Metastable State of Ultracold and Quantum
Degenerate Ytterbium Atoms:
High-Resolution Spectroscopy and Cold Collisions
by

Atsushi Yamaguchi

UNI VE

KY O

T

ITY
RS

FO

KYOTO

U

ND

JAPAN

97

O

8
ED 1

A dissertation in the Physics department of the
Kyoto University

March, 2008

Abstract
The study of laser-cooled two-electron atoms is one of the most interesting research
fields in atomic physics. In particular, the unique characteristics of the metastable
3 P state of two-electron atoms have recently attracted attention both for their ap2
plications and study of their intrinsic characteristics.
In this thesis, I present experiments for studying new aspects of the metastable statesin two-electron atoms. First, unique collisional properties of metastable Yb[3 P2 ]atoms at ultralow temperatures are revealed in detail. In previous studies, evaporative cooling of metastable atoms in a magnetic trap was unsuccessful in forminga Bose-Einstein condensate (BEC) due to trap loss caused by strong spin-flip collision processes. We overcame this difficulty by employing, instead, an optical trap
in which atoms in every magnetic sublevel of the 3 P2 state can be trapped. We
successfully achieved a number density of 2 × 1013 cm−3 , which is larger than that
achieved in a previous study by an order of three [1], at a temperature of 2 µK
with a phase space density of 0.01. We also measured a large two-body inelastic
collision rate in a far-off-resonance trap, which we interpret as fine-structure changing collisions in this ultracold temperature regime. Although a recent experimentwith magnetically trapped Ca atoms studied multichannel collisions between 3 P2
atoms and discussed the possibility of the fine-structure changing process [2], we
believe that our study is the first definite experimental measurement of this process
between 3 P2 atoms.
The other important achievement in this study is the successful observation of theultranarrow magnetic quadrupole 1 S0 ↔3 P2 transition. We first developed a novel507-nm laser source. By tightly locking the laser frequency to a high-finesse external optical cavity, we stabilized the laser frequency and reduced the linewidth toless than 1 kHz. Using this laser source, we observed the 1 S0 ↔3 P2 transition in
Yb bosonic (174 Yb) and fermionic (171 Yb, 173 Yb) isotopes. High-resolution spectroscopy of ultracold atoms and BECs was performed using this ultranarrow transition. The frequency shift and broadening due to the mean field energy of a BEC
was observed. Furthermore, we determined the polarizabilities of all the magnetic
sublevels of the 3 P2 state. Using an optical frequency comb, we also measured
the optical frequency of the magnetic-field-insensitive 1 S0 ↔3 P2 (m = 0) transitionin 174 Yb, which is one of the candidates for next-generation time and frequency
standards [3].

CONTENTS

2

Contents
1 Introduction

5

2 Laser cooling and trapping of neutral Yb atoms
2.1 General properties of neutral Yb atoms . . . . . . . . . . . . . . . . . . . .
2.2 Laser cooling and imaging of Yb atoms . . . . . . . . . . . . . . . . . . . .
2.2.1 Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Evaporative cooling in a FORT . . . . . . . . . . . . . . . . . . . .
2.2.4 Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 399 nm (1 S0 ↔1 P1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 556 nm (1 S0 −3 P1 ) - MOT . . . . . . . . . . . . . . . . . . . . . .
2.4.3 404 nm (1 S0 ↔3 D2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 770 nm (3 P2 −3 S1 ) and 649 nm (3 P0 −3 S1 ) - Repumping . . . . .
2.4.5 532 nm - FORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
13
14
15
15
17
20
20
20
25
26
28
28

3 A diode laser system for spectroscopy of the ultranarrow magnetic
quadrupole transition in ytterbium atoms
30
3.1 External cavity with diffraction grating . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Littman vs Littrow . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2 ECLD at 1014 nm - Littman configuration . . . . . . . . . . . . . . 35
3.2 Tapered amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 PPLN crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 ULE cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Optical excitations
4.1 Multipole expansion of an electromagnetic field . . . . . . . . . . . . . . .
4.2 Matrix elements and selection rules for E1, E2, M1, and M2 transitions . .
4.2.1 Interaction between atoms and a laser field . . . . . . . . . . . . . .
4.2.2 E1 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 M1 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 E2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 M2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44
44
48
48
50
52
53
55

CONTENTS

4.3

3

Light shift – Far Off Resonance Trap (FORT) . . . . . . . . . . . . . . . . 56
4.3.1 Numerical estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Line shift and broadening
5.1 Doppler shift and recoil shift . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Temperature shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Collision shift and broadening . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Other shifts and broadenings . . . . . . . . . . . . . . . . . . . . . . . . . .

60
60
62
64
65

6 Inelastic collisions in optically trapped ultracold metastable ytterbium 67
6.1 Optical trapping of Yb atoms in the 3 P2 state . . . . . . . . . . . . . . . . 68
6.1.1 Preparation and detection of atoms in the 3 P2 state in the FORT . 68
6.1.2 Polarizabilities of the 3 D2 state . . . . . . . . . . . . . . . . . . . . 74
6.1.3 Magnetic field compensation . . . . . . . . . . . . . . . . . . . . . . 77
6.1.4 Optical trapping of atoms in all magnetic sublevels of the 3 P2 state
in a FORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1.5 Trap frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1.6 High density ultracold metastable atoms . . . . . . . . . . . . . . . 83
6.2 Collisional properties between metastable atoms . . . . . . . . . . . . . . . 85
6.2.1 Two body collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Elastic and inelastic collision rates under thermal equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.3 Fine structure changing collisions . . . . . . . . . . . . . . . . . . . 90
6.3 Effect of the black body radiation . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 Optical trapping of Yb atoms in the 3 P0 state . . . . . . . . . . . . . . . . 93
7 Observation and application of the ultranarrow 1 S0 −3 P2 transition95
7.1 Estimation of the resonance frequency . . . . . . . . . . . . . . . . . . . . 97
7.1.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.2 Frequency measurement of the 3 P2 −3 S1 (770 nm) and3
P0 −3 S1 (649 nm) transitions by an optical frequency comb . . . . 100
7.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Observation of the ultranarrow 1 S0 −3 P2 transition . . . . . . . . . . . . . 105
7.2.1 Fermions: 173 Yb [1 S0 (F =5/2) ↔ 3 P2 (F =5/2)]
and 171 Yb [1 S0 (F =1/2) ↔ 3 P2 (F =5/2)] . . . . . 105
174
7.2.2 Boson:
Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Selective excitation to all magnetic sublevels of the 3 P2 state in 174 Yb . . . 110
7.4 Measurement and control of ac polarizabilities . . . . . . . . . . . . . . . . 112
7.4.1 Control of the polarizability . . . . . . . . . . . . . . . . . . . . . . 115
7.5 Magnetic resonance imaging of ultracold atoms . . . . . . . . . . . . . . . 117
8 High-resolution spectroscopy of a BEC
119
8.1 Mean field energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 BEC in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

CONTENTS

8.3

4

BEC in 1D optical lattice potential . . . . . . . . . . . . . . . . . . . . . . 128

9 Frequency measurement of the clock transition in 174 Yb
133
9.1 FC 8003 – Menlo Systems GmbH . . . . . . . . . . . . . . . . . . . . . . . 134
9.1.1 frep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1.2 fCEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.1.3 Position of frep and fCEO . . . . . . . . . . . . . . . . . . . . . . . . 136
9.2 Experiment and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.2.1 Rb clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10 Summary and outlook
142
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A Calculation of the magic wavelength of the 3 P2 state

145

B Basic theories on an SHG ring cavity
146
B.1 Theory of an SHG technique . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B.1.1 Wave equation for a nonlinear medium . . . . . . . . . . . . . . . . 146
B.1.2 Phase matching condition . . . . . . . . . . . . . . . . . . . . . . . 148
B.1.3 CPM and NCPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.1.4 Conversion efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.1.5 LBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.2 Design of the Ring Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2.1 ABCD matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2.2 Examples of ABCD matrices . . . . . . . . . . . . . . . . . . . . . . 155
B.2.3 ABCD law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.2.4 Ring cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.5 Enhancement factor . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.6 Numerical computations for 1111.3→556 nm, and 798→399 nm system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
C Iodine spectroscopy at 507 nm

163

D Spherical harmonics and vector spherical harmonics

166

5

Chapter 1
Introduction
The technique of laser cooling and trapping of neutral atoms has progressed rapidly inrecent years. One of the major breakthroughs in this field is the achievement of the
Bose-Einstein condensate (BEC) in dilute gasses of alkali metals in 1995 [4, 5, 6]. BEC
has enabled us to study fundamental quantum mechanics experimentally in a macroscopic
system, and thus numerous fascinating experiments have been performed worldwide using
BEC.
At the same time, laser cooled two-electron atoms, including alkaline-earth metals(e.g., Ca and Sr) and Ytterbium (Yb), have attracted increasing interest in various fieldsof atomic physics due to their unique features, which alkali metal atoms do not possess[7, 8]. In particular, the unique characteristics of metastable 3 P2 atoms have recently
attracted attention, both for their applications and for the study of their collisional properties [9]. These atoms are different from the more commonly studied alkali metal atoms
because collisions between 3 P2 atoms are intrinsically anisotropic. Recent theories haveinvestigated the effects of this anisotropy, including its interplay with magnetic field effects, which enable novel control of the scattering length [10], and multichannel collisions
due to strong coupling among the partial waves of relative motion [11]. Also, the magnetic
dipole dipole interaction between 3 P2 atoms is nine times larger than that between alkali
metal atoms. This has led to theoretical predictions such as novel quantum phases anduse in quantum information systems [12, 13].
On the other hand, studies on extremely narrow optical transitions in two-electron
atoms between the metastable state and the ground state have also progressed rapidly.
For example, in metrology, studies on an ultraprecise atomic clock using such ultranarrowtransitions have been pursued to achieve an uncertainty of 10−16 and beyond with the aimof producing an optical lattice clock [8]. In quantum information science, an optical twolevel system with large coherence times of the metastable state in two-electron atoms isuseful as a well-defined quantum bit (qubit), which can be easily converted to a flying qubitof a photon. Sideband cooling using narrow transitions to cool a neutral atom down to thevibrational ground state of a trap is also possible [14, 15]. In addition, the combination of
the ultranarrow 1 S0 ↔3 P2 (m = ±2) transition and its Zeeman shift enables ultraprecise
measurement of a magnetic field and high-resolution spatial addressing of atoms, forexample, atoms confined in optical lattice potentials in a magnetic field gradient.

6

In this study, I present experiments for studying new aspects of the metastable 3 P2state in two-electron atoms. To study the fascinating possibilities of 3 P2 atoms, several groups have performed laser cooling and trapping of two-electron atoms in the 3 P2state. For example, in order to investigate unique collisional properties, laser coolingof metastable Ca[3 P2 ] and Sr[3 P2 ] atoms has previously been performed; however, theatomic temperature was relatively high (∼ a few millikelvin). In this study, we developed
a new method to obtain ultracold and dense 3 P2 atoms in a trap. Using such atoms, weinvestigated unique collisional properties in the ultracold temperature regime. As for theultranarrow transitions, while previous studies mainly focused on its application to thenext-generation frequency standards, we demonstrated high-resolution spectroscopy of ultracold atoms and BECs using an ultranarrow optical transition in Yb. In the following
text, I will briefly introduce the experiments performed in this work.
Unique collisional properties of metastable Yb[3 P2 ] atoms at ultracold temperatures
First, unique collisional properties of metastable Yb[3 P2 ] atoms at ultralow temperatures
are revealed in detail. Previously, several laboratories have realized laser cooling andtrapping of metastable two-electron atoms. Ca and Sr atoms decaying to the 3 P2 statefrom the 1 P1 state, which is the upper state in the 1 S0 ↔1 P1 magneto-optical trap (MOT)transition, have been successfully trapped in a magnetic trap [16, 17, 1]. Also, a MOT
operating on the 3 P2 ↔3 D3 transition has been used to load a magnetic trap [18]. In spiteof the success of these approaches, evaporative cooling of 3 P2 atoms in a magnetic trap torealize a BEC was unsuccessful due to trap loss caused by strong multichannel collisional
processes. More recently, a similar large inelastic collision rate in Ca[3 P2 , mJ =2] wasobserved at temperatures of a few millikelvin [2].
The loss induced by multichannel collisions in a magnetic trap can be overcome byemploying an optical far-off-resonance trap (FORT) instead of a magnetic trap. The
FORT wavelength can be chosen so that atoms in every magnetic sublevel of the 3 P2 state
can be trapped with the same strength. As a result, although multichannel collisions canstill occur, they will not lead to trap loss. In this study, we developed a new method
to prepare ultracold and dense 3 P2 atoms in a FORT. Unlike previous methods, we firsttrapped Yb[1 S0 ] atoms in a FORT and performed evaporative cooling. Then, we optically
excited Yb[1 S0 ] to the 3 P2 state to obtain ultracold trapped Yb[3 P2 ] atoms. Using thismethod, we achieved a number density of 2 × 1013 cm−3 at a temperature of 2 µK witha phase space density (PSD) of 0.01. Our newly achieved number density is larger thanthat achieved in a previous study by an order of three [1].
While the trap loss due to the multichannel collisions must be suppressed in our FORT,
we still observed a large two-body inelastic collision rate. Thus, we deduce the existenceof a different inelastic collisional process, which we interpret as fine-structure changing
collisions in this ultracold temperature regime. Although fine-structure changing inelastic
collisional properties have previously been investigated for Mg[3 Pj ], O[3 Pj ], Sc[2 Dj ], andTi[3 Fj ] colliding with closed-shell atoms [19, 20, 21, 22, 23], they had not been seen incollisions between 3 P2 atoms. While a recent experiment on magnetically trapped Ca

7atoms studied multichannel collisions between 3 P2 atoms and discussed the possibilityof the fine-structure changing process [2], we believe that our study is the first definite
experimental measurement of this process between 3 P2 atoms.
High-resolution spectroscopy of ultracold atoms and BECs using the ultranarrow 1 S0 ↔3 P2 transition
The other important achievement in this study is the successful observation of the ultranarrow magnetic quadrupole 1 S0 ↔ 3 P2 transition (507 nm) in Yb bosonic (174 Yb) andfermionic (171 Yb, 173 Yb) isotopes.
We first developed a 507-nm ultranarrow-linewidth laser system, which consists of an
extended cavity laser diode, a tapered amplifier, and a periodically poled lithium niobate
nonlinear crystal. In order to observe the ultranarrow transition, the laser linewidth was
reduced below 1 kHz by tightly locking it to a high-finesse optical cavity. Due to its
ultranarrow linewidth, this transition had never been observed prior to this study. Hence,
we also estimated the transition frequency using an optical frequency comb. Using thedeveloped laser system and the estimated transition frequency, we observed the 1 S0 ↔3 P2transition for the first time.
High-resolution spectroscopy of ultracold atoms and BECs was demonstrated using
this transition. The polarizabilities of all magnetic sublevels of the 3 P2 state were determined with high precision. Using an optical frequency comb, we also measured thefrequency of the magnetic-field-insensitive 1 S0 ↔ 3 P2 (m = 0) transition in 174 Yb, which
is one of the candidates for next-generation atomic frequency standards [3]. However,
the highlight of this study is the detection of the mean field shift of a BEC using thisultranarrow optical transition. We observed, not only the large mean field shift in a BEC,
but also the change in the lineshape, which reflects the density distribution of a BEC ina trap. As a result, we successfully determined a12 (the scattering length between atoms
in the 1 S0 state and the 3 P2 state) from the observed spectrum. Furthermore, we also
performed spectroscopy of condensates in 1D optical lattice potentials and observed themean field shift due to on-site interaction at each site.
Thesis outline• Chapter 2: We will introduce all important aspects of laser cooling and trapping of
Yb atoms in this study. Details about laser systems will be also presented.
• Chapter 3: This chapter deals with the ultranarrow laser system developed in thisstudy in order to observe the ultranarrow 1 S0 ↔3 P2 transition.
• Chapter 4: Overview of the excitation of atoms by a laser field such as E1, E2, andM2 transitions. All of these transitions play a key role in this study. The calculationof the light shift is also presented.
• Chapter 5: The spectral shifts and broadenings that we have to consider whenanalyzing the data obtained in this study are discussed.

8• Chapter 6: Optical trapping of 3 P2 atoms and investigation of unique collisional
properties are presented. A new method involving the trapping of atoms in a FORTand estimation of elastic and inelastic collision rate constants are discussed.
• Chapter 7: Details about our first observation of the 1 S0 ↔3 P2 transition are described. Estimation of the resonance frequency and measurement and control of thepolarizability of the 3 P2 state are also included.
• Chapter 8: Spectroscopy of a BEC using the ultranarrow 1 S0 ↔3 P2 transition isreported. The detected mean field shift will be shown and analyzed in detail. In
addition, the spectroscopy of condensates in 1D optical lattice potentials is reported.
• Chapter 9: This chapter presents the frequency measurement of the 1 S0 ↔3 P2 (m =0) “clock” transition. Some basic properties of our frequency comb are described.
• Chapter 10: A summary of this study and some future prospects are presented.
• Appendices: Short but important topics related to this study are briefly summarized, including the estimation of the magic wavelength of the 3 P2 state, a novel laser
system based on narrow iodine spectra, and a new stable laser system to obtain the
MOT beam.

9

Chapter 2
Laser cooling and trapping of neutral
Yb atoms
In this Chapter, I will present details of laser cooling and trapping of Yb atoms. First,
general properties of Yb atoms will be introduced. Then, basic characteristics of our
Zeeman slower system, MOT, FORT, and imaging system will be presented. Finally,
details about laser systems will be described.

2.1

General properties of neutral Yb atoms

Laser cooling and trapping of Yb atoms have been yielding a fundamental progress inatomic physics. After the first realization of a BEC of 174 Yb in 2003, BECs of 176 Yb
and 170 Yb and Fermi degeneracies of 173 Yb and 171 Yb have been successfully realizedin our laboratory [24, 25, 26, 27]. As shown in Table 2.1, rich kind of stable isotopesof Yb (five bosons and two fermions) enables a variety of mixtures such as Bose-Bose,
Bose-Fermi and Fermi-Fermi gasses and even mixtures of three species. Seven otherunstable isotopes are also known. The scattering length which governs atomic collisions
at ultracold temperatures has been precisely measured via photoassociation spectroscopy[28, 29, 30] and estimated for all pairs of isotopes using the mass scaling law [31]. The
other significant feature of Yb is the existence of ultranarrow optical transitions. The socalled “clock transition” (1 S0 ↔3 P0 ) is one of the primary candidates of next-generation
frequency standards and ultraprecise atomic clocks whose stability has already reached3×10−16 [32]. The clock transition has been observed in fermionic isotopes (171 Yb and173Yb) and a bosonic isotope (174 Yb) [33, 34, 35]. Novel sideband cooling technique using
the clock transition in fermionic isotopes has been also proposed [15]. In addition, Yb isregarded as one of the good candidates for the test of time-reversal symmetry violation[36] and the study of atomic parity nonconservation [37].
Yb is a rare earth metal. The ground state electronic configuration1 is [Xe]4f 14 6s2 . Its
atomic mass is 173.04. The melting point and the boiling point are 1097 K (824 ◦ C) and1700 K (1427 ◦ C), respectively. At room temperature, Yb is very stable. Because of two
1

1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f 14 6s2

2.1. GENERAL PROPERTIES OF NEUTRAL YB ATOMS

10valence electrons, the structure of energy levels are similar to that of alkaline-earth metals,
that is to say, Yb has singlet and triplet series in energy levels as shown in Fig.2.1 and 2.2.
Isotope shifts in the 1 S0 ↔1 P1 and 1 S0 ↔3 P1 transitions are listed in Table 2.2. According
to the calculation in [38], the saturated vapor pressure of Yb is 3×10−21 Torr at room
temperature, which means that there exists no Yb atoms in air at room temperature. We
heat the Yb oven up to 375◦ C in the experiment.

Table 2.1: Natural abundance and nuclear spin of Yb [39].
Atomic Mass

Natural Abundance (%)

Nuclear Spin

174

31.8

0

172

21.9

0

173

16.12

5/2

171

14.3

1/2

176

12.7

0

170

3.05

0

168

0.13

0

2.1. GENERAL PROPERTIES OF NEUTRAL YB ATOMS

11

14 ns

(6s6p)1P1

770 nm

5.7 ns3D2(5d6s)

3S (6s7s)1649 nm

460 ns3P2

399 nm3P (6s6p)
1877 ns404 nm3P
0507 nm
556 nm(6s2)1S0

Figure 2.1: Low lying energy levels of Yb
Table 2.2: Yb isotope shifts of the 1 S0 ↔1 P1 transition at 399 nm [41] and 1 S0 ↔3 P1
transition at 556 nm [42] relative to the resonance frequency of 174 Yb.
1Atomic Mass

S0 ↔1 P1

1

S0 ↔3 P1

Isotope Shift (MHz)Atomic Mass

Isotope Shift (MHz)176−509.3173(5/2−7/2)−2386

173(5/2−5/2)−253.4

171(1/2→1/2)

−2132

0

176

−955

173(5/2−3/2)

516.0

174

0

172

533.3

172

1000

173(5/2−7/2)

588.0

170

2287

171(1/2−3/2)

832.4

173(5/2−5/2)

2312

171(1/2−1/2)

1153.7

168

3655

170

1192.4

171(1/2−3/2)

3805

168

1887.4173(5/2−3/2)

3806

174

2.1. GENERAL PROPERTIES OF NEUTRAL YB ATOMS(6p2)3P2(6p2)3P1(6p2)3P0(6s8s)1S0(6s8s)3S1(6s6d)1D2(6s6d)3D3(6s6d)3D2(6s6d)3D1(6s7s)1S0(6s7s)3S1(5d6s)1D2

44760.37cm-1
43805.42cm-1
42436.91cm-1
41939.9cm-1
41615.04cm-1
40061.51cm-1
39966.09cm-1
39838.04cm-1
39808.72cm-1
34350.65cm-1
32694.692cm-1

532nm
800nm
1064nm

27677.665cm-1(5d6s)3D3

25270.902cm-1(5d6s)3D2

24751.948cm-1(5d6s)3D1

24489.102cm-1(6s2)1S0

12(6s6p)1P1

25068.222cm-1(6s6p)3P2(6s6p)3P1(6s6p)3P0

19710.388cm-1
17992.007cm-1
17288.439cm-1

0cm-1

Figure 2.2: Energy levels in Yb in the unit of cm−1 [40]. Energy between each levels
and the (6s2 )1 S0 ground state is reflected as an amplitude of the interval in the verticaldirection. The positions of typical wavelength for a FORT (532nm, 800nm and 1064 nm)
are also shown for the (6s2 )1 S0 and (6s6p)3 P2 states.

2.2. LASER COOLING AND IMAGING OF YB ATOMS

2.2

13

Laser cooling and imaging of Yb atoms

In this section, each cooling stage from an atomic oven to a BEC will be presented indetail [43, 44, 7]. We use the 1 S0 ↔1 P1 and the 1 S0 ↔3 P1 transitions for the Zeeman
slower and MOT transitions, respectively. Important values of these two transitions are
summarized in Table 2.3.

Table 2.3: Important values of cooling transition 1 S0 ↔1 P1 and 1 S0 ↔3 P1 [40].
1

S0 ↔1 P1

1

S0 ↔3 P1

Unit

Wavelength (in air)

λ

398.8

555.6

nm

Lifetime

τ

5.7

877

nsec

Natural linewidth

Γ
1
=
2π
2πτ

27.9

0.181

MHz

Saturation Intensity

Isat =

πhc
3λ3 τ

57

0.14

mW/cm2

Doppler limit temperature

TD =

670

4.35

µK

18

1.4

cm/sec

1.04

1.49

Doppler limit velocity

~Γ
2kB
r
kB TD
vD =
m

g-factorg

Absorption cross section

σab =

3λ2
2π

76

147

10−15 m2

Recoil frequency

νR =

~k 2
4πm

7.2

3.7

kHz

2.2. LASER COOLING AND IMAGING OF YB ATOMS

14

Figure 2.3: Schematics of Zeeman slower and MOT laser beams in vacuum chamber.

2.2.1

Zeeman slower

To slow down atoms emitted from a Yb oven at 375 o C, a Zeeman slower system isused. Basically, the resonance frequency of atoms in an atomic beam is shifted due tothe Doppler effect depending on the atomic velocity. In a Zeeman slower system, such
frequency shifts are compensated by the gradually varied external magnetic field.
There exist two kinds of Zeeman slower configurations which are referred to as theσ+ and σ− configurations. In the σ+(−) configuration, amplitude of the magnetic field
decreases (increases) from the oven to the MOT region. The important difference between
these two configurations is that the frequency of the slower beam. It is far red-detuned incase of the σ+ configuration while it is almost resonant in the σ− configuration. The σ+
configuration has an advantage of small leak of the magnetic field into the MOT region.
However, in this work, the MOT transition (1 S0 ↔3 P1 ) is different from the transition
used for the Zeeman slower (1 S0 ↔1 P1 ). In such a case, atoms in a MOT may be blew out
by the slower beam in the σ+ configuration. Thus the σ− configuration is better thanthe σ+ one2 . The 399 nm slowing laser are detuned by about 500MHz from the resonanceand applied to the atomic beam from the opposite direction as shown in Fig.2.3.
Cleaning of coated windows
Since we irradiate a slowing laser from the opposite direction of the Yb atomic beam,
the input window is gradually coated with Yb atoms. As this coating becomes thick, theslowing laser is scattered at the window, which leads to decrease of the efficiency of the
Zeeman slower system. To clean up the Yb coating, we have blew away Yb atoms into
the vacuum chamber by irradiating the focused green (532 nm) CW laser3 . The intensityof I∼ 2.5 kW/cm2 (P =1 W, ω = 5 µm) is strong enough. This method enables us torecover the efficiency of the Zeeman slower system without opening the vacuum chamber.
2
3

To overcome this problem, the so-called zero-crossing configuration is available.
We can also use the green (532 nm) pulsed laser.

2.2. LASER COOLING AND IMAGING OF YB ATOMS

2.2.2

15

MOT

Atoms slowed by the Zeeman slower system are trapped in a magneto-optical trap (MOT).
A MOT consists with three pairs of two circularly polarized couterpropagating lasers andthe quadrupole magnetic field. We use the narrow 1 S0 ↔3 P1 transition for a MOT (λ= 556 nm, Γ/2π = 181 kHz) whose saturation intensity Isat = 0.14 mW/cm2 and the
Doppler limit temperature TD =4.4 µK (see Table 2.3).
When we load 174 Yb atoms in a MOT, detuning and intensity of MOT beams are −1.3
MHz (−7 × Γ/2π) and 45 Isat (' 6 mW/cm2 , P =10 mW and ω0 = 1 cm), respectively.
The MOT magnetic field gradient is 1.3 G/cm for x and y direction (2.6 G/cm for z
direction). After about 10 s loading, we can typically collect 2×107 atoms in a MOT.
In order to increase atomic density, to realize better mode matching with a FORT
potential and to carry out the additional cooling, a compressed MOT scheme is used.
After the loading, the magnetic field gradient is increased to 14 G/cm in 200 ms for x andy direction (28 G/cm for z direction). Then, the intensity of the MOT beam is decreased
to 4 Isat for 100 ms to cool atoms. Detuning of the MOT beam is not changed before andafter the compressed MOT. As a result, just after the transfer of atoms from a MOT toa FORT, typical number and temperature of atoms are 1×106 and 50 µK, respectively.

2.2.3

Evaporative cooling in a FORT

Since Yb atoms in the ground state do not possess electric spins, the magnetic trap can
not be used as successfully used in typical BEC experiment of alkali metal atoms. Instead,the optical trap, which is also called a Far Off-Resonance Trap (FORT), is used.
We chose 532 nm as the FORT wavelength for two reasons. One is that, at thiswavelength (532 nm), very high power (∼10 W) and stable lasers are now commerciallyavailable. The second reason is that the theoretical estimation shows that the trap potential made by tightly focused 532 nm laser is deep enough (∼1 mK) to trap cold Yb
atoms in the ground state.
Plain evaporation
Figure 2.4 shows how atoms in a FORT reach thermal equilibrium after the transfer ofatoms from a MOT to a FORT. η = U0 /kB T is the ratio of the trap depth U0 and thetemperature of atoms in a trap. kB is the Boltzmann constant.
After the cooling due to plain evaporation in a FORT for about 1.5 s, the systemreaches thermal equilibrium where cooling due to elastic collisions equilibrate with heating
due to inelastic collisions (three body collisions). Under thermal equilibrium, we foundηeql = 14 for Yb[1 S0 ] which corresponds to γ = σel /σin = 2200 (see section 6.2). Here,σel and σin are the scattering cross section of elastic and inelastic (three body) collisions.
In general, the condensation by evaporative cooling is thought to be straightforward if
γ > 104 and impossible if γ < 10 [22]. Thus, γ = 2200 of Yb is not bad for us to realize
a BEC by evaporative cooling [7].

2.2. LASER COOLING AND IMAGING OF YB ATOMS

16

Figure 2.4: Plain evaporation. Temperature of Yb[1 S0 ] atoms and η in a FORT are plotted
as a function of holding time. After the transfer of atoms from a MOT to a FORT, atoms
are cooled by plain evaporation in 1.5 s and then the system reaches thermal equilibrium.

Trap lifetime of atoms in a FORT
Figure 2.5 shows the typical trap lifetime of atoms in a single FORT. Trap depth is about
700 µK and the temperature of atoms is about 50 µK. Non-exponential trap loss up to 1 s
is the three body inelastic trap loss due to the high atomic density just after the transfer
of atoms. The solid line is a fit of the data after 1 s by an exponential function. A trap
lifetime determined by one body collisions (collisions between Yb and a background gas)is 15 s.
Forced evaporation in a crossed FORT
In order to realize a BEC, increasing an atomic number density is crucial. To this end, we
use a crossed FORT configuration. One of two FORT beams enters the MOT region along
the horizontal direction (perpendicular to the gravity) and the other along the verticaldirection (parallel to the gravity).
First, trap depth of the horizontal and vertical FORT is about 700 µK and 10 µK,
respectively. Thus atoms are trapped mainly in the horizontal FORT at the beginning. As
we decrease the trap depth of the horizontal FORT in order to induce forced evaporative
cooling, atoms gather in the crossed region through elastic collisions. At the same time,
their phase space density increases. Finally, a BEC is achieved. During forced evaporative
cooling for 6 s, trap depth of the vertical FORT is kept constant at 10 µK.

2.2. LASER COOLING AND IMAGING OF YB ATOMS

17

1⋅106

Number of atoms

8⋅105

6⋅105

4⋅105

0

1

2

3

4

5

6

Time (sec)Figure 2.5: Trap lifetime of atoms in a single FORT. Trap potential is about 700 µK andthe temperature of atoms is about 50 µK. After the inelastic loss due to the three bodycollisions up to 1 s, one body collisions govern the trap lifetime. The solid line is a resultof a fit by an exponential function. One body decay time is measured to be 15 s.

Technical issues
In a crossed FORT configuration, two tightly focused FORT beams must be perfectly
crossed at the trap region. However, there is instability of pointing of the FORT beam
due to a heating effect by RF power at an AOM. AOMs are used to control the FORT
power. In order to get rid of this instability, we put a copper plate under the AOM
and flow water inside it. Temperature of the water is stabilized at 15◦ C by a chiller. In
addition, we wait for about an hour after starting the time sequence of evaporative coolingto make everything thermally equilibrium.
In addition, since the FORT beam is tightly focused, it is rather difficult to correctly
guide in a MOT region. Thus when we first find the FORT beam in a MOT region, we
sometimes use the resonant light (399 nm or 556 nm) instead of 532 nm. If the resonant
beam hits the MOT, we can easily recognized it in a CCD image on a TV monitor. Then,we put the FORT beam in the same optical path. In addition, since the trap depth ofthe vertical FORT is shallow, we should use cold atoms (< 10 µK) to align the vertical
FORT.

2.2.4

Absorption imaging

Ultracold atoms and a BEC in a vacuum chamber are observed by absorption imaging.
The imaging laser which is resonant to the strong transition 1 S0 ↔1 P1 is irradiated to
an atomic cloud and then detected by a CCD camera after some imaging optics. The
magnification is designed to be 2.25 [45] by using proper achromatizing lenses. Since2.2. LASER COOLING AND IMAGING OF YB ATOMS

18

atoms absorb the imaging laser, they can be detected as a shadow on a CCD image
whose intensity and size have information about the number of atoms and their spatial
distribution, respectively.

Number of atoms
The number of atoms N can be derived from a CCD image by using the equation,N =−tS X Pi,j − Di,jln,
σab i,j
Fi,j − Di,j

(2.1)

where S is the area of each pixel, t is the magnification ratio of the imaging system, σabis the photon scattering cross section. Pi,j , Fi,j and Di,j are the signal intensity on a
CCD camera at a pixel (i, j) with atoms (Probe), without atoms (Flat) and without both
atoms and probe beam (Dark), respectively. For the closed two-level system, σab is given
by
µ
¶µ
¶
1
1
3λ2
σab (δ, s) =,(2.2)
2π 1 + (2δτ )2
1+s
where λ is the wavelength of the transition, δ is the detuning of the probe laser and τ isthe lifetime of the excited state4 . s is the saturation parameter which is defined by
s=

I
Isat

,

Isat =

hπc
,
3τ λ3

(2.4)

where I is the laser intensity, Isat is the saturation intensity and c is the speed of light.
In our experiment, the laser intensity of a probe beam is much lower than the saturation intensity of the 1 S0 ↔1 P1 transition (I < 0.1 Isat for P = 100 µW and ω0 = 1 mm)
and the detuning is zero due to the frequency locking (δ = 0). As a result, the photon
scattering cross section becomesσab (0, 0) =

3λ2
.
2π

(2.5)

Temperature - Time of flight (TOF) technique
Temperature T of atoms can be determined by measuring how fast atomic cloud expands
after turning off the trap potential. Assuming that the initial density distribution and thevelocity distribution of atoms in a trap can be approximated by the Gaussian function and
4

Full Width of the Half Maximum (FWHM) of the natural linewidth is given by
µ
¶
1
γ
,
γ=
.
δωnat = γ, δνnat =
2π
τ

(2.3)

2.2. LASER COOLING AND IMAGING OF YB ATOMS

19

the Maxwell-Boltzmann distribution, respectively, the density distribution after turning
off the trap potential is given by
r
·
¸
N
r2
kB T
n(r, t) =
exp −
,
(2.6)
, σv (T ) =
2
2
3/2
2
2
3/2
2
2
(2π) (σ0 + σv t )2(σ0 + σv t )mwhere N the number of atoms, σ0 the initial width of density distribution, kB the Boltzmann constant, and m the atomic mass. In the experiment, we obtain the optical density(OD) which corresponds to the integration of n(r, t) along the propagation direction (z)of the probe laser, that is,·¸qx2 + y 2
OD(T, t) = OD0 exp − 2, σ(T, t) = σ02 + σv (T )2 t2 .
(2.7)
2σ (T, t)
As a result, taking the density distribution at two (or more) different times, t1 and t2 , σv
can be determined, i.e., T is given byµ ¶ 2
m σ (T, t1 ) − σ 2 (T, t2 )
T =
.
(2.8)
kB
t21 − t22

150m

Typical absorption images for thermal atoms and a BEC are shown in Fig. 2.6.

Figure 2.6: Typical TOF images. (Top): Thermal atoms. (Bottom): BEC. TOF time is1 msec to 10 msec (from left to right) with the interval of 1 msec.

2.3. VACUUM SYSTEM

2.3

20

Vacuum system
270

110

1

60
18
MOT

Ion pump
30 l/s4
Oven

Zeeman slower

Ion pump
30 l/s
TMP

Figure 2.7: Schematics of vacuum systems. Unit: mm.
Our vacuum chamber is schematically shown in Fig. 2.7. Two ion pumps (PST030,
ULVAC, 30 l/s) are installed at the beginning of the Zeeman slower and behind the
MOT region. In order to keep good vacuum at the MOT region, the differential pumping
technique is used. The MOT region is well separated from an atomic oven by a thin pipe (φ= 4 mm, l = 6 mm). In addition, vacuum of the oven region is kept by a turbo-molecularpump (TMP).

2.4

Lasers2.4.1

399 nm (1 S0 ↔1 P1 )Zeeman slower laser
The slower beam at 399 nm is obtained by a frequency doubling unit (WAVE TRAIN,
Laser Analytical Systems Inc.) which has an LBO crystal (CRYSTECH, CPM, Type-I)5
inside a bow-tie ring cavity with oxygen flow. Oxygen extends the lifetime of the LBO
crystal. The fundamental light is generated by a Ti:sapphire laser (∼1 W at 800 nm,
MBR-110, Microlase Optical Systems) which is excited by a green laser (Millennia Xs,
Coherent Inc., 532 nm, 10 W). The output power is typically ∼50 mW at 399 nm. About
the frequency locking, details will be discussed later in this chapter.
Instead of this SHG system, we also have succeeded in slowing atoms by using a blue
laser diode (Nichia Corporation) which is stabilized by the injection locking method. But
in this work, we did not use the injection lock system at all.

5

Crystal parameters: θ = 90◦ , φ = 31.9◦ and Brewster angle θB =58.17◦ , dimension 4×4×12 mm.
Details about the SHG cavity are discussed in Appendix B.

2.4. LASERS

21

Imaging laser
For absorption imaging, we do not need much laser power since the 1 S0 ↔1 P1 transition
is strong (Isat = 57 mW/cm2 ). The frequency of the imaging laser, however, should bestable both in a short term and in a long term. Details about the frequency stabilization
are discussed here.
We use a GaN blue laser diode (NLHV500E, Nichia Corporation). By using an external
cavity laser diode (ECLD) system [46] with the Littrow configuration, the laser linewidthis narrowed to about 10 MHz. Details about the Littrow configuration will be discussed
in the next chapter. For the purpose of absorption imaging of Yb atoms, frequency of
the imaging laser is required to be scanned for ∼2 GHz without any mode hops to cover
all Yb isotopes (see Table 2.2). Figure 2.8 shows fluorescence spectra observed by our
imaging laser. Except for 168 Yb (natural abundance: 0.13 %), we can clearly identify
signals corresponding to all stable isotopes. These spectra are obtained by irradiating
the imaging laser with a narrow Yb atomic beam in a reference oven from the orthogonal
direction in order to eliminate a Doppler width as much as possible. This atomic beamis used for frequency locking for a long term.
The natural linewidth of the 1 S0 ↔1 P1 transition is γ/2π = 29 MHz. Hence, 10
MHz may be narrow enough. For the stable imaging, however, the narrower linewidth isdesirable. In addition, since the imaging laser is also used to stabilize the other external
cavity (transfer cavity), further frequency narrowing has been carried out.
To this end, we have constructed the optical feedback system [47, 48, 49] which isschematically shown in Fig. 2.9. We pick up about 10 % of the ECLD output and put it
into the additional confocal cavity (Mirrors: LASEROPTIK GmbH, Curvature = 10 cm,
R = 99.5 %, PZT: NEC Tokin, AER 13.6×10×20−D15). Only when the laser frequency
is resonant to the confocal cavity, the reflection light can return to the LD and the laser
frequency is narrowed due to the additional optical feedback. In such a case, since thelaser frequency is drawn to the cavity frequency, the square shape of the transmission
spectrum of the confocal cavity is observed when we scan the laser frequency as shown in
Fig. 2.10.
In order to lock the laser frequency to the confocal cavity, the length between the
ECLD and the confocal cavity is modulated by a PZT-φ attached to a mirror in front of
the confocal cavity by f = 40 kHz. The error signal is obtained by putting the modulatedtransmission signal of the confocal cavity into a lock-in amplifier and applied to the PZT-φ.
The PZT-φ is also used to adjust the phase of optical feedback. In addition, the distortionof the cavity transmission signal is detected by taking its third-order derivative. This isexperimentally realized by taking error signal by a lock-in amplifier with 3f (= 120 kHz)
reference frequency. The 3f error signal is applied to the PZT-G inside the ECLD. By
simultaneously applying the 1f and 3f electric feedback, the optical feedback is kept for
more than one hour as shown in Fig. 2.12.
Moreover, we stabilize laser frequency to the Yb atomic spectrum for the long term
stability. The laser frequency of the present optical feedback system is modulated at 40
kHz. Thus the fluorescence signal which is obtained by irradiating a Yb atomic beamwith this modulated light is also modulated at 40 kHz. Since the lifetime of the 1 P1 state2.4. LASERS

22(5.5 ns) is much shorter than the modulation frequency, we can obtain the error signal by
putting the modulated fluorescence signal into a lock-in amplifier as shown in Fig. 2.11.
This error signal is applied to the PZT-C at the confocal cavity.
Other candidates
We also construct some laser systems for a more stable and powerful blue laser source.
One of the candidates is a waveguide PPLN crystal (NGK Insulators, Ltd.) instead of
the present LBO system. Since we do not need the optical cavity, the waveguide PPLN iseasy to handle. So far, the larger blue output power than that of the present LBO system
has not been stably obtained.
The other candidate is using the MOPA system. We construct an ECLD at 800 nm(Eagleyard) and the output about 20 mW is amplified to about 1 W by a tapered amplifier(Eagleyard). Then it is put into a bow-tie cavity in which the PPKTP nonlinear crystal
is installed. This system is now under development in our group and preliminary about
50 mW blue light is obtained.

Fluorescence (Arb. Units)

2.0
174
1.5

173 (5/2-7/2)
173 (5/2-3/2)
172

1.0
176
0.5

171 (1/2-3/2)
173(5/2-5/2)

171 (1/2-1/2)
170

0.0
-1

-0.5

0
0.5
Frequency (GHz)1

Figure 2.8: Fluorescence signal (1 S0 ↔1 P1 ) obtained by the present imaging laser. Signals
corresponding to all isotopes except for 168 Yb (natural abundance: 0.13 %) are observed
without a mode hopping.

2.4. LASERS

23



Sig

Sig
Lock-in
f 3
amp.
Ref
Error sig
(f = 120 kHz)Ref

f = 40kHz+

Lock-inamp. Sig

Error sig

Servo for 1f
Servo for 3f

PZT-C

PD

PZTServo

Yb fluorescence

PZT-G
To experiment

Laser
diode

Figure 2.9: Optical feedback system for absorption imaging.

Figure 2.10: Typical transmission signal of the cavity in a optical feedback system. The
laser frequency is drawn to the reference cavity because of the optical feedback system
which results in a square shape.

2.4. LASERS

24

2

Fluorescence
Error signal

Amplitude (arb.units)

1.5

174Yb

1
0.5
0
-0.5
-1
-200 -150 -100-50
0
50 100Frequency (MHz)

150

200

Figure 2.11: Error signal of the atomic spectrum. Since the lifetime of 1 P1 state (5.5 ns)
is much shorter than the modulation frequency (40 kHz), we can obtain the error signal
by using a fluorescence signal.

Cavity Transmission

1.4

Amplitude (V)

1.2
1
0.8
0.6
0.4
0.2
0
0

15

30

45
60
Time (minutes)

75

90

Figure 2.12: Due to simultaneous feedback of 1f and 3f error signal, the frequency locking
survives for more than 90 minutes. Lack of some data is due to a trivial technical problemof data logging.

2.4. LASERS

2.4.2

25

556 nm (1 S0 −3 P1 ) - MOT

Dye laser
A MOT beam at 556 nm is obtained by a ring dye laser (Coherent 899) excited by Ar
ion laser (∼5 W). Rhodamin 560 (Exciton) is used and it is circulated by a circulator(RD1000, Radiant Dyes Laser & Accessories GmbH). This circulator is made of stainlesssteel (brass free). We found that brass in the circulator system decreases the lifetime ofa circulated dye.
Laser frequency should be narrowed below the natural linewidth of the MOT transition(γ/2π = 182 kHz). Frequency of the dye laser is locked to a high finesse ULE (Corning)cavity. The error signal obtained by the Pound-Drever-Hall method [50] is applied both toan EOM inside the ring cavity and to an AOM out side the cavity. The EOM suppresses
the fast varying component of the frequency fluctuation [52] and the AOM suppressed
the slowly varying part. As a result, the linewidth is suppressed below 100 kHz. The
ULE cavity is placed in a vacuum chamber whose temperature is stabilized to reduce thelong term drift of the resonance frequency. So far, the frequency drift is suppressed to 70
kHz/s.
Transfer cavity - 556 nm and 800 nm
To stabilize the laser frequency of the Zeeman slower laser for a long term, we use thetransfer cavity. First, we stabilize the transfer cavity (RG-91T, Burleigh) to the 556 nm
laser which is stabilized to the ULE cavity. Then, the fundamental light of the slower
beam at 800 nm is locked to the transfer cavity. As a result, the long term frequency
stability of the ULE cavity is transferred to the frequency of the slower beam via thetransfer cavity.
Zeeman modulation
At the beginning of this work when we did not have a ULE cavity, we stabilized thefrequency to the Yb atomic spectrum by a Zeeman modulation technique. Though thesystem is now totally replaced by the ULE cavity, the Zeeman modulation technique isbriefly introduced here since I developed the system.
For the purpose of the frequency stability of the MOT beam for a long term, we lockthe frequency to the atomic spectrum of the 1 S0 ↔3 P1 transition. To make the error
signal, the Zeeman modulation method is used. We apply both static B0 and modulated
magnetic field B1 sin ωmod t (3 kHz) to the Yb atomic beam. These magnetic fields induce
the modulated Zeeman shift ∆E on the |mJ | = 1 magnetic sublevels of the 3 P1 state.
Here ∆E is given by∆E = mJ µB g(B0 + B1 sin ωmod t).

(2.9)

where g is a g-factor (1.49 for the 3 P1 state). As a result, the intensity of the fluorescence
detected by a PMT from the atomic beam is also modulated. By putting this signal into
a lock-in amplifier, a steep error signal is obtained as shown in Fig. 2.13. The static

2.4. LASERS

26

magnetic field is used to tune the locking point and to adjust the detuning of the MOT
laser.
1

Amplitude (arb.units)

172

174

176

0.5
0
-0.5
-1
-1.5
-1

-0.5

0
0.5
Detuning (GHz)1

1.5

Figure 2.13: Error signals of the 1 S0 ↔3 P1 spectra obtained by the Zeeman modulation
method. Three peaks corresponds to isotopes 172 Yb, 174 Yb and 176 Yb.

Fiber laser system
The MOT laser at 556 nm is obtained by a dye laser. In order to use a dye laser, we(students) have to work very hard; we must change dye and clean up the cavity almost
every week, use Ar ion laser and so on. Thus, the new laser source at 556 nm which isstable and maintenance-free is strongly desired.
The double wavelength of the MOT laser is 1111.3 nm. This wavelength is close to thewavelength 1.3 µm which is known as zero-dispersion wavelength of the optical fiber. Since
the 1.3 µm laser is recently well used in the field of optical broadband communications,many stable commercial lasers around 1 µm are now available. Additionally, we find thata nonlinear crystal LBO can convert 1111.3 nm to 555.6 nm efficiently by a proper ringcavity. Then, we decided to construct a new SHG system.
We use a commercial fiber laser (Koheras or Keopsys) at 1111.3 nm whose linewidthis below 100 kHz. Using this new system, we can obtain more than 400 mW green laser
and its frequency locking survives for more than ten hours! In Appendix B, details about
this SHG system are described.

2.4.3

404 nm (1 S0 ↔3 D2 )A 404-nm laser is used to excite atoms to the 3 P2 state via the intermediate 3 D2 state(see section 6.1). The laser system is completely same as that used in the imaging system

2.4. LASERS

27

stabilized by the optical feedback technique. The linewidth is narrowed to about 1 MHz
for 0.5 s.
For the long term stability, we use the transfer cavity as shown in Fig. 2.14. First,
we stabilized the transfer cavity to the imaging laser which is stabilized to the Yb atomic
spectrum. Then, the 404 nm laser is stabilized to the transfer cavity. The double passed
AOM between the transfer cavity and the optical feedback system is used to tune thelaser frequency. In the experiment, we first find the resonance signals using the atomic
beam in a reference oven as shown in Fig. 2.15 and then irradiate it with atoms in a trap.


















.$"!/01"
!"
#$

%&',-,*+

2345678394:;<=%&'())*+

.$"!/01"



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Figure 2.14: Transfer cavity system to stabilize the 404 nm laser for a long term.

174Yb

173Yb(5/2

172Yb

9/2)
171Yb(1/2

3/2)

176Yb
173Yb(5/2
173Yb(5/2

3/2)

7/2)
173Yb(5/2

5/2)

170Yb

Figure 2.15: The 1 S0 ↔3 D2 spectra obtained by our laser system.

2.4. LASERS

28

Beam profiler
When we excite atoms in a FORT by the 404 nm laser, its focal point should be coincidewith that of the FORT laser (trap region) whose beam waist is 15 µm. To this end, thebeam profiler (BP, DataRay Inc., Beam R) is used. We put a glass plate (no coat) infront of the input window of the vacuum chamber and let the partially reflected FORT
light go into the BP. BP shows us a radius of the input light. Thus by moving the stageof BP, we can set the BP at the focal point of the FORT laser. Then, at the focal pointof the FORT laser, we put the 404 nm laser into the BP and adjust the focal point of 404
nm laser.

2.4.4

770 nm (3 P2 −3 S1 ) and 649 nm (3 P0 −3 S1 ) - Repumping

In this work, repumping lasers at 770 nm (3 P2 −3 S1 ) and 649 nm (3 P0 −3 S1 ) are used
to return atoms in the 3 P2 state to the ground state where we can apply the absorption
imaging.
Both wavelengths are obtained by laser diodes. As for 770 nm, we use the commercial
product (6200, New Focus Inc). As for 649 nm (Toptica), we construct the ECLD systemin the Littrow configuration.
A laser galvatron of Yb (L2783-70NE-Yb, Hamamatsu Co.) is used for spectroscopyof these two wavelength. In the laser galvatron, the metastable 3 P2 and 3 P0 states havesome population due to strong collisions between hot Yb atoms or recombination of Ybions. In order to avoid the Doppler broadening, we carry out the saturation spectroscopy.
Typical spectra are shown in Fig. 7.5 and Fig. 7.6.
At the beginning of this work, we stabilized both lasers to Yb atomic spectra. But
the system was not so stable. We then have constructed a new system as schematicallyshown in Fig. 2.16. First, the transfer cavity is stabilized to the MOT laser (556 nm)
which is stabilized to the ULE cavity. Then, both the 649 nm and the 770 nm lasers
are stabilized to the transfer cavity. In order to spatially well separate the 556 nm and
649 nm, a prism is used. Since we lock three colors 556 nm, 649 nm and 770 nm to thesame cavity, the mirror reflectivity is reduced to 95 % but is constant from 500 nm to 800
nm (Lattice Electro Optics). The cavity transmissions for all lasers before and after thelocking are also shown in Fig. 2.16.

2.4.5

532 nm - FORTAs the FORT lasers, we use two diode-pumped solid state lasers (Verdi-V10, CoherentInc.). High power fundamental laser at 1064 nm obtained at Nd:YVO4 crystal is converted
to 532 nm by the LBO crystal. The maximum power is 10W.

2.4. LASERS

29

770nm

649nm
Double Path

AOM

AOM

AOM

Saturated
Absorption
Spectroscopy

556 nm649 nm
770 nm:Mirror:QWP:HWP

To Experiment

Prism

Dichroic: Mirror: Photo
EOM

Optical Fiber

Dye Laser

556nm

Detector: Fiber

Coupler

12MH

Figure 2.16: Three color lock. The transfer cavity is stabilized to the MOT laser which isstabilized to the ULE. Then, the 649 nm and 770 nm lasers are stabilized to the transfer
cavity.

30

Chapter 3
A diode laser system forspectroscopy of the ultranarrowmagnetic quadrupole transition in
ytterbium atoms


/4

AOM

DBM

ULE

12
MHz



To ULE

/4

Servo

FI
40dB

EOM

ECLD
25mW@1014nm

Vibration isolation table

FI
60dB

FI
40dB
20mW

TA

300mW

PPLN

To Experiment
10mW@507nmFigure 3.1: Apparatus used for generating the 507-nm narrow-line laser. FI, Faraday
isolator; AOM, Acousto-optic modulator; DBM, Double-balanced mixer; φ, phase shifter.
In this chapter, we describe the development of a compact diode-laser based system
to generate 507-nm laser for the purpose of a spectroscopy of the 1 S0 ↔3 P2 transition in
Yb atoms. In previous studies, lasers around 507 nm is produced by, e.g., a frequency
doubled Yb:YAG disk laser [53] and frequency doubled single-frequency diode laser [54,55]. The narrowest linewidth is about 100 kHz [55]. In this study, in order to observe theultranarrow transition, the laser linewidth has been reduced to less than 1 kHz by tightly

3.1. EXTERNAL CAVITY WITH DIFFRACTION GRATING

31

locking it to a high finesse optical cavity. We developed an ultranarrow-linewidth laser
system which consists of an extended cavity laser diode (ECLD), a tapered amplifier, anda MgO-doped periodically poled lithium niobate nonlinear crystal (PPLN). In addition,
since the transition linewidth of iodine molecules becomes narrowest around 507 nm, oursimple and compact system could be a high performance and transportable frequency
reference. Toward this application, we also demonstrate laser spectroscopy of iodine
molecules over the range of 50 GHz at 507 nm, which is presented in Appendix C.

3.1

External cavity with diffraction grating

Gain-width of recent solitary laser diodes reaches about 100 nm. It is sure that thelaser linewidth can be narrowed by putting an external cavity around such laser medium.
However, the additional external cavity also produces many longitudinal modes. Thus,
if we just put two mirrors in front of and behind the medium, the laser operates withmulti-modes. This problem can be overcome by using a dispersive external cavity. The
diffraction grating is used to this purpose.
Diffraction grating diffracts incident lights to some discrete directions determined by
the so-called grating equation. The direction which satisfies the reflection law to theincident angle is defined as the 0th order diffracted direction. From this angle to outerangle, 1st, 2nd... orders are defined. Usually, we let the first order diffracted light go backto the laser medium, so that the backward facet of an LD and a grating surface construct
the external cavity. The direction of the 1st order light strongly depends on the laser
wavelength. As a result, many longitudinal modes caused by the external cavity could be
suppressed to the desired one (sometimes a few modes). This results in the single-mode
operation of the LD. In such a configuration, the 0th order diffraction is used as an output
coupler.
For now, as for the external cavity which consists of mirrors and diffraction gratings, two configurations are well used, that is, Littman and Littrow configurations. Bothmethods have advantages and disadvantages. In the next section, I will discuss them.

3.1. EXTERNAL CAVITY WITH DIFFRACTION GRATING

3.1.1

32

Littman vs Littrow

Littman

Littrow

MirrorLD Lens

LD Lens

Grating

Grating

Figure 3.2: Littman and Littrow configurations

Littrow configuration
In the Littrow configuration, the 1st order diffracted light returns to the LD and 0thorder direction is used as an output coupler. In this configuration, the backside facet ofan LD and the diffraction grating surface make the external cavity (I assume that thefront facet of an LD is AR coated). In order to change the laser wavelength, we should
move the grating back and forth by a PZT element attached to the grating.
The advantage of this method is that we can expect a strong feedback because we use
the diffraction grating once whereas Littman configuration uses twice. Strong feedback
may contribute to the frequency narrowing. Sometimes this feedback is too strong. Sincestrong feedback may damage the LD chip, we have to be careful when we choose thegrating [56]. The other advantage is that we can easily make this system and make itcompact. Thermal stability is one of the key point of the stable LD oscillation. If thesystem is compact, controlling the temperature becomes easier.
On the other hand, disadvantages also exist in the Littrow configuration. One of themis the change of the output direction which results from tuning of the laser wavelength.
When we rotate the grating to change the wavelength, the 0th order direction is also rotated together. This change is negligibly small for usual experiments. However, it may be
a big problem for long optical path without an optical fiber, position-sensitive experimentand so on. The other disadvantage is that sometimes the selectivity of longitudinal modes
by grating is not good enough for a stable and widely tunable single-mode operation. To
avoid this problem, you can put a thin etalon between an LD and a grating to suppress
residual longitudinal modes. This is, however, not common because additional optics are3.1. EXTERNAL CAVITY WITH DIFFRACTION GRATING

33

not desirable. Instead the common method is to use a non-AR coated LD. In this case,the front facet and back facet of the LD play a same role as a thin etalon.
Littman configuration
Unlike the Littrow configuration, 1st order diffracted light goes to the additional tuning
mirror in the Littman configuration. It returns to the grating and the 1st order diffracted
light goes back to the LD. Hence, the LD back facet and an additional mirror construct
the external cavity.
One of the advantages of the Littman configuration is its small frequency passband.
In this configuration, we can select an arbitrary incident angle to the grating. Hence,
we can choose a grazing incidence to use as many grooves on the grating as possible.
Since the optical passband of grating is determined by how many grooves are interactedwith the incident light, using a wide area of the grating surface is desirable. In addition,
the grating surface is used twice in one round trip. Thus, this effect becomes doubled.
Usually, in the Littman configuration, the frequency passband is small enough for an LD
to operate in a single-mode and be tuned for a wide area of wavelength without any otheradditional optics. The other advantage is that output direction does not change at allwhen we tune its wavelength. In the Littman configuration, the cavity length can bechanged by moving not the grating but the tuning mirror. Hence the 0th order direction
doesn’t change at all.
Additionally, Littman found the special configuration which enables us to simultaneously change the cavity length and the angle of the tuning mirror without breaking thegrating equation [57]. The principle is as follows.

MirrorLD
Grating
Pivot

Figure 3.3: Littman configuration
Figure 3.3 is the Littman configuration. The point at which three lines – grating
surface, tuning mirror surface and back facet of LD – are crossed is called a pivot point.

3.1. EXTERNAL CAVITY WITH DIFFRACTION GRATING34

In this configuration, as long as you rotate the tuning mirror around the pivot point, laser
frequency never experiences the mode hops, so that extremely wide tuning range could
be realized in principle. The reason can be explained as the following.
Two equations must be satisfied in this configuration.
λ
N
2
d
λ =
(sin θ0 + sin φ),
m

L =

(3.1)
(3.2)

where L is the cavity length, N is a positive integer, λ is the laser wavelength, d isgroove spacing, m is the diffraction order, and θ0 and φ are incident and diffracted angle,
respectively. (3.1) is the ordinary equation which determines the longitudinal modes.
(3.2) is known as the grating equation. As illustrated in Fig. 3.3,
L = lf + lp sin φ.

(3.3)

By substituting (3.3) to (3.1), the following equations are obtained.
2(lf + lp sin φ)Ndλ =(sin θ0 + sin φ).
m

λ =

(3.4)
(3.5)

Here, when we scan the wavelength, variables are only λ and φ and all of the otherparameters (lf , lp , θ0 , d, N , and m) are constant. If these constants satisfy the following
two equations
Nd
,
2m
Nd
=
sin θ0 ,
2m

lp =

(3.6)

lf

(3.7)

then both (3.4) and (3.5) are always satisfied for arbitrary λ and φ. Hence, we can tune
the laser wavelength continuously by changing φ without any mode hops.
The disadvantage of the Littman configuration is the complexity of the system. We
have to put an additional tuning mirror and properly arrange every element which candefine the correct pivot point.
We chose the Littman configuration for two reasons. First, we planned to use a tapered
amplifier which is very sensitive to the input direction of the seed laser. We considered thatchange of the output direction in the Littrow configuration may be crucial. In our different
work, however, we have succeeded in using a TA with ECLD in the Littrow configuration.
Thus, in practice, both configurations can be used. Second, we want to narrow the laser
linewidth as much as possible by the ECLD. Then the Littman configuration is better as
discussed above.

3.1. EXTERNAL CAVITY WITH DIFFRACTION GRATING

3.1.2

35

ECLD at 1014 nm - Littman configuration

1014nm LD ( Eagleyard ) is mounted
in a collimation tube(LT230P-B, Thorlabs)HR@1014nm mirror (50mm 12.5mm t 5mm)
on LEES mount (LM1 4025 6 )Anamorphic
prism pairs(Thorlabs, PS871-B)

Temperature of beryllium copper base is stabilized
separately from LD.

Ruled Grating 1200g/mm with Au coating
2.5mm 50mm t 9.5mm ( Optometric )Figure 3.4: Picture of the developed ECLD system in the Littman configuration. Laser
diode is stabilized by the external cavity which consists of a diffraction grating and amirror.
Figure 3.4 is a picture of our ECLD system. We first constructed an ECLD system at1014 nm in the Littman configuration. The laser source is a ridge waveguide GaAs semiconductor laser diode (Eagleyard Photonics, EYP-RWE-1060-10020-0750-SOT01-0000).
Due to the anti-reflection coating, the tuning range expands from 960 nm to 1080 nm.
The laser diode is installed in a aluminum mount. The polarization is TE mode. The
output beam is collimated by an aspheric lens (f = 4.5 mm, Thorlabs, C230TM-B) whichis also installed in an aluminum mount with a Teflon tape so that the screw of the lens
mount can be stably installed.
About 15 % of the diode output is diffracted by a grating with Au coating (Optometrics, 1200 grooves/mm) and retroreflected by a mirror (LASEROPTIK GmbH, 250
mm×12.5 mm×t5 mm). A laser (linear) polarization is parallel to the graves of the diffraction grating. The mirror is glued to a stable ”LEES” mirror mount (LINOS Photonics,
LM1-4025-6). As a result, a 12 cm-long extended cavity is constructed. A total feedback
to the laser diode is only 2 %, which is smaller than the typical feedback rate (∼ 60%)
in a usual ECLD system. The reason is that the AR-coated Eagleyard LD chip seems tobe easily damaged by the optical feedback [56]. The position of the mirror determines
the laser frequency. It can be tuned by a PZT actuator placed in a leaf spring on which
the mirror is mounted. The supporting point of the leaf spring corresponds in position
to the pivot point of the Littman configuration. The distorted beam shape of the laser3.2. TAPERED AMPLIFIER

36

diode is shaped by a pair of anamorphic prisms (Thorlabs, PS871-B). Thus, we can scan
the frequency over more than 30 GHz without mode hopping. The linewidth is reduced
to a few hundred kilohertz.
Since the cavity length is relatively long in the Littman configuration, temperaturestability of all components is crucial. The temperature of a laser diode and a base of
the ECLD system is separately stabilized at the same temperature by Peltier elements(Melcor) and temperature controller (Thorlabs, TED200) [58]. In addition, beryllium
copper (CuBe) alloy was chosen for the material of the base because CuBe has a goodthermal conductivity, good elastic modulus (for a leaf spring), and a small rate of thermal
expansion. In Fig. 3.5, the main properties of CuBe are compared to those of phosphor
bronze and aluminum. We can make the elastic modulus of CuBe better by annealing,
though we did not carry out. Note that since the powder of CuBe may be harmful to ourhealth, we have to pay careful attention when we fabricate it. After fabrication, metallic
plating of CuBe may be desirable to safely handle it in the experiment.

Material





Beryllium copper()
Phosphor bronze(
Aluminum()

  Longitudinalelastic modulus()

Thermalconductivity()

Rate of thermal
expansion
(
)

127.5

0.56

18

98.1

0.13

16.8

70

0.57
kN/mm2 

cal/cm sec

23.5
10-6/



Figure 3.5: Basic parameters of CuBe are compared to those of Cu and Al which arecommonly used as a base of the ECLD system.

3.2

Tapered amplifier

Adaptor and mount
Since the 20 mW diode output is too weak to generate enough power at 507 nm by using
the PPLN nonlinear crystal, we use a tapered amplifier (m2k-laser GmbH, TA 1030 1000).
Its polarization is TE mode. Thus the polarization of the seed light should be parallelto the epi-layer (the wing of the TA chip) as shown in Fig. 3.6(Left). We use a halfwave plate to precisely tune the polarization. We put a 60 dB optical isolator (Isowave,

3.2. TAPERED AMPLIFIER

37

I−98−SD−5) between the ECLD and the TA to protect the ECLD from the backward
emission of the TA.
Slot for TA not to rotate

Figure 3.6: (Left): Input and output direction of the C-mount type tapered amplifier.
(Right): The adaptor for a TA chip developed in this work.
We first developed a TA adaptor as shown in Fig. 3.6(Right). Our TA adaptor ismade of Cu and its temperature is detected by a thermistor. The anode and cathodeparts are separated by an insulator (Teflon).
Next we constructed a mount for the TA adaptor, an input coupler and an output
coupler. One of the important points to operate a TA chip is the stability of the positionof the lens mount of the input coupler. Since the aperture size of a TA chip is 2 µm
in height and 4 µm in width, small position shift of the input lens due to, for example,
the thermal expansion of the mount is sometimes crucial. We found that this problem
can be overcome by putting all elements on a same plate as shown in Fig. 3.7. To makethe system stable, we got rid of adjustment parts as much as possible and we fixed all
movable parts by epoxy after adjustment. Thus, finally, all components are completely
fixed to the mount.
The seed light and output amplified light are collimated by aspheric lenses whose
effective focal length are f =4.5 mm (Thorlabs, C230TM-B) and f = 2.8 mm (Thorlabs,
C390TM-B), respectively. We use two LEES stable mirror mounts to adjust the input.
As a result of these efforts, the seed laser has been coupled to the TA chip for more than
one month without any adjustment.
We also constructed a current controller for the TA based on [56]. The current driver
module whose maximum current is 2.5 A (Thorlabs, LD3000) is controlled by the additional circuit. With the operating current of 2.5 A and the seed power of 20 mW, thetapered amplifier provides 600 mW at 1014 nm (Fig. 3.8).
Alignment of the seed light
When a large current is applied to a TA chip, breaking the input coupling of the seed
light may damage the TA chip. We should carry out the alignment with low operating
current (500 mA ∼ 1 A). Also note that the output beam shape of the TA chip strongly3.2. TAPERED AMPLIFIER

38

9cm

5cm

Figure 3.7: Design of the TA mount. The key point is putting all elements on one plane.

Figure 3.8: Amplified power at a tapered amplifier with 20 mW seed light as a functionof the TA operating current.

3.2. TAPERED AMPLIFIER

39

depends on the operating current. As a result, when one makes an optical path using a
TA output, one should operate the TA current at the value you want to use (not the low
current).
The followings are the method to align a seed light to a TA chip obtained from ourtrial and error.
1. Tune the xy (normal plane to the input laser) position of the input coupler lens fora TA backward emission to locate at the center of the input coupler.
2. Tune the input coupler focus by seeing a far side image of TA backward emission.
3. Tune two coupling mirrors to make the optical path of the seed laser to correspondto the TA backward emission.
4. Tune coupling mirrors by seeing the amplified laser power.

OutputJig for output coupler

Input

Figure 3.9: By using a handmade jig, the output coupler of the TA system was carefully
adjusted by observing the amplified beam shape.

Beam shaping
A beam divergence of TA output along the horizontal axis is different from that along
the vertical axis. Basically, we first collimate the horizontal axis by the output coupler
and then collimate the vertical axis by using cylindrical lenses. However, we observed ainterference pattern in the collimated beam. As a result, in order to obtain the proper
beam radius and good beam shape, we needed several optics as shown in Fig. 3.10. Moreinvestigations will be required to understand the mechanism of this interference pattern.

3.3. PPLN CRYSTAL



40



    



 
 
   "#$#
#$  
!%!
#  %###Figure 3.10: Several convex lenses and cylindrical lenses were required to make the beam
shape of amplified light clean.

3.3

PPLN crystal

By using the SHG (Second Harmonic Generation) technique, we obtained 507-nm laser
light. We use a bulk PPLN – Periodically Poled (MgO doped Congruent) Lithium Niobate– crystal (HC Photonics) which converts the wavelength of 1014 nm to 507 nm. The quasi
phase matching (QPM) period of our PPLN crystal is 5.97 µm and QPM temperature isabout 40 ◦ C. The dimension is 50×3×0.5 mm (see Fig. 3.11) and both input and output
facets are AR coated both for 1014 nm and 507 nm.
We developed a crystal oven as shown in Fig. 3.11. Since the conversion efficiency
strongly depends on the crystal temperature, we have to uniformly heat the crystal.
Hence, the oven is made by Cu and covers whole crystal (top cover is not shown in Fig.
3.11). Additionally, the oven is covered by thermal insulators (foamed polypropylene)
and its temperature is controlled by Peltier elements.
After the 40 dB optical isolator (Isowave, I−98T−5−H) which protects the TA chipfrom the back scattering at the PPLN input surface, we put the fundamental light (1014nm) into the crystal. Since the polarization must be vertical, we adjust the polarizationby using a half wave plate in front of the PPLN crystal. In addition, we have to makethe Rayleigh length1 long enough to cover the crystal length 5 cm. Present parametersin our system are the followings: beam waist w0 = 86 µm, the Rayleigh length zR =2.3
cm, and the beam radius at the surface of PPLN = 86 µm. According to the data sheet
provided by the company, the average conversion efficiency is expected to be 2.6 % W−1
cm−1 , i.e., 13 % W−1 for our 5 cm crystal. However, so far, we have realized only 3 %.
The reason of such a bad conversion efficiency has not been clear yet. Bad spatial modeof TA output may be one of the reasons. Investigations are in progress.

1

Rayleigh length zR is given byzR =where w0 is the beam waist, λ is the wavelength.

πw02
,
λ

(3.8)

3.3. PPLN CRYSTAL

41

Oven for PPLN
PPLN crystal

5cm

SHG power (a.u.)

Figure 3.11: (Left) PPLN crystal used in the developed system. (Right): Developed ovento stabilize the temperature of the PPLN crystal at phase matching temperature.

PPLN temperature ()

Figure 3.12: Dependence of output power of SHG on the PPLN temperature.

3.4. ULE CAVITY

3.4

42

ULE cavity

To achieve further reduction of the laser linewidth, we lock the laser to a high-finesse
Fabry-Pérot cavity by the Pound-Drever-Hall technique [50]. To this end, a small partof the diode output (< 1 mW) is phase modulated by an electro-optic modulator (EOM)
driven at 12 MHz and sent to the cavity. The finesse and the free spectral range (FSR)
is 50 000 and 1.5 GHz, respectively, at 1014 nm. The electric feedback is applied to thelaser current through a servo circuit whose gain bandwidth expands over 3 MHz. As aresult, the laser linewidth is reduced to less than 1 kHz which is estimated by the residual
width of the error signal after the frequency locking.
In order to isolate the cavity from the environmental noise such as thermal and acoustic
fluctuations, the cavity spacer is made of an ultra-low expansion (ULE) glass (Corning)and placed in a vacuum chamber whose vacuum is kept below 10−8 Torr by an ion pump
(Varian, StarCell, 20 l/s). Figure 3.14 shows how to enclose the high-finesse cavity in a
vacuum chamber. The cavity is supported by two viton O-rings at Airy points (see Fig.
3.14) at which we can support the cavity so that the both input side and output side
can be maximally parallel to each other. Moreover, in the vacuum chamber, the cavity is
doubly covered with copper plates coated with Au in order to achieve a uniform heatingby black body radiation. The temperature of the vacuum chamber is also stabilized by
Peltier units almost at the zero-expansion temperature of the ULE glass through thecopper plate placed at the bottom of the chamber. The vacuum chamber is covered
with thermal insulators (foamed polypropylene) in order to thermally isolate it from theenvironment. As a result, the long-term drift is reduced to 0.4 Hz/s at 507 nm measured
by the 1 S0 ↔3 P2 (m = 0) transition in 174 Yb (see Fig. 3.13). Furthermore, by placing thecavity on a bench top vibration isolation platform (Minus K Technology, 350BM-1), we
isolate the cavity from the floor vibration at frequencies above 0.5 Hz. The double-passed
acousto-optic modulator (AOM) between the EOM and the cavity is used to tune thelaser frequency. In order to cover all of the FSR (=1.5 GHz) of the cavity, we use acombination of 350 MHz and 190 MHz modulators. After the frequency locking, the laser
frequency is scanned by changing the RF frequency applied to AOMs.
Frequency offset of 507 nm (kHz)

14
12
10
8
6
4
2
0
-2
0

50

100

150
200
Time (min)

250

300

350

Figure 3.13: Stability of the ULE cavity was measured as the shift of the resonancefrequency of the 1 S0 ↔3 P2 (m = 0) transition in Yb.

3.4. ULE CAVITY

43

Airy points
L

ULE
cavityaa = 0.5774 L
L: total length, a: interval between two points

Handmade gauge

Metal Bulb
(VAT, 54024-E02)

TMP(Turbo Molecular Pump)(ULVAC )Ion pump(Varian, StarCell,
20 L/sec)

Vacuum chamber(VIC international)

Figure 3.14: How to install a ULE cavity in a vacuum chamber.

44

Chapter 4
Optical excitations
In this chapter, theories of optical excitations are described. First, we introduce the multipole expansion of an electromagnetic field and then see how each terms induce multipole
transitions in atoms. Calculations of transition matrix elements of atoms interacting witha laser field yield selection rules. We derive the selection rules of the electric dipole (E1),electric quadrupole (E2), magnetic dipole (M1) and magnetic quadrupole (M2) transitions, all of which play a key role in this work except for the M1 transition [51].

4.1

Multipole expansion of an electromagnetic fieldBasic algebras
We first introduce some basic algebras required in the following discussions. When we
carry out a multipole expansion, vector spherical harmonics YJ,L,M (θ, φ) are used as abasis which are defined by
1−L−M

YJ,L,M (θ, φ) = (−1)√

2J + 1

+1 µ
X
q=−1

L
1 J
M − q q −M¶
YL,M −q (θ, φ)êq ,(4.1)

where YL,M is the spherical harmonics and ê q (q = −1, 0, +1) are the unit vectors in
spherical basis which are related to the three Cartesian unit vectors1 by1ê+1 = − √ (êx + iêy ),21ê−1 = √ (êx − iêy ).
2ê 0 = êz ,(4.3)

In (4.1),
µ
1

J
k J0
−M q M 0¶(4.4)

The inverse relations are1êx = − √ (ê+1 − ê−1 ),2iêy = √ (ê+1 + ê−1 ),2êz = ê0(4.2)

4.1. MULTIPOLE EXPANSION OF AN ELECTROMAGNETIC FIELD

45

is a Wigner 3j-Symbol which vanishes unless it satisfies the following three conditions.
µ
¶
j1 j2
J
m1 m2 −M1.
2.
3.

− |j1 | ≤ m1 ≤ |j1 |, −|j2 | ≤ m2 ≤ |j2 | and − |J| ≤ M ≤ |J|m 1 + m2 = M|j1 − j2 | ≤ J ≤ j1 + j2 (triangular inequalities).

(4.5)

As we can see later, these restrictions determine the selection rules of atomic transitions.
As a consequence of the orthogonality relation of the spherical harmonics2 , vector sphericalharmonics satisfy the following orthogonality relationZ
YJ,L,M · YJ 0 ,L0 ,M 0 dΩ = δJ,J 0 δL,L0 δM,M 0 ,(4.7)
where dΩ = sin θdθdφ is an element of solid angle.
Relations between Cartesian and spherical coordinates3 , inner product and cross productof two vectors in spherical basis are given by
A = Ax eˆx + Ay eˆy + Az eˆz= −A+1 eˆ−1 + A0 eˆ0 − A−1 ê+1
+1
X
A·B=
(−1)q Aq B−q

(4.9)

q=−1

= −A+1 B−1 + A0 B0 − A−1 B+1(4.10)A×B= i(A0 B−1 − A−1 B0 )ê+1 + i(A−1 B+1 − A+1 B−1 )ê0 + i(A+1 B0 − A0 B+1 )ê−1 .
(4.11)
Bessel functions jn (x) also play an important role in the discussion of selection rules which
satisfy the following identities,dn+1
jn (x) + jn (x)xdz
ndjn+1 (x) = jn (x) − jn (x),xdzjn−1 (x) =2Z(4.13)Z∗
YL,M(θ, φ)YL0 ,M 0 (θ, φ) sin θdθdφ ≡3(4.12)∗YL,M(θ, φ)YL0 ,M 0 (θ, φ) sin θdΩ = δL,L0 δM,M 0(4.6)

The relations between Ax,y,z and A−1,0,+1 are
1
A±1 = ∓ √ (Ax ± iAy ),2

A 0 = Az

(4.8)

4.1. MULTIPOLE EXPANSION OF AN ELECTROMAGNETIC FIELD

andjn (x) 'xn(2n + 1)!!

(x ¿ 1)46(4.14)For the convenience in later calculations, some spherical harmonics and vector spherical harmonics are listed in Appendix D.
Multipole expansion
Let us consider a plane electromagnetic wave whose vector potential A is given by4
A± (r, t) = êkλ e±i(k·r−ωt) ,(4.15)

where k is the wave number vector which is parallel to the wave propagating direction,êkλ is the polarization (unit) vector, ω = ck and c is the speed of light. In the following
discussion, the time dependent factor in (4.15) is not important and thus it is eliminated.
First, we expand (4.15) in a series of vector spherical harmonics
X
A± (r) =A±(4.16)
J,L,M YJ,L,M (θr , φr ),J,L,M

where θr and φr are the angular coordinates of r. A±
J,L,M can be calculated from (4.15),(4.16) and (4.7),
Z
Z
X
±
±ik·r
(YJ,L,M (θr , φr ) · êkλ )edΩ =AJ 0 ,L0 ,M 0 YJ 0 ,L0 ,M 0 (θr , φr ) · YJ,L,M (θr , φr )dΩJ 0 ,L0 ,M 0

Z

∴ A±
J,L,M

=

(YJ,L,M (θr , φr ) · êkλ )e±ik·r dΩ.

(4.17)The plane wave can be expanded in a series of spherical Bessel functions,e±ik·r = 4π

∞ X
l
X

∗
(±i)l jl (kr)Yl,m(θk , φk )Yl,m (θr , φr )(4.18)

l=0 m=−lwhere θk and φk are the angular coordinates of k.
By substituting (4.18) into (4.17) and performing integration, we can describe A± as
X
A± (r) = 4π(±i)L (YJ,L,M (θk , φk ) · êkλ )(jL (kr)YJ,L,M (θr , φr )).
(4.19)
J,L,M

Here, due to the restrictions (4.5) of a Wigner-3j symbol, possible values of L are J −1, J, J + 1 and possible values of M are J, J − 1, · · · − J. In other words, only three
independent unit vectors YJ,J−1,M , YJ,J,M , and YJ,J+1,M are possible. When we consider
the interaction between atoms and a laser field, it is more convenient to form mutually
4

The amplitude of the vector potential is not considered (assumed to be 1) since it doesn’t play anyrole here. It will be introduced in the next section.

4.1. MULTIPOLE EXPANSION OF AN ELECTROMAGNETIC FIELD

(−1)

(0)

47

(+1)

orthogonal unit vectors YJ,M , YJ,M , and YJ,M which correspond to longitudinal (−1)
and transverse (0, +1) polarizations of a laser field. Relations between these two bases in
spherical coordinates were given by Akhiezer and Berestetsky [59] in the form5rr
J
J + 1 (+1)(−1)
YJ,J−1,M (θ, φ) =JJ,M (θ, φ) +Y (θ, φ)2J + 12J + 1 J,M
YJ,J,M (θ, φ) = Y0J,M (θ, φ)r
r
J + 1 (−1)J(+1)
YJ,J+1,M (θ, φ) = −YJ,M (θ, φ) +YJ,M (θ, φ).
2J + 12J + 1Thus, we can rewrite (4.19) in the multipole expansion
A± (r) = 4π

∞ X
J
+1
X
X

(µ)

(µ)

(±i)J−µ (YJ,M (θk , φk ) · êkλ )aJ,M (r),(4.20)

J=1 M =−J µ=0

where6
TE mode(MJ transition)(0)
aJ,M (r)(0)= jJ (kr)YJ,M (θr , φr )TM mode(EJ transition)¶µpdjJ (kr)jJ (kr) (−1)(1)(+1)
jJ (kr) +YJ,M (θr , φr ) + J(J + 1)
YJ,M (θr , φr ).
aJ,M (r) =
d(kr)
kr
kr
(4.21)(−1)

The longitudinal term aJ,M is not included in (4.20) since it is parallel to the k and always
k · êkλ = 0. Note that, in (4.20), the polarization (ekλ ) and propagation direction of thelaser field (θk and φk ) are included in the expansion coefficient.
5

Inverse relations are
Longitudinal mode(k k) :
r
r
J
J +1
(−1)
YJ,M (θ, φ) =YJ,J−1,M (θ, φ) −YJ,J+1,M (θ, φ)
2J + 12J + 1
Transverse (TM and TE) mode :Y0J,M (θ, φ) = YJ,J,M (θ, φ)
r
r
J +1
J
(+1)
YJ,M (θ, φ) =YJ,J−1,M (θ, φ) +YJ,J+1,M (θ, φ).
2J + 12J + 1

6

Here we use following relations between spherical Bessel functions.
dn+1
jn (x) +jn (x)xdx
ndjn+1 (x) = jn (x) −jn (x)xdx

jn−1 (x) =4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
48(µ)

We should also remember that aJ,M satisfies the transversality condition∇ · aµJ,M (r) = 0∴ aµJ,M (r) · p = p · aµJ,M (r).

(4.22)In addition, the wave vector k must be always orthogonal to the polarization vectorêkλ (= −λ+1 ê−1 + λ0 ê0 − λ−1 ê+1 ), i.e., k · êkλ = 0. Thus, the following equation must be
always satisfied.
(λ−1 − λ+1 ) sin θk cos φk − i(λ−1 + λ+1 ) sin θk sin φk + λ0 cos θk = 0.

(4.23)

4.2

Matrix elements and selection rules for E1, E2,
M1, and M2 transitions

4.2.1

Interaction between atoms and a laser field

First of all, we derive matrix elements for the E1, E2, M1, and M2 transitions which
determine their selection rules.
A Hamiltonian describing the system of atoms and an electromagnetic field interactingwith them is given by
H = (Hrad + Hatom ) + Hint
(4.24)
where
Hrad =

X1~ωk (â†kλ âkλ + )2
kλ(4.25)

is the Hamiltonian for a free field. kλ represents the mode of the field, k and λ are thewave vector and the polarization (λ = 1, 2) respectively and ωk = ck. âkλ and â†kλ areannihilation and creation operators, respectively, for a photon with wave vector k andpolarization vector êλ .
X³ p2 ´i
Hatom =
+V0
(4.26)
2mi
is the atomic Hamiltonian for every electron i. V 0 contains all terms which is necessary
to define the atomic state (interaction between nucleus and electrons, repulsion among
electrons, and so on...).
If we take the most dominant term in the interaction Hamiltonian Hint , it can be given
by
³ e ´
p·A
(4.27)
Hint ∼=
mc
where p and A represent electron momentum and vector potential of external electromagnetic field, respectively [51].
In our calculations, the Hamiltonian H may be written as
H = H0 + Hint

(4.28)

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
49

where H0 = Hrad + Hatom . We regard the second term Hint as a perturbation. Thus what
we have to do first is to examine details of Hint .
In the Schrödinger representation and in SI unit, the vector potential for plane wave
A is described as,sX~c2êkλ [âkλ eik·r + â†kλ e−ik·r ](4.29)
A(r) =2V²0 ωkkλwhere V is the volume of the considered system. With (4.27) and (4.29), Hint can be
written asr
X³ e ´~
Hint =(êkλ · p)[âkλ eik·r + â†kλ e−ik·r ](4.30)
m
2V
²
ω
0 k
kλ
(+)

(−)

≡ Hint + Hint
where
(−)
Hint

=

X³ e ´
kλ

(+)
Hint

=

m

X³ e ´kλm(4.31)rr~(êkλ · p)âkλ eik·r
2V ²0 ωk(4.32)~(êkλ · p)â†kλ e−ik·r .
2V ²0 ωk(4.33)In the following calculations of the excitation rate or the trap depth of a FORT, matrix
elements of these Hamiltonians |hb, n0kλ |Hint |a, nkλ i|2 play a major role. Here |ai and |bi
are the atomic state vectors and n0kλ and nkλ represent the occupation number of kλ modeof the external field.
√√
By using âkλ |nkλ i = nkλ |nkλ − 1i and â†kλ |nkλ i = nkλ + 1 |nkλ + 1i, the square of
matrix elements can be modified as(−)|hb, nk0 λ0 |Hint |a, nkλ i|2 = |hb, nk0 λ0 |Hint |a, nkλ iδ(nk0 λ0 − 1, nkλ )(+)+hb, nk0 λ0 |Hint |a, nkλ iδ(nk0 λ0 + 1, nkλ )|2
(−)

(+)

= |hb; nkλ − 1|Hint |a; nkλ i|2 + |hb; nkλ + 1|Hint |a; nkλ i|2 .
(4.34)
Two terms in the last equation can be easily calculated by (4.32) and (4.33).
Substituting (4.20) to (4.32), (4.33), and (4.34), we have+1
³ e ´r ~n
XX
kλ
(µ)
(µ)
(−)
4π
iJ−µ (YJ,M (θk , φk ) · êkλ )hb|aJ,M · p|ai,hb; nkλ − 1|Hint |a; nkλ i =m
2V ²0 ωk J,M µ=0s+1³ e ´ ~(n + 1) X X
kλ
(+)
(µ)
(µ)
hb; nkλ + 1|Hint |a; nkλ i =4π(−i)J−µ (YJ,M (θk , φk ) · êkλ )hb|aJ,M · p|ai.
m
2V ²0 ωkJ,M µ=0(4.35)

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
50

The transition induced by the TM (transverse magnetic) field is called ”EJ transition”.
It corresponds to the case where µ = +1 in (4.20). In a similar manner, when µ = 0, i.e.,
the TE (transverse electric) field induces the transition, it is called ”MJ transition”. In
the following, we will see the E1, E2, M1, and M2 transitions (J=1, 2 and µ = 0, 1) indetail. All of these transitions (except for the M1 transition) play an important role inthis work.
Here, we introduce a useful equation. By using a commutation law between theposition r and the momentum p ([x, p2 ] = 2i~px and so on), we findµ ¶i~[r, Hatom ] =p,(4.36)mwhich leads tohb|p|ai =

³m´
³ i~
m´

hb|[r, Hatom ]|ai(Eb − Ea )hb|r|aii~= imωk hb|r|ai,=(4.37)

where ~ωk = Eb − Ea . For an N -electron system, we consider r = ΣN
j=1 rj , where rj is the
position of the electron j.

4.2.2

E1 transition

The E1 transition is induced by the term in (4.35) with µ = +1 and J=1. With the aidof (4.21), (4.37) and j1 (kr) ' kr/3 (when kr ¿ 1),
4π

+1
X

(+1)

(+1)

(ekλ · Y1,M (θk , φk ))hb|a1,M · p|ai

M =−1

'

Ã

+1
X

(+1)
4π
(ekλ · Y1,M (θk , φk ))hb|
M =−1(4.38)!
√2 (1)
2 (−1)
Y+Y· p|ai
3 1,M
3 1,M

+1
imωk X(+1)= 4π √(ekλ · Y1,M (θk , φk ))êM · hb|r|ai.
6π M =−1(4.39)The matrix element is given byr(−)
hb; nkλ − 1|Hint |a; nkλ iE1(±)

hb; nkλ ± 1|Hint |a; nkλ iE1

r

+1

8π X
(+1)
(ekλ · Y1,M (θk , φk ))êM · hb|r|ai,(4.40)
3 M =−1sr+1~(nkλ + 1)ωk 8π X(+1)(ekλ · Y1,M (θk , φk ))êM · hb|r|ai.
= ie
2V ²0
3 M =−1= ie~nkλ ωk
2V ²0(4.41)

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
51

Since r is the first rank irreducible tensor, the Wigner-Eckart theorem7 gives
µ
¶
J
1 J0
(1) 0
0
0
J−M
hα, J, M |rq |α , J , M i = (−1)
hα, J||r(1) ||α0 , J 0 i.
−M q M 0(4.45)

Due to the restriction (4.5) of a Wigner-3J symbol, the selection rule is given by∆J = 0, ±1∆M = 0, ±1,

Ja + Jb ≥ 1,

πa = −πb ,(4.46)

The conservation of parity requires πa πem πb = +1 where πa , πb , πem are parity of theinitial and final states and the multipole electromagnetic field, respectively. Parity of anelectromagnetic field is defined as the parity of the magnetic field B = ∇ × A. From
(4.20), parity of the TM and TE fields is determined by YJ,L,M and given by (−1)L .
In the case of the E1 transition, parity of the TE field Y±1
1,M with J = 1 is given byJ
πE1 = (−1) = −1 which leads to (4.46).
(1)
For the E1 transition, Y1,M are given byo√√
1 1 n √
(1)
Y1,±1 = √3 6(1 + cos2 θk )ê±1 ± 6 3 cos θk sin θk e±iφk ê0 + 18 5 sin2 θk e±2iφk ê∓1
4π 12(4.47)r3
3 1(1)√ sin2 θk ê0 +Y1,0 =cos θk sin θk (e−iφk ê+1 + eiφk ê−1 ).
(4.48)
2 6π
16π
As an example, let us consider the case where k k ez (θk = 0) and ekλ = e+1 . This
corresponds to the situation where a laser field propagates along the z direction and it iscircularly polarized. Sincer3(+1)
Y1,±1 (0, φk ) =
ê±
8π
(+1)
Y1,0 (0, φk ) = 0,7

Wigner-Eckart Theorem:µhα, J, M |Tq(k) |α0 , J 0 , M 0 i = (−1)J−M

J
−M

k
q

J0
M0

¶
hα, J||T (k) ||α0 , J 0 i(4.42)(k)where Tq is an irreducible tensor operator and J,M and J 0 , M 0 are angular momentum quantum numbers. α and α0 are additional quantum numbers required to completely identify the states.
hα, J||T (k) ||α0 , J 0 i is called ”reduced matrix element” and
µ
¶
J
k J0
(4.43)−M q M 0
is called ”Wigner 3j-Symbol”.
It can be described by using a Clebsch-Gordon coefficient(j1 , j2 , m1 , m2 |j1 , j2 , j, m).
µ¶
0(−1)k−J+MJ
k J0√(J, k, −M, 1|J, k, J 0 , −M 0 )(4.44)=−M q M 0
2J 0 + 1

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
52

when k k z, only the ∆M = ±1 transitions (σ+ and σ− transitions) are possible and the∆M = 0 transition (π transition) is impossible for any laser polarization. In addition,
due to the circular polarization ê+1 only M = +1 term survives and the matrix element
is given byr~nkλ ωk(−)(1)
hb; nkλ − 1|Hint |a; nkλ iE1 = ie
hb|r+1 |ai,
(4.49)2V ²0s~(nkλ + 1)ωk(+)
(1)
hb|r+1 |ai,
(4.50)
hb; nkλ + 1|Hint |a; nkλ iE1 = ie2V ²0
which means that the transition which satisfies ∆M = 1 occurs.

4.2.3

M1 transition

The M1 transition is induced by the term in (4.35) with µ = 0 and J=1. With the aid of(4.21), (4.37) and j1 (kr) ' kr/3 (when kr ¿ 1)8 ,1
krY1,1,M (θr , φr ) · p
3k= −i √(r × êM ) · p
24πkêM · (r × p)
= i√
24π
µ ¶
~
k
= i √ eˆM ·
L .
2
6π

(0)

a1,M (r) · p 'Since L is related to the magnetic momentum operator byµ ¶³ mc ´
~
µL ,
L=−
2
e

(4.52)(4.53)

the matrix element is given byr(−)
hb; nkλ − 1|Hint |a; nkλ iM1= −i~nkλ ωk
2V ²0

r

s
(+)

hb; nkλ + 1|Hint |a; nkλ iM1 = i

+1

8π X
(0)
(ekλ · Y1,M (θk , φk ))êM · hb|µL |ai,
3 M =−1~(nkλ + 1)ωk
2V ²0

r

(4.54)+1X

8π(0)(ekλ · Y1,M (θk , φk ))êM · hb|µL |ai.
3 M =−1(4.55)

8

In addition, next property of the scalar triple product is used.
a · (b × c) = b · (c × a) = c · (a × b).

(4.51)

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
53

Since µL is the first rank irreducible tensor, the selection rule is given by∆J = 0, ±1∆M = 0, ±1,

Ja + Jb ≥ 1,

π a = πb ,(4.56)

where πM1 = 1 is used. The M1 transition is allowed between states which has same
parity. It is, then, sometimes used to induce the RF transition between the magnetic
sublevels in a same J state.

4.2.4

E2 transition

The E2 transition is induced by the term in (4.35) with µ = +1 and J=2. With the aidof (4.21), (4.37) and j2 (kr) ' (kr)2 /15 (when kr ¿ 1),
!
Ã
√
6
1
(1)
(−1)
(+1)
(kr)Y2,M (θr , φr ) +(kr)Y2,M (θr , φr ) · p
a2,M (r) · p '
5
15
r
1
=
(kr)Y2,1,M (θr , φr ) · p
15 (1) (1)r±1 p±1(M = ±2)


³
´
r

1

(1)
(1)
(1) (1)1√ r±1 p0 + r0 p±1(M = ±1)=(4.57)
2³
´
20π 

1
(1) (1)(1) (1)(1) (1)(M = 0) √ r−1 p+1 + 2r0 p0 + r+1 p−1 ,6where r(1) and p(1) are the first rank irreducible tensor. Due to the transversality (4.22),(1) (1)(1) (1)(1) (1)(1) (1)
pq0 rq + pq rq0 = rq pq0 + rq0 pq . Thus we find
(1)

(1)

hb|rq(1) pq0 + rq0 p(1)
q |ai
³m´ n
o
(1)
(1)
=
hb|rq(1) [rq0 , Hatom ]|ai + hb|rq0 [rq(1) , Hatom ]|aii~ ³ ´
³m´ n
o
m
(1)
(1)
(1) (1)
(1)
(1)
= 2Ea
hb|rq rq0 |ai −hb|rq Hatom rq0 |ai + hb|rq0 Hatom rq |aii~i~(4.58)(1)(1)
hb|pq0 rq(1) + p(1)q rq 0 |ai

=

³m´ n

(1)
(1)
hb|[rq0 , Hatom ]rq(1) |ai + hb|[rq(1) , Hatom ]rq0 |aioi~ ³ ´
o
³m´ n
m
(1)
(1)
(1)
= −2Ebhb|rq(1) rq0 |ai +hb|rq(1) Hatom rq0 |ai + hb|rq0 Hatom rq(1) |ai .
i~i~(4.59)
From the summation of (4.58) and (4.59) and the transversality condition (4.22) is
(1)

(1)

(1)

(1)
hb|rq(1) pq0 + rq0 p(1)q |ai = imωk hb|rq rq 0 |ai.

(4.60)

4.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
54

In addition, according to the definition of a tensor product of two irreducible tensor
operators 9 ,
(2)

(1)

Q±2 = (r±1 )2√ (1) (1)(2)
Q±1 =2r±1 r0o
2 n (1) (1)(2)(1)
Q±0 = √ r+1 r−1 + (r0 )2 .
6

(4.61)

Using (4.57), (4.60) and (4.61), the matrix element of the E2 transition issr+2e~nkλ ωk3 4π X
(−)
(+1)
(2)
hb; nkλ − 1|Hint |a; nkλ iE2 = −(ekλ · Y1,M (θk , φk ))hb|QM |ai,c
8V ²0
5 M =−2s(+)

hb; nkλ + 1|Hint |a; nkλ iE2 =~(nkλ + 1)ωk3
8V ²0

e
c

r

(4.62)
+2
X

4π
(+1)
(2)
(ekλ · Y1,M (θk , φk ))hb|QM |ai.
5 M =−2(4.63)(2)

Since QM is the second rank irreducible tensor, the selection rule is given by∆J = 0, ±1, ±2,∆M = 0, ±1, ±2,

Ja + Jb ≥ 2,

πa = πb .

(4.64)

When we desire to carry out the ∆M = m transition, we should choose proper k(1)(incident direction) and ekλ (polarization) which does not eliminate Y2,m · ekλ . For the(1)
E2 transition, Y2,M are given by the following equations.
re±iφk
5 ±2iφk(1)
2
cos θk sin2 θk ê0
Y2,±2 (θk , φk ) = ∓ √esin θk (7 − 5 sin θk )ê±1 −
16π
40π
r
5 ±3iφk 3∓esin θk ê∓1
96
©
ª
e±iφk
5 cos3 θk(1)
ê±1 ∓ √sin θk 3 − 2 sin θk (5 sin2 θk − 4) ê0
Y2,±1 (θk , φk ) = √
40π
20π
r
5 ±2iφkcos θk sin2 θk ê∓1
e
+
8π
r
r
3 −iφk3(1)
Y2,0 (θk , φk ) =e(5 sin θk cos2 θk )ê+1 +cos θk (1 + 4 sin2 θk )ê0
80π
40π
r
3 ±iφk−e(5 sin θk cos2 θ)ê−1 .
(4.65)
80π
9
(K)

VQ

0

(K)

= [A(k) B(k ) ]Q
µ
X
√
0
k
2K + 1(−1)−k+k −Q
=
q
0
q,q

k0
q0

K
−Q

¶
(k)

A(k)
q Bq 04.2. MATRIX ELEMENTS AND SELECTION RULES FOR E1, E2, M1, AND M2
TRANSITIONS
55

For example, when the incident direction of the excitation laser is parallel to the z axis,
i.e., θk = 0,(1)

Y2,±2 (0, φk ) = 0(4.66)

5(1)
Y2,±1 (0, φk ) = √
ê±1
40π
r
3
(1)
ê0 .
Y2,0 (0, φk ) =40π(4.67)(4.68)(1)

Since the linear polarization along the z direction is impossible when k k z, Y2,0 (0, φk ) ·ekλ ≡ 0. As a result, only the ∆M = ±1 transition is possible in this case.

4.2.5

M2 transition

The M2 transition is induced by the term in (4.35) with µ = 0 and J=2. In the following,
we carry out calculations in case of M = ±2, which can be easily applied to the case of
M = ±1 and 0. With the aid of (4.21), (4.37) and j2 (kr) ' (kr)2 /15 (when kr ¿ 1),
µ 2¶
k
(0)
a2,±2 (r) · p 'r2 Y2,2,±2 (θr , φr ) · p
15ro
5 (1) n (1) (1)
k2
(1)
= ±
r± r±1 ê0 − r0 ê± · p
15 4π
r
k2
5 (1)
(1)
= −i
r (r × ê±1 ) · p
15 4π ±1µ¶µ 2¶r2mc
k
5 (1) (1)
= −i
r µ ,
(4.69)e15
4π ±1 ±1(2)

where (4.53) is used. Here, we define a second rank irreducible tensor Tq which is a tensor(1)(1)
product of two irreducible first rank tensor rq (position) and µq (magnetic moment)
operators and described by
(2)

T±2

(2)

T±1

(2)

T0

(1) (1)= r±1 µ±1
1 (1) (1)(1) (1)= √ (r±1 µ0 + r0 µ±1 )2
1 (1) (1)(1) (1)(1) (1)= √ (r+1 µ−1 + 2r0 µ0 + r−1 µ+1 ).
6

By carrying out same calculations for M = ±1 and 0, the matrix elements of the M2
transition are given by
s
µ ¶
+2
X
i ~nkλ ωk3 4 √
(0)
(2)
(−)
5π
(ekλ · Y1,M (θk , φk ))hb|TM |ai,hb; nkλ − 1|Hint |a; nkλ iM2 = −
c
2V ²0
15
M =−2
(4.70)

4.3. LIGHT SHIFT – FAR OFF RESONANCE TRAP (FORT)s(+)
hb; nkλ + 1|Hint |a; nkλ iM2

i
= −
c

~(nkλ + 1)ωk3
2V ²0

µ

4
15

¶

√

+2
X

5π

56

(0)

(2)

(ekλ · Y1,M (θk , φk ))hb|TM |ai.

M =−2

(4.71)(2)

As in the case of E2 transition, since TM is a second rank irreducible tensor, the selection
rule is given by∆J = 0, ±1, ±2,∆M = 0, ±1, ±2,

Ja + Jb ≥ 2,

πa = −πb .

(4.72)(0)

For the M2 transition, Y2,M are given by the following equations.
r´³√
5 ±iφk(0)
Y2,±2 =esin θk2 cos θk ê±1 ± sin θk e±iφk ê0(4.73)
16π
r
o
√
√
1 5 n
(0)
Y2,±1 = ∓(3 cos2 θk − 1)ê±1 ± 6 cos θk sin θk e±iφk ê0 − 3 sin2 θk e±2iφk ê∓
4 6π(4.74)r
1 15(0)
cos θk sin θk (e−iφk ê+1 + eiφk ê−1 )Y2,0 = −(4.75)
4 π
One of the important points in these equations is that in order to observe the ∆M = 0
transition we should let θk 6= 0 and π/2.
Let us consider, as an example, the case where θk = π/4 and φk = 0 and the polarization is parallel to the y axis, i.e., êkλ k êy which satisfies the condition k · êkλ = 0.
Then,
r
5
(0)
(4.76)êkλ · Y2,±2 (0, φk ) = i
16π
r
1 5 √
(0)
êkλ · Y2,±1 (0, φk ) = ±( 3 − 1)(4.77)
8 6π
r
i 15
(0)
.
(4.78)êkλ · Y2,0 (0, φk ) = −8 π
Thus all of the ∆M = 0, ±1, ±2 can be observed while the relative transition strengths
are different.

4.3

Light shift – Far Off Resonance Trap (FORT)The evaluation of transition matrix elements also enables the estimation of light shift, i.e.
the trap depth of a FORT.
When atoms are in a laser field, according to the perturbation theory, the laser field
causes a shift of an atomic energy level (∆E). This is called “light shift” and can be
described as
X |hb|Hint |ai|2.
(4.79)∆E =Ea − Ebb(6=a)

4.3. LIGHT SHIFT – FAR OFF RESONANCE TRAP (FORT)

57

Here we assume that Hint describes a dipole interaction between atoms and a laser field,|ai is the level considered and |bi is the energy levels which connect to a through theinteraction Hint . Ea and Eb are the energies of |ai and |bi, respectively.
We can utilize this property for trapping atoms in space because ∆E in (4.79) can be
written by the intensity of the laser field I(r),e2 I(r) h X
ωab i
2
.
U (r) ≡ ∆E(r) =|hb|r|ai| 2²0 ~cωab − ω 2

(4.80)b(6=a)

Atoms are Trapped!
00Potential for atoms U(r)Potential for atoms U(r)This equation shows that we can trap atoms because a position-dependent potential U (r)makes the force F(r) to atoms,
F(r) = −∇U (r).
(4.81)

Atoms run away...

00Position

Focal Point

Lens

Position

Focal Point

Lens(a) U(0) < 0(b) U(0) > 0

Figure 4.1: Whether we can trap atoms or not depends on the sign of the light shift.
Note that the sign of this potential could be both positive and negative, depending on
the sign of parentheses [] in (4.80). If it is negative (−), the potential depth is lowest at a
focal point and we can trap atoms. But if it is positive (+), we can not trap them because
the focal point is the most unstable point for atoms (Fig.4.1). Hence, before trying totrap 3 P2 atoms in a FORT, it is better to calculate the trap depth or its sign at least. In
the following discussion, the approximate calculation will be carried out.

4.3. LIGHT SHIFT – FAR OFF RESONANCE TRAP (FORT)

4.3.1

58

Numerical estimation

From (4.79) and (4.31)
X |hb|Hint |ai|2∆E =b(6=a)

Ea − Eb=

X |hb|H (+) + H (−) |ai|2
int

int

Ea − Ebb(6=a).

(4.82)

By using (4.40), light shift can be calculated as∆E(r) =

Xhb(6=a)=i|hb; n − 1|Hint |a; ni|2|hb; n + 1|Hint |a; ni|2+
(Ea + n~ω) − {Eb + (n − 1)~ω} (Ea + n~ω) − {Eb + (n + 1)~ω}
(−)

(+)

X h |hb; n − 1|H (−) |a; ni|2intb(6=a)

Ea − Eb + ~ω

(+)

oi
1
1
+
2²0 V
Ea − Eb + ~ω Ea − Eb − ~ωb(6=a)
X h n~ωc e22ωab i=|ê · hb|r|ai|2 2
V 2²0 ~c
ωab − ω 2b(6=a)
X h e2 I(r)ωab i|ê · hb|r|ai|2 2
,
=
²0 ~c
ωab − ω 2'

X h e2 ~ωn|hb; n + 1|Hint |a; ni|2 i+
Ea − Eb − ~ω|ê · hb|r|ai|2n(4.83)b(6=a)

where I(r) = n~ωc/V and n + 1 ' n because the photon number n is very large in thepresent system. ê is a polarization vector (unit vector) of the FORT beam and r is parallelto the direction of an induced dipole moment. We use a linearly polarized laser beam.
Thus, if we let the polarizing direction be z-axis10 ,(1)ê · hb|r|ai = hb|r0 |ai.

(4.84)e2 I(r) X h
ωab i
(1)
|hb|r0 |ai|2 2.
²0 ~cωab − ω 2

(4.85)The light shift is given by∆E(r) =b(6=a)

It is very
P difficult to numerically calculate this equation because we have to consider
all levels ( b(6=a) ) which connect with the 3 P2 state through the interaction Hint .
Here, instead, we will carry out the calculation by using the information about a few
levels given in [60]. Of course, this is not good enough to determine whether we cantrap 3 P2 atoms in a FORT or not. However in order to estimate the higher limit of thepotential depth, it is worth carrying out.
By using the Wigner-Eckart theorem,(1)

hb|r|3 P2 i = hb, J, M |r0 |3 P2 , 2, M 0 = (0, ±1, ±2)i
µ
¶
J
1 2
J−M
= (−1)
hb, J||r(1) ||3 P2 , 2i.
−M 0 M 0
10

Details of tensor analysis for E1 transition will be discussed in 2.3.5.

(4.86)

4.3. LIGHT SHIFT – FAR OFF RESONANCE TRAP (FORT)59

In our experiment, only the case of M − M 0 = ∆M = 0 has to be considered becausea Wigner 3j-symbol doesn’t vanish only when M − M 0 = ∆M = 0. In this case, (4.85)
becomes
µ
¶2
e2 I0
ωab
J
1 2
∆E =
|ha, J||r(1) ||3 P2 , 2i|2 2(4.87)
0²0 ~c −M 0 M
ωab − ω 2
where I0 is the intensity at the focal point.
The theoretical values of reduced matrix elements are listed in Table 4.1 [60]. Our
experimental parameters are I0 = 2.1 × 1010 W/m2 (532 nm, 6.4 W, beam waist=14 µm).
Table 4.1: Reduced matrix elements of Yb [60][atomic unit:a0 ]3
3

D1 (5d6s)

P2 (6s6p)

3

D2 (5d6s)

0.60

3D3 (5d6s)

2.39

1

D2 5d6s)

6.12

3

S1 (6s7s)

0.38

5.05

D2 (5d6s)

3

Table 4.2: Light shift [MHz]3

D1 (5d6s)

3

D2 (5d6s)

3

D3 (5d6s)

1

S1 (6s7s) Total

3

P2 (6s6p)(M =±2)

−

0.50

1.31

0.0226−

1.84

3

P2 (6s6p)(M =±1)

0.0222

0.125

2.10

0.00565

7.65

9.90

3

0.0296

0

2.36

0

10.2

12.6

P2 (6s6p)(M =0)

According to this estimation, light shifts are positive for all magnetic sublevels (1.84,
9.90 and 12.6 MHz). This means that we can not trap any atoms in the 3 P2 state.
However, in this study, we succeeded in trapping 3 P2 atoms in a FORT at 532 nm. Thismeans that the light shift for the 3 P2 state is negative and that the influence from upper
levels we did not include in above calculations is very large.

60

Chapter 5Line shift and broadening
Laser spectroscopy is one of the strongest techniques to understand the microscopic systems such as atoms, molecules and ions. Absorption and emission spectra include richinformation about the atomic internal structure and interaction between atoms and an
external field. Usually, such effects appear as line shifts and broadenings of spectra. Thus,in order to extract information as much as possible from the spectrum, we have to understand the mechanism of the line shift and broadening, which is the main purpose of thischapter.

5.1

Doppler shift and recoil shift

Let us consider the relativistic description of energyp
E = (pc)2 + (m0 c2 )2 ,(5.1)

where p and m0 are the atomic momentum and rest mass and c is the speed of light.
Considering the case where an atom in the initial state (momentum pi ) is excited to thefinal state (momentum pf ) by absorbing a photon (~k), the conservation of energy andmomentum arepphνL + (pi c)2 + (m0 c2 )2 = (pf c)2 + (m0 c2 + hν0 )2(5.2)
pi + ~k = pf ,(5.3)
where νL and k denote the frequency and wavenumber vector of the excitation laser andν0 is the resonance frequency between states i and f. Substituting (5.3) into (5.2) andtaking some leading terms,
µ 2¶
vi
(~k)2
−
hν0 + · · · ,(5.4)
hνL = hν0 + vi · (~k) +2m0
2c2
where vi is a velocity vector of atoms in the initial state. Due to the second, third andforth terms, the resonance frequency is shifted from ν0 and broadened. In the following,details of these terms will be discussed.

5.1. DOPPLER SHIFT AND RECOIL SHIFT

61First-order Doppler shift - Doppler broadening
Let us consider that the laser (frequency νL ) is irradiated to an atom which moves withthe velocity vi . Due to the Doppler effect, atoms absorb the photon whose frequency is,in the rest frame,νLνL = ν0 + (ek · vi ),(5.5)
cwhere ek denotes the unit vector of k. This is the second therm in (5.4), which is known
as the first-order Doppler shift. However, due to the linear dependence on v, the firstorder Doppler shift are observed not as a frequency shift but as a broadening. Under
thermal equilibrium conditions, the velocity of atoms is governed by the Maxwell velocity
distribution," µ ¶#
r
2
1
vk
2kB T
p(vk )dvk = √ exp −
dvk ,
vp =
,
(5.6)
vp
m
vp π
where vp is the most probable velocity and vk = ek · vi . Thus substituting (5.5) into (5.6)
and integrating it by vk , the lineshape I(ν) is described as"µ¶2 #c νL − ν0I(νL ) = I0 exp
,
(5.7)
vp ν0
and its full width at half maximum issµδνFWHM = 2ν0

2kB T
mc2

¶
ln 2.

(5.8)

For the 1 S0 ↔3 P2 transition (507 nm) in 174 Yb, the Doppler width (FWHM) is given byp(5.9)
δνFWHM [kHz] = 32.0 × T [µK].
Second-order Doppler shift
The next frequency shift in (5.4) related to the atomic velocity vi is the fourth term, which
is known as the second-order Doppler shift (SODS). Due to its quadratic dependence onvi , the SODS leads to a frequency shift. For the 1 S0 ↔3 P2 transition (507 nm) in 174 Yb
at 1 µK (vi = vp = 1 cm/sec), it is
µ 2¶
vi
ν0
δνSODS =2c2= 3.1 × 10−7 Hz.
(5.10)
Thus, the SODS can be safely neglected in this work. In general, the SODS yield alarge influence to the high-energy transition in light atoms such as the UV transition inhydrogen [61].

5.2. TEMPERATURE SHIFT

62

Recoil shift
The third term in (5.4) corresponds to the frequency shift caused by the momentum ofa photon ~k received by an atom. It is called a recoil shift. For the 1 S0 ↔3 P2 transition(507 nm) in 174 Yb, it is given by~k 2
4πm= 4.423 kHz.

δνrecoil =5.2(5.11)

Temperature shift

When atoms are confined in a trap, vibrational energy levels of the trap are occupied,depending on the Maxwell-Boltzmann distribution at an atomic temperature T . In such acase, the peak position of the spectrum does not correspond to the bottom of the trap but
to the peak position of the Maxwell-Boltzmann distribution in a trap [62]. As a result,
the resonance frequency slightly shifts except for the case of the magic wavelength. Sucha spectral shift which is referred to as a temperature shift was introduced in [63] and isbriefly summarized here.
According to the Fermi’s golden rule, the absorption spectrum is given by
X
I(ω) =
ρi |hf |V |ii|2 δ(ω − ωf,i ),(5.12)
i,f

where ρi is the distribution function for the trapped atoms, |ii = |gi |ni and |f i = |exi |miare the initial and final states of an atom. |nig = |nx , ny , nz ig and |miex = |mx , my , mz iex
describes the vibrational levels of an atom in a harmonic trap. Let the incident directionof the excitation laser be x. Then the interaction V between an atom and an excitation
laser can be given byX
V =(|exi Tex exp (iκx) hg| + H.c.) ,(5.13)
exwhere Tex describes the transition operator (rank 2 irreducible tensor in case of the M2
transition) and κ describes the wave number of the excitation laser.
Atoms in the ground state is in a certain vibrational level of a harmonic trap. Suchatoms can be excited to one of the vibrational levels in a harmonic trap. Then, strictly
speaking, the spectral shape I(ω) is obtained by summing the overlap integral over possible
vibrational levels in (5.12). However, when the number of atoms N is large as in thisexperiment N ∼105 , such calculations are not realistic.
We think of the semiclassical approximation which is valid when the temperature of
atoms is so high that the semiclassical approximation is valid. In this approximation, we
can label atoms by the radius vector r and momentum vector p and the density of statesin the six-dimensional phase space is h−3 . Then, the number density of atoms ρ(r, p, T )in the phase space is1
1,(5.14)
ρ(r, p, T ) = 3 β(Hg −µ)h e±1

5.2. TEMPERATURE SHIFT

63

where + and − correspond to the Fermi-Dirac distribution and the Bose-Einstein distribution, respectively. For the present purpose, it is true that we have to consider only
the Maxwell-Boltzmann distribution. However, for the future convenience, we derive thespectral shape for all distributions. Hg describes the energy of an atom in a harmonic
trap and given by
Hg,ex (r, p) =M Ω2g,ex (x2 + λy y 2 + λz z 2 )
p2
+
,
2M
2

(5.15)

where M is the atomic mass, Ωg,ex is the trap frequency for the ground state and the excited state, respectively. The dimensionless parameters λy and λz describe the anisotropyof the trap. Substituting (5.13), (5.14) and (5.15) into (5.12) and considering the energy
conservation gives
Z
I(ω) = A ρ(r, p, T )δ(ω − ωr,p )d3 rd3 p,(5.16)M (Ω2ex − Ω2g )(x2 + λy y 2 + λz z 2 ) px κωr,p =++ ωrec + ω02~M
where A = |Tex |2 and ωrec = ~κ2 /M is the recoil energy of the excitation laser.
The delta function in (5.16) can be described as
µ ¶ ·
µ ¶½
¾¸
M Ω2g
M
M
2
δ(ω − ωr,p ) =δ px −
ω − ωmax +
mex r
,
κ
κ
2~

(5.17)(5.18)

where ωmax = ω0 + ωrec and mex = 1 − Ω2ex /Ωg . Thus we can perform the integration ofpx and let dpy dpz = p cos φdpdφ, we find
¶Z
µ
Z ∞
pdp
2πM
3
"
#
dr
, (5.19)
I(ω) =
ff2

3
βM Ω2
M Ω2
hκ
β
βM
g
g 2
2
0
ω−ω
+
m
r
−
r
+βµ
±1
( 2M
)p2 − −( 2κ
ex
)
max
2
2~
2
e
where the integration of p is known as a polylogarithm Lin (x) which satisfies
Z ∞k s dk= ∓Γ(s + 1)Li1+s (∓eµ ),k−µ ± 1e
0(5.20)

where Γ(x) is the gamma function. When s=0, Γ(1) = 1 and Li1 (x) = − ln(1 − x). Thus
the lineshape is given by
 Z
"
"
¶ ##
µ
∞
2 2

m
y

ex

y 2 dy± ln 1 ± exp βµ − y 2 − p ω − ωmax +



β~
0




(+: Fermi, − : Bose)
A0 (λy , λz )×I(ω) =(5.21)"
16πα6 Ω6g p5/2 
¶2 #
µ
Z ∞

2

mex y

2

exp
βµ
−
y
−
p
ω
−
ω
+
y 2 dy

max


β~
0



(Maxwell)5.3. COLLISION SHIFT AND BROADENING

5.3

64

Collision shift and broadening

Interactions between atoms leads to the line shifts and broadenings, which is known
as the collision shift and broadening. Studies of such shift and broadening have been
considerably improved [64, 65, 66, 67, 68] because the collisional line shift is a major partin the inaccuracy of the present time and frequency clocks such as Cs or Rb fountains[69, 70].
For bosons at very low temperatures, only s-wave scattering occurs which can becharacterized by the scattering length a. For a nondegenerate gas of bosons, the meanfield energy due to elastic collisions between atoms shift the energy level of the atom,which is given by
µ
¶
4~n
a11 ,
(5.22)∆E = hmwhere n is the density and a11 is the s-wave scattering length of the collision between
state 1 atoms. When atoms are excited to the state 2, the energy level of the 2 state isalso shifted due to the atomic interaction between atoms in the 1 state and 2 state. This
interaction can be characterized by the scattering length a12 and the shift of the energy
level of the 2 state is given by replacing a11 with a12 in (5.22). As a result of shifts ofenergy levels both of the 1 state and 2 state, the resonance frequency between these statealso shifts, which is give by the difference between them,∆ν12 = (a12 − a11 )

4~n1
,
m

(5.23)

where n1 is the density of atoms in the 1 state. Since we do not consider the interaction
between atoms in the 2 state, (5.23) is valid for the weak excitation in which the 2 state
atoms interact only with the 1 state atoms. More rigorous and general description has
been traditionally given using the quantum Boltzmann equation, in which the collision
shift ∆νcol and the broadening ∆γcol are described by
2~ X
nj (1 + δ1j )(1 + δ2j )(a2j − a1j ),
m j
2~ X
=
nj (1 + δ1j )(1 + δ2j )(a21j + a22j )k,m j∆νcol =

(5.24)∆γcol(5.25)

where δab is the Kronecker delta
p and ~k is the relative momentum of the colliding atoms.
~k = mvrel /2 where vrel = 4 (kB T )/(πm) is the mean relative velocity. Note that weneglect the inelastic contributions between atoms in the state 1. As for Yb in a FORT,
only possible inelastic process is the three body recombination, which is negligibly small[24]. In this study, we have to consider only the case in j = 1 in (5.24) and (5.25). Thus,
the collisional shift and broadening for Yb atoms at 1 µK is given by
4~(a12 − a11 )n1m= 1.4494 × 10−15 (a12 − a11 )[nm] × n1 [cm−3 ]∆ν[kHz] =(5.26)

5.4. OTHER SHIFTS AND BROADENINGS

4~ 2(a + a212 )km 11p= 1.554 × 10−17 (a211 + a212 )[nm2 ] × n1 [cm−3 ] × T [µK].

65∆γ[kHz] =(5.27)For the typical parameters n1 = 5×1013 cm−3 , a11 = 5.53 nm, a12 = −33 nm and T = 1µK, the collision shift and broadening become 2.8 kHz and 0.87 kHz, respectively. Thus
both are relatively small if we compare them to other shifts and broadening. However,collisional shift and broadening plays a significant role in a BEC spectrum.

5.4

Other shifts and broadenings

Interaction time broadening
Since the interaction time between atoms and an excitation pulse is finite, it also yields aspectral broadening. Let us consider the square excitation pulse given by½
E0 cos ω0 t (−τ /2 < t < τ /2),(5.28)
f (t) =
0(otherwise)
where E0 is the amplitude, ω0 is the laser frequency and τ is the interaction time. Fromthe inverse Fourier transformation of f (t), the power spectrum (of the positive frequency
component) is given by
µ
¶2
E0 τ
sin2 [(ω − ω0 )τ /2]
2|F (ω − ω0 )| =.
(5.29)
2[(ω − ω0 )τ /2]2
The full width of half maximum (FWHM) of this spectrum is given by
0.8859.
(5.30)
τ
Thus the laser frequency is broadened due to the finite interaction time τ . This is theinteraction time broadening. For example, a square excitation pulse for 1 ms has theinteraction time broadening of 886 Hz. In this work, the pulse width of the excitation
FWHM= 89 Hz). Hence, it is negligible.
laser is at least 10 ms (∆νint
FWHM
∆νint
=

Saturation broadening
Let us consider N two-level atoms. By using the Einstein coefficients A and B, the stationary state including the spontaneous emission and stimulated emission and absorptionof the radiation field ρ(ν) is given by(Bρ(ν) + A)N2 = Bρ(ν)N1 ,(5.31)

where N1 and N2 are the number of atoms in the ground and excited state, respectively.
Here we introduce the saturation parameter S defined by
S=

2B
ρ(ν).
A

(5.32)

5.4. OTHER SHIFTS AND BROADENINGS

66

From (5.31), the population N2 is written as
N2 =

S
N.
2(1 + S)(5.33)

When we consider the atomic transition, the frequency dependence of absorption B isgiven by the Lorentzian function. Hence,
S=

1 2
( 4πτ
)
,
1 2
Isat ( 4πτ ) + (ν − ν0 )2I

Isat =

πhc
3λ3 τ

(5.34)

where τ is the lifetime of the excited state, I and ν are the intensity and frequency of theexcitation laser. Substituting (5.34) into (5.33), we havep
( γ2 1 + I/Isat )2
I/Isatp
N2 =,(5.35)
2(1 + I/Isat ) ( γ2 1 + I/Isat )2 + (ν − ν0 )2
where γ = 1/(2πτ ). This is a Lorentzian function with the broadened linewidthpγ 0 = γ 1 + I/Isat .

(5.36)0Thus
p the observed linewidth γ becomes broader than the original linewidth by a factorof 1 + I/Isat which is known as saturation broadening (or power broadening).

67

Chapter 6
Inelastic collisions in opticallytrapped ultracold metastableytterbium
Optical trapping of dense and ultracold metastable 3 P2 atoms has been successfully
demonstrated and their collisional properties in a FORT have been investigated.
Previously, several laboratories have realized laser cooling and trapping of 3 P2 atoms
in a magnetic trap. In spite of successes of trapping of 3 P1 atoms, evaporative cooling of
3
P2 atoms in a magnetic trap turned out to be unsuccessful due to trap loss caused bystrong multichannel collision processes. The loss induced by multichannel collisions in amagnetic trap can be overcome by employing a FORT. Thus we have first demonstrated
the optical trapping of Yb[3 P2 ] atoms in every magnetic sublevel. Also, the trap frequency
for the atoms in the 3 P2 state in a FORT was measured by using a parametric resonancetechnique. Whether or not we can trap Yb[3 P2 ] in a FORT at 532 nm was unknown
before this study because the precise calculation of transition strength between the 3 P2state and a lot of upper levels which include relativistic effects is difficult for heavy atoms
such as Yb. Moreover, during the course of the experiment, the light shift of magnetic
sublevels M = ±1, ±2 of the 3 D2 state was also determined. These experimental resultsmust be valuable for the precise theoretical calculations of Yb which is important, forexample, to understand the ultraprecise atomic clock using the clock transition between
the ground state and the metastable state in Yb.
By exciting pre-cooled Yb[1 S0 ] atoms in a crossed FORT, we have successfully obtained
high density ultracold metastable Yb[3 P2 ] atoms . These atoms are prepared at a densityof 2×1013 cm−3 and at a temperature of less than 2 µK. Although in principle a PSD
could be increased by further evaporative cooling of the Yb[3 P2 ] atoms to achieve a BEC,
we found this impossible due to the large trap loss.
We, then, next measured the trap lifetime of atoms in the 3 P2 state trapped in a
FORT and quantitatively measured the inelastic collision rate constant. Considering
the suppression of multichannel collisional loss in a FORT which can trap atoms in every
magnetic sublevel of the 3 P2 state with zero magnetic field, the observed inelastic collision
rate is anomalously large. As a result, a different inelastic collision process – fine-structure6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

68

changing collisions – is strongly suggested. The previous theoretical works revealed detailsof fine-structure changing transitions in collisions of Mg[3 Pj ], O[3 Pj ], Sc[2 Dj ], and Ti[3 Fj ]atoms with closed-shell atoms at a high temperature. However, there has not been no
theoretical work on the fine-structure changing collisions between atoms in the 3 P2 state at
ultralow temperature achieved in the present work. As our experimental results suggest,
further theoretical investigation is warranted.

6.1

Optical trapping of Yb atoms in the 3P2 state

6.1.1

Preparation and detection of atoms in the 3 P2 state in the
FORT

Basic procedure
Our typical procedure to prepare and detect atoms in the 3 P2 state is summarized in Fig.
6.1. At the first stage A, we prepare cold 174 Yb atoms in the ground state in a FORT by
the method described in Chap. 2 and briefly summarized here. With about 10 s loading,we typically collect 107 Yb atoms in the ground state at a temperature below 50 µK by
the MOT using the narrow intercombination transition 1 S0 ↔3 P1 . These atoms are thentransferred to a single or crossed FORT created by focused diode-pumped solid state 10
W-lasers at 532 nm. After carrying out evaporative cooling by gradually reducing the trap
depth, we typically have 106 atoms at a temperature of 30 µK in a single FORT. When
we use a crossed FORT configuration and perform evaporative cooling, a temperature of
atoms reaches below 1 µK.
At the next stage B, we optically excite atoms to the 3 P2 state in a FORT. We use
the intermediate 3 D2 state for the excitation (transition linewidth: 346 kHz [71]) and thesubsequent spontaneous decay to the 3 P2 state with the lifetime 460 ns of the 3 D2 state.
From the 3 D2 state, about 10% of atoms can decay to the 3 P2 state by the spontaneousemission. Although other atoms decay to the 3 P1 state, they immediately decay to theground state with the lifetime 875 ns of the 3 P1 state and are re-excited to the 3 D2 state.
Thus, after iterating this excitation cycle for several times, we can transfer all atoms to
the 3 P2 state typically within 5 ms. In order to efficiently excite atoms, we stabilize ablue GaN laser diode (404 nm) by an external cavity laser diode system combined with
an optical feedback technique (see section 2.4.3). The resulting linewidth is narrowed to 1
MHz for 0.5 s. The excitation laser is superposed with the FORT laser (see Fig. 6.2) andthe peak intensity at atoms reaches 140 W/cm2 . Just after the excitation, we irradiate ashort blast pulse which blows remaining atoms in the ground state by using the strong1
S0 ↔1 P1 transition.
The 1 S0 ↔3 D2 transition (404 nm) is the electric quadrupole (E2) transition whose
properties are summarized in Table.6.1. In this experiment, as shown in Fig. 6.2(top),
we set ~e404 k z (or ~e404 ⊥ z) and ~ek k x, where ~e404 and ~ek = ~k/|~k| are the polarizationand the wavenumber vectors of the 404 nm excitation laser. Thus atoms are excited tothe 3 D2 , m = ±1 and m = ±2 in case of ~e404 k z and ~e404 ⊥ z, respectively due to the6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

TOF

FORT
(532 nm)(A)(B)

Excitation(404 nm)(C)
Holding time t

Re-pumping(649 & 770 nm)
Blast

Blast & Imaging(399 nm)
(A)
3D

3S
1

(B)
3D

2
3P
2
3P
1

1S
0

69

2

404
nm
1S
0

3S
1

Imaging

(C)
3D

3P
2
3P
1

2

1S
0

3S
1

3P
2
3P
1

(D)
(E)

(D)
3D

2

1S
0

1P
1

(E)
399
nm

1S
0

649
nm

3S
1
770
nm
3P
2
3P
1
3P
0

(A)
(B)
(E)


m

Figure 6.1: Experimental procedures to excite atoms in the 3 P2 state. Absorption imagesof atoms in the ground state at stages (A), (B) and (E) are also shown. (A): Yb atoms
trapped in a MOT are transferred to the FORT and then are excited to the 3 P2 state in
the FORT via the intermediate 3 D2 state and subsequent spontaneous decay (3 D2 →3 P2 )(B). Since all atoms are now in the 3 P2 state, no atoms are observed in the absorption
image (B) of atoms in the ground state. (C): After the excitation, metastable atoms areleft in the FORT during the holding time t. (D): In order to measure temperature andthe number of atoms by using an absorption imaging, metastable atoms are repumped to
the ground state by 770 nm (3 P2 ↔3 S1 ) and 649 nm (3 P0 ↔3 S1 ) laser pulses of 100 µs
and then atoms are detected (E).

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

 

 

70

         









         

Figure 6.2: (Top) Polarizations of the excitation laser (404 nm) and FORT laser are shownfor two cases used in this experiment. (Bottom) Relative ratio of transition probability via
spontaneous emission from the 3 D2 state to the 3 P2 state are shown for two configurations
indicated above.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

71

selection rule (see Chapter 4). Subsequently, atoms decay to the magnetic sublevels of
the 3 P2 state |m|=0, 1, and 2 with the transition ratio shown in Fig. 6.2(bottom).
At the stage C, we hold metastable atoms in a FORT for a time t. Then, at the stage
D, in order to measure the temperature and the number of atoms in the 3 P2 state witha time-of-flight (TOF) technique, we rapidly repump all the metastable 3 P2 atoms to theground state after a TOF time. A 770 nm (3 P2 ↔3 S1 ) resonant pulse excites atoms fromthe 3 P2 state to the 3 S1 state, from which all the 3 P2 , 3 P1 , and 3 P0 states are populated
through spontaneous decay. By simultaneous application of a 649 nm (3 P0 ↔3 S1 ) resonant
pulse, all atoms return through the 3 P1 state to the ground state where we can use anabsorption imaging technique using the strong cyclic 1 S0 ↔1 P1 transition at the stage
E. Since 100 µs duration for the repumping procedure is short enough compared with atypical TOF time, we can safely regard that the observed atomic distribution preciselyreflects that of atoms in the 3 P2 state.
Table 6.1: Important values of the 1 S0 ↔3 D2 transition [71].
1

S0 ↔3 D2

Wavelength

λ

404

nm

Lifetime

τ

460

ns

Natural linewidth

Γ
1
=
2π
2πτ

346

kHz

Saturation Intensity

Isat =

πhc
3λ3 τ

0.69

mW/cm2

Doppler Broadening at 1 µK (FWHM)

δνD

40

kHz

Excitation rate (1 S0 ↔3 D2 )Let us calculate the excitation rate of the 1 S0 ↔3 D2 (E2) transition. From the Fermi’s
golden rule, the average transition rate Rab from one state |ai to the other state |bi is
given by
Z ∞
2π 3
1
2
δ(ω − ωab )g(ω)dω(6.1)
Rab = 2 |h D2 |Hint | S0 i|~
0where g(ω) is the normalized Lorentzian lineshapeg(ω) =

γ/2π.
(ω − ωab )2 + γ 2 /4

(6.2)

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE72

Here γ/2π is the linewidth of the (laser or atomic) spectrum. Substituting a squared
matrix element in (4.62) for this equation and using the relation I = n(~ω)c/V where I
is a laser intensity, we have|h3 D2 |Hint |1 S0 i|2 =ω 2 e2
I|h3 D2 |Q(2) |1 S0 i|2 .
16²0 c3(6.3)

As for the matrix element h3 D2 |Q(2) |1 S0 i, we have to consider the magnetic sublevels of
the 3 D2 state. In other words, we should calculate
h3 D2 , J = 2, M |Q(2) |1 S0 , J 0 = 0, M 0 = 0i.

(6.4)

We can use the Wigner-Eckart theorem to factor (6.4) because Q(2) is an irreducible tensor.
For any possible transitions from the 1 S0 state to the 3 D2 state, the matrix element takes
the same value because
µ
¶2
1
2
2
0
= .
(M = −2, −1, 0, 1, 2).
(6.5)−M q(= M ) 0
5
Thus the matrix element can be calculated as³ 1´2|h3 D2 , J = 2, M |Q(2) |1 S0 , J 0 = 0, M 0 = 0i|2 = √ h3 D2 , 2||Q(2) ||1 S0 , 0i .
5

(6.6)

The transition rate Rab is obtained by (6.1) and (6.6),
Rab (ω) = I´2
2π ω 2 e2 ³ 1 ´³ 1 3
(2) 1
√
h
D
,
2||Q
||
S
,
0i
.
2
0
~2 8²0 c3 πγ
5

(6.7)

For numerical computations of the transition rate Rab , a reduced matrix element is the
only unknown parameter. It is theoretically calculated in [71] for the 1 S0 →3 D2 transition.
1√ h3 D2 , 2||Q(2) ||1 S0 , 0i ' 0.5(4)a20
5(theory)(6.8)

In this study, the peak intensity of the 404 nm excitation laser I(= 2P/(πw02 )) is 140
W/cm2 (P = 2 mW, w0 = 30 µm) and the linewidth of the 1 S0 ↔3 D2 transition is 1/γ= 460 ns. Therefore the excitation rate at the resonant frequency can be computed as
Rab = 66 kHz.
By using this result, let us simulate the excitation process. The rate equations aredNS (t)dt
dND (t)dt
dNP1 (t)dt
dNP2 (t)dt= −Rex NS (t) + RP1 NP1 (t)(6.9)= Rex NS (t) − RD ND (t)(6.10)= 0.88RD ND (t) − RP1 NP1 (t)(6.11)= 0.12RD ND (t)(6.12)

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE73

where NS (t), ND (t), NP1 (t) and NP2 (t) are the number of atoms of the 1 S0 , 3 D2 , 3 P1 andP2 states, respectively. Constant RP1 , RD and Rex are 1/RP1 = 875 ns, 1/RD = 460
ns and Rex = 55 Hz1 . 0.88 and 0.12 are the branching ratio of the 3 D2 →3 P1 (0.88) and3
D2 →3 P2 (0.12) transitions.
The results of numerical computations are shown in Fig. 6.3 as a function of excitation
time. According to this result, we can excite all atoms in the ground state to the 3 P2state in 3 ∼ 4 ms, which is almost consistent with an experimental result.
3

1.0

1S0
3P2

Population

0.8
0.6
0.4
0.2
0.0
0

1

2

3

4

5

6

7

8

9

10

Excitation time (ms)Figure 6.3: Numerical simulations of the excitation 1 S0 ↔3 P2 through the 3 D2 state. 3 ∼4 ms is required to excite all atoms, which is consistent in an experimental result.

1

The effect of saturation broadening is included.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

6.1.2

74

Polarizabilities of the 3 D2 state



 



 


 




Figure 6.4: Time sequence for the measurement of polarizabilities of the 3 D2 state.
In a strong laser field, the energy levels of atoms shift (see section 4.3). Due to thislight shift, the resonance frequency of atoms in a FORT is shifted from the unperturbed
one. This is because the light shifts induced to atoms in the ground state and in the
excited state are, in general, different from each other 2 . The coefficient α|ai of the lightshift to laser intensity I for the state |ai which is defined by∆E|ai =

α|aiI,
4

(6.13)

is called polarizability.
We determined the polarizability of the 3 D2 state. The resonance frequency hν of the1
S0 ↔3 D2 transition in a FORT is given byIhν = hν0 − (α|3 D2 i − α|1 S0 i ),4

(6.14)

where ν0 is the unperturbed resonance frequency. We measured dependence of shift of theresonance frequency on the FORT-laser intensity, which corresponds to the measurementof α|3 D2 i − α|1 S0 i in (6.14). Since the polarizability of the ground state α|1 S0 i can be
precisely estimated, we can determine α|3 D2 i .
The time sequence of the measurement is schematically shown in Fig. 6.4. Afterloading atoms in a single FORT, excitation spectra were taken for various FORT powers.
Frequency of the 404 nm excitation laser was scanned by changing the radio frequency(RF) applied to the AOM which compensates the frequency difference between the transfer cavity and the excitation laser (see section 2.4.3). Polarizabilities of two different
magnetic sublevels (M = ±1, ±2) of the 3 D2 state were measured by choosing the proper
polarization of the 404 nm excitation laser (see Fig. 6.2).
2

By choosing a so-called “Magic wavelength”, we can cancel the difference of the light shift between
the ground state and the excited state. Such technique is crucially important in the frequency standard
and atomic clock experiment using neutral atoms in a FORT.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

75

Experimental results
The experimental results are shown in Fig. 6.5 for magnetic sublevels M = ±1 and ±2of the 3 D2 state. The excitation spectrum for different FORT powers are shown andfrequency shifts are clearly observed. By fitting Gaussian functions (solid lines) to thedata, we determined the resonance frequency of each peak. The shift of the resonancefrequency are summarized in Fig. 6.6. According to (6.14), both lines should coincide
at zero FORT power in zero magnetic field. In Fig. 6.6, two independently fitted lines
are shown. One can find that two lines coincide at zero FORT power, which proves thatthe experimental results are consistent with (6.14). Using the polarizability of the groundstate, the polarizability of the 3 D2 state are determined as listed in Table 6.2.

Table 6.2: Polarizabilities of the 3 D2 state. FORT laser is linearly polarized at 532 nm (ω0= 15 µm). The listed value for the 1 S0 state is calculated by the information of 1 S0 ↔1 P1
and 1 S0 ↔3 P1 transitions and contributions of other upper levels are neglected.
State

Polarizability (Hz· cm2 /W)

3

D2 (M = ±2)+183

D2 (M = ±1)

−23

1

S0

−24

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE76

FORT = 3.2 W
1.6 W
0.6 W

FORT = 3.2 W
1.6 W
0.6 W

Figure 6.5: Measurement of the polarizability of the 3 D2 state. Atoms are excited to(Left) M =±1 (Right) M =±2 of the 3 D2 state. The resonance frequency shifts due to thedifferent polarizabilities between the ground state and the 3 D2 (M =±1 and ±2) state.

1S

0

3D ,
2

|m| = 1|m| = 2

Figure 6.6: Shift of the resonance frequency in Fig.6.5 are summarized. Resonance frequency are plot as a function of the FORT powers. Solid lines are linear fits to data whose
slopes describe the difference of polarizabilities.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

6.1.3

77

Magnetic field compensation

Multichannel collisions in a magnetic field
3P , m2+2
EZeemanSpin flip

+1

3P ,
2

m = (0,Trap loss

0

Spin flip


1,

2)

-1

Energy-2

PositionWith B field

Without B field

Figure 6.7: (Left): Multichannel collisions in a magnetic field could lead to a trap loss
in a FORT. Atoms can obtain the kinetic energy corresponding to the Zeeman splitting
via multichannel collisions. (Right): Such trap loss process can be suppressed in zeromagnetic field.
In a FORT which can trap atoms in all magnetic sublevels, we can keep trappingof the 3 P2 atoms even if multichannel collisions occur (Fig. 6.7(Right)). However, if alarge magnetic field exists in a trap region, the multichannel collision could lead to a trap
loss. This is because atoms may obtain large kinetic energy corresponding to a Zeeman
splitting ∆EZeeman between two potentials for two different spin states. If this kinetic
energy is larger than the trap depth, atoms could escape from the trap as shown in Fig.
6.7(Left). We, then, first measured the residual magnetic field at the trap region andcanceled it by additional coils.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

78

Experiment

Evaporative
Cooling
10 msec
FORT

0.3msec
Excitation
556 nm0.1msec
Imaging

Figure 6.8: Measurement of the residual magnetic field. Zeeman splittings of the 3 P1 state
were used. In order to remove the light shift, we turned off the FORT laser during theexcitation.
To measure the magnetic field at the trap region, the Zeeman splitting of the 1 S0 ↔3 P1
transition (g = 1.49, 2.08 MHz/G) was used. The experimental procedure is schematicallyshown in Fig. 6.8. After evaporative cooling, we irradiated an excitation laser pulse (556nm). To remove the light shift, the FORT laser was turned off during the excitation.
After the excitation, 10 ms was required to open the mechanical shutter for an imaging
system.
Figure 6.9 is the observed spectrum before canceling the residual magnetic field. The
Zeeman splitting of abut 2 MHz was detected, which corresponds to the magnetic fieldof 0.9 G. Then, by adjusting currents of correction coils for x, y, z axes, we cancelled
the residual magnetic field step by step as shown in Fig. 6.10. As a result, the residual
magnetic field has been removed below 0.03 G which corresponds to a trap depth of 3 µK.
Since the measurement of trap loss of atoms discussed later was carried out in a trap of
193 µK, we regard the trap loss due to the multichannel collisions should be suppressed.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATEm = -1m=079

m = +1Figure 6.9: Zeeman splitting of the 3 P1 state due to the residual magnetic field of 0.9 G.
The number of atoms in the ground state is plotted as a function of the frequency offsetof the excitation laser (556 nm).

x axisy axisz axisx axisFigure 6.10: Residual magnetic field was compensated step by step by using compensationcoils for x, y, z axes. Positions of the resonance frequency are plotted as a function ofcurrent of compensation coils.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

6.1.4

80

Optical trapping of atoms in all magnetic sublevels of the
3
P2 state in a FORT

To begin with, we had to know whether or not we can trap the Yb[3 P2 ] atoms in a FORTat 532 nm. We confirmed that atoms in all magnetic sublevels of the 3 P2 state can be
optically trapped in a FORT at 532 nm. W e compared the number of atoms in a FORT
before and after the excitation of 1 ms to detect the untrapped state.
Due to the selection rule of the E2 transition, we can selectively excite atoms to the3
D2 (m = ±2)→3 P2 (m = ±1 and ±2) and 3 D2 (m = ±1)→3 P2 (m = 0, ±1,and ±2) bychanging the polarization of the excitation laser (see also Fig. 6.2). We compared thenumber of atoms in these two cases. First, atoms were excited to the 3 D2 (m = ±2) state.
As shown in Fig. 6.11(Left), we have not observed any trap loss after the excitation.
This means that the 3 P2 (m = ±1 and ±2) states are trap states in the FORT at 532 nm.
Similarly, by exciting atoms in the 3 D2 (m = ±1) state, we confirmed that 3 P2 (m = 0)state is also trap state (Fig. 6.11(Right)). Hence, we concluded that all magnetic sublevelsof the 3 P2 state are trap states. As discussed later, we also see that this result is consistent
with the direct measurement of the polarizability of the 3 P2 state using the 1 S0 ↔3 P2 directtransition.

3D , m=2



m=

FORT Polarization

2


2

3D
2

m=
3D , m=
2

1 0

1

2

Atoms



Polarization of theexcitation laser[1S0 3D2 (404 nm ]



M=

2

1

0

3P
2

1

Atoms
M=

1


2

1

3D


0

2

3P

2

Figure 6.11: The number of atoms in a FORT (532 nm) before and after the excitation of
atoms to the 3 P2 state are compared to detect the untrapped state. The number of atoms
in the ground state (blue square) and both the 1 S0 and 3 P2 state (red circle) are shown.
No significant trap loss was observed, which indicates that all levels are the trap state.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE

6.1.5

81

Trap frequency

!




   

"#$
 
!
 


 
 
 

Figure 6.12: Time sequency of measurement of the trap frequency of the FORT for the3
P2 atoms
In the previous experiment, we confirmed that all magnetic sublevels of the 3 P2 states
are trap states. Then, we next measured the trap depth.
In order to determine the trap depth of a FORT, parametric resonance is commonly
used. We regard a FORT potential as a harmonic potential. The atomic motion in aharmonic trap can be described byẍ(t) = −ω02 x(t),(6.15)

where x(t) is the position and ω0 is the characteristic frequency of a trap. Let us shakethe trap with the frequency ωe , that is,ẍ(t) = −ω02 (1 + h cos(ωe t))x(t)(6.16)

where h is the modulation amplitude. This is known as a Mathieu differential equation.
When the shaking frequency ωe satisfies
ωe =

2ω0
,
n

(6.17)

where n is a natural number, this system is known to become unstable due to the resonancebehavior. In such a case, atoms in a trap are strongly shaken and heated, which resultsin the trap loss. We are able to determine the trap frequency via (6.17).
The trap frequency for the radial direction ωr in a FORT is given bys
4U0,(6.18)
ωr =mw02

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE82

where m is the atomic mass and w0 is the beam waist. The trap depth U0 is proportional to
the FORT laser power P (see (4.85)). In other words, from the (6.18), the trap frequency
ωr2 is proportional to the laser power P (∝ U0 ). Thus, we can experimentally realize (6.16)
by modulating the FORT power P by the AOM at frequency ωe .
The experimental time sequence is shown in Fig. 6.12. After exciting atoms to all of
the magnetic sublevels of the 3 P2 state in a FORT, the FORT power P was modulated by
an AOM. The resonance frequencies were detected both as increasing atomic temperature(i.e., distribution width) and the number of atoms in a trap.
Experimental results

12

number of atoms
Width

5

10

4

8

3

6

2

4

1

2

0

Width (µm)

Number of atoms (x104)

6

0
0

2

4

6

8

10

12

14

16

18

20

Modulation Frequency (kHz)

Figure 6.13: Parametric resonance. The number of atoms (red) and distribution widthof the atomic cloud (black) are plotted as a function of modulation frequencies. Tworesonance signals are clearly observed at 8.6 and 4.3 kHz.
Typical experimental results are shown in Fig. 6.13. The width of the atomic cloud
which corresponds to the atomic temperature and the number of atoms are plotted as afunction of modulation frequencies. Two resonance signals corresponding to n = 1 and 2
in (6.17) are clearly observed at 8.6 and 4.3 kHz, respectively, while the trap frequencyof the 1 S0 state in this condition is 3.9 kHz. Note that to confirm the resonance observed
at 8.6 kHz corresponds to n = 1 in (6.17), we took data up to 20 kHz (> 2×8.6 kHz). If
it is n = 2, the other peak should appear around 17.2 kHz.
The resonance frequency is determined by fitting a Lorentzian function to the widthof atomic clouds. Since all the magnetic sublevels of the 3 P2 state are populated in thismeasurement and in fact we experimentally confirmed that the atoms in every sublevel6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE83

are trapped, their trap frequencies can be thought to coincide with each other within
the resolution of this measurement. This result is consistent with the measurement ofpolarizabilities of magnetic sublevels of the 3 P2 state using the ultranarrow 1 S0 ↔3 P2transition in our different work, which indicates that the difference of the trap frequency
between magnetic sublevels in this measurement is less than 1 kHz. Thus, in the following
discussion, we consider that the trap depth of all magnetic sublevels of the 3 P2 state aresame. With this information about the trap, we can also estimate a number density.

6.1.6High density ultracold metastable atoms

Now we know the trap depth of our FORT. Thus, what we should aim next is a BEC in
the metastable state.
We prepared ultracold and dense ground state Yb[1 S0 ] atoms in a crossed FORT at adensity of 2×1014 cm−1 and at a temperature of less than 0.7 µK. During excitation fromthe ground state to the 3 P2 state through the intermediate 3 D2 state, the atoms suffer fromheating due to the spontaneous decay of the 3 D2 →3 P1 →1 S0 transitions. Nevertheless, wehave successfully prepared Yb[3 P2 ] atoms in density n = 2 × 1013 cm−3 , temperature T =1.8 µK and PSD = 0.01. This is the highest density and lowest temperature ever achieved
for the 3 P2 atoms.
In principle, PSD could be increased by performing further evaporative cooling of theYb[3 P2 ] atoms. We, however, found that it was not possible due to a large trap loss. As
discussed in the next section, there is a large inelastic collisional process between Yb[3 P2 ]atoms in a FORT which we identify as fine-structure changing collisions.
Cross dimensional relaxation measurement
Before proceeding to the next section, I discuss the elastic collision rate of Yb[3 P2 ] atoms
at this ultracold temperature (2 µK).
In general, since the trap loss is caused both by elastic and inelastic processes, it
is difficult to separately measure one of them. Monroe et al. developed a technique toindependently measure the elastic binary collision rate, which is called cross-dimensional
relaxation measurement [72]. The basic idea is the following: First, we prepare atoms
whose axial temperature is different from the radial one in a trap by some way. Then, we
measure the thermalization time τ due to the elastic collisions between two dimensions in a
trap. According to the Monte Carlo simulation, atoms collide 2.7 times to be thermalized[72], that is to say,n̄βel τ = 2.7,
(6.19)
where n̄ is the average density. βel = σel v̄ is thep elastic collision rate constant, where σelis the elastic collision cross section and v̄ = 4 (kB T )/(πm) is the mean relative velocity.
Thus measuring the average density and the thermalization time, we can independently
estimate the elastic collision rate βel .
We first prepared atoms in the crossed FORT which were well thermalized and thenheated the atomic cloud along the x direction by using the excitation of atoms to the 3 P2state. We measured Tx and Ty as a function of holding time after the excitation. In Fig.

6.1. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P2 STATE
3.5

Tx
Ty

3
Temperature (µK)

84

2.5
2
1.5
1
0.5
0
0

5

10

15

20

25

30

35

Holding time (msec)

Figure 6.14: Cross dimensional relaxation measurement. Atoms are first heated to x
direction by the excitation laser. We could not observe any difference between Tx andTy 2 ms after the excitation (2 ms), which indicates that the system reaches thermal
equilibrium within 4 ms at this ultracold temperature and high density.

6.14, Tx and Ty are shown as a function of time t after the excitation. Even 2 ms after
the excitation of 2 ms, no significant difference between Tx and Ty was observed. Then we
conclude that the system reaches the thermal equilibrium within 4 ms and thus the lower
limit of the elastic collision rate is estimated to be 9×10−12 cm3 s−1 at the temperature of 2
µK. Assuming that the σel does not depend on T in the range of the present measurement,
this result is consistent with the estimation discussed in the following section. Note thatthe relaxation due to the anharmonicity of the trap is measured not to be less than 50
ms in our different work [26].

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

6.2

85

Collisional properties between metastable atoms

As discussed in the previous section, further evaporative cooling of Yb[3 P2 ] atoms in a
FORT to reach a BEC turned out to be difficult because the trap loss is too large. Since
atoms in all magnetic sublevels can be trapped in a FORT and then the trap loss due tomultichannel collisions is suppressed, this indicates that there is another large inelastic
loss process. In this section, details of the trap loss mechanism are discussed.

6.2.1

Two body collisions
106

180

140
120

5

10

100
80
60

104

Temperature (µK)

Numer of atoms

160

40
20
0

20

40

60

80

100

0
120

Holding time (msec)
Figure 6.15: Measurement of the trap loss of 3 P2 atoms in a FORT. Non-exponential
decay is caused by the trap loss due to two-body collisions. Constant temperature during
this measurement indicates that the system reaches thermal equilibrium.
We measured the trap loss of the Yb[3 P2 ] atoms from a single FORT. In Fig. 6.15, thenumber of the Yb[3 P2 ] atoms in a FORT are plotted as a function of time after loadingof atoms in every magnetic sublevels of the 3 P2 state. Note that the (presumably state
independent) one-body trap loss lifetime due to background gas collisions is measured to
be 15 s (Fig. 2.5) which is much longer than the observed trap lifetime. A significant
feature is the non-exponential decay of atom number along the constant temperature of
41 µK. Our model for the trap loss which includes a combination of one-body loss andtwo-body loss isdN= −ΓN − β 0 N 2 ,(6.20)dt

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

86

where N is the number of atoms, Γ is the one-body loss rate and β 0 is the measured
two-body atom loss rate. The solid line in Fig. 6.15 is a fit of (6.20) to the trap loss data.
β 0 is related to the density related (volume independent) two-body loss rate coefficient β
by β = β 0 Veff , where Veff is the effective volume of atoms given in the following discussion.
Effective volume
Here we consider the effective volume of atomic cloud in a FORT. Since a FORT potential
is obtained by a focused Gaussian beam, the potential U (r, z) is given by,
·
¸
w02
2r2
U (r, z) = −|U0 | 2
exp − 2,
(6.21)w (z)w (x)where r and z are radial and axial coordinates and U0 is the trap depth. Also,
s
µ ¶2
z
πw02
2
w(z) = w0 1 +,, zR =zR
λ(6.22)

where w0 is a beam waist, zR is a Rayleigh length, and λ is the laser wavelength. Let theradius of the atomic cloud be ρ(z). Then by considering the balance between the trap
potential and the thermal energy of an atomic cloud, we have·¸
w02
2ρ(z)2−|U0 | 2
exp − 2
= −|U0 | + kB T.
(6.23)w (z)w (x)Thus, using the parameter η (= U0 /(kB T )) which is the ratio of the trap depth U0 to thetemperature T (kB the Boltzmann constant), ρ2 (z) is given by
· 2
µ
¶¸
w (z)
w2 (z)
12
ρ (z) = −ln
1−.
(6.24)
2
w02
η
Atoms can be trapped within the region in which ρ2 (z) > 0 is satisfied. Thus the edge ofthe atomic cloud along the axial direction zmax is given byr
1.
(6.25)
zmax = zR
η−1
Since the volume of the atomic cloud can be well approximated by a cylinder with thesize of the FORT beam,
µ
¶r
η
1
2
2
.
(6.26)
Veff = πρ (0)(2zmax ) = πω0 zR ln
η−1
η−1
From the measured trap frequency, the trap depth is found to be U0 /kB =193 µK for all
magnetic sublevels. Since atomic temperature is 41 µK, we can determine η = 4.7. From
the FORT parameters w0 =15 µm and zR = 1.3 mm, we can determined the density related
two-body loss rate β = 1.3(4)×10−11 cm3 /sec. Note that β consists of both inelastic βinand elastic βel collision rates. By using a theoretical model for evaporative cooling, we
will next separately estimated βin and βel .

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

6.2.2

87

Elastic and inelastic collision rates under thermal equilibrium conditions
Elastic collisions(evaporative Cooling)

Inelastic collisions(Heating)

Figure 6.16: Loss of atoms in a trap is caused by the elastic and inelastic collisions.
While elastic collisions lead to cooling of atoms (evaporative cooling), inelastic ones leadto heating. Finally, both effects equilibrate and system reaches the thermal equilibrium.
In addition to the non-exponential trap decay, the other important feature in Fig. 6.15
is the constant atomic temperature throughout the 120 ms holding time at T = 41 µK. In
general, the trap loss due to two body collisions can be classified by elastic and inelastic
processes. The elastic collisions lead to evaporative cooling and inelastic collisions leadto heating since not only hot atoms but also cold atoms can be escaped from the trap via
inelastic collisions (Fig. 6.16). The observed constant temperature along with the large
trap loss of atoms indicates that the system reaches thermal equilibrium. In other words,
the cooling effect due to the elastic collisions equilibrates with the heating effect due tothe inelastic collisions. Under the thermal equilibrium condition, the relation between theobserved two-body decay rate β and the elastic βel and inelastic βin collision rate can beseparately described. This model was first developed by deCarvalho and Doyle [73] andis briefly summarized here.
First, we assume that the density distribution is given by
¸
·
U (r)
,
(6.27)
n(r) = n0 exp −kB T
where n0 is the peak density. This description is valid in case of the infinitely deep trap.
If the trap depth is finite, (6.27) is still valid when η > 4 [73] which is satisfied in ourcase. This approximation is called “large-η approximation”.

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

88

We next introduce the collisional effective volume. When a single inelastic scattering
leads to loss of one atom in a trap, the inelastic collision rate Ṅin is given byṄin = −(n0 v̄rel σin )(Λeff n0 )(6.28)
pwhere Λeff is the collisional effective volume, v̄rel = 4 (kB T )/(πm) is the mean relative
velocity between atoms, σin is the inelastic scattering cross section. On the other hand,under thermal equilibrium conditions,0Ṅ = Veffṅ0 ,(6.29)

00
where Veffis the effective volume defined by N = n0 Veff
. By substituting (6.29) into
(6.28), we haveµ¶
Γeff(ṅ0 )in = −v̄rel σin n200Veff= −βin n20(6.30)

Similarly, by using f the probability of elastic collisions which result in trap loss of one
atom,ṅ0 = (ṅ0 )in + (ṅ0 )el= −(βin + f βel )n20
= −βn20(6.31)

Using the ratio of elastic to inelastic scattering rate γ ≡ σel /σin ,
βin =

1
β,
fγ + 1βel =γ
β.
fγ + 1(6.32)

One of the significant conclusions in [73] is that f and γ in (6.32) depends only on η underthermal equilibrium conditions. We next calculate f and γ.
We assume that the trap is a linear elliptical shape whose long and short axes correspond to axial and radial direction of the FORT potential, respectively. Then the potential
is given bysµ ¶µ ¶2 µ ¶2 µ ¶2
2
r
z
r
z
+
,
+
≤ 1,
(6.33)U (r, z) = U0
Rz
Rr
Rz
Rr
where Rz and Rr denote the axial and radial length of the trap, respectively. However,by using Rzz = ρz and Rrr = ρr and using the polar coordinate instead of the cylindricalcoordinate, the potential (6.33) is equivalent to U (r) = U0 r (0 ≤ r ≤ 1). We, then, canapply the discussion in [73] to our system.
The total energy of atoms E in the trap is given by
E = N Eave ,(6.34)

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

89

where N is the number of atoms in a trap and Eave is the average energy of an atom in a
trap. By differentiating (6.34) by t under thermal equilibrium conditions,
Eave =Ė
Ṅeva Eeva + Ṅine Einef γEeva + Eine==,fγ + 1ṄṄeva + Ṅine(6.35)

where Ėave = 0 due to thermal equilibrium conditions and Ṅeva /Ṅine = f γ. From (6.35),
γ is written as
Eine − Eaveγ=.
(6.36)
f (Eave − Eeva )Under the large-η approximation, f , Eave , Eeva and Eine are described by only η as given
in [73] and shown in Fig. 6.17 for η > 4. For η = 4.7 in this experiment, f = 0.099 andγ = 2.2. As a result, the elastic and inelastic collision rate constant is given by
βel = 2.3(6) × 10−11
βin = 1.0(3) × 10−11

cm3 s−1
cm3 s−1(6.37)

for the temperature of 41 µK.
This inelastic collision rate constant is enormously large. Since the trap loss due tomultichannel collisions was suppressed in a FORT, the trap loss observed in this measurement must be due to a different physical mechanism which we interpret as fine-structurechanging collisions in this ultracold temperature regime.

0.16

4.6

0.14

4.4

0.12

4.2

11
10

Eave/kBT

0.08
0.06

Eeva/kBT

4.0

0.10

3.8
3.6
3.4

0.04

3.2

0.02

3.0

0.00

2.8
5

6

7
η

8

9

10

9
8
7
6
5

4

5

6

7
η

3.3

8

9

10

4

5

6

4

5

6

7
η

8

100

3.2
3.1
3.0
2.9

γ

4

Eine/kBT

f

12

10

2.8
2.7
2.6
2.5

1
4

5

6

7

η

8

9

10

7
η

8

9

10

Figure 6.17: All of f , Eave , Eeva , Eine and γ are determined by only η.

910

6.2. COLLISIONAL PROPERTIES BETWEEN METASTABLE ATOMS

6.2.3

90

Fine structure changing collisions

Energy1718.381 cm-1
3P + 3P2
2

3P + 3P
2
1 or 0Interatomic distance

3P
2
3P
1
3P
0

2421.949 cm-1

Figure 6.18: Fine structure changing collisions. If the potential curve of 3 P2 −3 P2 pairsintersect with that of 3 P2 −3 P1 or 3 P2 −3 P0 at some point, one of the colliding 3 P2 atoms
can transit to other state of the fine structure (3 P1 or 3 P0 ). In such case, the atom obtains
the energy difference as its kinetic energy.
In Fig. 6.18, the mechanism of fine structure changing collisions are schematicallyshown. If the potential curve of the 3 P2 −3 P2 pairs intersect with that of the 3 P2 −3 P1
or 3 P2 −3 P0 curves at some point, one of the colliding 3 P2 atoms can transit to otherstate of the fine structure (3 P1 or 3 P0 ) via collisions. In such a case, the atom obtains theenergy difference as its kinetic energy. The energy difference between fine structure levels
is much larger than the trap depth, such atoms immediately escape from the trap.
In our setup, atoms in every magnetic sublevel can be trapped in a FORT. Hence, thetrap loss due to the multichannel collisions in the 3 P2 state which were essential obstacles
in a magnetic trap is expected to be suppressed. Thus the existence of fine-structurechanging collisions is strongly suggested. The previous theoretical works revealed detailsof fine-structure changing transitions in collisions of Mg[3 Pj ], O[3 Pj ], Sc[2 Dj ], and Ti[3 Fj ]atoms with closed-shell atoms at a high temperature. However, there has not been no
theoretical work on the fine-structure changing collisions between atoms in the 3 P2 state
at ultralow temperature achieved in the present work. While the recent experiment ofmagnetically trapped Ca atoms studied multichannel collisions between 3 P2 atoms anddiscussed the possibility of the fine-structure changing process [2], we believe that ourwork is the first definite experimental measurement of this process between 3 P2 atoms.
For the further understanding of the observed large inelastic collision rate and possibilitiesof the fine structure changing collisions, a quantitative theory on collisional properties of
3
P2 atoms is highly desirable.

6.3. EFFECT OF THE BLACK BODY RADIATION

6.3

91

Effect of the black body radiation

For Strontium atoms in the metastable 3 P2 state, scattering rate of photons from theblack-body radiation (BBR) at room temperature is large [17]. We investigate the effectof black body radiation on the metastable Yb atoms. This calculation was first done by
Mizoguchi [74] in our group. The BBR effect is an important information in this study.
Thus it is again briefly introduced here.
The stimulated transition rate Rst by BBR is given by
Rst = Bρ(ω),(6.38)

where B is the Einstein B coefficient and ρ(ω) is the density of the photon energy of theblackbody which is well known the Planck formula,ρ(ω) =~ω 31.
~ω
π 2 c3 e kB T − 1(6.39)On the other hand, the B coefficient is related to the A coefficient (spontaneous emission)by
π 2 c3B=
A.
(6.40)~ω 3
According to these three equations, we have
Rst =

A
e

~ω
kB T

−1

.

(6.41)

To calculate the spontaneous emission rate A, let us consider the Fermi’s golden rule
which gives the (spontaneous and stimulated) emission rate
Wem =

2π
+
|hb|Hint|ai|2 δ(ωba − ωk ),~2(6.42)

where |ai and |bi are the initial and final state. In the real system, some modes dω are
included in a range of frequencies dωk . Considering the number of modes per unit energy
interval in a large cubic V ,V
k 2 dkdΩk ,(6.43)
dN =(2π)3where dΩk = sin θk dθk dφk . From the (6.42),
2πdN|hb|Hint |ai|2
2
~
dωk
2
Vω
=
|hb|Hint |ai|2 dΩk .
232(2π) c ~dWem =(6.44)

We consider only E1 and M1 transitions. By integrating the matrix element of E1 andM1 transitions given in section 4.2, we can findZ~ω 8
|hb|HR+ |aiE1andM1 |2 dΩk = (n + 1)
π|hb|M|ai|2 ,(6.45)
2V ²0 36.3. EFFECT OF THE BLACK BODY RADIATION

92

where M denotes er and µL for E1 and M1 transitions, respectively. Since the term 1
within (n + 1) corresponds to the spontaneous emission (the other corresponds to stimulated emission), integrating (6.44) and using (6.45) lead toµ
A =Jb 1 2−Mb q Ma0¶2

ω3|hb||M||ai|2 .
3π²0 ~c3(6.46)

Here we separate the dependence of the magnetic sublevels which is included in a Wigner3j symbol. Finally, from (6.41) and (6.46), the photon scattering rate in BBR which does
not include the information about the magnetic sublevels is
ω3
Rst = ~ω|hb||M||ai|2 .
3
e kB T − 1 3π²0 ~c1(6.47)In Fig. 6.19, scattering rates of the (6s6p)3 P2 →(6s6p)3 P1 (M1), (5d6s)3 Dx (x=1,2,3)(E1), (6s7s)3 S1 (E1) and (5d6s)1 D2 (E1) transitions are shown. We used theoretical valuesof reduced matrix elements given in [60]. Note that Fig. 6.19 does not include theinformation about the magnetic sublevels. If one wants to see it, one just has to multiply
the square of a Wigner-3j symbol which is always less than 1. As you can see in Fig. 6.19,
the photon scattering rate of the (6s6p)3 P2 state in BBR at room temperature is lessthan 10−4 Hz (10000 s) which is much longer than the lifetime 15 s. As a result, we canconclude that the effect of the BBR radiation can be negligible in case of the metastable
Yb[3 P2 ].
(5d6s) 3D3(5d6s) 3D2(5d6s) 3D1(6s6p) 3P1(5d6s) 1D2(6s7s) 3S1

Figure 6.19: Photon scattering rate of the 3 P2 state in the BBR radiation field. Scattering
rates from the 3 P2 state are shown as a function of temperature. At room temperature(300 K), BBR effects in the 3 P2 state are negligibly small ( < 10−4 Hz).

6.4. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P0 STATE

Optical trapping of Yb atoms in the 3P0 state
770nm

6.4

93

404nm

Optical Lattice

Hold time

FORT

TOF

5msec
Excitation(404 nm & 770 nm)
0.1msec
Repumping(649 nm & 770 nm)
0.1msec

150 m

Imaging

Figure 6.20: Trapping of Yb[3 P0 ] atoms in a FORT. (Top) By irradiating the 770-nm
laser in addition to 404-nm excitation laser during the excitation, the 3 P0 state becomes adark state. To suppress the trap loss by excitation lasers, tight confinement in 1D opticallattice potentials was used. (Bottom) Absorption image of trapped Yb[3 P0 ] atoms in a
FORT.
We report optical trapping of Yb[3 P0 ] atoms in a FORT at 532 nm. Recently, theP0 state in two-electron atoms attract interests from the viewpoint of novel frequency
standards and atomic clocks using the 1 S0 ↔3 P0 “clock” transition. A collision shift is
one of the main issues toward such applications. It is, then, important to investigate
the collisional property between Yb[3 P0 ] atoms. In this study, we have successfully trapYb[3 P0 ] atoms in a FORT and measured trap frequency by the parametric resonancetechnique.
Figure 6.20 shows experimental procedures. By irradiating the 770-nm laser (3 P2 ↔3 S1 )in addition to the 404-nm laser, the 3 P0 state becomes a dark state. However, excitation
cycles for atoms to be transferred to the 3 P0 state lead to heating and trap loss of atoms
in a trap. To overcome this problem, we made use of the tight confinement of the (1D)
optical lattice potentials during the excitation and successfully trap atoms in the 3 P0 state
in a FORT (see Fig. 6.20).
In addition, we measured the trap frequency by the parametric resonance technique.
Observed resonances are shown in Fig. 6.21. Although the number of atoms is small,
we could observe resonance signals and confirm the dependence on a FORT power. The
3

6.4. OPTICAL TRAPPING OF YB ATOMS IN THE 3 P0 STATE

94

estimated trap potentials are 130 µK for parameters P = 6 W, w0 = 15 µm and λ = 532
nm.

12

The number of atoms (3P0)
Width

4.0

11
10

3.5

9

3.0

Width (µ m)

The number of atoms (x104)

4.5

8

2.5

7

2.0

6

1.5

5

1.0

4
3

0.5

2

0.0
0

1

2

3

4

5

6

7

8

9

10

Modulation frequency (kHz)9The number of atoms (3P0)
Width

7.0

8
7

6.0

6

5.0

5

4.0

4

3.0

3

2.0

2

1.0

1

0.0

Width (µ m)

The number of atoms (x104)

8.0

0
0

1

2

3

4

5

6

7

8

9

10

Modulation frequency (kHz)

Figure 6.21: Measurement of the trap frequencies of Yb[3 P0 ] atoms in a FORT by theparametric resonance technique. Shift of the resonance signal for two different FORT
power (Top: 6.4 W, Bottom: 1.6 W) was observed.

95

Chapter 7
Observation and application of theultranarrow 1S0 −3 P2 transition
We have successfully observed the ultranarrow magnetic quadrupole 1 S0 ↔ 3 P2 transitionin Yb bosonic (174 Yb) and fermionic (171 Yb, 173 Yb) isotopes using the developed laser
system at 507 nm in Chapter 3.
Due to its ultranarrow linewidth, this transition had never been observed before thisstudy. In this chapter, we first describe the experiment to estimate the resonance frequency using an optical frequency comb. Using this transition, high-resolution spectroscopy of ultracold atoms and a BEC has been demonstrated. We have selectively
excited atoms to all of the magnetic sublevels of the 3 P2 state. Also, polarizabilities of all
magnetic sublevels of the 3 P2 state were precisely measured. We also demonstrated themagnetic resonance imaging (MRI) experiment in which atoms in a trap were addressedby using a magnetic field gradient. Details about the spectroscopy of a BEC will be presented in the next Chapter. Basic properties of the 1 S0 ↔3 P2 transition are summarizedin Table 7.1 and 7.2.

Table 7.1: Magnetic moment of the (6s6p)3 P2 state in isotopes. F is the total angularmomentum and µB is a Bohr magneton.
Bosons

171

173

Yb

Yb

F

2

3/2

5/2

1/2

3/2

5/2

7/2

9/2

µ/µB

3

2.7

3

−1

0.3

1.3

2.2

3

96

Table 7.2: Important parameters of the magnetic quadrupole 1 S0 ↔3 P2 transition in Yb[60, 75].
1

Boson
WavelengthλLifetimeτ

Natural linewidthSaturation intensity

15

πhc
3λ3 τ

Recoil frequency
Doppler broadening (FWHM)

Fermion507.35

Γ
1
=
2π
2πτ

Isat =

S0 ↔3 P2δνDFWHM

10.6

1.1

Unit
nm

6.3 (171 Yb)s

7.2 (173 Yb)s

25 (171 Yb)mHz

22 (173 Yb)mHz

2.5 (171 Yb)

×10−8 mW/cm2

2.2 (173 Yb)

×10−8 mW/cm2

4.4
p32.0 × T [µK]

kHz
kHz

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

7.1

97

Estimation of the resonance frequency

Since no one had ever observed the 1 S0 ↔3 P2 transition before our experiment, we did
not know the resonance frequency. It is rather difficult to find an ultranarrow resonancewithout any information about its resonance frequency. Thus we first roughly estimatedthe resonance frequency fest of the 1 S0 ↔ 3 P2 transition (507 nm).

7.1.1

Basic idea
3S

1

f1
649nm

f2
770nm

3P
3P

  
2

fest= f0+ f1- f2

0

f0

fest
507nm

578nm

1S

!"

0

Figure 7.1: The desired resonance frequency fest can be estimated from three known
frequencies. f0 was measured in [34] and f1 and f2 were measured in this work.
As shown in Fig. 7.1, combining three resonances of f0 (1 S0 ↔ 3 P0 , 578 nm), f1 (3 P0↔ S1 , 649 nm), and f2 (3 P2 ↔ 3 S1 , 770 nm), we can calculate fest by
3

fest = f0 + f1 − f2 .

(7.1)

The absolute frequency f0 of the so-called “clock” transition (1 S0 ↔3 P0 ) in fermion isotopes
171
Yb and 173 Yb was measured by the NIST group [34]. In order to calculate the desired
frequency fest , we then measured frequencies of the f1 (3 P0 ↔ 3 S1 , E1, 649 nm) and f2 (3 P2↔ 3 S1 , E1, 770 nm) transitions for fermionic isotopes by an optical frequency comb.
For bosonic isotopes, although the absolute frequency of the clock transition f0 had
not been presented anywhere at the moment, we can estimate fest by using a fact that theisotope shift of all 3 P states are similar to each other in Yb. To see this, let us consider
the isotope shift between 174 Yb (boson) and 171 Yb (fermion) as an example (Fig. 7.3). As
shown in Fig. 7.2, isotope shifts of the 3 Px ↔3 S1 (x=0,1,2) transitions are almost same

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

98

with each other within a several hundred MHz. By considering the isotope shift of the3
P2 ↔3 S1 and 3 P1 ↔3 S1 transitions between 171 Yb and 174 Yb shown in Fig. 7.3, we canderive following relations.
∆4 + ∆3 ' ∆4 + ∆2
∴ ∆3 ' ∆2 ,(7.2)

where ∆2 , ∆3 , and ∆4 are isotope shifts of the 3 P1 , 3 P2 and 3 S1 states, respectively. The
resonance frequency of the 1 S0 ↔3 P2 transition in boson isotopes (174 Yb in this example)
can be estimated by
171

∴

174

fest −174 fest = ∆1 + ∆3
' ∆1 + ∆2 (∵ (7.2))= 171 f3 −174 f3

fest =171 fest − (171 f3 −174 f3 ).

(7.3)

where 171 f3 −174 f3 , the isotope shift of the 1 S0 ↔3 P1 transition between 171 Yb and 174 Yb
was already measured in [42].
Isotope shifts of 3Px

3S transition
1Yb

3P (649 nm)
0170-524(21)-543.0(15)171-345(25)-326.3(20)

172

0

173

185(38)174

413(20)176805(17)

3P (680 nm)
1

0

3P (770 nm)
20196.0(5)425.3(3)801851.0(6)(MHz)

Isotope shifts in Mg (MHz)3P -3S0
1

3P -3S
1
1

3P -3S2
1

25Mg-24Mg-210(36)-201(21)-204(7)26Mg-24Mg-396(6)-390(5)-390(7)

Figure 7.2: (Top) Isotope shifts of the 3 Px ↔3 S1 (x=0,1,2) transitions in Yb are almost
same with each other within several tens of megahertz [76, 77, 78]. This characteristic
may be common in two-electron atoms. (Bottom) For example, isotope shifts in Mg atoms
coincide with each other within a several megahertz [79].

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

3S

174Yb

1

4

171Yb

3P

171Yb

2

3

174Yb
171Yb

2

1S

99

174Yb

3P

1

174Yb

0

1

Figure 7.3: Similar isotope shifts (∆2 ' ∆3 ) between 3 Px (x=0,1,2) states enable us
to estimate the resonance frequency 174 fest (unknown) from 171 fest (known) and alreadymeasured isotope shifts 171 f3 −174 f3 of the 1 S0 ↔3 P1 transition by 174 fest = 171 fest −(171 f3 −174 f3 ).

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

7.1.2

100

Frequency measurement of the 3 P2 −3 S1 (770 nm) and3
P0 −3 S1 (649 nm) transitions by an optical frequency comb

Since both the 3 P2 −3 S1 (770 nm) and 3 P0 −3 S1 (649 nm) transitions are strong electricdipole (E1) transition with natural linewidth of 12 MHz, spectroscopy of these transitions
are relatively easy. To this end, we have to first make population in the metastable 3 P0
and 3 P2 states. We use an opto-galvano cell (Hamamatsu Photonics, L2783) where Yb
atoms are emitted by an electric discharge in Neon buffer gas of about 10 Torr. Because
the electric discharge is a rather violent process, some Yb atoms are populated in the 3 P2
and 3 P0 state in the interaction region.
We performed saturation spectroscopy of metastable atoms as schematically shownin Fig. 7.4. The pump laser was chopped (3 kHz) by a mechanical chopper to get ridof the background level and the signal was obtained by a lock-in amplifier. Resonancefrequency was measured by an optical frequency comb (FC8003, Menlo Systems GmbH).
The frequency comb is generated by a femtosecond Ti:sapphire laser (Gigajet 20, GigaOptics GmbH) with a repetition rate of 824 MHz. Femtosecond pulses are injected
into a highly nonlinear photonic crystal fiber which expands the comb range to an octave
spanning. Then, the carrier-envelope offset frequency is derived from a beat frequency
in the self-referenced interferometer. Both the repetition rate and the carrier-envelope
offset frequency are locked to the commercially available Rb frequency standard (PRS10,
Stanford Research Systems). Laser frequency is measured as an RF beat signal between
the measured laser and the nearest frequency comb. Details about the frequency comb
will be presented in Chapter 9.
Because the accuracy of the measurement of tens of megahertz was good enough forthe present purpose, we measured frequency of the beat signal not by a frequency counterbut by a spectrum analyzer. We tuned the laser frequency at the top of resonance signals
obtained in saturation spectroscopy, left it there and measured the frequency of the RFbeat signal.
Observed spectra are shown in Fig.7.5 (3 P0 −3 S1 , 649 nm) and Fig.7.6 (3 P2 −3 S1 , 770nm). All resonance signals were identified by reference to the isotope shift and hyperfine
splitting previously measured in [77, 78]. Solid lines in Fig.7.5 and 7.6 are fits to data bysummation of corresponding Lorentzian functions, which well agree with observed spectra.
For fermions (171 Yb and 173 Yb), hyperfine splittings can be described by [80]
ν = ν0 + A

3K(K + 1) − 4I(I + 1)J(J + 1)K+B
2
8I(2I − 1)J(2J − 1)

(7.4)

where ν is the resonance frequency, ν0 is the unperturbed frequency (resonance frequency
when hyperfine interaction doesn’t exist), A and B are hyperfine constants, J is the electric
angular moment, I is the nuclear spin , F = I + J and K = F (F +1) − I(I +1) − J(J +1).
Hyperfine constants A, B and ν0 relative to that in 174 Yb are shown in the bottom of Fig.
7.5 and 7.6 for the relevant transitions.

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

 
HWP

101

Light
Chopper
Photo
Detector

HWP

Yb Hollow
CathodeECLD
649 nm, 770 nm
Figure 7.4: Saturation spectroscopy. A pump beam was chopped (3 kHz) for the purposeof removing the background and the resonance signal was obtained by the lock-in amplifier.
After tuning the laser frequency at the top of the resonance, the laser frequency wasmeasured by an optical frequency comb.

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

102

174Yb

172Yb

176Yb
171Yb
(1/2

3/2)

173Yb
(5/2

5/2)

170Yb
171Yb

173Yb

(1/2 1/2)
173Yb

(5/2

(5/2

3/2)

7/2)

Isotope shift

170

171Yb 3S
1
[A=6847.7MHz, B=0MHz]-937(41)

173Yb 3S
1
[A=-1892MHz, B=-0.2MHz]

F=3/2
F=3/2

171

-758(45)

3424MHz172-413(20)

6847MHz

173-228(58)

174

0

176

392(37)(MHz)

6624MHz
F=5/2
1892MHz

F=1/2

4732MHz
F=7/2

171Yb 3P ( F = 1/2 )0

173Yb 3P ( F = 5/2 )0Figure 7.5: Spectra of the 3 P0 ↔3 S1 transition (649 nm) obtained by the saturation spectroscopy using discharged Yb atoms. The solid line is a fit to data by a summation ofLorentzian functions whose peak positions are identified by the results of the frequency
measurement and previously measured isotope shifts shown below [77, 78]. Dotted line in
the level diagram indicates “center” frequency listed in the table by the side.

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

103

172Yb
174Yb
173Yb
173Yb
(5/2 5/2)(1/2 3/2)
176Yb

173Yb
(7/2

171Yb
(3/2

1/2)

173Yb
(9/2

170Yb

7/2)

7/2)

173Yb

171Yb

(5/2

(3/2

7/2)

3/2)

173Yb 3S
1
[A=-1892MHz, B=-0.2MHz]

Isotope Shift

170
171

-968.3(18)-425.3(3)173-229.3(8)

F=5/2

F=3/2
3424MHz

1892MHz

6847MHz4732MHz
F=7/2

F=1/2

F=1/2

F=5/2

0
425.7(3)(MHz)

6139MHz

2672MHz

F=3/2

4008MHz

F=5/2

F=3/2

176

F=3/2
6624MHz-751.6(23)

172

174

171Yb 3S
1
[A=6847.7MHz, B=0MHz]

171Yb 3P
2
[A=2672.27MHz, B=0MHz]

F=7/2
F=9/2

4420MHz
1893MHz
942MHz3378MHz173Yb 3P
2
[A=-742.71MHz, B=1342.3MHz]

Figure 7.6: Spectra of the 3 P0 ↔3 S1 transition (770 nm) [78].

7.1. ESTIMATION OF THE RESONANCE FREQUENCY

7.1.3

104

Results

Results of the estimation are shown in Fig. 7.7. The resonance frequency of the 1 S0 ↔3 P0
transition in bosonic isotopes and f1 in 170 Yb could not be measured in this experiment.
Then they were derived by assuming same isotope shifts with that of the 1 S0 ↔3 P1 transition.
Major problem of this estimation is the large pressure shift due to buffer gas (Neon)in an opto-galvano cell. Pressure broadening and shift of the 3 P0 ↔3 S1 transition wasmeasured in [81](3 P0 ↔3 S1 (649 nm) in Neon: 2.5(±0.3)×10−9 s−1 cm3 (broadening) and−0.5(±0.1)×10−9 s−1 cm3 (shift))1 . However, experiment for the 3 P2 ↔3 S1 transition has
not been reported. We assumed same values for the 3 P2 ↔3 S1 transition. This may leadto a large error in this estimation. We, however, regarded that the resonance frequency
must be located within a hundred megahertz around these estimated values. In fact, wehave successfully found the resonance.

f0 (578nm)

f1 (649nm)

f2 (770nm)

f est(507nm)

170518296.2954(14)461867.918(37)389259.442(33)590904.860(50)171518295.8366

461868.091(35)389259.690(23)590904.326(42)172518295.0089(20)461868.425(33)389259.973(24)590903.550(41)173518294.5768

461868.620(49)389260.220(24)590903.066(55)174518294.0215(19)461868.855(37)389260.412(24)590902.554(44)176

518293.6238(24)461869.272(37)389260.849(23)590902.138(44)(GHz)

Figure 7.7: Estimation of the resonance frequency of the 1 S0 ↔3 P2 transition. f0 for 171 Yb
and 173 Yb were measured in [34].

1

These value was measured to the buffer gas number density.

7.2. OBSERVATION OF THE ULTRANARROW 1 S0 −3 P2 TRANSITION

7.2

105

Observation of the ultranarrow 1S0 −3 P2 transition

Based on the information of the resonance frequency estimated in the previous section, we
found the resonances. The 1 S0 ↔3 P2 transition is the M2 transition. The Rabi frequencyΩ can be described by [82]p
 0.204 I [W/cm2 ]Ω [kHz] = 2π ×
,
(7.5)
p

2
2.280 I [W/cm ]where I is the intensity of the excitation laser. In this experiment, the maximum intensityof the excitation laser at atoms is 40 W/cm2 . If we make use of the theoretical transition
strength given in [60, 75], (7.5) indicates the Rabi frequencies ∼ 2π× 1 kHz for bosonicisotopes and ∼ 2π× 10 kHz for fermionic isotopes.
We first used fermionic isotope 173 Yb in a compressed MOT to find the resonancedue to its relatively large transition strength caused by the hyperfine mixing effect [75]which is further described in the following discussion. In addition, we artificially broaden
the spectrum [83, 34] to find the resonance as easy as possible. First, we used relatively“hot” (several hundred microkelvin) atoms in order to utilize a Doppler broadening. For
example, 230 kHz can be expected for 50 µK. Second, we rapidly switched on and off the
MOT laser during the excitation in order to utilize a broadening due to the ac Stark effectinduced by the MOT laser. Third, the broadening due to the Zeeman effect induced by
the MOT magnetic field gradient (∼ 14 G/cm) was also used. With the help of these,
we have succeeded in observing the resonances. Fig. 7.8 shows the number of atoms in
the MOT until we found the resonance. The MOT loading time was reduced to 1 ∼ 3 s
to search the resonance as fast as possible. Frequency of the excitation laser was scanned
at 200 kHz (507 nm) intervals and irradiated to atoms for 300 ms. The resonance canbe detected as the decrease of the number of atoms in a MOT. This is because the MOT
does not work for metastable 3 P2 atoms and the lifetime of the metastable 3 P2 state is 15
s which is long enough for atoms to fall from the MOT region by gravity. In this study,
the transitions we found are the 1 S0 (F =5/2) ↔ 3 P2 (F =5/2) transition in 173 Yb, the1
S0 (F =1/2) ↔ 3 P2 (F =5/2) transition in 171 Yb, and 1 S0 ↔ 3 P2 transition in 174 Yb.

7.2.1

Fermions: 173 Yb [1 S0 (F =5/2) ↔ 3 P2 (F =5/2)]
and 171 Yb [1 S0 (F =1/2) ↔ 3 P2 (F =5/2)]

Hyperfine mixing effect - induced E1 transition
The linewidth of the 1 S0 ↔3 P2 transition in Yb fermionic isotopes is expected to be slightly
broader than that of bosonic isotopes due to the hyperfine mixing (HFM) effect. The
HFM effect is caused by the interaction HHFI between a nuclear spin I and a total electric
angular momentum J. Regarding HHFI as a first order perturbation, the mixed 3 P20 stateis described as¯3 0® X h3 P2 , F, mF |HHFM |αi¯ P2 , F, mF =
|αi ,
(7.6)
E(α) − E(3 P2 )α7.2. OBSERVATION OF THE ULTRANARROW 1 S0 −3 P2 TRANSITION

106

Figure 7.8: Road to the resonance. The number of atoms in the MOT is plotted as afunction of time. Frequency of the excitation laser was scanned at 200 kHz (507 nm)
intervals. After 4.2 hours, the number decreases because we made the MOT loading timeshorter in order to accelerate the sequence. Finally, seven hours later, early in the morning(at 3:50 am on Sunday, March 4th, 2007), we found the resonance.

where α is all possible states. The matrix element of the E1 transition between the mixed
3 0
P2 state and the 1 S0 ground state ish3 P20 , F, mF |rq(1) |1 S0 , F 0 , mF 0 i =X h3 P2 , F, mF |HHFI |αiαE(α) − E(3 P2 )hα|rq(1) |1 S0 , F 0 , mF 0 i.

(7.7)

Since α includes some states which are connected to the ground state via the E1 transition(e.g. (6s6p)3 P1 ), this matrix element does not vanish in some cases. This means that, if
atoms have a nuclear spin, the 1 S0 ↔3 P2 transition can occur through the E1 transitionin addition to the M2 transition.
As for Yb atoms, fermionic isotopes 173 Yb and 171 Yb have nuclear spin 5/2 and 1/2,
respectively, but all bosonic isotopes do not have a nuclear spin (see Table 2.1). As a result,
we can expect the HFM effect only for fermionic isotopes. Note that the selection rulesof the E1 transition induced by the hyperfine mixing is same as that of the E1 transition.
For example, the 1 S0 (F =5/2)↔3 P2 (F =5/2) transition via HFM-E1 is allowed, but the1
S0 (F =5/2)↔3 P2 (F =1/2) transition is not since ∆F =2 > 1 which is not allowed in theE1 transition (see Fig. 7.9).
To exploit the spectral broadening induced by the HFM effect in fermionic isotopes2 ,we chose the 1 S0 (F =5/2)↔3 P2 (F =5/2) transition in 173 Yb as a first target. Fig. 7.10
shows the observed spectrum whose spectral width is broadened to ∼ 1 MHz due to several
effects mentioned above. Using this transition, we next carried out the spectroscopyof ultracold fermionic isotopes which were cooled by evaporative cooling. The residual
magnetic field split the 3 P2 (F =5/2) state split into six Zeeman sublevels, one of which
2

In order to find other two resonances(1 S0 (F =5/2)↔3 P2 (F =3/2), 1 S0 (F =5/2)↔3 P2 (F =7/2)), we
had to use two AOMs to compensate the frequency interval between the atomic resonance and the ULE
resonance. The 1 S0 (F =5/2)↔3 P2 (F = 5/2) required only one AOM, which was technically much easierthan others.

7.2. OBSERVATION OF THE ULTRANARROW 1 S0 −3 P2 TRANSITION

107

is shown in Fig. 7.11. The Zeeman splitting of the ground state 1 S0 (F =1/2) is about
three orders of magnitude less than that of the 3 P2 state because the nuclear spin is muchsmaller than the electric spin. Then the Zeeman splittings of the 3 P2 state were observed.
After finding the resonance in 173 Yb, we could find the resonance in the other fermionicisotope 171 Yb with relative ease. The reason is that the frequency interval (isotope shift)between them can be estimated with the accuracy of ∼1 MHz by using Eq. (7.3) and theinformation in [34], Fig. 7.5 and Fig. 7.6. We tried to find the 1 S0 (F =1/2)↔3 P2 (F =5/2)
transition in which the HFM effect does not contribute (see Fig.7.9). The observed spectrum is shown in Fig. 7.11.

3P

3P

F=1/2

2
F=3/2
F=5/2

2
F=5/2

F=7/2
F=3/2

1S

0

F=9/2

F=1/2

171Yb

1S , F=5/2
0
173Yb


  
Figure 7.9: Hyperfine mixing can occur between levels which satisfy the selection rule of
the E1 transition. Possible transitions via the HFM-E1 transition are described by thicklines. Other transitions are possible only through the M2 transition.

7.2. OBSERVATION OF THE ULTRANARROW 1 S0 −3 P2 TRANSITION













108

!"!#$%"&'(

173Yb 1S (F=5/2)
0

3P (F=5/2)
2

Figure 7.10: The first spectrum of the 1 S0 (F =5/2) ↔ 3 P2 (F =5/2) transition in fermion
isotope 173 Yb. The spectral width was artificially broadened in this measurement.







173Yb

1S (F=5/2)
0

3P (F=5/2)
2

171Yb

1S (F=1/2)
0

3P (F=5/2)2

Figure 7.11: (Left): Spectrum in ultracold 173 Yb. The residual magnetic field split the3
P2 (F =5/2) state into six Zeeman sublevels. One of them is shown here. (Right): The
1
S0 (F =1/2)↔3 P2 (F =5/2) resonance in 171 Yb. The HFM effect does not work in thistransition.

7.2. OBSERVATION OF THE ULTRANARROW 1 S0 −3 P2 TRANSITION

7.2.2

109

Boson: 174 Yb

Finding the 1 S0 ↔3 P2 transition in bosonic isotopes is crucially important. First, it enables the coherent excitation of a BEC in the ground state to the metastable 3 P2 state.
Furthermore, we have the ultranarrow magnetic-insensitive 1 S0 (m=0)↔3 P2 (m=0) transition in bosonic isotopes. This kind of transition is well known as a “clock” transition.
In fact, an atomic clock using this transition has been proposed in [3].
Since our group has realized a BEC of 174 Yb in the previous work [24], we searched theresonance in 174 Yb among five stable bosonic isotopes of Yb. By irradiating the excitation
laser to atoms in a compressed MOT for 3 s, we have successfully found the resonancewhich is shown in Fig.7.12. In the following sections and chapters, experiments using the1
S0 ↔3 P2 transition in 174 Yb will be presented in detail.









174Yb 1S
0
3P
2

Figure 7.12: The 1 S0 ↔3 P2 transition in 174 Yb. Atoms in a compressed MOT were used.
Spectrum was artificially broadened to make it easier for us to find the resonance, which
results in ∼1 MHz spectral width.

7.3. SELECTIVE EXCITATION TO ALL MAGNETIC SUBLEVELS OF THE 3 P2
STATE IN 174 YB
110

7.3

Selective excitation to all magnetic sublevels of
the 3P2 state in 174Yb

The 3 P2 state in a bosonic isotope 174 Yb has five magnetic sublevels corresponding to m
= 0, ±1 and ±2. If we consider the high-resolution spectroscopy using this transition,
the selective excitation of atoms to the arbitrary magnetic sublevel of the 3 P2 state isinevitable. For example, in order to measure the absolute frequency of the clock transition,
we need to observe well separated 1 S0 ↔3 P2 (m=0) transition. When we measure thepolarizability of each magnetic states, each magnetic sublevels must be well distinguished.
Hence we have demonstrated the selective excitation of atoms to any magnetic sublevels.
The experimental procedure and configuration of lasers and the magnetic field areshown in Fig. 7.13. Atoms are cooled by evaporative cooling in a crossed FORT and
excited in the FORT. Atomic temperature is about 1 µK. As an external magnetic field,
we used the residual magnetic field at the crossed FORT region.
Fig. 7.14 shows the obtained spectra. Fig. 7.14(a) is the spectrum obtained by roughly
scanning the frequency of the excitation laser, in which five components are included with
the interval of ∼1.5 MHz corresponding to the residual magnetic field of 0.7 G. The relative
signal strength of each components depends on the selection rule of the M2 transition. As
already discussed in 4.2, the selection rule depends on the polarization of the excitation
laser, the FORT polarization, and the external magnetic field. Fig. 7.14(b)−(f) aredetailed spectra of each magnetic sublevels. The spectral width are almost consistent
with the Doppler width.

Evaporative
Cooling ( 6 s)
Horizontal
FORT

TOF
(5 10 ms)
50 ms

Vertical
FORT
Excitation
507 nm
0.1 msImaging

Figure 7.13: (Left): Time sequence of the experiment and (Right): configuration of lasers
and the magnetic field.

7.3. SELECTIVE EXCITATION TO ALL MAGNETIC SUBLEVELS OF THE 3 P2
STATE IN 174 YB
111

-1

0

+2

-2
+1

(a) 174Yb(b) 3P2, m=+2(c) 3P2, m=+1(d) 3P2, m=0(e) 3P2, m=-2(f) 3P2, m=-1

Figure 7.14: Selective excitation to five magnetic sublevels of the 3 P2 state in 174 Yb. (a):The spectrum obtained by roughly scanning the frequency of the excitation laser. The
red solid line is a fit by five Gaussian functions with experimental parameters such as thepolarization of the excitation laser, the FORT polarization, and the external magnetic
field. (b)−(f): Detailed spectra of each magnetic sublevels.

7.4. MEASUREMENT AND CONTROL OF AC POLARIZABILITIES

7.4

112

Measurement and control of ac polarizabilities

The ac polarizability is the important information when we use a FORT since it determines
the trap depth of the FORT. In general, the polarizabilities of the ground state and theexcited state are different, except for the case of the magic wavelength. Thus the resonancefrequency of atoms in a FORT depends on the FORT power. Usually, in order to measure
the ac polarizability, parametric resonance technique was used as we previously did insection 6.1.5. However, thanks to the ultranarrow linewidth of the 1 S0 ↔3 P2 transition,
we can obtain much precise information than that obtained from the parametric resonanceexperiment. In a FORT, the resonance frequency ν can be described asIν = ν0 − (αP,m − αS ),
4

(7.8)

where ν0 is the transition frequency between unperturbed atomic states, I is the FORT
intensity, αP,m and αS are the polarizabilities of magnetic sublevel m of the 3 P2 state andthe 1 S0 ground state, respectively. We measured the dependence of the frequency shiftof the resonance of each magnetic sublevels (|m| = 0, 1, 2) on a FORT power. Since
the ac polarizability of the ground state αS can be precisely estimated, we can derive
the polarizability of the 3 P2 state. Time sequence of the experiment and configuration oflasers are shown in Fig. 7.15. In this experiment, the magnetic field was aligned almost
parallel to the polarization of the horizontal FORT.

Evaporative
Cooling ( 6 s)
Horizontal
FORT

TOF
(5 10 ms)
50 ms

Vertical
FORT
Excitation
507 nm
0.1 msImaging

Figure 7.15: (Left): Time sequence of the experiment and (Right): configuration of lasers
and the magnetic field.

7.4. MEASUREMENT AND CONTROL OF AC POLARIZABILITIES

113

Temperature shift
When we determine the center frequency of each spectrum, we have to consider thetemperature shift as discussed in section 5.2. To do so, we have to know the value ofΩg,ex which can be determined by the polarizability. In the present case, we alreadyknow the rough value of the polarizability of the 3 P2 state from the parametric resonanceexperiment as shown in section 6.1.5. Thus, we first assumed the polarizability of the
3
P2 state and fit the data to derive the temperature shift in which the free parameter isthe polarizability α1 , the center frequency and the signal strength. After compensating
the temperature shift, we could estimate the polarizability. In this experiment, all of thederived polarizability were well agreed with the fitted polarizability.
In the Fig. 7.16, three spectra corresponding to the 1 S0 ↔3 P2 (|m|=2) transition forthree different FORT powers are shown as an example. The solid lines are the fit of databy the (5.21). In order to measure the temperature of atoms, the TOF technique wasused. The arrows on the horizontal axis describe the unperturbed resonance frequencyof the spectrum shown with the same color. The center frequencies ν0 estimated by thisfitting are plotted as a function of the FORT powers in Fig. 7.17 with data corresponding
to |m| = 0 and 1. Using the known value of the polarizability of the ground state αS , thepolarizabilities of all of the magnetic sublevels of the 3 P2 state are determined which arelisted in Table 7.4. In addition, from these observations, we can conclude that atoms in
all the magnetic substates of the 3 P2 state can be optically trapped by a FORT at 532
nm, which is consistent with the result of the parametric resonance and also importantfor future studies of the 3 P2 atoms.

Table 7.3: Polarizability of the 3 P2 state (B is almost parallel to the eFORT ).
|m|=0
αP,m

|m|=1

|m|=2

Unit

41.8±0.3 40.6±0.3

38.3±0.1

mHz(mW/cm2 )−1

7.4. MEASUREMENT AND CONTROL OF AC POLARIZABILITIES

114

Normalized atom number

1
0.8
0.6
0.4
0.2
0
0

50

100
150
200
Frequency (kHz)

250

300

Figure 7.16: Due to the temperature shift (see 5.2), the observed resonance frequency
is slightly shifted from the true resonance frequency. In this figure, three spectra corresponding to three different FORT powers 56, 146, and 250 mW are shown. Solid lines
are the fit of (5.12) to the data. Each arrow indicates the position of the true resonancefrequency for spectra in the same color.

0Frequency offset (kHz)

-50
-100
-150
-200
-250
|m| = 0|m| = 1|m| = 2

-300
-350
0

50

100
150
200
FORT power (mW)

250

300

Figure 7.17: Measurement of the polarizability of every magnetic sublevel of the 3 P2 state.
The resonance frequencies determined by considering the temperature shift are plotted asa function of FORT powers. Solid lines are the linear fit to the data based on (7.8).

7.4. MEASUREMENT AND CONTROL OF AC POLARIZABILITIES

7.4.1

115

Control of the polarizability

Five magnetic substates of Yb[3 P2 ] in a FORT can be mixed with each other by the external magnetic field. As a result, their eigenvalues depend both on the FORT polarizationand on the direction of the external magnetic field. One of the significant phenomena isthat, even if m = 0, the energy level (light shift) can be shifted by the external magnetic
field in a FORT. This technique was previously utilized as an ac Stark shift cancellation
method using the 3 P1 state in Sr [84].
We performed the same measurement as that in the previous section by rotating
the external magnetic field. As an example, Fig. 7.18 shows how the magnetic field“insensitive” resonance (1 S0 ↔3 P2 , m = 0) can be easily shifted just by rotating theexternal magnetic field. We also measured the polarizability of the 3 P2 (m=0) state incase of B k eFORT and B ⊥ eFORT as schematically shown in Fig. 7.19(Top). Fig.
7.19(Bottom) shows the frequency shift of the resonance for two cases. Here, one cansee the clear controllability of the polarizability of the 3 P2 (m=0) state. In particular, weshould mention to our successful realization of making the polarizability of the 3 P2 (m=0)
state almost same as that of the ground state. This would enable us to perform ultrahighresolution spectroscopy of atoms in an optical lattice [8].

  174Yb, 1S
0

3P (m = 0)0

Figure 7.18: The resonance frequency of the magnetic field insensitive 1 S0 ↔3 P2 (m=0)
transition can be changed in a FORT just by rotating the external magnetic field. This
results from the mixing of eigenstates of atoms in a FORT which is induced by the external
magnetic field.

116Frequency offset (kHz)7.4. MEASUREMENT AND CONTROL OF AC POLARIZABILITIESFigure 7.19: Change of the polarizability of the 3 P2 (m=0) state. (Top) Configuration oflasers and the external magnetic field are schematically shown. (Bottom) Polarizability isclearly changed by rotating the external magnetic field. In particular, when B is almost
perpendicular to eFORT , the situation is similar to the magic wavelength situation where
the ac Stark shifts are almost the same for the 1 S0 and 3 P2 (m = 0) states.

7.5. MAGNETIC RESONANCE IMAGING OF ULTRACOLD ATOMS

7.5

117

Magnetic resonance imaging of ultracold atoms

Magnetic resonance imaging (MRI) is well known as the technique to visualize the human body mainly for the medical purpose. Such MRI system usually uses the magnetic
resonance of a hydrogen nuclear spin in a body, that is to say, the nuclear magnetic resonance (NMR). In addition, by applying the magnetic gradient, the information about theposition of the NMR signal can be obtained.
In atomic physics, the MRI technique has also been developed mainly in the experiment using alkali metals. In such experiments, a microwave pulse induces the transition
between two different hyperfine spin states in atoms which are addressed in a magnetic
field gradient. Sodium atoms in a magnetic trap was first investigate by using the MRItechnique [85]. One of the recent important experiments is the imaging of the Mott insulator shells. By using the spin changing collisions, the MRI technique have enabled
the observation of the shell structure of the Mott insulator [86]. More recently, spatiallyresolved microwave spectroscopy of a trapped Fermi gas with resonant interactions has
been demonstrated [87].
In this work, we developed the basic technique of the MRI. The basic idea is that
atoms in the 3 P2 (m=+2) state in a magnetic gradient can be energetically distinguishedby the ultranarrow 1 S0 ↔3 P2 (m=+2) transition. In general, the transition linewidth is toolarge to distinguish atoms in a realistic magnetic field gradient. In other words, the sizeof the atomic cloud is too small to be addressed with the realistic magnetic gradient. To
the contrary, since the natural linewidth of the 1 S0 ↔3 P2 transition is extremely narrow,
we can distinguish atoms in a small region such as a FORT.
For example, let us imagine that the typical magnetic field gradient generated by the
MOT coils of 10 G/cm is applied to the atomic cloud. The Zeeman shift of the 3 P2 (m=+2)
state is 4 GHz/m in such a magnetic field gradient. The most narrowest laser linewidth
ever achieved reaches below 1 Hz [88], then we assume that we can use the excitation laser
whose linewidth is 1 Hz. Then, in principle, the spectral resolution can be expected to be
0.25 nm. In other words, we can energetically distinguish two atoms which are separated
to 0.25 nm. Furthermore, so far, this resolution is limited only by the technical reason(laser linewidth). The natural linewidth of the 1 S0 ↔3 P2 transition is about 10 mHz whichis smaller than the above estimation by a factor of two. In addition, the magnetic field
gradient which is much larger than 10 G/cm is easily obtained. Hence, the extremely
high spatial resolution can be expected even with the present experimental technique inthis field.
The basic principle of this experiment is schematically shown in Fig. 7.20. Whenthe flat magnetic field is applied to the atomic cloud, the Zeeman shift induced in the3
P2 (m=+2) state is same for all atoms in a cloud. Thus, the spectral width is mainly
determined by the Doppler width. On the other hand, when the magnetic gradient isapplied, the Zeeman shift induced to the atoms depends on the position, which leads
to the broadening of the spectrum of the 1 S0 ↔3 P2 (m=+2) transition in addition to the
Doppler width.
We have used the ultracold atoms whose temperature is 0.5 µK. After evaporative
cooling, we have applied the magnetic gradient by turning on the MOT coil. Fig. 7.20

7.5. MAGNETIC RESONANCE IMAGING OF ULTRACOLD ATOMS

118

shows the obtained spectra before and after applying the magnetic field gradient. We
can clearly observe the spectral broadening due to the MRI effect. We should also notethat the resonance signal became drastically weak as the magnetic gradient increased.
This is reasonable because the number of atoms which are resonant with the excitation
laser at a certain frequency decreases due to the position dependent Zeeman shift. Thus,
ideally, we should have used the same excitation time for two cases and detected how
the position dependent excitation in atomic cloud occurred in order to demonstrate the
MRI. Unfortunately, the resolution of our present imaging system is not high enough forsuch a purpose. Instead, we made the excitation time longer to obtain the same signal
strength for both conditions. In this measurement, the excitation time is much longer
than the trap frequency. Thus, atoms passed over the resonant region for many times andwere excited. The larger the magnetic gradient became, the narrower the resonant region
became. Then, atoms had to pass over for many times to be excited, which resulted in
the longer excitation time.
In Fig. 7.20, spectra taken before and after turning on the MOT coils are compared.
Since we could not know where atoms were in the magnetic field gradient, it is not possible
to quantitatively analyze these spectra. Nevertheless, spectral broadening induced by themagnetic gradient has been clearly observed.
This technique of spatial addressing has great potential for future experiments, suchas addressing of the individual site of the optical lattice potentials which must be a key
technique for the quantum computation experiment [82, 13] and the construction of thequantum simulator [45].



  

     Figure 7.20: Demonstration of the magnetic resonance imaging. The magnetic field gradient applied to atoms leads to spectral broadening of the magnetic field sensitive ultranarrow 1 S0 ↔3 P2 (m = +2) transition.

119

Chapter 8
High-resolution spectroscopy of a
BEC
One of the significant properties of a BEC is its large mean field energy. The mean fieldenergy U0 results from the interatomic interaction energy. At low energies where only thes-wave scattering occurs, U0 of a BEC is described by
U0BEC =

4π~2
na,
m

(8.1)

where m is the atomic mass, n is the number density and a is the s-wave scattering length.
Thus, the large number density n of a BEC results in its large mean field energy.
Previously the large mean field energy of a BEC was, for example, used to identify ahydrogen BEC [89]. The BEC phase transition has been usually confirmed by the sudden
appearance of a dense central core inside a thermal cloud. However, that was not easy
with the hydrogen BEC since taking absorption images to see the spatial distribution ofhydrogen atoms was difficult due to technical reasons. Then, the large mean field shift of
the resonance frequency in a two-photon transition was used to identify a hydrogen BEC.
In experiments with alkali metals, the mean field energy has also played an importantrole. For example, the collision shift in the clock transition |F, 0i↔|F + 1, 0i in sodium
atoms was accurately measured by using the large mean field energy of a BEC [90]. One of
the recent remarkable applications of the mean field energy is the direct observation of theMott insulator shell structure [91]. In 3D optical lattice potentials, it was predicted thatthe system should separate into Mott insulator shells with different occupation numbers
for sufficiently strong interatomic interactions. By using RF spectroscopy, Campbell et al.
detected the shift of the resonance frequency of the clock transition due to the layered
structure of the Mott shells with occupancies from n = 1 to n = 5. Since the mean
field energy is proportional to the number density (see (8.1)), the discrete frequency shiftcorresponding to the number of atoms in each site was successfully observed.
It is true that all of these experiments are excellent, but there also exist some limitations. First, the resolution of RF spectroscopy to detect the mean field energy in alkali
metals is currently limited by the finite interaction time broadening. Atoms in two different states feel different mean field energies because of the difference of the scattering

120

length between a11 and a12 where axy is the scattering length between atoms in the states
x and y. This difference leads to the mean field shift ∆ν BEC described by∆ν BEC =2~n(a12 − a11 ).
m

(8.2)

In case of alkali metals, a12 − a11 is small. For sodium atoms as an example, a12 − a11 =(3.15 − 2.71) = 0.44 nm which results in the frequency shift of ∆ν = 250 Hz for thenumber density of n = 1×1014 cm−3 . The interaction time broadening is more than 500
Hz. Thus in order to achieve the higher resolution, we have to use longer excitation RF
pulses which is so far difficult due to, for example, the short trap lifetime of atoms in sucha system [90]. Second, addressing atoms by using position-dependent Zeeman shift in astronger magnetic field gradient to pursue the higher spatial resolution [86] may affectalkali metals in the ground state. This is because the total angular momentum F is nolonger the good quantum number under the strong magnetic field (Paschen-Back effect).
The ultranarrow 1 S0 ↔3 P2 transition1 in Yb has great possibilities to overcome such
obstacles. First, according to the result of this work, the mean field shift of this transitionin Yb is much larger than that of alkali metals as described in Table 8.1. The mean field
shift of the 1 S0 ↔3 P2 transition is larger than others by an order of one. In this work, thislarge mean field shift enabled us to successfully observe not only the frequency shift but
also the distortion of the lineshape due to the mean field energy of a BEC which has neverbeen observed in alkaline metals. Second, the ground state of Yb is spinless. Hence any
strong magnetic field gradient for spatial addressing of Yb atoms by using the Zeeman
shift of the 3 P2 state does not affect atoms in the ground state at all.
It is sure that Yb has two ultranarrow optical transitions, i.e., not only the 1 S0 ↔3 P2transition but also the 1 S0 ↔3 P0 “clock” transition. However, the 1 S0 ↔3 P2 transition has
advantages in some points. In bosonic isotopes of Yb, the 1 S0 ↔3 P0 transition is strictly
forbidden. While a novel technique to realize it in bosonic isotopes by using a strong
magnetic field has been recently invented, the transition strength is still very weak [33, 35].
Thus, considering spectroscopy of a BEC, the much stronger (but narrow enough for highresolution)1 S0 ↔3 P2 transition is desirable. In addition, since the 1 S0 ↔3 P0 transition is
the magnetic insensitive transition, addressing of atoms by the magnetic field gradient is
impossible, while the 3 P2 state has four magnetic field sensitive sublevels: 3 P2 (m = ±1
and ±2).
In this study, we have demonstrated the detection of a mean field energy of a BEC byusing the ultranarrow 1 S0 ↔3 P2 optical transition in Yb. We have observed not only thelarge mean field shift due to a BEC, but also the change of the lineshape which reflected
the density distribution of a BEC in a trap. Finally, we have successfully determined a12
from the observed spectrum. In the following, we will first introduce a mean field energyof a BEC. Then, the spectral lineshape expected in spectroscopy of a BEC in a trap will
be discussed. After presenting the experimental method, we will show the experimental
results and discuss the observed spectrum in detail.
1

The natural linewidth of 10 mHz corresponds to the energy resolution of 0.5 pK (= 5 × 10−13 K)
which is the extremely high-resolution.

8.1. MEAN FIELD ENERGY

121

Table 8.1: Frequency shift by the mean field energy of a BEC for the number density ofn = 1×1014 cm−3 .
23

Na
Rb
174
Yb
87

8.1

a12 (nm)
3.15
5.19
−33

a11 (nm)
2.71
5.32
5.53

∆ν BEC (Hz)
Ref.
241
[90]
−19
[91]
−2792
[30] and this work.

Mean field energy

Pseudo potential
When two neutral atoms come so close to each other that atoms can see atomic nucleus,
the interatomic interaction is repulsive. Otherwise, atoms must collapse and cannot stably
exist. The length scale of this repulsive core is, in general, less than 1 nm. On the other
hand, there is the attractive interaction known as van der Waals force between neutral
atoms. The reach of the van der Waals force is about 10 nm. As a result, the net force
between neutral atoms is sum of the repulsive force by a hard-sphere and the attractive
force by the van der Waals force. Whether the net force becomes repulsive or attractive
depends on atomic species2 . In dilute neutral atomic gasses, the average interatomic
length is 100 nm. Thus, the effect of the complicated realistic interatomic potential
within 10 nm can be described only by a phase shift δl of the scattering wave function
before and after the scattering. This can be seen in the radial part of the scattering wave
function Rkl (r) in the r → ∞ limit:
Rkl (r) '1πsin(kr + δl − l).
kr
2

(8.3)

In particular, when the only s-wave scattering can occur at low energies, the s-wave phaseshift is given byδ0 = −ka,(8.4)
where k = |k| is the wavenumber vector of the scattering wave and a is the scattering
length. According to the (8.3), δ0 shifts the origin of Rkl (r) by a. Thus, the scattering
wavefunction ψ(r) for the s-wave scattering in the center-of-mass coordinate satisfies the
Schrödinger equation(∇2 + k 2 )ψ(r) = 0 (r > a)ψ(r) = 0 (r ≤ a).

(8.5)
(8.6)

This indicates that when we consider the s-wave scattering, we can regard the interatomic
potential as a hard-sphere potential with radius a.
2

In case of 174 Yb, it is repulsive.

8.1. MEAN FIELD ENERGY

122

In general, it is difficult to solve (8.5) due to the boundary condition (8.6). However,
Huang and Yang [92] proved that, for s-wave scattering, (8.5) and (8.6) are equivalent to
the equationtan(ka)∂δ(r) (rψ(r)).
(8.7)(∇2 + k 2 )ψ(r) = 4πk∂r
At low energies,tan(ka)1
2= a + (ka)2 a + · · · .
k2
3
When we consider the s-wave scattering, the effect of the terms a3 (p-wave) and higher
can be neglected. As a result, (8.7) becomes~2 ∇ψ(r) + U (r)ψ(r) = Eψ(r)
m
4πa~2
∂
U (r) =
δ(r) r.
(8.8)
m∂r
Thus the interatomic interaction potential at low energies (s-wave scattering) is given by(8.8) which is known as a pseudo potential.
−

Thomas-Fermi approximation
Using the pseudo potential, the Hamiltonian H for N atoms in a trap Vext (r) at low
energies is given by
¾
N ½ 2
X
~
4πa~2 X∂
2
H=−
∇ + Vext (ri ) +δ(ri − rj )
rij .
(8.9)
2m
m
∂r
ij
i=1
i<j
In a BEC, all atoms occupy the same one-particle wavefunction φ(r). Thus we canintroduce ψ(r) as a wavefunction of a condensate which is defined by√ψ(r) = N φ(r).
(8.10)
Then, the density of a condensate n(r) can be described byn(r) = |ψ(r)|2 .

(8.11)

Now we can present the energy E of the system and the total atom number N as follows3 .
¸
Z · 2
~
1 4πa~22
2
4
E(ψ) =|∇| + Vext (r)|ψ(r)| +|ψ(r)| dr
2m
2 m
Z
N =
|ψ(r)|2 dr.
(8.13)
3

Here, we use the fact that ψ(r) doesn’t diverge at r = 0. In such a case, the pseudo potential becomesU (r) =

4πa~2
δ(r).
m

This is because
U (r)ψ(r) =4πa~2∂
δ(r) (rψ(r))
m∂r(8.12)

8.1. MEAN FIELD ENERGY

123

Minimizing the energy E with the binding condition (8.13) by the method of Lagrange
multipliers gives us the time-independent Gross-Pitaevskii equation:
·
¸
~2 2
4πa~2
2
−
∇ + Vext (r) +|ψ(r)| ψ(r) = µψ(r)(8.14)
2mm
where µ is the chemical potential. Here the third term in the left-hand side is called mean
field energy.
When we treat a BEC with a large number of atoms, the mean field energy is muchlarger than the kinetic energy. Then the kinetic energy term can be safely neglected. Thisis known as Thomas-Fermi approximation. In this approximation, the Gross-Pitaevskii
equation becomes
½
¾
4πa~2
2
Vext (r) +|ψ(r)| = µψ(r),(8.15)
mand then the density distribution is given µ¶−1
4π~2 a(µ − Vext (r)) (µ > Vext (r))mn(r) = |ψ(r)|2 =0(µ ≤ Vext (r))(8.16)

This implied that the density distribution of a BEC in a trap is the inverse of the trap
shape.
Line shape of the excitation spectrum of BEC in a trap
Let us consider the excitation of atoms in a BEC in a trap. We assume that the excitation
is so weak that the excited atoms interact only with the atoms in the ground state. Thus,
the interaction between atoms in the excited state is neglected.
According to the (8.14), the effective potentials V eff for a BEC in the ground state |gi
and an atom in a excited state |ei in a trap are
4π~2 aggn(r) = µm
4π~2 ageeff
V|ei (r) = Vext (r) +
n(r),
m

eff
V|gi
(r) = Vext (r) +(8.17)

where a12 denotes the scattering length between atoms in the state |1i and |2i. In Fig.8.1,
these effective potentials are schematically shown in case of age < 0.

=
=

µ
¶
∂ψ(r)
4πa~2
δ(r)ψ(r) + δ(r)r
m
∂r
2
4πa~
δ(r)ψ(r).
m

8.2. BEC IN A HARMONIC TRAP

124

The lineshape of the excitation spectrum of a BEC in a trap can be derived by calculateff
ing the transition probability from the flat potential V|gi(r) = µ to individual vibrationalefflevels in V|ei (r). The resulting lineshape is given by [93]s15h(ν − ν0 )I(ν) =
4n0 ∆U1−h(ν − ν0 )
,
n0 ∆U

∆U =

4π~2
(age − agg ),m(8.18)

where ν0 is the unperturbed resonance frequency (including the recoil shift in our experiment), n0 is the peak number density. The lineshape Eq. (8.18) is schematically shownin Fig. 8.1(Right). Reflecting a mean field energy, the spectrum is drastically shifted to
the red side from the original position in case of age < 0. If age > 0, it is shifted to theblue side. The asymmetry of the lineshape reflects the anomalous density distribution of
a BEC in a trap.
3P
2

Frequency

1S
0

SignalFigure 8.1: (Left): Effective potentials for a BEC in a harmonic trap are schematicallyshown both for the 1 S0 state and the 3 P2 state in case of a12 < 0. Reflecting the density
distribution of atoms in a trap, the effective potentials are modified as described in Eq.
(8.17). (Right): Excitation spectrum (Eq. (8.18)) between these two effective potentials
is shown. Due to the mean field energy, the resonance frequency shifts and the lineshape
becomes distorted.

8.2

BEC in a harmonic trap

In Fig. 8.2, the time sequence and configurations of lasers and the magnetic field areschematically shown. After evaporative cooling for 6 s, atomic temperature reaches 1.2,
0.5, 0.3 µK and a BEC in this measurement. This is done by reducing the final FORT
powers depending on the desired atomic temperature. Then the excitation laser is irradiated for 50 ms. The intensity of the excitation laser is less than I = 100 mW/cm2 . We

8.2. BEC IN A HARMONIC TRAP125

use the magnetic-field-insensitive 1 S0 ↔3 P2 (m = 0) transition for spectroscopy. After the
TOF time t, the absorption images are taken.
Evaporative
Cooling ( 6 s)
Horizontal
FORT

TOF
(5 10 ms)
50 ms

Vertical
FORT
Excitation
507 nm
0.1 msImaging

Figure 8.2: (Left): Time sequence for spectroscopy of a BEC. (Right): Configurationof the polarization of the FORT, the magnetic field, and the incident direction of theexcitation laser in this measurement are schematically shown.
The magnetic field is applied to almost vertical to the polarization of the horizontal
FORT beam. Using thermal atoms, the light shift in this configuration was measured inadvance as shown in Fig. 8.3. The temperature shifts have been already compensated(removed) by using the (5.12). As a result, the ratio of the trap depth between the3
P2 (m = 0) and 1 S0 states is determined to be U3 P2 (m=0) /U1 S0 = 1.02 ± 0.01 which is very
close to the magic wavelength condition.
Figure 8.4 (Top) shows the observed spectra for thermal atoms at three different
temperatures and a BEC. In this figure, only the frequency drift of the excitation laser
due to the instability of the ULE cavity was compensated. Such drift was estimated fromthe difference of resonance frequencies in two data taken at two different times under thesame conditions (∆νLS in Table 8.2). We assumed that the frequency of the excitation
laser linearly drifted during this measurement. Solid lines in spectra of thermal atoms(T =1.2, 0.5, and 0.3 µK) are fit of (5.21). All spectra are equally shifted by the recoil
shift.
A significant feature in Fig. 8.4(Top) is the large shift of the resonance frequency
after the BEC transition. This is clearly caused by sudden increase of the atomic density
due to the BEC transition. Also, the spectral lineshape is distorted, which reflects thecharacteristic density distribution of a mean-field limited BEC in a harmonic trap. All
these features can be well described by (8.18) (solid line in the BEC spectrum). The
scattering length a11 has been accurately measured to be 5.53 nm via photoassociation
spectroscopy in our laboratory [30]. Thus, the free parameter in (8.18) is only a12 . By
fitting (8.18) to the observed spectrum, we could estimate that a12 = −33(±10) nm.
Using a12 determined in this study, we now can estimate the collision shift ∆νcol for8.2. BEC IN A HARMONIC TRAP126

0Frequency offset (kHz)

-5
-10
-15
-20
-25
-30
-35
-40
0

50

100
150
200
250
FORT power (mW)

300

350

Figure 8.3: Measurement of the light shift in the configuration of Fig. 8.2 (Right). Temperature shifts in these data have been already removed.

thermal atoms4 which is listed in Table 8.2. In addition, the position of atoms in a trapslightly shifts from the focal point of the FORT laser due to the gravity. This also leads
to the shift of the resonance frequency estimated as ∆νgr in Table 8.2. After compensated
all frequency shifts5 mentioned above except for the mean field shift in a BEC spectrum,
we again compare the BEC spectrum with those of thermal atoms as shown in Fig. 8.4(Bottom). Here, one can clearly see the frequency shift caused by the mean field energyof a BEC. As predicted in (8.18), the upper edge of the BEC spectrum coincides with theunperturbed (no collision shift) resonance frequency.
In conclusion of this section, we have successfully detected the mean field energy of a
BEC by using an ultranarrow 1 S0 ↔3 P2 transition in Yb. From the fitting of the lineshape,
we can also determine the s-wave scattering length between atoms in the 1 S0 state andthe 3 P2 state. Now the mean field shift corresponds one-to-one with the number density
n in (8.18). Also, the s-wave scattering length a12 = −33 nm is very large, compared tothat of alkali metals (see Table 8.1). Furthermore, by performing the spectroscopy in amagnetic field gradient, it is possible to address atoms by the position-dependent Zeeman
shift. As a result, it is well within our ability to obtain the high resolution density profile
of atomic clouds at any point in a trap by using the 1 S0 ↔3 P2 transition in Yb. Such atechnique must be a powerful tool in the studies of, for example, the phase separation in
the mixture of two (or more) degenerate gasses and the density structure of condensatesin 1D, 2D and 3D optical lattice potentials.
4

Note that the collision shift is twice of the Eq. (8.2). In general, the collision shift is given by∆νcol = g2 (0)2~n(a12 − a11 ),m(8.19)

where g2 (0) is the equal point value of the second order correlation function. g2 (0) = 2 − (nBEC /n)2 fora uniform Bose gas in thermal equilibrium [94, 95].
5
The broadening due to the finite size of the condensate which implies a distribution of momenta [96]is estimated to be about 10 Hz in our case. Hence, it is neglected.

127

Normalized lineshape (a.u.)

8.2. BEC IN A HARMONIC TRAP-60

1.2 µK
0.5 µK0.3 µK
BEC
-40

-20

0

20

40

Normalized lineshape (a.u.)Frequency (kHz)-60

1.2 µK
0.5 µK0.3 µK
BEC
-40

-20

0

20

40

60

Frequency (kHz)

Figure 8.4: Spectroscopy of ultracold 174 Yb atoms and a BEC in a FORT by using the 1 S0
↔ 3 P2 (m=0) transition. Solid lines in spectra of thermal atoms are fit of the lineshape of(5.21) to the data. The solid line in the BEC data is a fit of (8.18) to the spectrum. (Top):
Only the ULE drift of the excitation laser was compensated. (Bottom): All frequency
shifts (see Table 8.2) except for the mean field shift in a BEC spectrum are compensated.
Here one can clearly see the mean field shift of a BEC.

8.3. BEC IN 1D OPTICAL LATTICE POTENTIAL

128

Table 8.2: Experimental parameters for data in Fig. 8.4. T : atomic temperature, ω̄: thegeometrical average of the trap frequency, n: number density, ∆νcol : collision shift, ∆νLD :light shift, and ∆νgr : frequency shift by the gravity.
T (µK)
1.2
0.5
0.3
BEC

8.3

N
1.2×105
5.9×104
4.1×104
2.9×104ω̄/2π (Hz)
610
420
370
140

n(cm−3 )5.6×1013
4.4×1013
6.8×1013
4.8×1014∆νcol (kHz)
−3.1
−2.5
−3.8
−12.8

∆νLS (kHz) ∆νgr (kHz)
−5.9
0.0
−2.9
0.1
−2.0
0.1
−1.4
0.2

BEC in 1D optical lattice potential

Using optical lattice potentials, we can tightly confine atoms in a small region within thelength scale of the optical wavelength. When such atoms are excited by the laser along
the confinement axis, the laser field felt by atoms is phase modulated by the oscillation
frequency. As a result, sidebands appear in the excitation spectrum. This kind of tightconfinement is known as the Lamb-Dicke confinement which is characterized by the LambDicke parameter [97]ss
δ = k507

~
=
2mΩg

ER507
,
~Ωg

(8.20)

where k507 and ER507 (= 0.21µK for 507 nm) are the wavenumber and the recoil energy ofthe excitation laser. Ωg denotes the trap frequency of the ground state. Thus, the smallerδ means the tighter confinement.
A significant feature of the Lamb-Dicke confinement is that the carrier component
doesn’t suffer from the Doppler effect, which enables the Doppler free spectroscopy. Thus
spectroscopy of thermal atoms and a BEC in optical lattice potentials enables us toprecisely measure the mean field shift and broadening of the spectrum without sufferingfrom the Doppler broadening. In addition, since the tight confinement of lattice potentials
increase the number density at each site, the effect of the mean field energy become larger
than that in usual harmonic traps.
In Fig. 8.5, the time sequence of the experiment and configurations of lasers andthe magnetic field are schematically shown. After evaporative cooling to make a BEC,
1D optical lattice potential was ramped up to 9.6ER for 100 ms. Since we observed theinterference pattern between BEC arrays, we regard that a BEC was adiabatically loaded
into optical lattice potentials. Then the excitation laser was irradiated for 50 ms alongwith the optical lattice laser. By turning off all lasers and letting atoms freely expand in
the TOF time t, we took absorption images.
The observed spectrum is shown in Fig. 8.7. First, we can clearly identify two peaks.
These peaks correspond to the |1 S0 , 0i → |3 P2 , 0i and |1 S0 , 0i → |3 P2 , 1i transitions, where|a, ni denotes the state of atoms in the electric state a and the nth vibrational levels of
the trap. In this measurement, the Lamb-Dicke parameter is (δ =) 0.4. Assuming that

8.3. BEC IN 1D OPTICAL LATTICE POTENTIAL

129

the trap depth of the 1 S0 state is same as that of the 3 P2 state, the ratio of the transition
strength between the |1 S0 , 0i → |3 P2 , 0i and |1 S0 , 0i → |3 P2 , 1i transitions is given by [97]P00 : P01 = 1 : δ 2 = 1 : 0.16,(8.21)

which is consistent with the observed signal strengths.
To analyze the width of the observed spectrum, we need to know the density distribution of condensates in each site. To this end, based on the discussions in [98], we assume
that the wave function of a BEC in 1D optical lattice potentials is given byXΦ0 (r) =fk (y)Φk (r⊥ ),(8.22)
k=0,±1,···,kM

where k labels each site and 2kM + 1 is the number of lattice sites. fk (y) and Φ(r⊥ ) are
the wave function along the axial (confinement, y-axis) direction and the radial direction22(xz-plane), respectively. Here we assume the Gaussian function fk (y) = e−(y−kd) /2σ
where d is the lattice interval and σ characterizes the width of the condensates along theconfinement direction in each well. σ is given by σ = d/(πs1/4 ) with s = ((~ωy )/(2ER532 ))2 ,ωy the trap frequency along the confinement direction and ER532 the recoil energy of the
FORT laser (ER532 = 0.194 µK for 532 nm). We assume that the confinement along theaxial direction is so strong that σ is not affected by interaction energies in each site.
On the other hand, the radial direction should be treated by the Thomas-Fermi approximation. Hence the wave function of the kth site in the radial direction is give by
¶
µ
2
r⊥
m
2
µk 1 −,(8.23)|Φk (x, z)| = √(R⊥ )2k
2 2π~2 aq
2where µk and (R⊥ )k = 2µk /mω(⊥)
are the chemical potential and the Thomas-Fermi
radius of the kth condensate,
respectively. Here, m is the atomic mass, a is the s-wave√
scattering length, r⊥ = x2 + z 2 and ω(⊥) is the radial trap frequency of the envelope
harmonic potential. µk satisfies
12
2µk = mω(y)d2 (kM− k 2 ),
2

(8.24)

where ω(y) denotes the axial trap frequency of the harmonic
potential (not the lattice trapPfrequency). From the normalization condition N =Nk , the kM is give by
s
µ
¶1/5
15
a d
2~ω̄
√ N
.
(8.25)kM =2mω(y)d2 8 π aho σ2Here, ω̄ = (ω(y) ω(⊥))1/3 is the geometrical average of the trap frequency of the harmonicp
trap, aho = ~/mω̄ is the oscillator length. The number of atoms in each site is give by

15 N
Nk =
16 kM

µ

k2
1− 2
kM

¶2
.

(8.26)

8.3. BEC IN 1D OPTICAL LATTICE POTENTIAL

130

In this work, the parameters are the followings: ω(y) = 2π ×140 Hz, ω(⊥) = 2π ×99 Hz,
d = 266 nm, N = 5×104 , a = 5.53 nm [30], a/aho = 7.3×10−3 , s = 9.6 and σ/d = 0.18.
As a result, kM = 13 and the chemical potential µk in each site is shown in Fig. 8.6(Top).
In this figure, we compare the chemical potential at each site with ~ω⊥ and ~ω(y) . In Fig.
8.6(Top-Left), ~ω(y) À µk for every site. Thus we can justify the approximation that we
can determine the Gaussian width by neglecting the two-body interaction at each site.
Similarly, ~ω⊥ < µk for all k justifies the application of the Thomas-Fermi approximation.
Using the Eq. (8.23), the atomic peak density (r = 0) at each site could be determined
as shown in Fig. 8.6(Bottom). Now we can calculate the spectral shape considering thatthe observed spectrum is summation of lineshapes Eq. (8.18) in each lattice site whose
atomic density is given by Fig. 8.6(Bottom). Since we determined the a12 from the
BEC spectroscopy in the previous section, there are no free parameters. The result of
this calculation is shown in Fig. 8.7 as a solid line which well agrees with the observed
spectrum.
As a result, we can conclude that we have succeeded in spectroscopically observing theon-site interaction at each site of the 1D optical lattice potentials by using an ultranarrow1
S0 ↔3 P2 transition in Yb. Since in the experiments using the optical lattice potentials,
the hopping energy and the on-site interaction energy are the important parameters. The
technique developed in this work enables us to precisely measure the on-site interactionenergies in such experiments. Also, the density distribution of mixtures of condensates in
optical lattice potentials can be precisely observed. Moreover, by additionally using the
MRI technique, we can observe the density distribution of any lattice site you want tosee.

Evaporative
Cooling ( 6 s)
Horizontal
FORT

TOF
(5 10 ms)
100 ms

50 ms

VerticalFORT
1D Lattice

9.6ER

Excitation
507 nm
0.1 msImaging

Figure 8.5: (Left) Time sequence and (Right) configurations of lasers and the magnetic
field for 1D optical lattice experiment are shown.

8.3. BEC IN 1D OPTICAL LATTICE POTENTIAL

Chemical potential (×10−1µK)

Chemical potential (×10−1µK)

1.0

µk
Axial

14
12
10
8
6
4
2
0
-15

-10

-5

0

5

131

10

µk
Radial

0.8
0.6
0.4
0.2
0.0
-15

15

-10

-5

Lattice site

0

5

10

15

Lattice site

Peak density (×1014 cm−3)

8
7
6
5
4
3
2
1
0
-15

-10

-5

0

5

10

15

Lattice site

Figure 8.6: (Top): Chemical potentials of condensates in each site are compared to theenergy of motion along the axial axis (Left) and the radial axis (Right). (Bottom):Calculated atomic density at each site are shown.

8.3. BEC IN 1D OPTICAL LATTICE POTENTIAL

132






266 nmExcitation (507 nm)





Normalized lineshape

1
0.8
0.6
0.4
0.2
0
-100

-50
0
50
Frequency offset (kHz)100

Figure 8.7: (Top): Experimental situation in spectroscopy of a BEC in 1D optical latticepotentials is schematically shown. The tight confinement of lattice potentials leads tothe increase of the separation between vibrational levels of a trap. (Bottom): Observed
spectrum. In addition to the carrier spectrum corresponding to the |1 S0 , 0i→|3 P2 , 0itransition, the heating sideband |1 S0 , 0i→|3 P2 , 1i was also successfully observed. The
solid line is the calculated spectrum assuming the mean field broadening by “mini” BECs
in each lattice site.

133

Chapter 9
Frequency measurement of the clocktransition in 174Yb
The 1 S0 ↔3 P2 (m = 0) transition is ultranarrow and insensitive to a magnetic field. It havebeen one of the candidates of the frequency standards using the optical transition [3]. We
measured the frequency of the clock transition by using an optical frequency comb. In
this chapter, we first introduce some basics of a frequency comb. Then the frequency
measurement of the clock transition will be presented.
Frequency is one of the most accurately measured physical quantities [99]. Since theoptical frequency f ∼ 106 GHz is much larger than the present electronic frequency
counters, it is impossible to directly measure the optical frequency. Hence, techniques toconvert an optical frequency to the radio frequency where we can use accurate frequency
counters have been developed for a long time such as harmonic frequency chains [100],interval bisection [101] techniques and so on. However, the breakthrough is a frequency
comb technique invented by Udem et al. in 1999 [102].
Frequency comb is based on the mode locked femtosecond laser pulses. In the femtosecond laser cavity with cavity length L, a pulse is circulating with the round trip timeT = vg /L (for the bow-tie cavity) where vg is the group velocity of pulses. As a result,the output pulses are generated with the repetition rate frep = 1/(2πT ) (∼ 1 GHz). The
group velocity vg which is the velocity of the pulse envelope is, in general, different fromthe phase velocity vp which is the velocity of the carrier wave. The phase of a carrier
wave in each pulse, then, shifts by ∆φ from neighbor pulses.
In a frequency domain, femtosecond pulse trains are equally spaced frequency combs.
The interval between each comb is given by frep . The phase shift ∆φ appears as so called
carrier envelope frequency offset fCEO (< frep ). Thus the frequency of the nth comb f isgiven byf = nfrep + fCEO .
(9.1)
Let us consider the beat signal (fbeat ) between the frequency comb and an external laser
flaser . Considering a beat signal from the nearest n0 th comb, the frequency of the externallaser can be written as
f = fCEO + n0 frep + fbeat .
(9.2)

9.1. FC 8003 – MENLO SYSTEMS GMBH

134

Since f locates somewhere between two frequency combs, fbeat is always smaller thanfrep . Thus, all frequencies fCEO , frep and fbeat are in a radio frequency region and can be
accurately measured by frequency counters. Also, it is possible to determine the integern0 (∼106 ) by using a common wavemeter. As a result, the absolute frequency f of theexternal laser can be accurately measured by (9.2).
When we use the frequency comb, we first have to stabilize (determine) frep and fCEO
in (9.2). To this end, using the 10 MHz frequency standard based on a Coordinated
Universal Time (UTC) is desirable.

FC 8003 – Menlo Systems GmbHmeasuredlaser

Detector

frep

HWP

Gratingfceo

Ti:S Crystal

Detector

Prism
pair

HWP

Dichroic
Mirror

fb

HWP

SHG

Detector

HWP

9.1

HWP

Photonic Crystal Fiber

AOM

Ti:S Pulse Laser

Verdi V-8
: mirror: PBS: lens: glass plate: concave mirror

Figure 9.1: Schematics of the frequency comb system (FC 8003, Menlo Systems GmbH)used in this study.

9.1. FC 8003 – MENLO SYSTEMS GMBH

135

Figure 9.1 shows the optical frequency comb system1 used in this work (FC 8003, Menlo
Systems (GmbH)). A femtosecond cavity is designed to make frep lie around 825 MHz.
A photonic crystal fiber [103, 104] enables the octave spanning of the comb region which
is necessary to detect the fCEO . The following optics are designed to detect and stabilize
the frep and fCEO which is explained below.
We note that in stead of the frequency standard based on a UTC which is required tostabilize frep and fCEO , we used the Rb frequency standard (PRS10, Stanford Research
Systems) which will be referred as “Rb clock” in this thesis. To estimate the accuracy of
the Rb clock, we compared the Rb clock signal with the UTC(NICT) after the measurement.

9.1.1

frep

Figure 9.2: Block diagram of the RF system to stabilize frep to the Rb clock.

Detection
frep usually lies between a few 10 MHz and a few GHz. Thus it can be determined bydirectly measuring the pulse interval with a fast photodiode.
Stabilization
To stabilized the frep to the Rb clock, the RF system described in Fig. 9.2 is used.
A double-balanced mixer mixes the frep and Rb clock signal and converts the frep (∼820 MHz) down to ∼ 20 MHz. Next, it is additionally mixed with the DDS (Direct
Digital Synthesizer) signal. By using a PLL (Phase-Locked Loop), the error signal can beobtained to lock the downconverted signal to the DDS signal. Since frep = L/(2πvg ), thefeedback signal is applied to the PZT attached to one of mirrors in a femtosecond laser
cavity2 .
1This frequency comb belongs to NICT, Tokyo. We moved the system from Tokyo to Kyoto before
the measurement.
2
The original PZT driver installed in FC 8003 was very noisy. We replaced it by the PTZ driver
M-2647 (MESS-TEK, Japan).

9.1. FC 8003 – MENLO SYSTEMS GMBH

136

In the frequency measurement, frep was adjusted by the DDS frequency by several Hz
in order to place the beat frequency between the comb and the measured laser at theappropriate RF frequency for the frequency counter (20 ∼ 30 MHz in this system).

9.1.2

fCEO

Figure 9.3: Block diagram of the RF system to stabilize fCEO to the Rb clock.

Detection
Since fCEO is caused by the phase difference in each pulse, we require the interferometer.
Thanks to the photonic crystal fiber, the comb contains an optical octave. Let us consider
the beat signal between the second harmonics of the nth comb fn and the 2nth comb f2nwhich can be written as
2 × fn = 2nfrep + 2fCEO
f2n = 2nfrep + fCEO .
(9.3)
Thus 2 × fn − f2n = fCEO . The second harmonic of a comb is obtained by a BBO crystal
and interfered with the original comb laser. A beat signal is detected by a fast photodiode.
The prism pair in Fig. 9.1 is used to adjust the time delay of one arm of the interferometer.
Stabilization
Figure 9.3 shows the RF system to lock the fCEO to the Rb clock. fCEO is always stabilized
at 80 MHz by locking it to Rb clock signal (10 MHz × 8 ). An error signal obtained bya PLL is feedbacked to the RF power used to adjust the pump power of the femtosecond
laser [105].

9.1.3

Position of frep and fCEO

When we use a frequency comb, four kinds of positions of each frequency are possible
as shown in Fig. 9.4. To determine whether the measured frequency f is upper side

9.2. EXPERIMENT AND ANALYSIS

137
Case 2.

Case 1.
measured laser
fb

fceo

fceo

frep
n-1

0

fb

comb
n

n+1

n-1

0

n

n+1

frequency

Case 3.

Case 4.

fceo

0

n-2

n-1

fb

fceo

n

0

fb

n-1

n

n-2

Figure 9.4: Four kinds of positional relationship of f , frep , and fCEO are possible.

or lower side of the comb, we change the DDS frequency by a few Hz. If f is upperside of the comb (Case 1 and 2 in Fig. 9.4), the beat frequency fbeat becomes small by
increasing the frep and vise versa. Similarly, as for the fCEO , if it is +80 MHz (Case 3
and 4 in Fig. 9.4), the beat frequency fbeat becomes small as fCEO increases by changing
the RF power applied to the AOM. In order to determine the number of comb n in (9.2),
we use the wavemeter WA-1500 (Burleigh/EXFO). Further information about the comb
measurement is described in [106].

Experiment and analysis
1.4

1.4

1.2

1.2

Number of atoms (x105)

Number of atoms (x105)

9.2

1.0
0.8
0.6
0.4
0.2
0.0
-100

1.0
0.8
0.6
0.4
0.2

-50

0

Frequency offset (kHz)

50

100

0.0
-100

-50

0

50

100

Frequency (kHz)

Figure 9.5: Typical spectra of the 1 S0 ↔3 P2 (m=0) transition.
We excite atoms in a FORT whose wavelength is not a magic wavelength. Thus, the9.2. EXPERIMENT AND ANALYSIS

138

transition frequency suffers from the light shift depending on a FORT power. To remove
such effects, we measured the transition frequency at three different FORT powers andextrapolated the unperturbed resonance frequency at zero FORT power. Note that the
weak magnetic field was applied to separate the clock transition from other magnetic
sensitive transitions. Also, one of the FORT lasers of the crossed FORT configuration
was turned off just before the excitation to eliminate the additional light shift.
Figure 9.5 shows typical spectra of the clock transition. The number of atoms in
the ground state is plotted as a function of the frequency offset of the excitation laser(507 nm). The spectral width is determined almost by the Doppler width. The atomic
temperature is about 1 µK in this measurement. We chose 11 frequencies around theresonance and measured the laser frequency by a comb for 100 s at each frequency. Since
our typical experimental procedure (loading, cooling and excitation of atoms) takes 10 s,
more than 10 CCD images were taken in order to measure the number of atoms.
The center frequency is determined as follows. We first fit the summation of the sinefunction and the Gaussian function to the data. In some data, we found the slight shiftof the background level due to the instability of the initial number of trapped atoms. As
shown in Fig. 9.5, such a shift can be well described by the sine function. This may
be caused by the sine-shape frequency shift of the MOT laser for a long term which isstabilized to the ULE cavity.
In order to carry out the weighted fitting of data, we convert the error in the x axis
σx (frequency) to that in y axis σy by taking [107]sµ¶2df (x)2σy (equiv) = σy +
σx ,
(9.4)
dx
where f (x) is a fitting function. Then, we used the weighted least-square fit. Here, notethat we have to consider the temperature shift, that is to say, we use not the Gaussianfunction but the (5.21). However, the ratio of the trap depth of the ground state tothat of the excited state in mex is required which is not known before the fitting. Then
we regards mex as one of the fitting parameters. Since we could separately estimate themex from the fitting results, we evaluated the consistency between these two values andestimated mex .
We also compensated the collision shift by (5.26) and the recoil shift by (5.11). Using
the scattering length between atoms in the ground state and the 3 P2 state a12 obtainedby the BEC spectroscopy (see section 8.2), the collision shift is estimated to be 2∼3 kHz
in the present case. In addition, when the trap depth is shallow, we have to consider theeffects of the gravity (= 0.2 µK/µm) while it was relatively small (about 1 kHz) in thismeasurement.
Finally, considering all of these frequency shifts, the unperturbed frequency is determined by the linear weighted least-square fit as shown in Fig. 9.6. The transition
frequency is determined to be
590 902 342 562 ± 3 ± 60 kHz.

(9.5)

Here The first error results from the statistical uncertainties originating from fitting of thedata to find a resonance frequency, linear fitting of the observed resonance frequencies,

9.2. EXPERIMENT AND ANALYSIS

139

Frequency - 590 902 342 562 kHz

and the frequency counting. The second error is associated with uncertainties of the Rb
frequency standard (10 MHz) used for the optical frequency comb which is estimatedbelow.
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
0

5

10

15 20 25 30 35
FORT power (mW)

40

45

50

Figure 9.6: Resonance frequencies are shown as a function of the FORT power. Eachresonance frequency was measured by an optical frequency comb.

9.2.1

Rb clock
UTC (NICT)
time

Local oscillator

Splitter

TIC

Rb-clock

Figure 9.7: Block diagram of the DMTD system.
In order to evaluate the measured frequency, we estimated the accuracy of the Rb clock(PRS10, Stanford Research Systems). To this end, we compared the Rb clock with the9.2. EXPERIMENT AND ANALYSIS

140

UTC(NICT) by a DMTD (Dual Mixer Time Difference) system. The mechanism of the
DMTD system is schematically shown in Fig. 9.7. It has two inputs and one output. The
reference signal, UTC(NICT)-10 MHz, is put into the input A. The measured signal, theRb-clock, is put into the input B. Then, the DMTD system outputs the phase difference∆φ = (tB − tA )νA where tB − tA is the time interval between the input A and B. By
observing ∆φ for a long time and if ∆φ is
1. constant, νA = νB
2. decreasing, νA < νB
3. increasing, νA > νB .
Thus, we can determine whether the Rb clock frequency is larger or smaller than the
UTC(NICT). Qualitatively, the difference can be described by
µ
¶
1
1
∆t =
−
× ν0 × T,
(9.6)
νA νB
where ν0 = 10 MHz, T is the total measurement time and ∆t is the change of (tA − tB )during T . Since νA ∼ νB ∼ ν0 , this equation can be written as
µ ¶
∆t
ν0 ,
(9.7)
∆ν =
T
where ∆ν = νA − νB . For example, let us consider the case where the ∆φ decreases
from 0.7 to 0.4 during the measurement time T = 8×104 s (about one day). Then, since∆t = (0.7 − 0.4) × 10−7 = 3 × 10−8 s, we can state that the measured frequency is larger
than the UTC(NICT) by 4 µHz. In general, the result is described by ∆ν/ν0 as a functionof measurement time.
Figure 9.8 shows the stability of Rb clock when we change the magnetic field andtemperature around it. Based on this measurement, we evaluate all possible environmental
errors (temperature, magnetic field) and estimated the error of 60 kHz.
Finally, in Fig. 9.9, we show the Allan deviation between Rb clock and UTC(NICT)
signal. The stability is limited by the stability of Rb clock (∼ 10−12 at 100 s). In orderto improve the accuracy of the frequency measurement, we should link the Rb clock to
the GPS signal for a long term frequency stability and may use the hydrogen maser forthe short term frequency stability.

9.2. EXPERIMENT AND ANALYSIS

141


    Time (x 105 sec)
Figure 9.8: Frequency instability of Rb clock against the change of the external magnetic
field and temperature. Rb clock was compared to the UTC(NICT) signal by the DMTD
system. Magnetic field up to 1 G was applied and the temperature was changed from 15
to 35 ◦ C.

Allan deviation

10-10

10-11

10-12

10-13 0
10

101

102
103
Averaging time (s)104

Figure 9.9: Allan deviation between Rb clock and UTC(NICT). The stability is limitedby the stability of Rb clock (∼ 1012 at 100 s).

142

Chapter 10
Summary and outlook
10.1

Summary

Optical trapping of 3 P2 atoms and their unique collisional properties
We investigated the unique collisional properties of 3 P2 atoms. In previous studies, evaporative cooling of 3 P2 atoms in a magnetic trap was unsuccessful due to the large trap loss
caused by multichannel collisions in a magnetic trap. Therefore, we decided to trap 3 P2
atoms in an optical trap (FORT), which can trap 3 P2 atoms in every magnetic sublevelwith the same strength. We can expect that, although multichannel collisions can still
occur, they will not lead to the trap loss.
To this end, a new technique to prepare ultracold and dense 3 P2 atoms in a FORT
was developed in this study. We first prepared pre-cooled 1 S0 atoms in a FORT and then
optically excited such atoms to the 3 P2 state. By this new method, we achieved a number
density of 2 × 1013 cm−3 at a temperature of 2 µK with a phase space density (PSD) of
0.01. Our newly achieved number density is larger than that in a previous study by anorder of three [1].
While a trap loss due to the multichannel collisions must be suppressed in our FORT,
we still observed large trap loss due to two-body inelastic collisions. Thus, we deduce theexistence of a different inelastic collisional process, which we interpret as fine-structurechanging collisions in this ultracold temperature regime. The previous theoretical studies
revealed details of fine-structure changing transitions in collisions of Mg[3 Pj ], O[3 Pj ],Sc[2 Dj ], and Ti[3 Fj ] atoms with closed-shell atoms at a high temperature. However,there has not been any theoretical study on the fine-structure changing collisions between
atoms in the 3 P2 state at ultralow temperatures achieved in the present study. Whilea recent experiment on magnetically trapped Ca atoms studied multichannel collisions
between 3 P2 atoms and discussed the possibility of the fine-structure changing process [2],we believe that our study is the first definite experimental measurement of this process
between 3 P2 atoms. For further understanding of the observed large inelastic collision
rate and possibilities of the fine structure changing collisions, a quantitative theory on
the collisional properties of 3 P2 atoms is highly desirable.

10.2. OUTLOOK

143

High-resolution spectroscopy of ultracold atoms and BECs using the ultranarrow 1 S0 ↔3 P2 transition
The other important achievement in this study is the successful observation of the ultranarrow magnetic quadrupole 1 S0 ↔ 3 P2 transition (507 nm) in Yb bosonic (174 Yb) andfermionic (171 Yb, 173 Yb) isotopes.
Previously, studies on ultranarrow transitions in two-electron atoms have aimed entirely at ultraprecise frequency standards. In contrast, this study showed how powerful
the ultranarrow transition is as a tool for the spectroscopy of ultracold atoms and BECs.
We first developed a 507-nm ultranarrow-linewidth laser system. In order to observe
the ultranarrow transition, the laser linewidth was reduced to less than 1 kHz by tightly
locking it to a high-finesse optical cavity. This transition had never been observed prior
to this study. Hence, we estimated the transition frequency using an optical frequency
comb. The developed laser system and the estimated transition frequency enabled us toobserve the 1 S0 ↔3 P2 transition for the first time.
High-resolution spectroscopy of a BEC was thus demonstrated. We successfully detected the mean field shift of a BEC using this ultranarrow optical transition. We observed
not only the large mean field shift in a BEC, but also the change in the lineshape, whichreflects the density distribution of a BEC in a trap. As a result, a12 (the scattering length
between atoms in the 1 S0 state and the 3 P2 state) was successfully determined from theobserved spectrum. Furthermore, we performed spectroscopy of condensates in 1D opticallattice potentials and observed the mean field shift due to the on-site interaction at eachsite. Since, in the experiments using optical lattice potentials, the hopping energy andthe on-site interaction energy are important parameters, the technique developed in thisstudy enabled us to measure the on-site interaction energies with high precision in such
experiments.
Furthermore, the polarizabilities of all the magnetic sublevels of the 3 P2 state were
determined with high precision. Using an optical frequency comb, we also measured thefrequency of the magnetic-field-insensitive 1 S0 ↔ 3 P2 (m = 0) transition in 174 Yb, which
is one of the candidates for next-generation atomic frequency standards [3].

10.2

OutlookThe 3 P2 state and the ultranarrow transition between the 1 S0 and the 3 P2 states havegreat potential for future studies in atomic physics.
As observed in this study, it is true that a large inelastic collision rate leads to a short
trap lifetime of 3 P2 atoms even in a FORT. However, to overcome the trap loss due toinelastic collisions, we can use 3D optical lattice potentials or ultracold fermions, which
do not collide with each other at ultracold temperatures. In particular, as for metastable3
P2 bosons in 3D optical lattice potentials, the number of atoms finally becomes 1 or 0
per site due to strong two-body inelastic collisions. In such a situation, since no collision
shift occurs, it is an ideal system for an atomic clock or for precise measurement.
Using the magnetically sensitive 1 S0 ↔3 P2 (m = ±1, ±2) transitions in a magnetic field
gradient, we can address atoms with ultrahigh spatial resolutions. In addition, considering10.2. OUTLOOK

144

the large scattering length a12 of Yb, we can probe the density distribution of atoms at
any point in a trap with higher precision, compared to experiments using alkaline metal
atoms. Thus, for example, we may be able to probe the density distribution of a phaseseparated mixture of different kinds of condensates. It will also be possible to measure theon-site interaction energy at any site of optical lattice potentials. Since, in experiments
using optical lattice potentials, the hopping energy and the on-site interaction energy areimportant parameters, such a technique must therefore be a powerful tool in experimental
studies of Bose-Hubbard models using optical lattice potentials.
Quantum computing using the dipole-dipole interaction of 3 P2 atoms between neighboring sites in optical lattice potentials has been proposed [82, 13]. Now, this is experimentally well within our capability [106]. Using the 1 S0 ↔3 P2 transition, we can initialize,
manipulate, and read any qubit (atom) in optical lattice potentials.
The technique of controlling the polarizability of the 3 P2 state by an external magnetic
field, as demonstrated in section 7.4.1, also has great potential. We may be able to realize
Stark shift cancellation by only rotating the magnetic field. It may also be possible torealize a situation where Stark shift cancellation is satisfied both for the 3 P0 state andthe 3 P2 state simultaneously. This could be done by adjusting the external magnetic fieldat the magic wavelength of the 3 P0 state. Such a situation must be an ideal system tocompare two atomic clocks under the same circumstances or to test the time-varying finestructure constant α, [108] because no systematic errors exist in such a system.
In ion trap experiments, cooling of an ion by using an ultranarrow transition with a
combination of quenching processes is a key technique [14]. In the case of neutral atoms,
sideband cooling using the ultranarrow 1 S0 ↔3 P2 transition seems to be impossible due tothe different trap shape between the states used for cooling and quenching. However, we
may be able to use a repumping laser, which can immediately quench the 3 P2 atoms to theground state (see Chapter 6). If the repumping time is considerably shorter than the trap
frequency, it may be possible to make the heating effect negligibly small, thus allowing
the sideband cooling mechanism to work. In addition, Reichenbach and Deutsch haverecently proposed a novel sideband cooling technique using the 1 S0 ↔3 P0 clock transitionand hyperfine states in a 171 Yb fermionic isotope [15]. Such a technique may be applied
to the 1 S0 ↔3 P2 transition.
As for the frequency measurement demonstrated in Chapter 9, we need to further
improve the accuracy. First, the Rb clock should be linked to a GPS signal for a longterm stability. To improve the short-term stability, the Rb clock may be replaced by,for example, a hydrogen maser system. Precise measurement of the absolute transition
frequencies of hyperfine states of the 3 P2 state in fermionic isotopes in the kilohertz range
will enable us to probe the nuclear magnetic octupole moments [109].
Furthermore, according to [115], the spectral width of I2 molecules becomes narrowest
around 507 nm (∼ 40 kHz [HWHM]). Thus, the 507-nm laser source developed in thisstudy has great potential as a new portable frequency standard using the narrow iodine
spectra around 507 nm. We demonstrated the spectroscopy of I2 molecules (Appendix
C).

145

Appendix A
Calculation of the magic wavelengthof the 3P2 state

Potential depth (a.u.)

For future experiment, we estimated the magic wavelength of the 3 P2 state. The calculation seems not to be reliable in the visible region because we have to include effects
of a lot of upper states. Here, we show the result above 900 nm where such effects
may be small. The calculation was completely same as that in section 4.3. The matrix elements between the (6s6p)3 P2 state and the following states are considered [60]:(5d6s)3 D1 , (5d6s)3 D2 , (5d6s)3 D3 , (5d6s)1 D2 , (6s7s)3 S1 . The linearly polarized FORT andzero magnetic field are assumed. In Fig. A.1, the calculation results are shown. The
magic wavelength is estimated to be 965 nm for the 3 P2 (|m| = 1) state and 1030 nm forthe 3 P2 (|m| = 0) state. In this region, there is no magic wavelength for the 3 P2 (|m| = 2)state.

965 nm1030 nm

Figure A.1: Estimation of the magic wavelength of the 3 P2 state for linearly polarized
FORT and zero magnetic field.

146

Appendix B
Basic theories on an SHG ring cavity
The MOT beam at 556 nm is obtained by a dye laser in our laboratory. In order to use
a dye laser, we (students) have to work very hard; we must change dye and clean up thecavity almost every week, use Ar ion laser and so on. Thus, the new laser source at 556
nm which is stable and maintenance-free is strongly desired.
The double wavelength of the MOT beam is 1111.3 nm. This wavelength is close to
the wavelength 1.3 µm which is known as zero-dispersion wavelength of the optical fiber.
Recently the 1.3 µm laser is widely used in the field of optical broadband communications.
Thus, many stable commercial lasers around 1 µm are now available.
We use a commercial fiber laser (Koheras or Keopsys) at 1111.3 nm. Its linewidth isbelow 100 kHz. In addition, we find that a nonlinear crystal LBO can convert 1111.3 nm
to 555.6 nm efficiently by the proper ring cavity. We can obtain more than 400 mW green
laser and its frequency locking scurvies for more than ten hours.

B.1

Theory of an SHG technique

B.1.1

Wave equation for a nonlinear medium

Let us consider a certain medium in which both averaged charge density and current
density are zero. If the medium is illuminated by a laser beam, atoms and molecules
inside the medium are polarized. Let the density of these induced electric dipoles be
P(r). The electrostatic potential at r generated by P(r) can be written as
Z
³ 1 ´
1
0
P(r ) · ∇dV 0 ,(B.1)
φ(r) =
0
4π²0 V|r − r |where V 0 is the volume of considered system. By using³ P(x0 ) ´ ∇0 · P(r0 )
³ 1 ´
0
0
∇0
=
+
P(r
)
·
∇
|r − r0 ||r − r0 ||r − r0 |and Gauss’s divergence theorem
I
Z
A(r) · n(r)dS =∇ · A(r)dV
S

V

(B.2)

(B.3)

B.1. THEORY OF AN SHG TECHNIQUE

147

where n(r) is the normal vector at r on S, we can transform (B.1) intoZZ
1
1∇0 · P(r0 ) 0−P(r0 ) · n0 0φ(r) = −dV+dS .
4π²0 V |r − r0 |4π²0 S |r − r0 |(B.4)

On the other hand, if we assume that, instead of P(r), some charges are distributed inside
V and on the surface S and their distributions are given by ρd (r) in V and wd (r) on S,
the potential can be described asZZ01
ρd (r0 )1
0
0 wd (r )φ(r) =dV +∇dS 0 .
(B.5)
0
0
4π²0 V |x − x |4π²0 S |r − r |Comparing (B.4) to (B.5), we haveρd (r) = −∇ · P(r).

(B.6)

This equation shows that we can regard induced electric dipoles P(r) as the charge distribution ρd (r). From the Maxwell equation (∇ · (²0 E) = ρ), we have∴

∇ · (²0 E) = ρd= −∇ · P.
∇ · (²0 E + P) = 0,

(B.7)
(B.8)
(B.9)

where D ≡ ²0 E + P is called dielectric flux density .
In general, P is proportional to E. However, in some mediums like nonlinear crystals
we are considering, the nonlinear components of P plays an important role. In that case,P can be divided into a linear part PL and nonlinear parts PNL , that is to say,
P = PL + PNL= ²0 χ(1) · E + ²0 χ(2) : EE + · · · ,(B.10)(B.11)

where χ(1) is a linear electric susceptibility and χ(k) (k ≥ 2) are nonlinear electric susceptibilities. The electric susceptibility χ(k) is generally a tensor of rank k + 1.
Here we are interested in the second term (²0 χ(2) : EE) in (B.11) because this term
generates the second harmonic light. “:” means applying two vectors E and E to thetensor of rank 3 (χ(2) ). Thus χ(2) : EE is a vector. If we represent χ(2) : EE by using theircomponents, it is written as(2)(χ(2) : EE)i = χijk Ej Ek(B.12)

where subscripts are contracted. Hence the Maxwell equations in a medium are given by∇·D = 0∂B
∇×E = −∂t∇·B = 0
∂D
∇×H =
∂t

(B.13)(B.14)(B.15)(B.16)

B.1. THEORY OF AN SHG TECHNIQUE

148

We neglect the current density in (B.16) because we assumed that the current density is
0. Since∇ × (∇ × E) = ∇(∇ · E) − 4E = − 4 E³ ∂B ´∂∂ 2D∇× −= −µ0 (∇ × H) = −µ0 2 , (∵ (B.16))
∂t
∂t
∂t

(B.17)(B.18)

by taking the rotation of (B.14) and using D = ²0 E + P, we have
∂ 2E
∂ 2 PNL
4E = ²µ0 2 + µ0
,
∂t
∂t2

(B.19)

where ² = ²0 (1 + χ(1) ). This is the basic wave equation for a medium which includes
nonlinear polarization PNL .
If a nonlinear crystal is illuminated by the electric field Ein , the nonlinear polarization
PNL is induced in the crystal because PNL = ²0 χ(2) : Ein Ein . The physical meaning of(B.19) is that if PNL exists in the crystal, a certain electric field Eout defined by (B.19)
can exist and this electric field Eout is the second harmonics we want to generate.

B.1.2

Phase matching condition

In the following discussion, details of the calculations of (B.19) are shown. Let us assume
that the incident electric field to a crystal isEin (r, t) = E0in ei(k·r−ωt) .

(B.20)

We assume that E0in doesn’t depend on r because if we compare Ein to the generated
electric field, we always use strong Ein for nonlinear crystals. Because PNL = ²0 χ(2) : Ein Ein ,PNL can be written as
PNL = ²0 (χ(2) : E0in E0in )ei(2k·r−2ωt) .

(B.21)In order to examine how the second harmonics will be generated and grow up in the
nonlinear crystal by PNL , let us assume that the expected output electric field Eout isgiven by0

Eout (r, t) = E0out (r)ei(k ·r−2ωt) ,(B.22)

Notice that k0 does not necessarily correspond with 2k because of the difference of theirindices of refraction in the crystal. For simplicity, we assume that the polarization of Eoutis parallel to x-axis. z-axis is parallel to k. In this case, Eout and PNL are simplified to0

Exout (z, t) = Ex0,out (z)ei(k z−2ωt)
PxNL (z, t)=(2)²0 (χxjk Ej0,in Ek0,in )ei(2kz−2ωt)(B.23)(B.24)

B.1. THEORY OF AN SHG TECHNIQUE

149

Substituting (B.23) and (B.24) for (B.19) gives us
0,out
d2 Ex0,out
0 dEx
(z)
+
2ik
(z) − k 02 Ex0,out (z) = −4²µ0 ω 2 Ex0,out (z)
2
dz
dz
0
(2)
−4²0 µ0 ω 2 (χxjk Ej0,in Ek0,in )ei(2k−k )z (B.25)

If PxNL (z, t) does not exist, the amplitude of Exout (z, t) must be constant everywhere.
In other words, by imposing the condition that if PxNL (z, t) ≡ 0 then Exout (z, t) =Ex0,out (0)e−2iωt , we havek 2 = 4²µ0 ω 2 .

(B.26)In addition, we assume that Ex0,out does not change within the length of ∼ λ(wavelength),that is to say,¯ d2 E 0,out ¯¯ dE 0,out ¯
¯
¯
¯
¯
x
(z)
¿
(B.27)¯¯¯k x (z)¯.
2dz
dz
From (B.26) and (B.27), (B.25) can be transformed todEx0,out
2µ0 ω 2(2)(z) = i²0 (χxjk Ej0,in Ek0,in )ei∆kz
dz
k
2µ0 ω 2 i∆kz
≡ i
p0 e
,
k

(B.28)(B.29)(2)

where p0 = ²0 (χxjk Ej0,in Ek0,in ) and ∆k = 2k − k 0 . This equation can be easily integrated.
Let the length of the crystal L,
Z L
dEx0,out 0 00,outEx (L) =(z )dz(B.30)dz 00
2µ0 ω 2 p0 ei∆kL−1=(B.31)k∆k
2µ0 ω 2 p0 ³ sin(∆kL/2) ´ i∆kL/2= −Le(B.32)k∆kL/2
2µ0 ω 2 p0 L ¯¯ sin(∆kL/2) ¯¯∴|Ex0,out (L)| =(B.33)¯¯k∆kL/2
Note that the relation between incident power Pin and generated power Pout is describedasPout ∝ Pin2

(B.34)

because p0 includes square of |Ex0,in |.
In Fig.B.1, |Ex0,out (L)|2 are shown as a function of (∆kL)/2. The condition to obtainthe maximum second harmonics power is∴∆k2k − k 0
L =
L=0
2
2
2k − k 0 = 0(B.35)(B.36)

B.1. THEORY OF AN SHG TECHNIQUE

150

1.4

∆k=0

1.2

|E|

2

1
0.8
0.6
0.4
0.2
0
0

2

4

6

8

10

∆kL/2

Figure B.1: Phase matching condition is satisfied at ∆k=0. The vertical axis is normalizedby (2µ0 ω 2 p0 L)/k.

where k 0 and k are the wave vectors of the induced polarization and the output electricfield respectively. This condition is called “Phase matching condition”. k 0 and k can be
described by the index of refraction n(ω),
2ω
k 0 = n(2ω)
cωk = n(ω) .
c(B.37)(B.38)

Thus the phase matching condition can be given as∴2ω{n(2ω) − n(ω)} = 0c(B.39)n(2ω) = n(ω),(B.40)

which means that the nonlinear crystals for SHG have to be the same refraction index
for both ω and 2ω. For the purpose of this, birefringent crystals are used. The specificcharacteristic of a birefringent crystal is that the crystal has two different refraction indicesfor each orthogonal polarization.
As an example, we will consider an uniaxial crystal which has one optic axis (so called
c-axis). In an uniaxial crystal, the beam whose polarization is orthogonal to the c-axis
is called “ordinary beam” and the beam whose polarization is parallel to the c-axis iscalled “extraordinary beam”. Let the refraction indices for ordinary and extraordinary
beams no (θ, ω) and ne (θ, ω) respectively, where θ is the angle between c-axis and wave
vector. The nonlinear crystals can be classified by the magnitude relation of ne (θ, ω) andno (θ, ω). If ne (θ, ω) > no (θ, ω), the crystal is called a “positive uniaxial crystal”. On theB.1. THEORY OF AN SHG TECHNIQUE

151

other hand, if no (θ, ω) > ne (θ, ω), it is called a “negative uniaxial crystal”. For a positive
uniaxial crystal, if one chooses proper θ,no (2ω1 ) = ne (ω1 )(B.41)

is satisfied. In this case, the input beam should be the extraordinary beam and the output
beam should be the ordinary beam, which is described as (e, e, o). This configuration iscalled “Type I phase matching”. The configuration (e, o, e) or (e, o, o) is called “Type IIphase matching”.

B.1.3

CPM and NCPM

Since both ne and no depend on θ, we have to tune the angle of a crystal so that the phasematching condition is satisfied. This method is called “Critical phase matching”(CPM).
It is true that we can generate the second harmonics by CPM, but CPM has one defect.
If θ 6= 90◦ , the direction of the Poynting vector S does not correspond to the direction of
the wave vector k for the extraordinary beam in the crystal. Thus the ordinary and theextraordinary beams separate as they propagate in the crystal, which leads to the loss ofthe conversion efficiency. This effect is called “walk off effect”.
To overcome the walk off effect, “Non-critically phase matching”(NCPM) is used.
The birefringence strongly depends on temperature in some crystals. Hence, the phasematching condition can be satisfied at θ = 90◦ by tuning a crystal temperature. Since nowalk off effect exists at θ = 90◦ , high conversion efficiency can be realized.
For the purpose of just generating SH, CPM is good enough if one can find a proper
crystal. However, if one needs high power second harmonics, NCPM is desirable althoughadditional temperature control system is required.

B.1.4

Conversion efficiency

In general, we use a CW Gaussian beam to generate a second harmonic. In this case,
Eckardt et al. [110] showed that the conversion efficiency ηCW is given byηCW =2ω 2 d2eff Lk
Pout (2ω)=
h(B, ξ)Pin2 (ω)π²0 c3 n3
d2 L= 16π 2 eff2 3 h(B, ξ)²0 cn λ0

(B.42)(for Guassian beam)(B.43)

where Pin (ω) and Pout (2ω) are the powers of the incident fundamental(after surface losses)
and the output second harmonic(before surface losses), c is the speed of light, ω is theangular frequency of incident fundamental, deff is an effective nonlinear coefficient, L
is a crystal length, and k is the magnitude of the fundamental wave vector of incident
fundamental inside the crystal. In the last equation, k = 2πn/λ0 and ω = 2πc/λ0 where
λ0 is the wavelength of the fundamental incident are used. h(B, ξ) is a function known as
the Boyd and Kleinman focusing factor first introduced in [111]. Here, B and ξ are calledB.1. THEORY OF AN SHG TECHNIQUE

the walk-off parameter and the focusing parameter, respectively and defined as
√
Lk
B = ρ
2
L
λ0 L
ξ =
=
,
2
kw0
2πnw02152(B.44)(B.45)

where ρ is the birefringent walk off angle, w0 is the √1/e amplitude radius at the beam
waist. Also, if ξ ¿ 1(weak focusing condition) and πw0 /ρ > L (birefringent walk off
aperture length > crystal length), by assuming the perfect phase matching, h(B, ξ) canbe approximated ash(B, ξ) ' ξ(1 −

t2
t4
t6
+
−
+ · · ·)
12 120 1344

(B.46)√
where t = 2B 2ξ. According to [111], it is proved that h(B, ξ) takes its maximum valueat ξ=2.84 and B=0. In NCPM case, B=0 is always satisfied. Then we should design thecavity to make ξ close to 2.84.

B.1.5

LBO

Considering that generated 556 nm light by SHG technique will be used for MOT, theoutput power is expected to be 200∼300 mW at least. Therefore, NCPM is suitable forour experiment because it does not suffer from the loss of efficiency caused by the walk-off
effect. We chose an LBO crystal because it can generate 555.6 nm light from 1111 nm
light by Type-I NCPM according to [112] and also has a high damage threshold. Details
about the light source of 111 1nm will be discussed in the next section.
Lithium triborate(LiB3 O5 , LBO) is a very popular and widely used nonlinear crystal.
LBO is a negative biaxial crystal. According to the catalogue of CRYSTECH Inc., an
LBO crystal has the following specific characteristics.
• Broad transparency range from 160 nm to 2600 nm• Relatively large effective SHG coefficient (about three times that of KDP)
• High damage threshold (18.9 GW/cm2 at 1053 nm)• Type-I and Type-II Non-Critical Phase Matching (NCPM) over a wide wavelengthrange
We bought two LBO crystals (3×3×7 mm and 3×3×10 mm) from KASTON Inc.(China).
Both sides of crystals are Anti-Reflection coated for 1111nm and 555.6nm light. The
damage threshold of the AR coat is 300∼400 MW/cm2 and reflectivity R of the AR coat
is R < 0.2 % at 532 nm light.

B.2. DESIGN OF THE RING CAVITY

B.2

153

Design of the Ring Cavity

The easiest method to generate the second harmonics is to put a nonlinear crystal at the
focal point of the fundamental light. It is true that the second harmonics will be producedby this method. However it may be very weak due to small conversion efficiency. Thus,in general, the optical cavity is used to accumulate the power of fundamental light andwe put a nonlinear crystal into the cavity. A Fabry-Pérot resonator and a ring typeresonator is generally used as an optical cavity. For the purpose of generating second
harmonics, a ring type resonator is commonly used. The reason is that the fundamental
light propagates in one direction in the ring type resonator and the generated second
harmonics in nonlinear crystal also travel in the same direction. On the contrary, thefundamental travels back and forth between two mirrors in a Fabry-Pérot resonator. As
a result of this, the second harmonics are generated in both opposite directions. In order
to use generated second harmonics as much as possible, this is not convenient1 . We, then,decided to use a ring cavity.
In order to design a ring cavity for the SHG, the following two procedures are required.
First, we have to find best parameters related to reflectivities of cavity mirrors whichmake ξ close to 2.84. ξ is determined by the Rayleigh length zR of the fundamental
laser at the nonlinear crystal. For the calculation of zR , ABCD matrices are used [113].
Next, by considering the loss rate on the surfaces of mirrors and crystals, the “impedance
matching condition” is examined. By using all these parameters, we can estimate the
SHG conversion efficiency ηSHG and then predict the theoretically expected SHG power.

B.2.1

ABCD matrices









Figure B.2: Definition of the ray vector.
In order to design the optical system, ABCD matrices are often used [113, 114]. In
this method, every optical element is described by a 2×2 matrix and assembles of such
optics can be represented just by multiplying ABCD matrices of each elements. The mostpowerful point of this method is that the change of a Gaussian beam can be traced by
1

There exists some methods to efficiently pick up the second harmonics generated in a Fabry-Pérot
cavity by using their polarization.

B.2. DESIGN OF THE RING CAVITY

154

using a famous ABCD law. Hence, when we design the optical cavity, this method is often
used to calculate the beam waist, the total cavity length for the stable cavity operationand so on.
There exist two kinds of ABCD matrices. This makes the situation a little complicated.
The difference is the treatment of the refractive index. In the following discussion, these
two methods will be used parallel. One will be introduced as Kogelnik [113] and the otheris Yariv [114]. As shown in Fig.B.2, a light ray can be defined by
µ
¶
µ ¶
x
x
[Kogelnik],[Yariv],(B.47)nθθwhere x is the position, θ is the angle and n is the refractive index of the medium. This
vector is called a “ray vector”. When this light ray propagates through some optics, rayvector changes. ABCD matrices describe the relationship between the input ray vector
and the output one.
µ
¶
µ
¶µ
¶
xout
A B
xin
=
[Kogelnik](B.48)
nout θout
C D
nin θin
µ
¶
µ
¶µ
¶
xout
A B
xin
=
[Yariv],(B.49)
θout
C D
θin

B.2. DESIGN OF THE RING CAVITY

B.2.2

155

Examples of ABCD matrices

ABCD matrices are already known for many optical components we usually use. In thissection, I will introduce some ABCD matrices required for our subsequent calculations.
· Propagation d in n medium











Figure B.3: Propagation d in n medium
From Fig.B.3 , the next relation must be satisfied.
θout = θin ,xout = xin + dθin

Then the ABCD matrix becomes
¶µ
¶
µ
¶
µ
xin
xout
1 d/n
[Kogelnik]
=
nθin
nθout
0 1
µ
¶
µ
¶µ
¶
xout
1 d
xin
=
. [Yariv]
θout
0 1
θin
· Entering from n1 medium to n2 medium  
Figure B.4: Entering from n1 medium to n2 medium(B.50)(B.51)

B.2. DESIGN OF THE RING CAVITY

156

According to the Snell’s law,nin θin = nout θout ,xout = xin .

(B.52)

Then,
µ

¶
µ
¶µ
¶
xout
1 0
xin
=
[Kogelnik]
nout θout
0 1
nin θin
µ
¶
µ
¶µ
¶
xout
1
0
xin
=
. [Yariv]θout
0 nin /nout
θin(B.53)· Reflection by curved mirror whose curvature is R








Figure B.5: Reflection by curved mirror whose curvature is R
From Fig.B.5,
tan θout = tan(π + θin − 2φ) = tan(−θin + 2φ)∴ θout = −θin + 2φ
θin − φ =xin.
R

(B.54)(B.55)

By substituting (B.55) to (B.54),
θout = θin −

2
xin
R

xout = xin .
Then the ABCD matrix is derives as follows.
¶
¶
µ
¶µ
µ
1
0
xin
xout
[Kogelnik]
=
nθin
nθout
−2n/R 1
¶
¶
µ
¶µ
µ
1
0
xin
xout
. [Yariv]
=
θin
θout
−2/R 1(B.56)(B.57)(B.58)

B.2. DESIGN OF THE RING CAVITY

B.2.3

157

ABCD law

There exists a famous and very useful law between ABCD matrices and q-factor of Gaussian beam. This is know as the “ABCD law”. Note that because of the difference ofthe definition of the base for ABCD matrices in Kogelnik and Yariv representation, theABCD law is also different.
Generally, we can consider the ABCD matrix as a black box. However the next two
equations are always valid.
½
xout = Axin + Bnin θinnout θout = Cxin + Dnin θin[Kogelnik](B.59)½
xout = Axin + Bθin
θout = Cxin + Dθin
[Yariv]
Let us introduce the optical ray R which is the vector whose direction is perpendicular to
the wave front. By assuming small θ, we can obtain the next equation.
Rout =

xout
Axin + Bnin θin
= ³
1 ´
θout
(Cxin + Dnin θin )nout
nout (ARin + Bnin )[Kogelnik]=
CRin + Dnin
(B.60)

Rout =

Axin + Bθin
xout
=
θout
Cxin + Dθin
ARin + B=[Yariv]CRin + D

As you can easily confirm, R and q-value for Gaussian beam are converted by opticalelements in a same manner. For example, both satisfy the lens equation, and so on. Thismeans that, we can replace R by q included in above equations.
³q ´
in
A
+B
³q ´
out
in ´
³n
=
[Kogelnik]
qin
nout
+D
C
nin
Aqin + B
qout =[Yariv](B.61)
Cqin + D
This is the “ABCD law”.

B.2. DESIGN OF THE RING CAVITY

158

Kogelnik’s ABCD matrices satisfy one more useful relation2 . If we consider the opposite propagation with same optical path described by same ABCD matrices, next two
equations must be satisfied.
µ
¶
µ
¶µ
¶
xout
A B
xin
=
[from left to right]
nθout
C D
nθin
¶
¶
µ
¶µ
µ
A B
xout
xin
[from right to left]=
C D−nθout−nθinThis can be possible only for Kogelnik’ base. If we use Yariv’s base, the second equation
is not always satisfied. We can easily see this difference by considering the light which
enters medium n2 from n1 .
Applying the inverse matrix to the second equation and comparing two equations witheach other, we get
A = D,AD − BC = 1.
(B.62)
Let me repeat that this is not always valid for Yariv’s base.
According to the definition of q-factor of Gaussian beam,
11
λ= − i 2.
qrπw(q−factor)(B.63)

where r is the curvature and w is a spot size.
Let us consider designing the optical cavity by using ABCD matrices. In the stable
optical cavity, a certain ray from some point surely return to the same point with sameq-factor. This means that, qout = qin in (B.61). Therefore by constructing the total
ABCD matrix and solving the quadratic equation, you have complex number q describedby ABCD. According to (B.63), you can estimate the beam curvature from real part ofq, and the imaginary part can be used to calculate the beam waist.

2

Yariv’s ABCD matrices also sometimes satisfy the same relation, but not always. Kogelnik’s ABCD
matrices always satisfy this relation.

B.2. DESIGN OF THE RING CAVITY

B.2.4

159

Ring cavity

Let us consider a ring cavity as an example. With the aid of ABCD matrices introducedin the previous sections, the total ABCD matrix for the ring cavity of Fig. B.6 can be
described as3

Figure B.6: The ring cavity for SHG. L is the half length of a nonlinear crystal.
µ

1 L
0 1

¶µ

¶µ
¶µ
¶µ
¶
1 0
1 L2
1
0
1 L1 + 2L3 cos α
0 1/n
0 1
−2/R 1
0
1
¶
¶µ
¶µ
¶µ
µ
1 L
1 0
1 L2
1
0
[Yariv]
×
0 1
0 n
0 1
−2/R 1(B.64)

where R is the curvature of M4 and M3 and n is the refractive index of an LBO crystal.
Note that the direction of the fundamental light is perpendicular to the surface of anonlinear crystal and that next equation must be satisfied from the geometrical restriction.
L1 = 2L3 cos α − 2(L + L2 ).

(B.65)

One of the possible designs is shown in Fig. B.7 with some parameters. Calculating theconversion efficiency ηSHG by using these parameters, (B.43) and parameters in Table.
B.1 gives us
ηSHG = 1.4 × 10−4 . [W−1 ]B.2.5(B.66)

Enhancement factor

In [110], Eckardt et al. showed that the ratio of the fundamental power incident on thering cavity P1 and the fundamental power reflected from the cavity Pr is written as√√( r1 − rm )2
Pr=,(B.67)√
P1(1 − r1 rm )2
3

Starting point is the center of the SHG crystal.

B.2. DESIGN OF THE RING CAVITY

160
M1 (Flat Mirror)

M2 (Flat Mirror)
Collimation Lens (f = 15mm)

LBO Crystal(Crystal Length: 7mm ~373K)

PZT

Lock System

SHG Beam(300mW 556nm)

Incident Beam
(1W 1111nm)

MonitorOven
M4 (Concave Mirror R = 50mm)

M3 (Concave Mirror R = 50mm )M1 - M2140mm
M2 - M3100mm
M3 - M458.0mm
Total
398mm
Incident Angle 8 degree
Beam Waist in the Crystal 33.9µm
Rayleigh Length 3.0mm
Boyd-Kleinman Coeff. ξ 0.73

Thermal Isolator (Teflon)[Mirror Coating] ( λ1 =1111nm λ2 = 556nm )Left Side

Right SideM1

R = 99.7% @ λ1

AR @ λ1

M2

N/A

100% @ λ1

M3

R = 100% @ λ1

M4

AR @ λ2

AR @ λ1
R = 100% @ λ1
& AR @ λ2 (Dichroic Mirror)

Figure B.7: One of the possible designs. The calculated output power is 300∼400mW.

where r1 is the power reflection coefficient of the cavity mirror M1 (see Fig.B.7), rm iscalled a cavity reflectance parameter and defined as
rm = T tSG r2 ,(B.68)

where tSG (= (1 − ηSHG Pc )) is the crystal transmission. Pc is the circulating fundamental
power just inside the first mirror M1. r2 is the power reflection coefficient of the ringcavity mirrors M2, M3 and M4. T is the single-pass power transmission coefficient which
does not include Pc and r2 .
From (B.67) and (B.68), we can achieve Pr =0 by choosing proper r1 which satisfies
r1 = rm . This condition is called “impedance matching condition”. When this conditionis satisfied, the maximum Pc is achieved. Therefore we have to carefully choose thereflectivity of M1.
To determine Pc for given P1 , we can use the next equation given in [110].
1 − r1Pc=.
√
P1(1 − r1 rm )2
Since rm is a function of Pc , we can calculate Pc as a function of r1 .

(B.69)

B.2. DESIGN OF THE RING CAVITY

B.2.6

161

Numerical computations for 1111.3→556 nm, and 798→399
nm system

Now we can estimate the expected SHG power of designed system. For the purpose ofnumerical calculations, we use (B.43). In TableB.1, required parameters are listed. I
also calculated the theoretically expected values for the LAS system. LAS is used in ourlaboratory to convert 798 nm to 399 nm.

Table B.1: Parameters for SHG systems in our laboratory(1111.3 nm→555.6 nm and 798
nm→399 nm). λ0 and P0 are the wavelength and incident power of fundamental light, n
is the refractive index for the fundamental light, Lcrystal is the crystal length, zR and w0
are the Rayleigh length and the beam waist of the fundamental light at the crystal, deffis the effective nonlinear coefficient, h is the Boyd-Kleinman coefficient, and ηSHG is the
SHG conversion efficiency.

Fiber laserLAS

λ0 (nm)1111.3798

P0 (W)

1

1

n

1.56

1.61

Lcrystal = 2L (mm)712zR (mm)3.0

10.5

w0 (µm)

33.9

51.6

deff (pm/V)

1.24

0.75

h(B, ξ)

0.73

0.36

ηSHG (×10−4 W−1 )

1.4

1.1

B.2. DESIGN OF THE RING CAVITY

500

T=99.5%
T=99.0%T=98.5%
T=98.0%450400
SHG Power (mW)

162

350
300
250
200
150
100
50
0
95

96

97

98

99

100

M1 Reflection (%)
Figure B.8: Theoretically expected SHG power for 1111.3→556nm SHG system[NCPM,
Type-I]. P0 =1W, w0 = 33.9µm, Lcrystal (Crystal length)=7mm.

350

SHG Power (mW)

300
250

T=99%
T=98%
T=97%
T=96%

200
150
100
50
0
95

96

97
98
M1 Reflection (%)

99

100

Figure B.9: Theoretically expected SHG power for 798→399nm SHG system[CPM, TypeI]. P0 =1W, w0 = 51.6µm, Lcrystal (Crystal length)=12mm.

163

Appendix C
Iodine spectroscopy at 507 nm

Figure C.1: Ultranarrow “compact” laser system at 507 nm developed in this study. Allcomponents are installed on an optical table of 90×120 cm.
According to the [115], the spectral width of iodine molecules becomes narrowest
around 507 nm. Thus, our compact laser system (see Fig. C.1) has possibility of afuture high performance transportable frequency reference. We therefore demonstratespectroscopy of iodine molecules using the developed laser system.
We first performed absorption spectroscopy of iodine molecules. Observed spectra
over 50 GHz are shown in Fig. C.2 . The amplitude is normalized so that distortions

164of the signal due to power fluctuations are removed. The numbers labeled to some ofthe spectra in Fig. C.2(Bottom) correspond to those in [116]. In addition, the detailed
spectrum of one of the lines in No. 1388 resonance in Fig. C.2(Bottom) was obtained via
saturation spectroscopy as shown in Fig. C.3. The spectrum in Fig. C.3 is broadened bysaturation broadening and residual Doppler broadening due to the slight misalignment
between the pump and probe lasers in saturation spectroscopy. Further development of
the system toward high-resolution spectroscopy of iodine molecules using, for example,
the modulation transfer technique [117, 118] is now in progress.

I2 Cell

PD1

PD2

1394

1393

1391
1392

507nm

1388
1389

1387

1390

PPLN

1385
1386

Normalized absorption

1014nmFrequency (GHz)
Figure C.2: Absorption spectroscopy of iodine molecules over the 50 GHz around 507 nm.
Some of resonances are identified to those in [116] and labeled with the corresponding
numbers.

165

 
I2I CellCell2(0(0 ) )  
PD1
PD1



!"#  Figure C.3: Detailed spectra of one of the lines in No.1388 resonance in Fig. C.2 taken
by saturation spectroscopy.

166

Appendix D
Spherical harmonics and vector
spherical harmonics
Some spherical harmonics and vector spherical harmonics used in this work are listed
below 1 .
(D.2)
Spherical harmonics rL YL,M (θ, φ)r
1
Y0,0 (θ, φ) =
4π
r
r
3 (1)
3 (1)
rY1,0 (θ, φ) =r0 ,rY1,±1 (θ, φ) =
r
4π
4π ±1
r
µ
¶
5 3
r2
(1) 2
2
(r0 ) −
r Y2,0 (θ, φ) =4π 2
3rr r ³ ´
5 √ (1) (1)5 3 (1) 22
2
r Y2,±1 (θ, φ) =3 r0 r±1 ,r Y2,±2 (θ, φ) =r
4π
4π 2 ±1r
7 1 (1)(1)
r0 (5(r0 )2 − 3r2 )r3 Y3,0 =
4π 2
r r
r r
7
3
7 15 (1) (1) 2(1)(1)
r3 Y3,±1 =
r± (5(r0 )2 − r2 ),r3 Y3,±2 =r (r )
4π 8
4π 2 0 ±r r
7 5 (1) 33(r )r Y3,±3 =4π 2 ±

1(1)

First rank irreducible tensor rq (q = −1, 0, +1) are given by1(1)
r±1 = ∓ √ (x ± iy),2

(1)

r0 = z.

(D.1)

Vector spherical harmonics YL,J,M (θ, φ)r
1
Y0,1,0 =(Y1,−1 ê+1 − Y1,0 ê0 + Y1,1 ê−1 )3

Y1,0,±1 = Y0,0 ê±1 ,r
Y1,1,±1 = ±Y1,0,0 = Y0,0 ê01(Y1,±1 ê0 − Y1,0 ê±1 ) Y1,1,0 =2

r

1(Y1,1 ê−1 − Y1,−1 ê+1 )
2

r

√
√
1
Y1,2,±1 =(Y2,0 ê±1 − 3Y2,±1 ê0 + 6Y2,±2 ê∓1 )
10
r
√
1 √
Y1,2,0 =( 3Y2,−1 ê+1 − 2Y2,0 ê0 + 3Y2,1 ê−1 )10r

1
Y2,1,±2 = Y1,±1 ê±1 , Y2,1,±1 =(Y1,0 ê±1 + Y1,±1 ê0 )2r
1(Y1,−1 ê+1 + 2Y1,0 ê0 + Y1,1 ê−1 )Y2,1,0 =6r

1 √( 2Y2,±2 ê0 − Y2,±1 ê±1 )
3
r
√
1 √
( 3Y2,0 ê±1 − Y2,±1 ê0 − 2Y2,±2 ê∓1 )Y2,2,±1 = ∓6r
1
Y2,2,0 =(Y2,1 ê−1 − Y2,−1 ê+1 )2
Y2,2,±2 = ±

r

√
√
1 √
( 2Y3,1 ê−1 − 3Y3,0 ê0 + 2Y3,−1 ê+1 )
7
r
√
√
1 √
Y2,3,±1 =( 3Y3,0 ê±1 − 8Y3,±1 ê0 + 10Y3,±2 ê∓1 )
21
r
√
√
1
Y2,3,±2 =(Y3,±1 ê±1 − 5Y3,±2 ê0 + 5Y3,±3 ê∓1 )21

Y2,3,0 =BIBLIOGRAPHY

168

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