical lattice
was the observation of the transition from the superfluid to the Mott insulator phase, withbosonic atoms [Gre02]. This experiment demonstrated also the possibility to describe cold
atoms in optical lattice with a relative simple Hamiltonian, the Bose-Hubbard model, andthus reconnect to the idea of quantum simulation to solve solid-state problems. From thatexperiment, several works have reported a non standard Hubbard model [Dut14] or theexperimental realization of an antiferromagnet in the Hubbard model, with a quantum gas
microscope [Maz16], offering the potential to answer the regime of the doped Hubbard
model.
Recently, a new class of atomic species has been brought to quantum degeneracy, namely
the magnetic lanthanides, which triggered a huge interest in particular in the prospectof quantum simulation because of the distinct properties of these atoms. In particular,
the lanthanides have the largest magnetic moment in the periodic table and thus bringan additional ingredient to the system, implementing long-range and anisotropic dipoledipole interactions (DDI) between the particles. Lanthanide experiments open fascinating
possibility to study the impact of the DDI on few and many body quantum physics. The
ERBIUM experiment in Innsbruck (AT) produced the first BEC and dFg of erbium (Er)atoms. The combination of dipolar quantum gases and optical lattices provides a powerfulplatforms to study strongly correlated quantum system beyond standard Bose-Hubbard model.
Recently, the group realized an extended Bose Hubbard model featuring both onsite andoffsite interactions [Bai12] in dealing with the long-range nature of interaction in dipolarbosonic atoms.
In this thesis we investigate the preparation of a spinor quantum gas of Er atoms. Preparing a deterministic spin mixture can be particularly challenging for a high-spin system as
Er. The ground state of Er has a total angular momentum J=6 (F=19/2) giving rise to 13(20) different spin states for bosons (fermions). At our typical magnetic field strength, theZeeman splitting for the bosonic isotopes between adjacent spin states is equal; hence a
deterministic preparation of one particular spin state is not possible with standard methods
such as radio-frequency (RF) coupling. In this work, we show how the quadratic Zeeman
shift enables us, in the fermionic case, to obtain a deterministic spin preparation with RF
pulse or RF sweep. We in particular demonstrate and experimentally implement a versatileIntroduction

3

preparation of a mixture of two spin states. Several unexplored phases are predicted to occur
with a spin mixture of highly magnetic atoms in a deep lattice, e.g. stripe phase [Maz17].
Before searching for such new phases in our experiment, to characterize the system we load
the fermionic isotope into a deep lattice, and we extract the onsite interaction by lattice
modulation spectroscopy measurements between the lowest spin states. Additionally, asa step towards the predicted phases, we present a method to obtain a deterministic spin
preparation and a single-spin-state control, which exploits the tensorial ac-Stark shift andcan be implemented for both fermionic and bosonic isotopes. For this aim, I developed anExternal Cavity Diode Laser (ECDL) source, emitting close to a narrow transition at 631 nm,
together with an optical setup that allows different schemes for spin manipulation. This
study provides new elements to increase the knowledge of our system and opens the door toinvestigate the stability of ordered magnetic phases.
The structure of this thesis is the following:
Chapter 1 introduces the main properties of Er, based on its high magnetic moment andthe long-range and anisotropic nature of the interactions.
Chapter 2 describes the ERBIUM experimental apparatus and then focuses on the methods
adopted to realize a spinor gas of Er.
Chapter 3 investigates a system of fermionic Er atoms into a deep optical lattice andpresents the consecutive measurements of the onsite interaction, and the extracted scattering
length, between the two lowest spin states.
Chapter 4 introduces the theory and calculations of the AC-Stark shift, then it describes
the realization of a laser source at 631 nm and the related optical setup developed during mythesis work. Finally, it presents an overview of different methods to achieve a deterministic
spin preparation with light and the manipulation at the single-spin-state level in Er.

Chapter 1
Erbium properties
Erbium (Er) is an element of the periodic table, and it is one of the so called rare earth
elements (REE). The adjective "rare" is much more indicative of the history related to the
erbium’s discovery (1788) rather than its real properties, since for many years these elements
could only be found in a miner in Ytterby (Sweden). Erbium is present on the entire earth
crust with a concentration of 3.4 ppm [Lid10]. Nevertheless, the price of pure erbium ishigh for several reasons. First, it is difficult to find REE at high concentration in a single
location; second they are not found as pure elements but mixed together and finally, due tothe monopoly of market run by China1 .
Erbium has 5 stable bosonic isotopes and 1 stable fermionic isotope. The masses of suchisotopes range from 162 amu to 170 amu. Table 1.1 shows the abundances and statistics ofstable Er isotopes.
Table 1.1: Relative abundances and type of quantum mechanical statistics for all stableerbium isotopes.
isotope

162 Er

164 Er166 Er

167 Erabundance
statistics

0.14%
boson

1.61%
boson

33.6% 23.0%
boson fermion

168 Er

170 Er

26.8%
boson

15.0%
boson

The high abundance of the fermionic isotope paves the way for studying dipolar fermionic
quantum gases. This chapter is devoted to the basic properties of erbium. The first section
introduces the electronic configuration and the energy spectrum with the description of theprincipal atomic transitions used in the experiment. The second section is dedicated to
the Zeeman splitting induced by an external magnetic field, which is fundamental for the
1 China has the largest rare earth reserves in the world; the industrial reserves occupy first place (76%) and

it is the major producer of REE for the world market.

4CHAPTER 1. ERBIUM PROPERTIES

5

understanding of the spin preparation and manipulation. Finally, the last section introduces
the interaction properties of Er atoms.

1.1Electronic configuration and energy spectrum

In the ground state, the electronic configuration of Er is written as below:(1s2 2s2 2p6 3s2 3p6 3d 10 4s2 4p6 4d 10 5s2 5p6 ) 4 f 12 6s2 ,(1.1)

where the terms in the round brackets represent the electronic configuration of xenon. In fact
it is often abbreviated as[Xe] 4 f 12 6s2(1.2)
According to the Madelung’s rule, this electronic configuration presents an open 4f shell
shielded by a completely filled 6s. For this reason, it is usually referred as a submerged-shell[Jen91] and it is expected to give rise to exciting unexplored physical scenarios. The motivation behind this has to be searched in the anisotropy of the electronic density distributionof the 4f shell, which leads to a large magnetic moment. Only the incompletely filled shell
contributes to the quantum number of the ground state, which can be obtained using Hund’srule and LS (Russel-Saunders) coupling scheme. The 4f shell has 7 mℓ states filled by only
12 electrons, which arrange each other to maximize the total spin S = 2 × 1/2 = 1 and theorbital angular momentum L = 2 + 3 = 52 .

mℓ

−3

−2

−1

0

+1

+2

+3

ms

↑↓

↑↓

↑↓

↑↓

↑↓

↑

↑

Using the spin-orbit coupling3 the total angular momentum quantum number can be calculated as J = S + L and the ground state, denoted by 2S+1 LJ in the Russell-Saunders notation,reads:
ground state : 3 H6 .
2 In the LS coupling the spins of the single electrons have to be added to get the total spin S and the individual

orbital angular momenta as well, to obtain the orbital angular momentum L. Only then L and S couple to givethe total angular momentum J. It is assumed that spin-spin interaction > orbit-orbit interaction > spin-orbit
interaction.
3 The LS coupling scheme is applicable only for the ground state. Due to the heavy mass, the spin-orbit
interactions get important with respect to the Coulomb interactions and the JJ-coupling scheme has to be used
for the other levels.

CHAPTER 1. ERBIUM PROPERTIES

6

This ground state has high angular momentum, which will affect the interactions properties.

1.1.1

Energy spectrum

Figure 1.1 shows the energy spectrum of erbium up to 25000 cm−1 . The complexity arises

Figure 1.1: Energy level scheme of erbium up to an energy of E = 25000 cm−1 . The full level
scheme is presented in [Fri14]. States with even (odd) parity are shown in red (black). The
three most important transitions for the ERBIUM experiment are at a wavelength of 401 nm(blue), 583 nm (yellow) and 631 nm (red) . The blue transition is used for the Zeeman slower(ZS), transversal cooling (TC) and imaging. The yellow transition is used for creating themagneto-optical trap (MOT), whereas the red transition will be used for spin manipulation.
from the electronic configuration, which due to the submerged shell nature, allows theelectrons to be excited to higher energy levels in many different ways. Indeed, not only thevalence electrons of the 6s shell can be excited, but also the valence electrons of the 4f shell.
Focusing on the electric dipole transitions, it is possible to find different lines available forlaser cooling and ultracold atom applications, whose linewidths range from Hz to MHz. The
most important transitions for our experiment are shown in Fig. 1.1.

CHAPTER 1. ERBIUM PROPERTIES

7

The blue line in Fig. 1.1 at 401 nm involves the transition of one electron from the 6s
orbital to the 6p orbital in a singlet state. This is a strong transition with a broad linewidth(30 MHz) convenient4 for the first stage of cooling, based on transversal cooling and Zeeman
slower. Moreover, this light is used for the imaging of the atomic cloud.
The yellow transition at 583 nm is an intercombination line, where one of the electrons of
the 6s orbital gets promoted in the 6p state, but this time in a triplet state. This line is weakerwith respect to the blue line because, since involves a change of the spin, is semi-forbidden.
The narrow linewidth of about 190 kHz is ideal for realizing the magneto-optical trap (MOT)
light. As described in [Fri12], the MOT capture velocity with this linewidth is sufficiently
high to trap atoms coming from the Zeeman slower and reach temperature as low as 10µK.
The red transition at 631 nm involves the excitation of an electron coming from the 4f
shell. Due to the completely filled 6s shell, the electrons of the 4f shell are shielded. Verynarrow transitions can be found with electrons coming from the 4f shell and arriving at the
5d shell (also shielded by the 6s). The transition at 631 nm has a linewidth ≃ 20 kHz but
linewidths as narrow as 2 Hz are present in the spectrum. Part of the work of this thesis isfocused on the realization of a laser source, emitting at this wavelength, which can be used
for a deterministic control of the atomic spins, see Chapter 4.

1.1.2

Hyperfine structure

Among the erbium isotopes, bosons have a zero nuclear spin, I = 0, whereas the fermions
have I = 7/2, which couples with the total angular momentum J leading to the hyperfine
structure[Bra83]. For the 167Er isotope, the already complex energy level scheme becomeseven more complicated due to the removal of the degeneracy caused by the hyperfine splitting
expressed as
3/2C(2 C + 1) − I(I + 1)J(J + 1)
∆Eh f s = AC + B,(1.3)
2I(2I − 1)J(2J − 1)
where A and B are the hyperfine structure constants and C = 12 (F(F + 1) − J(J + 1) − I(I + 1)),
with F the total angular momentum quantum number. For erbium, F = 19/2 gives eight
hyperfine states from F = J + I = 19/2 to F = J − I = 5/2.
4 A broad transition is usually suited for application in which a large number of scattering events are needed

as transversal cooling, Zeeman slower and imaging.

CHAPTER 1. ERBIUM PROPERTIES

1.2

8Interaction with magnetic fields

The Zeeman effect consists in the separation of the spectral lines due to an external magnetic
field (B). In particular the lines are splitted due to the interaction of the B field with themagnetic moment of the atoms. The goal of this section is to explain the behavior of atomic
lines in presence of an external magnetic field and how this changes in the bosonic andfermionic case.

1.2.1

Magnetic moment

For bosons the atomic magnetic moment µ is given by:µBoson = gJ µB

p

J(J + 1),

(1.4)

where gJ is the Landé g-factor and µB is the Bohr magneton. The component along thedirection of the magnetic field isµBoson (z) = −mJ gJ µB ,(1.5)

where mJ is the magnetic quantum number and it can assume the following 2J + 1 values:[−J, −(J − 1) . . . 0 . . . (J − 1), J] .

(1.6)

For the fermionic isotope, which has an hyperfine structure, the total angular momentum J
couples with the nuclear momentum I and the total angular momentum F has to be considered
to calculate the magnetic moment:µFermion (z) = −mF gF µB .

(1.7)

It is clear that the Landé g-factor is needed in order to obtain the magnetic moment.

1.2.2

Landé g-factor

In the LS-coupling, the Landé g-factor for the ground state readsJ(J + 1) − L(L + 1) + S(S + 1)
gJ = 1 + (gS − 1),
2J(J + 1)(1.8)

CHAPTER 1. ERBIUM PROPERTIES

9

where gs ≈ 2.00232 is the gyromagnetic ratio. Using J = 6, L = 5 and S = 1, we find for the
ground stategJ (J = 6, L = 5) = 1.167053.
(1.9)
On the other hand, the most precise experimental value is [Fri14].
gJ = 1.163801(1).

(1.10)The discrepancy between the two values can be explained by different reasons [Jud61]. First,
deviation from totally L-S coupling. Second, the relativistic corrections necessary due tothe kinetic energy of the electron in higher shells, and finally diamagnetic effects. The
magnetic moment can now be estimated for the ground state of bosonic erbium isotopes(J=6) considering a spin polarized sample in mJ = −6
µAtom (z) = −6.982806(6) µB .

(1.11)

This incredibly high value is not common in the periodic table and after terbium, dysprosium
and holmium is the highest value one can find.
For the excited states, the j j-coupling has to be considered. In the case of j j-coupling,
first the orbital angular momentum l and the spin s of each electron couple to form a total
angular momentum j, then all the single-electron j sum up to the total angular momentum of the atoms J. Two particular cases of this coupling are J1 J2 and J1 j that are often
used for the excited states of lanthanide atoms. In this scheme, all the inner electrons up
to the 4f-shell couple together in the standard LS-coupling giving out J1 , and then couple to the outer lying electrons via J1 J2 , if these couple among themselves via LS (asfor [Xe]4 f 12 (3H6 )6s6p(1P1 ))5 , or via J1-j if the 6s electronic shell remains filled (as for[Xe]4 f 11 (4I13/2 )5d6s2 )6 . In the case of j j-coupling, an estimate of the Landé g-factor canbe obtained using the following relation
J(J + 1) + J1 (J1 + 1) − J2 (J2 + 1)+
2J(J + 1)
J(J + 1) + J2 (J2 + 1) − J1 (J1 + 1)
+ gJ2,
2J(J + 1)

gJ =gJ1(1.12)

where J2 = j in case of J1 j-coupling. Table 1.2 gives the Landé g-factors found for the
ground state and the three excited states shown in Fig. 1.1. The most precise values are takenfrom [Mar78], where the three corrections are taken into account.
5 This is the electronic configuration of the 401 nm light shown in Fig. 1.1
6 This is the electronic configuration of the 631 nm light shown in Fig. 1.1

CHAPTER 1. ERBIUM PROPERTIES

10

Table 1.2: Landé g-factor gJ of the ground state and the excited states of bosonic erbiumincluding the values with corrections.
electronic configuration

Wavelength(nm)

gJ
value from Eq. 1.12

gJ
value from Ref.[Mar78]

401
583
631

1.167
1.143
1.215
1.090

1.164
1.160
1.195
1.070

[Xe]4 f 12 6s2[Xe]4 f 12 (3 H6 )6s6p(1 P1 )[Xe]4 f 12 (3 H6 )6s6p(3 P1 )[Xe]4 f 11 (4 I13/2 )5d6s2

In the case of fermionic 167Er, since the nuclear spin I is not zero, the magnetic momentof the nucleus couples to the magnetic field produced by the electrons of the atom. As aresult, the gyromagnetic ratio has to be calculated according to the following relation:
gF = gJ

F(F + 1) − I(I + 1) + J(J + 1),
2F(F + 1)(1.13)

where gJ is given by Eq. (1.12). Using the most precise experimental value for gJ , it ispossible to estimate the gF factor and the magnetic moment for the ground state of thefermionic isotope:
19 19( 2 + 1) − 72 ( 72 + 1) + 6(6 + 1)
= 0.735032(1),19( 192 + 1)(1.14)µFermion (z) = −6.982804(4) µB ,(1.15)

gF = 1.163801 2

where Eq. (1.7) was used considering the fermionic spin polarized ground state (F = 19/2;
mF = −19/2).

1.2.3

Zeeman splitting

By applying an external magnetic field, the degeneracy of the energy levels is removed
creating several magnetic-sublevels. Let’s consider two separetely cases: bosonic andfermionic atoms of erbium.
For bosonic erbium isotopes, the lack of nuclear spin allows a straightforward argumentation. The energy level with total angular momentum J splits into 2J + 1 levels with magnetic
quantum number mJ . The effect of the external magnetic field is given by the following termin the Hamiltonian:
ĤBBosonic = µB (gS Ŝ + gL L̂) · B.
(1.16)

CHAPTER 1. ERBIUM PROPERTIES

11

From which, for standard magnetic fields used in ultracold atom experiments7 , the Zeeman
energy shift is linear and it reads:∆EmJ = gJ mJ µB B.

(1.17)

Considering two different consecutive spin states, e. g. mJ = −6 and mJ = −5, the degeneracy
present at B = 0 G is now removed and the splitting is of 1.628879 MHz for each Gauss ofmagnetic field applied8 . The same splitting is also present for all adjacent spin states∆Em j′ − ∆Em j′ +1 = ∆Em j′ +1 − ∆Em j′ +2 .

(1.18)In the fermionic case (167 Er), the hyperfine structure modifies the Zeeman interaction
due to the nuclear spin:
ĤBFermionic = µB (gS Ŝ + gL L̂ + gI Î) · B.

(1.19)In the case of weak magnetic fields, where the Zeeman energy shift is much smaller than thehyperfine scale, the Zeeman interactions can be treated as a perturbation in the |F, m f ⟩ basis.
In this case the Zeeman energy shift can be approximated to be linear:∆EZ (B) = mF gF µB B.

(1.20)

Considering two different adjacent spin states, e.g. mF = −19/2 and mF = −17/2, thedegeneracy present at B = 0 G is now removed and the magnetic sublevels are separated by
1.025409 MHz for each Gauss of magnetic field applied.
For higher B fields9 , I and J fully decouple, leading to the Paschen-Back effect. In
this framework an exact diagonalization of the Hamiltonian is needed. Figure 1.2 showsthe effect of the B field on the hyperfine ground state F = 19/2. In this plot it is hard to
notice the quadratic dependence of the Zeeman splitting at high B fields. On the other
hand, if we consider the three adjacent spin states: e.g. mF = 19/2, 17/2 and 15/2, then the
splitting between the first two ∆E19/2−17/2 with respect to the following two ∆E17/2−15/2
7 The range in which the Zeeman energy shift is linear in the case of I = 0 depends on the energy shift

given by the fine structure, which are large for lanthanides. In fact the result of Eq. (1.17) can be used for Bh̄≪ ∆ f s µB
gJ ≈ 500 T since ∆fs ≈ h × 1 PHz. The energy scales to be compared are the Zeeman splitting and thefine structure splitting.
8 To obtain this result, the g value was taken from expression (1.10).
J
9 The range in which the Zeeman energy shift is linear, in the case of I ̸= 0, depends on the hyperfine
energy shift which are much smaller than the fine ones. Indeed, the result of Eq. (1.20) can be used for Bh̄≪ ∆hfs µB
gF ≈ 5G since ∆hfs ≈ h × 1 GHz.

CHAPTER 1. ERBIUM PROPERTIES

12

m

F

1 7 /2

1 5 /2-3 5 0 0

1 3 /21 1 /2-3 5 5 0

E / ħ(M H z )1 9 /29 /2

7 /2-3 6 0 0

5 /2-3 6 5 0

1 /2

3 /2

-1 /2

-3 7 0 0-3 /2-3 7 5 0

-7 /2

-5 /2

-9 /2

-1 1 /2-1 3 /2-3 8 0 0-1 5 /2-1 7 /2-3 8 5 0-1 9 /2-3 9 0 0
0

5

1 0

1 5

2 0

M a g n e tic F ie ld ( G )Figure 1.2: Effect of the Zeeman splitting on the F=19/2 manifold. The resulting states areevaluated from exact diagonalization of the Hamiltonian.
becomes unequal due to the quadratic dependence. This can be visualized in Fig. 1.3 fromthe difference of the two splitting: ∆E19/2−17/2 − ∆E17/2−15/2 .

1.3

Interactions properties

Despite cold gases are dilute systems, the inter-particle interactions are crucial in determining
the scattering properties. The two-body interactions in erbium can not be described using only
the contact interaction [Lah09], but, due to the high magnetic moment, also the dipole-dipole
interaction (DDI) plays a fundamental role. In the following, short-range contact interaction
and the long-range dipole-dipole interaction are briefly introduced.

1.3.1

The contact interactions

Two atoms at a separation distance r feel an interaction potential U(r) which, in a first orderapproximation, is the so-called van der Waals potential
C6
UvdW = − 6 ,r(1.21)

CHAPTER 1. ERBIUM PROPERTIES

13

1 2

1 0

D iffe r e n c e in k H z

8

6

4

2

0
0

5

1 0

1 5

2 0

B (G )

Figure 1.3: Effect of quadratic Zeeman splitting on the difference of the separation between
adjacent spin states ∆EmF+2 −mF+1 − ∆EmF+1 −mF .
where C6 is the van der Walls coefficient. This potential results in a short-range and isotropic
character. For a classical picture of the scattering between two particles, consider Fig. 1.4,
in which two atoms are colliding with incident momentum p and impact parameter b. The
angular momentum reads
L = |r × p| = bp = h̄l,(1.22)
where l represents the partial wave number. As a condition for the scattering to be relevant,the impact parameter has to be much smaller than the interactions range r0 , leading to thefollowing condition
2πr0b ≪ r0 → l ≪,(1.23)
λdB
where λdB = hp is the de Broglie wavelength. In the limit of low collisional energy, i. e. low
momentum p, the only solution is a s − wave collision (l = 0). In this situation, the van der
Waals potential can be approximated by the pseudo-potentialh̄2 a
Ucontact (r) = 4π
δ (r),
m(1.24)

CHAPTER 1. ERBIUM PROPERTIES

14

Figure 1.4: A scattering process of two neutral atoms. The two colliding atoms have arelative velocity of 2v and they are approaching each other with an impact parameter b which
determines their relative angular momentum.
where δ (r) is the Dirac delta function and the only important parameter is a, the s-wave
scattering length.

1.3.2

The magnetic dipole-dipole interactions

In dipolar gases, beside the contact interaction, one has to consider DDI. For two atoms witha magnetic moment µ1 and µ2 pointing in the direction e1 and e2 , respectively, the DDI readsasµ0 µ1 µ2 (e1 · e2 )r2 − 3(e1 · r)(e2 · r)
UDDI (r, e1 , e2 ) =,(1.25)
4π
r5
where |r| is the distance between the atoms and µ0 the magnetic permeability of free space.
By applying a magnetic field, the dipoles align themselves along the quantization axis givenby the direction of the external field. The interaction between two polarized atoms can be
simplified and described by only two parameters: the distance r and the angle θ between r
and the quantization axis,µ0 µ1 µ2 1 − 3 cos2 θ
UDDI (r, θ ) =.
4π
r3(1.26)The result is a tunable interaction whose tunability is given by the angle θ . In fact, theterm (1 − 3 cos2 θ ) can assume values ranging between −2 and 1, going from an interaction
maximally attractive when θ = 0° (head-to-tail configuration) to maximally repulsive when
θ = 90° (side-by-side configuration). It is worth noticing that the DDI can also be tuned tozero for the magic angle θ = 54.7°. Due to the θ dependence and the scaling as r−3 , DDI
are anisotropic and long-range in a 3-dimensional system. In the presence of DDI, the elastic
scattering processes between atoms can radically change. In the following we will only

CHAPTER 1. ERBIUM PROPERTIES

15

briefly discuss why this can affect the physics of spinor gases. In a contact-type collision(s-wave), which is spin-independent, atoms exit the scattering with no net relative angularmomentum. After the interaction their total spin is conserved, but not necessarily the spin of
the single atom (flip-flop).
On the other hand, in a dipole-dipole-type of collision, due to the long-range nature,
atoms do not preserve the relative angular momentum, but rather only the total angularmomentum. Thus, the total spin is not conserved.

Chapter 2
Production of spinor Er quantum gases
A system composed of BECs or dFgs, occupying different Zeeman sublevels is usuallyreferred as “spinor” quantum gas [Sta13].
The aim of this chapter is to describe the techniques adopted to realize an ultracold spinor
erbium gas. In the first and second section we give an overview of the vacuum apparatus
and main procedure for the production of a degenerate quantum gas. Finally, the last section
describes the techniques usually adopted to generate a spinor gas.

2.1

Experimental setup

An overview of the machine is shown in Fig. 2.1. In the figure two different pressure regions
are highlighted. The High Vacuum (HV) section maintains a pressure of about 4 × 10−9 mbar
while the Ultra High Vacuum (UHV) section maintains a pressure of 1 × 10−11 mbar. The twoparts are connected to each other by a differential pumping tube, which provides a pressure
gradient of up to three orders of magnitude. We now briefly review the main production
steps.
Erbium travel in the high vacuum section A commercial diffusive oven, where solid
pieces of Er evaporate at a temperature of 1100 ◦C, realizes an erbium atomic vapor. Then,from the vapor, a set of apertures, forming the so-called hot lip and usually heated at
1200 ◦C, produces the collimated atomic beam. Atoms exiting the oven are additionally
optically collimated by a 2D transversal cooling stage made by two retro-reflected laser
beams, operating at 401 nm (broad transition of erbium). Figure 2.1 displays in blue the two
beams, crossing each other at an angle of 90°. This stage enhances the atomic flux in thelongitudinal direction. The atomic beam travels then towards the Zeeman slower.

16

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

17

Figure 2.1: The erbium apparatus (picture taken from[Fri14]). The figure points out thepresence of two sections. The HV section on the right is used to realize a collimated atomic
beam while the UHV section to slow down the atoms and finally reach the degeneracy
conditions in the main chamber.
Slowing down, cooling and trapping in the Ultra High Vacuum section Along thedifferential pumping tube, connecting the HV to UHV section, atoms are cooled down using
the Zeeman slower. The basic idea is to use combination of Zeeman effect and large radiative
pressure from laser light to reduce the velocity of the atoms down to the MOT capture
velocity. This is implemented by shining a 401 − nm beam collinear and counterpropagating
to the atomic beam. Due to the broad linewidth of the 401nm transition, a large numberof scattering events happen during the travelling time of the atoms in the Zeeman Slower.
During one of this event, the atom by absorbing a photon, receives a momentum kick inthe direction of light propagation while its internal state is promoted to the excited state of
the 401nm-transition. The atom decays back to its internal ground state by spontaneously
emitting a photon isotropically in space. Thus, due to the high number of events, the averagechange of the momentum given by the emission vanishes, resulting in a overall slowing
process. The reduction of the velocity makes the light off resonant due to the Doppler shift.
Several coils are positioned to create an inhomogeneous magnetic field which leads to a
space-dependent Zeeman shift. This compensates the Doppler shift and keeps the atoms
always resonant with the light.
When atoms approach the main chamber, shown in yellow in Fig. 2.1, they are alreadyslow enough to be trapped into the MOT. The MOT combines laser cooling with a position
dependent force to trap the atoms. The idea consists in the use of two counter-propagating
laser beams, for the three directions, of momentum kL and −kL with frequency ωL detuned

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

18

Figure 2.2: (a) Simplified 1D version of a MOT (picture taken from[Fri14]) for a two-level
atom with J=1 excited state and J=0 ground state that coincide with the z axis. The B field
gradient splits the excited state in the magnetic sublevels. For J=1 only three sublevels arepresent and the atom can be excited in these levels with the right polarization. Exploting σ +and σ − polarized photon only the two level mJ = ±1 are involved in the transition. In the
picture the spatial effect of reducing the detuning is also illustrated.
from the atomic resonance (ωa ). Similar to the Zeeman slower case, the resonant condition
for the beam of momentum kL is met for atoms whose velocity v is such that kL · v = ωL − ωaand the absorption of photon from this beam is most likely for atom with this velocity. Then,by detuning ωL < ωa (red detuning), the atoms are always most likely to scatter light fromthe beams against which they propagate kL · v < 0 and the net force is always opposite to
the velocity and null if the atoms are at rest (friction force). This simple configuration is nota trap. The atoms, indeed, feel a force that is velocity dependent but not space dependent.
Thus, the configuration with six beams can be seen more as a optical molasses rather than a
trap for the atoms. To correct for this and obtain a trap, the three retro-reflected laser-beams
are combined to a quadrupole magnetic field and a special choice of light polarization ismade [Raa87] (MOT). Figure 2.2 shows a simplified 1D sketch of the mechanism of a MOT,
approximating the atom to a two-level system with J=0 and J’=1 ground and excited states.
The quadrupole field induces a position-dependent Zeeman shift for the different magnetic
sublevels (quantum number m′ ), cancelling at z = 0. The two counterpropagating beams arechosen of polarization σ± so that they selectively excite the atom to m′ = ±1, respectively.
Thus, atoms displaced to the right with respect to the central position of the trap absorb moreprobably photons coming from the beam σ − polarized propagating towards the left, andvice-versa for a displacement to the other side of the trap. Hence, they always receive a kick(due to the linear momentum carried by photons) opposite with respect to their displacement.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

19

The result is a tunable trap, since it is also possible, after trapping the atoms, to reduce thedetuning of the MOT beams with respect to the transition and compress the atomic cloud (seeblue arrows in Fig. 2.2). In the case of Er, several optical transitions can in principle be used
to generate a MOT. Our experiment uses the 190 kHz linewidth transition at 583 nm. Thish̄Γnarrow linewidth leads to a Doppler temperature TD = 2kof 4.6 µK which enables a direct
Bloading of the optical dipole trap (ODT) from the MOT. For heavy atoms as erbium, thegravitational sag associated with the use of large detuning of the MOT light makes that the
sample get automatically spin polarized in the lowest spin states mF = −19/2 or mJ = −6
for the fermionic and bosonic case, respectively. The choice of narrower transitions can help
to reach very low temperatures but a trade off has to be chosen because also the capture
velocity will dramatically be reduced.

2.1.1

Reaching quantum degeneracy

From the MOT, we load the atoms into a single-beam optical dipole trap (ODT). Matching
the aspect ratio of the dipole trap to the atomic cloud in the MOT via a scanning system,
see Refs. [Bai12; Ahm05], enables to achieve a maximum loading efficiency of 35%. The
scanning system allows a change of the horizontal beam waist by rapidly scanning thefrequency of an AOM, whose first order is used for the ODT. After loading, atoms aretransfered into a croosed ODT and cooled down to the degeneracy using evaporative cooling[Ket96]. The idea of cooling by evaporation consists of removing the high-energetic atoms
from the sample and to allow the remaining atoms to rethermalize to a lower temperature via
elastic collisions. This process requires the atoms to collide and that the elastic collision rate(enabling rethermalisation) is much larger than the inelastic one (limiting the lifetime of thesample). This is achieved in cold atom experiments by truncating the confining potential to acertain depth U, which we slowly decrease. Reaching a degenerate quantum gas is not only
a matter of temperature, rather, the important parameter is the phase space density (PSD),
defined as:
3
PSD = nλdB,
(2.1)
where n is the peak density of the gas and λdB is the De Broglie wavelength. For a thermal
cloud, atoms follow the Maxwell-Boltzmann energy distribution and the peak density n canbe calculated from the total atom number N, once the trap frequencies and the temperature T
are known, by using this expression:n = Nω 3m
2πkB T

3
2

,

(2.2)

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

20

where ω = (ωx ωy ωz )1/3 is the geometrical mean trap frequency, kB is the Boltzmann constantand m is the mass of the chosen isotope. The condition to reach the phase transition from athermal sample to a degenerate quantum gas is PSD > 1 [Pit16]. It is important to carefully
choose the parameters of the evaporation ramp (of the ODT depth) and the trap frequencies
ωx,y,z , in order to remove only the hottest atoms. Indeed, it is important that the ramp is slow
enough, and the density high enough (and thus its elastic collision rate) so that the system
has time to retermalise. It is also necessary to have a good ratio between elastic and inelastic
collisions, since the latter will lead to atom losses. Furthermore, increasing the PSD requires
to compete the decreases of temperature with respect to the atom number.
In our lab the first Bose-Einstein condensate and degenerate Fermi gas of erbium was
obtained in 2012 and 2013, respectively, exploiting the evaporative cooling technique [Aik12].
Indeed, in the case of dipolar gases, a significant elastic cross section persists at low temperatures also for a Fermi gas due to the universal character of dipolar scattering [Bon09].

2.2

Overview on spin preparation

The peculiarity of spinor Er gases derives essentially from two characteristics. First, highspin systems as Er can lead to the possibility of investigating disparate magnetically orderedstates, such as [San06; Lia12] for bosons and stripe-phase [Maz17], d-wave superfluidity[Bar04] for fermions. Second, DDI plays an important role in driving process where thespin is not conserved. Several difficulties need to be overcome to reach these unexplored
quantum states, e.g. deterministic control of the population of the Zeeman sublevels. Thus,
it is helpful to reduce the complexity of the system from 13 (20) spin states for bosonic(fermionic) isotopes, to an effective spin 1/2 or spin 1 system. This can be realized by
exploiting the Zeeman effect or AC-stark shift, which depends nonlinearly on the different
magnetic sublevels. In addition, a good control of the B-fields is often necessary, making
the study of these systems even more challenging. For this purpose the AC-noise has to be
reduced and the DC value stabilized with high care. In the following subsections, we give an
overview of the techniques usually utilized to realize a spin mixture in the experiment. The
techniques introduced are two: RF pulse and the Landau-Zener Sweep.

2.2.1

Direct radio-frequency excitation of the Zeeman sublevels⃗0 ,Consider a sample of Er atoms in the lowest spin state interacting with a static field B⃗1 cos(ωt), along the x-axis. The radioparallel to the z-axis, and a radio-frequency field B
frequency (RF) field can be decomposed along z in a superposition of a counterclockwise

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

21

3 0 0

m J = 6m J = 5m J = 4m J = 3m J = 2m J = 1m J = 0m J = -1
m J = -2m J = -3m J = -4m J = -5m J = -6

E n e rg y (M H z )

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

5

1 0

1 5

2 0

2 5

3 0

M a g n e tic f ie ld ( G )Figure 2.3: Zeeman splitting up to 30 G for the ground state J=6 of the bosonic isotopes. The
splitting gives rise to 13 magnetic sublevels. According to the formula (1.17) the splitting islinear in the magnetic field range usually used in ultracold-atoms experiments.
and clockwise rotating field and it drives the spin flip1 . The discussion will be divided
in two cases, bosons and fermions, and rely on the use of basis change, considering therotating frame of frequency ω. The first step is to understand the bare states obtained byremoving from the Hamiltonian the off-diagonal coupling terms between field and atom. In
the next step, the effect of the coupling is switched on reinserting the off-diagonal elements.
Diagonalizing the resulting Hamiltonian it is possible to obtain the new eigenvalues andeigenstates, which represent the state of the atom in presence of the RF coupling.
For bosons, the B0 field splits the ground state J = 6 in 13 different m j -states, according
to the Eq. (1.17). Figure 2.3 shows the resulting effect. The splitting induced by the magnetic
field is linear2 and hence, a RF pulse that couples two adjacent spin states unavoidably
couples all the spin states. To fully understand the Hamiltonian for 13 spin states, it is usefulto start from the standard textbook spin- 12 system [Coh11]. In the laboratory frame the
1 Different configurations, as for example positioning the antenna on the z-axis, can effectively drive thetransition. This is the configuration usually used in our experiment, where one exploits the presence of all thepolarizations in the near field regime. In fact, only generating radio-frequencies above 300 MHz, one can reachthe far field in a realistic distance antenna-atoms of 1 meter.
2 As described in Chapter 1, due to the lack of nuclear spin, there is no quadratic regime accessible formagnetic field usually used in ultra-cold experiment.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

22

Hamiltonian of the atom reads:
Ĥ = ω0 Sˆz + 2Ω1 cos(ωt)Sˆx ,(2.3)

where Si = 2h̄ σi with i = x, y, z and σi are the Pauli matrices defined by:
σx =

!
0 1
,
1 0

σy =

!
0 −i
,
i 0

σz =

!
1 0
.
0 −1

(2.4)

The first diagonal term in Eq. 2.3 represents the Larmor precession at frequncy ω0 inducedby the B0 field, whereas the second one is the term given by the interaction with the RF field,
in which Ω1 = −γB1 is the Larmor precession frequency. According to the rotating-wave
approximation3 , the terms that oscillate in the opposite direction with respect to the Larmor
precession of the spin are neglected. Thus, the Hamiltonian in frame rotating at ω reduces toĤ = δ Sˆz + Ω1 Sˆx
or in the matrix form:
h̄
Ĥ =
2

δ Ω1
Ω1 −δ

(2.5)

!
,

(2.6)

where δ = ω − ω0 is the detuning between the RF frequency and the Larmor frequency.
For bosonic erbium, the Hamiltonian in the laboratory frame results in:Ĥ = µB gJ (ω0 )Jˆz + 2Ω1 cos(ωt)Jˆx ,(2.7)

which gives, in the rotating frame of frequency ω and expressed in the basis of the eigenvectors, |mJ ⟩, of Jz
δ−6 Ω1
0 ... 0
 Ω1 δ−5 Ω1 . . . 0 h̄ 
0 Ω1 δ−4 . . . 0 
H= 
(2.8)

,
2 .
.
.
.

.
..
..
. . ..  ..
0
0
0 . . . δ6
where the coupling between not adjacent states is equal to zero, because of the selection
rules for magnetic dipole transitions, and δmJ = ω − mJ ω0 . Before obtaining the eigenvalues
and eigenvectors of the complete Hamiltonian, it is interesting to have a look at the bare3 In this approximation, the terms in the Hamiltonian which oscillate rapidly are neglected. This is a valid

approximation when the applied field is near resonance with an atomic transition. In this case the fast oscillatingterm is the counterpropagating one with respect to the precession of the spin.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

23

states in the rotating frame shown in Fig. 2.4a, obtained using Ω1 = 0. Switching on theatom-field coupling, as shown in Fig. 2.4b, lifts the degeneracy between the eigenstates andthe crossings become avoided crossings. The new states are a mixture of all the sublevels
magnetic states.
In the experiment, we apply a RF coupling at a given detuning and coupling, for a given
time and detect the population of the different mJ states. After optimizing the RF pulse
duration, detuning and strength we find that it is possible to achieve a population of few spinstates but a deterministic and reproducible population of the latest two is not achievable.
Furthermore, it is not possible to clearly see Rabi oscillations, because the high number ofstates makes the situation particularly complex and susceptible to noise. The applied RF
starts to populate the level m j = −5 and by iteration, it fills the others, up to the highest mJ .
This can be seen in Fig. 2.5 where, for a thermal sample, a RF is applied for different pulse
durations. For a very long pulse this leads to an occupation of all the spin states.
On the contrary, in the fermionic case, the preparation of a given spin state can be better
controlled. It is in particular possible to achieve a deterministic population of the first two
spin states in a reproducible way. This takes advantage of the fact that the nuclear spin leads
to a non-negligible quadratic Zeeman shift (see section 1.2), even for magnetic fields of a few
Gauss. In this case the degeneracy of all spin states for δ = 0, observed in the bosonic case(Fig. 2.4a), is removed and it is possible to achieve a deterministic population of the first two
spin states in a reproducible way, above few Gauss. The bare states and the coupled states,
shown in Fig. 2.6, can be found after including the quadratic effect, induced by a magnetic
field of 40 G, as an additional quadratic detuning on the diagonal term of the Hamiltonian.
The states in presence of coupling exhibit avoided crossings at various detuning δ ̸= 0. In
this way, when following the lowest energy state and increasing the detuning from δ < 0 to
δ > 0, one encounters 19 avoided crossings.
Although the RF is a selective resonant phenomenon, a careful analysis is needed tounderstand the quadratic effect necessary to avoid to populate other states, in particular the
one necessary to drive the transition to the state mF = −17/2, without coupling the statemF = −15/24 . Figure 2.7 a) shows an illustration of the effect of the quadratic Zeeman shift.
At a given magnetic field, the Zeeman effect results in an energy splitting between the different
mF states. The linear effect leads to a dominant term, independent of the mF → mF + 1
transitions, while the quadratic effect effectively detunes each of these different transitions
by ∆m , increasing (quadratically) with B0 . The relevant ω for the spin state preparation
would be the one resonant with the −19/2 → −17/2 transition but detuned, thanks to a large
enough B0 , from the −17/2 → −15/2. The necessary detuning ∆m between the different
4 The following consideration can be applied also for the bosonic case.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

24

25

mJ=-6

20

mJ=-5

E / h (kHz)

15

mJ=-4

10

mJ=-3

5

mJ=-1

mJ=-2

0

mJ=0

-5

mJ=2

mJ=1
mJ=3

-10

mJ=4

-15

mJ=5
mJ=6

-20
-25

-20

-15

-10

-5

0

5

RF detuning (kHz)

10

15

20

(a)
25

mJ=-6

20

mJ=-5

J

E / h (kHz)

15

mJ=-4

10

mJ=-3

5

mJ=-1

mJ=-2

0

mJ=0

-5

mJ=2

mJ=1
mJ=3

-10

mJ=4

-15

mJ=5
mJ=6

-20
-25

-20

-15

-10

-5

0

5

RF detuning (kHz)

10

15

20

(b)

Figure 2.4: In (a) the 13 bare states (i.e. for vanishing RF coupling Ω1 ) of the atom in therotating frame, for the bosonic ground state, become degenerate when the RF is resonantwith the Zeeman splitting between adjacent spin states. In (b) switching on the coupling(off diagonal terms of the Hamiltonian (2.8)), the bare states are mixed and the crossing an
avoided one.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES
(a)

(b)

(c)

(d)

mJ

25

(e)
200

+6
+5
+4
100

+3

+1
0

0
-1

position (µm)

+2

-2
-3

-100

-4
-5
-6
RF off90

210

2005 tRF (µs)

-200
0.0

0.5
1.0
norm. atom number

Figure 2.5: In the figure the result of applying a RF pulse for different time durations isshown in (a) (b) (c) (d) (picture taken from[Fri14]). In (e) the normalized atom number forthe 2005 µs pulse is plotted versus the position. The spin position dependence is the result of
the Stern Gerlach procedure.
Zeeman transition, necessary to have a robust spin preparation, usually results mostly fromthe magnetic noise present in the system. In particular, for our experiment, a typical noise for
the B field is on the mG level. As it can be seen from the schematic illustration in Fig. 2.7b), a magnetic field noise is directly converted in a different energy splitting of the levels
mainly via the linear Zeeman effect. To quantify this effect, for 1 mG the distance between
adjacent spin states varies up to 1 kHz. This not only reduces the RF coupling but also could
cause a population of the higher spin states. To avoid population of the mF = −15/2 thedetuning δ has to be much higher than 2 kHz (δ ≫ 2 kHz). This condition can be verified inour experiment working at 20 G.
In addition, it is important to take care of the frequency spectrum of the RF signal. Indeed,
when a very short pulse is used, the spectrum can be Fourier-limited. Particular shapes for theintensity of the RF pulse can be used to improve the Fourier spectrum, as for example the soCHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

26

mF =19/2

500mF =17/2
mF =15/2
mF =13/2
mF =11/2

0

mF =9/2

E/h (kHz)mF =7/2
mF =5/2
mF =3/2-500

mF =1/2
mF =-1/2
-540

-540

-1000 -550

-550

-560

-560

-570

-570

-1500 -580

avoided crossing

mF =-5/2
mF =-7/2
mF =-9/2
mF =-11/2
mF =-13/2-580

mF =-15/2

crossing

mF =-17/2

-590

-590
-390

-2000

mF =-3/2

-400

-380

-300

-370

-200

-360

-100

0

-390

-380

100

200

-370

300

-360

mF =-19/2

400

RF detuning (kHz)
Figure 2.6: The 20 bare states of the system atom plus radiation, for the fermionic isotope in
the F=19/2 manifold, are plotted as a function of the RF detuning. The zero indicates the
mF =-19/2 mF =-17/2 resonance. The presence of a quadratic effect makes the adjacent states
resonantly coupled at different positions of the RF frequency. The two insets show a zoom of
the dashed area. In the right inset, the bare states of the two lowest magnetic sublevels crosseach other in absence of RF coupling. In the left inset the bare states become dressed state
and the crossing an avoided one.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

27

Energy splitting(B)|m + 2⟩∆mω|m + 1⟩ωB0

ε B0 + ε

B

|m⟩
(a)

(b)

Figure 2.7: (a) When the frequency ω matches the energy difference between |m⟩ and |m + 1⟩,
the next state |m + 2⟩ is not coupled due to the quadratic effect, which results in an effective
detuning ∆m . (b) The magnetic field noise, depicted as a sinusoidal wave on the x-axis, leads
to a variable energy splitting for adjacent spin states according to their different slope.
called Blackman pulse [Mon10; Cha13]. Figure 2.8 compares a windowing5 of a Blackman
waveform6 to the standard rectangular one, in the time and in the frequency domain.
Although the main lobes’ width is slightly larger in the case of Blackman pulse, therelative sidebands attenuation is much bigger and equal to −58 dB in comparison to the only
−13 dB of the rectangular one. In particular, regarding the RF coupling of higher states inpresence of a certain detuning δ , it could be possible to reduce the RF power, leaked at thatparticular detuning, using an ad-hoc choice of the function parameters.

2.2.2

Landau-Zener sweep

To make the spin preparation more robust with RF, the Landau-Zener passage can be executed[Mew97]. This consists in realizing a frequency sweep from the positive to negative RF
detuning. The idea is to reduce the influence of the B field noise on the final state and inparticular to decrease the coupling to higher spin states. Let’s consider the previous casewith a S = 1/2 system described by the Hamiltonian of Eq. (2.6). The eigenstates, obtained
5 A window function is a function whose value is 0 outside an interval. Windowing of a simple waveform isuseful since often it is interesting to look at the spectral content if a function is a applied only during a limited
time period. Indeed, windowing of simple waveform like sin ωt causes its Fourier transform to develop spectral
leakages at frequencies different from ω.

6 The expression used to realize the analysis of the Blackman is w(t) = a − a cos 2πt + a cos 4πt
0
1
2
N−1
N−1
with a0 = 0.42, a1 = 0.5, a2 = 0.08

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

28Time domain
1

RectangularBlackman

0.9
0.8

Amplitude

0.7
0.6
0.5
0.4
0.3
0.2
0.1

0

1

2

3

4

5

6
104

Samples(a)Frequency domain
RectangularBlackman-20

Magnitude (dB)

-40
-60
-80
-100
-120
-140
-160
0

1

2

3

4

5

6

7

8

9

Normalized frequency(b)

Figure 2.8: (a) A pulse with a rectangular amplitude in time and frequency domain iscompared to the Blackman shape in (b). It is possible to notice the power leaked out atdifferent frequency with respect to the central one can be reduced with a particular timedependence of the amplitude of the pulse.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES
2

29mS =-1/2(blue)

1.5

m =1/2(yellow)
S

1

E/h

0.5
0
-0.5
-1
-1.5
-2
-30

-20

-10
0
10
RF detuning

20

30

Figure 2.9: The result of the diagonalization of the Hamiltonian for a two-level system. The
dressed state presents a gap. The colours gives an indication of the possibility of measuremS = 1/2 in yellow or mS = −1/2 in blue. Only far off resonance the dressed states matchthe bare states whereas they are mixed states across the resonance. As detailed in the main
text, due to the presence of a gap, an adiabatic sweep of the RF frequency following the blackarrow allows a passage from the mS = −1/2 in to mS = 1/2 state.
by diagonalizing the Hamiltonian, contain superposition of mS = −1/2 and mS = 1/2.
Figure 2.9 shows the two mS states, where the solid lines indicate the new states dressed by
the RF, whereas the dotted lines are referred to the bare states. Far off resonance, the coupledstates matches the bare states mS = ±1/2. The method to reliably change the state of theatom, starting from mS = −1/2 to mS = 1/2, consists in an adiabatic rapid passage(ARP).
According to the adiabatic theorem, as stated by Max Born and Vladimir Fock: "A physicalsystem remains in its instantaneous eigenstate if a given perturbation is acting on it slowly
enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian’s
spectrum."[Bor28]. Following this idea and starting with a spin polarized system in themS = −1/2, one can apply an off resonance RF and slowly ramp the detuning across theresonance. In Fig. 2.9 this corresponds to follow the dotted arrow. In this way all the atoms
are transferred from the mS = −1/2 to the mS = 1/2 state, if the passage is adiabatic7 .
Figure 2.6 shows the energy states in presence of RF coupling for the fermionic Er isotope.
Two different types of crossing are present. One type refers to a crossing between energy
7 To verify the adiabaticity condition we need to compare the two energy scales that are playing a role. In

particular the time duration of the ramp (t) has to satisfy the relation 1t ≪energy splitting between the coupled states.

Egap
h , where Egap is the minimum

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

30
mF =19/2
mF =17/2
mF =15/2-583.85

mF =13/2
mF =11/2-583.9mF =9/2

E/h (MHz )-583.95

mF =7/2-584

mF =5/2-584.05

mF =1/2

mF =3/2
mF =-1/2-584.1mF =-3/2
mF =-5/2-584.15mF =-7/2-584.2mF =-9/2
mF =-11/2-584.25mF =-13/2-584.3mF =-15/2

-300

-250

-200

-150

-100

RF detuning (kHz)-50

0mF =-17/2
mF =-19/2

Figure 2.10: Applying the RF sweep in an adiabatic way the atoms will remains in theireigenstates. Following the black arrow, in this dressed state picture, results in obtaining a
population up to the highest spin states. Instead following the red arrow leads to a populationof the lowest spin states.
levels with zero-coupling, which were imposed in the Hamiltonian for all the not-adjacentspin states. The other type is instead the avoided crossing between adjacent spin states,
which now results in a gap. Only adjacent spin states can be coupled by the RF, according
to the magnetic dipole transition selection rules. The presence of a gap can be exploited toeffectuate the adiabatic rapid passage. Starting with a spin polarized sample in mF = −19/2,
it is possible to effectuate the RF sweep in both directions with completely different results.
Figure 2.10 displays a particular region of Fig. 2.6. The two sweep paths are highlighted withdotted arrows of different colors and allow disparate spin preparations.
Starting from an off resonance RF from negative detuning, the RF is not driving any
transition up to the first zero crossing. At that point, the population of the −19/2 is transferredto the −17/2 state. Afterwards, the next crossing is the one that represents the coupling of−17/2 with −15/2. Since there is population in the −17/2 this will populate the higher
spin state, leading to the coupling of all the spin states. Then it is possible to end up with apopulation in the higher mF = 19/2 spin state. A complete transfer in the higher spin states
is challenging since it requires to maintain the adiabaticity condition over the 19 avoided
crossing present in the path. In a non-adiabatic crossing the system will not follow the groundstate of the Hamiltonian and it will mix the different energy states. This path can be chosenCHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

31

to achieve a population in the highest spin states, although the purity of the sample and the
reproducibility will be affected.
Starting with negative detuning and following the red path, leads to a deterministic andreproducible population of the lowest spin states mF = −19/2 and mF = −17/2. During
the sweep of the RF, the same number of crossings as before are present, but this does not
influence the population of the magnetic sublevels because these are empty states. The onlychanges in the population is due the last avoided crossing. When this is crossed, it is possible
to end up with a mixture of the only two lowest spin states.
To get the population of the different sublevels during the RF sweep, the time evolutionof the wave function has to be evaluated according to the following relation:|ψ(t)⟩ = e−i

R Ht

h̄ dt

|ψ(0)⟩(2.9)

where ψ(0) is the wave function at t = 0, which is initialized in the lowest spin state.
Figure 2.11 and 2.12 show the time evolution of the mF state population during the two
sweeps. The parameters of the sweep were chosen to optimize the spin population of thehighest spin states, sweeping from positive to negative detuning, and to obtain a 50-50
mixture of the lowest two spin states, sweeping in the other direction.
1

mF =+19/2
mF =+17/2

0.9

mF =+15/2
mF =+13/2

0.8

Normalized population

mF =+11/2

0.7

mF =+9/2
mF =+7/2

0.6

mF =+5/2
mF =+3/2

0.5

mF =+1/2
mF =-1/2

0.4

mF =-3/2
mF =-5/2

0.3

mF =-7/2

0.2

mF =-9/2
mF =-11/2

0.1
0
-300

mF =-13/2
mF =-15/2

-200

-100

0

100

RF detuning (kHz)

200

300

mF =-17/2
mF =-19/2

Figure 2.11: Normalized population of the spin states during the sweep as a function of the
RF detuning. The not complete transfer of the population in the highest spin state is theresult of a not completely adiabatic crossing of the states. This is obtained from the result ofa simulation where the black arrow of Figure 2.10 is followed.

CHAPTER 2. PRODUCTION OF SPINOR ER QUANTUM GASES

32

1
0.9

Normalized population

0.8
0.7m F =-19/2

0.6

m F =-17/2

0.5

m F =-15/2
m F =-13/2

0.4
0.3
0.2
0.1
0
-15

-10

-5

0

5

10

15

RF detuning (kHz)

Figure 2.12: Normalized population of the spin states during the sweep as a function of the
RF detuning. The not complete transfer of the population in the mF = 17/2 spin state is theresult of a not completely adiabatic crossing of the states. This is obtained from the result of
simulation where the red arrow of Figure 2.10 is followed.
This technique is usually used in our lab to produce 50-50 mixture of fermions with better
reproducibility with respect to the simple RF pulse.
It is also possible to use this technique to prepare spinor bosonic gases, but due to the lackof quadratic Zeeman effect, a pure sample is not achievable since all the levels cross at thesame RF.
In Chapter 4 a promising spin preparation and a protocol for a single spin state manipulation is proposed, which can be implemented and used for fermionic and bosonic isotopes aswell.

Chapter 3
Experiment on spin-1/2 system: dipolar
fermions in a lattice
The previous chapter discussed a method to reduce the complexity of a spinor quantum gasof the fermionic erbium isotope from 20 to 2 spin states. The main step was to decouple
the spin states and realize an effective spin-1/2 system exploiting the quadratic Zeeman
effects at high B fields (B > 20 G). In this way, the complex situation in which all the
spin states are coupled can be broken down into smaller solvable problems. A spin-1/2
system in the periodic potential of an optical lattice is described by the Hubbard model. In
the case of erbium, its large magnetic moment requires to modify the Hubbard model byadding the effect of the DDI; the resulting model is known as an extended Hubbard model[Dut14]. Several unexplored phases are predicted to occur with long-range interactions in
the extended Hubbard model of dipolar spinor gases, e.g [Bar12; Maz17] . In the prospectof future investigations of the rich physics of dipolar spinor gases, in this chapter, we start
to characterize our system with a measurement of the onsite interaction between the twolowest spin states of 167Er loaded into a three-dimensional optical lattice. From the onsite
measurements we can understand the interaction between the two spin states. This chapter
consists of three sections. The first section introduces the basic concept of optical lattices
and the lattice adopted in our experiment. The second section describes the Fermi-Hubbard
model and the effects induced by the presence of DDI in the extended version. Finally, inthe last section the experimental procedure adopted to investigate the onsite interaction is
explained and the result analyzed.

33

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

3.1

34

The optical lattice: basic concepts

A one-dimensional optical lattice (1D) can be realized by simply overlapping two counterpropagating laser beams with the same wavelength λ and with a constant relative phase.
This results in a standing wave whose period is λ /2. Thus, the atoms feel a periodic
potential as a consequence of the effects of interference and Stark-shift. In the same way, athree-dimensional (3D) lattice geometry is obtained with the use of three pairs of counterpropagating beams, each pair being of different frequencies. Ultracold atoms in opticallattices are widely used to simulate solid state problems in which the lattice is free of
impurities and atoms can be easily manipulated [Blo05] . Furthermore, this artificial crystal
made of light is highly tunable. It is indeed possible to change the lattice geometry andits characteristics by modifying the laser beam parameters. For example, by changing the
wavelength of one pair of counter-propagating beams, one can modify the spacing between
the lattice sites in the direction of propagation of the two lattice beams. Additionally, tuning
the intensity of the laser beams leads to a variation of the lattice depth, which determines thestrength of the different terms of the Hubbard Hamiltonian, both the kinetic (tunneling) andthe interparticle interactions. This tunability gives the possibility to carefully investigate time
dynamics and phase transitions [Blo12].
In the ERBIUM experiment, two lasers running at 532 nm and 1064 nm create a threedimensional(3D) optical lattice. In particular, two beams coming from the 532-nm lightsource and one from the 1064-nm light source are retro-reflected to obtain the 3D array oflattice sites. The 2 pairs of 532nm beams are purposedly detuned by 220 MHz (so that they
do not interfere). Taking the laboratory reference system, in which z is the axis of gravity
and x − y is referred as horizontal plane, the three lattice standing waves are orthogonal toeach other and propagate along x, y and z. The resulting potential acting on the atoms can be
written as
2−2 y +z2

Vlattice (x, y, z) = −Vx e

wx

2

2

2

2

−2 x +z2

sin (kx x) −Vy e

wy

2

2

2

−2 x +y2

sin (ky y) −Vz ewzsin2 (kz z), (3.1)

where Vη are the three amplitudes induced by the combined effect of interference and Stark
shift, kη are the wavevectors and the Gaussian envelopes (with waist wη ) result from theGaussian profile of the laser beams. Since the atomic cloud is usually small compared withthe beam waist, only the central part of the optical lattice is occupied. Thus, the trapping
potential can be approximated as an harmonic confinement superimposed to an homogeneousCHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

35

EnergyPosition

Figure 3.1: Sketch of the potential (3.2) along one direction i=x,y or z resulting from theinterference. The harmonic confinement is enhanced to show the effect of the Gaussianenvelope confinement.
periodic lattice potential, as illustrated in Fig. 3.1:
Vlattice (x, y, z) = −m∑ Vη sin2(kη η) + 2 ωη2 η 2 ,


(3.2)

η=x,y,zwhere the trapping frequency ωi are given by:
4
ωx2 =

m

Vy Vz
+
w2y w2z!
(3.3)

and ωy2 , ωz2 can be obtained from Eq. (3.3) with cyclic permutations [Gre03]. The horizontal
lattice beams produce a separation of 266 nm between the lattice sites on the x and y direction,
while on the vertical direction the spacing is of 532 nm. The maximum lattice depth is given
by four times the single beam trap depth and it is usually expressed in recoil energy (Er ).
Due to the different wavelengths of the lattice beams in the horizontal plane with respect tothe vertical direction, the recoil energies are different and equal to2
Erhor h̄kx,y== 2π · 4221 Hz,h̄
2mErver h̄kz2== 2π · 1055 Hz,
h̄
2m

(3.4)

where m is the atomic mass of the fermionic isotope. With the current setup the maximum
lattice depths achievable are (Ex ,Ey ,Ez ) = (30,30,100) Er . We usually work at the conditions
in which the two beams, propagating in the x and y direction, have the same intensity. This
gives Vx = Vy .

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

3.2

36

Fermi-Hubbard model

The Hubbard model was first introduced to describe electrons in solids, i.e spin 1/2 fermions,and it was intensively studied to solve solid-state problems [Hub63; Ani91]. In the lastyears, this model was applied to ultracold atoms including new features, e.g the bosonicstatistics (Bose-Hubbard model) or with the presence of a long-range interaction [Dut14],
namely the DDI, which can behave as a model of the long-range Coulomb interactions which
occur between electrons. To derive the Fermi-Hubbard Hamiltonian, as a starting point, one
can consider the Hamiltonian resulting for a system of spin- 12 fermions, whose interaction
potential can be written as′

V (r − r ) =
4π h̄2 asδ (r − r′ ) ≡ gδ (r − r′ ).
m

(3.5)

This is the pseudo-potential approximation of the van der Waals potential, referred as contact
interaction in Chapter 1, which features a short-range and isotropic character. Using theformalism of the second quantization, the Hamiltonian reads:
Ĥ =

Z

#




2
h̄
dr ∑ Ψ̂†σ (r) − ∇2 +Vlattice Ψ̂σ (r) + g Ψ̂†↓ (r)Ψ̂†↑ (r)Ψ̂↑ (r)Ψ̂↓ (r) , (3.6)
2m
σ"where σ refers to the two spins (↑,↓), Ψσ and Ψ†σ are the fermionic annihilation and creation
field operators, respectively, which have to satisfy the following fermionic anti-commutation
relation:{Ψ̂σ (r), Ψ̂†σ (r′ )} = δσ σ ′ δ (r − r′ ).
(3.7)
In the Hamiltonian, the first term describes the kinetic and potential energy of the single
particle in the lattice, whereas the second term takes into account interactions between two
atoms. In this case, only the s-wave type interactions between two different spin states (↑ ↓)are taken into account, since the s-wave collisions vanish for identical fermions (↑ ↑ , ↓ ↓)according to the Pauli’s principle and relation (3.7). The field operators can be expanded
into a basis of Bloch functions (φn,k ). These are the eigenfunctions of the single-particle
Hamiltonian where one considers only the first term of Eq. (3.6). This expansion reads:Ψ̂(r) = ∑ ĉn,k φn,k ,

(3.8)

n,k

where ĉn,k is the annihilation operator of the particle in the band number index n with quasi
momentum k. If the degenerate Fermi gas is loaded adiabatically in a deep lattice, theCHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

37

population of second and higher bands can be neglected, since the large bandgap prevents thehigher bands to be populated [Pfa97]. In this case, since the Bloch functions are periodic in
the momentum space, it is useful to expand them in Fourier series. Then the field operators
are given byΨ̂(r) = ∑ b̂i wi (r).
(3.9)
iHere, b̂i is the annihilation operator of the particle localized at the i-th site and wi are theassociated Wannier functions defined by:wi (r) = ∑ e−ik(r−ri ) φk (r).

(3.10)k

Substituting Eq. (3.9) in Eq. (3.6), one finds the Fermi-Hubbard Hamiltonian:H =−∑ ti j b†i,σ b j,σ +U ∑ b†i↑b†i↓bi↓bi↑ .

<i, j>,σ|{z(3.11)

i=1} |tunneling from site j to i{z

on site interaction}

The first term represents the tunneling from one lattice site labeled j to one neighbor labeled
i, whereZh̄2 ∇2∗+V (r)]w j (r).
(3.12)
ti j = − drwi (r)[−2mWhereas, the second term gives the onsite interaction between two different spin states in thesame lattice site, withZ
4π h̄asdr |wi ( r)|4 .
(3.13)U=m
The aim of the next section is the measurement of the onsite interaction between the mF =−19/2 and the mF = −17/2 spin states.
As already underlined in Chapter 1 erbium has an high magnetic moment. Thus, the interatomic potential (3.5) has also a second contribution, which is long-range and anisotropic.
The total interaction potential of Eq. (3.5) has to be modified as the sum of the contact
interaction term and the dipole-dipole term
V (r − r′ ) = gδ (r − r′ ) +µ0 µσ µσ ′ 1 − 3 cos2 θ.
4π|r − r′ |3(3.14)

Following the same steps as before, for a system with long-range interactions expressedby the potential (3.14), one obtains the extended Fermi-Hubbard Hamiltonian. Limiting
the offsite interaction terms to nearest neighbors and neglecting the terms which contain aCHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

38

negligible overlap of the Wannier functions from different lattice sites, one obtains:
H =−∑<i, j>,σ|V′∑ ∑′ b†jσ ′ b†iσ biσ b jσ ′ ,2 <i,j> σ ,σ} |{z}ti j b†i,σ b j,σ +U ′ ∑ b†i↑ b†i↓ bi↓ bi↑ +

{z

i=1

} |

tunneling from site j to i{z

on site interaction(3.15)

near neighboring interaction

where the new term derives from the dipole-dipole interactions between two near neighboring
atoms (NNI), and it readsµ0 µσ µσ ′
V=
4π

Z

Z

dr

1 − 3 cos2 θr−r′
2
w j (r′ ) .
′3|r − r |d r′ |wi (r)|2(3.16)

It is worth noticing that even if the second terms is already part of the Hamiltonian in
Eq. (3.11), in (3.15) U ′ = U +UDDI contains the anisotropic dependence of DDI. The term
UDDI strongly depends on the aspect ratio of the Wannier functions and on the dipole
orientation, and it can be written asµ0 µσ µσ ′UDDI =
4π

Z

Z

dr

′

2 1 − 3 cos

dr |wi (r)|

2θ

r−r′
|r − r′ |32

wi (r′ ) .

(3.17)

In fact, the weight between attractive and repulsive dipole-dipole interactions in UDDI relieson the shape of the Wannier function and it can become equal for a symmetric shape, resultingin UDDI = 0. Additionally, for a non-symmetric case it can be tuned, at a fixed amplitudeof the magnetic fields, rotating the dipoles. In the Hamiltonian (3.15), we neglect the terms
driving flip-flop dynamics. This is a valid approximation since, for the onsite measurements
presented in this thesis, we work under conditions where these processes are forbidden by
the energy conservation1 .

3.3

Measurement of the interaction properties

To probe the onsite interaction between two spin states in the lattice, it is possible to measure
the resonant frequency that drives the process where, starting from single occupancy of theatoms in a lattice site, one of the atom tunnels to an already occupied site. It is possible
to completely neglect the possibility that two atoms in the same lowest lattice band and in
the same spin state share the lattice site, due to Pauli blocking. The following subsections
1 For the typical magnetic field values in our experiment the quadratic ac-Stark shift, induced by the lattice,(see Chapter 4) and the quadratic Zeeman effect (see Chapter 1) do not compensate each others. Hence, theenergy splittings between adjacent magnetic sublevels are not degenerate thus leading to negligible flip-flop
dynamics.

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

39

describe the experimental sequence usually adopted in our experiment to investigate theonsite interaction and the result of the measurements.

3.3.1

Experimental sequences

The experimental sequence starts with the realization of a degenerate Fermi gas at a typicaltemperature of 0.2 T /TF in the ODT, where TF is the Fermi temperature. The sample is spinpolarized in the lowest magnetic sublevel mF = −19/2. The atoms are adiabatically loaded
into the 3D lattice using an exponential ramp to increase the power of the three retro-reflectedlattice beams, afterwards the ODT is linearly ramped down in 10 ms. The final lattice beam
powers result in a lattice depths of (Ex ,Ey ,Ez ) = (20,20,60) Er . The loading parameters are
optimized to achieve population only in the lowest band of the lattice. Although several
energy bands exist, the large bandgap allowed to obtain population almost only in the lowest
lattice band. Thanks to the quadratic Zeeman effect several spin mixture can be achieved
with a RF sweep (see Chapter 3). In this case, we work at a magnetic field of 40 G andthe parameters of the sweep, i.e. power, radio-frequency duration and frequency range, areoptimized to obtain, in a reproducible way, a spin mixture with 75% of the atoms in thelowest spin state mF = −19/2 and 25% in mF = −17/2.
Figure 3.2 shows how, varying the time duration, it is possible to optimize the populationof the higher spin states. The frequency range of the sweep is tuned from 75 kHz to −75 kHzwith respect to the frequency resonance of the two lowest spin states, both in the experiments
and in the simulations, performed along the lines of Chapter 2. The simulated results are
obtained considering the time duration and frequency range of the sweep implemented in theexperiment, and optimizing as only parameters the off-diagonal Rabi frequency terms of the
Hamiltonian (2.8). It can be noticed that a RF sweep duration of 400 µs produces a sample
of (| − 19/2 >, | − 17/2 >) = (75, 25)%. In this set of measurement we thus use a sweep of
400 µs.
After the sweep, we ramp down the magnetic field to 0.6 G, where we assume to reach a
Mott insulator of unit filling per lattice site, assumption that we a-posteriori confirm at theend of this measurement. The measurement of the interpsin as is then based on parametricheating lattice spectroscopy [Stö04; Kol06]. To resonantly create double occupancy of thelattice sites with two different spin states, the amplitude of the horizontal lattice beams are
sinusoidally modulated in phase, at a frequency νm with a typical total amplitude of 20%
for 500 ms as illustrated in Fig. 3.3. At the resonant condition U = hνm , a maximum of
doublons2 are created.
2 Two different spin states in the same lattice site are also referred as doublon

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

40

1.2m F =-19/2
m =-17/2

1

F

Normalized populationm =-15/2
F

0.8

0.6

0.4

0.2

0

-0.2
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time duration RF sweep (ms)Figure 3.2: Population of the higher spin states as a function of the time duration of the RF
sweep. The sweep was performed at a magnetic field of 40 G with a total RF range of 150
kHz. The shaded regions represent the experimental data, whereas the thicker lines are thesimulated results.

EnergyPosition
EnergyPosition

Figure 3.3: In the upper part a schematic illustration of the parametric heating. The twocolors represent two different spin states. The lattice depth is changing by 20%. In the lower
case, doubly occupied lattice sites are produced as an effect of the parametric heating.

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

3.3.2

41

The onsite interaction

The resolution of the imaging setup implemented in our experiment does not allow us to lookdirectly at the doubly occupied sites; yet we can evaluate the number of doubly occupied sites
from the atom losses. Indeed, there are several loss mechanisms that can shorten the lifetimeof a doubly occupied site, e.g light induced two-body losses or three-body recombinations
caused by the spatially overlap of the Wannier functions. The experimental sequence isrepeated modulating the lattice at different frequencies νm . The resulting atom number isplotted as a function of the lattice modulation frequency (νm ) in Fig. 3.4. The two colors
indicate the different dipole orientations. We perform this measurement for two different
polarization directions, in one case the dipoles are oriented along the z axis whereas in theother in the x-y plane, making an angle of 45° with x and y. A resonance is clearly visible
in both the modulation scans indicating that, when the modulation frequency matches theenergy gap given by the onsite interaction U, doubly occupied lattice sites occur and thelosses are enhanced. This process is independent of the sign of the onsite energy; thus fromthe measurements we can only estimate the absolute value.
Indeed we observe a shift between the resonances for the different dipole orientations,which results from the anisotropic character of the DDI associated with an anisotropic
Wannier function. Applying a Gaussian approximation to the Wannier function, it is possible
to express the anisotropy in terms of the onsite aspect ratio (AR) as AR = lz /lx,y , where thelx,y,z are the harmonic oscillator lengths for one lattice site along the x,y and z directions.
In our experiment the depths of the two green lattices are equal and as a result lx = ly . The
anisotropy is guaranteed adopting a lattice with different lattice depths in the z direction
with respect to the x − y plane. This gives an AR of 1.5 for the lattice used. For this AR, we
expect a shift of the onsite energy U ′ with respect to U of UDDI = −400(30) Hz, in the casewhere the dipoles are oriented along the vertical lattice beams, and of UDDI = 200(15) Hz,
when the dipoles are oriented in the x − y plane, where the errors are obtained assuming a
5% uncertainty on the lattice depths.
We can use the dipole-dipole interaction as a tool to understand the sign of the total onsite
U ′ . Since when the dipoles are in the x − y plane (red points) the curve in Fig. 3.4 is shiftedto higher frequency; this means that the onsite interaction is repulsive and the sign of U ′positive. The measured values3 resulting from a Gaussian fit of the resonances are
Dipoles along z: U ′ = 2.49(4) kHz(3.18)

3 The measurements of the onsite interaction are done at a magnetic field of 0.6 G where we do not observe

close by Feshbach resonances.

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

42

D ip o le a lo n g z
D ip o le in th e x - y p la n e

N o r m a liz e d a to m

n u m b e r

1 .0 00 .9 50 .9 00 .8 5

0 .8 00 .7 5
0

1

2

3

4

5

M o d u la tio n fr e q u e n c y ( k H z )Figure 3.4: Normalized atom number as a function of the modulation frequency. When themodulation frequency matches the energy gap, given by the onsite interaction U, doubly
occupied lattice sites are formed and as a result the losses are enhanced. The two colors
indicate the different dipole orientations at a magnetic field amplitude of 0.6 G. The black
points indicate an orientation of the dipoles along the vertical infrared lattice beam (z axis).
Whereas, for the red points the dipoles are oriented in the plane of the horizontal lattice (x-yplane), at almost 45° between the two lattice beams. The clear shift of the resonant position
is induced by the angle dependency of UDDI .

CHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

(a)

43

(b)

Figure 3.5: In (a) the initial configuration where all the lattice sites are occupied by one
atom. The two colors represent the two spin states, whereas the arrows indicate the dipolesorientation. In (b) a double occupied site is formed as a result of the parametric heating of
the green lattice beams. It is possible to notice that after the hopping process one neighborinteraction is lost.

Dipoles in the x − z plane: U ′ = 3.00(5) kHz(3.19)The measured difference is 510(90) Hz and it is consistent with the expected value of600(45) Hz. So far, we have neglected the effect of the NNI on the predicted value of theresonance frequency. On the other hand, this is known to have a contribution [Bai12]. Since
the lattice spacing in the horizontal plane is half of the spacing along the vertical direction,
one can assume that the NNI interactions are non negligible only in the two-dimensional
green lattice and one can see the experiment as an ensemble of 2D lattice planes decoupledfrom each other. The 2D lattice is illustrated in Fig. 3.5. In the production process of adoubly occupied site with two different spin states, the initial configuration has each atom
sitting in a different lattice site, thus the NNI is the only energy contribution. When a doubly
occupied site is reached, the two atoms in the same lattice site feel an interaction given by
the onsite U, and the NNI contribution coming from the now empty lattice site is lost, i.e.
one V ′ . Therefore, the resonant frequency (νm ) driving the hopping is more precisely givenby the following condition:h̄νm = U ′ −V ′ ,(3.20)
where the anisotropic character is present not only in U ′ but also in V ′ . In the 2D lattice, with
266-nm spacing, the dipole-dipole interaction assumes a side-by-side configuration whichleads to a repulsive interaction, when the dipoles are oriented in the vertical direction, orCHAPTER 3. EXPERIMENT ON SPIN-1/2 SYSTEM

44

a head-to-tail configuration, when the dipoles are in the green lattice plane with an angleof 45° with respect to the two green lattice beams. The resultant corrections, according to
Eq. (1.26), are of 30 Hz and −15 Hz for the two cases respectively, which lie within our error
bars. Finally we can evaluate the scattering length as for the interspin interaction between
the two lowest spin states from the relation (3.13), this results:
as = 126(3)a0(3.21)

where a0 is the Bohr radius.
The strong repulsive onsite energy present in both configurations is much larger than thetunneling rate J and the thermal energy T (U ≫ J, T ). This yields a Mott insulating phasein our lattice, where each atom sits in a lattice site and the double occupancy is suppressed
without the parametric heating. This means that repeating the same experiment, in which
the lattice is loaded adiabatically after preparing the spin mixture, we obtain the same initialstate.

Chapter 4
Advanced Spin-preparation with light
In this chapter we discuss different approaches to obtain a deterministic spin preparation withlight. Almost all of them rely on the quadratic Stark shift of the magnetic sublevels inducedby laser light. The chapter is divided in three sections. In the first section, we introduce
the AC-Stark shift theory and the relative calculations are explained. The second section
describes the realization of a laser source adopted to implement the spin preparation. Finally,
the last section explains all the different implementation schemes of the laser source.

4.1

AC-Stark shift

When an atom interacts with light, the electric field E induces an electric dipole moment P
on the atom. Furthermore, if the electric field oscillates at a frequency close with respect
to an atomic transition, it induces an effective modification of the atomic levels involvedin the transition, e.g. energy shift and broadening or narrowing of the spectral line. The
induced shift of the atomic line can be calculated using the Stark shift theory, which will be
introduced in this section. Using such a theory, we also carry out several calculations using
an appropriate light wavelength. Section 4.1.2 shows the relative results.

4.1.1

Stark shift: theory

In the case of spherically symmetric atoms, such as alkali atoms, the polarizability is definedas the ratio between the induced dipole moment P and the electric field E, which induces the
dipole moment
P = αE,(4.1)

45

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

46

where α is the complex scalar polarizability. It is possible to calculate from the polarizability
the interaction potential energy and the scattering rates [Gri00]:
1
1
Udip (r) = − ⟨pE⟩ = −ℜ(α)I(r),22ε0 cΓsc (r) =

(4.2)

Pabs
1
=
ℑ(α)I(r).
h̄ωh̄ε0 c(4.3)

The complex polarizability is usually calculated via second-order perturbation theory [Lan72]
in the limit of far-detuned frequency with respect to any atomic transition. This limit requiresthat the detuning between the laser frequency and the atomic transition has to be much larger
than the linewidth. This means that the detuning has to be also large enough to avoid that the
AC-Stark shift energy scale becomes comparable with the fine structure splitting1 . Under
these conditions, specifying a state with |β J⟩ where J is the total angular momentum and βsummaries all other quantum numbers, it is possible to obtain the complex polarizability after
defining for all the atomic levels a complex energy E|β ′ J ′ ⟩ . The real part ℜ(E|β ′ J ′ ⟩ ) represents
the energy difference from the ground state, whereas the imaginary part ℑ(E|β ′ J ′ ⟩ ) is related
to the inverse lifetime of the excited level γβ ′ J ′ [Lep14]
E|β ′ J ′ > = Eβ ′ J ′ − Eβ J − ih̄

γβ ′ J ′
.
2

(4.4)

Following [Li17; Lep14] this gives:1α=3(2J + 1) β∑′J′|⟨β ′ J ′ |d|β J⟩|2 |⟨β ′ J ′ |d|β J⟩|2+
E|β ′ J ′ ⟩ − h̄ω
E|β ′ J ′ ⟩ + h̄ω!
.

where |⟨β ′ J ′ | |d| |β J⟩| are the reduced transition dipole moments. Assuming Eβ ′ J ′ + h̄ω ≫h̄γβ ′ J ′ /2 and Eβ ′ J ′ − h̄ω ≪ h̄γβ ′ J ′ /2 it is then straightforward to isolate the real and imaginarypart:
Eβ ′ J ′ − Eβ J ⟨β ′ J ′ ∥ d ∥β J⟩22ℜ[α(ω)] =
,
(4.5)
2
3(2J + 1) β∑′J′
Eβ ′ J ′ − Eβ J − h̄2 ω 2
2Eβ ′ J ′ − Eβ J + h̄2 ω 2
12′ ′ℑ[α(ω)] =hi2 h̄γβ ′ J ′ β J d ∥β J⟩ .
∑
3(2J + 1) β ′ J ′2Eβ ′ J ′ − Eβ J − h̄2 ω 2(4.6)

In contrast, the response of non-spherically symmetric atoms, such as erbium, to an electromagnetic field is more complex. The reasons for this are essentially two and both related to
1 For fermions hyperfine splitting.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

47

the electronic configuration. First, lanthanides, as already underlined in the Chapter 1, have acomplex energy spectrum which has to be taken into account with a sum over states formulaover all the dipole allowed transitions. Second, Er atoms can be seen as an anisotropic
medium, where the asymmetry comes from the highly anisotropic electronic wavefunctionof the 4f-electrons. Due to these reasons, the polarizability in these cases is a tensor:
αxx αxy αxz
α = αyx αyy αyz 
αzx αzy αzz
The result is a polarization and a magnetic-sublevel dependent polarizability. For convenience
the polarizability tensor matrix is usually decomposed in three parts: αscal represents the
diagonal elements and hence describes the response in the same direction of the appliedfield, αvect represents the anti-symmetric part of the off-diagonal elements and together withthe symmetric part of the off-diagonal elements αtens , describes the response in different
directions with respect to the applied field. The interaction potential energy equals [Vex11]:

MJ
1
ell
I(r) ℜ[αscal (ω)] + A cos θk ℜ[αvect (ω)] +UmJ (r; θ p , θk , A; ω) = −
2ε0 c
2J

2
2
3MJ − J(J + 1) 3 cos θ p − 1
×ℜ[αtens (ω)] ,(4.7)+
J(2J + 1)
2
where A = u × u∗ , u is the polarization vector, mJ are the magnetic sublevels, θk is the angle
between the direction of propagation of the light beam with the quantization axis, definedby the direction of the magnetic field for dipolar atoms, and θ p is the angle between thepolarization vector and the quantization axis. The angles are displayed in Fig. 4.1, where themagnetic field is oriented in the z direction. The decomposition of the polarizability tensor isgiven by:
2ωJJ ′ |⟨J| |d| |J ′ ⟩|2
αscal (ω) =,(4.8)
2 − ω 2)3h̄(2J + 1)(ωJJ′
s
J+J ′ +1

αvect (ω) = (−1)

sαtens (ω) = (−1)J+J′

6J(J + 1)(2J + 1)()
1 1 1 ωJJ ′ |⟨J| |d| |J ′ ⟩|2,2 − ω 2)h̄(ωJJJ J J′′()
1 1 2 ωJJ ′ |⟨J| |d| |J ′ ⟩|2
40J(2J − 1),
2 − ω 2)3(J + 1)(2J + 1)(2J + 3) J J J ′h̄(ωJJ′(4.9)(4.10)

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

B

48

x

z

y

θk

θp

~
E

Figure 4.1: Representation of the angle notation used to define the polarizability. The B field
is along z and defines the quantization axis. The light is linearly polarized and it propagatesalong x. θk is the angle between the direction of propagation of the light and the quantization
axis, whereas θ p is the angle between the polarization vector and the quantization axis.
where the curly brackets are used to indicate the Wigner 6J-symbols.
The result of Eq. 4.7 points out that the anisotropy of the polarizability in erbium atoms candisappear for a proper choice of the angle. In particular the vectorial part is always zero forlinear polarization (since u is a real vector and hence A = 0) and the tensorial one choosing
the angle θ p = 54.7°. This definition is also consistent with alkali atoms where no quadraticmagnetic sublevel and polarization dependency is found since J=1/2 makes the tensorial
contribution always zero.
The photon-scattering rate is even more strongly anisotropic [Lep14]. It can be calculated
from
n
1
mJ
ΓmJ (r; θ p , θk , A; ω) = −I(r) ℑ[αscal (ω)] + A cos θk ℑ[αvect (ω)] +
2ε0 c
2J

2
2
3mJ − J(J + 1) 3 cos θ p − 1+
×ℑ[αtens (ω)] .
(4.11)
J(2J + 1)
2
To obtain a spin preparation exploiting light, different methods can be implemented. As
already underlined in Chapter 2, the main difficulties to obtain a reproducible spin preparation
with the use of RF in erbium is due to the presence of a high number of states which become
all coupled together, for bosons. This problem can be overcome with fermions preparing thespin at field larger than 20 G through the quadratic Zeeman effect. Also preparing the spinstates with other methods as a Landau-Zener sweep or using excitation via excited states,

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

49

Table 4.1: Summary of the characteristics of the chosen transition. It is worth noticing thatthe gJ value of the excited state is different with respect to the one of the ground state.
Wavelength(nm)

Linewidth(kHz)

gJ valueΓ(s−1 )

631.04

11.6

1.07

8.587×104

as for Raman scheme or Stimulated Raman Adiabatic Passage (STIRAP) will not provide
a pure preparation if particular care is not taken in the choice of the used wavelength. In
fact, the main idea for a light scheme relies on the possibility of removing the degeneracy
induced by the coupling between the different spin states. This is feasible due to the quadratic
dependence of the magnetic sublevels of the tensorial term that is present in the AC-Stark
shift of Eq. (4.7). Hence, it is crucial to choose a wavelength for which the tensorial partis suitable to obtain enough detuning when the coupling is switched on. In addition, it isnecessary that the scattering rate is sufficiently low to avoid heating of the system. For thesereasons and for others that will be clear following the discussion, the transition at 631 nm
of erbium was the one finally chosen. The main characteristics of the chosen transition arelisted in Table 4.1.

4.1.2

Ac-Stark shift: calculation at 631 nm

To compute the total Ac-Stark shift we use the formalism of the previous section 4.1.1. The
following calculations are realized for the bosonic isotopes but similar result are obtained
also for the fermionic one. Using a sum over Eq. (4.7) for the 1284 lines, all the known
dipole-allowed transitions of erbium were considered. In addition, when possible, thetheoretical matrix elements were replaced with experimentally measured values as for the
631-nm transition [Ban05]. Figure 4.2 shows the calculated total energy shift for different
light polarizations as a function of the detuning from the transition. For simplicity, in the
figure we show only the four lowest spin states. Below each plot a schematic diagram ofthe shift is shown for a fixed positive detuning. Among the three light polarizations, theσ − gives the largest AC-Stark shift. On the other hand, for a deterministic spin preparation
we are interested in the differential light shift between adjacent spin states. Thus, for thethree light polarizations, we need to compare the difference between two adjacent spin states∆EmJ − ∆EmJ +1 with the following two ∆EmJ +1 − ∆EmJ +2 . Figure 4.3 shows the resulting
differential light shift as a function of the light detuning. Finally, the quadratic effect on the
different magnetic sublevels, coming from the tensorial term, is directly visible. Table 4.2
summarizes the results for a fixed detuning of 2 GHz. In particular, it is possible to notice

-200

-150

-4

-5

ΔE

0

-4

ΔE

Detuning (GHz)mJ= -5-2

mJ= -4

2

4

6

B=0

mj=-6
mj=-5
mj=-4
mj=-3

ac-Stark shift (kHz)
-6

ΔE

-200

-150

-100

mJ= -6

-6

-50

0

50

100

150

200

-4

-5

ΔE

mj=-6
mj=-5
mj=-4
mj=-3

0

-4

ΔE

Detuning (GHz)mJ= -5-2πmJ= -4

2

4

B=0

6

-200

-150

-4

-5

ΔE

mj=-6
mj=-5
mj=-4
mj=-3

mJ= -6

-6

-50

0

50

100

150

200

-100

-6

ΔE

ac-Stark shift (kHz)

0

-4

ΔE

Detuning (GHz)mJ= -5-2σ−2mJ= -44

B=0

6

Figure 4.2: Total light shift for σ + π σ − polarization as a function of the detuning for an interval of 12 GHz around the resonance.
For this calculations the intensity is set to 1 W/mm2 and only the first four magnetic sublevels are plotted. Below each plot a schematic
diagram of the shift of the levels is depicted in red. Without the light at a magnetic field B = 0 all the states (blue) are degenerate.

mJ= -6

-6

-50

0

50

100

150

-100

-6

ΔE

ac-Stark shift (kHz)

200

σ+

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT
50

0

Detuning (GHz)

2

4

6

-4

-2

0

Detuning (GHz)

2

4

6

-8
-10
-6

-10
-6

-10
-6

-4

-2

0

2

4

6

8

-6

-2

(6-5)-(5-4)

10

-8

π

-8

-4

-2

0

2

4

6

8

10

-6

-4

(6-5)-(5-4)

ac-Stark shift (kHz)

-6

-4

-2

0

2

4

6

8

σ-

ac-Stark shift (kHz)

-4

-2

0

Detuning (GHz)

σ+

2

4

(6-5)-(5-4)

6

Figure 4.3: Differential shift ∆EmJ =−6 − ∆EmJ =−5 − ∆EmJ =−5 − ∆EmJ =−4 obtained with an intensity I = 1 W/mm2 is plotted versus thedetuning. It is worth to notice the similar behavior with respect to the total AC-stark shift of Fig. 4.2. The quadratic effect is directly
visible. The π polarized light is the one in which the biggest quadratic effect is obtained.

ac-Stark shift (kHz)10

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT
51

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

52

that the quadratic effect is not constant over the detuning range analyzed, but assumes almost
the same shape of the total shift. Considering Eq. (4.8) and Eq. (4.9), the ratio between
the tensorial term and the scalar one, for this transition, is almost constant and equal toαtens /αscalar = 0.3. To notice, this ratio results particularly big in this narrow linewidthtransition. For dysprosium a ratio of the same order was also experimentally confirmed forthe same type of transition at 741 nm [Kao17].
Table 4.2: Summary of the differential AC−Stark shift for different polarizations. The πpolarized light gives the absolute bigger value. The data were calculating at a detuning of 2
GHz from the resonance, with an intensity I = 1 W/mm2 .
δ = 2 GHz; I = 1 W/mm2
Differential AC−Stark shift(kHz)

σ+
-4

π
7

σ−
-4

Even if larger quadratic effects can be achieved from the simulation going towards theresonance, one should always take care of two different effects. First, the calculations are
only valid in far-detuned condition δ ≫ Γ, which for the 631-nm light requires δ ≫ 116 kHz.
Second, the AC-Stark shift is only half of the story, it is important to look at the scattering
rates to avoid that an excessive scattering overheat the system out of the degeneracy condition.
Figure 4.4 shows the scattering rates, obtained with an intensity of 1 W/mm2 from the sum of
all the known states of Eq. (4.11), as a function of the detuning and for different polarizations.
It is possible to notice the different scaling with respect to the AC-Stark shift. The potential
and the AC-Stark shift scales as I/∆, whereas the scattering rates as I/∆2 . Therefore,
increasing the intensity and the detuning will lead to the desired shift with lower scatteringand hence, lower heating.
Beside the beneficial narrow-linewidth character of the 631-nm atomic line, the gJ factorof the excited state is 7% smaller with respect to the one of the ground state. This, according
to Eq. (1.17), results in a different slope of the magnetic sublevels of this excited state withrespect to those of the ground state, allowing several possibilities for the spin preparation, as
explained in section 4.3.

4.2

Realization of a laser source at 631 nm

As described above, the 631-nm line has different advantages from the atomic prospective.
Not so lucky is instead the commercial side. The solutions available are usually expensive or
have low output power. The main reason is that finding a diode lasing at this wavelength isnot straightforward and often very extreme temperatures are required to reach the desired
wavelength. For this reason, and in addition due to the fact that we luckily found a diode

0
-6

-4

-2

0

Detuning (GHz)

2

4

6

0
-6
-4

-2

0

Detuning (GHz)

2

4

6

0
-6

50

150

200

250

300

350

400

450

500

100

mj=-6
mj=-5
mj=-4
mj=-3

50

π

50

150

200

250

300

350

400

450

500

100

mj=-6
mj=-5
mj=-4
mj=-3

Scattering Rate (1/s)

100

150

200

250

300

350

400

450

σ-

Scattering Rate (1/s)-4-2

Detuning (GHz)

0

σ+

2

4

mj=-6
mj=-5
mj=-4
mj=-3

6

Figure 4.4: The scattering rates obtained from the calculation using an intensity of 1 W/mm2 is plotted as a function of the detuning for
an interval of 12 GHz around the resonance. From the left to the right the result for different polarizations σ − π σ + is shown. It is
possible to notice that the photon-scattering rates is strongly dependent on the polarization and that the use of σ + light induces lessheating.

Scattering Rate (1/s)

500

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT
53

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

Grating
Screws torotate theflexureCylindricalflexureDiodeholder54

MirrorSpace foranadditional
cavityScrews forverticalalignment

Figure 4.5: The laser design used for realizing the source emitting at 631 nm. Four peltierelements on the basement are used to thermally stabilize the whole body. An additionalpeltier is used to stabilize the temperature of the diode housing. This is a Littrow enhanced
configuration. The rotating cylindrical part depicted in blue allows a simultaneous rotationof the grating and of the mirror. The resulting output beam has a fixed direction parallel tothe one coming from the diode. Design from [Kir15]
lasing at the right wavelength, we decided to realize the laser source. In this section theexperimental setup and the development of the laser source are discussed. First, the chosen
design is briefly discussed. Second, the diode laser is presented and characterized. Finally,
the optical setup used in the experiment is shown and explained in detail.

4.2.1

Design of the laser system

Diode lasers are generally compact light sources that, using as external cavity with opticalretroreflection, can easily achieve linewidths in the range of 200 kHz [Rie04]. The designused in this thesis is a compact, robust Littrow-type external cavity diode laser (ECDL),
already tested and discussed in [Kir15]. Here we briefly summarize the characteristics.
The design is illustrated in Fig. 4.5. It is often referred as enhanced Littrow configuration.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

55

output beam
Mirror

Laser

Lens

Laser

Lens

grating(a) Littrow configuration

Grating(b) Littman–Metcalf configuration

MirrorLaser

Fixed output beam

Fixed output beam

Lens

Grating(c) Littrow enhanced configuration

Figure 4.6: Tunable external-cavity diode lasers in (a) Littrow, (b) Littman–Metcalf and (c)Littrow enhanced configuration. In the Littrow configuration the selection of the wavelengthis obtained rotating the grating. This rotation causes a variation in the direction of the beam.
In the Littman–Metcalf configuration a fixed output beam is obtained using an additional
mirror that is rotated to select the wavelength. The fixed output beam is obtained from thesecond order diffracted beam from the grating. In the Littrow enhanced configuration, thegrating and mirror, parallel to each others, are rotated simultaneously. The resulting fixed
beam is parallel to the beam coming from the laser and is due the 0 order of the grating.
In a standard Littrow, a grating is used as wavelength-selective optical element to tune
the light wavelength, retroreflecting a portion of the laser beam into the diode chip itself.
The configuration, which includes diode, lens and grating, as standard Littrow, has themain disadvantage that the beam direction changes when the grating is rotated. Having
a fixed direction of the output beam is often convenient for many applications. For thisreason, other configurations were developed as the Littman-Metcalf or the Littrow enhanced.
For comparison the three configurations are illustrated in Fig. 4.6. In the Littman-Metcalf
configuration the first order beam from the grating is not directly retro-reflected on thediode chip but it is reflected on an additional mirror. This mirror reflects the beam again
on the grating and finally back to the chip. By rotating the mirror it is possible to tune thewavelength. The second order of the grating is a fixed beam that can be used for the opticalsetup. This configuration tends also to offer a smaller linewidth due to the double use of thegrating. The main drawback is that the zero-order reflection of the beam is lost, resulting
in a lower power output compared to the simple Littrow. Finally, the Littrow-enhanced

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

56

configuration allows to obtain an almost fixed beam with the same power of the simple
Littrow configuration, combining the main advantages of the other two configurations. Thisis realized with the simultaneous rotation of the grating and of an additional mirror. The latter
is usually positioned parallel to the grating to additionally obtain a beam traveling parallelto the original one coming from the diode. In this configuration, when the system gratingplus mirror is rotated, the beam is slightly laterally displaced on the mirror by a quantity∆x ≃ 2L∆θ , where L is the distance between mirror and grating and ∆θ is the grating angle.
For a tuning range of 1 GHz, typical for tuning through an atomic resonance of different
isotopes in Er, the displacement is on the order of 100 nm for L ≃ 1 cm. This is negligible forour experimental uses, in which the light will be fiber coupled before aligned on the atoms.
The whole body of the laser is made of aluminum and thermally stabilized by 4 peltiers wired
in series. Passive stability and acoustic immunity were obtained using a CAD program whereelastic properties could be simulated.

4.2.2

Diode laser characterization

The diode chip2 is supplied with an home-made low noise current driver and it is mounted
in a commercial collimation package provide by Thorlabs (LT230P-A). The collimation
package is covered by a thin copper foil and inserted in an alluminum diode holder isolated
from the whole body by a plastic insert. One side of a peltier element is glued with highthermo-conductive glue on the top of the diode holder and the other side of the peltier isglued to a copper wire which is connected to the main body. The overall system, copper
wire plus main body, serves as a heatsink for the peltier. Two temperature-sensing elements3
are used to thermally stabilize with an home-made proportional-integral (PI) control loop
feedback. A PI continuously calculates an error signal as the difference between a set value
and a measured one and applies a correction proportional to the difference and takes into
account the previous value with the integrative part. The integrative part is not necessarily
zero when the error signal is zero. This makes it possible to maintain the error signal zeroeven when the action of the proportional part would be zero. In this way, the current flowing
on the peltier is instantaneously chosen from the PI to make the error signal zero and tomaintain this value over time.
Once the temperature is correctly stabilized, we characterize the diode laser. Figure 4.7
displays the output power as a function of the injection current for the free running diode at a2 HL63163DG is an AlGaInP from USHIO with a maximum optical output power of 100 mW.
3 The two elements are made of semiconductor that displays large increase of resistance in proportion tosmall diminution in temperature. For this reason are usually referred as Negative Temperature coefficient(NTC).

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

57

Figure 4.7: Power output versus injection current of the diode. The measurements set is
obtained at a temperature of 20°C.
fixed temperature. The threshold current is at 60 mA and the laser provides maximum powerof 100 mW at 158 mA. For a fixed current (80 mA), the power of the emitted light decreases
when the temperature of the diode is increased4 , see Fig. 4.8.
To measure the wavelength, we coupled the light into a fiber, which it is then connected
to a commercial wavemeter from the company High Finesse. To investigate the wavelength
tunability, the temperature of the diode holder was stabilized to different values. Figure 4.9
reports the induced variation of the wavelength. Changing the temperature of the diodetunes the frequency of the laser for two main reasons. First, the gain curve depends on the
wavelength. Second, the optical path length of the internal cavity of the diode changes withthe temperature. The three jumps in the plot indicate hopping from one longitudinal mode to
the next mode (≃ 0.4 nm) induced by a shift of the gain curve. The erbium line at 631.04 nm
can be obtained also in free running at a temperature of 24.8°C.
For a fixed temperature, we also investigate the current dependence of the wavelength,
since the current affects again the temperature of the diode and the carrier density (causinga small change in the index of refraction of the material). This is shown in Fig. 4.10. It is
worth noticing that the resulting tuning curve looks almost equal with respect to the previous
one. The main effect is induced by the temperature variations.
After characterizing the free running laser, we aligned the retroreflection from the gratingof the Littrow configuration. To obtain a good alignment the current of diode was set near
4 This effect is induced by the increase of not radiative process, i.e. thermal electron-hole recombination.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

58

Figure 4.8: Power output versus temperature of the diode. The measurement set is obtained
at a fixed current of 20 mA.

Figure 4.9: Laser output wavelength versus temperature of the diode holder. The current
is fixed at 80 mA during the measurements. The continuous tuning behavior indicates thevariation of the optical length of the cavity where the mode emitted longitudinally is notchanged. Three mode jumps are also present indicating that the peak of the gain medium has
shifted so far that the laser jumps to another mode. It is possible to notice that the erbium
line is in the free running regime of the diode at a temperature of 24.8°C.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

59

Figure 4.10: The wavelength reported from the wavemeter is registered for different currents.
The current was always chosen to achieve a stable single mode emission. It is possible tonotice the presence of the typical mode jump of the free running diode.
the threshold, since the laser power is more sensitive to optical feedback. The brightest spot
ensures that the feedback is correctly retro-reflected. The alignment can then be optimized
maximizing the output power. In our case the feedback is roughly 20% and this induces apower around 3-5 times higher than the power of the free running laser when the feedback is
well aligned and the current is at the threshold. The wavelength range, which can be achieved
rotating the cylindrical part, is (628 − 633) nm. The angle of the grating was adjusted toobtain approximately the Er line wavelength. The fine adjustment can be realized applying
a voltage to the piezoelectric elements. The turn on characteristic (output power versus
injection current) in the ECDL was repeated and the result is compared to the free runningcase in Fig. 4.11.
It is possible to notice that the threshold is 3 mA lower with respect to the free runninglaser as a result of the external optical feedback. Choosing the "right" amount of feedback
fraction depends on the specific diode. It is important to verify that the lasing frequency in
the ECDL is fixed by the angle of the grating. To check this, the wavelength was registeredfor different temperatures, repeating the measurements of Fig. 4.9 but in the external cavity
configuration. Fig. 4.12 shows the correspondent measurements. It is possible to notice thatthe same variation of temperature (≃ 5 ◦C) results in a total variation of 0.1 nm while, in thefree running configuration, is of the 1 nm order. The diode is now forced to operate at the
optical feedback wavelength [Lau02].

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

60

Figure 4.11: Power output versus injection current of the diode. The black points indicate thedata of the free running diode, whereas the data in red are for the ECDL where the grating
provides a feedback in the diode. Both measurements set are obtained at a temperature of
20°C. It is possible to notice the typical shift of the threshold, expected in the ECDL, fromthe relative zoom in green.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

61

Figure 4.12: The wavelength is measured for different currents. It is possible to notice thatthe wavelength variation is much smaller with respect to the free runnning case.

4.2.3

Experimental setup

After testing the ECDL, we build the optical setup, as illustrated in Fig. 4.13. This setup
has three main sections. First, a prism pair reduces the astigmatism of the beam then, twocommercial faraday isolators5 give an high isolation (-35/dB each ) and allow a stable useof the laser6 . Using a combination of three polarizing beam splitters and a λ /2 wave plate
it is possible to select the power of the three sections. To achieve a long-term stability thelaser is locked to an ultra low expansion (ULE) cavity through the Pound Drever Hall (PDH)
technique. The sidebands are generated through a fiber coupled EOM7 . Details on the lock-in
system are in Appendix A. From the 0th order of an acusto optic modulator (AOM) we
realize the beam for the second section. The beam, which has a total power smaller than
200 µW, is fiber-coupled to check the wavelength via a commercial wavemeter. Finally, in
5 Thorlabs IO-3-633-LP.
6 The setup was initially tested using only one isolator. The incredible high sensitivity of the diode to theadditional optical feedback was making the use of it almost impossible. In particular during the scan of thepiezo the power of the laser was distributed in different longitudinal mode. This high response to reflection ismainly due the low finesse and the small length of the internal cavity. As a result, the number of photons incavity is low so that the laser frequency can be easily perturbed. As discussed in [NIS98] "this unique feature
can be both a blessing and a curse". In our case after adding a second isolator the laser could be stably used.
7 Jenoptick PM 635.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

62

PBS
4

ULE Cavity

PDrefl

Laser

Prism
Pair

Faraday
Isolator

Lock to the
ULE cavity

2
Experiment

PBS
PBS
2

PBS

Wavemeter
4
L

Wavelength
check

Experiment

Fibers fortheexperiment

Figure 4.13: The optical setup is divided in three sections for the three different purposes.
The power is divided between the sections using two half wave plates and three polarizing
beam splitters ( λ2 and PBS respectively in the picture). The section underlined in blue isused to lock the wavelength of the laser to an ultralow-expansion cavity. This is achieved
using a fiber coupled electro-optic modulator (EOM in the picture) to generate sidebandson the laser. The reflection from the cavity is sent to a photodetector (PDrefl). The section
underlined in yellow is used to check wavelength of the laser using a commercial wavemeter.
The 0th order of an AOM is fiber coupled and sent to the wavemeter. Finally, the section
underlined in green is used to make two beams ready for the experiment. In particular two
AOMs are used to stabilize the intensity of the two beams and to realize a tunable frequency
shift between them.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

63

Figure 4.14: The overall efficency, diffraction efficency of the AOM plus fiber coupling
efficiency is plotted versus the driving frequency of the AOM. It is possible to notice thatusing a double pass an increase of the bandwidth is obtained. The double pass beam is infact not displaced when the frequency is changed. Since the fiber coupling remain almost thesame the shape is mainly caused by the bandwidth of the AOM.
the last section two AOMs prepares the beams for the experiment. In one AOM, the +1
order, shifted in frequency of 110 MHz, is fiber coupled and sent to the experiment while,for the other AOM a double pass is realized using the -1 order. In this way two beams witha frequency difference of 3 times the AOM modulation frequency are obtained. Fig. 4.14
shows the frequency tunability of the double pass and of the single pass, where the overall
efficiency, given by the diffraction efficiency of the AOM plus the fiber coupling efficiency, isconsidered. The FWHM spectral bandwidth is of 68 MHz for the double pass and of 8 MHz
for the single pass it is possible to tune the frequency difference between the two experimental
beams of 78 MHz. The role of the two AOMs is not only to allow for a controllable frequency
shift on the MHz region but also to stabilize in intensity the beam. Dynamically changing theradio frequency power, which drives the AOM, it is in fact possible to change the fraction of
the beam that is diffracted in the first order. The intensity seen by the atoms can be stabilized
sampling a portion of the light sent to the experiment on a photodiode. The photodiode gives
a voltage output proportional to the intensity of the beam. Using a reference voltage it ispossible to obtain an error signal. A PI system is then used in combination with a voltage
attenuator to change the RF power that drives the modulation.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT|b⟩

ω1 σ +64|b⟩

ω2 σ +

ω2 π

ω1 σ +

|d⟩
|c⟩
|a⟩

|c⟩
|a⟩

(a) t=0(b) t=0

Figure 4.15: Optical pumping and depumping scheme. On the left a scheme to populate an
adjacent spin state is shown. It consists of two resonant laser beams with σ + and π polarized
light. On the right side it is shown the effect of two σ + polarized frequencies. It is possible
in this way to skip the following spin state.

4.3

Overview on spin preparation/manipulation with light

The setup explained in the previous section was developed for different implementation
schemes. As the quadratic Zeeman effect, also the quadratic light shift effect removes thedegeneracy between the magnetic sublevels. This allows us to use a simple RF pulse or an
adiabatic passage sweep to prepare a spin mixture, as we do for fermions, also for the bosonicisotopes. In addition, different schemes involving a higher excited state can be applied. In
the following an overview on the variety of existing techniques to produce a spin preparation
is given. We focus on the bosonic case. The techniques introduced are three: an opticalpumping scheme, a Raman transition and a stimulated Raman adiabatic passage. They areillustrated using the formalism of the bosonic isotopes since in this case it is not possible toexploit the quadratic Zeeman effect. Finally, it is shown that exploiting the light it is also
possible to manipulate the energy of the single spin state.

4.3.1

Optical pumping scheme

The optical pumping scheme relies on the use of resonant light combined with a magnetic
bias field. The transition at 631 nm is a transition from J = 6 to a J = 7 state. This meansthat in the excited state 15 Zeeman sublevels are present and that all types of polarization candrive a transition to the excited state. This method depends strongly on the characteristic of
the transition and in particular it is reliable thanks to the following two different factors: thedifferent Landé g-factor of the excited state with respect to the ground state and the narrow
linewidth of the excited state. Thus the resulting Zeeman splitting of the magnetic sublevels

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

65

of the ground state is different with respect to the magnetic sublevels of the excited state. The
idea is shown in Fig. 4.15. Using one beam π polarized, atoms can be pumped from the initialstates (J, mJ ) = (6, −6) to the excited state (J, mJ ) = (7, −6) . Using also a σ + polarized
beam, there are two different processes that can lead to a relaxation. Spontaneous emission
from the excited state (J, mJ ) = (7, −6) can lead to a population of the (mJ ) = (−6, −5)magnetic sublevels of the ground state, with a different branching ratio. This process isslow in our case due to the long lifetime of the excited state, thus can be neglected. The
other process is the stimulated emission as a result of the coupling induced by σ + light.
This process transfers the population from (J, mJ ) = (7, −6) to the state (J, mJ ) = (6, −5).
The atoms are now not anymore transferred by the radiation. In fact the light results notresonant for the coupling of (J, mJ ) = (6, −5) and (J, mJ ) = (7, −5), due to the different
slope of equation (1.17) induced by the different Landè g-factor. In addition the quadratic
spin dependent light shift, resulting from the use of resonant light, helps in removing thedegeneracy. In addition as depicted in Fig. 4.15 (b) it is possible to obtain as final state J = 6,mJ = −4 using the two resonant frequencies σ + polarized. Due to the finite linewidth of
the 631-nm line, the degeneracy is removed only at a magnetic field of 10 mG. From thisfield the detuning ∆ = µB (g′J − gJ )B is much greater than the linewidth of the transition Γ8 .
In the ERBIUM experiment a reliable spin preparation using this method need particularcare because of the heating rate induced by the use of resonant light and because of the laser
frequency noise. In fact only with a B field of 10 G the different resonances are separated by
1 MHz. This means that both the long term stability and the linewidth of the laser have to be
below 100 kHz.

4.3.2

Raman scheme

The Raman scheme relies on the use of an off-resonant light to populate the target state.
Figure 4.16 shows a schematic diagram of two different schemes. Two laser beams are
simultaneously used to couple the initial state to the final state. The frequency difference
between the two beams has to be equal to the magnetic sublevels separation which is now
given by the Zeeman splitting and the differential light shift. Once again, to avoid couplingwith all the spin states, the beam detuning (∆1 ) has to be chosen to achieve a sufficiently
large quadratic Stark shift and low heating. In addition it is important that the frequency ofthe beams is stable to avoid a change in the quadratic effect. Furthermore, the frequency
difference of the beams has to be stable, below the differential quadratic shift induced. These
conditions, using the setup in Fig. 4.13, are easily obtained since the laser beam is frequency
8 If the laser linewidth is greater than the atomic linewidth it is necessary to consider the laser linewidth as Γ.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

ω1

|b⟩

|b⟩

∆1

∆1

ω2 π

σ+

ω1

66

ω2 σ +

σ+

|d⟩
|c⟩
|a⟩

|c⟩
|a⟩

(a)

(b)

Figure 4.16: Two different Raman schemes. On the left, a scheme to populate an adjacentspin state is shown. It consists of two off resonant laser beams with σ + and π polarized light.
The detuning ∆1 has to be carefully chosen to obtain the desired quadratic effect. On theright side it is shown the effect of two σ + polarized frequencies. It is possible in this way toskip the following spin state.
locked to a ULE cavity and the frequency difference of the beams, since generated by thesame laser, is only caused by the modulation RF the drives the AOMs9 . In the case (a) ofFig. 4.16, one of the beams is σ + polarized, whereas the second is π polarized. In this case
it is possible to transfer the population from the mJ = −6 to the mJ = −5. In the case (b),both beams are σ + polarized10 and the population can be transferred from the mJ = −6 tothe mJ = −4.

4.3.3

Stimulated raman adiabatic passage

Another technique to transfer population between the magnetic sublevels is the stimulated
Raman adiabatic passage (STIRAP). In its simplest version, this technique allows a coherent
transfer of the population between two quantum states using an intermediate state. The threestates are coupled together by two radiation fields switched on with a temporal delay. The useof a second optical field introduces the possibilities of exploiting the quantum interference inamplitude of the transition. Using a counterintuitive order for the time dependence of theintensity of the two beams an adiabatic transfer between the two states is possible. The main
benefits of STIRAP are two. First, the spontaneous emission from the usually radiatively
9 The frequency that drives the AOM is generated trough two independent DDS that are automatically phaselocked since they share the same reference clock.
10 In this case using a bias field that realize a splitting on the MHz range for adjacent spin states it is possible
to drive one AOM with both frequency removing the need of aligning two beam on the atoms.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

67

|b⟩

|b⟩

|b⟩

∆1

∆1

∆1

ω2

ω2
ω1

∆2
|a⟩

ω1

∆2

|c⟩
|a⟩

(a) t=0

∆2

|c⟩

|c⟩

|a⟩
(b) t=t1(c) t=tf

Figure 4.17: A Stirap scheme. In temporary order from the left to the right. Using thiscounterintuitive order for the time dependence of the intensity of the two beams an adiabatic
transfer between the two states |a⟩ and |c⟩ is obtained. The intensity of the laser pulses has adelayed Gaussian shape. In (a) only the laser that couples the final state and the intermediate
one is on. In the intermediate step both the lights reach the same intensity. In this step thetransfer of the population started. In the final step the laser that couples the initial state andthe intermediate one is gradually switched off. The best results are usually obtained in the
double resonance condition ∆1 = ∆2 = 0.
decaying intermediate state is prevented despite the fact that the radiative lifetime of theintermediate state can be several orders of magnitude shorter than the length of the laser
pulses. Second, as in the case of the adiabatic passage, it is robust against small fluctuationsof the experimental conditions e.g magnetic field noise, intensity and frequency noise of
the laser. The idea is drawn in Fig. 4.17c. As starting point, the intermediate state of theRaman scheme is coupled to the final magnetic sublevel with a Gaussian-shape pulse labeledfrom now on as pump beam (P). This coupling will not drive any transition of the population,
which remains unchanged. These two states have to be now considered dressed by theradiation and as result these special Autler-Townes states do not interact with other opticalfields present in the system. Then, while reducing the intensity of the probe beam, a second
beam labeled Stokes (S), which couples the initial state to the intermediate one, is switchedgradually increasing the intensity. In the beginning this field will not drive any transition
to the two Autler-Townes states. Later, both the S and P laser beams are with comparable
intensities. This stage is usually referred as adiabatic passage phase [Vit01]. Since on
the two-photon resonance condition, one of the eigenstate of the Hamiltonian results in asuperposition of the only initial and final state, where the weight is given by the ratio of the
two Rabi frequencies involved, the population can be transferred in an adiabatic passage if
an adequate length and delay of the two Gaussian pulses is chosen in such a way that, toCHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT

68

conclude the transfer, only the probe laser beam results on and slowly switched off. Thiscounter intuitive coupling scheme is complicated by different factors in using Er. First,
this three level population transfer will result in a multistate chain STIRAP-like populationtransfer and the necessary condition to realize the transfer to the higher magnetic sublevels
is the existence of an eigenstate of the multilevel Hamiltonian that connects the initial state
to the final one of the chain. Multistate STIRAP, formed by the Zeeman sublevels of J=2
system, has been demonstrated experimentally [Vit01]. Finally the different Landé g-factorof the excited states removes the resonant condition for the STIRAP to higher spin states.
Whereas a transfer in the highest magnetic sublevels could be difficult using the 631-nm
light, the quadratic magnetic sublevels dependence of the Stark shift leads to a decouple ofhigher magnetic states and an effective three level STIRAP scheme could work to transfer the
population between the first two magnetic sublevels. In the setup the two intensity Gaussian
profiles needed for the pulse can be realized dynamically changing the RF power of theacousto-optic modulators.

4.3.4

Single spin state manipulation

One of the more interesting features is that the applied laser field induces light shifts ,which
depends on mJ , i.e. on the magnetic sublevels. The different Landé g-factors can beadditionally exploited to obtain a controllable shift of almost only one spin state in the ground
manifold. The intensity of the beam, which can be dynamically modified11 in the setup, canlead to a dynamical change of the energy of the coupled spin state. To achieve this effect the
wavelength of the laser beam is locked close to a resonance between one magnetic sublevelof the ground state and the respectively one of the excited state according to the polarizationof the laser beam. It is possible in this way to obtain a much bigger shift in the quasi-resonant
states with respect to the other magnetic sublevels. The obtained AC-Stark shift can becalculated taking care of the different Landé g-factors. The resulting shift using an intensityof the laser beam of 1 W/mm2 with a bias magnetic field of 30 G is plotted as a functionof the detuning of the laser beam and it is shown in Fig. 4.19. In this way it is possible toprevent the population of the coupled level. To obtain a control over a single spin state, therequirements are even more stringent with respect to the spin preparation. To have a stableenergy of the spin level working at 30 Gauss requires to have both long term stability and alinewidth of the laser below 50 kHz. Due to the high scattering rates of this quasi-resonant
light the intensity used and the time duration has to be carefully chosen. This techniquecould be useful to follow spin phase protocol as for example to realize stripe phase [Maz17]
11 Changing the RF power that drives the AOM.

CHAPTER 4. ADVANCED SPIN-PREPARATION WITH LIGHT|b′ ⟩|d ′ ⟩

|c′ ⟩
∆2

|a′ ⟩
∆1

69

ω1 σ +
ω1 σ +|d⟩ω1 σ +|c⟩|b⟩|a⟩Figure 4.18: Four magnetic sublevels for the ground and the excited state are displayed. Only
one spin state of the ground manifold is resonantly coupled. The two Autler-Townes states|c⟩ |d ′ ⟩ receive now a shift that is much bigger with respect to the one induced in the otherstates. It is possible to dynamically change the shift changing the intensity of the beam.

E mJ=-6
E mJ=-5100

E mJ=-4

AC-Stark shift (MHz)E mJ=-3

50

0

-50

-100

-70

-65

-60

-55

-50

-45

-40

Detuning (MHz)Figure 4.19: Total light shift for σ + polarization versus detuning for an interval of 8 MHz
around the resonance. It is possible to distinguish the different spin states resonances.

Conclusion
The aim of my master thesis was to developed an optical setup which allows to obtain adeterministic spin preparation in the context of the ERBIUM experiment and to start theinvestigation of the rich physics behind spinor quantum gases [Sta13].
In the ground state Er has a total angular momentum J=6 (F=19/2) giving rise to 13(20) different spin states for bosons (fermions). For bosons, the splitting induced by an
external magnetic field B is linear and hence, a RF pulse that couples two adjacent spin states
unavoidably couples all the spin states. These makes the spin preparation not straightforward.
In the contrary, in the case of fermions, the magnetic moment of the nucleus couples to themagnetic field produced by the electrons of the atom, resulting in a quadratic splitting of themagnetic sublevels with the external B field. In this way, a RF pulse can couple only twomagnetic sublevels. Working at magnetic fields higher than 20 G, we achieved a deterministic
spin preparation of the lowest two spin states using a RF pulse or a RF sweep. To optimize
the sweep parameters and to get further understanding we simulated the RF transition with an
Hamiltonian in the rotating frame approximation that qualitatively describes the experimental
results.
Additionally, once we understood how to obtain a stable spin mixture, we loaded theatoms in a deep three-dimensional lattice, where several unexplored phases are predicted tooccur with a spin mixture of highly magnetic atoms, e.g [Maz17]. In the ERBIUM experiment
the lattice is created retro-reflecting 2 laser beams operating at 532 nm and one operating at
1064 nm. In particular, two beams coming from the 532-nm and one from the 1064-nm lightsources are retro-reflected to obtain the 3D array of lattice sites. The beams cross each others
orthogonally. In this lattice we investigated the interactions between the lowest spin states by
estimating the onsite interaction U. We showed that this quantity preserves the anisotropic
character of the dipole-dipole interactions of Er. We measured this value for two different
dipole orientation and find a good agreements with the theory. Additionally, as a step towards
the predicted phases, we present a method to obtain a deterministic spin preparation and asingle spin state control, which exploits the tensorial ac-Stark shift and can be implemented
for both fermionic and bosonic isotopes. For this aim, we developed an External Cavity
70Conclusion

71

Diode Laser (ECDL) source, emitting close to a narrow transition at 631 nm, together with
an optical setup that allows different schemes for spin manipulation. The laser was intensity
stabilized using the acusto-optic modulator technology and frequency stabilized using thePound-Drever-Hall technique. This study provides new elements to increase the knowledgeof our system and opens the door to investigate quantum magnetism and the stability ofordered magnetic phases in a well-controlled manner.

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Appendix A
Lock of the laser system
Ultra-stable laser sources are fundamental tools in ultracold atoms experiments. Even more
if the laser is adopted to realize a reliable and reproducible spin preparation. The idea of laser
locking is to use a stable optical cavity as a reference to reduce frequency fluctuations or
long-term drifts and hence to transfer the stability of the cavity to the laser. The requirements
is a tunable laser source, which allows to dynamically change the laser frequency in sucha way to match the resonant mode of the cavity. In the last years the necessity of stable
resonators for several research topics, e.g. atomic clock [Blo14], drove the developmentof optical cavities more and more accurate and stable [Kes12]. In our experiment, we use
an ultra-low expansion (ULE) cavity1 with optically-contacted mirrors [Rie12; Pat17] In
our case, and in general with diode laser in Littrow configuration, the tunability is usuallyachieved by varying the current or rotating the grating. In particular, the former is used toresponds to fast fluctuations (> 1 kHz) of the wavelength. The latter controls the angle of thegrating, as shown in figure tot, and respond to slow fluctuations (up to 1 kHz).
The locking scheme relies on the coupling between cavity and laser light. The light istransmitted by the cavity only if the laser frequency matches one of the Airy peaks [Sve10],in such a way that, for a Fabry-Perot cavity, two times the length of the cavity is a integer
multiple of the wavelength. If the laser operates off resonance with respect to the cavity
transmission peaks, the light is reflected out from the cavity. Thus, one can try to lockthe laser using the transmitted signal from the cavity and stabilize the laser. Since bothtransmission and reflection are symmetric with respect to the resonance, one needs to lockthe laser at one side of this resonance, where a change of intensity can be directly converted
to a frequency change. This was how lock was performed before the Pound Drever Hall
(PDH) technique. One of the disadvantageous of this technique is that power fluctuations are
not decoupled from frequency fluctuations. Indeed, if the laser power changes, the feedback
1 Made of a special glass the has low thermal expansion coefficient.

79

APPENDIX A. LOCK OF THE LASER SYSTEM

80

PBS

Lock to the
ULE cavity

4

ULE Cavity

PDrefl

Laser

Prism
Pair

Faraday
Isolator

AC amplifier

2

Mixer
Lowpass filter

RF generator

PBS

Current feedback

PDtrans2

PBS

Piezo feedback
AmplifierFigure A.1: Level occupation given in the equations (2.3) and (2.5) as function of theenergy ε for the Bose-Einstein (blue), Fermi-Dirac (red) statistics. For comparison also theMaxwell-Boltzmann (yellow) is included.
response is the same as if the laser frequency drifts out. Thus, to improve a lock realized withthis technique, the laser beam sent through the cavity has to be intensity stabilised [Tro78].
Futhermore, using the transmitted light increases the shot noise.
Today, the standard technique adopted to achieve a long-term stability is the PDH. In thistechnique one exploit the use of the reflected light, where the signal on the photodiode canbe reduced to zero and the resulting lock can achieve better performance. Since the reflectedsignal from the cavity is symmetric around the resonance, one should look at the derivative.
This is easily achieved by slightly modulating the phase or the frequency of the laser. The
variation of the reflection induced by the modulation tells us on which side of the resonancethe laser is sitting. Furthermore, it gives us the possibility to realize an error signal, which
can be easily sent to the laser electronics control as a feedback.
Figure A.1 shows the layout used in the lab, where the electronics adopted for the lockis included to fully understand the working principle of the laser locking. The EOM in the
picture works as a phase modulator. The signal coming from the reflection of the cavity is sent
to a photodetector through the combination of a λ /2 waveplate and a polarizing beam splitter.
The signal from the detector is then amplified and sent to a commercial mixer2 together withthe same signal that drives the EOM. This latter is generated from a DDS and splitted by acommercial splitter3 . The mixer multiplies both signals in such a way that the result, after an
2 Minicircuit
3 Minicircuit ZFSC-2-1W+

APPENDIX A. LOCK OF THE LASER SYSTEM

81

0.15

0.1

Voltage (V)

0.05

0

-0.05

-0.1

-0.15
-30

-20

-10

0

10

20

30

Frequency (MHz)Figure A.2: Level occupation given in the equations (2.3) and (2.5) as function of theenergy ε for the Bose-Einstein (blue), Fermi-Dirac (red) statistics. For comparison also theMaxwell-Boltzmann (yellow) is included.
additional low pass filter which removes the signal at twice the modulation frequency, is asignal whose value is zero on resonance since the reflection is zero and it changes sign on
the two sides of the resonance. This represents our error signal. Figure A.2 shows the error
signal obtained using a fast phase modulation4 . The error signal is additionally amplified
and sent to the PI system to generate the final feedback for the piezo and for the current.
Locking both, one finally obtains the desired result. The error signal gives also additional
informations regarding the linewidth of the cavity or of the laser. The error signal is fittedwith the following relation [Mar07]:Γ∆Ω sin(φ )Γ(Γ2 + ∆2 + Ω2 ) + cos(φ )Ω(Γ2 − ∆2 + Ω2 )S(∆) = A + B ·,(Γ2 + ∆2 ) (Γ2 + (∆ + Ω)2 ) (Γ2 + (∆ − Ω)2 )(A.1)

where A takes into account an eventually offset, B is the amplitude of the error signal, Γis the linewidth, ∆ is the detuning of the laser with respect to the cavity resonance, Ω isthe modulation frequency that drives the EOM, and φ takes into account not perfect phasematching between the reflected signal on the photodiode and the RF modulation sent to the
4 Although one could think that with high modulation frequency the reflection intensity would not follow themodulation and the technique would not work. One can easily demonstrate with a quantitative model [Bla98;
Nic13] that the not only high modulation frequency can be used but also that the performance of the lock arealso improved.

APPENDIX A. LOCK OF THE LASER SYSTEM

82

mixer. From the fit one can obtain an approximate estimation of the linewidth. This quantity
has to be referred to which one between the laser and the cavity has the largest linewidth.
From the fit the linewidth(Γ) results Γ = 199(4) kHz. The measured value is consistent with
our prediction on the linewidth of both laser [Pat83] and cavity [Rie12]. Thus, one canconclude that both the long-term stabilization and the narrowing of the linewidth can be
expected from the cavity and that the laser system is ready to be used in the experiment.

Acknowledgements
First, I would like to thank my supervisor Francesca Ferlaino. She gave me the possibility
to do the master thesis in the exciting environment of Innsbruck where I joined a fantastic
group in the ERBIUM experiment. I would like to thank her for the continuous support andthe time spent in correcting my thesis.
Thanks also to my supervisor in Pisa, Donatella Ciampini. During the course of atom
optics she shared with us her knowledge and she was able to transmit to me the passion for
the field of ultracold quantum gases.
Many thanks to Manfred J. Mark for the patience in answering the never-ending questions.
A big thanks goes to the ERBIUM team, Simon Baier, Jan Hendrik Becher, Lauriane Chomaz
and in particular to Daniel Petter who were always available to clarify my doubts during
the lab experience and for correcting my thesis. I would like to thanks them also for giving
me all the instruments to conclude my project and making the atmosphere in the group sofriendly and light. Thanks to Emil Kirilov for the always fruitful discussions and for realizing
such a nice laser design. I would like to thank Giulia Faraoni for the long conversation about
physics and not.
Thanks to Elisa and Lorenzo for organizing the industry production of burned home-madejam and limoncello. They were always available when I need to unplug the brain.
Finally, I would like to thanks Claudia for always support me and stand by my side and tobe the center of my frame of reference.

83