Dissertation

All-optical selective formation of ultracold moleculesin the rovibrational ground state

Kiyotaka Aikawa

The University of Tokyo
Department of Applied Physics
February, 2011

Abstract
This thesis describes the experimental realization of a novel approach for producing ultracold
molecules in the rovibrational ground state. Photoassociation followed by stimulated Raman
adiabatic passage (STIRAP) enabled us to achieve ultracold 41 K87 Rb molecules in the rovibrational ground state within 100 ms, which was faster by more than two orders of magnitude thanprevious methods based on magnetoassociation. The presented all-optical method opened up a
new avenue for manipulating a wide variety of molecular species including diamagnetic atoms,
which cannot be manipulated via previous methods.
We ﬁrst describe the experimental procedure for producing and detecting ultracold weakly
bound 41 K87 Rb molecules via the photoassociation of laser-cooled 41 K and 87 Rb atoms. The
apparatus consisted of a vacuum chamber, optical setups for laser cooling, photoassociation
lasers, a pulse laser for ionization, and ion-detecting devices. Combined with time-of-ﬂight mass
spectrometry, resonance enhanced multi-photon ionization (REMPI) enabled us to detect KRb
molecules in a speciﬁc vibrational level.
The crucial step towards realizing ultracold molecules in the rovibrational ground state wasan appropriated choice of the intermediate state for STIRAP. We found 10 vibrational levels(v ′′ = 41–50) of the (3)1 Σ+ state via depletion spectroscopy for weakly bound molecules. A
detailed analysis enabled us to construct an accurate potential energy curve of the (3)1 Σ+ state
as well as to identify the symmetry of the observed state as (3)1 Σ+ . Another candidate for theintermediate state, the (2)3 Σ+ state, was also studied for 17 vibrational levels (v ′′ = 39–55). In
each vibrational level, we observed structures of the order of 1 GHz, which were understood asrotational structures. In addition, we found that hyperﬁne structures due to the nuclear spin of
87 Rb played an important role in the levels with a low rotational quantum number (J ′ < 3).
Based on the spectroscopic studies, we developed a stable light source for STIRAP at two
wavelengths (900 nm and 650 nm) which could connect weakly bound molecules and the rovibrational ground state. Diode lasers referenced to an ultralow expansion (ULE) cavity served
as master lasers, which had a short-term linewidth below 10 Hz and a long-term stability better
than 20 kHz. Other lasers, a diode laser and a dye laser, were then locked to master lasers andwere utilized for STIRAP experiments. By changing the oﬀset frequencies between the master
lasers and the slave lasers, we could scan the slave lasers over a few GHz with keeping thestability of the master lasers. The tunability and the stability realized in this setup were one of
the key elements for realizing the STIRAP transfer of photoassociated molecules.
The most important issue for reaching the rovibrational ground state was a new spectroscopic
method, which enabled us to precisely determine the transition frequencies from photoassociatedmolecules to the excited state (3)1 Σ+ . Further, this new method enabled us to observe therovibrational ground state via two-photon dark resonance spectroscopy, which played a crucialrole for determining the transition frequency as well as the transition strength between the
excited state and the rovibrational ground state. Finally we demonstrated the selective formationof ultracold molecules in the rovibrational ground state via a STIRAP transfer with an eﬃciency
of more than 70 %. The produced molecules were directly detected via REMPI ionization. The
STIRAP transfer into the rovibrational ground state opened a new possibility for precisely
measuring the hyperﬁne structures of weakly bound molecules in unexplored region with anaccuracy of 10 kHz.

謝辞
本論文は東京大学工学系研究科物理工学専攻井上研究室における博士課程での研究をまとめたものです。本研究を進めるに当り、非常に多くの方々から助言や助力を頂きました。この場を借りて、お世話になった方々へ厚くお礼を申し上げます。
グループリーダーである井上慎准教授には、実験の詳細から、研究の進め方、研究発表の仕方や論文の書き方に至るまで、研究全般に渡って多くの事柄を指導して頂きました。物理の様々な分野に関する豊かな知識と、それに基づく的確な洞察には多々学ぶものがありました。極低温極性分子という興味深いテーマを与えて頂き、自由に研究できる環境を整えて頂いた点に深く感謝いたします。
立ち上げ当時助教であった岸本哲夫特任准教授（現電気通信大学）には、研究を立ち上げていく上で重要となる様々なポイントを余すところなく教えて頂き、大変お世話になりました。技術的な側面から研究の方向性の決定に至るまで、実験家とはどうあるべきか、という姿勢を学ばせて頂きました。また、電気通信大学に移って以降も助言や助力を頂きましたことに感謝致します。
小林淳助教には、E1 グループ立ち上げ当初からレーザーを初めとした実験装置の扱いを一から丁寧に教えて頂きました。また、特に本研究において重要な役割を果たしている、ULE
共振器の真空槽製作に多大な貢献をして頂いた点に改めて感謝致します。私が E2 に移って以降も実験結果の解釈や装置の詳細についての議論に毎日のように付き合って頂いたことは、
単に本研究の成果を得るという以上に大きな糧となりました。
共に E2 グループ立ち上げに関わってきた赤松大輔博士（現産業技術総合研究所）には、本研究に関わる実験内容ばかりでなく、他グループの研究、技術的な問題など多岐に渡ってほぼ毎日議論に付き合って頂き、大変感謝しております。本研究を完成させることができたのは、様々な物事に対して一から謙虚に考える姿勢を教えて頂いたお陰であると言っても過言ではありません。
ERATO 上田マクロ量子プロジェクトのプロジェクトリーダーである上田正仁教授には、ERATO という素晴らしい研究環境を提供して下さったこと、また研究内容に関する鋭い指摘・洞察を頂いたことに深く感謝致します。また、研究者としてどのような視点を持つべきか、という点についていくつかもの示唆を頂き、大変ためになりました。
ERATO 上田マクロ量子プロジェクトのグループリーダーである上妻幹男准教授（東京工業大学）には、幅広い分野に渡る興味深い話題を教えて頂いたお陰で、視野を大きく広げる
ことができました。実験から理論に至るまでの幅広い知識に基づく議論から、多くのものを学ばせて頂きました。また、分光の実験においては波長計を、STIRAP の光源開発においてはスペアナを貸して頂き、大変助かりました。改めて感謝致します。
ERATO 上田マクロ量子プロジェクトのグループリーダーである向山敬特任准教授（電気通信大学）には、実験の詳細に関して多くの有用な助言を頂くと共に、研究内容に関しても様々な鋭い指摘を頂き、大変お世話になりました。また、向山グループから光学機器から電子機器に渡るまで種々の機器をお貸し頂いたお陰で、迅速に研究を進めることができました。
改めて感謝致します。
ERATO 上田マクロ量子プロジェクトの研究員である堀越宗一博士、福原武博士、竹内誠
博士、武井宣幸博士、学振研究員であった衛藤雄二郎博士、および向山グループの博士課程の学生であった稲田安寿氏には、日々の議論を通じて、実験の詳細から他グループの研究状
況に至るまで、様々な新しい知識・情報を教えて頂きました。自分自身の研究テーマとは異なる研究分野についても見識を得ることができたのは、同じ居室内で日々こうした方々と話をできたお陰です。
ERATO 上田マクロ量子プロジェクトの研究員である Pascal Naidon 博士には、本研究に関して、特に理論的な側面から多くの有用な助言をして頂き、大変助かりました。基底状態の超微細構造に関する計算をして頂いたことは、本研究の大きな支えとなりました。
清水富士夫教授（電気通信大学）には、研究上の助言を頂いたばかりでなく、本研究において重要な役割を果たす色素レーザーの色素を提供して頂きました。改めて感謝致します。
香取秀俊教授には、本論文の審査員を努めて頂いたばかりでなく、特に Raman 光源開発に当って、非常に重要な助言を頂きました。この場を借りて改めて感謝致します。
本論文の審査員を務めて頂きました五神真教授、古澤明教授には、審査において多くの貴重なご助言を頂きましたことを感謝致します。
向山グループで博士課程の学生である中島秀太氏には、特に三体の物理を中心として日々議論を交わし、多くの知見を頂いたことに感謝致します。また、ERATO プロジェクト内の唯一の同期として幾つもの会議に共に参加できたことは良い思い出となりました。
井上研究室の修士課程に在籍し、同じ E2 グループとして実験を行ってきた林正泰氏、および大麻浩平氏には、実験作業を通じて多くの示唆を頂きました。その丁寧な仕事にいつも助けられてきました。改めて感謝致します。
井上研究室において修士課程の学生であった野田開氏、藤掛陽輔氏、山陸芳之氏、および現在修士課程の学生である加藤宏平氏、斉藤祐介氏には、E1 グループにおける実験の詳細を度々教えて頂きました。また、上妻グループの修士課程の学生である野口篤史氏には、上妻グループでの実験について詳しく教えて頂き、興味深い知見を得ることができました。
井上研究室の学部学生であった森圭介氏、田野岡祐介氏、新栄拓人氏、福岡健太氏には、実験装置の製作や条件出しなど多くの実験作業を手伝って頂きました。
物理工学専攻工作室の川端氏、佐藤氏には研究の全般に渡って必要となった様々な部品の製作に際し、多くの助言・助力を頂きましたことを感謝致します。
ERATO 事務室の事務参事である佐藤敏彦氏、技術参事である榎本陽一氏、技術員である
国府田愛氏、および事務員である塩田美奈子氏には、研究を進める上で不可欠となる物品の購入から事務手続きや論文校正の手続きに至るまでの多くの作業を行って頂き、大変お世話になりました。滞りなく研究を進めることができたのは、こうした方々の尽力のお陰です。
この他、ミーティングや議論を通じて研究活動を支えて頂いた方々に、改めてお礼を申し上げます。

Contents
1 Introduction
1.1 Cold atoms to cold molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Photoassociative formation of ultracold molecules . . . . . . . . . . . . . . . . . .
1.3 This study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
5
6

2 Experimental setup
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Coil setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 MOT coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Feshbach optimized photoassociation coils . . . . . . . . . . . . . . . . . .
2.4 MOT lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Master lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Injection locked slave lasers . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Tapered ampliﬁers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Photoassociation laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Transfer cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Amplifying system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Pulsed dye laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Ti: Sapphire laser and dye laser . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Ion detection and time-of-ﬂight mass spectrometry . . . . . . . . . . . . . . . . .
2.9 Computer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1 Timing control with a LabView program . . . . . . . . . . . . . . . . . . .
2.9.2 Data acquisition with a LabView program . . . . . . . . . . . . . . . . . .
2.9.3 Laser control and wavemeters . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Calibration of laser frequencies with a cavity . . . . . . . . . . . . . . . . . . . .

8
8
8
10
10
10
11
12
16
20
25
25
28
30
32
33
34
34
35
36
38
39

3 Production and detection of photoassociated molecules
3.1 Preparation of dense and cold sample of 41 K and 87 Rb atoms . . . . . . . . . . .
3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Steady-state MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Compressed MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Photoassociation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Scanning over a wide range . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Rotational structures and symmetries . . . . . . . . . . . . . . . . . . . .
3.2.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Ionization spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Estimation with Franck-Condon factors . . . . . . . . . . . . . . . . . . .
3.3.2 Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
41
42
43
48
48
49
51
51
52
52
57

1

4 Molecular spectroscopy for a Raman transfer
60
4.1 Excited states for an intermediate state . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 The (3)1 Σ+ state: Direct potential ﬁt analysis . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Ionization spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Depletion spectroscopy and rotational analysis . . . . . . . . . . . . . . . 63
4.2.3 Parity-selective spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.4 Measurement of the binding energy . . . . . . . . . . . . . . . . . . . . . . 67
4.2.5 Transition strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.6 Constructing an accurate potential: Direct potential ﬁt analysis . . . . . . 72
4.2.7 Theoretical consideration on the saturation broadening in depletion spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 The (2)3 Σ+ state: Rotational and hyperﬁne analysis . . . . . . . . . . . . . . . . 80
4.3.1 Triplet molecules in the ground state . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Depletion spectroscopy with triplet molecules . . . . . . . . . . . . . . . . 83
4.3.3 Parity-selective spectroscopy with singlet molecules . . . . . . . . . . . . . 85
4.3.4 Understanding observed spectra with a three-level model . . . . . . . . . 87
4.3.5 Hyperﬁne structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.6 Possible intermediate levels . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.7 Theoretical treatment of the rotational and hyperﬁne structures of triplet
molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Raman lasers
103
5.1 Required performance and the design . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Development at 795 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.1 ECDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Servo circuit for locking an ECDL . . . . . . . . . . . . . . . . . . . . . . 108
5.2.3 EOM for PDH locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.4 Interferences in an optical ﬁber . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.5 Digital optical phase-locked loop . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Development at 901 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 ULE cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.2 ECDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.3 Low noise photo-detector for PDH locking . . . . . . . . . . . . . . . . . . 127
5.3.4 Phase noise cancellation in an optical ﬁber . . . . . . . . . . . . . . . . . 128
5.3.5 Improvements on optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.6 Improvements on electronics . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.7 Analog optical phase-locked loop . . . . . . . . . . . . . . . . . . . . . . . 140
6 Coherent transfer into the rovibrational ground state149
6.1 Diﬃculties associated with photoassociated molecules . . . . . . . . . . . . . . . 150
6.2 SpIDR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Two-photon dark resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4 STIRAP transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.5 Coherence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6 Studies with the rovibrational ground-state molecules . . . . . . . . . . . . . . . 180
6.6.1 Molecular transitions from the rovibrational ground state . . . . . . . . . 181
6.6.2 Transition strength from a weakly bound level . . . . . . . . . . . . . . . 183
6.6.3 Precision spectroscopy of weakly bound molecules . . . . . . . . . . . . . 186
6.7 Theoretical description of a two-photon dark resonance and a STIRAP transfer . 188
6.7.1 General discussions for a three-level system . . . . . . . . . . . . . . . . . 188
6.7.2 Two-photon dark resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.7.3 STIRAP transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

2

7 Conclusion and outlook

198

A Publication: Toward the production of quantum degenerate bosonic polar
molecules, 41K87Rb
A1
B Coherent transfer of photoassociated molecules into the rovibrational groundstate
B1
C Predicting and verifying transition strengths from weakly bound molecules C1

3

Chapter 1

Introduction
1.1

Cold atoms to cold molecules

Since the realization of quantum degeneracy with neutral atoms [1–3], cold atoms have been
one of the cutting edges in the broad range of physics. As a novel many body system which
can be studied via the direct imaging of wavefunctions, they opened a route to precisely study
quantum phenomena known in condensed matter physics, such as Bardeen-Cooper-Shreeferstate [4], Superﬂuid-Mott insulator transition [5], Anderson localization [6, 7], Tonks-Girardeau
state [8, 9], and Berezinskii-Kosterlitz-Thouless transition [10]. Further, cold atoms enabled us
to investigate a wide variety of novel quantum states such as Spinor BEC [11], Repulsively
bound pairs [12], Eﬁmov states [13], spontaneous symmetry break down in spin textures [14],and phase-transition in a coupled cavity-BEC system [15].
These striking results have been relied mainly on two advantages in tunability: interaction
between atoms can be tuned via Feshbach resonances whereas tunable periodic potentials can beobtained with an optical lattice. A Feshbach resonance is a kind of scattering resonance whicharises as a result of the resonant coupling with a bound state [16]. When the molecular state has
a diﬀerent magnetic moment from that of the atomic state, the amount of coupling is dependent
on a magnetic ﬁeld, enabling us to tune the interaction only with external ﬁelds [17–19]. An
optical lattice is usually formed as a retro-reﬂected optical trap, which has a periodic structurewith half the wavelength of the laser. It allows us to investigate many body physics in a periodic
environment similar to electron systems in solids as well as to realize tunable low dimensional
systems. In addition, an optical lattice has an important application in frequency standards as
an atomic clock [20].
The prominent achievements obtained with cold atoms provoked several new directions. Oneof them is cold molecules: a quantum matter with an anisotropic dipole-dipole interaction is
expected to show unprecedented behaviors, such as crystalline phase [21, 22], checkerboard, andsupersolid phases [23]. Cold molecules have been of great interest also because of their possible
applications in quantum simulation of a lattice-spin model [24], quantum computation [25], theprecision measurements of the electron electric dipole moment (eEDM) [26] and of the electronto-proton mass ratio [27, 28] and ultracold chemistry [29, 30]. Thorough reviews on the current
status and the applications of ultracold molecules are found in Ref. [31–33].
It is not straightforward to achieve cold molecules with a temperature below 1 mK. Lasercooling, which gives a good starting point for obtaining quantum degenerate atoms, is diﬃcult
for molecules due to their complex internal structures. Although laser cooling of molecules
in one dimension has recently been realized for a special molecular species [34], it is still amajor challenge to obtain three-dimensionally cold molecules via laser cooling. There have been
two main strategies for obtaining cold molecules. One is direct cooling and the other is theassociation of cold atoms. We brieﬂy overview these methods.
In direct cooling methods, molecules are cooled through collisions with buﬀer-gas atoms[35, 36] or decelerated after supersonic expansion by converting kinetic energy to the Stark
4

shift [37]. In addition, there is another scheme, which does not cool molecules but captures
low velocity components in room temperature molecules [38]. With these methods, molecules
have been trapped at a milli-Kelvin regime [39–42].These methods have an advantage in theirability to deal with a wide variety of molecular species, but thus far they have been limited in
the lowest available temperature (∼1 mK).
In contrast, the association of atoms relies on photoassociation [43, 44] or magnetoassociation at a Feshbach resonance [45] of laser-cooled or evaporatively cooled atoms. This method
enables us to fully take advantage of cooling schemes developed for atoms; hence it is suited forachieving a quantum degenerate gas of molecules. One obstacle in this method is that produced
molecules have a high vibrational energy because photoassociation and magnetoassociation can
only couple atoms with weakly bound molecular state. The vibrationally excited molecules havean extremely low dipole moment, and readily decay to lower vibrational levels via collisions
with each other. Thus, for further applications, it is imperative to remove the large vibrationalenergy (4000–6000 cm−1 for alkali diatomic molecules) without heating, which can be realized
via a two-photon transition from the weakly bound state to the lowest rovibrational level.
The scheme based on a two-photon transfer of weakly bound molecules was ﬁrst demonstrated for RbCs by Yale group [46]. In this work, weakly bound molecules were prepared
via the photoassocition of laser-cooled atoms and were transferred into the vibrational groundstate via stimulated emission pumping. Although this experiment did not intend to achieve aquantum degenerate gas, there are two important aspects which have been shared with later
experiments. First, they demonstrated a general framework of a two-photon transfer of weakbound molecules. Second, they showed the existence of an excited state which had large wavefunction overlaps with both weakly bound state and the vibrational ground state. In order toobtain a quantum degenerate gas of molecules from quantum degenerate atoms without loosing
phase space density, one should employ a coherent method that connects a single quantum statewith another quantum state. The combination of magnetoassociation and Stimulated Raman
adiabatic passage (STIRAP) [47–49] is an ideal method for this purpose, because both magnetoassociation and STIRAP enable a unit transfer eﬃciency in principle. Here all of the degrees
of freedom in a molecule, hyperﬁne, rotation, and vibration, can be controlled. With the fully
coherent method, molecules in their rovibrational ground state have been achieved for 40 K87 Rb
at JILA [50] and for Rb2 [51] and Cs2 [52] at Innsbruck. The realization of the rovibrational
ground-state molecules have opened a new avenue for studying chemical reactions at a temperature near quantum degeneracy, including the observation of a universal reaction rate [29] andthe suppression of reactions in a 2D optical lattice with an induced dipole moment [53].
Now that the method for achieving the ground-state molecules is established, we are ina position to investigate new physics with ultracold polar molecules. Here the stability ofmolecules is crucial. The experiments at JILA have been dealing with KRb, which is reactive
even in the rovibrational ground state due to a collisional decay into the lower energy levelof K2 and Rb2 [29, 54]. In this point of view, half of the alkali diatomic molecules, LiNa,
LiK, LiRb, LiCs, and KRb, are reactive, whereas the other half, NaK, NaRb, NaCs, KCs, and
RbCs, are not reactive. Moreover, recent theoretical studies on the possible collisional reaction
channels, 2KRb→KRb2 +K, K2 Rb+Rb, have shown that these channels are endothermic for any
combinations of alkali atoms [55–57]. Thus, we can expect that the latter half of alkali diatomic
molecules shown above are stable against collisions and are ideal for investigating new physics
with dipolar interaction.

1.2

Photoassociative formation of ultracold molecules

The STIRAP transfer of Feshbach molecules is a promising way towards quantum degenerategases of alkali diatomic molecules. However, it is not available for all molecular species: thus far,magnetic Feshbach resonances are available only for atomic species with electronic spin. Thus,
another scheme is required for dealing with molecules including diamagnetic atoms. For exam5

ple, alkali-alkaline earth molecules, such as SrRb, are of great interest because they can simulate
a lattice spin model [24]. Alkaline earth diatomic molecules, such as Sr2 , are a promising playground for measuring the time variation of the electron-to-proton mass ratio [27]. The mostgeneral way to form these molecules is photoassociation: one-photon excitation to an excitedmolecular state followed by spontaneous emission forms weakly bound molecules. Photoassociation have been observed at a much wider temperature range for a much more atomic species
than magnetoassociation. Since photoassociation is an important starting point in this study,
we brieﬂy review previous studies herein, focusing on the creation of ultracold molecules. A
thorough review on photoassociation is found in Ref. [44].
The photoassociative formation of ultracold molecules was proposed in 1987 [58], but it
had not been observed until 1998. The ﬁrst observation was made for Cs2 molecules produced
in a magneto-optical trap (MOT) [59]. This experiment had most of the important features
shared among the later experiments. Photoassociated molecules were ionized by an intense
nanosecond pulse laser. The molecular ions were accelerated to an ion detector by a highelectric ﬁeld. The molecular signal was separated from the atomic signal with a diﬀerence intime-of-ﬂight (TOF). Similar experiments have been demonstrated for other molecular species
as well, including K2 [60], Rb2 [61–63], KRb [64–66], RbCs [67], NaCs [68], and LiCs [69].
In the later experiments, a resonance enhanced multi-photon ionization (REMPI) scheme was
employed for detecting molecules in a speciﬁc vibrational level. Combined with a CW laser, thestate-selective detection of photoassociated molecules enables us to carry out the rotationallyresolved spectroscopy of excited states, which we call depletion spectroscopy throughout thisstudy after its ﬁrst implementation [70].
Another direction in photoassociation experiments is to associate quantum degenerate atoms.
In contrast to the experiments in a MOT, the low temperature of atoms made it possible tocouple an atomic level with a weakly bound level via a Raman transition without distributing
over multiple rovibrational levels [71, 72]. In these experiments, spontaneous Raman scatteringof the photoassociation lasers limited the lifetime of produced molecules to approximately 1
ms. An important step towards the eﬃcient state-selective formation of deeply bound molecules
was the observation of a coherent dark state between atoms and molecules, in which a small
fraction of weakly bound molecules was produced [73]. A coherent dark state between atoms
and molecules was also observed in a MOT of sodium [74].
A breakthrough in the photoassociation experiments was the formation of RbCs molecules inthe vibrational ground state by Yale group [46]. Since then, ultracold molecules in the vibrational
ground state has been obtained in two methods. The ﬁrst method was the direct photoassociation to a deeply bound excited state followed by spontaneous decays to the rovibrational groundstate [75]. With a careful selection of excited states, they also observed a predominant formationin the v ′′ = 0, J ′′ = 2 level. The second method relies on optical pumping with a broadband
femtosecond laser [76]. By removing the frequency components which can excite molecules inthe vibrational ground state, they realized an eﬃcient transfer of photoassociated molecules tothe vibrational ground state.
These pioneering works have shown that ultracold molecules in the vibrational (and in somecases also in the rotational) ground state can be obtained from photoassociated molecules.
However, a state-selective transfer to the rovibrational ground state has yet to be achievedfor photoassociated molecules. For a wide variety of the intriguing applications of ultracold
molecules, it is crucially important to populate them in a speciﬁc rovibrational level.

1.3This study

In this study, I show that a rotationally and vibrationally pure sample of ultracold 41 K87 Rb
molecules can be directly produced in a MOT via photoassociation followed by a STIRAP. Thisis the ﬁrst realization of a STIRAP transfer of photoassociated molecules into a deeply boundlevel. Although the scheme of this study itself is similar to that of previous experiments with6

Feshbach molecules, the situation is quite diﬀerent from them due to the incoherent nature ofphotoassociation. Special attentions on various topics were required for reaching this end. In
this thesis, I cover the entire topics on how we converted laser-cooled 41 K and 87 Rb atoms toultracold 41 K87 Rb molecules in the rovibrational ground state without omitting technical details.
In addition to the introduction in the chapter 1, ﬁve main parts comprise this thesis. I brieﬂysummarize the contents of each section herein.
We construct an experimental setup for producing and detecting weakly bound 41 K87 Rb
molecules from laser-cooled atoms. The setup consists of the UHV chamber, the laser systems,
the ion detection system, and the computer control system (chapter 2).
In order to improve the low density and the high temperature of atoms in the steady-state
MOT, we apply time-dependent laser cooling, which enhanced the photoassociation rate by
three orders of magnitude. The optimum wavenumbers for photoassociation and ionization aredetermined such that molecules in each vibrational level can be clearly observed (chapter 3).
Depletion spectroscopy for photoassociated molecules enables us to identify excited statesused as an intermediate state for a STIRAP transfer. For the (3)1 Σ+ state, we obtain thevibrational energies and rotational constants of the v ′ = 41–50 levels, allowing us to construct
an accurate potential curve of the (3)1 Σ+ state [66, 77]. For the (2)3 Σ+ state, we observe therotational and hyperﬁne structures of the v ′ = 39–55 levels, which were clearly understood witha simple analytical model (chapter 4).
For ensuring an eﬃcient Raman transfer, we develop a new two-color light source with alinewidth of below 10 Hz and a mode-hop-free range of more than 2 GHz. The setup consistsof two master lasers and two slave lasers at ∼650 nm and at ∼900 nm. The master lasers arereferenced to a single ULE cavity, whereas the slave lasers are locked to the master laser via
optical phase-locked loop (OPLL) (chapter 5).
We develop a new spectroscopic method, SpIDR spectroscopy, which enables us to measure
the precise frequency of the two-photon resonance and the transition strength to the rovibrational
ground state via a two-photon dark resonance. By directly monitoring the population in thevibrational ground state with REMPI ionization, we conﬁrm that the rovibrational ground-state
molecules are formed via a STIRAP transfer with an eﬃciency of more than 70 %. The STIRAP
transfer into the rovibrational ground state precisely reveals the hyperﬁne structure of weakly
bound molecules in an unexplored region (chapter 6) [78].

7Chapter 2

Experimental setup
2.1

Overview

In this chapter, we describe the experimental apparatus for producing and detecting ultracold
41 K87 Rb molecules via the photoassociation of laser-cooled 41 K and 87 Rb atoms. The setup
is based on the previous studies [67, 79]. Roughly speaking, the experimental procedure is asfollows. First, atoms are cooled and trapped in a magneto-optical trap (MOT). Since atoms in a
steady state MOT are dilute and hot, they are further cooled and compressed by time-dependent
MOT (compressed MOT; C-MOT). After the compressing stage of 40 ms, a photoassociation
laser is irradiated for 10 ms, during which atoms absorb a photon resonant to an excited molecular state. Finally the excited molecules decay to weakly bound molecules in the ground state via
subsequent spontaneous decays. A pulsed dye laser is then irradiated to ionize the ground state
molecules via resonance enhanced multi-photon ionization (REMPI). The produced molecularions are detected at an ion detector (micro-channel plates; MCP). Although atomic ions are also
produced via multi-photon ionization, they are separately detected by means of time-of-ﬂight
mass spectrometry.
For realizing these processes, we need a wide variety of apparatuses, including vacuum components, electronics, and optical elements. Fig. 2.1 shows the schematics of the experimental
apparatus. It consists of an ultra-high vacuum (UHV) chamber, magnetic coils, lasers for coolingatoms, lasers for photoassociation, a pulsed laser for ionization, MCP for detecting ions, a CCD
camera for imaging atoms, and computer systems for controlling the experimental procedure.
In the following sections, we discuss them in detail.

2.2

Vacuum chamber

Fig. 2.2 shows the vacuum chamber for the experiments. For detecting ions at an ion detector,
ions should be accelerated by an electric ﬁeld to the ion detector. Here the electric ﬁeld is appliedby adding a high voltage (2 kV) to the ion detector itself. We employed a metal chamber rather
than a glass cell in order to ensure that an appropriate electric ﬁeld is applied between the MOT
position and the ion detector. Although additional ﬁeld plates can make an appropriate electricﬁeld at the MOT position, we did not employ this scheme for having as large optical accesses
as possible.
In addition to normal six MOT ports (four horizontal ports and two vertical ports), we neededa lot of view ports in this experiment. Both K and Rb atoms are supplied from dispenser sources.
They were installed on a mini port which is the nearest to the MOT position. In the opposite ofthis mini port, we set a CCD camera for monitoring the MOT. The remaining mini ports weredevoted to imaging of atoms and to irradiating a photoassociation laser and a pulsed laser and.
We installed both micro-channel plates (MCP; Hamamatsu, F1551-01) and Channeltron (Burle,4869) for ion detection because we did not know which one was better. Although Channeltron

8Computer 3(Laser control)

Spectroscopy
Laser

Pulse
Laser
Ion detection
systemVacuumchamber

Coils

Photoassociation
Laser

CCD
CameraMOT
LasersComputer 1(Timing control)

Computer 2(Data acqusition)

Figure 2.1: Overview of the experimental apparatus. It composed of a vacuum chamber, MOT
lasers, coils, spectroscopy lasers, a pulsed laser, an ion detection system, a photoassociation
laser setup, and computer control systems. Lights are indicated with red arrows while electronic
signals are indicated with black arrows.

MOT

CCD
Pulse laser
MOT

Channeltron
ICF70
Ion pump(75l/s)Dispenser 117mm

MCP
MOT
MOT

Probe

Figure 2.2: Vacuum chamber for producing and detecting ultracold KRb molecules (top view).
In addition to six MOT ports, there are three ICF70 ports and ﬁve ICF34 ports which enable
us to have large optical accesses. Atoms are supplied form dispensers. The pulsed laser for
ionization and the photoassociation laser are on the same optical axis, but are incident on the
opposite view ports.

9

oﬀered a high gain, it had been broken in a year (it began to loose vacuum seriously when an
electric ﬁeld was applied). Thus, ﬁnally we decided to use MCP for later experiments.
The whole chamber was evacuated with an ion pump (Varian, Diode 75 L/s, operated at
5 kV). The pressure in the vacuum as measured with an ion gauge was 4×10−10 Torr when
we did not operate experiments, whereas it was increased to 6×10−10 Torr when we turned ona high voltage for MCP. Presumably outgases from the MCP slightly increased with the highvoltage. For 41 K, we used a home-made dispenser with enriched 41 K (97 %), which was based on
Ref. [80]. The Rb dispenser was a commercial one (SAES). Even if we did not do experiments,
we always run 3 A to both dispensers such that atoms attached to the wall of the chamber did
not completely detach. We increased the current to 4 A for operating experiments. With theoﬀset current of 3 A, we could recover the MOT within an hour, whereas we required a day ormore to recover the MOT without the oﬀset current. In this way, the dispensers could supply
atoms required for experiments almost constantly for more than 3 years.

2.3

Coil setup

The control of magnetic ﬁelds is one of the essential parts of this experiment. For realizing ahigh photoassociation rate suﬃcient for further experiments, it is imperative to precisely remove
stray magnetic ﬁelds with compensating coils as well as to compress the density of atoms via
increasing the magnetic ﬁeld gradient with normal MOT coils.

2.3.1

MOT coils

The MOT requires anti-Helmholtz coils which can make a magnetic ﬁeld gradient of at least
15G/cm. In addition, a larger magnetic ﬁeld gradient should be applied for compressing atoms
in a C-MOT stage. We prepared a set of coils by winding polyimideamide coated cupper wires
with a diameter of 1.5 mm by 240 times. This coil set could make 45 G/cm with a current of
10 A, which was suﬃcient for compressing both K and Rb atoms.
The current for the coils was controlled digitally via a solid state relay (crydom, D1D40)
whereas the magnitude of the current was manipulated via an external control of the power
supply (Kikusui, PAN 35-10A) (Fig.2.3). The digital control could turn oﬀ the magnetic ﬁeld
within 500 µs whereas the analog control could change the current within a few tens of ms, which
was limited by the power supply. A resistor was connected in parallel with the coil such that
an electromotive force in the MOT coil during the C-MOT process could immediately decay.
Further, a diode was installed in series for protecting the power supply.
In addition to the normal MOT coil, we required compensating coils for precisely adjusting
the magnetic ﬁeld at the MOT position because the metal chamber was magnetized by the highmagnetic ﬁeld at the C-MOT process. We added approximately 0.2 G in each direction, which
was determined such that the MOT of both K and Rb survived at low magnetic ﬁelds (1 G/cm
for K and 0.5G/cm for Rb). Details on adjusting the MOT will be given in section 3.1.

2.3.2

Feshbach optimized photoassociation coils

We installed another Helmholtz coils for Feshbach optimized photoassociation (FOPA) [81] inorder to enhance the photoassociation rate. The increase in the molecular wavefunction neara Feshbach resonance results in an increase in the photoassociation rate by several orders [18,82, 83]. Although it has not been used yet, we brieﬂy discuss the details herein for future
experiments utilizing FOPA.
In the FOPA, a homogeneous magnetic ﬁeld is applied during photoassociation. For 41 K87 Rb,
there are several Feshbach resonances among several spin channels below 100 G [84,85]. Thus, it
is suﬃcient to apply a low magnetic ﬁeld for enhancing photoassociation rate. Our experimental
apparatuses are operated at 10 Hz synchronizing the pulsed laser. During the repetition time of10

Figure 2.3: Circuit diagram for MOT coils. The solid state relay enables switching oﬀ themagnetic ﬁeld within 500 µs. A diode is put in series for protecting the power supply from theelectromotive force arising from a sudden switch oﬀ of the current. The resistor in parallel withcoils is installed such that the electromotive force should decay immediately.

1Ω

0.4G/A

47mF
100V

250V
0.25A

IGBT
Figure 2.4: Circuit diagram for FOPA coils. The large capacitor accumulates charge from thepower supply and releases it during the photoassociation process of 10 ms.
100 ms, the photoassociation process takes 10 ms. In most cases, a power supply takes a longtime (a few tens of ms) for increasing the current. Hence, we constructed a special coil setup
which could increase a homogeneous magnetic ﬁeld within 500 µs and keep it almost constantfor 10 ms.
Fig. 2.4 shows the circuit diagram for FOPA coils. Inside the MOT coil, we placed two coils
winded by 24 times with an inner diameter of 6 cm in the Helmholtz conﬁguration. The key
idea is to release charge accumulated in a large capacitor: charge is accumulated from the power
supply during 90 ms of charging time, whereas it is released during 10 ms of the photoassociation
process. With this additional coil setup, we could ramp up the current within 500 µs and keep
it almost constant during 10 ms for the photoassociation process.

2.4

MOT lasers

Our MOT lasers are based on semiconductor lasers. Fig.2.5 shows the schematics of our MOT
laser system. External cavity diode lasers (ECDLs) locked to the atomic resonance (master
laser) are ampliﬁed with injection locked slave lasers and tapered ampliﬁers. In the following
11

10mW
3mW

1mW

ECDL(767nm)

160mW

Slave

TA

K cooling

Slave

TA

K repumping

Slave
3mW

10mW
10mW

ECDL(780nm)
ECDL(780nm)

5mW

Slave

150mW

Rb repumping
25mW

TA

160mW

Rb cooling

Figure 2.5: MOT laser system for K and Rb. The MOT lights are all obtained from semiconductor lasers. We had one master laser, three slave lasers, and two tapered ampliﬁers for K
whereas two master lasers, one slave laser and one tapered ampliﬁer for Rb. Master lasers andslave lasers are placed on the same table, whereas tapered ampliﬁers are placed on bread boards.
sections, we discuss each laser in detail, mainly focusing on the important aspects for operatingthem appropriately.

2.4.1

Master lasers

ECDLs, or grating-stabilized diode lasers, are widely used for cold atom experiments as low cost,
tunable, narrow-band laser sources [86, 87]. It takes advantage of the fact that a semiconductor
laser has a wide gain spectrum of the order of 10 nm, allowing us to tune the wavelength overin this wide range just by changing the angle of the grating. Moreover, tuning the cavity lengthwith a piezo module makes it possible to continuously scan over a few GHz (and in some cases
more than 100 GHz). When the temperature of the laser diode is appropriately stabilized, an
ECDL can be stably locked to an atomic resonance and can be utilized for a wide variety of
applications such as spectroscopy and laser cooling.
We employed the Littrow conﬁguration since we needed a high power but a limited scanning
range was necessary for experiments (a few GHz). Fig. 2.6 shows the schematic (a) and image(b) of our ECDL, which is slightly modiﬁed from the design in Ref. [88]. A mirror was added
after the grating such that the output beam did not move largely with a change in the angle of
the grating. In this type of ECDL, the diﬀraction angle is used for rough tuning on the orderof 1 nm, whereas the piezo module (Noliac, CMAP09; capacitance, 440 nF) on the back side of
the grating is used for ﬁne tuning. Our standard grating for the near-infrared (NIR) light was agold coated 1800 lines/mm holographic grating for UV range (Edmund, 43774). The diﬀraction
eﬃciency at the grating is 3 or 5 % for a S polarized light which is dependent on the direction of
the grating, whereas it is more than 20 % for a P polarized light. We have long been using theS polarization for obtaining a high power, but recently we have changed it to the P polarization(this was readily done by rotating the LD with respect to the grating) because a high opticalfeedback allows a much more stable operation, especially in terms of mode-hop-free range.
The laser diode (LD) for ECDL was a commercial anti-reﬂection (AR) coated LD (Eagleyard,
EYP-RWE-790) for both K (767 nm) and Rb (780 nm). This LD oscillated at 30–40 mA andgave approximately 50 mW with a current of 100 mA. In addition, the AR coating on thisLD ensured a stable single-mode operation. This was in contrast to a normal LD: an ECDLwith a non-AR coated LD was normally stable at a speciﬁc current, which was sensitive to thetemperature of LD. Despite these advantages, this LD had a serious problem in its lifetime. In
our environment, the AR coating on the LD had a short lifetime of 1∼1.5 year. Thus, we had
to exchange the LD once in 1∼1.5 year even if the power did not decrease largely.
12Mirror mount(Newport U100-P)

1cm
Grating mount

Piezo module(Single layer)

Collimation tube
(Thorlabs LT230P-B)

Holographic grating(1800 lines/mm for UV)
Laser diode

Adapter(Newport UPA-PA1)

Right angle mirror(Gold coated)(a) Schematic of the structure(b) Image
Figure 2.6: The ECDL for MOT lasers. (a) Schematic of the structure. (b) Image. The black
package attached to the LD is a commercial laser diode socket including a protection circuit(Thorlabs, SR9B).

13

Function
generator

15MHz
4.5V

Power
splitterPhaseshifterServocircuitPhoto-detector

Gas cell

Mixer
EOM
Current
sourceECDL
AOM(80MHz)
Optical table

Piezo feedback

Figure 2.7: Schematics of the locking scheme for a master laser. A diﬀerentiated signal of thesaturated absorption is obtained by the PDH method, which is integrated and feed-backed tothe piezo module of the ECDL. The AOM before the gas cell is required only for K in order toshift the locking point.
The current for laser diodes was controlled by a commercial LD controller (Thorlabs, LDC202C).
Although it has a low current noise (below 5 µA), we found that the current noise broadened
the linewidth of the ECDL to 1 MHz. By adding a RC low pass ﬁlter (R = 22 Ω, C = 2.2
mF electrolytic and 15 µF tantalum in parallel) to the current controller, we could reduce thelinewidth to ∼300 kHz. This linewidth was measured by monitoring the beat note between two
ECDL averaged over a few 100 ms and was presumably limited by the mechanical vibration of
the mirror mount.
The temperature of the laser diode (LD) was not directly controlled. We stabilized thetemperature of the mirror mount with a peltier module under the mirror mount. This scheme
works well if the stabilized temperature was close to room temperature. However, when thestabilized temperature was diﬀerent from room temperature by more than several degrees, thetemperature of the LD was inﬂuenced by the variation in room temperature, causing a slow driftof the oscillation frequency and making the lock unstable. For controlling the temperature, weused a commercial thermoelectric controller (Thorlabs, TED200C).
Hereafter we describe the scheme for locking master ECDL lasers to the atomic resonance.
Fig. 2.7 shows the schematics of the locking scheme for a master ECDL. The master lasers were
locked to the D2 line via Pound-Drever-Hall (PDH) method [89] with a modulation frequencyof 15 MHz. A commercial electro-optic modulator (EOM) (Thorlabs, EO-PM-15-C1), which
was resonant at 15 MHz, was employed for phase modulation. After passing through a heatedgas cell (∼70◦ C for both K and Rb), the phase modulated light was detected via a home-madephoto-detector (Fig. 2.8). The signal from the photo-detector was mixed with an oscillator
driving the EOM at a double balanced mixer, generating a diﬀerentiated saturated absorption
signal. The error signal was ﬁnally integrated in a servo circuit (Fig.2.9) and was feed-backed
to the piezo module after being ampliﬁed by 15 times with a commercial high voltage ampliﬁer(Matsusada, HPZT-0.15). The feedback bandwidth was limited to 1 kHz due to the high voltage
ampliﬁer. We did not use current feedback since locking with the piezo module gave a suﬃciently
stable operation.
The hyperﬁne splitting in the ground state of 41 K was only 254 MHz; hence, we generated
both cooling and repumping light from a single master laser. In contrast, we prepared two
master lasers for the cooling and repumping light of 87 Rb since the hyperﬁne splitting was as14

Figure 2.8: Circuit diagram of the home-made high speed photo-detector.

Figure 2.9: Circuit diagram of a servo circuit for locking a master laser.

15

Intensity
controlFrequency
control
L

VCO

9dBm

I

Mixer
R
0dBm

RF amplifer
30dBm

AOM
Figure 2.10: Setup for driving AOM. A home-made driver controlled the power and frequencyof the radio frequency (RF) signal to the AOM with analog voltages. The small RF signal
obtained from a voltage controlled oscillator (VCO) is ampliﬁed with a high power RF ampliﬁer(Mini-Circuits, ZHL-1-2W).
large as 6.8 GHz.
The frequency of the MOT light was controlled via an acoustic-optic modulator (AOM)
inserted between a master laser and a slave laser. Our standard setup for driving AOM was
illustrated in Fig. 2.10. The intensity and frequency were controlled via a home-made driver(Fig. 2.11). In general, we had two AOMs for a single MOT light: one was inserted before
a slave laser for tuning the frequency whereas the other was inserted after a tapered ampliﬁer
for controlling the intensity of the MOT light. Fig.2.12 and Fig. 2.13 show the whole scheme
for generating MOT frequencies for K and Rb, respectively. The repumping light of Rb was
obtained directly from a master ECDL because a small power was suﬃcient (∼10 mW).

2.4.2

Injection locked slave lasers

The role of slave lasers is to amplify the power of master lasers and to obtain a suﬃciently highpower for tapered ampliﬁers (10–30 mW). Since the frequency of the light is not inﬂuenced byslave lasers, they have a simple structure (Fig.2.14). The collimation tube (Thorlabs, LT230PB) including a laser diode (Eagleyard, EYP-RWE-790; the same as that of master lasers) is
placed in an aluminum mounting. The temperature of the mounting is stabilized within 0.1 K
via a peltier module under the mounting, which is controlled by a home made thermoelectric
controller (Fig.2.15).
The seed light for a slave laser was injected through the side port of the isolator placed justafter the slave laser. The typical power of the seed light was 0.5–1 mW for K laser (767 nm). In
contrast, a higher power (3–5 mW) was required for Rb laser (780 nm) since the gain proﬁle ofthe laser diode was centered at 760–770 nm. When the seed light was not suﬃcient, an ampliﬁed
spontaneous emission (ASE) increased. Further, in some cases, the original oscillation mode of
the LD itself appeared, which was due to the degraded AR coating on the LD (the LD emitting
more than 30 mW gradually lost their AR coating within a few months and became unstable inour environment).
There were two points always we had to take care. First, the current should be set to a
speciﬁc value which gave a stable oscillation. Fig.2.16 shows the variation of the output powerof a slave laser with respect to the LD current. At a low current below 50 mA, the powerincreases almost linearly. However, at a high current, we observe several points where the power
decreases with increasing the current. At these points, the power of the carrier light decreases

16Figure 2.11: Circuit diagram of the AOM driver. The intensity and frequency of the RF can becontrolled either manually or externally. An integrator for intensity stabilization is also includedin the circuit.

Double-pass(+220MHz)

Single-pass(-80MHz)

Single-pass(+220MHz)
Double-pass(+200MHz)
F=2

F=2

F=1

F=1

Cooling

Repumping

Figure 2.12: Frequency control for K MOT. The master laser for 41 K is oﬀset-locked to the
The frequencies of the cooling and repumping lights are controlled via +110 MHz
AOMs in a double-pass conﬁguration placed before slave laser. In order to control the intensityof the MOT lights, a single-pass -80 MHz AOM is placed after the tapered ampliﬁer for thecooling light, whereas a single-pass +220 MHz AOM is placed after the tapered ampliﬁer forthe repumping light.
39 K line.

17

Double-pass(+220MHz)

Single-pass(-110MHz)

Single-pass(+80MHz)

F=2

F=2

F=1

F=1

Cooling

Repumping

Figure 2.13: Frequency control for Rb MOT. The cooling light passed through a double-pass
AOM (+110 MHz) and a single-pass AOM (-110 MHz). The repumping light passes through asingle-pass +80 MHz AOM. The frequency of the cooling light is controlled via the double-pass
AOM. The single-pass AOMs are used for controlling the intensity for both the cooling lightand the repumping light.

Figure 2.14: Image of a slave laser. The laser diode is mounted in a collimation tube. The
temperature of the tube is stabilized through aluminum mounting with a peltier module underthe mounting.

18

Figure 2.15: Circuit diagram of the thermoelectric controller for slave lasers. The output of acommercial controller (Omron, E5CN) is ampliﬁed with a mirror circuit.
40)Wm( rewoP

30

20

10

0
0

20

40

60

80

100

Current (mA)

Figure 2.16: Current characteristics of a slave laser for K. The output power of a slave laser isplotted with respect to current. There are several bad currents at which an ASE component in
the output power increases. For a stable oscillation, the current should be set to one of the local
maximum points.
whereas an ampliﬁed spontaneous emission (ASE) increases (2.17). The measured fraction of theASE component was approximately 10 % with respect to the carrier component. In addition, theoriginal oscillation mode of the LD itself appeared at these bad currents when the AR coating
was damaged. A stable oscillation was ensured only at a current where the power increased withincreasing the current, which was just before the bad current.
The second important point is the temperature stabilization. Since the bad current isstrongly dependent on the temperature, the temperature should be suﬃciently stable. Although
we stabilized the temperature of the aluminum mounting, a large variation in room temperaturesometimes caused a small change in the temperature of the mounting, resulting in an unstableoscillation. This was particularly became a problem when the AR coating of the LD was almost
completely damaged. For reproducible experiments, it is recommended to install a heat isolation
system such that the temperature of the LD is never inﬂuenced by room temperature.

19

0.15
ASE fraction
2.0)V( egatlov DP

0.10

1.5

1.0
0.05

noitcarf ESA

Carrier light

0.5

0.0

0.00
40

50

60

70

80

90

100

Current (mA)

Figure 2.17: Ampliﬁed spontaneous emission of a slave laser for Rb. The ASE component in
the output power of a slave laser is measured by monitoring the slave laser with a Fabry-Perotcavity (FSR, 1GHz; ﬁnesse, 400). The ASE fraction and the peak height of the carrier light areplotted with respect to current.

2.4.3

Tapered ampliﬁers

A tapered ampliﬁer enables us to amplify the slave laser (10–30mW) to 600–800 mW which is
required for capturing a large number of atoms. We bought only a tapered ampliﬁer chip in
the C-mount type (m2k laser, TA-0765-1000 for K and TA-0785-1000 for Rb) and mounted it
on a home made mounting made of cupper. Fig. 2.18 shows our design of the mounting for atapered ampliﬁer. The temperature of the mounting was stabilized to ∼ 20◦ C with a peltier
module and a home made controller similar to that for slave lasers (Fig.2.19). The mounting
was ﬁnally included in a box with AR coated windows for avoiding dusts from damaging thetapered ampliﬁer.
In order to ensure a good thermal contact and an eﬃcient heat release, the peltier module
was attached to the mounting and the pedestal with graphite sheets. Thermal grease was not
used because it damages the facets of tapered ampliﬁers. Spacers made of ABS resin, which has
a low thermal expansion coeﬃcient, were inserted between the mounting and the pedestal forkeeping the distance between them constant. Without these spacers, the height of the mounting
varied due to a change in room temperature, causing the variation in the output power after
passing through an optical ﬁber of more than 10%. With spacers, the power was stable within
5%.
The alignment of the collimation lenses were performed with two mirror mounts as shownin Fig.2.20. With three tuning knobs, we could align the lens tube in three directions. The
input lens was adjusted to obtain a maximum output power with a good spacial mode close
to the TEM00 mode at a far ﬁeld. The output beam was adjusted such that the output beam
was collinear in the vertical direction. The remaining divergence in the horizontal direction
was compensated with the cylindrical lens just after the output collimations lens. Although
this method could not control the position and the inclination of the tube independently, we
could obtain a good spacial mode without having another stage for independent controls at
wavelengths below 800 nm.
Fig.2.21 shows the dependence of the output power of a tapered ampliﬁer for K on the powerof the seed light and current. Although a high power was obtained just by raising the current,
we usually set the current to 1.5–1.6A for enlarging the lifetime and increased the power of theinjecting light when a high power was required. Another important issue for ensuring a long
lifetime was to connect protection diodes near the tapered ampliﬁer chip (Fig.2.22), which was
similar to that given in Ref. [86]. In this way, we could operate a tapered ampliﬁer for more
20Collimation lens holder
OutputOutput collimation lens(Thorlabs C390TM-B)

Screw
Heatsink(Copper)

Top view
Tapered amplifier chip

Input collimation lens(Thorlabs C230TM-B)

Input

Peltier module

Side view
Pedestal
Optical table(Breadboard)

Heat

Figure 2.18: Design and image of the mounting for a tapered ampliﬁer. The tapered ampliﬁer
is attached to the cupper mounting. The temperature of the mounting is kept constant witha peltier module below the mounting. Two collimation tubes for focusing the seed light andcollimating the output beam are placed on the same mounting. The heat of the tapered ampliﬁer
is released to an optical table through the cupper pedestal below the peltier module.

21

Figure 2.19: Circuit diagram of a thermoelectric controller for tapered ampliﬁers. The output
of a commercial controller (Omron, E5CN) is ampliﬁed with a mirror circuit. A transistor isadded to the TEC for slave lasers (Fig.2.15) for dealing with a high current required for coolingtapered ampliﬁers.

Figure 2.20: Alignment of the collimations lenses. Two mirror mounts enables aligning thecollimations lenses in three directions.

22

500

400
)Wm( rewoP

300

200

100

(a)
0
0.0

0.5

1.0

1.5

2.0

Current (A)
600

(b)

500
)Wm( rewoP

400
300

200

2.3A
2.0 A

100

1.8 A
0
0

2

4

6

8

10

12

14

Seed power (nW)

Figure 2.21: The output power of a tapered ampliﬁer. (a) Dependence on current.. (b) Dependence on the power of the seed light23

Figure 2.22: Circuit diagram of the protection circuit for laser diodes. A backward diode isconnected for protecting a laser diode from a reverse voltage whereas four forward diodes areconnected for protecting a laser diode from a high forward voltage.
than three years without a signiﬁcant decrease in power.
The output lights from tapered ampliﬁers were carried through a polarization maintaining
single mode (PMSM) ﬁber to an optical table mounting the vacuum chamber. In this way, we
avoided the bad spatial mode of tapered ampliﬁers from inﬂuencing the MOT. In addition, we
inserted a PMSM ﬁber between a slave laser and a tapered ampliﬁer such that we could placetapered ampliﬁers apart from an optical table for master and slave lasers. The tapered ampliﬁer
and optical elements for coupling the input beam to the tapered ampliﬁer and coupling theoutput beam to a ﬁber were mounted on a single breadboard, constituting an amplifying systemwith a ﬁber input and a ﬁber output. Fig.2.23 shows the schematic of the amplifying system.
An AOM serving as the fast switch of the MOT light was also included in the system. The
typical eﬃciencies at each stage after the tapered ampliﬁer were 90% at the isolator, 80% at theAOM, and 65% for coupling to a ﬁber. The typical power which could be used for experiments
was thus 250–300 mW.
Fig.2.24 shows the schematics of the optical setup for generating six beams for dual species
MOT. The polarization beam splitters (PBS) were inserted after the ﬁbers for avoiding the
ﬂuctuation in polarization from ﬂuctuating the balance of MOT lights. The cooling lights for
K and Rb were combined at special dichroic mirrors fabricated by Sigma Koki, which wastransmissive at 780 nm (=90%) and reﬂective at 767 nm (=85%). The repumping lights for K
and Rb were also combined at dichroic mirrors independently. The cooling and repumping lights
were then expanded, overlapped at a PBS, and split to six beams. In order to adjust the balancein each direction independently, the power was ﬁrst split into three beams, which were further
split to two beams and incident on the chamber. The power for the vertical direction was twice
the power for the horizontal direction. Although we did not use dual-wavelength waveplates but
those at 767 nm for splitting the lights including both 767 nm and 780 nm, we did not have aproblem in adjusting the intensity balance: we could obtain a suﬃciently dense and cold cloudsof K and Rb atoms at the same position (see the section 3.1). However, the condition of atoms
will be further improved by inserting dual-wavelength waveplates and adjusting the intensity
balance for K and Rb independently.
We found that the ASE in the MOT light originating mainly from the ASE of slave lasers
destroyed a part of photoassociated molecules, presumably due to excitation to the 4P + 5S
molecular states followed by predissociation. In order to suppress the inﬂuence of the ASE
on molecules, we inserted a transmissive grating (Edmund optics, 48589; diﬀraction eﬃciency=90%) before combining K and Rb lights at dichroic mirrors.

24

Cylindrical (vertical)
f=100mm

Cylindrical (horizontal)
f=30mmCylindrical (vertical)
f=50mmInput fiber
λ/2

λ/2
f=500mmTaperedamplifier

Isolator

f=200mm

Photodiode(Power stabilization)
Output fiber

AOM

Breadboard(300x450 mm)

Figure 2.23: Schematic of the amplifying system. The two cylindrical lenses after the input ﬁber
match the mode of the input beam to that of the tapered ampliﬁer. The waveplate after thetapered ampliﬁer is adjusted such that a maximum power is obtained. The cylindrical lens after
the tapered ampliﬁer compensates the horizontal beam divergence. The waveplate after theisolator is required for rotating the polarization by 45◦ , ensuring that the light is not ellipticallypolarized at mirrors.

2.5

Photoassociation laser

The photoassociative formation of ultracold molecule requires a high laser intensity on the orderof 103 W/cm2 . We had to prepare at least 100 mW since the laser should irradiate the wholeatomic clouds with a diameter of approximately 500 µm. In most of the previous experiments, thephotoassociation laser has been supplied from a Ti:Sapphire laser. However, we had only one Ti:
Sapphire laser, which was devoted to spectroscopy for ﬁnding transitions from photoassociatedmolecules Thus, we constructed a high power laser system based on semiconductor lasers. In
the section 2.5.1, we describe the locking scheme for realizing a tunable laser. In the section
2.5.2, we overview the amplifying system and important points for a proper operation.

2.5.1

Transfer cavity

Before carrying out experiments, we did not have any information on the precise frequency ofthe photoassociation resonance. Previous studies on another isotopes, 39 K85 Rb [65, 79], gave
a rough information that molecules could be formed at a detuning of ∼10 cm−1 from the Rb
D1 line (795 nm). Therefore we prepared a laser setup that could tune over 1 nm at 795 nm(Fig.2.25). An ECDL at 780 nm (hereafter we denote it as a cavity-locking laser) was oﬀsetlocked to the Rb cooling master laser. This laser was used for locking a home-made transfer
cavity (coating over 600–900 nm; ﬁnesse, 100), to which a distributed feedback (DFB) laser(Toptica, #LD-0795-0020-DFB-1) was locked. By changing the oﬀset frequency of the ECDL
locking the cavity, we could tune the cavity; thereby we could tune the DFB laser continuously.
The beat note between the Rb cooling master laser and the cavity-locking laser was detectedwith a high speed photo-detector (Hamamatsu, G10447-54; bandwidth, 8 GHz; transimpedance,
6 kΩ) and ampliﬁed by a limiting ampliﬁer (GigOptix, iT3011E; gain, 42dB; bandwidth, 1025

Rb coolingK repumping

K cooling

λ/2

Rb repumping

Bottom
Top

λ/2
λ/2

λ/2

Horizontal 1
λ/2

Horizontal 2
Horizontal 3
Horizontal 4
Figure 2.24: Schematics of the optical setup for MOT beams. For clarity, the K lights are givenin blue lines and the Rb lights are given in red lines. The lights for K and Rb are ﬁrst combined
at a dichroic mirror for cooling and repumping independently. The cooling and repumping lights
are then overlapped with a PBS, after which they are split to six beams.

26

Photo-detector
λ/4Current

Transfer cavity
PZT

DFB
λ/4
Photo-detector
λ/2

λ/2 λ/2

ECDL(780nm)Photo-detector

PZTfrom Rb cooling light
Figure 2.25: Schematics for the optical setup of a tunable photoassociation laser at 795 nm. An
ECDL at 780 nm was oﬀset-locked to the Rb cooling master laser. This laser locks a transfer
cavity to which a DFB laser is locked.
GHz). The ampliﬁed beat signal was compared with a direct digital synthesizer (DDS) signal
in a DDS chip with phase-locked loop (Analog Devices, AD9858). A limiting ampliﬁer wasnecessary because the prescaler for dividing the frequency by 8 required square-wave signals
at low frequencies. The charge pump output of AD9858 for controlling a voltage controlled
oscillator was integrated in a servo circuit given in Fig.2.9 and feed-backed to the piezo moduleof the cavity-locking laser. The oﬀset-locking was later replaced to optical phase-locked loop
(OPLL) with a double balanced mixer [90] because it allows much more stable locking (thedetails on OPLL will be given in the section 6.2).
The reﬂection of the cavity-locking laser from the cavity was integrated and feed-backed tothe piezo module attached to one of the mirrors of the transfer cavity, whereas the reﬂectionof the DFB laser was integrated and feed-backed to the current of the DFB laser. Interferenceﬁlters were inserted before photo-detectors for detecting the phase-modulated lights such thatboth lights did not inﬂuence each other.
The DFB laser enabled us to tune the wavelength over 1 nm without mode hopping. This
continuous tunability was particularly beneﬁcial for ﬁnding the resonance in the early stage ofexperiments. The frequency could be linearly tuned with both current and temperature. Fig.
2.26 shows the dependence of the wavelength of the DFB laser on temperature and current.
The sensitivity was 25 GHz/K in temperature and 9.7 GHz/mA in current. Since the oscillation
wavelength of a DFB laser was solely determined by temperature and current, both temperatureand current should be stable enough. We employed the combination of a commercial temperaturestabilization package for a laser diode (Toptial, ColdPack) and a commercial thermoelectric
controller (Thorlabs, TED200C). The current was controlled via a commercial current controller(Thorlabs, LDC201CU). Although we did not directly measure the linewidth of the DFB laser,
it appeared to have a broader linewidth than our ECDL when we monitored it with a cavity(Fig.2.27). This is presumably due to the current noise of the current controller. Since wefeed-backed the integrated signal to the current modulation port of the current controller, we
could not insert a low pass ﬁlter with a low cutoﬀ frequency after the current controller. As
shown in Fig. 2.25, we inserted a low pass ﬁlter with a cutoﬀ frequency of 200 kHz mainly for

27

(a)

795.6
)mn( shtgnelevaW

795.4
25 GHz/K

795.2
795.0
794.8
794.6

25

30

35

40

45

Temperature (°C)

795.9

(b)

)mn( htgnelevaW

795.8
9.7 GHz/mA
795.7

795.6

25

30

35

Current (mA)

Figure 2.26: Tuning the wavelength of the DFB laser. (a) Dependence on temperature. (b)Dependence on current. The frequency of the DFB laser can be linearly tuned with bothtemperature (25.1 GHz/K) and current (974 MHz/mA).
reducing high frequency noises. The linewidth was further broadened during an operation for 1
year. Hence, we replaced the DFB laser with an ECDL after we found the precise frequenciesof the photoassociation resonances.

2.5.2

Amplifying system

The amplifying system consisted of a slave laser and a tapered ampliﬁer. The setup itself wassimilar to that of MOT lasers. However, there were two problems which should be noted when
it was used for generating a photoassociation laser.
The ﬁrst problem arose from the current characteristics of the slave laser, which was described in the section 2.4.2. Fig.2.28 shows the current characteristics of the slave laser forphotoassociation. This laser was operated with a commercial AR-coated laser diode (Eagleyard,
EYP-RWE-0840). We observed a much larger variation in the power when the current waschanged. Further, we found that the output power was also strongly dependent on the wavelength of the DFB laser (Fig.2.29). These behaviors presumably indicated an incomplete AR
coating. We observed a periodic behavior with a spacing of 24 GHz. Thus, it was imperative totake care of the current of the slave laser when the frequency of the DFB laser was scanned fora wide range or changed by more than 20 GHz.
The second problem was that the ASE in the tapered ampliﬁer for amplifying the slave laser
could completely kill photoassociated molecules. It had a much larger inﬂuence on molecules

28Cavity transmission (arb.units)

1.0
0.8

DFB laser

0.6

ECDL

0.4
0.2
0.0
1.5

1.6
1.7
Frequency (GHz)

1.8

1.9

Figure 2.27: Cavity transmission signal of the DFB laser and an ECDL. The left peak is the
DFB laser whereas the right peak is the cavity-locking laser. Although the spectral width islimited by the low ﬁnesse of the cavity, the DFB laser appears to have a broader linewidth thanthe ECDL.
1210)Wm( rewoP

8
6
4
2
0
40

50

60

70

80

90

100

Current (mA)

Figure 2.28: Current characteristics of the slave laser for photoassociation. The oscillation
becomes unstable at several points where the power decreases with increasing the current. The
current should be set to one of the local maximum points.
than the ASE in the MOT light since the intensity was higher by four orders of magnitude.
The ASE in the photoassociation light originated from the tapered ampliﬁer itself rather than
the slave laser because the emission spectrum of the tapered ampliﬁer was centered at 775 nm,
which was far from the wavelength for photoassocition (795–796 nm). We inserted a grating forremoving the ASE. Thus, the typical power of the photoassociation light was 180 mW, which
was focused to a beam waist of 190 µm in the vertical direction and 330 µm in the horizontal
direction.
The photoassociation laser was ﬁrst roughly aligned to the center of the MOT by blowing
oﬀ Rb atoms with the photoassociation laser resonant to the Rb D1 line. A ﬁne alignment was
carried out by monitoring the ion counts of KRb.

29

60

50
)Wm( rewoP

40

30

20

7.8

8.0

8.2

8.4

8.6

8.8

9.0

Wavenumber of DFB laser - 12570 (cm

9.2

9.4

-1

)

Figure 2.29: Dependence on the frequency of the seed light. The output power of the slavelaser for photoassociation is measured with respect to the wavenumber of the seed light. A large
variation is observed when the seed light is scanned over a wide range.

2.6

Pulsed dye laser

A pulsed dye laser (Continuum, ND6000) had been used throughout the entire experiments forionizing molecules. Fig.2.30 shows the internal structure of the pulsed laser. It consisted ofan oscillator and two stages of ampliﬁers, which had diﬀerent concentrations. Pumped by adouble Nd:YAG laser at 532 nm (Continuum, Surelite III; repetition, 10 Hz; pulse width, 7 ns;
maximum power, 420 mJ), the dye laser could supply an intense pulse with a power of 10–30
mJ and a linewidth of 0.1 cm−1 . The ﬂash lamps in the Nd:YAG laser were exchanged once inseveral months when the pulse counts reached approximately 15 million counts from the ﬁrst
use of the lamps. The output beam had a diameter of 8 mm, which was further narrowed to 1
mm just before it was incident on the chamber. The output power was readily controlled via
controlling the pump power with a Q-switch delay (Fig.2.31). In most cases, we only requireda few 100 µJ for ionization. A too intense power led to the light-induced desorption of atoms
coating the vacuum chamber, yielding undesirable oﬀset ion signals. In contrast, too weak pump
power led to an unstable oscillation which caused a large power ﬂuctuation. We typically set
the Q switch delay to 350–380 µs and controlled the intensity via neutral density ﬁlters. For the
reduction of the oﬀset ion signals arising from the chamber, it was also important to avoid thelight reﬂected at the view ports from hitting the chamber by entering the pulsed laser exactly
perpendicular to the surface of the view port.
In contrast to a CW dye laser, the pulsed dye laser was so robust that we did not require anyalignment in the daily experiments. However, the output power signiﬁcantly decreased with anoperation of a few tens of hours when it was operated with the mixture of Rhodamine 640 and
Rhodamine 610 dyes, which can be used at a wavelength range of 595–620 nm; thus, we had
to exchange the dye once in three days. In the later experiments, we found that a mixture of
DCM and Kiton Red 620 dyes gave a much more stable power in a much wider wavelength range(595-665 nm). This combination could be used for more than one month without a signiﬁcantdecrease in the power. Further, the wide tunability was especially important for the STIRAP
experiments in which operations at three largely separated wavelengths were required (604 nm
for monitoring photoassociated molecules, 603 nm for SpIDR spectroscopy of excited states, and
649 nm for detecting the rovibrational ground-state molecules). Hereafter we brieﬂy describe
the way to exchange an old dye solution to a new dye solution for a mixture of DCM and Kiton
Red 620, because it was one of the important processes for operating the experiments.
First the old dye solution should be completely removed from the tanks of oscillator andampliﬁer. The tanks are cleaned by circulating pure methanol several times. Then we prepare a30

Figure 2.30: Image of the pulsed dye laser. A weak seed light produced in the oscillator is
ampliﬁed at the preampliﬁer and ampliﬁer stages.

12Power (mJ)

10
8
6
4
2
0
200

250
300
Q-switch delay (µs)

350

Figure 2.31: The output power of the pulsed dye laser operating with DCM and Kiton Red 620
at 649 nm. The power can be controlled with the Q-switch delay of the pump laser.

31

3.0
Power (mJ)

2.5
2.0
1.5
1.0
0.5
0.0
600

620
640
Wavelength (nm)660

Figure 2.32: Emission spectrum of the pulsed dye laser. With a mixture of DCM and Kiton
Red 620, the pulsed dye laser can cover a wide range of 595-665 nm.
new dye solution by solving 100 mg of DCM in 300 mL of methanol and 80 mg of Kiton Red 620
in 100 mL of methanol separately. Kiton Red 620 is readily solved just by stirring with a spoon
whereas DCM requires stirring in a ultrasonic bath for a few tens of minutes. DCM can be used
at 608–665 nm whereas Kiton Red 620 can be used at 590–620 nm. Since required oscillations
are mainly obtained with DCM, the concentration of DCM is ﬁrst determined. The new DCM
solution is added ﬁrst to the ampliﬁer tank while we monitor the output power. Without adding
dyes to the oscillator tank, the output power is only an ASE from the dyes. We typically adjust
the dye concentration of the ampliﬁer such that we obtain 0.6 mJ of ASE at a Q-switch delayof 300 µs, which is empirically determined. Then the DCM solution is added to the oscillator
tank until the power at 660 nm is saturated. This is because an oscillation at a long wavelength
requires a high concentration. Finally the Kiton Red 620 solution is added such that the power
at 600 nm is saturated. Fig.2.32 shows the emission spectrum of the dye laser adjusted in this
way.

2.7

Ti: Sapphire laser and dye laser

Spectroscopy for ﬁnding transitions from photoassociated molecules was carried out with acommercial Ti:Sapphire laser (Sirah, Matisse-TX). This laser was later converted to a dye laser(Sirah, Matisse-DX) for generating a STIRAP light by changing the internal conﬁguration. Here
we brieﬂy describe the operation of the Ti:Sapphire laser and the dye laser.
The Ti:Sapphire laser was pumped by a double Nd:YAG laser (Spectra Physics, Millenia;
maximum power, 10 W). We mainly operated this laser in the range 850-920 nm where we could
obtain a power of up to 1 W. The short-term linewidth was as narrow as 20 kHz, which was
ensured by locking the laser to a reference cavity placed after the laser. Although the frequencyof this laser was ﬂuctuated over a several MHz within a second, it did not matter because
spectroscopy was carried out in a saturated region where the spectral width was more than 10
MHz. The large mode-hop-free scanning range of this laser (∼50 GHz) was quite beneﬁcial forﬁnding a lot of unknown transitions. Further, this laser was so stable that we almost did not
require daily alignments; once in a few weeks was suﬃcient.
The pump laser for the dye laser was identical to that for the Ti:Sapphire laser. We operated
the dye laser with DCM, which could supply a power of 600 mW in the range 625–660 nm. The
dye laser was not as stable as the Ti:Sapphire laser: we required alignments for recovering
the power once in a few days. Further, a slightly bad alignment yielded undesirable frequency
modes, which made locking unstable. As a result of alignments for obtaining a stable oscillation

32

Single channel(Diameter 12µm)

Chamber surface

Anode
Positive ion
Preamplifier

e-

-2kV

-0.1kV

GND

Figure 2.33: Schematic diagram for operating an MCP. A high voltage applied to the MCP has
roles of accelerating ions to the MCP as well as amplifying the signal. The small voltage isapplied to the anode such that multiplied electrons are accelerated to the anode.
in a single mode, the direction of the output beam was changed and accordingly we had torealign the coupling of the laser to a reference cavity as well as to the subsequent optical pathfor experiments. The dye exchange was required only once in several months.
For the use of the dye laser as a STIRAP light, we stabilized its frequency to anotherstable laser with a double-pass AOM operated at 80 MHz placed just after the reference cavity.
Simultaneously a slow integrated signal was feed-backed to the dye laser itself through a LabView
control box such that the AOM stayed at approximately 80 MHz. In this way, we could scan
the laser for more than several GHz without suﬀering from the small scanning range of AOM.
Although the original linewidth of the dye laser was as narrow as 100 kHz, the limited feedback
bandwidth of the AOM (∼300 kHz) yielded larger regulation sidebands than for semiconductor
lasers (details will be given in section 5.3).

2.8

Ion detection and time-of-ﬂight mass spectrometry

We had both an MCP (Hamamatsu, F1551-01; gain, 7×106 at 2 kV) and a Channeltron (Burle,
4869; gain, 1×108 at 3 kV) for detecting atomic and molecular ions. As already stated in thesection 2.2, we decided to use the MCP because the Channeltron was broken in a one-year
operation. The MCP had a lifetime of more than three years in our environment. Fig. 2.33
shows the schematic diagram for detecting positive ions with the MCP. The ions collided the
MCP are converted to secondary electrons, which are further multiplied by more than 106 times
in the MCP. Although the conversion eﬃciency of ions to electrons is not known for our MCP,
it should be a few tens of %, which is the detection eﬃciency for Channeltron. The multiplied
signal is further ampliﬁed by 200 times with a home-made preampliﬁer (Fig.2.34). This is aslightly modiﬁed version of the ampliﬁer described in Ref. [91]. Although the preampliﬁer hasa bandwidth of 60 MHz, an additional low pass ﬁlter (cutoﬀ frequency, 2 MHz) was inserted
before ampliﬁcation for suppressing high frequency noises in the MCP signal.
Ions are produced at the MOT which is separated from the MCP by 20 cm. The time of
ﬂight (TOF) is dependent on the mass and the charge of ions. Hence, we can observe atoms
and molecules separately at a diﬀerent TOF (Fig. 2.35). The observed TOF was between 20 µs
and 40 µs, which were much larger than we expected. Assuming a homogeneous electric ﬁeld
between the MOT and the MCP, we should obtain an electric ﬁeld of ∼80 V/cm (see Fig.2.2
in the section 2.2). This should result in a TOF of ∼8 µs for 41 K87 Rb molecular ions. The
unexpectedly large TOF indicates an inhomogeneous electric ﬁeld between the MOT and the33Figure 2.34: Circuit diagram of the preampliﬁer for the MCP signal. The signal is ampliﬁed with
two stage non-inverting ampliﬁers. Low pass ﬁlters are inserted for suppressing high frequency
noises.

K

0.2

+

Rb

+

+

KRb

)stinu.bra( langiS

0.1

0.0

-0.1

0

10

20

30

Time of flight (

40

ms)

50

60

Figure 2.35: Time of ﬂight spectrum. 41 K+ , 87 Rb+ , and 41 K87 Rb+ are simultaneously observed.
The atomic ions are not produced by the direct ionization of atoms in a MOT, but by thephotodissociation of KRb molecules with a pulsed laser (see section 3.3.3 for further discussions).
The leftmost large signal arises from the pulsed laser hitting the MCP.
MCP, and an electric ﬁeld much smaller than 80 V/cm at the MOT position. These behaviors
could be understood as an inﬂuence of the closely lying upper and lower surfaces of the chamber.
Although the TOF time was longer than that of previous experiments in other groups (a fewµs), we did not have any problem about it. The long TOF time allowed us to use a low passﬁlter for suppressing high frequency noises in the signal and achieve a high signal-to-noise ratio.

2.9

Computer system

The sequences of experiments were mainly controlled by a personal computer (PC). In addition,
we had another three computers for acquiring data, monitoring laser frequencies with wavemeters, and controlling some of the lasers (Table 2.1). Here we describe the entire computer system
for the experiments.

2.9.1

Timing control with a LabView program

In order to obtain suﬃciently high density and low temperature clouds of K and Rb simultaneously, we required both the analog and the digital control of lasers and magnetic ﬁelds. The
details on this topic will be given in the section 3.1. We utilized a slightly modiﬁed version of
the LabView program developed by Prof. T. Mukaiyama for controlling the entire system witha repetition rate of 9 Hz (Fig.2.36). The computer for operating this program was devoted only34

Figure 2.36: LabView program for controlling the experiments. This program enabled us tocontrol the experiments in both analog and digital manners.
to this purpose because it required a high capacity of processing. Fig.2.37 shows the schematic
diagram for the timing control of the experiments. In the early stage of the experiments, pulses
for driving the apparatuses were obtained from the pump laser for the pulsed dye laser. Later
we found that the pulses from the pulsed laser had a jitter of several µs, which made it diﬃcult
to carry out STIRAP experiments. Thus, a stable pulse generator (Tektronix, AFG3022) was
employed for driving the experiments in the later experiments.

2.9.2

Data acquisition with a LabView program

The TOF signal from the MCP ampliﬁed at a preampliﬁer was acquired with a high-speed
digitizer (National Instruments, PCI-5112) installed in a computer. Triggered by the pulse generator driving the pulsed laser, this digitizer could acquire two channels of data. An additional
channel was important throughout the entire experiments: in the spectroscopy experiments,
we simultaneously acquired a cavity transmission signal used for calibrating the frequency of

Table 2.1: Computers for operating the experiments. 4 computers are used for controlling andmonitoring the entire experiments.
PC name
PurposeInstalled devices&software
Molecule1
Digital control
National Instruments, PCI-6534
Analog control
National Instruments, PCI-6733
Molecule2
Imaging of atoms
Princeton Instruments, PIXISTOF acquisition
National Instruments, PCI-5112
Molecule3
Pulsed laser control
Continuum, ND6000
Ti: Sapphire (Dye ) laser control
Sirah, Matisse-TX(DX)
Wavelength (CW)HighFinesse, WS-7
Molecule4
Wavelength (Pulse)
HighFinesse, WS-6
Molecule5
Data analysis
Mathworks, MATLAB

35

Pulse generator (9Hz)
delay
delay

LabView(Timing control)delay

MOT lasers
Photoassociation laser
magnetic fields

Flash
lamp

LabView(Data acquisition)

Q-switch

Pulse laser

Figure 2.37: Schematic diagram for controlling the time sequences. MOT lasers and magnetic
ﬁelds are controlled via a LabView program triggered by a pulse generator which drives thepulsed laser. The Q-switch delay of the pulsed laser is manually controlled.
the scanned laser, whereas in the STIRAP experiments we acquired a transmission signal of
the ultralow expansion (ULE) cavity for ensuring that STIRAP lasers were locked during theexperiments.
In our experiments, the TOF signal obtained at each cycle corresponds to one data point
in a spectrum. Since we had to deal with a large amount of data points (∼10000 for a single
spectrum), we did not save all the TOF signal but saved the signals of atomic and molecularions integrated over a short time (2–4 µs) with a LabView code (Fig.2.38). Here the ion counts
are given in the unit of 10−8 V s and 1 ion count indicates approximately 5 ions arriving at
the MCP. In order to avoid a large ground noise due to the pulsed laser from adding oﬀset
noises in the acquired data, we subtracted the signal at a long TOF from each ion signals before
integration. The data processing (subtraction and integration) was carried out in the programat each cycle. The computer for data acquisition was also devoted only to this purpose because
it required a high capability of processing.
The signal-to-noise ratio of the acquired signal was mainly limited by four factors. First,
there were random junk pulses in the TOF spectrum. We have not speciﬁed the origin of
these pulses, but they appeared to originate from light hitting the MCP as well as atoms andmolecules in the chamber. Second, oﬀ-resonant two photon ionization at a high pulsed laser
power generated oﬀset ion counts. This was dependent on the wavelength of the pulsed laser;
it was negligible at some wavelengths. Third, the gain of the MCP ﬂuctuated from shot toshot by a few tens of %, which was a typical behavior of electron multipliers. Fourth, thepower of the pulsed laser also ﬂuctuated from shot to shot by approximately 10 %., which wasdependent on the pump power of the pulsed laser. The former two factors constituted the noise
ﬂoor of the obtained spectra, determining the signal-to-noise ratio of the upward spectra whereions increased at speciﬁc frequencies. This was the case for ionization spectroscopy, SpIDR
spectroscopy, and two-photon STIRAP spectroscopy. In contrast, the latter two factors were
the origin of the ﬂuctuation in the oﬀset ion counts, determining the signal-to-noise ratio of thedownward spectra where resonances were detected as a decrease of the oﬀset ion counts. This
was the case for depletion spectroscopy and two-photon dark resonance spectroscopy.

2.9.3

Laser control and wavemeters

The pulsed dye laser and the Ti:Sapphire/Dye laser were controlled via a laptop computer withaccompanying softwares. In the same laptop computer, a commercial wavemeter (HighFinesse,
WS-7; accuracy, 30 MHz) with four channels input was installed for monitoring the STIRAP

36

Figure 2.38: LabView program for data acquisition. The TOF signal is exhibited in the panel.
The ion counts at K, Rb, KRb, and Rb2 peaks are integrated and saved to a ﬁle. 1 ion count inthis program indicates approximately 5 ions arriving at the MCP.

37

600mm200mm

solenoid shutter
Probe light

CCD

f=150mmFigure 2.39: Imaging system for K and Rb. The density and temperature of atoms are measured
via absorption imaging. The absorption images of atoms are recorded with a CCD camera.
lasers and the photoassociation laser. Further, the wavelength of the pulsed laser was monitored
on another laptop computer via another wavemeter (HighFinesse, WS-6; accuracy, 300 MHz).
With the computer controlling and monitoring system, we could carry out experiments sittingin front of the computers. Monitoring wavelengths with these wavemeters on the computers
had another advantage: these wavemeter were equipped with a long-term monitoring program,enabling us to check that laser frequencies had been stable during the experiments.
The accuracy of the WS-7 wavemeter was excellent because it could be calibrated with aaccompanying reference He-Ne laser. Calibrations were carried out once in a month, which was
suﬃcient for the experiments. The day-to-day reproducibility was within a few tens of MHz. In
contrast, the WS-6 wavemeter was deviated by approximately + 650 MHz, which was conﬁrmed
by monitoring an atomic resonance. However, this deviation was not a problem in our casebecause the WS-6 was used for monitoring the pulsed laser with a linewidth of 0.1 cm−1 .

2.10

Imaging system

In the early stage of the experiments, we required an imaging system for measuring the densities
and the temperatures of both K and Rb atoms such that we could obtain dense and cold atoms
suﬃcient for the photoassociation of KRb at exactly the same position. Fig.2.39 shows the
setup for the absorption imaging of K and Rb. A fast solenoid shutter was inserted before the
CCD camera (Princeton Instruments, PIXIS) for avoiding the ﬂuorescence from the MOT fromhitting the CCD camera during the long exposure time. This shutter was opened after the MOT
light was turned oﬀ.
In each measurement, the spatial distribution of optical density (OD) was obtained by taking
three images: one with atoms and a probe light, one without atoms and with a probe light, andone without atoms and a probe light. Assuming a Gaussian distribution, we derived the peakdensity of atoms. The temperature of atoms T was determined by taking several images with
diﬀerent TOF t, based on the following relation.
σ 2 (t) = σ02 +kB T 2tm(2.1)

where m was the mass of an atom. When the density exceeded ∼5×1010 cm−3 , we detuned
the frequency of the probe light such that the OD was lower than 3. This was important for
obtaining a correct image because a too high OD indicated an extremely low transmission at
the center of the clouds, which could not be measured correctly.

38

Cavity transmission

6

4

2
3
0
2

1

)stinu.bra( stnuoc noI)stinu.bra( langis ytivaC

Ion counts (raw)

0
0

5

10

15

20

´

Point-numbers ( 1000)

Figure 2.40: Depletion spectrum for KRb molecules and cavity transmission signals. Cavity
transmission signals of the Ti: Sapphire laser are simultaneously acquired during the scan of the
Ti: Sapphire laser and are used for calibrating the nonlinearity of the scan.

2.11

Calibration of laser frequencies with a cavity

The transfer cavity for locking the photoassociation laser was also used for monitoring andcalibrating the frequency of the Ti: Sapphire laser in the spectroscopy experiments. Although
the Ti: Sapphire laser was always locked to a reference cavity, a scan over a wide range (∼1
cm−1 , or 30 GHz) took a long time (∼1 hour), during which the Ti: Sapphire laser could drift.
Thus, we could not ensure that the frequency was linearly swept during the scan. In order tocompensate this nonlinearity, we acquired the cavity transmission signal of the Ti: Sapphirelaser when we acquired the TOF signal of atoms and molecules. Fig. 2.40 shows a typicaldepletion spectrum for KRb molecules with cavity transmission signals.
Since data points were ensured to have an equal spacing in terms of time, the number ofpoints counted from the ﬁrst point (hereafter we note this number as point-number) was ﬁrst
assigned to each data point including both spectra and cavity signals. The point-number served
as one ruler. Then the point-numbers at the cavity transmissions were obtained by ﬁtting thecavity signals with a Lorentzian function. Here we had two rulers: point-numbers and thenumber of transmission peaks, which we call cavity-numbers. The cavity-number was ensured
to have an equal spacing in terms of the frequency of the Ti: Sapphire laser. Assuming that thescan was smooth, we could assign a cavity-number to each data point via a spline interpolation.
Fig.2.41 shows an example of the relation between point-numbers and cavity-numbers. Finallyeach data point could have a frequency measured from the ﬁrst cavity transmission signal bymultiplying the free spectral range (FSR) of the cavity.
For achieving accurate spectra, it was imperative to use a precisely measured value of the
FSR of the cavity. We measured it in the following manner. First we locked an ECDL to one of
the transmission peaks of the cavity. Next we locked the Ti: Sapphire laser to the neighboringtransmission peak and measured the beat note between the ECDL and the Ti: Sapphire laser.
While we locked the ECDL, the same procedure was repeated at the other transmission peakseparated from the ECDL by 11. Thus, the FSR could be obtained by dividing the frequency
diﬀerence of the two measurements by 10. The ﬁrst measurement of the beat note did not givean accurate measurement because it was dependent on the alignment of lasers to the cavity:
the cavity we used was a confocal cavity, whose degenerate spatial modes resulted in the low
sensitivity to the alignment.
However, there was still a problem not circumvented in this method: the drift of the oﬀset
of the error signal due to the residual amplitude modulation in the electro-optic modulator(EOM) used for PDH locking [92]. We measured the FSR with the above method for an hourand obtained an averaged value of 997.372(50) MHz. The drift of the cavity was within several

39

´)0001 ( srebmun-tnioP

20

15

10

5

0
0

10

20

30

40

50

60

Cavity-numbers

Figure 2.41: Relation between point-numbers and cavity-numbers. Interpolation with a spline
function is a good approximation because these two numbers are almost linear with each other.
The point-numbers indicate the number of data points from the beginning and are equal in termsof time, whereas the cavity-numbers indicate the number of transmission peaks and are equal interms of the frequency of the Ti: Sapphire laser. Although the cavity-numbers are much fewer
than the point-numbers, each data point can be assigned a non-integer cavity-number.
transmission peaks throughout the experiments, indicating a change in the FSR of a few tensof kHz, which was smaller than the error in the FSR measurements. The estimated error due
to the error in the FSR measurements was 1.5 MHz for spectroscopy over 30 GHz, in which theerror was multiplied by 30. Another source of the error could be the error in the ﬁtting to cavity
transmission signals. We estimated this error as below 200 kHz. These errors were suﬃciently
small for the current study, since the typical width of the spectrum was on the order of 100
MHz.
For measuring the FSR of the cavity, we also tried to use the Cs D2 line as a frequency
reference. In this case, the cavity transmission signal was recorded with the saturated absorption
signal obtained by scanning the Ti: Sapphire laser. For calibrating the cavity signals, we used
the largest signals, F = 4 → F ′ = 4 + F ′ = 5 (crossover) and F = 3 → F ′ = 2, which wereseparated by 8714.574 MHz [93]. Fitting to both signals (Lorentzian for cavity signals and tilted
Lorentzian for saturated absorption signals) enabled us to determine the FSR of the cavity.
However, the FSR obtained with this method had a large error of 0.5 MHz. This was because
the large width (20–30 MHz) of the saturated absorption signal as well as the Doppler broadened
absorption serving as a tilted oﬀset made the ﬁtting incorrect.
The relative accuracy of the measurements in a single scan was thus limited to approximately
3 MHz in most of the experiments. In contrast, the reproducibility of the obtained spectra
between scans was determined by the wavemeter (WS-7): the accuracy of the wavemeter (±30
MHz) determined the absolute accuracy of the measured spectra.

40

Chapter 3

Production and detection ofphotoassociated molecules
Nobody in our group had an experience on producing molecules via photoassociation and detecting them. We had information on experiments only through the published papers of previous
studies by other groups. In addition, we had to deal with molecules including 41 K, for whichlaser cooling is ineﬃcient.
We ﬁrst focused on preparing dense samples of 41 K and 87 Rb atoms because the high density
is crucial for the photoassociative formation of molecules. Next we had worked on ﬁndingtransitions for photoassociation and ionization. In the early stage of the experiments, we did notknow the photoassociation resonances as well as the ionization wavenumbers. By scanning bothlasers randomly, we coincidently found both resonances which could produce and detect KRb
molecules. After we found the resonances, it was easy to characterize each process and optimize
the conditions such that we could achieve a large number of molecules. We also studied the
produced molecules themselves: it was important to assign the vibrational number of produced
molecules for further investigations.

3.1

Preparation of dense and cold sample of 41 K and 87 Rb atoms

3.1.1

Overview

The photoassociative formation of ultracold molecules in a MOT has often been carried out witha dark spontaneous optical trap (dark-SPOT) technique [94] for increasing the density of atoms
and enhancing the photoassociation rate. With this technique, a dark spot is placed in the centerof the repumping beam, thereby making a high density (=1011 cm−3 ) sample of atoms in thelower hyperﬁne level which does not scatter the MOT light. However, it is not straightforward
to obtain a dense and cold sample of 41 K atoms in this type of MOT. Due to the high rate for
optical pumping of the upper hyperﬁne level (F ′ = 2) to the lower hyperﬁne level (F ′ = 1), we
need almost the same intensity of repumping light as cooling light for obtaining a large numberof atoms. As a result, placing a spot in the repumping light causes a leakage of atoms from the
MOT. Another problem arises from the scattering of the high intensity repumping light, which
results in a high temperature of atoms (∼3 mK).
In order to achieve a high density and a low temperature for both 41 K and 87 Rb, we employed
a compressed MOT (C-MOT) technique [95], in which the MOT condition were modiﬁed suchthat we could obtain a dense and cold sample of atoms. We could keep this condition only
for a short time (a few tens of ms) because atoms did not feel a strong capturing force in thiscondition and slowly expanded. However, the short time was suﬃcient for producing molecules
via photoassociation.
The basic strategy for adjusting the conditions was as follows. We ﬁrst characterized thesteady-state MOT of Rb and K such that we could obtain a large number of atoms with as high41

a density as possible. This characterization was made in the early stage of the experiments,
where we had approximately 3 times larger numbers of atoms for both Rb and K than later
experiments. The alignment of the MOT and the compensation of the residual magnetic ﬁeld
were also carried out at this point. These adjustments were important because a C-MOT
required both good beam balances and low stray magnetic ﬁelds in each direction.
After the conditions of the steady-state MOT were determined, we investigated the C-MOT
in terms of the frequency and intensity of light and the magnetic ﬁeld gradient. Although thecondition of the MOT lights for K and Rb could be tuned independently, the magnetic ﬁeld
gradient was shared between these species. In our experience, K atoms were more diﬃcult tocool and compress than Rb atoms. Hence, we ﬁrst optimized the condition for K, after which
we adjusted the frequency and intensity of the Rb light.

3.1.2

Steady-state MOT

We ﬁrst describe our method for aligning the MOT lasers and compensating stray magnetic
ﬁelds. In general, K atoms were sensitive to beam balances whereas Rb atoms were sensitive
to magnetic ﬁelds. Thus, beam balances were precisely adjusted by monitoring the K MOT
whereas magnetic ﬁelds were precisely adjusted by monitoring the Rb MOT. Note that we did
not have independent tuning knobs for K and Rb because waveplates were shared.
The MOT beams were ﬁrst aligned such that they passed through the center of the viewports. The beam balances were then roughly adjusted by measuring the center intensity of
the MOT light with a power meter such that the intensities of two counter-propagating beams
coincided within 3 %. Here it was quite important to obtain two counter-propagating beams by
dividing a single beam because then we could tune the balance with a single waveplate. Next
we adjusted the current to the compensating coils such that Rb MOT survived even at a low
magnetic ﬁeld. When the stray magnetic ﬁelds were well cancelled, we could observe the Rb
MOT at a magnetic ﬁeld gradient of 0.5 G/cm. The ﬁnal step was to tune the beam balance
precisely such that the K MOT survived at a low magnetic ﬁeld. With good beam balances,
we could observe the K MOT at a magnetic ﬁeld gradient of 1 G/cm. In some cases, theseprocedures should be repeated until both beam balances and magnetic ﬁelds were well adjusted.
Although these adjustments gave both good beam balances and low stray magnetic ﬁelds, wefound that too good alignment resulted in a large ﬂuctuation of number of molecules. Reﬂected
at a view port, a beam could interfere with its counterpropagating beam. This interference
pattern was quite sensitive to the vibration of mirrors and chambers, inducing a large intensity
ﬂuctuation at the MOT position. The K MOT was especially sensitive because the MOT light
was close to the resonance in the C-MOT process. In contrast, the Rb MOT was not sensitive
because the MOT light was largely detuned from the resonance in the C-MOT. We could avoid
this instability by slightly misaligning one of the beams in each direction (Fig.3.1).
Fig.3.2 shows the energy level diagrams of both the ground and the excited states of K and
Rb. Although these species have the same nuclear spin of 3/2, the smaller nuclear dipole momentof 41 K results in a much smaller hyperﬁne splitting for 41 K than for 87 Rb. The diﬀerence inthe hyperﬁne structure signiﬁcantly inﬂuences the eﬃciency of laser cooling: a high density andlow temperature sample of 87 Rb atoms are readily obtained, whereas for 41 K the high opticalpumping rate from F = 2 to F = 1 requires a high intensity of the repumping light.
We studied the number and the density of the MOT for both Rb and K with absorption
imaging in terms of frequencies of the MOT light. The frequency of the repumping light for Rb
was kept on the resonance F = 1 → F ′ = 2 because we did not observe a large dependence.
Further, the intensities of the MOT lights were kept at the maximum available values: 50
mW/cm2 for Rb cooling, and 4 mW/cm2 for Rb repumping, 30 mW/cm2 for K cooling, and30 mW/cm2 for K repumping in total intensities. Under these values, the number was almost
linearly decreased whereas the densities were not signiﬁcantly changed for both K and Rb. The
magnetic ﬁeld gradient was determined to be 9 G/cm, which was between the best value for K

42

Ion counts (arb.units)

12

Before alignment
After alignment

10
8
6
4
2
0
0

50

100
Data points

150

200

Figure 3.1: Number ﬂuctuation caused by interferences. Ion counts of KRb molecules is plotted.
(a) Before misalignment. (b) After misalignment. One point corresponds to 0.1 s. In (a), theﬂuctuation in number is dominated by the instability of the C-MOT of K, whereas in (b) the
ﬂuctuation in number is dominated by the variation in the gain of MCP from shot to shot aswell as the intensity ﬂuctuation of the pulse laser.
(4.5 G/cm) and that for Rb (13.5 G/cm) in terms of number of atoms.
Fig.3.3 shows the number and the density of the steady-state Rb MOT with respect to thedetuning of the cooling light. We decided to set the detuning as -20 MHz where we could obtainthe largest number because the density of Rb was readily increased in the C-MOT stage. We
obtained 1 × 109 atoms with a density of 5 × 1010 cm−3 .
Fig.3.4 shows the number and the density of the steady-state K MOT with respect to thedetuning of the cooling light. With diﬀerent detuning of the repumping light, we observed
diﬀerent behaviors. In the Raman condition, in which the frequency diﬀerence between thecooling light and the repumping light was identical to the splitting between F = 1 and F = 2,
a large number was obtained but the density signiﬁcantly decreased and the temperature wasquite high (more than ∼10 mK) at this condition. Therefore, we avoided this condition. We
also studied the dependence on the repumping frequency (Fig.3.5). If we avoided the Raman
condition, we could obtain a good condition which could give a large number and a high density
at a detuning of 20–30 MHz. On the basis of these measurements, we decided the frequencies of
the cooling and the repumping light as -28 MHz and -20 MHz, respectively. At this condition,
we could obtain 5×108 atoms with a density of 2 × 1010 cm−3 .
Although these characterizations were made independently for K and Rb, we observed a large
loss of atoms due to the coexistence of K and Rb atoms. Typically the K MOT was decreased
by 20 % due to Rb atoms whereas the Rb MOT was decreased by 50 % due to K atoms. These
values were dependent on the condition of the MOT.

3.1.3

Compressed MOT

The C-MOT processes consisted of two stages: compression and cooling, which were required
for achieving a good condition for K. In the compression stage, the density of K was increased
by increasing the magnetic ﬁeld gradient whereas that of Rb was increased by lowering therepumping intensity. Since atoms were heated during this process, we set the cooling stage, inwhich atoms were cooled by decreasing the total intensity as well as by increasing the detuningof Rb cooling and decreasing the detuning of K cooling. Fig. 3.6 shows the parameters for thedual-species C-MOT of 41 K and 87 Rb. With these procedures, K atoms were compressed by 40
times (5×109 → 2 × 1011 cm−3 ) and cooled by 5 times (2 → 0.4 mK), whereas Rb atoms werecompressed by 40 times (1×1010 → 4 × 1011 ) and cooled by 4 times (400 → 100µK). Fig.3.7
43

13MHzCooling266MHzCooling

Repumping

F=2

Repumping

F=2
6.83GHz

254MHz
F=1

F=1(a) 41K(Γ=6.1MHz)(b) 87Rb(Γ=6.1MHz)

Figure 3.2: Energy level diagram and laser cooling transitions. (a) 41 K. (b) 87 Rb. Both species
have a nuclear spin of 3/2, resulting in the identical structure. The smaller hyperﬁne splittingof 41 K makes the laser cooling of 41 K less ineﬃcient than 87 Rb; the repumping light with almost
the same intensity as the cooling light is required to pump back atoms in F = 1 to F = 2 dueto the absence of a closed transition (F = 2 → F ′ = 3 for 87 Rb).

8

´
7
1.0

0.5
4

01
3-

5

mc

´

6

)

9

) 01 ( smota fo rebmuN

1.5

01 ( ytisneD

NumberDensity

3

0.0
-30

-25

-20

-15

-10

Detuning (MHz)Figure 3.3: Characterization of the Rb MOT. The number and the density of the Rb MOT areplotted with respect to the detuning of the cooling light.

44

0.8

1.0

0.5

0.2

01

´

0.4

01 ( ytisneD

´

1.5

0.6

)

9

) 01 ( smota fo rebmuN

NumberDensity(a) Repump detuning = -10 MHz
0.0

0.0
-40

-30

-20

-10

Cooling detuning (MHz)
3.0

1.5

´
1.0

1.0

0.5

01

´

1.5

)

9

) 01 ( rebmuN

2.0

01 ( yrtisneD

2.5

0.5
0.0

(b) Repump detuning = -20 MHz

-40

-35

-30

0.0
-25

-20

Cooling detuning (MHz)Figure 3.4: Characterization of the K MOT: dependence on the detuning of the cooling light.
(a) At the repumping detuning of -10 MHz. (b) At the repumping detuning of -20 MHz. The
resonance is taken as the F = 2 → F ′ = 3 transition. The number and the density of the K
MOT are plotted with respect to the detuning of the cooling light for diﬀerent detuning of therepumping light.

45

2.0

´

1.5

0.6

0.5

0.2

01

´

1.0
0.4

)

9

) 01 ( smota fo rebmuN

0.8

01 ( ytisneD

1.0(a) Cooling detuning = -28 MHz
0.0

0.0
-20

-15

-10

-5

0

Repump detuning (MHz)
2.5

Number

2.0

1.6

1.5

1.0
0.8

01

´

1.2

01 ( ytisneD

´

2.0

)

9

) 01 ( smota fo rebmuN

Density0.5(b) Cooling detuning = -36 MHz

-25

-20

-15

-10

0.4
-5

0

Repump detuning (MHz)Figure 3.5: Characterization of the K MOT: dependence on the detuning of the repumping light.
(a) At the cooling detuning of -28 MHz. (b) At the cooling detuning of -36 MHz. The resonanceis taken as the F = 1 → F ′ = 2 transition. The number and the density of the K MOT areplotted with respect to the detuning of the repumping light for diﬀerent detuning of the coolinglight.

46

40ms
Rb Repump intensity(mW/cm2)
Rb Cooling intensity(mW/cm2)

3.6
0.026
0

47
29

Rb Cooling detuning -18
(MHz)

-15

40

B-field(G/cm)K Cooling detuning
(MHz)

-42
24

12

K Repump intensity(mW/cm2)
K Cooling intensity(mW/cm2)

0

27
0.050
0

29
1.1
-4.0

0

-30
30

PA laser(W/cm2)
Pulse laser(W/cm2)

0
3 106
0

time
Figure 3.6: Parameters for the dual-species C-MOT of K and Rb. In the compression stage of
30 ms, K atoms are compressed by increasing the magnetic ﬁeld gradient. Simultaneously Rb
atoms are compressed and cooled by decreasing the repumping intensity. In the cooling stage of
10 ms, K atoms are eﬃciently Doppler-cooled by the cooling light hitting the middle of F ′ = 3
and F ′ = 2. In contrast, Rb atoms are cooled via polarization gradient cooling.
shows the images of K and Rb atoms after the C-MOT.
Before we had succeeded in observing the photoassociated KRb molecules, we took care of
the position as well as the density and the temperature of both species. In a normal MOT, theposition of K and Rb should coincide at the center of coils. However, we observed that K atoms
were moving in the C-MOT process. One reason for this behavior was that the change in themagnetic ﬁeld gradient generated induced voltage in compensating coils, causing the oscillationof the position of mainly K atoms, which is lighter than Rb atoms. We installed resistors in series
with the compensating coils such that induced voltage immediately consumed in the resistors.
Another reason for the displacement of K atoms was the slight imbalance of the cooling light.
Since we had to reduce the detuning of the cooling light close to the resonance (−4 MHz fromthe F = 2 → F ′ = 3 transition) in the cooling stage, K atoms were quite sensitive to thebalance between two counter-propagating beams. We found that the position of K atoms could
be precisely controlled by adjusting the detuning of cooling light.
After we could observe KRb molecules, we optimized the C-MOT process not by the absorption imaging of atoms, but by monitoring the ion count of KRb molecules itself. What we
47

19mm

87Rb

2 108
4 1011cm-3
100µK

1 108
2 1011cm-3
400µK
41K

Figure 3.7: Absorption images of 41 K and 87 Rb atoms after the C-MOT.
found there was that the control of the intensity of the Rb repumping light and the detuning of
the K cooling light was especially important. Without C-MOT, we could not see any molecules
at all, even if other conditions (the alignment and the frequency of both the photoassociation
laser and the pulse laser) were perfectly adjusted.
Although the C-MOT was a crucial process for producing photoassociated 41 K87 Rb molecules, it reduced number of atoms in the MOT by a factor of 3 compared to the steady-state
MOT. This was because the MOT could not capture atoms eﬃciently during the C-MOT process, which occupied 40 % of the one cycle (100 ms). With a lower repetition rate, we could
obtain a larger number of molecules.

3.2

Photoassociation spectroscopy

An appropriate choice of the photoassociation line is one of the essential aspects of the experiments. The number of produced molecules in the ground state is strongly dependent on thephotoassociation line. Although we found approximately 20 resonances, only a few resonancesamong them could produce a large number of molecules suﬃcient for subsequent applications,such as depletion spectroscopy and the STIRAP transfer into the rovibrational ground state.
Another important issue is to identify the symmetry of the photoassociation level because itinﬂuences the symmetry of the produced weakly bound molecules. In this section, we present
photoassociation spectra and discuss important points for further experiments.

3.2.1

Scanning over a wide range

The long-range potentials of the S+P dissociation of heteronuclear alkali dimers are dominatedby C6 potentials. The C6 values of these potentials are close to that of the ground-state potentials [96–102]. Hence, the excited-state molecules have almost the same size as the ground-state
molecules at the same binding energy. In the present experiments, we produced molecules witha biding energy of ∼10 cm−1 by hitting a photoassociation level detuned from the Rb D1 lineby ∼10 cm−1 .
The photoassociation lines in the low detuning region (12569–12579 cm−1 ) were ﬁrst foundby scanning the DFB laser. Later a Ti: Sapphire laser became available and was used for
48

14

12
)stinu.bra( stnuoc noI

10

8

6

4

2

0
12560

12565

12570

12575
-1

Photoassociation wavenumber (cm)

Figure 3.8: Photoassociation spectrum. The ion counts in the v ′ = 21 level of a3 Σ+ are plottedwith respect to the wavenumber of the Ti: Sapphire laser. Approximately 20 resonances arefound in this region.
scanning over a wide range. Fig.3.8 shows the wide range spectrum obtained with a Ti: Sapphirelaser. Since we were mainly interested in the produced molecules, we did not assign the entire
photoassociation spectrum. Instead, we focused on the resonances which could produce a large
number of molecules in the ground state. We found three large resonances at 12570.13 cm−1 ,12572.10 cm−1 , and 12561.45 cm−1 , which were assigned as Ω = 0+ , Ω = 1, and Ω = 1,
respectively. The assignment was carried out based on their rotational structures (see section
3.2.2). These three lines have been used throughout the experiments.

3.2.2

Rotational structures and symmetries

Photoassociation levels have a clear rotational structure with a rotational constant of several
100 MHz. The structure is readily understood as an ordinary J(J + 1) progression. Here J
indicates the total angular momentum given as J = L + S + R, where L is electronic angularmomentum, S is electronic spin, and R is rotational angular momentum. The selection of therotational level inﬂuences the symmetry of the produced molecules in two ways. First, both theground-state molecules and the excited-state molecules have deﬁnite parities. The parity of the
produced molecules in the ground state should have an opposite parity to the photoassociationlevel. Second, the rotational quantum number of the ground-state molecules should be determined by the selection rule on J, ∆J = 0, ±1. The selectivity on the rotational level in thephotoassociation process enabled us to choose the rotational level of the produced ground-state
molecules. This possibility was particularly beneﬁcial for the parity-selective spectroscopy of
the intermediate states for STIRAP (see chapter 5).
Fig.3.9 shows the rotational structure of the Ω = 0+ state, whereas Fig.3.10 shows that ofthe Ω = 1 state. The symmetry is labeled by the projection of J on the molecular axis, Ω,and not by that of L nor S because spin-orbit interaction (∼100 cm−1 ) is much larger thanelectronic interactions (∼10 cm−1 ) in the weakly bound levels of the excited states. Given therotational structure, the symmetry is readily distinguished because of the relation J ≥ Ω. An
additional symmetry on the parity for the Ω = 0 state ((−1)J for Ω = 0+ and (−1)(J+1) forΩ = 0− ) is distinguished with either the distribution of line intensities or the spin componentof the produced molecules: the Ω = 0+ state has the most strong line of J = 1 and forms both
singlet and triplet molecules, whereas the Ω = 0− state has the most strong line of J = 0 and
49

5

J=1
)stinu.bra( stnuoc noI

4

3

J=2
J=0

2

J=3

1

J=4

0

12570.1

12570.2

12570.3-1

Photoassociation wavenumber (cm)

Figure 3.9: Rotational structure of a photoassociation level of Ω = 0+ . A clear rotational
structure with J = 0–4 is observed. The J = 1 line is the strongest because this level can becoupled to the s-wave collisional channel.

J=1)stinu.bra( stnuoc noI

4

2

J=2

J=3
J=4

0

12561.4

12561.5

12561.6-1

Photoassociation wavenumber (cm)

Figure 3.10: Rotational structure of a photoassociation level of Ω = 1. Each rotational line
splits into multiple lines due to hyperﬁne interaction, although they are not clearly resolved.
forms only triplet molecules. The strongest line should have an odd parity with a rotational
number of J = 0 or J = 1 because collisions in the ground state are dominated by s-wave dueto the Wigner threshold law. The spin component of the produced molecules is determined by
the potentials in the Hund’s case (a) relevant to the state: the Ω = 0+ state is a mixture of(2)1 Σ+ and (1)3 Π, whereas the Ω = 0− state is a mixture of (2)3 Σ+ and (1)3 Π. In general,photoassociation levels have both rotational and hyperﬁne structures of the same magnitude.
As a result of mixing among L, S, R, and the nuclear spin I, the rotational structure has no
more a simple J(J + 1) progression and each rotational line splits into multiple lines. This is
not the case for the Ω = 0 level: the hyperﬁne structure of the Ω = 0 state is negligible to ﬁrst
order.
When we assigned the symmetry of the photoassociation levels, we had to take care of thehyperﬁne ghost of photoassociation lines arising from diﬀerent collision channels (Fig.3.11).
In the C-MOT process, atoms were predominantly pumped to the lower hyperﬁne level, F = 1,
by decreasing the intensity of the repumping light. However, a part of the atoms were remaining
in the upper hyperﬁne level, F = 2, because the repumping light was not inﬁnitely small. As
a result, we observed additional two photoassociation lines at higher frequencies by 254 MHz50

14)stinu.bra( stnuoc noI

12

F =1+F
K

=1

Rb

10
8
F =2+F
6

K

4

Rb

=2

F =1+FK

Rb

F =2+F

=2

K

Rb

=1

2
0
12571.9

12572.012572.1

Photoassociation wavenumber (cm-1)

Figure 3.11: Hyperﬁne ghosts of photoassociation lines. Three photoassociation lines appear
from a single photoassociation level because there are four possible collision channels, FK =1 + FRb = 1, FK = 2 + FRb = 1, FK = 1 + FRb = 2, and FK = 2 + FRb = 2. The ﬁrst channel isthe strongest because atoms are predominantly pumped to the F = 1 level during the C-MOT
process.
(the hyperﬁne splitting in the ground state of 41 K, taken from Ref. [103]) and by 6834 MHz (thehyperﬁne splitting in the ground state of 87 Rb, taken from Ref. [104]) as ghosts of the main line.
The line intensities of both ghosts were approximately 10 % of the main line, indicating that
∼10 % of atoms were remaining in the F = 2 level. The former one was particularly confusing in
the assignment because the separation was close to the rotational constants. The latter one wasalso confusing when we determined the binding energy of the produced molecules by comparing
a photoassociation spectrum with a depletion spectrum for the same photoassociation level (seesection 4.2.4). The ghost arising from the highest collision channel FK = 2 + FRb = 2 was notobserved for most of the photoassociation levels.

3.2.3

Characterization

The information on the saturation intensity in a photoassociation process is important because it
limits the maximum rate for the production of molecules. In the early stage of the experiments,
we studied the power dependence of the yield of the molecules for a photoassociation level at asmall detuning of 2.66 cm−1 (Fig.3.12). The ion counts saturate at approximately a power of
200 mW, corresponding to a peak intensity of 130 W/cm2 . This value is valid only at a small
detuning. Although we used more deeply bound levels for the later experiments, we did notstudy the dependence on the power for these levels; the maximum power for the photoassociation
laser was limited to ∼200 mW, which would be smaller than the saturation power for the deeply
bound levels.

3.3

Ionization spectroscopy

Ionization spectroscopy of photoassociated molecules with the pulse laser, which was based onresonance enhanced multi-photon ionization (REMPI), constituted the crucial part of the experiments. The REMPI ionization enabled us the state-selective detection of produced molecules.
Hence, we could identify the vibrational quantum numbers of produced molecules by analyzing
the ionization spectra. The result was important for further experiments, such as the spectroscopic analyses of excited states and the manipulation of internal states, because it gave us
essential information on the initial state. In this section, we discuss how we identiﬁed in which
vibrational levels molecules were produced via photoassociation.
51

Power (mW)
0

100

200

300

400

500

16
)stinu.bra( stnuoc noI

12

8

4

0
0

50

100

150

200

250

300

2

Intensity (W/cm )Figure 3.12: Power dependence of the photoassociation of molecules. Ion counts in the v ′′ =23 level of a3 Σ+ are plotted with respect to the power of the photoassociation laser. The
photoassociation rate saturates at a power of approximately 100 mW. For this study, the Ti:
Sapphire laser is focused to a beam waist of 310 µm. The saturation of ion counts indicates thesaturation of photoassociation and is not due to the saturation of the MCP, which occurs at
much higher ion counts (> 100).

3.3.1

Estimation with Franck-Condon factors

Although the REMPI detection of photoassociated KRb molecules had already been reported
in the previous studies for another isotopic combination, 39 K87 Rb [65, 79], they gave us a roughinformation that molecules should be observed at an ionization wavelength of approximately 600
nm. There were no good potentials which could accurately predict the transition wavelength.
In order to obtain a thorough understanding on the REMPI detection for the 41 K87 Rb case, weroughly estimated the transition strengths for various excited states as Franck-Condon factors
calculated from ab initio potentials. Since REMPI relies on the resonant excitation to an excitedstate followed by an oﬀ-resonant excitation to a molecular ion state, calculations for the ﬁrst
excitation probability are suﬃcient for judging whether transitions can occur or not.
Fig.3.13 shows the potential energy curves of KRb [105] and of KRb+ [106]. These potentials
are obtained by ab initio molecular orbital calculations. In the spectral range 14000–17500 cm−1 ,which could be covered by our dye laser, we could expect excitations to the (4)1 Σ+ and the(5)1 Σ+ states for singlet molecules, and to the (4)3 Σ+ and the (3)3 Π states for triplet molecules.
Fig.3.14 shows the calculated Franck-Condon factors obtained from the ab initio potentials.
The calculations were carried out with a freely available code, LEVEL [107]. In this calculation,
we used experimentally determined accurate potentials for the ground states (X1 Σ+ and a3 Σ+states) [108].
The calculations gave us the following information: for singlet molecules, the (4)1 Σ+ stateshould be observed in the range 16400–17000 cm−1 , whereas the (5)1 Σ+ state should not be
observed below 17500 cm−1 . Further, for triplet molecules, the (4)3 Σ+ state should be observedin the range 16000–16400 cm−1 , whereas the (3)3 Π state should be observed in the range 16800–
17500 cm−1 . This calculation served as a good guide for assigning the observe spectra.

3.3.2

Assignments

Ionization spectroscopy was carried out in the range 16100–16800 cm−1 which could be coveredby the pulsed dye laser operating with a mixture of Rhodamine 610 and Rhodamine 640 dyes.

52

1.8
1 +

(5) Σ
3 +

(4) Σ

4

-1

Energy (×10 cm )

3

(3) Π

1.7

1.6

1 +

(4) Σ

1.5

(a) Intermediate levels
1.4
3

4

5

6
7
8
Internuclear separation (Å)

9

10

4.2
2 +

(2) Σ

4.0

+

K +Rb

3.6

4

-1

Energy (10 cm )

3.8

3.4
3.2

+

K+Rb

3.0

2 +

(1) Σ

2.8

(b) Ionization levels

2.6
3

4

5

6
7
8
Internuclear separation (Å)

910Figure 3.13: Potential energy curves of KRb relevant to the REMPI detection of photassociatedmolecules. (a) ab initio potentials for the intermediate levels in the range 14000–18000 cm−1 .
(b) ab initio potentials for the ionization levels in the range 25000–42000 cm−1 . Mediated by
one of the intermediate levels, weakly bound molecule can be excited to one of the KRb+ states.

53

(4)

-3

10

(5)
-5

10

(4)

rotcaf nodnoC-kcnarF

-7

(3)

10

1

1

3

3

S
S
S
P

+

+

+

-9

10

10

10

10

10

10

10

-11

-13

-15

-17

-19

-21

15500

16000

16500

Transition wavenumber (cm

17000

17500

-1

)

Figure 3.14: Franck-Condon factors for the ionization near 600 nm. Franck-Condon factors
calculated with ab initio potentials are plotted with respect to the wavenumbers from the groundstate atomic level. Spin-orbit interactions are neglected in this calculation.
Fig. 3.15 shows the ionization spectra obtained by scanning the pulsed dye laser whereas thephotoassociation laser was locked to a speciﬁc resonance.
We could clearly observe the vibrational structure of the singlet molecules in the X1 Σ+ state
mainly in the range 16400–16600 cm−1 , whereas that of triplet molecules in the a3 Σ+ state
was clearly observed in the range 16100–16400 cm−1 . In other regions, the spectra were toocomplicated or too simple to be interpreted correctly. Another important issue for observing
the clear vibrational structures was the detuning of the photoassociation laser: at ﬁrst we had
been using the resonance near -2.66 cm−1 and -7.24 cm−1 from the Rb D1 line. In these cases,
photoassociation predominantly formed molecules in a few weakly bound levels. It was diﬃcult
to assign the spectra because they showed no speciﬁc features. With a photoassociation levelwith a large detuning of -9.0 cm−1 , we could obtain the clear signatures of both singlet andtriplet molecules.
Fig.3.16 shows the expanded view of the spectra representing the vibrational structure of
the ground state molecules in the X1 Σ+ state, whereas Fig.3.17 shows the vibrational structureof the ground state molecules in the a3 Σ+ state. We observed the v ′′ = 90–93 levels of X1 Σ+and the v ′′ = 20–24 levels of a3 Σ+ , which are in good agreement with a calculated spectrum
obtained from the experimentally determined potential curve. In the later experiments, thesinglet molecules in the v ′′ = 91 level were monitored at a pulsed laser wavenumber of 16543
cm−1 , whereas the triplet molecules in the v ′′ = 21 level were monitored at a pulsed laser
wavenumber of 16190 cm−1 . For better understanding of the weakly bound levels relevant to
the present study, a coupled-channel calculation including hyperﬁne interaction is shown in
Fig.3.18. This calculation was carried out by Dr. P. Naidon.

54

8

Rb 6D

(a)

Rb 11D

K 7D

Rb 13D Rb 8S

Rb 12Dv'=4)stinu.bra( stnuoc noI

6

v'=6
v'=7

v'=5

v'=8

v'=3
4

v'=9

2

0

16100

16200

16300

16400

16500

-1

Ionization wavenumber (cm16(b)

Rb 17DRb 15DRb 16D

16500

16550

)

14
)stinu.bra( stnuoc noI

12
10
8
6
4
2
0

16400

16450

Ionization wavenumber (cm

16600

16650

-1

)

Figure 3.15: Ionization spectra of photoassociated molecules. (a) In the range 16100–16500 cm−1
obtained with a photoassociation level of 12576.49 cm−1 . (b) In the range 16400–16650 cm−1
obtained with a photoassociation level of 12570.13 cm−1 . Ion counts of KRb are plotted withrespect to the wavenumber of the pulsed dye laser. In the ﬁgure (a), the vibrational numbersof the (4)3 Σ+ state are assigned on the basis of spectroscopic studies in the low wavenumber
region (section 6.4). In the ﬁgure (b), the expected vibrational levels of the (4)1 Σ+ state are
given in blue lines. The Rydberg states of both K and Rb atoms are also observed because theselevels yield extremely large ion counts, inﬂuencing the ion counts of KRb. The Rb 6D and Rb
8S states are observed due to the absorption of both the photoassociation laser and the pulsed
laser.

55

14)stinu.bra( stnuoc noI

1210

v''=918v''=936
4v''=90v''=92v''=942
0

16.536

16.54016.544

Ionization wavenumber (cm16.548x10

3

-1

)

Figure 3.16: Vibrational structure of photoassociated molecules in the X1 Σ+ state observed witha photoassociation level of 12570.13 cm−1 . The calculated vibrational structure of the X1 Σ+
state is also given in blue lines. Photoassociation forms molecules in the v ′′ = 90 − 93 levels of
X1 Σ+ . In the later experiments, molecules in the v ′′ = 91 are used for the STIRAP transfer aswell as the spectroscopy of excited states.

10)stinu.bra( stnuoc noI

8v''=23
6v''=22v''=254v''=21v''=262v''=20

0

16170

16180

16190

Ionization wavenumber (cm

16200
-1

)

Figure 3.17: Vibrational structure of photoassociated molecules in the a3 Σ+ state observed witha photoassociation level of 12576.49 cm−1 . The calculated vibrational structure of the a3 Σ+state is also given in blue lines. Photoassociation forms molecules in the v ′′ = 20 − 26 levelsof a3 Σ+ . In the later experiments, molecules in the v ′′ = 21 are used for the spectroscopy of
excited states because the are the most abundant with a deeply bound photoassociation level of
12561.45 cm−1 .

56

10

0

vX''=90

vX''=89

vX''=91

vX''=92

vX''=93

-2

10

-4

Singlet fraction

10

-6

10

-8

va''=22

va''=21

va''=25

va''=24

va''=23

10

-10

10

Triplet levels
Singlet levels

-12

10

-14

10

-16

10

-22

-20

-18

-16
-14
-12
-1
Binding energy (cm )

-10

-8

-6

Figure 3.18: Vibrational and hyperﬁne structures of the weakly bound levels of 41 K87 Rb. The
singlet fraction of each hyperﬁne level is plotted with respect to binding energy. The rotational
quantum number is l = 0. The v ′′ = 91 level of X1 Σ+ and the v ′′ = 22 level of a3 Σ+ is closely
lying, resulting in a strong hyperﬁne mixing.

3.3.3

Characterization

The spectra in the range 595–620 nm allowed us to estimate the minimum transition strength
which could ionize the weakly bound molecules via two-photon transitions. There should be
two limiting factors: the ﬁrst resonant transition and the second ionization process. We do nothave information on the ionization process. Thus, what we can consider now is about the ﬁrst
resonant transition. Although the intensity of the pulsed laser was much higher than saturation
intensity of the transition, ionized molecules could be observed only when the corresponding
Franck-Condon factor of the ﬁrst transition was higher than approximately 0.01. We guess that
this threshold is determined by the ratio between the linewidth of the pulsed laser and saturation
broadening. If we use the ab initio values for the transition dipole moments without FranckCondon factors, 0.5 ea0 for both (4)1 Σ+ ← X1 Σ+ and (4)3 Σ+ ← a3 Σ+ transitions [109], theestimated Rabi frequencies for these transitions with a typical pulsed laser intensity of 1×106
W/cm2 are several GHz. This is approximately the same as the linewidth of the pulsed laser.
With a lower intensity, the linewidth exceeds saturation broadening, resulting in an ineﬃcient
excitation at the ﬁrst resonant transition. The saturation broadening was also observed in the
spectrum as a spectral width. The observed width of the ionization spectrum was between
0.3–1 cm−1 in most cases, which should be understood as a sum of saturation broadening, thelinewidth the pulsed laser, and the rotational structures of both ground and excited states.
Fig.3.19 shows a typical dependence of ion counts on the pulsed laser power. We could obtainlarger ion counts with a larger power of up to a few mJ. However, the intensity of the pulsed
laser was kept to lower than 1 mJ in most of the experiments. There are two reasons for the
low power. First, we avoided the pulsed laser scattered at view ports from yielding junk pulses,
which degraded the signal-to-noise ratio of spectra. Second, we suppressed oﬀset ion counts due
to oﬀ-resonant multi-photon ionization. The ﬂuctuation of the oﬀset ion counts also degraded57

12)stinu.bra( stnuoc noI

10
8
6
4
2
0
0

1

2

3

4

Power (mJ)

Figure 3.19: Ion counts vs the power of the pulsed laser. The ion counts are saturated at a
power of a few mJ. In most of the experiments, the power is kept to lower than 1 mJ for a high
SNR of spectra.
the signal-to-noise ratio of spectra.
In the TOF spectrum, atomic ions were not always observed. Irradiating the pulsed laser
under the existence of the MOT light produced atomic ions of both K and Rb. This is because
atoms in the excited states have a higher energy and are readily ionized. A large atomic signal
was also observed for either K or Rb when the pulsed laser was resonant to the Rydberg states ofK or Rb through a two-photon transition. These behaviors were especially beneﬁcial for aligning
the pulsed laser such that it hit the MOT appropriately. However, the pulsed laser was normally
irradiated after the MOT lights were turned oﬀ for avoiding the MOT light from inﬂuencing
molecules. In this case, atoms in the MOT were not observed. Nevertheless, we observed atomic
ions of both K and Rb when the photoassociation laser produced molecules and the pulse wasresonant to a singlet transition ((4)1 Σ+ ← X1 Σ+ ). We interpreted this behavior as follows.
The pulsed laser can ionize molecules via REMPI ionization to the (1)2 Σ+ state, whereas it
can also excite molecules in the X1 Σ+ state to the (2)2 Σ+ state and dissociate them to atoms(Fig.3.13). Molecules in the a3 Σ+ state appeared not to have a transition to the (2)2 Σ+ state.
As a result, we could distinguish between singlet and triplet molecules only by monitoring the
TOF spectrum.
The temperature of photoassociated molecules was evaluated with a TOF after the MOT
lasers and the photoassociation laser were turned oﬀ. Fig.3.20 shows the variation in ion counts
with respect to the TOF. This plot indicates that molecules produced in a photoassociation laser
reaches the edge of the pulsed laser within several ms. Assuming the diameter of the pulsed
laser as 1 mm, we can roughly estimate the velocity of molecules as 10 cm/s, which corresponds
to 150 µK.

58

30)stinu.bra( stnuoc noI

25
20
15
10
5
0
2

4

6

8

Time of flight (ms)Figure 3.20: Time variation of ion counts after the MOT lasers and the photoassociation laser
are turned oﬀ. The plot indicates that the temperature of molecules is approximately 150 µK.

59Chapter 4

Molecular spectroscopy for a Raman
transfer
One of the crucial steps for realizing an eﬃcient transfer between a weakly bound level andthe rovibrational ground state is an appropriate choice of intermediate excited states. In thepresent study, we studied two promising candidates, the (3)1 Σ+ state and the (2)3 Σ+ state.
We expected that the same laser setup could be used for both states because these two states
have good intermediate states in approximately the same energy range. The (2)3 Σ+ state hadalready been known as a good intermediate state in the previous studies, which showed thatthe (2)3 Σ+ state could couple weakly bound triplet molecules with the vibrational ground state.
By contrast, the (3)1 Σ+ state had not been used as an intermediate state bridging a weakly
bound level and a deeply bound level. The (3)1 Σ+ state was particularly important in thepresent study because it could couple weakly bound singlet molecules with the rovibrational
ground state. Although the (2)3 Σ+ state was eventually not used as an intermediate state in
the present study, this state showed much more rich structures than the (3)1 Σ+ state, thereby
giving us a deep understanding on the molecular structure of the excited states. Therefore, we
present the spectroscopic studies on both the (3)1 Σ+ state and the (2)3 Σ+ state in this chapter.

4.1

Excited states for an intermediate state

Before presenting details on the present studies, we give a brief overview on the possible schemes
for connecting weakly bound levels with the rovibrational ground state. The excited state used as
an intermediate state for STIRAP should have large Franck-Condon overlaps with both weakly
bound levels and deeply bound levels. For KRb molecules, three promising schemes have been
proposed and one of them have already been experimentally demonstrated. Figure 4.3 shows
the relevant potential energy curves of KRb.
In the scheme (a), weakly bound triplet molecules are converted into the vibrational groundstate via the Ω = 1 level of the (2)3 Σ+ state. This scheme was originally proposed for RbCs
molecules [110] and experimentally demonstrated as stimulated emission pumping of photoassociated 85 RbCs molecules [46] into the vibrational ground state. Later this scheme was employed
for the STIRAP transfer of near-degenerate 40 K87 Rb molecules [50]. Hereafter we call thisscheme as triplet scheme.
The scheme (b) also begins with weakly bound triplet molecules, which can be coupled withthe rovibrational ground state via the Ω = 0+ level of the (1)3 Π state [111, 112]. Although asimilar scheme was used for producing Cs2 molecules in the rovibrational ground state [52], thisscheme has not been realized for heteronuclear molecules.
The scheme (c) involves only singlet levels: the initial state is weakly bound singlet molecules
and the intermediate state is the (3)1 Σ+ state. Although spectroscopic studies on the (3)1 Σ+
state has been reported [70,113,114], this scheme has not been realized before the present study.

60

(1)

P

1

10

(2)

S

3

+

1-

)

3

´

mc 01 ( ygrenE

5

a

+

S

3

0

S

1

X

2

4

+

6

(a)

8

10

12

Internuclear separation (Å)

Figure 4.1: Scheme (a) for a Raman transfer. The initial state is triplet molecules and theintermediate state is the Ω = 0 component of the (2)3 Σ+ state. The spin-orbit mixing of the(1)1 Π state into the (2)3 Σ+ state enables us to couple weakly bound triplet molecules to therovibrational ground state of X1 Σ+.

3

(1) Π

1 +

(2) Σ

3

-1

Energy (×10 cm )

10

5
3 +

aΣ
0

1 +

XΣ
2

4
6
8
10
Internuclear separation (Å)(b)
12Figure 4.2: Scheme (b) for a Raman transfer. The initial state is triplet molecules and theintermediate state is the Ω = 0+ level of the (1)3 Π state. The spin-orbit mixing of the (2)1 Σ+
state enables us to couple weakly bound triplet molecules to the rovibrational ground state of
X 1 Σ+ .

61

1 +

(3) Σ

3

-1

Energy (×10 cm )

10

5

0
1 +

XΣ
2

4
6
8
10
Internuclear separation (Å)(c)
12Figure 4.3: Scheme (c) for a Raman transfer. The initial state is singlet molecules and theintermediate state is the (3)1 Σ+ state.
Hereafter we call this scheme as all-singlet scheme.

4.2

The (3)1 Σ+ state: Direct potential ﬁt analysis

Previously spectroscopic studies on the (3)1 Σ+ state have been carried out for the most abundantisotopic combination, 39 K85 Rb [70]. An experimental Rydberg-Klein-Ree (RKR) potential curve
was reported from the potential minimum (∼ 9800 cm−1 ) up to 10400 cm−1 (energy is given withrespect to the ground atomic threshold 4S + 5S without hyperﬁne interaction throughout thischapter) [113,114]. The levels suited for an intermediate state are expected to lie at much higher
energies, 11400–11800 cm−1 , than the range covered by the RKR potential. Thus, additional
precision spectroscopy was required for ﬁnding the transitions of 41 K87 Rb molecules. Although
an ionization spectrum taken with a pulsed laser and a depletion spectrum near 11700 cm−1 for39 K87 Rb were given in Ref. [70], we did not utilize it for estimating transitions for 41 K87 Rb with
mass-scaling because there was a large uncertainty in the potential between the RKR curve andthe range studied in Ref. [70].
In this study, we ﬁrst took an ionization spectrum based on the two-color detection scheme(section 4.2.1). Guided by the ionization spectrum, we obtained depletion spectra with a CW
laser by monitoring ion counts in a speciﬁc vibrational level (section 4.2.2). With the analysis on
the rotational constants, we conﬁrmed that the observed states were deﬁnitely the (3)1 Σ+ state.
However, for some vibrational levels, near-lying states made the spectrum diﬃcult to assign. We
developed parity-selective spectroscopy which enabled us to observe rotational levels with botheven and odd parities separately (section 4.2.3). In this way, we could ensure that we observedv ′ = 41–50 levels of the (3)1 Σ+ state. Another important issue was to determine the binding
energy of the molecules in the ground state (the v ′′ = 91 level of X1 Σ+ ) used for spectroscopy(section 4.2.4). The information on the binding energy of produced molecules is necessary for
determining the energy of the observed levels with respect to the atomic threshold.
As an advanced study, we constructed an accurate potential energy curve of the (3)1 Σ+ statebased on the direct potential ﬁt (DPF) analysis, in which an inaccurate potential is modiﬁed
such that it reproduces observed spectra. The accuracy of the obtained potential was conﬁrmed
through transition strengths in the following manner. We ﬁrst formulated the representation
62

for saturation broadening of a depletion spectrum (section 4.2.7). The obtained new expression
allowed us to extract transition strengths for each vibrational level. The validity of this formulation was experimentally checked (section 4.2.5). We observed a large variation in transition
strengths among the vibrational levels of the excited states, which we attributed to the variation in Franck-Condon factors due to a large number of nodes in the wavefunction of weakly
bound molecules. This fact indicates that transition strengths from weakly bound molecules aregood measures of the wavefunctions in the excited states. We conﬁrmed that the observed large
variation in the transitions strengths was well reproduced with a modiﬁed potential (section
4.2.6).

4.2.1

Ionization spectroscopy

The two-color ionization spectroscopy in an infrared (IR) range was carried out in the samemanner as Ref. [70]. The photoassociated molecules were ﬁrst excited by the pulsed dye laser(LDS867, 840–895 nm) and were immediately ionized with a 532 nm green light for pumping thedye laser. The green light was delayed by 10 ns with respect to the IR pulse such that reversed
processes (excitation with the green pulse followed by ionization with the IR pulse) and twocolor oﬀ-resonant multi-photon processes (simultaneous absorption of the IR pulse and the green
pulse) did not make an excess ion counts. Fig.4.4 shows the obtained ionization spectrum in awide range. In order to suppress one-color oﬀ-resonant multi-photon ionization (three-photon
for the IR pulse and two-photon for the green pulse), we had to decrease the power to below
60µJ for the IR pulse and to below 40 µJ for the green pulse. As a result, the observed lines inthis range had smaller ion counts than in the orange–red range (Fig.3.15). With a high intensityof either the IR pulse or the green light, the oﬀ-resonant multi-photon ionization yielded a large
oﬀset counts, which surpassed most of the observed spectrum.
Although we could not observe a clear vibration progression in the excited state as reported in Ref. [70], a calculated spectrum showed a good agreement at several vibrational levels(Fig.4.4(c)). Here the calculated spectrum was obtained from an ab initio potential and wasshifted by a few wavenumbers. For each upper level, we observed a vibrational progression of
the X1 Σ+ state (v ′′ = 91–93), which was in good agreement with the previous observations in
the orange–red region (Fig.3.15).

4.2.2

Depletion spectroscopy and rotational analysis

The ionization spectroscopy enabled us to estimate resonant frequencies for a speciﬁc vibrational
level with an accuracy of several GHz. We scanned the Ti: Sapphire laser (power, ∼100 mW)
at the estimated frequencies while we monitored ion count in the v ′′ = 91 level of X1 Σ+ . The
Ti: Sapphire laser was continuously applied during the photoassociation process. When the
Ti: Sapphire laser was resonant to an excited state, ion counts decreased and a dip appearedin the spectrum. We call this spectroscopic method for photoassociated molecules as depletion
spectroscopy after its ﬁrst implementation in Ref. [70]. Due to its simplicity and robustness,
depletion spectroscopy has been frequently used throughout the present study as a powerful tool
for searching transitions.
Fig.4.5 shows a typical depletion spectrum of the (3)1 Σ+ state. For understanding on thisspectrum, the diagram of relevant molecular levels is shown in Fig. 4.6. 41 K and 87 Rb atoms are
ﬁrst excited to the Ω = 0+ , J ′ = 1 level by a photoassociation laser. Due to the even character ofs-wave collision, the excitation to the Ω = 0+ is the most intense for the J = 1 level, which has
odd parity. Subsequent spontaneous decays forms molecules distributed among the v ′ = 90–93
levels of X1 Σ+ and the v ′ = 20–24 levels of a3 Σ+ . Singlet molecules are rotationally distributed
among the J ′′ = 0, 2 levels which have even parity. With these molecules, we can observetransitions to the odd-parity rotational levels of the (3)1 Σ+ state if the selection rule on J issatisﬁed. In this way, we observe three transitions, J ′ = 3 ← J ′′ = 2, J ′ = 1 ← J ′′ = 2, andJ ′ = 1 ← J ′′ = 0.
63

(a))stinu.bra( stnuoc noI

3

2

1

0
11350

11400

11450

11500

-1

Ionization wavenumber (cm

11550

)

)stinu.bra( stnuoc noI

(b)
3

2

1

0
11550

11600

11650

-1

Ionization wavenumber (cm

11700

)

)stinu.bra( stnuoc noI

(c)
2

1

0
11460

11480

11500

11520

-1

Ionization wavenumber (cm)

Figure 4.4: Ionization spectrum in the range 11340–11750 cm−1 and calculated line positions.
(a) 11340–11550 cm−1 . (b) 11540–11740 cm−1 . (c) 11450–11520 cm−1 . The observed spectrum
is given in black dots, whereas calculated line positions for the v ′′ = 91, v ′′ = 92, and V ′′ = 93
levels are given in red, green, and blue lines. Ion counts are limited to approximately 1 dueto the low laser intensity for suppressing oﬀ-resonant multi-photon ionization. The observed
spectrum showed a good agreement with a calculated spectrum at several wavenumbers. The
calculated spectrum is obtained with ab initio potential and is shifted by a few wavenumbers.

64

Ion counts (arb.units)

30
25

J'=1←J''=0

20
15

J'=1←J''=2

J'=3←J''=2105011689.711689.811689.9-1
Depletion wavenumber (cm )11690.0

Figure 4.5: Depletion spectrum for the v ′ = 48 level of the (3)1 Σ+ state. Ion counts in thev ′ = 91 level of the X1 Σ+ state are monitored while a Ti: Sapphire laser is scanned. The Ti:
Sapphire laser is resonant to excited levels at the observed dips.

Photoassociationlevel (Ω=0+)2(3)1Σ+

1
0

4
3
2
1
0
3

K(F=1)+Rb(F=1)

2
1 Weakly bound X Σ0

1 +Figure 4.6: Rotational levels relevant to the depletion spectroscopy of the (3)1 Σ+ state. 41 K
and 87 Rb atoms are excited to the Ω = 0+ , J = 1 level by a photoassociation laser. The excitedmolecules spontaneously decay to the J = 0, 2, v ′′ = 91 level of X1 Σ+ , which are excited by the
Ti: Sapphire laser to the J = 1, 3 level of (3)1 Σ+ .

65

16

(3)

S

3

+

15

1-

)

14

3

P

3

13

´

mc 01 ( ygrenE

(2)

12

(2)

11

P

1

(3)

10

S

1

+

9
2

4

6

8

10

Internuclear separation (Å)

Figure 4.7: Potential energy curves dissociating to the K(4P)+Rb(5S) threshold. There are fourstates, (3)1 Σ+ , (3)3 Σ+ , (2)3 Π, and (2)1 Π. Spin-orbit mixing of other states with the (3)1 Σ+
state is negligible in the studied range 11400–11800 cm−1 because they are far away from thestudied range.
The entire spectroscopy was carried out for the v ′′ = 91 level of X1 Σ+ , but we also observed
depletion spectra for the v ′′ = 90 and v ′′ = 93 levels for conﬁrmation. The splitting between
J ′ = 1 ← J ′′ = 2 and J ′ = 1 ← J ′′ = 0 enabled us to extract the rotational constant of
the X1 Σ+ . Although the dip was not clear, we could obtain rotational constants of 0.0084(2)cm−1 for the v ′′ = 90 level, 0.0076(2) cm−1 for the v ′′ = 91 level, and 0.0060(2) cm−1 for thev ′′ = 93 level, where the errors were ﬁtting errors due to the broad spectral width. These values
are in good agreement with calculated values of 0.0085 cm−1 , 0.0077 cm−1 , and 0.0061 cm−1 ,respectively. These agreements indicate the correctness of the assignment to ionization spectra(section 3.3.2 and 4.2.1). The rotational constant of the v ′′ = 91 level is later measured precisely
as 0.0076646(17) cm−1 via SpIDR spectroscopy (section 6.2).
We carried out spectroscopy in the range 11400–11800 cm−1 and observed 10 vibrational
levels (v ′ = 41–50) with a spacing of ∼40 cm−1 . For each vibrational level, we observed rotational
structures similar to Fig. 4.5. These spectra gave us the information on vibrational energies,
rotational constants of both ground and excited states, and transition strengths as obtained
from the spectral widths. For some vibrational levels, ionization spectra were hardly observable
due to the weak power of the pulsed laser. Even in such cases, we could readily ﬁnd these levels
with depletion spectroscopy by assuming nearly constant vibrational spacing. The regularity ofthe (3)1 Σ+ state is due to a small perturbation from other electronic states, which is understoodas follows. The (3)1 Σ+ state dissociates to the K(4P)+Rb(5S) threshold, to which the (3)1 Σ+ ,(3)3 Σ+ , (2)1 Π, and (2)3 Π states dissociate (Fig.4.7). In general, the largest perturbation in thestates with electronic orbital angular momentum is spin-orbit interaction. For heteronuclear
diatomic molecules, the spin-orbit interaction couples only states with the same dissociationthreshold because the coupling is mediated by an atom in the excited state. Thus, the states
dissociating to K(4P)+Rb(5S) are not coupled with those dissociating to K(4S)+Rb(5P) via
spin-orbit interaction. The (3)1 Σ+ state should be coupled only with the (3)3 Σ+ or (2)3 Πstates, but they are far away by more than 1000 cm−1 , which is much larger than the spin-orbit
splitting of the K 4P state (∼58 cm−1 ). Hence, we can consider the (3)1 Σ+ state as a purely
singlet state.
The observed dips in depletion spectra were always smaller than half of the monitored ion
66counts. The dip appeared to be too shallow because the monitored ion counts were much larger
than the sum of the peak height for J ′ = 0 and J ′ = 2. Even if we apply our maximum powerof 500 mW, the depth of the dip did not change signiﬁcantly. This was not the case when we
took depletion spectra with other lower levels such as v ′′ = 90: with a maximum power we
could deplete the dip more deeply. We understood these behaviors as follows. The v ′ = 91
level of X1 Σ+ has a ∼10 % of triplet component due to the hyperﬁne mixing of the near-lying
v ′′ = 22 level of a3 Σ+ , which is separated by only ∼0.1 cm−1 . As a result, the v ′ = 91 level has
a hyperﬁne structure of the order of 10 MHz (at the moment of this experiment, these values
were not measured one, but were known only from coupled-channel calculations [115]). The
applied laser is not resonant for some hyperﬁne levels unless the Rabi frequency is larger thanthe hyperﬁne splitting. Since the transition dipole moment between X1 Σ+ and (3)1 Σ+ is low(< 1 ea0 [109]) we could obtain Rabi frequency of up to ∼10 MHz in our experiments. Thus,
molecules in some of the hyperﬁne levels were not eﬃciently excited. These considerations wereconﬁrmed by exciting molecules to the photoassociation level for measuring the binding energy(section 4.2.4). Here we could obtain a high Rabi frequency (> 20MHz) and deplete the ion
counts close to zero. In this way, we could indirectly know the existence of hyperﬁne splittingin the monitored level although we could not observe any splitting with depletion spectroscopy.
In the ﬁrst place, we focused on the rotational constants of the excited states. The rotational
constants are of particular interest since they are speciﬁc to symmetry (spin multiplicity S andthe projection of the electronic orbital angular momentum along the internuclear axis Λ) andare useful for identifying the symmetry of the observed levels. The vibrational energies also
become important when we construct an accurate potential which can reproduce experimental
observations (section 4.2.6).
There are ﬁve symmetries in the investigated range: (1)3 Π, (2)1 Σ+ , (2)3 Σ+ , (3)1 Σ+ , and(1)1 Π. These states have diﬀerent potential shapes, which result in diﬀerent sizes of moleculesand diﬀerent magnitudes of rotational constant. Although experimental potential curves are
not available in most cases, ab initio potentials give us a good estimation on these values.
Therefore, we can identify the symmetry of the observed levels just by comparing observedand calculated rotational constants. Fig.4.8 shows comparison between observed and calculated
rotational constants. All the observed 10 vibrational levels had rotational constants close to
the calculated values for the (3)1 Σ+ state. Thus, we concluded that the observed levels had
deﬁnitely a symmetry of (3)1 Σ+ .

4.2.3

Parity-selective spectroscopy

For some of the observed levels, another states were observed near the (3)1 Σ+ state, making thespectrum diﬃcult to be understood. In order to clearly assign these spectra and derive correct
rotational constants, we developed parity-selective spectroscopy. The basic idea is quite simple:
by changing the rotational levels for photoassociation, we can prepare molecules in diﬀerent
rotational levels with opposite parity and thereby obtain a spectrum for diﬀerent rotational levelsof the (3)1 Σ+ . By comparing the spectra for two diﬀerent rotational levels, we could deﬁnitely
assign the spectra. In the study of the (3)1 Σ+ state, spectra taken with a photoassociation levelof J ′ = 2 were compared with the spectra with a photoassociation level of J ′ = 1 (Fig.4.9). Here
we observed four transitions rather than three transitions: J ′ = 4 ← J ′′ = 3, J ′ = 2 ← J ′′ = 3,
J ′ = 2 ← J ′′ = 1, and J ′ = 0 ← J ′′ = 1. When we compared these two spectra, we coulddeﬁnitely extract a rotational constant and assign the observed lines. This method was employedalso for the study on the (2)3 Σ+ state, in which it was important to observe both even and odd
parity levels for understanding the spectra.

4.2.4

Measurement of the binding energy

The binding energies of the photoassociated molecules were measured by comparing a photoassociation spectrum with a depletion spectrum for the photoassociation level. Fig.4.10 shows com67

1

(1)

3.0
1-

)

1

3

(2)

P
S

3

(1)

+

1

(2)

+

+

Observed

2.6´

01 ( stnatsnoc lanoitatoR

2-

mc

(3)

2.8

P
S
S

2.4

2.2

2.0

1.8

1.6

11400

11500

11600

11700
1

Transition wavenumber from v"=91, X

S

11800

+

-1

(cm

)

Figure 4.8: Observed and calculated rotational constants for the molecular states in the energy
range 11400–11800 cm−1 . Observed and calculated rotational constants are plotted with respect
to the energy from the K(4S)+Rb(5S) asymptote. Observed values are in good agreement withcalculated values for the (3)1 Σ+ state.

Photoassociation with J'=1
Photoassociation with J'=240
)stinu.bra( stnuoc noI

30
1'

¬

1'

20

10
0'

¬

11466.6

0''
3'

¬

2''

2''

1''

¬

2'
0

¬

2'

¬

¬

1''

4'

3''

11466.7

11466.8

11466.9

Depletion wavenumber (cm3''11467.0-1)

Figure 4.9: Parity selective spectroscopy of the v ′′ = 42 level of the (3)1 Σ+ state. By exciting
atoms to a photoassociation level of J ′ = 2, we can prepare molecules in the J ′ = 1, 3 level andobtain a spectrum of four transitions, which is diﬀerent from the spectra with a photoassociationlevel of J ′ = 1. The shift of the spectra is proportional to a rotational constant. Thus, we canderive rotational constants correctly and assign the spectra deﬁnitely.

68parison between a photoassociation spectrum at 12570.13 cm−1 (Ω = 0+ ) and depletion spectra
for this level. We observed three lines, corresponding to J ′ = 3 ← J ′′ = 2, J ′ = 1 ← J ′′ = 2, andJ ′ = 1 ← J ′′ = 0 just as in the same way as for the (3)1 Σ+ because the (3)1 Σ+ state in Hund’scase (a) constitutes the Ω = 0+ state in Hund’s case (c). The measured binding energies of the
J ′′ = 2, v ′ = 91 level and the J ′′ = 2, v ′ = 93 level with respect to the atomic collision channelof FK = 1 + FRb = 1 were -12.454(1) cm−1 and -6.084(1) cm−1 , respectively. The calculated
binding energies for these levels based on an experimentally obtained potential were -12.4845
cm−1 and -6.0935 cm−1 , respectively. The observed errors of ∼900 MHz for the v ′ = 91 leveland of ∼ 300 MHz for the v ′ = 93 level should be attributed to the hyperﬁne mixing of thenear-lying levels of a3 Σ+ .
In this measurement, we found an interesting fact. Fig.4.11 shows depletion spectra takenwith molecules in the v ′ = 91 level, which were prepared with diﬀerent photoassociation levels.
Here the excited state had a symmetry of Ω = 1, which had both even and odd parity levels ina single rotational line. In addition, the selection rule on J did not hold due to the hyperﬁnemixing in the excite state. Thus, we observed four rotational levels in the excited state. Theinteresting feature appears in a rotational distribution in the ground state: the population ratioof J ′′ = 2 and J ′′ = 0 is dependent on the photoassociation level. Although intuitively J ′′ = 2should dominate because it has a larger number of sublevels, the J ′′ = 0 dominates for somephotoassociation levels. We did not understand this behavior, but it should be explained in termsof Franck-Condon factors including angular momenta. A similar behavior was also observed fortriplet molecules (section 4.3.1). What we found here is that it is important to take care of
the rotational distribution for further applications such as a STIRAP transfer into other levels
because a coherent transfer can couple only a single level with another single level.

4.2.5

Transition strengths

The analyses on rotational constants gave us clear understanding on the structure of observed
spectra. However, we could not explain the observed spectral widths: they were on the orderof 100 MHz, which were much larger than the expected Rabi frequencies. Further, we founda large variation in the widths among vibrational levels of the (3)1 Σ+ . We derived a theory
for giving a relation between the spectral widths of depletion spectra and the Rabi frequencies(given in the section 4.2.7). According to this theory, the full width at half maximum (FWHM)
2∆ is given by√2∆ ∼ 0.79Ω Γτ

(4.1)

where Ω is the Rabi frequency, Γ is the natural linewidth of the excited state, and τ is the duration of spectroscopy. Roughly speaking, the width increases not only with a high light intensity
but also with a long interaction time. The increase in the width is understood as saturation
broadening, in which a probe light induces considerable excitations. Before using this relation
for further analyses, we ﬁrst conﬁrmed the validity of this relation by observing the dependenceson both intensity (Fig.4.12) and interaction time (Fig.4.13). When we studied the dependence
on interaction time, the depletion laser was continuously turned on while the photoassociation
laser was turned on for a ﬁxed time. The width shows clear square-root dependences on bothintensity and interaction time, indicating that the obtained relation (4.1) is essentially correct.
However, this relation is expected to be violated when the Rabi frequency is too small or thewidth is close to the hyperﬁne splitting in the ground state. We conﬁrmed that this relation
was true at a width of down to a few tens of MHz.
The relation (4.1) gives a good estimation on the transition strengths. Although it is√notsuited for determining the absolute value of the transition strengths due to the factor of Γτ ,which is not known precisely in general, it enables us to systematically compare transition
strengths obtained in the same experimental setup. In our case, an interaction time was estimated as a few milliseconds from the temperature of the molecules and the beam diameter of6940(a) Photoassociation spectrum)stinu.bra( stnuoc noI

30

20

10

0
12570.0

12570.5

12571.0

Photoassociation wavenumber (cm

-1

)

30
)stinu.bra( stnuoc noI

25

20

15

10
(b) Depletion spectrum

5012582.5

12583.0
Depletion wavenumber (cm12583.5-1)

Figure 4.10: Comparison of a photoassociation spectrum (a) and a depletion spectrum (b) for thesame excited state. The binding energy of weakly bound molecules can be precisely measured bycomparing the diﬀerence in wavenumber between a photoassociation spectrum and a depletion
spectrum.

70

Depletion spectrum taken with a PA level of 12570.13 cm
Depletion spectrum taken with a PA level of 12572.10 cm

40

Depletion spectrum taken with a PA level of 12561.45 cm

-1
-1
-1

Photoassociation spectrum)stinu.bra( stnuoc noI

30

20

10

0
12.56140

12.56145

12.56150

12.56155x10

Photoassociation wavenumber (cm

3

-1

)

Figure 4.11: Depletion spectra with diﬀerent photoassociation levels. The three upper spectra
are depletion spectra with a diﬀerent photoassociation level, whereas the lowest spectrum isthe photoassociation spectrum for the same excited state. The depletion spectra are shiftedby -12.454 cm−1 , which corresponds to the binding energy of the v ′′ = 91, J ′′ = 2 level. The
rotational distribution depends on the photoassociation level. For the lower two cases, theJ ′′ = 0 level appears to be dominant.

500

400
)zHM( htdiW

300

200

100

0
0

10

20

30

40

50

Peak intensity (W/cm2)

Figure 4.12: Width of the depletion spectrum vs intensity. The widths of the depletion spectra
for the v ′ = 42 level are plotted with respect to the intensity of the depletion laser. The errors
indicate standard deviations in the ﬁts to the spectra.

71

200

)zHM( htdiW

150

100

50

0
0

2

4

6

8

Interaction time (ms)Figure 4.13: Width of the depletion spectrum vs interaction time. The widths of the depletion
spectra for the v ′ = 42 level are plotted with respect to interaction time. The errors indicate
standard deviations in the ﬁts to the spectra. Due to the expansion of the molecular cloud, aﬁt is appropriate only at a short interaction time.
the depletion laser. The natural width of the (3)1 Σ+ state was not precisely known, but it wasestimated as 2π × 300 kHz from an ab initio calculation [109]. In the following discussion, we
derive the√Rabi frequency from the observed spectrum by assuming τ = 2 ms, Γ = 2π × 300
kHz, and Γτ = 60.

4.2.6

Constructing an accurate potential: Direct potential ﬁt analysis

The rotational constants extracted from the observed spectra enabled us to identify the symmetry (section 4.2.3) and the derivation of the Rabi frequencies gave us understanding on how
the observed spectral widths were determined (section 4.2.5). These analyses raised a question:
was the ab initio potential suﬃciently accurate for describing all of the obtained results? The
answer was no. Both analyses showed a signature of the inaccuracy of the ab initio potential.
Fig.4.14 shows the expanded view of a comparison between observed and calculated rotational
constants. The rotational constants become smaller at higher energies because the size of molecules becomes larger at higher energies. This behavior is found in the observed values, butthey are systematically smaller than calculated values by ∼ 1 %, which are much larger thanerrors in ﬁtting. Deviations are also found in transition strengths (Fig.4.15). Observed transition strengths have a large variation among vibrational levels, which cannot be understood as avariation in Franck-Condon factors when calculations are carried out with ab initio potential.
We interpreted these deviations as indicating that the ab initio potential was deviated fromthe actual one. There are two possible reasons for these deviations. First, the potential minimum exceeds the minimum of the RKR curve by 97 cm−1 . Second, the outer turning point
is at a shorter internuclear separation than the RKR potential. The observations can not be
attributed to the small change in the vibrational quantum number, which corresponds to theenergy diﬀerence between the minima of the RKR and the ab initio potentials.
In order to obtain an accurate potential, we carried out direct potential ﬁt (DPF) analysis,
whereby a potential is iteratively modiﬁed until its eigenvalues coincide with those determinedfrom the experimental spectra [116]. The procedure for the analysis is as follows. With the aidof the phiFIT program code [117], we ﬁrst constructed an analytical Extended Morse Oscillator(EMO) potential of the (3)1 Σ+ state on the basis of (1) the RKR curve, (2) a few points fromthe inner curve of the ab initio potential, and (3) a few points around 12500 cm−1 from the outercurve of the ab initio potential. We did not use recently developed potential functions such as
the Morse/Lennard-Jones (MLJ) potential nor the More/Long-range (MLR) potential because
72

2.10
1-

)

S

+

Observed´

01 ( stnatsnoc lanoitatoR

2-

mc

1

(3)

2.08

2.06

2.04

2.02

2.00

1.98
11400

11500

11600

11700
1

Transition wavenumber from v"=91, X

11800

S (cm )+

-1

Figure 4.14: Comparison between the observed and calculated rotational constants for the(3)1 Σ+ state. The observed values are systematically smaller than the calculated values by∼1 %. Errors for observed values are 5×10−5 cm−1 , which are mainly due to ﬁtting errors.

Observed Rabi frequency (2π MHz)

15

10

5
500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)
0
0

5
10
Calculated Rabi frequency (2π MHz)

15

Figure 4.15: Comparison between the observed and calculated Rabi frequencies for the (3)1 Σ+
state. No relation is found between the observed and calculated values. The ab initio potential
does not have accuracy suﬃcient for explaining the variation in transition strengths among
vibrational levels.

73

the investigated molecular levels are deeply bound and do not require an accurate descriptionof the potential near the dissociation limit as obtained with the MLJ and the MLR potentials.
The potential curve of the ground state X 1 Σ+ was also required for the calculations. We used
an EMO potential ﬁtted to an accurate, experimentally determined potential [108]. Then, theanalytical potential of the (3)1 Σ+ state was modiﬁed to reproduce our data including both J ′ = 1
and J ′ = 3 levels by using the DPotFit program code [118]. A good convergence was achieved
when we modiﬁed the potential signiﬁcantly by manually moving the points from the ab initio
potential. This procedure was repeated until the eigenvalues of the potential were within 0.05
cm−1 of the observed levels, i.e., only 0.14% of the vibrational level spacing. Here we took carethat both vibrational energies and rotational constants were reproduced with the new potential.
The remaining deviations were presumably due to the incomplete analytical function used torepresent the potential. Table 6.1 lists the ﬁnal potential parameters. These parameters areused to represent the potential energy curve in the following form:

V (R) =
ϕ(R) =y(R, Re ) =Vmin + De (1 − e−ϕ(R)(R−Re ) )212∑
ϕi y(R, Re )ii=0
R3 − Re3.
R3 + Re3(4.2)

Fig. 4.16 shows the RKR potential, ab initio potential and modiﬁed potential. With the newpotential, rotational constants are reproduced within 0.3% of the observed values (Table 4.2).
We excluded the v ′ = 44 level from the analysis because the characteristics of this level areanomalous; a much larger transition strength, a much smaller rotational constant, and a muchlarger deviation in a vibrational energy are observed at this level. These features indicate that
this level is coupled to the (2)1 Σ+ state which is observed at ∼1 cm−1 above the v ′ = 44 level.
The most important implication of this analysis is that the variation in the transition
strengths with respect to the vibrational levels of the excited state can be accurately explainedon the basis of the new potential. Fig. 4.17 shows a plot of the Rabi frequencies derived fromthe observed spectra against those calculated from the corrected potential.
The calculated values are calibrated on the basis of our recent measurement of the transition
dipole moment between (3)1 Σ+ , v ′ = 41 and X 1 Σ+ , v ′′ = 91, 0.035(2)ea0 , determined via thedark resonance spectroscopy of the rovibrational ground-state molecules (see section 6.3).
Weakly bound molecules have more than 90 nodes in their wavefunction; hence, the FranckCondon factors are highly dependent on the wavefunction of the excited state (Fig.4.18). We
can consider the Franck-Condon factors as a kind of interferometer between two wavefunctionsof ground and excited states, which shows the accuracy of wavefunctions. Our results show that

Table 4.1: Parameters for EMO potential obtained via DPF analysis of the observed spectra.
The units are cm−1 for Vmin and De , Å for Re , and Å−1 for ϕi .
Parameter
Value
Parameter
Value
Vmin
9777.6963
ϕ5-0.804673587
De
3246.0363
ϕ63.00474008
Re
5.25904119
ϕ711.3877324
ϕ00.449794066
ϕ8-2.140717462
ϕ10.200265883
ϕ9-39.73110586
ϕ20.406840126
ϕ10-17.01400311
ϕ30.207884795
ϕ1150.93279647
ϕ4-0.476349301
ϕ1244.83137419

74

Table 4.2: Comparison between observed, ab initio, and modiﬁed values for vibrational energies(E) and rotational constants (B). ∆Eab and ∆Bab denote the diﬀerences between the observed
values and ab initio values, whereas ∆Emod and ∆Bmod denote those between the observed
values and the modiﬁed values. Vibrational energies are for the energy levels of the J ′ = 1 level.
Error bars in vibrational energies are 2 × 10−3 cm−1 , whereas those in rotational constants are
5 × 10−5 cm−1 . Observed vibrational energies are reproduced within 5 × 10−2 cm−1 , whereas
observed rotational constants are reproduced within 7 × 10−5 cm−1 . The vibrational numbering
for ab initio values is shifted by 2 because the potential minimum lies below that of the correct
potential.
v′
E (cm−1 )B (10−2 cm−1 )Obs.
∆Eab . ∆Emod (×10−3 ) Obs. ∆Bab (×10−2 ) ∆Bmod (×10−3 )41 11428.965
-1.5
2.5
2.067
1.9
-7.0
42 11466.644
-1.6
29.9
2.056
2.3
-1.7
43 11504.193
-1.8
42.6
2.05
2.4
-1.5
-1.9
164.2
1.99
7.8
52.6
44 11541.484
45 11578.919
-2.2
-11.4
2.038
2.4
-1.5
46 11616.036
-2.4
-28.0
2.037
1.8
-6.8
47 11652.982
-2.6
-37.3
2.027
2.2
-3.2
48 11689.751
-2.7
-37.5
2.017
2.5
0.3
49 11726.308
-2.7
1.2
2.008
2.9
2.5
-2.7
43.0
2.005
2.5
-1.4
50 11762.685

12.0

-1

Energy (10 cm )

11.5

3

11.0

10.5

modified
ab initio
RKR

10.0

9.5
4

5

6

7

8

Figure 4.16: Potential energy curves of (3)1 Σ+ state. The modiﬁed potential obtained in thepresent study is compared with the ab initio potential and the experimental RKR potential.
The shaded area denotes the region where depletion spectra were obtained.

75

Observed Rabi frequency (2π MHz)15
500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)
10

5

0
0

5
10
15
Calculated Rabi frequency (2π MHz)

Figure 4.17: Observed and calculated transition strengths. Observed values are determined fromthe spectra obtained via a depletion laser at 100 mW (red circle) and 500 mW (blue rectangle).
Calculations based on the modiﬁed potential can explain the variation in transition strengths
with respect to the vibrational levels in terms of the Franck-Condon factors. The point deviatedfrom other points corresponds to the v ′ = 44 level, which is expected to be mixed with otherstates.

1 +(3) Σ , v'=41
1 +X Σ , v'=91

Amplitude (arb.units)

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
4

5
6
Internuclear distance (Å)

78Figure 4.18: Wavefunctions of the v ′′ = 91 level of X1 Σ+ and the v ′ = 41 level of (3)1 Σ+ . Dueto the large number of nodes in the wavefunction of the weakly bound level, the Franck-Condon
factor between these two levels is sensitive to the inaccuracy of the potential curve of the (3)1 Σ+
state.

76

accuracies of 0.14% in vibrational level spacings and 0.3% in rotational constants are suﬃcient
to predict the Franck-Condon factors from weakly bound levels; these values are justiﬁed byconsidering the typical size of nodes in the radial wavefunction. On one hand, the weakly boundlevel v ′′ = 91 in the ground state X 1 Σ+ has an outer turning point of ∼10 Å and an innerturning point of ∼2Å in the internuclear distance. Within these two points, there are 91 nodes;
hence each node has a typical size of ∼0.1Å. Therefore, the required accuracy for representing
the wavefunction is ∼10−2 Å. On the other hand, in the present analysis, an accuracy of ∼0.3%
in rotational constants or ∼0.15% in internuclear distance is obtained for the (3)1 Σ+ statebecause the relation between the rotational constant B and the internuclear distance R is given
by B ∝ R−2 . Assuming the typical size of molecules in the v ′ = 41–50 levels of (3)1 Σ+ as
6Å, we can derive the accuracy of the modiﬁed potential in the radial direction as ∼10−2 Å;
this value is in good agreement with the required accuracy for representing the weakly boundlevel. Note that the hyperﬁne mixing of the a3 Σ+ state in the v ′′ = 91 level of X1 Σ+ is almost
negligible in the present consideration. This is because the mixing decreases the magnitude of thesinglet wavefunction by only 10 % and does not change the shape of the singlet wavefunction
signiﬁcantly. The expected errors due to the hyperﬁne mixing are thus within 10 % in the
Rabi frequency, which are smaller than the typical ﬁtting errors for the spectral widths. The
potential presented herein can enable an accurate prediction for other isotopic combinations of
KRb. Further, the present method for achieving an accurate potential and verifying its accuracy
can be extended to other molecular states that exhibit signiﬁcant spin-orbit mixing by evaluating
eigenvalues via coupled channel calculations including spin-orbit interaction.
The accurate potential enables us to predict the best intermediate for a STIRAP transfer.
Fig.4.19 shows the Franck-Condon factors between the (3)1 Σ+ state and the X1 Σ+ state. The
Franck-Condon factors for the least bound level are obtained by a coupled-channel calculation[115]. From this plot, we identiﬁed the best intermediate state as the v ′ = 41 level, which haslarge wavefunction overlaps with both the weakly bound level and the lowest rovibrational level.
The wavelengths for the Raman transition are 875 nm for the upward transition and 641 nm forthe downward transition. Another levels such as the v ′ = 38 level and the v ′ = 43 level are also
expected to be good candidates for the intermediate state.

4.2.7

Theoretical consideration on the saturation broadening in depletion
spectroscopy

We consider a general situation, as shown in Fig. 4.20, and we assume that radiative transitions
occur much faster than the time variation of the population in each molecular level because of
the photoassociative creation of molecules. We ﬁrst consider the evolution of the population in
the ground and excited states of a single molecule. The optical Bloch equation for this system
can be written as
ds
=
dt
dw
=
dt
ρf
eg
=
dt
ρf
eg ≡

s−w
2
s−w
(Γ2 + Γ1 )− 2ΩIm(ρf
eg )
2
Γ2
i
−( − i∆)ρfeg + wΩ22
ρeg e−i∆t−(Γ2 − Γ1 )(4.3)

where s = ρgg +ρee , w = ρgg −ρee , and |e > and |g > denote an excited state and a weakly boundlevel, respectively. ∆ and Ω denote the detuning frequency and the Rabi frequency, respectively.
The decay rates from the excited state to the initial weakly bound level and to other levels are
given by Γ1 and Γ2 , respectively. The photoassociation rate is denoted by AP A . The width ismuch larger than the Rabi frequency in the experiment; hence, we can assume that the time
evolution of s and w is much slower than that of ρfeg . Thus, we can set dρfeg /dt = 0 and obtain
77

1 +

0.1

Feshbach molecule to (3) Σ
1 +(3) Σ to the lowest rovibrational level

Franck-Condon factor

6
5
4
3
2

0.01
6
5
4
3
2

0.001
11200

11300

11400

11500
11600
-1
Energy (cm )

11700

11800

Figure 4.19: Franck-Condon factors between the (3)1 Σ+ state and X1 Σ+ state. The horizontal axis indicates the vibrational quantum number of the (3)1 Σ+ state. The v ′ = 36–49
levels are plotted. The Franck-Condon factors for the upward transition are obtained with acoupled-channel calculation which takes into account hyperﬁne interaction, whereas those forthe downward transition are obtained with an experimentally determined ground-state potentialof X1 Σ+ .

Excited state

PA level

∆

Ω
Γ2-Γ1

Γ1

APA

Atoms
Weakly boundlevel
Other levels

Figure 4.20: Energy levels relevant to the depletion spectroscopy of photoassociated molecules.
Weakly bound molecules are formed at a rate of APA . A laser having Rabi frequency of Ω detuned
from the resonance by ∆ excites ground state molecules. Excited molecules spontaneously decay
to the initial level and other levels at rates of Γ1 and Γ2 − Γ1 , respectively. We mainly consider
a situation where Γ2 > Γ1 is satisﬁed.

78

the following expression for ρf
eg :
ρf
eg =

iΩw
Γ2 − 2i∆(4.4)

Substituting Eq.(4.4) in Eqs.(4.3), we obtain alternative equations for s and w as
ds
=
dt
dw
=
dt

s−w
2
s−w
2Ω2 Γ2 w(Γ2 + Γ1 )− 2
2
Γ2 + 4∆2−(Γ2 − Γ1 )(4.5)
Here, we assume that the time taken by the mean value of the population ratio w/s to attaina constant value z after is greater than the typical time for radiative transitions. By using therelation w = zs, Eqs.(4.5) give following equations:
1−z
s
2
1−z
2Ω2 Γ2(Γ2 + Γ1 )
s− 2
zs
2
Γ2 + 4∆2

ds
=
dt
ds
=
z
dt

−(Γ2 − Γ1 )(4.6)
Substituting the ﬁrst equation in the second equation, we obtain a time-independent equation
for z:
4Ω2 Γ2z(4.7)(1 − z) [Γ2 (1 + z) + Γ1 (1 − z)] = 2
Γ2 + 4∆2This equation is readily solved, and it gives the following expression for z.
√(1 + k 2 )Γ22 + 2Γ1 Γ2 k − (kΓ2 + Γ1 )z=
Γ2 − Γ12Ω2k=
Γ22 + 4∆2

(4.8)

This expression is used in the following discussion. Next, we derive rate equations for the
population in the ground and excited molecular levels for the number of molecules:
dN
=
dt
Ne =
Ng =

APA − (Γ2 − Γ1 )Ne
1−z
N
2
1+z
N
2

(4.9)

where Ng and Ne denote the number of molecules in the ground and excited states, respectively,
and N = Ng + Ne is the total number of molecules. In these rate equations, a typical timescale
is of the order of 1 ms, and it is governed by APA . This is much longer than the typical timescale
for radiative transitions in most cases (≤1 µs). The time evolution of Ng is given by
dNg
1+z
1−z
=
APA −
(Γ2 − Γ1 )Ng
dt
2
2(4.10)

Thus, the solution for Ng is given by
Ng (τ ) =APA 1 + z
Γ2 − Γ1 1 − z

(
[
])
1−z
1 − exp −(Γ2 − Γ1 )τ
.
2

79(4.11)

This expression gives the line shape of a depletion spectrum for an interaction time τ . Assuming(Γ2 − Γ1 )τ ≫ 1, Eq.(4.11) gives
APA
Ng (τ ) →(4.12)
Γ2 − Γ1
on resonance (∆ = 0 and w → 0), whereas Ng at an inﬁnite detuning (∆ → ∞ and w → 1) isgiven asNg (τ ) → APA τ(4.13)
Ng at an inﬁnite detuning is much larger than Ng on resonance. Thus, the width of the spectrum
is determined by ﬁnding w such that it satisﬁes
1
Ng (τ ) = APA τ.
2(4.14)

Substituting Eq.(4.11) in Eq.(4.14) and rewriting the equation with a new variable x ≡ 1 − z(≪1), we obtain the following equation:[ x]) Γ − Γ2(21
1 − exp − (Γ2 − Γ1 )τ =
τ
x2
2

(4.15)An analytical solution of this equation is given byx==

4 + 2W(−2/e2 )(Γ2 − Γ1 )τ
1
× 3.18724...
(Γ2 − Γ1 )τ

(4.16)

where W is the Lambert W function. Thus, the parameter k in Eqs.(4.8) is given byk=

3.18724(1 − 1.59362/Γ2 τ )(Γ2 − Γ1 )τ − 3.18724

Assuming (Γ2 − Γ1 )τ ≫ 1, we obtain the following expression for the FWHM:√
2∆ ≈ 0.79Ω (Γ2 − Γ1 )τ

(4.17)(4.18)

When decays from the excited state to the initial state are negligible√ (Γ2 ≫ Γ1 ), we obtain asimple relation (4.1). It is diﬃcult to evaluate the numerical factor Γτ precisely; therefore, thewidth of a depletion spectrum cannot serve as an accurate measure of the transition strength.
However, the expression (4.1) enables us to systematically compare transition strengths fordiﬀerent vibrational levels.

4.3

The (2)3 Σ+ state: Rotational and hyperﬁne analysis

Previously, the RKR potential of the (2)3 Σ+ state of KRb was reported only for low vibrational
levels (v ′ = 0–13) in the range 9300–9900 cm−1 [119]. There have been no spectroscopic studies
in the range 10800–11500 cm−1 where the highest transition strength to the rovibrational groundstate is expected [115, 120].
In this study, we carried out depletion spectroscopy for photoassociated 41 K87 Rb moleculesin the range 10920–11440 cm−1 . The basic concept of the depletion spectroscopy of the (2)3 Σ+
state is identical to the study for the (3)1 Σ+ : a CW laser is applied to weakly bound triplet
molecules prepared via photoassociation. The resonances are detected as dips in the monitored
ion counts. However, observed spectra were much more complicated and diﬃcult to understandthan those of singlet states. We ﬁrst studied the structure of the ground-state triplet molecules,
which included large hyperﬁne structures (section 4.3.1). By scanning the Ti: Sapphire laser over80

a wide range, we found 17 vibrational levels with depletion spectroscopy for triplet molecules(section 4.3.2).
In the study for the (2)3 Σ+ state, it is imperative to identify the J ′ = 1 level of the Ω = 1
manifold in the observed spectra since only this level can be couple with the rovibrational groundstate. Another important issue is to take care not to be confused with the other triplet state,(1)3 Π, which can be distinguished from the (2)3 Σ+ state by the diﬀerence in rotational constants
and in transition strengths: the (1)3 Π state has larger rotational constants and smaller transition dipole moments [109]. For an appropriate assignment, we employed the parity-selective
spectroscopy introduced in the study of singlet states (section 4.3.3). The parity-selective spectroscopy allowed us to assign the observed spectra deﬁnitely. We ﬁt the spectra with a simple
analytical model and obtained spectroscopic constants for expressing the spectra. By comparing the obtained rotational constants with those calculated from ab initio potential, we could
conﬁrm that observed states had a symmetry of (2)3 Σ+ (section 4.3.4).
As an advanced study, we investigated the hyperﬁne structures observed in the low J ′ levelsof the Ω = 1 manifold (section 4.3.5). A simple theoretical model was introduced, which canclearly understand the observed structures as hyperﬁne interaction between electronic spin andthe nuclear spin of 87 Rb (the Fermi contact interaction). Further, we studied the hyperﬁne
structures of the (2)3 Σ+ state with SpIDR spectroscopy, whose detail will be introduced in
the section 6.2. The precision spectrum with SpIDR spectroscopy enabled us to observe thehyperﬁne structure arising from the nuclear spin of 41 K, which has a much smaller nuclear
dipole moment than 87 Rb. The theoretical framework required in the study for the (2)3 Σ+ stateis given in the section 4.3.7.

4.3.1

Triplet molecules in the ground state

In the present study, photoassociated triplet molecules are ionized by a pulsed laser in the
red region (16100–16400 cm−1 ). The ionization spectrum in this range (Fig.3.17) appears toindicate that the v ′′ = 22 level is the most abundant and is suited as an initial state for depletion
spectroscopy. However, instead of the v ′′ = 22 level or the v ′′ = 23 level, we used the v ′′ = 21level in the present study for the following two reasons. First, the pulsed laser for ionizing
the triplet molecules in the v ′′ = 22 level to the (4)3 Σ+ also ionizes singlet molecules in thev ′′ = 91 level of X1 Σ+ to the (4)3 Σ+ state. We found this anomalous behavior in the depletion
spectroscopy for the (3)1 Σ+ state. Fig.4.21 shows a comparison of the depletion spectra forthe v ′′ = 91 level of X1 Σ+ and for the v ′′ = 22 level of a3 Σ+ . We observed dips at the same
wavenumbers, but with half depths, indicating that approximately half of the ion counts formonitoring the v ′′ = 22 level of a3 Σ+ arises from the v ′′ = 91 level of X1 Σ+ . The v ′′ = 22 levelof a3 Σ+ lies just below the v ′′ = 91 level of X1 Σ+ by 0.07 cm−1 , which is roughly the same as
the linewidth of the pulsed laser. In addition, the v ′′ = 91 level has a 10 % of triplet component
due to the coincidental proximity of the v ′′ = 22 level of a3 Σ+ . Hence, the singlet moleculesin the v ′′ = 91 level are readily ionized to the triplet excited state, (4)3 Σ+ . In order to ensurethat we observe transitions from triplet molecule, we decided not to use the v ′′ = 22 level. Notethat the opposite process, in which triplet molecules are ionized to a singlet excited state, issuppressed. The v ′′ = 22 level splits into three hyperﬁne manifolds separated by ∼0.2 cm−1 .
The uppermost manifold is the most close to the v ′′ = 91 level of X1 Σ+ , but it includes a verysmall fraction of singlet component (Fig.3.18). The singlet component is dominantly includedin the lower manifold, which are separated by more than 0.2 cm−1 from the v ′′ = 91 level of
X1 Σ+ . Thus, this problem did not arise in the study for the (3)1 Σ+ state in the section 4.2.
The second reason why we selected the v ′′ = 21 level for the present study is also related tothe hyperﬁne structure. For observing a clear depletion spectrum, the initial state should not be
distributed over a large number of rotational and hyperﬁne levels. Although we also attempted
to observe a depletion spectrum for the v ′′ = 23 level of a3 Σ+ , the spectrum was distributed overmultiple levels and was hard to locate peaks. In general, hyperﬁne mixing is more signiﬁcant81

Spectrum taken with singlet molecules (v''=91)
Spectrum taken with triplet molecules (v''=22)

Ion counts (arb.units)

35

30

25

20

11689.611689.711689.811689.9-1
Depletion wavenumber (cm )11690.011690.1

Figure 4.21: Comparison of the depletion spectra obtained with the v ′′ = 91 level of X1 Σ+ andthe v ′′ = 22 level of a3 Σ+ . The singlet molecules are detected at 16543 cm−1 whereas triplet
molecules are detected at 16232 cm−1 . The excited state is the v ′ = 48 level of (3)1 Σ+ . The
same structure appears at the exactly same wavenumber, indicating that a part of the ion counts
in the v ′′ = 22 level of a3 Σ+ is composed of singlet molecules in the v ′′ = 91 level.
for more weakly bound levels. In this point of view, we should take a depletion spectrum withdeeply bound molecules. For these reasons, we decided to use the v ′′ = 21 level of a3 Σ+ as an
initial state for the present study.
The three hyperﬁne manifolds in the v ′′ = 21 level were experimentally observed via depletion
spectroscopy. We found that the distribution among the hyperﬁne manifolds was dependent on
photoassociation levels although we do not have a good explanation on this behavior. Fig.4.22
shows a comparison of depletion spectra taken with diﬀerent photoassociation levels. The shift
in the spectra indicates the hyperﬁne structures in the v ′′ = 21 level. These three levels canbe labeled with T ′′ = IRb + S as T ′ = 5/2, 3/2, and 1/2. Here we neglect coupling between
a rotational angular momentum and the other angular momenta because the rotational energy
of weakly bound molecules (∼0.02 cm−1 ) is much smaller than the hyperﬁne splitting of 87 Rb(> 0.2 cm−1 ). Hence, the rotational number of these levels is labeled by N ′′ = R′′ + L′′ (seesection 4.3.5 for further discussions on how we determined the labeling). The observed splitting
between T ′ = 5/2 and T ′ = 3/2 was 0.134(1) cm−1 and the splitting between T ′ = 3/2 andT ′ = 1/2 was 0.092(1) cm−1 . The errors are ﬁtting errors due to the broad spectral width.
These values are in good agreement with a coupled channel calculation which gives ∼0.13 cm−1and ∼0.1 cm−1 for each separation, respectively. The small hyperﬁne splitting induced by the
nuclear spin of 41 K is expected to be of the order of 10 MHz and is not resolved in depletion
spectra. Thus, the observed dips in the depletion spectra presented in this chapter should beinterpreted as those integrated over the small hyperﬁne structure. The hyperﬁne structure dueto the nuclear spin of 41 K will be clearly resolved with newly developed SpIDR spectroscopy inthe section 4.3.5.
For the present study, we used the T ′′ = 5/2, N ′′ = 2, v ′′ = 21 level of a3 Σ+ as an initialstate since only this level could be prepared without distributing over multiple hyperﬁne levels.
The other two levels, T ′′ = 3/2 and T ′′ = 1/2, were always accompanied with each other andwere not suited for investigating the structure of the excited states. The binding energy of
the T ′′ = 3/2, N ′′ = 2, v ′′ = 21 level was determined to be -16.478(1) cm−1 with respect tothe atomic threshold FK =1+FRb =1 by comparing a photoassociation spectrum and a depletion
spectrum for the photoassociation level (Fig.4.23). In this depletion spectrum, molecules were82

-1

PA at 12561.45 cm-1
PA at 12572.10 cm-1
PA at 12570.13 cm

Ion counts (arb.units)

20

15

10

5

0
11093.011093.5

11094.011094.5-1
Depletion wavenumber (cm )11095.0

Figure 4.22: Depletion spectra taken with diﬀerent photoassociation levels. The distribution
among the hyperﬁne structures is dependent on photoassociation levels. The rotational numberof the photoassociation levels are all J ′ = 1.
formed via the photoassociation level at 12572.10 cm−1 and were mainly in the T ′′ = 3/2 level.
We observed four transitions to the J ′ = 1–4 levels because the selection rule is not governed by
J = L + R + S but by T = J + IRb , which includes the nuclear spin of 87 Rb (further discussions
will be given in the section 4.3.5). The obtained binding energy is in reasonable agreement witha calculation for the v ′′ = 21, N ′′ = 2 level without hyperﬁne structures, which gives -16.4404
cm−1 . By subtracting the energy diﬀerence in the T ′′ = 3/2 level and the T ′′ = 5/2 level, we
determined the binding energy of the T ′′ = 5/2, N ′′ = 2 level to be -16.344(1) cm−1 .

4.3.2

Depletion spectroscopy with triplet molecules

The depletion spectroscopy was carried out by scanning the Ti: Sapphire laser over a wide range(∼10 cm−1 for ﬁnding a single vibrational level). Although we had an ionization spectrum inthe range 11350–11750 cm−1 obtained in the study for the (3)1 Σ+ , it was diﬃcult to identify the(2)3 Σ+ in the ionization spectrum: we could not ﬁnd a clear signature of the vibrational progressions in both the a3 Σ+ and the (2)3 Σ+ states. We guess that this is because the vibrational
spacing of the (2)3 Σ+ state is not constant due to strong spin-orbit mixing. With a high power(∼100 mW), we coincidently found strong transitions at ∼11657 cm−1 , for which the spectral
width of each rotational line was broadened to a few GHz due to the large transition dipole moment between a3 Σ+ and (2)3 Σ+ (∼5 ea0 [109]). Fig.4.24 shows a typical broadened spectrum.
The broadening was a good measure for distinguishing the (2)3 Σ+ state and the (1)3 Π statebecause the transition dipole moment between the a3 Σ+ state and the (1)3 Π state is ∼1 ea0 .
We searched for lower vibrational levels one by one down to 10920 cm−1 assuming a vibrational
spacing of ∼30 cm−1 , which is predicted with an ab initio potential. For some vibrational levels,
we had to scan for more than 10 cm−1 . The vibrational progression was quite irregular at ahigher energy than 11500 cm−1 . In the following studies, we focused on the vibrational levels ofv ′ = 39–55 in the range 10920–11440 cm−1 which is expected to have a large transition dipole
moment with the rovibrational ground state.

83

Ion counts (arb.units)

14
12

Depletion-1
Photoassociation, shifted by +16.478 cm

10
8
6

J'=1

4

J'=2

J'=3

2

J'=4

0
12577.912578.012578.1-1
Depletion wavenumber (cm )Figure 4.23: Comparison between a photoassociation spectrum and a depletion spectrum forthe same level. The initial state for the depletion spectrum is mainly in the T ′′ = 3/2 level. The
small dips appearing in the left side of the principal dips arise from the T ′′ = 1/2 level.

Ion counts (arb.units)

16
12
8
4
0
11402.011402.5
11403.0-1
Depletion wavenumber (cm )11403.5

Figure 4.24: Depletion spectrum taken with a high power (100 mW). The spectral width isbroadened to a few GHz. This broadening is a good measure of observing the (2)3 Σ+ staterather than (1)3 Π.

84

In this study, it is also imperative to distinguish between the Ω = 0 state and the Ω = 1
state. Roughly speaking, a single vibrational level of the (2)3 Σ+ splits into the two states, Ω = 1
and Ω = 0, due to spin-orbit interaction. In the early stage of this study, we could not deﬁnitely
identify these two states because of the lack of the knowledge on the rotational structures ofthese states. Although later we could assign the rotational structure, what we found at this point
was a set of manifolds including several lines, which we considered as Ω = 1 and Ω = 0. The
separation between these two states was between 0.5 and 8 cm−1 , which was strongly dependent
on vibrational levels. For some vibrational levels, either one was not found because of the large
separation. The two manifolds are shown in Fig.4.22. Understanding on these structures will
be discussed in detail in the section 4.3.4.

4.3.3

Parity-selective spectroscopy with singlet molecules

The observations of more than 20 vibrational levels with an expected broad width and a vibrational spacing of 20–35 cm−1 appeared to indicate that these states were the (2)3 Σ+ state.
However, it was necessary to extract rotational constants for deﬁnitely identifying the symmetry. The extraction of a rotational constant from the observed spectra was not as easy as thestudy for the (3)1 Σ+ state because there were much more lines, which should be understood byﬁtting with an analytical formula. We had four questions at this point. First, there are four
principal lines in the left state in Fig.4.22. This number is larger than expected with an ordinary
molecular transition because the selection rule on J should allow only three transitions. Second,
there should be some missing lines in the right manifold in Fig.4.22 if we assume an ordinary
J(J + 1) progression and a typical rotational constant of 0.02–0.03 cm−1 . The ratio of two
separations among the three lines is ∼2.1, implying that J ′ = 0, 2, 4 are observed and J ′ = 1, 3
are missing. We were not sure that we could assume missing lines without observing them.
Third, a small splitting was observed in the leftmost line of the left manifold in Fig.4.22. The
splitting is almost the same as expected rotational constants and is too small to be understoodas rotational splitting. These questions made it diﬃcult to assign the observed structures.
In order to clarify what we observed, we employed parity-selective spectroscopy, in which
depletion spectra for diﬀerent rotational levels are obtained with molecules in a diﬀerent rotational level (section 4.2.3). Just by changing the rotational level for photoassociation, we
could observe missing lines in the study for the (3)1 Σ+ (section 4.2.3). We ﬁrst attempted to
implement this spectroscopy with triplet molecules by using a diﬀerent rotational level (J ′ = 2)
for photoassociation. However, we obtained an unclear spectrum with a large number of small
peaks, which were diﬃcult to locate. Our guess for this behavior is as follows. The change inthe rotational level for photoassociation does not change the rotational level of the ground state,
but changes the distribution over the hyperﬁne levels of the ground state. The selection rule on
J does not hold for triplet molecules due to the strong hyperﬁne interaction.
Here we focused on the v ′′ = 91 level of X1 Σ+ , which has a 10% of triplet component due
to the hyperﬁne mixing of the v ′′ = 22 level of a3 Σ+ . The singlet molecules are ideal for the
spectroscopy of excited states because they have a deﬁnite parity and a small hyperﬁne structure.
In principle, 10 times larger power than for triplet molecules is expected to be enough to excite
these molecules to the (2)3 Σ+ state. Fig.4.25 shows a comparison of depletion spectra takenwith triplet molecules in the v ′′ = 21 level and singlet molecules in the v ′′ = 91 level. Here thesinglet spectrum is shifted by +3.89 cm−1 , which corresponds to the diﬀerence in binding energy
between these two levels. The spectrum for singlet molecules was obtained with a 10 times larger
power (100 mW) than that for triplet molecules. The fact that nearly identical spectra wereobserved with the diﬀerent initial states indicates that singlet molecules can be used for the
spectroscopy of the (2)3 Σ+ state. In addition, there is another important implication that the
triplet molecules have the same parity as the singlet molecules in the J ′′ = 2, v ′′ = 91 level. In
this way, we could determine the parity of the triplet molecules to be even.
With singlet molecules, the parity-selective spectroscopy was quite successful. Fig.4.26 shows

85

3 +

v=21 of a Σ1 +-1
v=91 of X Σ , shifted by +3.89cm

Ion counts (arb.units)

15

10

5

0
10993.6

10993.8

10994.010994.2 10994.4 10994.6-1
Depletion wavenumber (cm )10994.8

10995.010995.2

Figure 4.25: Comparison of depletion spectra taken with triplet molecules in the v ′′ = 21 leveland singlet molecules in the v ′′ = 91 level. The spectrum for singlet molecules is shifted by+3.89 cm−1 , which is a diﬀerence in binding energy between the singlet level and the triplet
level.
the comparison of two depletion spectra obtained with diﬀerent rotational levels (J ′ = 0 and
J ′ = 1) for photoassociation. The photoassociation with the J ′ = 1 level yields molecules in
the J ′′ = 2 level with even parity, whereas the photoassociation with the J ′ = 0 level yields
molecules in the J ′′ = 1 level with odd parity. The spectrum with the photoassociation level ofJ ′ = 0 is shifted by 0.0307 cm−1 , which corresponds to the energy diﬀerence between the J ′′ = 1
level and the J ′′ = 2 level.
The parity-selective spectroscopy enabled us to observe all possible levels. The previously
missing lines were also observed. We obtain ﬁve important implications from this ﬁgure. First,
the excited state has a deﬁnite parity as in the ground state. The upper line indicates levels
with odd parity and the lower line indicates levels with even parity. Second, the rotational
structure of the left manifold is labeled as J ′ = 0, 2, 4 for the even levels and J ′ = 1, 3, 5 forthe odd levels. Third, the right manifold has nearly degenerate two levels in each rotational
level, although the separation between these levels becomes larger at a higher rotational level.
Assuming an ordinary J(J + 1) progression, we can label these rotational levels as J ′ = 1, 2, 3, 4.
Fourth the left manifold corresponds to the Ω = 0− state since it begins with J ′ = 0 and has
a parity of (−1)J+1 . By contrast, the right manifold is the Ω = 1 state since it begins with
J ′ = 1. Fifth, the ﬁne structures observed in the J ′′ = 1, 2 levels of Ω = 1 are not associated torotational progression, but is associated to another kinds of interaction. Later we will discussthese structures in detail (section 4.3.5).
The parity-selective spectra with singlet molecules were obtained for approximately halfof the vibrational levels found in the section 4.3.2; once we found how we should assign the
spectra, we did not require taking the same spectra as obtained with triplet molecules. For
some vibrational levels of the (2)3 Σ+ state, the other states were coincidently found near the(2)3 Σ+ state. Fig.4.27 shows the (2)1 Σ+ state found between the Ω = 1 state and the Ω = 0
state of the (2)3 Σ+ state. In this case, the selection rule is the same as the study for the (3)1 Σ+
state. Hence, we observed only two transitions, J ′ = 1 ← J ′′ = 2 and J ′ = 3 ← J ′′ = 2 forthe odd spectrum and J ′ = 2 ← J ′′ = 1 and J ′ = 0 ← J ′′ = 1 for the even spectrum. The
derived rotational constant of the (2)1 Σ+ state was close to the ab initio value for the level near

86

10

J'=1 PA (J''=2, v''=91)-1
J'=0 PA (J''=1, v''=91), shifted by 0.0304cm (=4xB'')-

Ion counts (arb.units)

8

Ω=0

Ω=1

6

4

2

0
10990.010990.5-1
Depletion wavenumber (cm )10991.0

Figure 4.26: Parity-selective spectroscopy for the (2)3 Σ+ state obtained with singlet moleculesin the J ′′ = 1, v ′′ = 91 level. For comparison, the spectrum taken with singlet molecules in
the J ′′ = 2, v ′′ = 91 level is also given. Previously missing lines are clearly observed just bychanging the rotational state of the photoassociation level.
11189 cm−1 , but that for the level near 11090 cm−1 was largely deviated from the ab initio
value. These observations indicate that the former level has a strong (2)1 Σ+ character, whereas
the latter level has a strong (1)3 Π character due to spin-orbit mixing. For the latter case, thelabeling of the state should be Ω = 0+ instead of (2)1 Σ+ .
Fig.4.28 shows the Ω = 1 state of the (1)3 Π state found near the (2)3 Σ+ state. Although therotational structure of the (1)3 Π state is similar to that of the Ω = 1 state of the (2)3 Σ+ state,
the structure spreads over ∼0.5 cm−1 , which is larger than the typical value of 0.4 cm−1 for the(2)3 Σ+ state. This is because the (1)3 Π state has larger rotational constants (∼0.025 cm−1 ) thanthe (2)3 Σ+ state (∼0.019 cm−1 ). Similarly, we found that the left manifold in the Fig.4.27(b)is not the (2)3 Σ+ state but the (1)3 Π state because it has a larger rotational structure than the(2)3 Σ+ state.

4.3.4

Understanding observed spectra with a three-level model

The parity-selective spectroscopy enabled us to deﬁnitely assign the observed rotational structures. The important information obtained from the rotational structures is rotational constants,
which are dependent on symmetries and vibrational levels. In the study on the (3)1 Σ+ state,
rotational constants were readily obtained just by dividing the separation between the J ′ = 3
peak and the J ′ = 1 peak by 10. However, the observed spectra for the (2)3 Σ+ include numerous
peaks, which make it diﬃcult to extract rotational constants in a simple manner.
For extracting a rotational constant from a spectrum for a single vibrational level, which
includes the rotational structures of both the Ω = 1 state and the Ω = 0 state, we employed athree-level model for a 3 Σ+ molecule [121, 122], which is brieﬂy given in the section 4.3.7. We
omitted the centrifugal distortion term −Dv′ R4 in the present study because it gives a muchsmaller correction for low J ′ levels than the rotational energy BJ 2 . The procedure for obtaining
rotational constants is as follows. First, the peaks included in the depletion spectra for both
triplet molecules and singlet molecules are extracted by ﬁtting with a Lorentzian function. Thenthese peaks are further ﬁtted with the formula in the three-level model (4.22), which has ﬁtting87Ion counts (arb.units)

30

J'=1 PA (J''=2, v''=91)
J'=0 PA (J''=1, v''=91)-1(a) At 11089 cm

25
20
15
10
5
0
11089.511090.011090.5-1
Depletion wavenumber (cm )Ion counts (arb.units)

20

J'=1 PA (J''=2, v''=91)
J'=0 PA (J''=1, v''=91)

11091.0-1(b) At 11090 cm

15
10
5
0
11188.011188.511189.011189.5-1
Depletion wavenumber (cm )11190.0

Figure 4.27: The (2)1 Σ+ state observed between the two manifolds of the (2)3 Σ+ state, Ω = 0
and Ω = 1. (a) At approximately 11189 cm−1 near the v ′ = 47 level of (2)3 Σ+ . (b) At
approximately 11090 cm−1 near the v ′ = 44 level of (2)3 Σ+ . The rotational constants for theselevels are 0.0196(1) cm−1 and 0.0229(1) cm−1 , respectively.

88

Ion counts (arb.units)

20

J'=1 PA (J''=2, v''=91)
J'=0 PA (J''=1, v''=91)

15
10
5
0
10954.5

10955.0 10955.5 10956.0 10956.5-1
Depletion wavenumber (cm )10957.0

Figure 4.28: The (1)3 Π state observed near the (2)3 Σ+ state. The three manifolds indicateΩ = 0 of (2)3 Σ+ , Ω = 1 of (2)3 Σ+ , and Ω = 1 of (1)3 Π from left to right, respectively.
parameters of electronic energy E, separation between Ω = 0 and Ω = 1 λ, rotational constantB, and spin-rotation coupling γ. Table 4.3 tabulates the ﬁtting parameters obtained in thepresent analysis. Fig.4.29 shows examples of the ﬁtted spectra for diﬀerent vibrational levels.
In general, there are two types of spectra depending on the sign of λ. These two spectra are
equally well understood with the three-level model.
Here we can systematically compare the rotational constants extracted from the observed
spectra with those obtained from ab initio potentials (Fig.4.30). Most of the observed values
were close to the calculated values for the (2)3 Σ+ state, indicating that observed states has
a dominant character of (2)3 Σ+ . The deviations from the calculated values are interpreted
as a spin-orbit mixing of the (1)3 Π state into the (2)3 Σ+ state. The spin-orbit interaction in
deeply bound KRb molecular states dissociating to the 4S+5P threshold is ∼100 cm−1 , whichis larger than the typical spacing between two vibrational levels of the (2)3 Σ+ state and the(1)3 Π state (a few tens of cm−1 ). As a result, the (2)3 Σ+ state and the (1)3 Π state are strongly
coupled with each other. For some vibrational levels, the spin-orbit coupling results in a nearly
equal admixture of the (2)3 Σ+ state and the (1)3 Π state, thereby making it diﬃcult to deﬁne adominant symmetry in Hund’s case (a). In this study, we assigned symmetries in Hund’s case (a)to the observed levels and introduced spin-orbit interaction as a perturbation which splits eachlevel. However, some levels are not well described with this description because this descriptionis accurate in the limit of weak spin-orbit interaction. When spin-orbit interaction is larger thanelectronic interaction, molecular states should be labeled only with Ω (Hund’s case (c)) [123].
Here we can consider that potentials in Hund’s case (a) are coupled by spin-orbit interaction
to form new potentials in Hund’s case (c). KRb can be considered to be in between these twolimits. Thus, the observed rotational constants are in some cases largely deviated from those forthe (2)3 Σ+ state. The observed values are always between the calculated values for the (2)3 Σ+
state and the (1)3 Π state. This fact indicates that the deviations should be interpreted as theamount of the (1)3 Π state in the observed states. The periodic behavior in the observed values
can be understood from the diﬀerence in the vibrational spacing of the (2)3 Σ+ and the (1)1 Πstate: these states become close with an approximately 120 cm−1 spacing since the (2)3 Σ+ state
has a spacing of ∼ 30 cm−1 while the (1)3 Π state has a spacing of ∼ 40 cm−1 .
In these discussions, we do not have to take into account the (1)1 Π state, which is also
included in the Ω = 1 state via spin-orbit interaction, because the spin-orbit mixing of the(1)1 Π state into the (2)3 Σ+ state is known to be smaller than a few % in this region [115, 120].
89

1 +

Ion counts in v"=91, X Σ (arb.units)(a) λ > 0

10
8
6
4
2
0

1 +

Ion counts in v"=91, X Σ (arb.units)10990.010990.5
10991.0-1
Depletion wavenumber (cm )(b) λ < 0

35
30
25
20
15
10
5
0
11218.011218.511219.0 11219.5 11220.0 11220.5-1Depletion wavenumber (cm )11221.0

Figure 4.29: Depletion spectra of the (2)3 Σ+ state taken with molecules in the v ′′ = 91 level ofX 1 Σ+ and ﬁtting to the spectra based on a three level model. (a) The spectrum for the v ′ = 41
level. The Ω = 1 state lies above the Ω = 0 state. (b) The spectrum for the v ′ = 48 level. TheΩ = 1 state lies below the Ω = 0 state. The red upper line indicates the spectrum taken withmolecules in the J ′′ = 2 level and shows molecular levels with an odd parity. The blue lower
line indicates the spectrum taken with molecules in the J ′′ = 1 level and shows molecular levels
with an even parity. The odd parity levels are shown in red lines while the even parity levels
are shown in blue lines. The spectrum taken with the J ′′ = 1 level is shifted by 0.03069 cm−1
which is 4 times the rotational constant of the v ′′ = 91 level..

90

3.2

3 +

(2) Σ

-2

-1

Rotational constant (10 cm )

3.0
1 +

2.8

(2) Σ

1 +

(3) Σ

2.6

3

(1) Π
1

(1) Π
Observed

2.4
2.2
2.0
1.8
10900

11000
11100
11200
11300
11400
1 +
-1
Transition wavenumber from v"=91 of X Σ (cm )Figure 4.30: Comparison between rotational constants calculated from an ab initio potential
and those extracted from observed spectra. Observed values are close to the ab initio values ofthe (2)3 Σ+ , which shows the strong (2)3 Σ+ character of observed states. The deviations fromthe ab initio values indicate spin-orbit mixing of the (1)3 Π state.

4.3.5

Hyperﬁne structures

Fitting based on the three level model gave us a systematic understanding on the observed
spectra, indicating that the observed states are dominated by the (2)3 Σ+ state. However, weobserved signiﬁcant deviations from the three level model in the low-J ′ (J ′ < 3) levels of theΩ = 1 manifold for all vibrational levels. Each rotational level should consists of a single level,
but we observed multiple levels for a single rotational level. Understanding on the observed
splitting for the J ′ = 1 level is of particular importance because this level couples weakly boundmolecules with the rovibrational ground state.
The observed splitting is of the order of 1 GHz, which is close to the hyperﬁne splitting of
87 Rb in the excited 5P state. Therefore, we interpreted the splitting as a hyperﬁne structure
and introduced nuclear spins into discussions. The theoretical framework for the rotational
structure including nuclear spins is given in the section 4.3.7. In this model, the Fermi contact
interaction terms are added to the three-level model. Here we take into account only nuclear
dipole moments and neglect quadrupole moments because the contribution of the quadrupole
moments is smaller than that of the dipole moments by one order of magnitude. Detailed
discussions on the hyperﬁne structure of triplet molecules are found in [124]. Although the
Hamiltonian for the present model is simple, there are no simple analytical expression for the
eigenvalues of the Hamiltonian. Hence, all the eigenvalues given in this section were obtained
numerically via a MATLAB code. The numerical evaluation of the eigenvalues is beneﬁcial for
identifying the parity of the eigenstates: we can readily identify the parity as a diﬀerence in theform of eigenvectors (see section 4.3.7 for details).
Fig.4.31 shows a calculated rotational and hyperﬁne spectrum for the Ω = 0 state and theΩ = 1 state. In these calculations, we only consider the nuclear dipole moment of 87 Rb because
it is much larger than that of 41 K. These calculated spectra give a qualitative explanation for thefeatures appearing in all the observed spectra. First, each rotational level splits into multiple

91

levels due to hyperﬁne interaction. Second, the splitting in the Ω = 0 state is small, whereas the
splitting in the Ω = 1 state is large. Third, the splitting in the Ω = 0 state is larger for larger
J ′ , whereas the splitting in the Ω = 1 state is approximately identical for each rotational level.
Fourth, the hyperﬁne splitting arises from coupling between rotational angular momentum andthe nuclear spin of 87 Rb, which have almost the same magnitude of energy. Thus, the hyperﬁnesplitting in the Ω = 1 state should be labeled with T ′ = J ′ + IRb . The three lines in the J ′ = 1
level of the Ω = 1 state can be labeled as T ′ = 5/2, 3/2, 1/2. In this way, the observed deviationsin the low J ′ levels are understood as hyperﬁne structures due to the nuclear spin of 87 Rb.
We can check whether the calculations quantitatively agree with the observed spectra.
Fig.4.32 shows a comparison between the observed spectra and the calculated spectra. Although
it is diﬃcult to see if all the observed levels are in agreement with the calculated levels due tonearly degenerate multiple levels and broad spectral widths, we ﬁnd a fairy good agreement between observations and calculations. The diﬀerence between odd-parity levels and even-parity
levels in the J ′ = 3 level is also reproduced. Thus, we concluded that the observed spectra were
well understood only by adding the Fermi contact interaction of 87 Rb to the three-level model.
The above discussions give a good understanding on the observed structures. However, one
question still remains unsolved: why such many lines are observed. In a textbook explanation,a selection rule for a molecular transition is given as ∆J = 0, ±1 which allows at most threerotational transitions. Nevertheless, we observed more than three rotational transitions in thepresent study. Here we summarize the observations for both ground and excited states.
1. In the spectra taken with triplet molecules in the v ′′ = 21 level, the J ′ = 0, 2, 4 levels wereobserved for the Ω = 0 state, whereas the J ′ = 1, 2, 3, 4, 5 levels were observed for Ω = 1
state.
2. The spectra taken with singlet molecules in the J ′′ = 2 level were almost identical withthose obtained with triplet molecules. However, the leftmost line observed for singlet
molecules disappeared in the spectra for triplet molecules.
3. In the spectra taken with singlet molecules in the J ′′ = 1 level, the J ′ = 1, 3 (and 5 weakly)
levels were observed for the Ω = 0 state, whereas the J ′ = 1, 2, 3, 4 (and 5 weakly) levels
were observed for the Ω = 1 state. The levels in the Ω = 1 state were slightly deviatedfrom those in the spectra obtained with singlet molecules in the J ′′ = 2 level.
These observations are clearly explained when we consider the parity and the good quantum
numbers for describing molecular states. We ﬁrst consider the parity of ground and excitedstates. In general, the rotational levels of the 1 Σ+ state and the 3 Σ+ state have parities asshown in Fig.4.33 [123]. With an electric dipole radiation (E1), transitions between two levels
with opposite parities are allowed. Thus, the parities of the levels relevant to the present study
are given as follows.
1. The parities of singlet molecules in the J ′′ = 1 level and those in the J ′′ = 2 level are odd
and even, respectively.
2. The three levels in the v ′′ = 21 level of a3 Σ+ have all even parity because they have thesame parity as the J ′′ = 2 level of 1 Σ+ .
3. The J ′ = 0, 2, 4 levels of Ω = 0 have odd parity because they are observed in the spectrawith singlet molecules in the J ′′ = 2 level. By contrast, the J ′ = 1, 3 levels of Ω = 0 haveeven parity. These interpretations agree with the general knowledge that the Ω = 0− state
has a parity of (−1)(J+1) where +1 indicates even and −1 indicates odd.
4. Each rotational level of Ω = 1 consists of two closely-lying (but non-degenerate) levels
with opposite parities. These two levels are independently observed in the spectra for the
J ′′ = 1 level of X1 Σ+ and those for the J ′′ 2 level of X1 Σ+ .
92

(a) Spectrum forΩ = 0 and Ω = 1

4

Ω=0

-1

Energy (cm )

3

2

1

Ω=1

0
0

2

(b) Spectrum near J'=1, 2 of

6

8

Ω=1

0.10

-1

Energy (cm )0.15

4
Total angular momentum F

0.05

0.00
0

2

4
Total angular momentum F

68

Figure 4.31: Calculated rotational spectrum including hyperﬁne interaction for the Ω = 0 state
and the Ω = 1 state of 3 Σ. (a) The entire spectrum including both Ω = 0 and Ω = 1. (b)Spectrum near the low rotational levels of Ω = 1. The Fermi contact constant for 87 Rb is set
as 0.023 cm−1 , whereas that for 41 K is set as 0. The Ω = 0 state has small hyperﬁne splitting,whereas the Ω = 1 state has large hyperﬁne splitting.

93

J'=3
6

T'=5/2
T'=3/2
T'=1/2

4
2

Ion counts (arb.units)

015

Total angular momentum F

J'=2

J'=1

10
5
0
11121.911122.011122.1-1
Depletion wavenumber (cm )Figure 4.32: Comparison between the observed spectra of the v ′ = 45 level (two lower curves)and the calculated spectra (upper bars). The odd parity levels are shown in red lines, whereas
the even parity lines are shown in blue lines. For comparison, the spectrum taken with triplet
molecules are also shown in black line. The Fermi contact constant for 87 Rb is set to 0.023 cm−1 .
The calculation agrees qualitatively with observed spectra and allows us to label each line.

K

0

1

2

3

4

5

0

1

2

3

4

5

1

0 1 2

1 2 3

2 3 4

3 4 5

4 5 6

1Σ +

J
3Σ +

J

Figure 4.33: Parity of the rotational levels of 1 Σ+ and 3 Σ+ states. The parity is deﬁnitelydetermined by the angular momenta J = L + R + S and N = R + L. The plus sign indicateseven parity and the minus sign indicates odd parity.

94

Next we consider the good quantum number for the ground and excited states. In the
ground state, weakly bound molecules have a low rotational energy of ∼0.01 cm−1 . This issmaller than electronic interaction and the hyperﬁne interaction of 87 Rb. Hence, we can thinkof the rotational angular momentum as decoupled from electronic and nuclear spins. The goodquantum numbers for describing weakly bound molecules are T ′′ = S ′′ + IRb and N ′′ (= R′′ ).
The singlet molecules used for spectroscopy have a deﬁnite spin of T ′′ = 3/2 and a rotational
number of J ′′ (= N ′′ ) = 2. The triplet molecules have three spin components, T ′′ = 5/2, 3/2, 1/2
as observed experimentally (section 4.3.1). In the present study, molecules in the T ′′ = 5/2 areused for spectroscopy. Regarding the rotational number, the triplet molecules can be N ′′ = 0 or
N ′′ = 2 because the photoassociation level is J ′ = 1. In contrast to the singlet case, we cannot
discriminate these two rotational levels as a diﬀerence in a depletion spectrum because only oneof these two levels was observed. We determined the rotational number of the triplet molecules
as N ′′ = 2 based on the coupled channel calculation. Note that J ′′ is not a good quantum
number for weakly bound triplet molecules because of the strong hyperﬁne interaction. In this
discussion, we omit the nuclear spin of 41 K because the magnitude of the nuclear dipole momentof 41 K is much smaller than that of 87 Rb.
The situation changes in the excited states. Even without the hyperﬁne interaction, therotational angular momentum is coupled to other angular momenta, electronic orbital angularmomentum and electronic spin, via spin-rotation interaction and spin-orbit interaction. Hence,
the hyperﬁne interaction induces a signiﬁcant coupling between rotational angular momentum
and the nuclear spin of 87 Rb. The good quantum number for describing the excited states isgiven by T ′ = L′ + S ′ + R′ + IRb (= J ′ + IRb ). The quantum number T should be considered
as a kind of spin which is shared between the ground state and the excited state and governs
the selection rule as ∆T = 0, ±1.
Now we can consider which transition should be observed in terms of parity and T . For
this purpose, Fig.4.31 serves as a good guide. In the spectra with triplet molecules, we shouldobserve the T ′ = 7/2, 5/2, 3/2 levels with odd parity, which indicates the J ′ = 0, 2, 4 levels
for the Ω = 0 state and the J ′ = 1–5 levels for the Ω = 1 state. In the spectra with singlet
molecules, we should observe the T ′ = 5/2, 3/2, 1/2 levels, which indicates the J ′ = 0, 2, and 4
levels (odd parity) and the J ′ = 1, 3 levels (even parity) for the Ω = 0 state and the J ′ = 1–4
levels for the Ω = 1 state. Fig.4.34 shows the summary of the observed levels. These are ingood agreement with our observations. The fact that the spectral widths for the Ω = 1 state
become narrower for larger J ′ is also understood as a decrease in the number of hyperﬁne levels
accessible from the T ′′ = 5/2 level. In addition, we can see a clear signature of the diﬀerence
in T between the singlet molecules (T ′′ = 3/2) and the triplet molecules (T ′′ = 5/2) in the
spectrum for the J ′ = 1 level of Ω = 1 (Fig.4.32). In the spectra with singlet molecules, theleftmost line, the T ′ = 1/2 level, was always observed, whereas it was never observed in the
spectra with triplet molecules.
Thus far, the nuclear spin of 41 K has been neglected. However, we can see its inﬂuence on the
spectra as a weak transition to the J ′ = 5 level observed in the spectra with singlet molecules,
which cannot be explained in terms of a selection rule on T . Another signature of the hyperﬁneinteraction due to the nuclear spin of 41 K appears in a depletion spectrum with a low power(Fig.4.35), which appears to indicate a hyperﬁne structure of the order of 10 MHz in the groundstate.
Here we apply SpIDR spectroscopy, which was originally developed for observing narrowtransitions to the (3)1 Σ+ state, also to the (2)3 Σ+ state. The details on SpIDR spectroscopy
will be given in the section 6.2. Due to a good signal-to-noise ratio achieved with SpIDR
spectroscopy, we could clearly resolve the hyperﬁne structures induced by the nuclear spin of
41 K in the excited state. Fig.4.36 shows a SpIDR spectrum for the J ′ = 1, Ω = 1, v ′ = 44 levelof (2)3 Σ+ obtained with singlet molecules in the J ′′ = 2 level. A calculated spectrum is also
shown. The simple internal structure of singlet molecules is suited for revealing the hyperﬁne
structure in the excited state (the hyperﬁne structure of the singlet molecules will be further
95

(a) λB(2)3Σ+Ω=1

Excited states(L = 1)

(b) λ

5
4
3
2
1

B

(2)3Σ+Ω=0

same vGround states(L = 0)4
32
1
0

same v

5(2)3Σ+
Ω=0

5

5

4
3
2
1
0

(2)3Σ+
Ω=1

4
3
2
1

5/2

(1)1Σ+
v = 91

2(1)3Σ+
v = 21 3/2
N=2

1

1/2

Figure 4.34: Summary of the observed levels. The levels with even parity are given in blue solidline, whereas the levels with odd parity are given in red dashed line.

Ion counts (arb.units)

40
30
20
10
0
1.0

1.1

1.2
1.3
1.4
1.5
Frequency - 332565.54 (GHz)

1.6

Figure 4.35: Depletion spectrum for the T ′ = 3/2, Ω = 1, v ′ = 44 level of (2)3 Σ+ obtained witha power of 100 µW. The intensity is 11 mW/cm2 . At a low intensity, the depletion spectrum
becomes asymmetric due to the hyperﬁne structure in the weakly bound level induced by the
nuclear spin of 41 K.

96

4

Ion counts (arb.units)

2.0

0

T'=5/2

T'=1/2
1.5
T'=3/2
1.0
0.5

Total angular momentum F

2

0.0
1.0

1.5
2.0
Frequency - 332448.7 (GHz)

2.5

Figure 4.36: Comparison between SpIDR spectrum for the v ′ = 44, J ′ = 1 level of (2)3 Σ+ takenwith molecules in the v ′′ = 91, J ′′ = 2 level of X 1 Σ+ (lower curve) and calculated spectra for
odd parity levels (upper bars). The Fermi contact constants for 87 Rb and 41 K are set to 698
and 26 MHz, respectively. Labeling is based on the total angular momentum F = J + IRb + IK .
Since most of the population in the v ′′ = 91, J ′′ = 2 level of X 1 Σ+ is in the F ′′ = 2 level,transitions to the F ′ = 1, 2, 3 levels are mainly observed. Although the T ′ = 1/2 line clearly
split into two levels, which arises from the nuclear spin of 41 K and are labeled as F ′ = 1 andF ′ = 2, hyperﬁne levels in the T ′ = 3/2 and T ′ = 5/2 lines are not completely resolved. The
rightmost line corresponds to a transition from the v ′′ = 91, J ′′ = 0 level to the T ′ = 1/2 level.
discussed in the section 6.6). The spectral resolution was limited by the natural linewidth ofthe (2)3 Σ+ state to ∼7 MHz. The two levels observed in the T ′ = 1/2 level are explained only
by taking into account the nuclear spin of 41 K and are labeled as F ′ = 1 and F ′ = 2, where
F = T + IK . Based on ﬁtting to the spectra, we determined the Fermi contact constants forthe v ′ = 44 level to be 698(10)MHz for 87 Rb and 26(5) MHz for 41 K. We found a deviation of∼20 MHz from the calculation in the T ′ = 3/2 level when the T ′ = 5/2 level and the T ′ = 1/2
level were used for ﬁtting. The similar deviations are also observed in the other vibrational
levels. In fact, the ratio of the separation between T ′ = 5/2 and T ′ = 3/2 to the separation
between T ′ = 3/2 and T ′ = 1/2 varied between 1.4 and 1.9 among the vibrational levels. If
we assume an ordinary T ′ (T ′ + 1) progression, this ratio should be 1.67. The main diﬀerence
among the vibrational levels is in the amount of spin-orbit mixing of the (1)3 Π. Hence, one ofthe possible reasons for this variation is the hyperﬁne interaction between the nuclear spin of
87 Rb and the angular momenta in the (1)3 Π state. Another possibility is the inﬂuence of thenuclear quadrupole moments, which were neglected in the present study.
The SpIDR spectroscopy enables us to resolve the hyperﬁne structure due to 41 K in the
triplet ground state as well. Fig.4.37 shows the SpIDR spectrum for the same excited state as in
Fig.4.36 taken with triplet molecules in the T ′′ = 5/2, N ′′ = 2, v ′′ = 21 level of a3 Σ+ . Although
the spectrum for high J ′ levels was too complicated to be understood clearly, the hyperﬁne
structure in the ground state was clearly resolved in the transition to the T ′ = 1/2, J ′ = 1 level.
The observed four levels are labeled as F ′ = 4, 3, 2, 1, respectively.

97

T'=3/2 ← T''=5/2
F''=4Ion counts (arb.units)

1.0

T'=5/2 ← T''=5/2
F''=4
F''=3

0.8
F''=3

F''=2

0.6

F''=2
F''=1

0.4

F''=1

0.2
1.0

1.2

1.4
1.6
1.8
2.0
2.2
Frequency - 332565.51 (GHz)

2.4

2.6

Figure 4.37: SpIDR spectrum between the Ω = 1, v ′ = 44 level of (2)3 Σ+ and the T ′′ = 5/2,
N ′′ = 2 level of a3 Σ+ . The hyperﬁne structure in the weakly bound triplet level due to the
nuclear spin of 41 K is resolved.

4.3.6

Possible intermediate levels

In contrast to the (3)1 Σ+ state, it is diﬃcult to estimate the transition strengths between the(2)3 Σ+ state and the rovibrational ground state of X1 Σ+ . In order to estimate the transition
strengths correctly, we need information on the R-dependent spin-orbit functions, where R
denotes the internulcear separation between K and Rb, as well as the potential energy curves ofthe (1)1 Π state, the (2)3 Σ+ state, and the (1)3 Π state. These functions can only be obtained
through ﬁtting to a large number of data obtained with spectroscopy over a wide range [110,125].
According to model calculations, the transition strengths between the (2)3 Σ+ state and therovibrational ground state are not strongly dependent on vibrational levels in the investigatedrange. Thus, we decided not to take care of it. Rather, the selection of the intermediate state
was determined by the other criteria mainly arising from experimental limitations (Fig.4.38).
In our Raman laser setup, semiconductor lasers are required for locking to the ULE cavity.
Thus, the existence of commercially available laser diodes is one of the crucial limitations on thepossible intermediate levels. We had an AR-coated laser diode for the upward transition, which
can cover the entire range (870–920 nm). By contrast, we did not have a AR-coated laser diodewhich can cover the entire range in the red region (638–664 nm). We prepared several kinds of
non-AR-coated laser diode with a tuning range of several nm, covering the ranges of 639–646
nm and 653–665 nm (section 5.3.2). Another important criterion is the spin component: it isbetter to select a vibrational level which exhibits a strong (2)3 Σ+ character. As we discussed
in the section 4.3.4, the spin component can be estimated with rotational constants. We could
not neglect the possibility that the assignment is not appropriate for a level with a too large
rotational constant, such as v ′ = 42, which might have a large (1)3 Π component.
In addition to these considerations, taking into account the fact that some optical components
used in the Raman laser setup can only be used in the range 600–900 nm (due to the range ofAR coating), we selected the v ′ = 44 level. The wavelengths for the Raman transitions are 901
nm for the upward transitions and 655 nm for the downward transition.

98

3.2

3 +

(2) Σ

-2

-1

Rotational constant (10 cm )

3.0
2.8

1 +

(2) Σ

1 +

(3) Σ

2.6

3

(1) Π
1

(1) Π
Observed

2.4

Best candidate
2.2
2.0
1.8
10900

11000
11100
11200
11300
11400
1 +
-1
Transition wavenumber from v"=91 of X Σ (cm )Large transition strength
LD for the infrared light
LD for the red light
Optics (can be shared between IR and Red)
Figure 4.38: Limitations on the selection of the intermediate state for the (2)3 Σ+ . A strong
limitation arises from the laser diodes in the red region. We had a few kinds of non-AR coatedlaser diode. Each diode can cover only several nm, yielding uncovered regions. In addition, we
avoided the vibrational levels with large rotational constants because they are expected to havea large (1)3 Π component.

99

4.3.7

Theoretical treatment of the rotational and hyperﬁne structures of
triplet molecules

In this section, we ﬁrst introduce an analytical expression for the rotational structure of a 3 Σmolecule on the basis of a three-level model, which is called the Schlapp’s expression. The obtained expression is used for analyzing the observed spectra and deriving rotational constants foreach vibrational level. Next we extend the three-level model to include the hyperﬁne interaction
between electronic spin and the nuclear spin of Rb by adding the Fermi contact term to the
Hamiltonian.
The rotational part of the Hamiltonian is given by()
Hrot =B Rx2 + Ry2[=B (J 2 − Jz2 ) + (L2 − L2z ) + (S 2 − Sz2 )+(L+ S− + L− S+ ) + (J+ S− + J− S+ ) + (J+ L− + J− L+ )](4.19)

where J = R + L + S is the total angular momentum, R is the rotational angular momentum,
L is the electronic orbital angular momentum, and S is the electronic spin.
For a 3 Σ state, there are three spin components labeled by Ω = Lz + Sz , i.e. Ω = 0, ±1.
When the rotational angular momentum is absent, the Ω = 1 level and the Ω = −1 levelare degenerate, whereas Ω = 0 and Ω = ±1 are non-degenerate due to spin-orbit interaction.
Therefore, we can write the electronic part of the Hamiltonian in a matrix form as


E
0
0
Hel =  0 E − 2λ 0 (4.20)
0
0E
where λ is introduced to represent the separation between the Ω = 0 level and the Ω = ±1 level.
The rotational Hamiltonian (4.19) can be expanded by the bases of Eq.(4.20):√
N J(J + 1)√+ B⟨L2⊥ ⟩−(B − γ/2) 2J(J + 1)
0√2
Hrot = −(B − γ/2) 2J(J + 1)
BJ(J + 1)−(B − γ/2) 2J(J + 1)
√+ ⟨L⊥ ⟩0−(B − γ/2) 2J(J + 1)
BJ(J + 1) + B⟨L2⊥ ⟩(4.21)
where γ is phenomenologically introduced for representing the coupling between N = R + S
and S. The term ⟨L2⊥ ⟩ arises from the L–uncoupling operator L2 − L2z and gives slightly
diﬀerent values between Ω = 0 and Ω = ±1. In this study, this diﬀerence due to the L–
uncoupling operator is considered to be included in the parameter λ. The eigenvalues of thetotal Hamiltonian given as a sum of (4.20) and (4.21) are analytically expressed as




ϵ1
ϵ2


ϵ3

= E + B⟨L2⊥ ⟩ + BJ(J + 1)√
= E + B⟨L2⊥ ⟩ + BJ(J + 1) + (B − λ) + (B − λ)2 + 4(B − γ/2)2 J(J + 1)√
= E + B⟨L2⊥ ⟩ + BJ(J + 1) + (B − λ) − (B − λ)2 + 4(B − γ/2)2 J(J + 1)

(4.22)The parities of the rotational levels of a 3 Σ+ state are strictly determined. They can readily judged from the form of the eigenstates. Here we write the three basis set function as|J, Ω; Lz , Sz ; J⟩=|J, 1; 0, 1⟩, |J, 0; 0, 0⟩, and |J, −1; 0, −1⟩. Due to the symmetry properties ofthese basis functions, the state with a form of |J, 1; 0, 1⟩ + |J, −1; 0, −1⟩ has odd parity for even
J and even parity for odd J. In contrast, the state with a form with |J, 1; 0, 1⟩ − |J, −1; 0, −1⟩
has even parity for even J and odd parity for odd J. The state with a form of |J, 0; 0, 0⟩ has
odd parity for even J and even parity for odd J.

100

Next we introduce the hyperﬁne interaction into discussions by adding the Fermi contact
term I · S to the three-level model. The Hamiltonian exclusive of electronic interaction can be
written as
Hrot + Hhf =B(Rx2 + Ry2 )+ ARb IRb · S + AK IK · S[
22) + (I2K − IKz)=B (F2 − Fz2 ) + (L2 − L2z ) + (S2 − Sz2 ) + (I2Rb − IRbz
+ ARb IRbz Sz + AK IKz Sz + B(L+ S− + L− S+ ) − B(F+ S− + F− S+ )− B(F+ L− + F− L+ ) + B(L+ IRb− + L− IRb+ ) − B(F+ IRb− + F− IRb+ )+ B(L+ IK− + L− IK+ ) − B(F+ IK− + F− IK+ ) + B(IK+ IRb− + IK− IRb+ )+(B + ARb /2)(IRb+ S− + IRb− S+ ) + (B + AK /2)(IK+ S− + IK− S+ )] .

(4.23)

where F = L + S + R + IRb + IK is the total angular momentum, IRb is the nuclear spinof 87 Rb, IK is the nuclear spin of 41 K, AK and ARb are the Fermi contact constants for 41 K
and 87 Rb, respectively. For simplicity, terms with L± are omitted in the following calculation
since other electronic states are far away in most cases. The slight diﬀerences in Ω = 0 andΩ = ±1 due to L± are included in λ as implemented for the three-level model. In the following
calculations, phenomenological spin-coupling parameters are introduced in the same manner as
the three-level model by modifying the rotational constant. Note that the raising and lowering
operator for F and its projection ΩF satisﬁes the following relation:⟨F, ΩF |F∓ |F, ΩF ⟩ = ~√(F ∓ ΩF )(F ± ΩF + 1)

(4.24)

This relation is similar to the case for J and Ω, but is opposite to the relation for otherangular momenta such as S and L.
Since Lz = 0 in our case, the bases to represent the Hamiltonian can be taken as | Sz , IRbz , IKz ;F, ΩF > where ΩF = Lz + Sz + IRbz + IKz is the projection of F along the molecular axis. The
number of bases is 10 for F = 0, 28 for F = 1, 40 for F = 2, 46 for F = 3, and 48 for F ≥ 4.
The eigenvalues of the total Hamiltonian Htot = Hel + Hrot + Hhf are numerically obtained inthis study because we did not ﬁnd an analytical expression. An example of the eigenvalues ofthis Hamiltonian is given in Fig.4.31. The parity of eigenstates is readily identiﬁed via the formof numerically obtained eigenvectors. If we deﬁne a variable p as p = 0 for an even parity andp = 1 for an odd parity, the eigenstate |F > has a form of∑|F > =
ci [|Sz , IRbz , IKz ; F, ΩF >i+(−1)F +p | − Sz , −IRbz , −IKz ; F, −ΩF >].

(4.25)

where i denotes the possible combination of Sz , IRbz , and IKz , and ci is a numerically obtained
eigenvector.

101

Table 4.3: Results of ﬁtting to observed spectra of the (2)3 Σ+ state. Fitting parameters arevibrational energy E, rotational constant B, spacing between Ω = 0 and Ω = 1 λ and spinrotation coupling γ, which are given in cm−1 . Here E denotes the energy level from the triplet
molecules. The spectra for the singlet molecules are shifted by +3.89 cm−1 in advance, which
is a diﬀerence in binding energy from the triplet molecules. In the rightmost column, residual
deviations normalized by the spectral width are shown. For the v ′ = 47 level, a simple BJ(J +1)
expression was used for ﬁtting since the Ω = 1 level was not found.
v’
E
B
λ
γ
Residual
55 11434.649 0.0183 0.7362 0.0037
0.218254 11402.440 0.0203 0.9836 0.0207
0.197753 11377.262 0.0196 -0.6236 0.0392
0.332352 11344.798 0.0191-1.11
0.01450.326951 11316.018 0.0189 0.5019 0.0043
0.247
50 11283.893
0.02-2.4628 -0.0134
0.524549 11254.760 0.0197 -0.7261 0.0082
0.441848 11221.953 0.0191 -1.1453 0.0099
0.108947 11191.936 0.0189
0.149146 11161.192 0.0217 0.3868 0.0064
0.618645 11125.787 0.0197 -0.893
0.00350.125444 11093.201 0.0199 -0.6105 0.0046
0.120343 11059.231 0.0212 -0.5682 0.0129
0.360742 11022.277 0.0241 1.7991 0.0158
0.482241 10994.596 0.0198 0.4872 0.0028
0.156940 10959.177 0.0215 0.5348 0.0038
0.517339 10923.023 0.0226 1.1855 0.0452
0.452

102

Chapter 5

Raman lasers
The laser lights for driving a Raman transition between two largely separated molecular statesrequires a much better coherence than those for laser cooling and photoassociation. The narrow
linewidth is one of the key issues for realizing a high transfer eﬃciency because the transfer
is based on keeping the population in a superposition state obtained with two lights. The
typical required linewidth for STIRAP lights is ∼1 kHz (see discussions in the section 6.7),
which is narrower than the linewidth of the MOT light and the photoassociation light by threeorders of magnitude. In addition, a long-term stability of ∼100 kHz is required for day-to-day
reproducibility. We did not have good knowledge and techniques for generating such stable
lights.
Hence, we developed our original two-color light source, which gives a linewidth of narrower
than 10 Hz and a long-term stability of better than 100 kHz for a few days. Another important
issue is the mode-hop-free range: our setup enables continuous scanning of more than 1 GHzwith keeping the narrow linewidth. In this chapter, we describe the technical details on our
setup.
The present setup is largely based on the various lessons obtained with our primary setup. In
the previous experiments in our group, a STIRAP transfer between two neighboring vibrational
levels was realized with an optical setup based on a transfer cavity. In this setup, the modehop-free scanning range was eventually limited to several tens of MHz by the scanning rangeof AOM. This scanning range was too narrow for ﬁnding unknown molecular transitions. In
addition, there was another problem related to an oﬀset frequency for bridging the large gap
between a cavity comb line and a molecular transition: the oﬀset frequency was obtained bygenerating sidebands with a radio-frequency of several 100 MHz. The problem was that themagnitude of the sideband was small and strongly dependent on frequency. As a result, we
needed to adjust the gain for locking at each frequency, making the experiments diﬃcult andineﬃcient. Further, the error signal was limited to ∼10 mV in some cases, thereby limiting therelative linewidth to ∼40 kHz.
In the present study, we pursued the narrow linewidth and achieved a much narrower
linewidth than lasers used in the previous studies [50–52]. One reason for this eﬀort is toknow the ultimate limit of the present setup. However, there is a more important implication
speciﬁc to our experiments. In the previous studies, molecules were optically trapped in a small
region of a several tens of µm. Thus, the STIRAP lasers could be tightly focused to obtain ahigh transition strength. In contrast, molecules were not trapped in our experiments, and werewidespread over ∼1mm. This means that we had to work with a much lower transition strength.
We aimed at achieving a high transfer eﬃciency by taking a long transfer duration rather thanby having a high transition strength. For this purpose, a longer coherence than required in
the previous studies became crucial. This is why we intensively made eﬀorts for achieving thenarrow linewidth.
We ﬁrst describe the required performance and the basic concept for the present experiments
and overview the details on the setup used in the previous studies (section 5.1).
103

The development was carried out in two stages. In the ﬁrst stage, we made an opticalsetup for testing the performance at 795 nm because the primary setup was prepared at thiswavelength. Most of the important aspects for reaching a narrow linewidth were obtained here.
By referencing an invar cavity, we achieved a linewidth of 30 Hz. We conﬁrmed the basic
performance of the setup consisting of two master lasers locked to a single cavity for making
stable frequencies and two slave lasers oﬀset-locked to the master lasers for realizing a large
mode-hop-free range (section 6.2).
In the second stage, we developed an advanced setup in both linewidth and stability. The
second-stage setup referenced an ultra-low expansion (ULE) cavity. Although the linewidthof the ﬁrst-stage setup was suﬃciently narrow, we observed a large ﬂuctuation of the center
frequency. This problem was solved in the second-stage setup, resulting in a beat note linewidthof below 1Hz. In addition, the method for locking slave lasers to master lasers was changed in
the second-stage setup, enabling much more stable locking and scanning over a much more wide
range (section 6.3).

5.1

Required performance and the design

Although the idea of STIRAP was suggested as a tool for population control more than 15 years
ago, thus far it had been primarily implemented with an intense pulsed laser (for reviews, see
Ref. [47–49]). Roughly speaking, a high transfer eﬃciency is ensured by changing the intensity
ratio of two lights suﬃciently slower than the Rabi frequency of the transition. Thus, there are
two directions for achieving a high transfer eﬃciency: to increase the intensity of the light, or
to slow the variation of the intensity ratio.
The ﬁrst direction is suited for a pulsed laser, whereas the second direction is suited for a
CW laser. In the early experiments, pulsed lasers had been employed because a Fourier-limited
pulsed laser with a nanosecond pulse width was already available at that time. However, the
STIRAP transfer with a pulsed laser can readily couple multiple hyperﬁne levels with separationsof the order of 1 MHz because the Rabi frequency reaches more than 100 MHz.
In terms of preparing a molecular ensemble in a single quantum state, it is important tosuppress the Rabi frequency and to take a long transfer duration. In this direction, the key
issue is to ensure a long coherence time; in principle, unit transfer eﬃciency is achieved with ainﬁnitely low intensity if we have an inﬁnitely long coherence time. This will be impossible inreality because of various processes which loose the coherence during the transfer. In practice,of great importance is the ﬂuctuation of laser frequencies, which can destroy the superposition
state produced with the lasers. Thus, there is an essential requirement that the linewidth of thelasers should be narrow.
We can roughly estimate the required linewidth by assuming a typical Rabi frequency of
1 MHz. For an eﬃcient transfer, the duration should be much longer than the inverse of the
Rabi frequency, indicating a required duration of ∼10 µs. The laser frequencies should be stable
during the transfer. Thus, the linewidth of the lasers should be narrower than 1 kHz. Anotherimportant decoherence mechanism is the ﬂuctuation of magnetic ﬁelds, which is usually of theorder of 1 mG. The magnetic ﬁeld of 1 mG induces a shift in energy levels of ∼1 kHz for
triplet molecules, ∼100 Hz for weakly bound singlet molecules, ∼ 1Hz for deeply bound singlet
molecules. Although we always have to take care of the stability of magnetic ﬁelds, it canbe sometimes neglected, e.g., when we deal with only singlet states. In such cases, there is apossibility of achieving a high transfer eﬃciency by taking a longer duration of more than 100µs even if the Rabi frequency is much lower than 1 MHz. In addition to the requirements fora successful transfer in a single experiment, we need stability over a long time for day-to-day
reproducibility. The stability required for STIRAP experiments is ∼100 kHz because the twophoton resonance has a width of ∼1 MHz, which is determined by the Rabi frequency and thetransfer duration. More detailed discussions on the estimation of the required performance will
be given in the section 6.7.3.
104

In the STIRAP experiments with molecular states, there is another important requirement:
tunability of the lasers. In contrast to atomic resonances, the resonant frequencies between two
molecular states are not precisely known before carrying out experiments. In particular, it isdiﬃcult to precisely locate a good intermediate state with large transition strengths. Thus, therequired tunability is ∼10 nm because a typical vibrational spacing is several nm. Apart fromthe tunability, which does not necessarily mean scanning in a single experiment, we also haveto take care of a mode-hop-free scanning range in a single experiment. Here mode-hop-freescanning indicates scanning without re-locking with keeping the narrow linewidth. The wide
mode-hop-free range of several GHz becomes important for ﬁnding a two-photon resonance,
which has usually an uncertainty of several GHz.
It is not straightforward to make a laser setup meeting these demands. For reference, we
review the laser setups and their performances in the three previous studies which transferredFeshbach molecules to the rovibrational ground state. In the experiments for 40 K87 Rb [50], thelights for STIRAP had wavelengths of 970 nm and 690 nm, which were derived from diode lasers.
The diode lasers were directly locked to a Ti: Sapphire frequency comb with oﬀset frequencies.
The frequency comb was referenced to a stable 1064 nm Nd: YAG laser locked to a stable cavity.
The short-term linewidth of the Raman lasers was a few kHz [126]. In the experiments for Rb2 ,the wavelengths for a STIRAP transfer into the triplet rovibrational ground state were 1018
nm and 994 nm. These wavelengths were derived from a Ti: Sapphire laser and a diode laser.
The setup was based on a transfer cavity locked to the Rb D2 line, which also covered the two
wavelengths. The Ti: Sapphire laser could be scanned over 2.4 GHz. The short-term relative
linewidths between the Ti: Sapphire laser and the diode laser were 20 kHz [127].
In the experiments for Cs2 , they required four wavelengths (1126 nm, 1006 nm, 1351 nm, and1003 nm) for a two-step STIRAP because there were no good intermediate state which can bridge
Feshbach molecules to the rovibrational ground state in a single step. Their setup for Raman
lasers was based on a combination of an Erbium ﬁber frequency comb and optical resonators.
Diode lasers were locked to optical resonators (ﬁnesse, ∼1000; covering range, 850–1510 nm).
The long-term drift of the cavities were monitored via the frequency comb and were feed-backed
to the piezo module in the cavities. The frequency comb was ﬁnally referenced to the 10 MHz
GPS-stabilized signal. The short-term linewidth of the Raman lasers was approximately 10 kHz.
The reason why did not directly lock the diode lasers to the frequency comb is to ensure a large
mode-hop-free scanning range, which was required for ﬁnding transitions with an uncertainty ofthe order of 10 GHz [128].
Now we discuss the best design for our experiment. Which of a frequency comb or a transfer
cavity should we use? This was one of the major questions in this study. Our ﬁnal answer
to this question was neither of them; instead, we decided to use a single ultralow expansion(ULE) cavity as a reference of both Raman lasers. In this consideration, the knowledge sharedin atomic clock experiments is quite insightful. Roughly speaking, a short-term stability (up to10 s) is achieved with a ULE cavity, whereas a long-term stability for more than 10 s is achievedby referencing an atomic clock. Recent advances in making a stable frequency reference enable
us to obtain a long-term stability of 1 kHz solely with a carefully temperature-stabilized ULE
cavity [129]. The stability of this order is suﬃcient for the present study. An optical setup with
only a ULE cavity has signiﬁcant advantages over previous methods. First, the setup is much
more simple and much more robust than a setup with a frequency comb or a transfer cavity.
Without locking to another reference, a ULE cavity can serve as a stable reference. Second, acavity can provide a narrower short-term linewidth than a frequency comb. The linewidth of alaser locked to a frequency comb is limited to a few kHz, unless the frequency comb is locked toanother reference. In contrast, a ULE cavity can simultaneously provide a short-term linewidthand a long-term stability suﬃcient for the present study.
In this study, two largely separated wavelengths were required (∼650 nm and ∼900 nm).
Normally a high-ﬁnesse cavity is fabricated at one wavelength and cannot cover over a few
100 nm. We purchased a specially ordered mirrors with a dual-wavelength coating at these105

wavelengths. Our cavity can proved high ﬁnesses (> 5000) over several tens of nm, enabling us
to select the best intermediate state among several vibrational levels.
A wide mode-hop-free scanning range required for this study was realized by adding otherlasers to the setup. We employed an optical phase-locked loop (OPLL) technique [90], whichallows us to keep the frequency diﬀerence between two lasers identical to the frequency of astable microwave source. In this study, a laser locked to the ULE cavity is called a master
laser, which serves as a stable frequency reference, whereas a laser locked to the master laser iscalled a slave laser, which is used for experiments. By sweeping the microwave frequency, we
can continuously scan the slave laser with keeping the narrow linewidth of the master laser.

5.2

Development at 795 nm

After we determined the basic design of the optical setup for Raman lasers, its performance wasﬁrst tested at 795 nm. On the basis of our standard ECDL for MOT lasers, we developed a new
design for ECDLs which were more stable against acoustic and vibrational noises than previous
ones (section 5.2.1). In order to reduce the linewidth, we focused on improving the servo circuit
for locking the ECDLs to a cavity, resulting in a linewidth of ∼30 Hz for master lasers (section
5.2.2). An EOM plays a crucial role in the PDH locking scheme. We developed a resonant
circuit for EOM with a Q-value of 64 at 18 MHz such that we could obtain a maximum error
signal with a minimum RF power. In addition, we developed a method for suppressing residual
amplitude modulation in a EOM which caused a variation in the oﬀset of error signals (section
5.2.3). We found that the center frequency can ﬂuctuate when the reﬂected light from the cavity
returned to a ﬁber carrying the light to the cavity. By adding an isolator, we could suppress theﬂuctuation (section 5.2.4). For carrying the narrow linewidth of the master lasers to slave lasers,
we employed a digital optical phase-locked loop (OPLL) at this point. We conﬁrmed that thebeat note linewidth between slave lasers was the same as that between master lasers (section
5.2.5).

5.2.1

ECDL

The ECDLs for Raman lasers are a modiﬁed version of our home-made ECDL for MOT lasers(section 2.4.1). There are two main modiﬁcations: mounting and a circuit for current feedback.
For suppressing the ﬂuctuation of oscillation frequency due to acoustic and vibrational noises,
the mounting is changed to a new design given in Fig.5.1. In the previous design, the ECDL
was mounted on a rubber sheet (Edmund optics, Sorbothane) and was ﬁxed by holding themounting from above. In this case, we could not tightly hold the mounting because the rubber
sheet readily broke. By contrast, the new mounting can be tightly ﬁxed with screws, enablingan eﬃcient damping of vibration. Although the modiﬁcation is slight, the new design realizes
much more stable locking against vibrations.
The second important modiﬁcation is the circuit for current feedback in a bias-tee conﬁguration given in Fig.5.2. For MOT lasers, the ECDL is locked only with the piezo feedback because
we do not need to reduce the linewidth. The high bandwidth of up to 4 MHz of the current
feedback enables reducing the linewidth of the ECDL, which is an essential requirement for
Raman lasers. In the new ECDL, high frequency signals are feed-backed to the current, whereas
low frequency signals are feed-backed to the piezo module. The current feedback plays rolesof reducing the linewidth to approximately 10 Hz and stabilizing the laser against vibration,
whereas the piezo feedback plays roles of reducing the linewidth below 10 Hz and compensating
a long-term drift (the DC component).
Another important issue for making a stable ECDL is to suppress the inﬂuences of winds
and vibrations on the cables for current and temperature stabilization. These inﬂuences are themain cause of the ﬂuctuation of the laser frequency in a free-running condition. For realizing106

Aluminum box
Mirror mount (LD mount)0

Peltier module

Sorbothane (Edmund optics, NT37-000)
G mat (Trusco, TR-GM40)
λ GEL (Taica, COH-4000)
Nylon washer (RS, 525-739)
Figure 5.1: A new design for the mounting of ECDL (side view). In the new design, the vibrationof the optical table is well isolated by tightly holding the mounting with screws. A special rubber
sheet with a good thermal conductance (Taica, λGEL) allows a heat transfer and a vibration
isolation simultaneously. The new design is much more insensitive to the vibration of the optical
table and acoustic noises.

Figure 5.2: Circuit diagram for the current feedback. After passing through a low pass ﬁlter for
suppressing current noises, the DC current is applied to a laser diode. The capacitors used asa low pass ﬁlter consist of a tantalum capacitor (10 µF) and an electrolytic capacitor (2.2 mF).
The inductor separates the current source and the servo circuit. The capacitor C consists ofmultiple capacitors and is included in the servo circuit. The diodes (1N4148) are always requiredfor protecting the laser diode from electric noises.

107Figure 5.3: Circuit diagram of the servo circuit. The external compensation capacitors of AD829(Ccomp for Pin 5) are set to be 15 pF for the ﬁrst stage and 55 pF for the second stage such thatboth ampliﬁers do not oscillate.
stable locking, the cables should be carefully ﬁxed to the optical table through rubber sheets
held with diﬀerent forces such that vibrations of various frequencies are eﬃciently damped.

5.2.2

Servo circuit for locking an ECDLThe numerous modiﬁcations mainly on the servo circuit enabled us to reduce the linewidth of our
ECDL to 30 Hz by three order of magnitude. The improvements are carried out by monitoring
the beat note between two independent lasers locked to the cavity with a ﬁnesse of 3000. The
ﬁnal design for the circuit determined at this wavelength is shown in Fig.5.3. Here we itemize
the important knowledge obtained at this point.
1. Roughly speaking, it is crucial to increase the gain of current feedback at low frequencies
for narrowing the linewidth. Here we have to increase the gain only at low frequencies
because the gain at high frequencies is determined such that the feedback does not causes
oscillation. However, a too high gain at low frequencies results in unstable locking because
the piezo feedback should dominate at suﬃciently low frequencies for stable locking. The
best proﬁle of the gain is thus determined as a compromise between the narrow linewidth
and the stability of locking.
2. By adding a slow integrator to the positive side of the normal integrator for current feed108

back, which is usually connected to the ground, we can increase the gain at low frequencies.
We call this integrator as low-frequency (LF) locking. The cutoﬀ frequency, below which
the gain is enhanced, is determined by the time constant of the slow integrator.
3. The error signal should be directly integrated without adding or subtracting any DC
voltage. In the previous design for locking MOT lasers, our servo circuit included an
oﬀset-subtracting circuit for removing the oﬀset of the error signal arising from the residual
amplitude modulation in an EOM. The noise in the DC voltage, which was obtained by
dividing ±15 V for operating op-amps, could readily broaden the linewidth to more than
100 Hz.
4. The capacitor comprising a bias-tee should be composed of several kinds of capacitors
connected in parallel. In general, a single kind of capacitor cannot cover a broad frequency
range. We use tantalum capacitors in series for covering low frequencies and ceramic
capacitors for covering high frequencies.
5. The polarity of the error signal is ﬁrst adjusted with the length of the RF cable such thatthe current feedback signal is appropriately feed-backed. Next the polarity of the piezo
feedback signal is selected with a switch in the servo circuit.

5.2.3

EOM for PDH locking

When we improved the servo circuit, we found that the linewidth was reduced by increasing
the error signal. This was because the linewidth was limited by the electric noise in the servo
circuit. Hence, we developed a high-Q resonant circuit for the EOM such that we could obtaina large error signal by increasing the phase modulation in an EOM. Here we did not employ
an easier option which ampliﬁed the RF signal for driving the EOM because then a part of
the RF signal contaminates the ground and ﬁnally can inﬂuence the performance of locking.
In fact, later we found that the RF leakage could be a problem (section 5.3.3). Fig.5.4 shows
our resonant circuit for operating a non-resonant EOM (Thorlabs, EO-PM-NR-C1) at 18 MHz.
The resonant frequency is determined by the inductance of the coil and is arbitrarily selectable.
In this study, the resonant frequency of 18 MHz was chosen such that it was suﬃciently higher
than the expected feedback bandwidth of a laser diode (∼4 MHz). Originally we intended toseparate the signal from other frequency components with a low pass ﬁlter after the double
balanced mixer (DBM) and with a high pass ﬁlter before the DBM. For that purpose, a higher
frequency was desirable. However, they were ﬁnally not used because they cause a phase delay
which can limit the bandwidth.
The resonant circuit had a Q-value of 64, which was 1.5 times higher than that of thepreviously used resonant EOM (Thorlabs, EO-PM-15-C1). Here we point out one important
notice for this circuit. In the resonant circuit, the home-made coil, winded on a toroidal core,
plays a crucial role for obtaining a high Q. The best material for the toroidal core depends onthe frequency: the permeability should be low at a high frequency. In our case, we employed
the product from Micrometals, T50-10, which has a permeability of 6 µ0 . Commercial inductors
including a high permeability material are not suited for the present purpose.
In addition to add a high-Q resonant circuit to the EOM, we took care that the RF signal was
not reduced by passing through a phase shifter for matching the phase of the RF signal between
the photo-detector and the local oscillator. The phase shifter we have been using (Mini-Circuits,
SPH-16+) gave rise to high frequency components due to some nonlinearity. We matched thephase of the RF signal by adjusting the length of the RF cables. This method enabled us to carry
the RF signal without any degradation, although we did not conﬁrm whether this modiﬁcation
improved the performance.
The residual amplitude modulation (RAM) in an EOM can be a signiﬁcant problem in the
PDH locking scheme [92]. The facets of the crystal of the EOM work as a kind of Fabry-Perrot
cavity and induce a frequency dependence of the sidebands in the EOM. As a result, the error
109

Figure 5.4: Circuit diagram for the resonantly driving the EOM. The circuit provides a Q-valueof 64 at 18 MHz.
signal has a ﬁnite oﬀset, which is sensitive to the temperature of the crystal. Although it does notdirectly inﬂuence on the linewidth, it induces a slow ﬂuctuation of the locked frequency, making
it diﬃcult to evaluate the linewidth correctly. In addition, a too large ﬂuctuation sometimes
makes the locking unstable. Thus, it is quite important to solve this problem.
It was pointed out that RAM was signiﬁcantly attenuated by tilting the beam with respect
to the crystal by a few degrees [92]. However, the oﬀset still remains in most cases. Our solution
to this problem comprises two parts: to stabilize the temperature of the EOM by surrounding itwith a polystyrene foam and to reduce the RAM by carefully adjusting the polarization of thelight before and after the EOM. We discuss the second part in detail here because the ﬁrst partis straightforward. In our case, the phase modulation in the EOM is valid only for a S-polarized
light. We found that the remaining oﬀset arose from a mismatch of the light polarization withrespect to the crystal. By carefully adjusting the polarization of light before and after the EOMwith half waveplates (and sometimes also with quarter waveplates) as well as by tilting the
crystal, we could cancel the residual oﬀset to below a thousandth part of the error signal overa few GHz at the desired wavelength. In this way, we could remove the inﬂuence of RAM on
the locking. The present method has an important implication that oﬀset-subtraction in a servo
circuit is no more required. This allows us to signiﬁcantly narrow the linewidth as mentioned
in the section 5.2.2.

5.2.4

Interferences in an optical ﬁber

The improvements on the servo circuit and the increase of the magnitude of the error signal
resulted in a narrow linewidth of 30 Hz. At this stage, we encountered another serious problem
that the center frequency was ﬂuctuating over several kHz within a few second. The linewidthappeared narrow, but we could not evaluate it because of the ﬂuctuation. Thus, the goal of
reducing the linewidth was shifted to suppressing the ﬂuctuation in the middle of the work.
Although the ﬂuctuation problem was not completely circumvented in the present study, themagnitude of the ﬂuctuation was reduced from the original value of several kHz to a few Hz
by three orders of magnitude. There were a lot of factors, and most of them were found andsolved in the development at 901 nm (section 5.3). In the development at 795 nm, we found thatthe largest contribution to the ﬂuctuation was induced by the reﬂected light from the cavity
returning to the optical ﬁber.
In our setup, we need an optical ﬁber for carrying the lights of the ECDL to a cavity. Insuch a situation, a part of the light reﬂected at the cavity can always reenter the ﬁber. The
cavity reﬂection is usually of the order of 10 % of the power incident on the cavity. Normallymost of the reﬂected light is extracted with a polarization beam splitter (PBS) for obtaining anerror signal. However, in our early setup, only half of the reﬂected light was extracted in orderto lock two lights to a single cavity independently (Fig.5.5). As a result, a part of the light
110

DBM
Photo-detector
λ/4
Isolator

CavityEOM
λ/4
Photo-detector

Isolator

EOM
DBM

Figure 5.5: Optical setup for measuring the linewidth betwee two lasers by locking them todiﬀerent comb modes of a single cavity. The two lasers are overlapped with a PBS and areincident on the cavity. Half of the reﬂected light is extracted for obtaining error signals, whereas
the other half is returned to the optical ﬁbers.
15
Voltage (mV)

10
5
0
-5
-10
-15
-4

-2

0
Time (ms)2

4Figure 5.6: An error signal including interferences in an optical ﬁber. The periodic modulationof the oﬀset is due to the interference in an optical ﬁber, which is quite sensitive to vibration of
the optical ﬁber. In addition, there is an interference between the cavity and the ﬁber.
reentered the ﬁber was reﬂected at the facets of the ﬁber, thereby causing an interference withthe original light, which is quite sensitive to the vibration of the ﬁber. Here the two facets of the
ﬁber and the cavity work as a kind of low-ﬁnesse cavity. Thus, the interferences can contributeto the error signal as weak resonances, ﬂuctuating the oﬀset of the error signal(Fig.5.6). The
oﬀset level is also sensitive to the cavity itself because the interference can also occur between
the cavity and the facet of the ﬁber.
The ﬂuctuations of the oﬀset of the error signal was the main reason for the ﬂuctuationsof the locked frequencies over several kHz. By taking care of the following three points, we
could suppress the ﬂuctuations to a few 100 Hz by one order of magnitude. First, the EOMshould be inserted after the EOM such that the phase-modulated light does not pass throughthe ﬁbers. Second, optical isolators should be inserted just after the optical ﬁbers such thatthe reﬂected light does not reenter the ﬁbers. Third, the vibration of the cavity should be
suppressed by placing heavy objects on the cables connected to the vacuum chamber for thecavity. In some cases, the interference between the cavity and the surface of the photodiodebecame problem, which could be circumvented by tilting the photo-detector. The linewidth
between two independent ECDLs obtained at this stage is shown in Fig.5.7.
Although large ﬂuctuations were still remaining at this stage, we concluded that the ob-

111

tained performance (a linewidth of 30 Hz and a ﬂuctuation of a few 100 Hz) would be suﬃcientfor STIRAP experiments. The knowledge obtained here that the interferences can cause theﬂuctuation of the oﬀset has been important throughout the eﬀorts to stabilize the frequency of
Raman lasers. This topic will be further discussed in the section 5.3.5.

5.2.5

Digital optical phase-locked loop

The optical phase-locked loop (OPLL) is a general technique for carrying the frequency of one
light to another light. We can think of the OPLL as a perfect copy of the frequency within
the feedback bandwidth. The OPLL is particularly advantageous when we want to set an oﬀset
frequency between two lasers. Fig.5.8 shows the schematic diagram of the OPLL. The stability ofa microwave source is much better than that of lasers, thereby enabling us to shift the frequency
without changing the linewidth. Thus, the OPLL is ideal for our purpose which requires both alarge mode-hop-free range and a narrow linewidth.
Although the original implementation of the OPLL was based on an analog (passive) component for detecting the frequency diﬀerence [90], recent advances in semiconductor technologies
provides the possibility to implement the OPLL with a digital (active) component. The advantage of the digital OPLL is its wide locking range: the digital PLL chip such as AD9858 (Analog
Devices) can detect whether the frequency is higher or lower than the reference microwave frequency even if the frequency is widely separated. Therefore, locking should be easier than theanalog OPLL, which requires locking near the reference microwave frequency.
With these expectations, we constructed a setup given in Fig.5.9. The main part of thecircuit was on the evaluation board of AD9858 available from Analog Devices. The beat note
between two lasers was detected with a home-made photo-detector based on a high-speed GaAs
PIN photodiode for optical communications (Hamamatsu, G10447-54; bandwidth, 8 GHz; transimpedance, 6 kΩ). The beat note was further ampliﬁed with a limiting ampliﬁer (GigOptix,
iT3011W; bandwidth, 10 GHz; gain, 42dB) because the prescaler requires a square wave for aproper operation particularly at low frequencies. The use of a limiting ampliﬁer is also advantageous for ﬂattening the frequency response of the photo-detector, which has a low output power
at high frequencies, and enabling scanning over a wide frequency range. The circuit for locking
an ECDL with the digital OPLL is given in Fig.5.10 whereas that for locking a Ti: Sapphirelaser with the digital OPLL is given in Fig.5.11.
We conﬁrmed that the linewidth between two lasers locked with the digital OPLL was within
1 Hz, indicating that they had the same frequencies (Fig.5.12 for two diode lasers and Fig.5.13
for a diode laser and a Ti: Sapphire laser). Further, we also conﬁrmed that the linewidth
between two slave lasers was the same as that between two master lasers as expected (Fig.5.14).
Thus, it became clear that the basic concept of having two master lasers and two slave lasers
worked well and the performance of the master lasers was not degraded at the slave lasers.
In terms of the linewidth, the digital OPLL was successful. However, we found three serious
problems in its usability. First, the bandwidth was limited to ∼500 kHz and we could notincrease it by using a diﬀerential gain in the servo circuit because the DC output signal could
not be integrated. As a result, the lock was unstable. The second problem was its narrow
scanning range (a few 100 MHz). A reduction in the error signal for avoiding the oscillation of
the piezo module resulted in a narrow scanning range because we could not control the gain andthe oﬀset DC voltage independently. The third and the most serious problem was the change in
the gain for the current feedback with a change in the DC voltage from AD9858 during a single
scan. This was presumably due to the variation in the capacitance of a coupling capacitor before
an integrator of a servo circuit, caused by the DC bias characteristics of ceramic capacitors.
We could not ﬁnd good methods for circumventing these problems. Although the presented
setup based on the digital OPLL could be used for scanning over ∼100 MHz, we ﬁnally gave up
to use it because we could not have a conﬁdence that the obtained stability was suﬃcient forlater experiments. Instead, we decided to use the analog OPLL, whose details will be discussed

112

Amplitude (dBm)-10

Span 4MHz
RBW 30kHz
Swp 9.054ms
Avg 20

-20
-30

(a)

-40
-50
1

2
3
Frequency - 1494 (MHz)4

Amplitude (dBm)

-10
-20
-30
-40

(b)

Span 100kHzRBW 1kHzSwp 203.7ms
Avg 10-50-60

Amplitude (dBm)

780

-20
-30

800
820
840
Frequency - 1496000 (kHz)860

FWHM = 29Hz

Span 5kHz
RBW 30Hz
Swp 582ms
Avg 3

(c)

-40
-50
-60
821

822
823
824
Frequency - 1496000 (kHz)825

Figure 5.7: Linewidth of two master ECDLs. (a) Span 4MHz. (b) Span 100 kHz. (c) Span 5 kHz.
The measured linewidth is 29 Hz, which is limited by the resolution bandwidth of the spectrum
analyzer. Due to the ﬂuctuation of the center frequency, we cannot measure the linewidth witha lower resolution.

113

Signal Generator

Phase detector

Servo circuit

Amplifierλ/2Laser2

Photo-detector
λ/2Laser1Figure 5.8: Schematic diagram of the optical phase-locked loop. The frequency diﬀerence between two lasers is detected with a photo-detector. After mixed with a microwave signal, thebeat note is feed-backed to the second laser such that the frequency diﬀerence between two lasers
is kept exactly the same as the frequency of the microwave.

ECDL1

Photo-detector
λ/2Limiting Amplifier
ECDL2
PZT

Current

Prescaler (1/16)
DIV
CP

Servo circuit

AD9858

IOUT(0-400 MHz)REFCLK

Figure 5.9: Setup for the digital OPLL with AD9858. Most of the circuits are on the evaluation
board of AD9858. The beat note between two lasers is divided by 16 at the prescalers andis compared with the RF frequency supplied from the direct digital synthesizer (DDS) on theboard. The charge pump output to the VCO is extracted and is used for locking the laser.
114

Figure 5.10: Circuit diagram for locking a diode laser via the digital OPLL.

115

Figure 5.11: Circuit diagram for locking a Ti: Sapphire laser via the digital OPLL.
in the section 5.3.7.

5.3

Development at 901 nm

The development at 795 nm provided a laser setup with a minimum performance for carryingout experiments. We proceeded to make a light source at wavelengths of ∼900 for the upward
transition nm and ∼650 nm for the downward transition which should be required in the actual
experiments. The main diﬀerence was the use of a ultra-low expansion (ULE) cavity mounted
on a special optical table, which gave a lower sensitivity on vibration and a smaller long-term
drift than our previous cavity (section 5.3.1). In the process for making the light source at
new wavelengths, we made various modiﬁcations on the setup developed at 795 nm, resulting
in signiﬁcant improvements on the performances. The modiﬁcations were made on the ECDL
(section 5.3.2), photo-detectors for the PDH locking (section 5.3.3), phase-noise cancellation ina ﬁber (section 5.3.4), optical setup (section 5.3.5), electronics (section 5.3.6), and the method
for optical phase-locked loop (OPLL) (section 5.3.7). With these improvements, we obtained
a linewidth of below 10 Hz, a long-term stability of better than 100 kHz, and a mode-hopfree scanning range of a few GHz. Further, the system was highly stable against acoustic andvibrational noises. Although the improvements were mainly carried out at 901 nm for the
upward transition we conﬁrmed that a nearly identical performance was obtained at 655 nm forthe downward transition as well.

5.3.1

ULE cavity

The cavity for the present study is a commercial ULE cavity available as an assembly of a spacer
and optically contacted mirrors from Advanced Thin Films. The spacer is 10 cm in length and116

Amplitude (dBm)(a)
Span 10kHz
Swp 605ms
RBW 100Hz

-20
-30
-40
-50
948

Amplitude (dBm)

-20

-40

950

952
954
Frequency - 1599000 (kHz)956

FWHM 0.8Hz

Span 100Hz
Swp 2.29s
RBW 1Hz

(b)

-60

-80
953.36953.38953.40953.42953.44Frequency - 1599000 (kHz)953.46

Figure 5.12: Beat note between the master and slave ECDLs. (a) Span 10 kHz. (b) Span 100
Hz. The feedback bandwidth is ∼1 MHz.

117

Amplitude (dBm)

-20
-30

(a)

Span 200kHz
Swp 407.4ms
RBW 1kHzAvg 10

-40
-50
850

900
950
1000
Frequency - 1599000 (kHz)Amplitude (dBm)

-10
-20
-30

(b)

Span 10kHz
Swp 605ms
RBW 100HzAvg 10

-40
-50

934

Amplitude (dBm)

-20
-30
-40

936
938
940
Frequency - 1599000 (kHz)

942

(c)
Span 100Hz
Swp 2.294s
RBW 1Hz
Avg 2

-50
-60
-70
937.62937.64937.66937.68Frequency - 1599000 (kHz)937.70Figure 5.13: Beat note between the master ECDL and the slave Ti: Sapphire laser. (a) Span
200 kHz. (b) Span 10 kHz. (c) Span 100 Hz. The feedback bandwidth for the Ti: Sapphirelaser is limited to ∼150 kHz. The sidebands at 50–100 kHz appear to arise from the feedback
characteristics of AD9858.

118

Amplitude (dBm)

-10
-20

(a)

Span 4MHz
Swp 9.054ms
RBW 30kHz
Avg 30

-30
-40
-50
1

2
3
Frequency - 644 (MHz)Amplitude (dBm)

-10

(b)

Span 200kHz
Swp 407ms
RBW 1kHzAvg 10

-20

4

-30
-40
-50
700

Amplitude (dBm)

-20
-30

750
800
850
Frequency - 646000 (kHz)Span 5kHz
Swp 582ms
RBW 30Hz
Avg 3

(c)

-40
-50
-60
799

800
801
802
Frequency - 646000 (kHz)803

Figure 5.14: Beat note between two slave lasers. (a) Span 4MHz. (b) Span 200 kHz. (c) Span
5 kHz. The measured linewidth is the same as that of master lasers. The linewidth of the slavelasers is not broadened with the digital OPLL.

119

λ/4

ULE cavity

ECDL
Piezo
Function Generator(Ramp)

Photo-detector
AC component
Photo-detector

Figure 5.15: Optical setup for measuring the dynamic response of a Fabry-Perrot cavity.
has a notched shape which has a low sensitivity to vibration [130]. The mirrors were speciallyfabricated for the present study. At the moment we ordered the cavity, we did not determine
which intermediate state should be used. If we use the (3)1 Σ+ state, good Flanck-Condon
overlaps are expected at 870–880 nm for the upward transition and 635–645 nm. By contrast,
good Flanck-Condon overlaps are expected at 890–910 nm for the upward transition and 650–660
nm for the downward transition if we use the (2)3 Σ+ state. These wavelengths ranges include
several vibrational levels. Thus, one of the crucial requirements for the cavity mirrors was thatthey could cover these entire range. In the previous study at 795 nm, we had knowledge thata ﬁnesse of 3000 was suﬃcient for obtaining a linewidth of narrower than 100 Hz. We speciﬁedthe mirror coating as a ﬁnesse of more than 3000 at 880–960 nm and 640–670 nm.
Although the loss of the mirrors was not speciﬁed, the measured ﬁnesse indicated that theloss did not limit the ﬁnesse. The ﬁnesse was measured at 901 nm and 655 nm by monitoring the
dynamic response of the cavity. Fig.5.16 shows the setup for measuring the dynamic response ofa Fabry-Perrot cavity [131]. This method is more simple than the ring-down measurement andis particularly suited for measuring the ﬁnesse of low-reﬂectivity cavity because we only haveto monitor the reﬂected light from the cavity with sweeping the frequency of the light. The
principle of measurement is quite simple: the reﬂection from the cavity includes a delayed lightaccumulated in the cavity. If the frequency of the light is kept sweeping, we can observe a beatnote between the reﬂected light and the light accumulated in the cavity. The time decay of thebeat note is a direct measure of the ﬁnesse of the cavity (Fig.5.16). The ﬁnesse is determined
to be 6000 at 901 nm and 10000 at 655 nm, which are in good agreement with the estimationfrom the reﬂectivity.
The ULE cavity was kept in a vacuum chamber for heat isolation (Fig.5.17). The designof the vacuum chamber was based on Ref. [129]. The cavity was kept in two nested aluminum
cylinders. The temperature of the outer cylinder was stabilized with stacked peltier modules suchthat the inner cylinder was kept at ∼6.6 ◦ C, which was near the zero-expansion temperature of
our cavity. The zero-expansion temperature was determined by monitoring the Cs D2 transition
at 852 nm. The long-term drift of the cavity during a few days was also measured with theCs D2 transition to be within 100 kHz. However, we observed a larger long-term drift of theorder of 100 kHz during a few months. The SpIDR spectroscopy, which will be introduced in
the section 6.2, allows us to measure the transition frequency of the upward transition to the(3)1 Σ+ state with a precision of 20 kHz. Fig.5.18 shows the measured oﬀset-frequency between
the master laser and the slave laser. Here the slave laser is resonant to the molecular transition
T ′ = 5/2, J ′ = 1, v ′ = 41 of (3)1 Σ+ ← F ′′ = 2, J ′′ = 0, v ′′ = 91 of X1 Σ+ , whereas the master120

PD signal (arb.units)(a) 901 nm
1

0

-1

0

2

4
Time (µs)

68PD signal (arb.units)(b) 655 nm
5

0

-5

0

2

4
Time (µs)

68Figure 5.16: Dynamic response of the ULE cavity. (a) Measurement at 901 nm. (b) Measurement
at 655 nm. The time decay of the beat note indicates that the ﬁnesse is 6000 at 901 nm and10000 at 655 nm.

121Ion Pump 20 L/s

260mm

200mm
ULE
AR-coated
Window

Viton
114mm
TEC

Graphite sheet

Optical table (MinusK)

Figure 5.17: Vacuum chamber for mounting the ULE cavity. The cavity is kept in two nested
aluminum cylinders for heat isolation. The AR-coated glass windows avoid the inﬂuence of thevariation in room temperature on the cavity. The temperature of the outer cylinder is stabilizedwith peltier modules which are attached to the base plate made of aluminum.
laser is resonant to the speciﬁc transmission peak of the cavity. We observed a linear increase in
the oﬀset frequency with a slope of 65 mHz/s, which is close to that of a previous study [129].
This drift is not due to a temperature variation, but is due to the aging (crystallization) of the
ULE glass itself. The drift of the cavity was suﬃciently smaller than the typical two-photon
linewidth of ∼ 1 MHz. Hence, we have only to check the drift of the cavity once in a few weeks.
The chamber was mounted on a passive optical table with a natural frequency of 0.5 Hz(MinusK, 250BM-1). The acceleration of the optical table directly inﬂuences on the linewidthof the lasers even if the cavity has a vibration insensitive shape. We measured the accelerationof the table with a sensitive seismic accelerometer (Wilcoxon, 731A/P-31) which can measure
an acceleration of 1 µg at a frequency of 0.1 Hz. Fig.5.19 shows the Fourier spectrum of themeasured acceleration. For comparison, the acceleration of an active optical table (Newport,
RS4000) measured with the same accelerometer is also shown in the ﬁgure. In contrast to thespectrum for RS4000, which showed a clear peak at the natural frequency, the spectrum for
Minus-K did not show any clear peak at the expected natural frequency of 0.5 Hz. Here theimportant point is that acceleration on the Minus-K table is smaller than that on the normal
optical table by one order of magnitude. Assuming the typical sensitivity of a notched cavity as
several tens of kHz/g [130], we can estimate the frequency ﬂuctuation of the cavity due to thevibration of the optical table to be within 10 Hz. This ﬂuctuation is suﬃciently lower than therequired linewidth of Raman lasers.

122Offset frequency (MHz)-1234.8-1235.0-1235.2

65(1) mHz/s-1235.4-1235.6-1235.8-1236.0
0

50

100
Day

150

200

Figure 5.18: Long-term drift of the ULE cavity measured by a molecular transition. The oﬀset
frequency between the master laser and the slave laser is plotted with respect to date. The
linear drift indicates that the ﬂuctuation due to the variation in room temperature is negligible.
The slope of 65 mHz/s is comparable to the previous studies.

5.3.2

ECDL

The design of the ECDL for 901 nm and for 655 nm was the same as that for 795 nm except for
the angle of the grating. The grating itself was the same: the holographic grating for UV range(Edmund optics, 43774) can cover a broad range of 300–1050 nm. The polarization of the laser
diode with respect to the grating was chosen such that the diﬀraction eﬃciency is the highest.
The diﬀraction eﬃciency was typically 20–30 %. For 901 nm, we employed a commercial ARcoated laser diode covering 870–920 nm (Eagleyard, EYP-RWE-940). This laser diode could
cover the entire range required for the present study. In the red region 635–660 nm, we could
not ﬁnd a good AR-coated laser diode. We employed a non-AR-coated laser diode covering 650–
660 nm (Roithner lasertechnik, ADL-65401TU) for 655 nm and a non-AR-coated laser diodecovering 635–645 nm (Opnext, HL6367DG) for 641 nm. The AR-coated laser diode gave a quite
stable operation with a broad mode-hop-free range: the oscillation does not strongly depend on
the current. In contrast, a stable oscillation of the non-AR-coated laser diode was obtained ata speciﬁc current. This current was strongly dependent on the temperature. As a result, the
ECDL in the red region could readily hops with a slight change in room temperature, whereas
the ECDL in the infrared region was quite stable even if room temperature varied.
In terms of narrowing the linewidth, it is better to prepare an ECDL with narrower linewidth
at the free-running condition. One of the possible reasons for broadening the linewidth of ECDL
is the current noise. As already described in the section 5.2.1, the current noise for the ECDL
is suppressed by a low-pass ﬁlter with a cutoﬀ frequency of a few Hz. In principle, this ﬁlter
can reduce the high frequency noises, which can broaden the linewidth, by several orders ofmagnitude. We checked whether we could improve the performance by supplying the current
from a better current source (HighFinesse, LCS02/6) which has a lower noise than our typicalcurrent source (Thorlabs, LDC202C). Fig.5.20 shows the setup for testing the inﬂuence of thecurrent source on the linewidth of ECDL. The linewidth was measured by monitoring the beatnote between two independent lasers. The ECDL1 was locked to the ULE cavity with bothcurrent and piezo feedbacks and was narrowed to below 1 kHz, whereas the ECDL2 was locked
to an invar cavity only with a piezo feedback. For locking only with a piezo feedback, we required
the invar cavity with a broader linewidth of ∼500 kHz. The piezo feedback could not narrow

123

10
Horizontal
VerticalAcceleration (µg)

4
2

1
4
2

0.1
4
2

(a) Minus-K

0.01
2

3

4 5 6

0.1

2

3

4 5 6

1
Frequency (Hz)

2

3

4 5

10

100
Horizontal
VerticalAcceleration (µg)

4
2

10
4
2

1
4
2

(b) R4000

0.1
0.1

1

10
Frequency (Hz)100

Figure 5.19: Acceleration of the optical table. (a) Passive table for mounting the ULE cavity(Minus-K). (b) Active table for mounting laser setups. The active table shows a clear peak at
the natural frequency (2–3 Hz), whereas the passive table shows no clear peak. The accelerationon the passive table is lower than that on the active table by one order of magnitude.

124

Thorlabs
LDC202C

Electric line
Light (901nm)

Low pass
filterCurrent
BW: 100kHz

High finesse
cavityECDL1
PZT
BW: 10kHz

Thorlabs
LDC202C

orIntegration
CircuitPBS PBSLow pass
filterPhoto
Detector
λ/2

HighFinesse
LCS 02/6

ECDL2
PZT
BW: 10kHz

Low finesse
cavityIntegration
CircuitFigure 5.20: Setup for testing the inﬂuence of the current source on the linewidth. The beat note
between two lasers is measured. The linewidth of the ECDL1 is narrowed to below 1 kHz and isused as a reference for measuring the linewidth of the ECDL2. For suppressing the ﬂuctuationof the center frequency, the laser2 is locked to an invar cavity only with a piezo feedback. We
can observe the linewidth of the laser2 operated with a current source of either LDC202C or
LCS02/6.
the linewidth because the bandwidth of the piezo feedback (∼10 kHz) was much slower than theoriginal linewidth of the ECDL. In this way, we could test if the current source had an inﬂuenceon the linewidth.
Fig.5.21 shows the measured linewidth of the laser2 with our conventional current source(LDC202C) and the current source with a lower noise (LCS02/6). We conﬁrmed that the
linewidth of the laser1 was narrower than 1 kHz by locking both lasers with a current feedback
as shown in Fig.5.21(c). In both cases, the linewidths were ∼30 kHz. We did not observe asigniﬁcant improvement by using the better current source, indicating that our conventional
current source ﬁltered with a low-pass ﬁlter gave a suﬃciently low-noise current. Thus, we
continue to use the conventional current source for the present study.
In the above measurement, the beat note was observed with an averaging time of 80 ms.
Without averaging, the linewidth appeared to be narrower than observed with averaging, although the center frequency was ﬂuctuating. These observations indicate that the linewidth of
our ECDL is mainly limited by the ﬂuctuation of the center frequency. The ﬂuctuation should
be considered to be faster than 10 kHz because it cannot be removed with a piezo feedback. The
possible origin of the ﬂuctuation is the mechanical vibration of the cavity comprising the laser
125

LCS 02/6
Fit, FWHM = 33kHz

Amplitude (dBm)-20

Span 1MHz

(a)
-30
-40
-50

-400

LDC202C
Fit, FWHM = 35kHz-20
Amplitude (dBm)

-200
0
200
Frequency - 3428730 (kHz)400

Span 1MHz

(b)
-30
-40
-50
-400

-200
0
200
Frequency - 3428032.372 (kHz)

400

-10
Amplitude (dBm)-20

Span 100kHz(c)

With current feedback
Fit, FWHM = 390Hz

-30
-40
-50
-60
-70
-40

-20
0
20
Frequency - 3427024.872 (kHz)40

Figure 5.21: Beat note between two ECDL. (a) Beat note with LCS02/6. (b) Beat note with
LDC202C. (c) Beat note when Laser1 and laser2 are narrowed with both piezo and current
feedbacks. The linewidth of the laser1 is narrowed to below 1 kHz, whereas the laser2 is locked
to an invar cavity with a piezo feedback for suppressing the ﬂuctuation. The beat note given inthis ﬁgure is the direct measurement of the linewidth of the laser2 because the linewidth of thelaser1 is suﬃciently narrow as shown in (c).

126

-60
Amplitude (dBm)

RF signal for driving EOM
-65
-70
-75
RBW 300 kHz
-80
0

10

20
Frequency (MHz)30

40

Figure 5.22: The noise spectrum of the previously used photo-detector for PDH locking. The
output from the photo-detector is directly measured with a spectrum analyzer without any light.
The noise spectrum indicate that the bandwidth is limited to approximately 15 MHz. The peakat 18 MHz indicate that the output of this photo-detector includes the RF signal leaked to theground.
diode and the grating. Another possibility is the inﬂuence of the ﬂuctuation of the refractive
index of air due to convection. We have not conﬁrmed which one is the dominant mechanism
for the ﬂuctuation of our ECDL.
In the present study, we required a tunability over tens of nm because we did not know
which intermediate level gives large transition strengths. The infrared light was readily tuned
over tens of nm owing to the AR-coated laser diode. However, we could not have the AR-coatedlaser diode in the red region. In order to cover a wide range near 650 nm, we prepared several
kinds of non-AR-coated laser diodes (Table 5.1).

5.3.3

Low noise photo-detector for PDH locking

The photo-detector for detecting a phase-modulated light is a crucial element in the PoundDrever-Hall (PDH) scheme. We always have to take care of the performance of the photodetector because it should ﬁnally limit the performance of locking, particularly in terms oflinewidth. Previously we have been using a home-made transimpedance ampliﬁer (Fig.2.8).
In the previous design, we had a problem that the bandwidth of ∼15 MHz was lower thanthe modulation frequency of 18 MHz used in the Raman laser setup. The bandwidth of atransimpedance ampliﬁer can be readily estimated from the noise spectrum (Fig.5.22). Roughly
speaking, the peak in the noise indicates the bandwidth of the transimpedance ampliﬁer. The
noise arises from the ﬁrst-stage op-amp and is strongly inﬂuenced by the design of the photodetector.

Table 5.1: Prepared laser diodes in the red region (∼650 nm). With these laser diodes, we cancover the ranges of 638–646 nm and 653–665 nm.
Supplier
Model
Covering range (nm)
TOPTICA
LD-0645-0080
643–646
Opnext
HL6363DG
638–642
Opnext
HL6378DG
641–643
Roithner lasertechnik ADL-65401TU
653–659
Roithner lasertechnik
QL65L6SA
659–665127

In order to ensure a wide feedback bandwidth, the bandwidth of the photo-detector should
be larger than the modulation frequency by at least several MHz. However, generally a highspeed transimpedance ampliﬁer has a high noise. One good solution to obtain a low-noise and ahigh-bandwidth ampliﬁer is to use an integrated chip specialized for a transimpedance ampliﬁersuch as SA5211 (Philips) or AD8015 (Analog √Devices). The typical characteristics of these chips
are 160–180 MHz in bandwidth and 2–3 pA/ Hz in equivalent input noise current. We did not
employ these chips because ﬁnally
we developed an ampliﬁer with a lower bandwidth (∼40 MHz)√and a lower noise (0.14 pA/ Hz) by using a general-purpose op-amp. We tested ﬁve kinds of
op-amps, AD8099 (Analog Devices), OPA847 (Texas Instruments), MAX4107 (Maxim), AD829(Analog Devices), LMH6624 (National semiconductor), among which AD8099 showed the best
performance. Fig.5.23 shows the ﬁnal design of the new photo-detector, while Fig.5.24 shows thenoise spectrum. For the correct operation of AD8099, the circuit should be composed of surface
mount chips in as small an area as possible. Here we brieﬂy describe the concept for designinga low noise ampliﬁer herein. For achieving a low equivalent input noise current, it is crucial
to have a high feedback resistance. The limitation on the resistance arises from the required
bandwidth: surface mount resistances have a typical capacitance of 0.2 pF, which can serve asa feedback capacitance and limit the bandwidth. Thus, the maximum allowable resistance isdetermined by the bandwidth and 0.2 pF. In addition, there is another limitation arising fromthe requirement for a stable operation as a transimpedance ampliﬁer. The feedback capacitanceshould be larger than a speciﬁc value such that an input noise is not ampliﬁed.
Constructing the circuit with surface mount components on a small area has another important eﬀect on the ﬂuctuation of the locked frequency. In the previous design, the circuit
was composed of normal-size components on a commercial universal board (Sunhayato, ICB88SEG). This circuit caught the RF signal leaked on the ground. Particularly the RF signal for
driving the EOM became a problem because it can constitute the oﬀset of the error signal. The
output RF signal depends on the shape of the RF cable, giving rise to a ﬂuctuation of the oﬀset
through the vibration of the cable. With the surface mount components and the home-madesmall circuit board, we could avoid the inﬂuence of the RF signals on the ground; thereby we
could suppress the ﬂuctuation of the center frequency.
In the new design, we did not install the second-stage ampliﬁer which could add noises. As
a result, the new photo-detector has a lower transimpedance (20 kΩ) than our previous design(100 kΩ). We tried to increase the signal by amplifying the RF signal just after the photodetector with an RF ampliﬁer because generally a good RF ampliﬁer has a lower noise ﬁgurethan an ampliﬁer with op-amps. However, in this attempt we found that an RF ampliﬁer was
not suited for amplifying an error signal. Due to the gain compression eﬀect, an error signal was
distorted and lost sensitivity to a small change in frequency. Finally we solved the problem justby increasing the power coupling to the cavity. The typical power incident on the cavity was∼200 µW, which was suﬃcient for achieving a linewidth narrower than 10 Hz. Another goodsolution will be to use an Avalanche photodiode.
√The obtained performance of 0.14 pA/ Hz is suﬃcient for the present study. Assuming
the typical photo-current of 100 µA, we can ensure a signal-to-noise ratio of more than eightorders at 1 Hz. This SNR is suﬃcient for achieving a linewidth of 1 Hz from an error signal
with a linewidth of 200 kHz. Although a single-stage ampliﬁer gives a suﬃciently low noise witha required bandwidth in the present study, a two-stage ampliﬁer can have a lower noise. The
possibility of a two-stage transimpedance ampliﬁer is discussed in detail in Ref. [132].

5.3.4

Phase noise cancellation in an optical ﬁber

The development at 795 nm gave us a narrow linewidth of 30 Hz, but left a signiﬁcant problemof the ﬂuctuation of the center frequency over a few 100 Hz. We suspected that this ﬂuctuation
was caused by the phase noise in an optical ﬁber. In our setup, the ULE cavity is mounted ona passive optical table, which is too small to mount all the lasers (three semiconductor lasers

128

(a) Circuit diagram

79L05

78L05
I

+15V

G

O

G

I

O

-15V

(b) Board layout (x 1)Upper: Image for etching
Lower: Layout(c) Image
Left: 40 MHz version Right: 240 MHz version

Figure 5.23: Design of the new photo-detector for PDH locking. (a) Circuit diagram. (b) Board
layout. (C) Image. A current signal from a photo-detector is converted to a voltage signal witha transimpedance ampliﬁer. The DC part of the output is separately obtained for monitoring
whether locking is successful. All the parts are mounted on a home-made surface mount circuit.
In addition to the components given in (a), the power supplies are ﬁltered with 15µF tantalum
capacitors.

129

-70
(a)
Amplitude (dBm)

-75
-80
-85
-90
-95
RBW 1 MHz
-100
0

50

100
Frequency (MHz)

150

(b)

-76
Amplitude (dBm)

200

-78
-80
-82
-84
-86
-88

RBW 300 kHz

-90
0

10

20
Frequency (MHz)30

40

Figure 5.24: Noise spectra of the new photo-detector for PDH locking. The noise spectrum√
indicates that the bandwidth is ∼40 MHz and the equivalent input noise current is 0.14 pA/ Hz
at 18 MHz, which is suﬃcient for achieving a narrow linewidth in our setup.

130

Spectrum Analyzer

Photo-detector
NPBS

PBS

Signal Generator

λ/4

λ/2

ECDL

MirrorRF amplifier

to cavity

AOM
PBS

Fiber (10 m)

NPBS

Figure 5.25: Optical setup for measuring the phase noise in an optical ﬁber. An AOM is inserted
before the ﬁber for measuring the inﬂuence of the ﬁber. A part of the light is returned at the
NPBS on the Minus-K table. The NPBS is required for matching the polarization of the returned
light to that of the ﬁber. The beat note between the original light and the returned light isdetected with a high-speed photo-detector. The frequency of the returned light is shifted bytwice the modulation frequency of the AOM from the original light before the AOM.
and a dye laser). Therefore, lasers are mounted on another optical table and the lights fromthese lasers are carried with optical ﬁbers with a length of 10 m. The vibration of the ﬁber canmodulate the frequency of the light passing through the ﬁber with the Doppler eﬀect because
the ﬁber serves as numerous mirrors.
In order to check if the ﬁbers have an inﬂuence on the ﬂuctuation of the frequency in our
setup, we measured the phase noise in the optical ﬁber with a setup shown in Fig.5.25. With thissetup, we could measure the phase noise in the ﬁber as a beat note between the original light andthe returned light. If no phase noises are added, the beat note should be the frequency-doubled
signal of the RF signal driving the AOM derived from a signal generator. However, we observedthat the beat note was broadened to a few 100 Hz mainly due to a controller for the pump lightof the Ti: Sapphire laser (Fig.5.26). This controller added an intense vibration on the rack towhich the ﬁbers were ﬁxed. When we turned oﬀ the controller, the broadening was negligible.
The important observation in this measurement is that the ﬂuctuation of the center frequency
in an optical ﬁber is usually negligible and the beat note shows a nearly symmetric shape. When
we touched the ﬁber with a hand, the beat note showed an asymmetric shape, indicating thatthe ﬂuctuation of the center frequency in a ﬁber is signiﬁcant only when the ﬁber is strongly
vibrated. Thus, we concluded that the observed ﬂuctuation of the center frequency was not due
to the phase noise in the ﬁbers.
Although the observed ﬂuctuation of the frequency was not associated to ﬁbers, we ﬁnallydecided to introduce a system for cancelling the phase noise in the ﬁbers. For ensuring thatthe ﬁbers did not have any inﬂuence even if we touched the rack for adjusting electronics, it
was important to completely cancel the inﬂuence of the ﬁbers. The principle of cancellation is
simple: we have only to modulate the frequency of the AOM such that the observed beat note
between the original light and the returned light is exactly the same as the frequency-doubled
signal of the RF signal driving the AOM [133]. The feedback was carried out via phase-locked
loop. Fig.5.27 shows the overall setup for cancellation. For this purpose, a 200 MHz photodetector was developed just in the same way as explained in the section 5.3.3. The servo circuit
for controlling the voltage controlled oscillator (VCO) operating the AOM is shown in Fig.5.28.
We tried two types of servo circuit: one was with P gain (a normal inverting ampliﬁer) and a
RC low pass ﬁlter (R = 470Ω, C = 100µF) after the ampliﬁer and the other was with P andI gain and without a low pass ﬁlter. Both enabled stable locking, but we chose the latter one
because it had a much larger feedback bandwidth (see Fig.5.29).
The beat note with cancellation is shown in Fig.5.29. We observed a signiﬁcant suppression
131

-60
(a)
Amplitude (dBm)

-70
-80
-90
-100
-110
-120
-4

-2
0
2
Frequency - 220000.050 (kHz)4

Amplitude (dBm)

-60
(b)

-70
-80
-90
-100
-110
-120
-1.0

-0.5
0.0
0.5
Frequency - 220000.010 (kHz)

1.0

Figure 5.26: (a) Span 10 kHz. (b) Span 2 kHz. The controller for the pump light of the T:
Sapphire laser broadens the frequency of the light in the ﬁber (blue line). Just by turning oﬀ
the controller, the sidebands at a few 100 Hz are suppressed (red line). When we touch the ﬁber
with a hand, the beat note is further broadened and becomes asymmetric (green line in (a)).

160MHzMixer

RF amplifier
160MHzPhoto-detector
NPBS

Crystal Oscillator

Servo Circuit
VCO
80MHz

PBS

λ/4

λ/2

ECDL

MirrorRF amplifier

to cavity

AOM
PBS

Fiber (10 m)

NPBS

Figure 5.27: Electronic and optical setup for cancelling the phase noise in a ﬁber.

132

Figure 5.28: Servo circuit for the ﬁber phase noise cancellation. The integrator in the ﬁrst stage
serves as a kind of loop ﬁler and is required for stable locking.
of the phase noises introduced in the ﬁbers at frequencies below a few tens of kHz. The noise
slightly increased at higher than 100 kHz due to the feedback bandwidth of the VCO (∼180
kHz). With this condition, shaking the ﬁbers did not have any inﬂuence on the beat note.
Thereby, we could ensure that any vibrations on the ﬁber did not inﬂuence the measurement of
the linewidth of the lasers.

5.3.5

Improvements on optics

The improvements on the optical setup around the ULE cavity constituted a crucial part in theeﬀorts for making Raman lasers. We made numerous modiﬁcations and determined the ﬁnal
design which represented the lowest ﬂuctuation of the center frequency. Fig.5.30 shows the ﬁnal
design of the optical setup on the Minus-K optical table. Most of the optical components, except
for isolators, are shared between red and infrared lights. Thus, this setup can readily be used forboth the STIRAP experiments (red and infrared lights) and the measurement of the linewidth(two red lights or two infrared lights). The improvements on the optical and electrical setup
were carried out by measuring the linewidth between two infrared lasers independently locked
to the same cavity.
This design includes four important intentions. A part of them is speciﬁc to a two-color
light source. First, the reﬂected lights from the cavity are extracted with an isolator such that
we can obtain a maximum available error signal for both lights. In the previous design at 795
nm, only half of the light incident on the cavity was extracted with a PBS. The isolators arealso important for suppressing the back-reﬂection to the optical ﬁber, which can cause a large
ﬂuctuation of the frequency. We found that an isolation of -40 dB was required for suppressing
the ﬂuctuation to below 10 Hz in our case.
The second point is to use a Glan-Thomson polarizer before an EOM instead of a PBS anda λ/2 waveplate. The ﬂuctuation of polarization can induce the ﬂuctuation of the oﬀset of theerror signal. Therefore, it is important to purify the polarization with a high-extinction ratio
polarizer. The waveplate is removed because we can roughly match the polarization with theangle of the ﬁber and precisely with the angle of the polarizer. The angle of the polarizer andthe waveplate after the AOM are adjusted such that the oﬀset of the error signal vanishes.
The third point is to avoid any interference between optical components after the EOM.
The interference can ﬂuctuate the center frequency through the vibration of optical components
because the interference eﬀectively works an additional cavity before the ULE cavity. Largeinterferences appear between the cavity and the isolator (on the both surfaces of the crystal andthe polarizer), between the cavity and the photodiode for PDH locking, and between the cavity
and the polarizer for overlapping two lights. In order to avoid any interference, all the opticalcomponents are tilted with respect to the optical axis such that the ﬂuctuation of the output RF
133

Amplitude (dBm)

-40
-60

Amplitude (dBm)

-100

-50

0
50
Frequency -160000 (kHz)(b) Span 20kHz

-20

100

w/o cancellation
with cancellation

-40
-60
-80
-10

Amplitude (dBm)

w/o cancellation
with cancellation(a) Span 200 kHz

-20

-20

-5

0
5
Frequency - 160000 (kHz)10

w/o cancellation
with cancellation(c) Span 1kHz

-40
-60
-80
-0.4

-0.2
0.0
0.2
Frequency - 160000 (kHz)

0.4

0.6

Figure 5.29: Beat note with phase noise cancellation. (a) Span 200 kHz. (b) Span 20 kHz. (c)Span 1 kHz. The phase noises within a few tens of kHz are eﬃciently cancelled.

134

ULE cavity

Vacuum chamber
Fiber
GlanTaylor
f = 500mm

λ/2

GlanTaylor

λ/2

f = 1000mm
NPBS

ND 0.6
λ/4

EOM
PD1

PD3

λ/2

PD4

λ/2

Glan-Thomson

Glan-Thomson

EOM
λ/2

Glan- λ/2
Taylor

NPBS

PD2
Fiber

λ/4
ND 0.6

f = 1000mm

Figure 5.30: Optical setup on the Minus-K optical table mounting the ULE cavity. The opticalsetup enables us to lock both red (∼650 nm) and infrared (∼900 nm) lights to the same ULE
cavity with a minimum ﬂuctuation of the center frequency. In addition, we can measure thelinewidth between two red lasers and between two infrared lasers by locking them to the same
cavity. Except for the isolators, all the optical components are shared between the red and theinfrared lights.

135

Amplitude (dBm)

-20
-30
-40
-50
-60
-70
-80
-100

-50
0
50
Frequency - 2995216011 (Hz)100

Figure 5.31: Beat note between two master lasers. Large sidebands at 50 Hz indicate large
ground noises.
signal of the photo-detector is suﬃciently small. Of a particular problem is an isolator: due to its
small aperture with respect to its long body, they tend to be almost vertical to the optical axis.
For tilting the isolator, we remove the input polarizer and place a tilted Glan-Taylor polarizer
separately before the isolator. By entering the light from the cavity side, the output polarizer isrotated such that the transmission is suppressed to below -40 dB of the incident light. Note thatthe polarizer should be a Glan-Taylor type and not a PBS: Glan-Taylor polarizer has a small
but ﬁnite ﬁeld of view in which a high extinction ratio is obtained. We also have to take care of
the transmitted light after the ULE cavity such that the transmission scattered or reﬂected atsome components does not hit the photo-detector. We observed a large frequency ﬂuctuation
just by placing a black paper after the ULE cavity.
The fourth point is to stabilize the intensity after the Glan-Thomson polarizer. The ﬂuctuation of polarization in the optical ﬁber leads to an intensity ﬂuctuation of a few % after thepolarizer. The intensity ﬂuctuation can inﬂuence on the center frequency presumably due to theasymmetric shape of the error signal. In our case, the intensity after the polarizer is stabilizedwith the AOM before the ﬁber to below 0.2 %; the AOM has two roles of cancelling both phaseand polarization noises in the ﬁber.
With these improvements, the ﬂuctuation of the center frequency is suppressed to ∼20 Hz
by approximately one order of magnitude.

5.3.6

Improvements on electronics

The improvements on the electronic setup constituted the ﬁnal part of developing master lasers:
the ﬂuctuation of the beat note frequency was suppressed to a few Hz with them. Further, thesmall ﬂuctuation of the center frequency allowed us to conﬁrm that the linewidth between two
independently locked lasers is below 1 Hz. We stopped the development at this point because
we reached the limit of the resolution limit of our spectrum analyzer. There are two major
improvements: AC line noises and the servo circuit. In addition to these major modiﬁcations,
we made another minor change: the DBM for obtaining an error signal was changed from thepreviously used one (Mini-Circuits, ZAD-1H+) to another one with a high LO-IF isolation (MiniCircuits, ZAY-1+). With this modiﬁcation, the low-frequency pedestal was slightly suppressed.
At ﬁrst, the AC line noises were observed as large sidebands at 50 Hz (Fig.5.31). We
found that switching power supplies for driving high-power RF ampliﬁers added large noises to
the ground. Finally we exchanged all the switching power supplies to series regulators. Thissigniﬁcantly reduced the ground noise at high frequencies of more than 10 kHz, but the sidebands
at 50 Hz were still remaining in the beat note spectrum.
136

A signiﬁcant improvement was obtained by connecting the ground of the servo circuits (±15
V) to the ground of the lasers (optical table) with plain stitch copper wires with a cross sectionof more than 100 mm2 . Previously the ground of the servo circuits and the ground of lasers
were connected with narrower cables (three cables with a cross section of 3.5 mm2 ), but they
were not suﬃcient. With this connection, the sidebands were suppressed by more than 10 dB.
In addition, the ﬂuctuation of the center frequency was suppressed to below several Hz. At thisstage, we recognized that the center frequency was strongly inﬂuenced by the ground noise.
The second and ﬁnal major modiﬁcation was about the servo circuits. Fig.5.32 shows theﬁnal design of the servo circuit. In order to suppress regulation sidebands and to make thelock more stable, we added a diﬀerential (D) gain. In practice, the D gain was obtained justby adding a capacitor in parallel with the variable resistor before the integrator. The D gainincreased the band width from 700 kHz to 2.4 MHz and suppressed the regulation sidebands
by 10 dB. The bandwidth was presumably limited by the length of the RF cable (6 m) and theoptical ﬁber (10 m).
At this stage, the linewidth was a few Hz and the ﬂuctuation was several Hz. We suspectedthat the broadening of the linewidth as well as the ﬂuctuation was due to an insuﬃcient gain at
low frequencies. When we directly connected the output of the servo circuits to piezo modules
and locked the lasers without high-voltage ampliﬁers, the linewidth was broadened to ∼10 Hz.
This indicated that the gain for the piezo feedback was not suﬃcient against electronic noises.
Here the problem was the signal-to-noise ratio, and we could not improve it just by increasing
the gain. The possible source of the noise was the ﬁrst op-amp for buﬀering. Therefore, we
directly integrated the error signal to obtain the piezo feedback signal. The current feedback
signal was buﬀered with an op-amp (AD829) because the low-frequency (LF) integrator requires
the buﬀering op-amp. In this conﬁguration, we had to precisely adjust the oﬀset voltage of thebuﬀering op-amp to zero: without this adjustment, we could not lock simultaneously with thepiezo feedback and the LF current feedback. Fig.5.33 shows the beat note of two infrared lasers
locked with the circuit given in Fig.5.32(a). The ﬁnal design provided a better performance: theﬂuctuation of the center frequency was suppressed by a factor of 2 and the sidebands at 50 Hz
were suppressed by 10 dB. Further, the beat note shows a much clearer spectrum at frequencies
lower than 20 Hz. Although a ﬂuctuation of a few Hz during ∼10 s is still remaining, now we
can claim that the linewidth of our lasers is below 1 Hz. Our conclusion obtained here is thatthe ﬁrst buﬀering op-amp in a servo circuit can be the source of the noise at low frequencies. By
directly integrating the error signal without buﬀering, we can obtain a clean signal suited for
obtaining a linewidth of narrower than 1 Hz. Note that the linewidth mentioned in this context
does not include the ﬂuctuation due to the vibration of the cavity itself. If we take into accountthe vibration of the Minus-K optical table, we cannot claim that our lasers have a linewidth of
below 1 Hz. The measurement of the acceleration on the table gives us an estimation that thefrequency ﬂuctuation during ∼10 s is below 10 Hz.
The above development has been carried out at 901 nm. We can expect a better performanceat 655 nm because the ﬁnesse of the cavity at 655 nm is higher than that at 901 nm. We conﬁrmedthat the obtained results are also valid at 655 nm by monitoring the beat note between an ECDLand a dye laser locked to the same cavity independently (Fig.5.34). In this measurement, thedye laser was used because we prepared only the dye laser as a slave laser. At 655 nm, weobserved a beat note linewidth of ∼3 Hz and a center ﬂuctuation of ∼10 Hz. Although thesevalues are worse than those at 901 nm, we expect that they are due to the imperfect stabilizationof the dye laser. Regarding the instability of the dye laser, there are several possible reasonsfor the worse performance. First, the dye laser has a large intensity noise, which cannot becompletely removed. The sidebands at ∼1 kHz is due to the residual intensity modulation used
for stabilization inside the dye laser. The intensity noise with a low frequency can ﬂuctuate
the center frequency. Second, the dye laser is stabilized with an AOM by using a servo circuit
similar to our previous design (Fig.5.35). The noise arising from the ﬁrst buﬀering op-amp will
limit the linewidth as well as increase the ﬂuctuation.
137

Figure 5.32: Circuit diagram of the servo circuit for Raman lasers. The piezo feedback signal
is obtained by directly integrating the error signal. In contrast, the current feedback signal
is buﬀered such that the low-frequency integrator works well. All the signals are carried withcoaxial cables. For suppressing high frequency noises, power cables are ﬁltered with ferrite cores.
The box is directly connected to the laser ground with plain stitch copper wires.

138

Amplitude (dBm)

-20

(a)

-30
-40
-50
-60
-3

Amplitude (dBm)

Span 6MHz
RBW 30 kHz
Avg 20

-20

-2

-1
0
1
Frequency - 1497.608852 (MHz)2

3

Span 100kHzRBW 1 kHzAvg 10

(b)

-40
-60
-80

Amplitude (dBm)

-40

-20

40

Span 10kHz
RBW 100 Hz
Avg 5

(c)

-40
-60
-80
-4

Amplitude (dBm)

-20
0
20
Frequency - 1497608.852 (kHz)

-20

-2
0
2
Frequency - 1497608.852 (kHz)4

Span 100Hz
RBW 1 Hz
Avg 0

(d)

-40
-60
-80
-40

-20
0
20
Frequency - 1497608852 (Hz)40

Figure 5.33: Beat note between two infrared lasers independently locked to the ULE cavity. (a)Span 6MHz. (b) Span 100 kHz. (c) 10 kHz. (d) Span 100 Hz. This ﬁgure represents the ﬁnal
performance of our Raman lasers.

139

Although this measurement was not a direct evaluation of the master ECDL at 655 nm, we
concluded that the master laser at 655 nm was suﬃciently stable as well. With this measurement,
we ﬁnished the development of the master lasers.

5.3.7

Analog optical phase-locked loop

Now that we have suﬃciently stable master lasers, the remaining task is to develop suﬃciently
stable slave lasers. The development at 795 nm made it clear that the digital OPLL is notsuited for the present study. Thus, we decided to employ an analog OPLL with a double
balanced mixer [90]. The setup is similar to the setup for the digital OPLL (Fig.5.36). The
principle of the operation is simple. In contrast to the digital OPLL, in which a frequency
diﬀerence between the beat note and a microwave signal is detected digitally by counting thenumber of pulses, a frequency diﬀerence is detected with a double balanced mixer (DBM). The
signal obtained with a DBM is feed-backed to the slave laser, ﬁnally giving rise to a constantphase.
The analog OPLL is quite successful for ECDL. Fig.5.37 shows the beat note between two
ECDLs. Here the remarkable points are the wider bandwidth and the lower pedestal than in thedigital OPLL (Fig.5.12). Roughly speaking, the signal obtained with a DBM can be precisely
controlled such that the gain characteristics of the feedback signals match the system, resulting
in a better performance than in the digital OPLL. The servo circuit for the analog OPLL isshown in Fig.5.38. The analog OPLL is possible only with the current feedback for a short term(∼10 s). In order to compensate a long term drift, the error signal is integrated and is feedbacked to the piezo module. The phase-advance circuit after the DBM enables us to extend the
bandwidth of the current feedback to ∼4 MHz. The wide bandwidth as well as the integrationof the piezo feedback signal makes the lock quite robust against acoustic and vibrational noises.
The obtained stability is comparable to that of master lasers.
The servo circuit is diﬀerent from that for locking the master laser to a cavity because of
the diﬀerence in the frequency dependence of the error signal. The cavity signal has a lower
sensitivity to a lower frequency component, whereas the OPLL signal has a nearly ﬂat frequency
response. Hence, a narrow linewidth is obtained without integrating the error signal. The
integration of the piezo feedback signal is solely for achieving stable locking.
The most important aspect of the slave laser is that it can be scanned for a wide range withkeeping the narrow linewidth. In our setup, the slave laser is scanned by sweeping the frequencyof the microwave signal with a constant frequency step of up to 100 kHz. The basic idea is to
shift the frequency just after the STIRAP transfer is carried out. Considering the bandwidthof the current feedback of 4 MHz, we can expect that the duration required to reach a stablefrequency is within 10 µs. This duration is much shorter than the one-cycle duration of 100
ms. In this way, we can realize a scan with keeping the narrow linewidth during the most of
the time. The present method has another advantage in the accuracy of the obtained spectrum:
the frequency step is precisely determined by the microwave source. The inaccuracy of thespectrum arises only from the long-term stability of the frequency of the master laser. In theprevious spectroscopic studies given in the chapter 4, the frequency measurement was carriedout by calibrating the frequency with a cavity. This method is suited for scanning an extremely
wide range of more than several tens of GHz, but it has various origins of uncertainty, such as
errors in ﬁtting to the cavity spectrum, errors in the interpolation with a spline function, anderrors in the measurement of FSR. The present method based on sweeping a microwave sourceis particularly suited for the study to observe a narrow structure with a width of below 1 MHz.
The microwave sources for the analog OPLL are N5181A (Agilent technologies; 100 kHz–6
GHz) and E4420B (Agilent technologies; 250 kHz–2 GHz). At ﬁrst, we prepared E4420B and
SMC-100A (Rohde&Schwartz; 9 kHz–1.1 GHz) for this purpose. However, both were not suited
for scanning over a wide range: the former one can scan only 400 points, whereas the latter one
sometimes stops to output microwave signals for ∼500 µs. Therefore, we purchased a new signal

140

Amplitude (dBm)

-10
-20
-30
-40
-50
-3

Amplitude (dBm)

Span 6MHz
RBW 30 kHz
Avg 10

(a)

-20

-2

-1
0
1
Frequency - 1497.747634 (MHz)2

3

Span 200kHzRBW 1 kHzAvg 10

(b)

-30
-40
-50
-60

Amplitude (dBm)

-100

-20

-50
0
50
Frequency - 1497747.634 (kHz)Span 5kHz
RBW 30 Hz
Avg 5

(c)

-30
-40
-50
-60
-2

Amplitude (dBm)

100

-20

-1
0
1
Frequency - 1497747.634 (kHz)2

Span 300Hz
RBW 3 Hz
Avg 0

(d)

-40
-60

-150

-100

-50
0
50
Frequency - 1497747458 (Hz)

100

150

Figure 5.34: Beat note between two red lasers independently locked to the ULE cavity. (a) Span
6 MHz. (b) Span 200 kHz. (c) Span 5 kHz. (d) Span 200 Hz. The dye laser is stabilized with
an AOM, whose bandwidth is limited to ∼400 kHz due to the beam diameter in the AOM.

141

Figure 5.35: Circuit diagram of the servo circuit for locking a dye laser to a ULE cavity. This isa modiﬁed version of the our previous design (Fig.5.3). This circuit is used only for measuring
the linewidth of the master laser at 650 nm.

142

ECDL1

Photo-detector
λ/2Limiting Amplifier
ECDL2
PZT

Current
R
I

Servo circuit

Double Balanced
Mixer
L
Signal Generator

Figure 5.36: Setup for the analog OPLL between two ECDLs. The beat note between the masterand the slave laser is detected with a high-speed photo-detector. After being compared witha microwave signal, the beat signal is feed-backed to the slave laser. Although we also trieda phase detector (Mini-Circuits, ZRPD-1), a mixer (Mini-Circuits, ZFM-2-S+) gave a larger
bandwidth (∼4 MHz).

143

Amplitude (dBm)(a) Span 10 MHz
RBW 100 kHz

-10
-20
-30
-40
-50

Amplitude (dBm)

0

-20

2

8

10

(b) Span 100 kHz
RBW 1 kHz

-40
-60

60
Amplitude (dBm)

46Frequency - 75 (MHz)

-20

80
100
120
Frequency - 79900 (kHz)140(c) Span 10 kHz
RBW 100 Hz

-40
-60
-80

Ampitude (dBm)

96
-20

98
100
102
Frequency - 79900 (kHz)104(d) Span 100 Hz
RBW 1 Hz

-40
-60
-80
1.00

1.02
1.04
1.06
Frequency - 79999 (kHz)

1.08

Figure 5.37: Beat note between the master and the slave lasers at 901 nm. (a) Span 10 MHz.
(b) Span 100 kHz. (c) Span 10 kHz. (d) Span 100 Hz. Compared to the digital OPLL, a wider
bandwidth and a lower pedestal are obtained with the analog OPLL.

144

Figure 5.38: Servo circuit for the analog OPLL of two ECDLs. For the current feedback signal,
the error signal is not integrated but only ampliﬁed.

145

ECDL

Photo-detector
λ/2
AOM

VCO

Limiting Amplifier

Dye
Prescaler (1/4)
Fast

Slow
Servo circuit

R
I

Double Balanced
Mixer
L
Signal Generator

Figure 5.39: Setup for the analog OPLL at 655 nm. A dye laser is locked to the master laser
with the analog OPLL via a double-pass AOM. In order to increase the scanning range, thefeedback signal for the VCO controlling the AOM is also feed-backed to the dye laser itself suchthat the AOM always stays at around 80 MHz. A prescaler for dividing the frequency of thebeat note by 4 is inserted such that our 2.2GHz synthesizer (E4420B, Agilent technologies) can
cover the entire range.
generator, N5181A. All the spectra obtained in the chapter 6 are taken by scanning either the
red laser or the infrared laser with this signal generator. While one laser is scanned, the otherlaser is locked with E4420B.
The stability of the analog OPLL enables us to scan the slave laser for more than 2 GHz
without mode-hopping. The scanning range is limited only by the mode-hop-free range of the
ECDL itself. The signiﬁcant improvement is due to the following points. First, the feedback
gain is almost constant during a wide-range scan. The problem of the DC bias characteristics
of ceramic capacitors does not arise here because there is only a small DC component before
the coupling capacitor for the current feedback. The variation of the gain arises only from thedecrease in the magnitude of the beat note signal at frequencies higher than a few GHz. Second,the DC voltage for sweeping the frequency of the ECDL with the piezo module is obtained byintegrating a small DC signal from the DBM. This is limited only by the maximum output
voltage of the op-amp.
A similar setup for the analog OPLL was also constructed at 655 nm (Fig.5.39). The main
diﬀerence is that the slave laser is not a diode laser but a dye laser. The dye laser is stabilizedwith a double-pass AOM (80 MHz) placed just after the dye laser via a voltage controlled
oscillator (VCO). In addition, a home-made prescaler (Hittite microwave, HMC365S8GE) isinserted for dividing the beat note frequency by 4 such that E4420B can cover more than 2
GHz. The servo circuit for controlling the VCO is shown in Fig.5.40. In contrast to the OPLL
between two ECDLs, there is a signiﬁcantly larger pedestal in the dye laser than in the diodelaser (Fig.5.41); this is due to the limited feedback bandwidth of the AOM (∼300 kHz).
The AOM itself has a scanning range of only several tens of MHz. In order to achieve a146

Figure 5.40: Circuit diagram of the servo circuit for the analog OPLL of the dye laser.
wide scanning range, the feedback signal to the VCO is also feed-backed to the dye laser itself.
Our dye laser has a software feedback system, which can stabilize the frequency of the dye laser
slowly (up to a few 100 Hz). The principle of the operation is as follows. When the frequency ofthe microwave source is shifted, the frequency of the RF signal for the AOM is ﬁrst shifted witha change in the voltage to the VCO. Then the dye laser is slowly shifted such that the frequencyof the AOM is kept constant. In this way, we can scan the dye laser for up to 4 GHz without
mode-hopping. The scanning range is not limited by the dye laser, which has a mode-hop-freerange of 50 GHz, but by the change in the magnitude of the signal from the photo-detector for
observing the beat note.

147

Amplitude (dBm)

-10
-20

875 nm (two ECDL)
641 nm (ECDL&Dye)(a) Span 10 MHz
RBW 100 kHz

-30
-40
-50
-4

Amplitude (dBm)

-10
-20

-2

0
Frequency (MHz)24875 nm (two ECDL)
641 nm (ECDL&Dye)(b) Span 2 MHz
RBW 10 kHz

-30
-40
-50
-60
-1.0

-0.5

0.0
Frequency (MHz)

0.5

1.0

Figure 5.41: Comparison of the beat note between the infrared light and the red light. (a) Span
10 MHz. (b) Span 2 MHz. This measurement was carried out at 875 nm for the infrared lightand at 641 nm for the red light. For both lights, the master lasers are locked to the ULE cavity.
Due to the limited bandwidth of the AOM, there is a larger pedestal in the OPLL of an ECDLand a dye laser than in the OPLL of two ECDLs.

148

Chapter 6

Coherent transfer into therovibrational ground state
In spite of the successful demonstrations on Feshbach molecules near quantum degeneracy [50–52], a STIRAP transfer of photoassociated molecules into the rovibrational ground state is notstraightforward. Before starting the present study, we had a number of questions arising fromthe completely diﬀerent experimental conditions. Can we obtain a high transfer eﬃciency witha much lower intensity than in the previous studies? Does a magnetic ﬁeld gradient as well
as a residual bias magnetic ﬁeld inﬂuence on the coherence? Should we need a spin-polarized
sample for an eﬃcient transfer? What is the optimum pulse duration? What is the optimum
intensity ratio between two lights? What initial state should we choose? Can we ﬁnd thetwo-photon resonance without using two-photon dark resonance spectroscopy? Is a power of
100 mW prepared for the downward transition is suﬃcient? Does the higher temperature ofphotoassociated molecules than that of Feshbach molecules become a problem? Which one ofthe (3)1 Σ+ state and the (2)3 Σ+ state is suited for the present study? Can we directly detect
the produced ground-state molecules with REMPI? What should limit the transfer eﬃciency?
The answers to most of these questions are obtained in the present study, although a part ofthem still remain unanswered.
Our previous results on the transfer between two adjacent vibrational levels gave us an
expectation of a successful transfer also into the rovibrational ground state. However, thepresent situations are completely diﬀerent from our previous study on weakly bound levels; inthe previous studies, we already had good knowledge on the transition strengths, the two-photon
resonance frequency, and the ionization wavenumbers. In addition, the previous condition, in
which the intermediate state was the photoassociation level, is easier to work with: the large
Flanck-Condon overlaps with weakly bound level ensure large transition strengths, whereas thebroad natural linewidth of ∼12 MHz makes spectroscopy easy.
In this chapter, we ﬁrst point out the diﬃculties associated with the STIRAP transfer ofphotoassociated molecules (section 6.1). In order to circumvent a part of the diﬃculties, we developed a new spectroscopic method, which we named Spontaneous-decay Induced Double Resonance (SpIDR) (section 6.2). The SpIDR spectroscopy enabled a high-resolution spectroscopyof photoassociated molecules in a single state. In addition, SpIDR spectroscopy enabled us toobserve a two-photon dark resonance, which was a key phenomenon for measuring the unknown
properties of the transition to the rovibrational ground state (section 6.3). Based on the measurement with a two-photon dark resonance, we carried out a STIRAP transfer. By scanningthe pulse laser, we found several wavenumbers which can ionize molecules in the rovibrational
ground state (section 6.4). This was also the direct conﬁrmation that we have succeeded in the
STIRAP transfer of photoassociated molecules. The study on the multiple-transfers gave us
estimation on the single-step eﬃciency as 73 %. Here we found that the coherence during thetransfer was a few kHz, which was signiﬁcantly larger than we expected from the linewidth of
the Raman lasers. Therefore, we studied the coherence properties of the two-photon transition149

(section 6.5). The possible origin of decoherence was not the laser linewidth, but the pedestal
arising from a bad regulation. Particularly the dye laser had a large pedestal, resulting in themain cause of decoherence. In the end, we present the applications of the produced molecules(section 6.6). The theoretical treatment of coherent phenomena in a three-level system is givenin the section 6.7, in which we derive useful expressions for analyzing the observed behaviors.

6.1

Diﬃculties associated with photoassociated molecules

The ﬁrst attempt to reach the rovibrational ground state was carried out with the triplet scheme:
we chose the v ′′ = 21 level of a3 Σ+ as an initial state and the v ′ = 44 level of (2)3 Σ+ as anintermediate state. This was because the absolute value of the transition strength could roughly
be estimated from the previous experiments on 40 K87 Rb [50]. In contrast to the triplet scheme,
the all-singlet scheme had an uncertainty on the transition strength between the (3)1 Σ+ state
and the rovibrational ground state when we started the experiment. There was a possibility
that the transition was too weak to be used for an eﬃcient transfer. We discuss this point indetail herein.
As already we discussed in the section 4.2.6, we had a good estimation on the Franck-Condon
factor based on our accurate potential. However, the Franck-Condon factor does not directlymeans a transition dipole moment. The transition dipole moment between two molecular states
d can be approximately written by the R-dependent transition dipole moment D(R) and theFranck-Condon factor f as√
d = D(R0 ) f(6.1)

where R0 is the internuclear distance at which the two wavefunctions of the ground and theexcited states have a maximum overlap. In our case, R0 ∼10 Å for the upward transition andR0 ∼4 Å for the downward transition. Here the problem is that an ab initio calculation on
D(R) gives a quite small value of below 0.1 ea0 at a short internuclear distance [109]. If thecalculation slightly deviates to a lower value, the expected transition dipole moment is nearlyzero even if the Franck-Condon factor is large. This is why initially we did not employ the
all-singlet scheme.
The triplet scheme has an advantage in its known transition strength. However, there wasanother problem in the triplet scheme. In our experiment, photoassociated molecules were nottrapped. Therefore, we had to carry out a STIRAP transfer immediately after the magnetic ﬁeld
for a MOT was turned oﬀ. In such a situation, a sudden change in the magnetic ﬁeld gradientinduced a current in the metal chamber as well as in the compensation coils, making a residual
magnetic ﬁeld of ∼0.1 G. Triplet molecular states can readily be inﬂuenced by these residual
magnetic ﬁelds with a sensitivity of ∼3 MHz/G. At ﬁrst, we expected a successful STIRAP
transfer based on the estimation that the energy splitting among magnetic sublevels of ∼0.3
MHz is smaller than the typical two-photon width of ∼1 MHz. However, we could not observeanything in spite of intensive eﬀorts. At this stage, we recognized that we should also think of
the inﬂuence of magnetic ﬁelds on the coherence during a two-photon transfer. In contrast to theprevious experiments on Feshbach molecules, in which a high magnetic ﬁeld was applied, there
were only residual magnetic ﬁelds of ∼0.1 G in our experiments. The Raman laser can couple
all the magnetic sublevels because the expected Rabi frequency is ∼1 MHz. In such a situation,
we can consider that the splitting due to magnetic ﬁelds corresponds to a broadening of thelinewidth of the Raman lasers of the order of ∼0.1 MHz. In other words, the magnetic ﬁelds can
induce spin precession and loose the coherence of the two-photon transition at a rate of ∼0.1
MHz. This fast decoherence makes the transfer eﬃciency quite low. With the consideration on
this possibility, we stopped to proceed the experiments in the triplet scheme and switched tothe all-singlet scheme.
In the singlet scheme, we had a wide variety of signiﬁcant diﬃculties. One of them was the150

Ion counts (arb.units)

25
20
15
10
5

100µW
200µW
500µW

0
-1000

-960
-920
-880
Offset frequency of L1 (MHz)Figure 6.1: Depletion spectra of the (3)1 Σ+ state obtained with a low laser intensity. Ion counts
in the v ′′ = 91 level of X1 Σ+ are plotted with respect to the oﬀset frequency between the master
laser and the slave laser of the infrared light. The initial rotational state is the J ′′ = 2 level.
When we reduce the intensity, the dip becomes small and ﬁnally disappears. The minimum
width observed is ∼10 MHz, which is much larger than the expected natural linewidth of ∼300
kHz.
broad width of the depletion spectrum. Although depletion spectroscopy for photoassociatedmolecules is robust and straightforward, the observed width was much broader than the expectednatural linewidth. The ab initio calculation on the transition dipole moment gives estimation on
the natural linewidth of the (3)1 Σ+ state as ∼300 kHz [109]. However, we never observed sucha narrow linewidth (Fig.6.1). When we reduced the intensity, the dip ﬁnally disappeared at a
relatively large width of ∼10 MHz. The broad width compared to the natural width should beinterpreted as saturation broadening. The saturation broadening can be a signiﬁcant problem
in a STIRAP experiment because we have to specify the frequency of the one-photon transition
with an accuracy of 1MHz, which is the expected width of the two-photon resonance.
The other signiﬁcant diﬃculties were the existence of a number of uncertain quantities. In
addition to the transition strengths, we did not know the precise frequency of the two-photon
resonance and the wavenumber of the pulsed laser for ionizing the ground-state molecules. The
absence of the knowledge on the transition strengths made it diﬃcult to determine the intensity
ratio between the red light and the infrared light as well as the duration of the STIRAP transfer.
The attempt to observe a STIRAP signal was thus the multi-dimensional search of the optimum
values for these parameters. The key process to circumvent these diﬃculties should be twophoton dark resonance spectroscopy (or electromagnetically induced transparency, EIT).
The theoretical descriptions on a two-photon dark resonance are given in the section 6.7.
Reviews on this topic are found in Ref. [134–136]. Here we describe the physical interpretationof the phenomenon. We consider a three-level system as shown in Fig.6.2. When the laser 2 ison two-photon resonance and the condition Ω2 ≫ Ω1 is satisﬁed, the population is trapped ina coherent dark state consisting of the initial state and the ﬁnal state, which does not scatterphotons. Thus, a two-photon dark resonance appears in a one-photon absorption spectrum
as a dip in which absorption is suppressed. Two-photon dark resonance spectroscopy enables
us to determine the precise frequency and the transition strength of the two-photon resonancewith monitoring the population in the initial state. In this process, we do not need to ﬁnd
the ionization wavenumber for the rovibrational ground state. In addition, a two-photon darkresonance can be observed without accurate information on the transition strengths. This meansthat we do not have to take care of the transfer duration and the intensity ratio, which are
speciﬁc to a STIRAP transfer. In this way, a two-photon dark resonance makes the situation
quite simple. Once it is observed, the remaining uncertainty is only the question on whether we
151

Excited state(3)1Σ+,Ωp
Laser1 (IR)Probe lightΩc
Laser2 (Red)Control light

Initial stateX1Σ+, =91, =0

Final stateX1Σ+, =0, =0

Figure 6.2: A three-level system relevant to the present study. The laser 1 is called a probe
laser, whereas the laser 2 is called a control laser. When the control laser is on two-photon
resonance, the population is kept in a dark state. As a result, the absorption of the probe laser
is suppressed.
can directly ionize the ground-state molecules.
Here is the largest problem in this study: a twp-photon dark resonance had not been observedfor photoassociated molecules. In the long history of more than 15 years of photoassociation,
coherent phenomena such as a two-photon dark resonance and STIRAP were not reported forphotoassociated molecules. Although a coherent coupling between an atomic state and a weakly
bound molecular state was reported [71–74], photoassociated molecules had been used only forthe one-photon spectroscopy of excited states and not for the spectroscopy of the ground state.
In spite of our intensive eﬀorts, we could not observe a two-photon dark resonance with thepreviously used method for observing one-photon transitions, depletion spectroscopy.
Why we cannot observe a two-photon dark resonance for photoassociated molecules? This
question had been at the center of this study. To our knowledge, nobody answered to thisquestion before this study. We found that the answer to this question is related to the saturation
broadening of the depletion spectrum. In order to clarify the points, we compare the propertiesof photoassociated molecules and Feshbach molecules for which two-photon dark resonances
have been observed.
Due to the coherent nature of a Feshbach resonance, which arises from the interaction between a single atomic Zeeman level and a single molecular Zeeman level, Feshbach molecules
are pure in terms of rotational, hyperﬁne, and Zeeman levels. Feshbach molecules in a shallow
optical trap with a typical depth of a few µK immediately escape from the trap once they absorb
a photon. The number of molecules can be precisely measured via absorption imaging. Thus,

Table 6.1: Comparison between photoassociated molecules and Feshbach molecules.
Photoassociated molecules
Feshbach molecules
Trap
Untrapped
TrappedDensity
Low (∼106 cm−3 )High (∼1012 cm−3 )Detection
Ionization
Absorption imaging
Internal states Rotational, Hyperﬁne, Zeeman
Pure (single)

152

Population in the initial state
(a)

(b)

(c)

Probe laser frequency
Figure 6.3: Schematic of the inﬂuence of saturation broadening on a two-photon dark resonance.
(a) The probe laser is suﬃciently weak. (b) The probe laser is moderately intense. (c) The
probe laser is intense. When the one-photon spectrum is broadened by saturation broadening,
the transparency peak arising from the two-photon dark resonance is smoothed out.
depletion spectroscopy on Feshbach molecules has a quite high signal-to-noise ratio for a single
quantum state. A very low intensity is suﬃcient for detecting the resonance.
The situation is completely diﬀerent for photoassociated molecules. The spontaneous decay
process of photoassociation distributes molecules over multiple vibrational, rotational, hyperﬁne,
and Zeeman levels. The photoassociated molecules can only be observed with ionization, which
can resolve vibrational structures but cannot resolve other structures. Thus, the ion counts always include signals from various quantum states. The resonance is detected as optical pumping
to other vibrational levels rather than an escape from a trap. Optical pumping is less eﬃcientthan escapes from a trap if the Franck-Condon factor of the transition is large. These aspects
result in a low signal-to-noise ratio for a single quantum state, requiring a rather high intensity
for observing the resonance.
Now we can give an answer to the question on why we cannot observe a two-photo darkresonance for photoassociated molecules. In the depletion spectroscopy for photoassociatedmolecules, the spectral width is always broadened due to saturation broadening. In such asituation, a two-photon transparency peak in the one-photon spectrum disappears (Fig.6.19).
Although we do not introduce a theoretical framework for explaining the phenomenon, we canintuitively understand this behavior just by considering the relation between a transparency at
a weak limit and an observed depth.
In summary, there are three major obstacles in our STIRAP experiments. In Fig.6.4, these
problems are schematically shown with a level diagram for the relevant molecular states. The
diﬃculties of observing a STIRAP signal in our experiment are summarized as follows. Due totheir low density, photoassociated molecules cannot be observed with absorption imaging. A
REMPI ionization with a pulsed laser is required for detecting photoassociated molecules. In

153

Excited state(3)1Σ+,

Wavemeter ~ 60 MHz> Two-photon width ~ 0.4 MHz
Transition strength is unknown

Spectral width ~ 10 MHz> Natural width ~ 0.3 MHz

KRb+
Initial state=91, =0

Unknown if the rovibrational
ground-state can be ionized

X1Σ+,Good wavelengths are unknown
Final stateX1Σ+, =0, =0
Figure 6.4: Three signiﬁcant problems towards the STIRAP transfer of photoassociated molecules. The ﬁrst problem is the broad width of depletion spectra (∼10 MHz) which is much larger
than the expected natural width (∼300 kHz). The second problem is the unknown parameterson the two-photon transition. The precise frequency of the two-photon resonance and the transition strength for the downward transition are not known. The third problem is about the
REMPI detection of molecules in the rovibrational ground state. The ionization wavenumber
as well as whether it can be eﬃciently ionized are not known.

154such a situation, the broad linewidth of the pulsed laser results in the simultaneous detection ofmolecules in multiple rotational and hyperﬁne levels distributed by spontaneous decays. Thus,
the depletion spectroscopy combined with REMPI ionization suﬀers from a bad signal-to-noise
ratio due to the large ﬂuctuation of the oﬀset ion counts. In order to observe a large depletion,a high intensity is required, thereby causing saturation broadening and making it diﬃcult toobserve a narrow natural linewidth of the (3)1 Σ+ state. The saturation broadening also makes
it diﬃcult to observe a two-photon dark resonance which is necessary for ﬁnding the two-photon
resonance and measuring the transition strength. Finally the lack of knowledge on the REMPI
ionization of the rovibrational ground-state molecules makes it diﬃcult to know whether atransfer is successful or not. Although in principle a STIRAP transfer should also be detected
as a decrease in the initial population, the large ﬂuctuation of the oﬀset ion counts will make itdiﬃcult to observe it.
In a word, the simultaneous ionization of multiple quantum states is at the center of the
problem.

6.2

SpIDR spectroscopy

The understanding on the underlying physics of the spectroscopy for photoassociated molecules
gave us a possible route towards the observation of a two-photon dark resonance. What weneed here is a method for observing a one-photon transition from photoassociated moleculeswith a suﬃciently low intensity. The problem of depletion spectroscopy lies in the ﬂuctuationof the population in the initial state. Therefore, we should observe a signal which appears as
a result of excitation. An intuitive quantity is the number of spontaneously emitted photons.
If we ﬁlter out the MOT light, the photoassociation light, and the infrared light for excitation,
an excitation can be eﬃciently detected via a photon counting device. In principle, this scheme
should work well, but this scheme requires a special lens system placed near the molecules foran eﬃcient collection of emitted photons. It was diﬃcult to install such a lens system in our
setup.
Here we reach the next possibility: if we can eﬃciently detect the molecules formed withspontaneous emissions, we should be able to use them for detecting the resonance (Fig.6.5).
Although the scheme of detecting spontaneously distributed molecules is similar to photoassociation spectroscopy, it is a non-trivial problem whether we can apply this scheme for detecting
deeply bound molecular states. In photoassociation spectroscopy, spontaneous emissions form
only several weakly bound levels which have high Franck-Condon factors with the photoassociation level. By contrast, deeply bound excited molecules can decay to a wide variety ofvibrational levels in the ground state. The typical Franck-Condon factor from an excited state
to the ground state is between 10−1 and 10−4 , indicating that only several % of the excitedmolecules can decay to a single vibrational level.
In order to judge if the above scheme is available for the spectroscopy of the (3)1 Σ+ state,
we calculated the Franck-Condon factors between the v ′ = 41 level and all the vibrational levelsin the X1 Σ+ state (Fig.6.6). Spontaneous decays to the triplet state can be neglected sincethe (3)1 Σ+ state is a purely singlet state. The calculation gives a rough estimation that theexcited molecules should dominantly decay to the v ′′ = 70–85 levels. In addition, we can see two
important points in this plot. First, the deeply bound levels of v ′′ = 70–85 are not populated
in the photoassociation process. Therefore, the ion signal for these levels does not have any
oﬀset count and is free from the ﬂuctuation. Second, the v ′′ = 70–85 levels have a much higher
Franck-Condon factors than the photoassociated molecules (v ′′ = 90–93). This means that theexcited molecules do not return to the initial state, enhancing the detection eﬃciency.
Note that this calculation does not directly indicate the expected population in the groundstate because of the R-dependence of the transition dipole moment. If we take into account thefact that the transition dipole moment is smaller at a short internuclear distance than at a large
internuclear distance by one order of magnitude [109], the population in the vibrational levels
155

Excited state(3)1Σ+,

Excited state(3)1Σ+,

KRb+

Monitor

KRb+
Spectroscopy

Spectroscopy
MonitorInitial state=91, =0

Initial state=91, =0

X1Σ+,

X1Σ+,

Deeply bound levelX1Σ+, =70-80(a) Depletion spectroscopy(b) SpIDR spectroscopy

Figure 6.5: Schematic comparison of two spectroscopic methods. (a) Depletion spectroscopy. (b)SpIDR spectroscopy. In depletion spectroscopy, the population in the initial state is monitored.
In such a situation, a large number of rotational and hyperﬁne levels constitute large oﬀset ion
counts and degrade the signal-to-noise ratio. In SpIDR spectroscopy, number of deeply boundmolecules formed by spontaneous decays from the excited state is monitored. A high signalto-noise is achieve in this scheme because the deeply bound levels are not populated before
excitations.

156

-1

Franck-Condon factors

10

-2

10

-3

10

-4

10

-5

10

0

20
40
60
80
1 +
Vibrational quantum number of X Σ

100

Figure 6.6: Franck-Condon factors between the v ′ = 41 level of (3)1 Σ+ and all the vibrational
levels of X1 Σ+ . For this calculations, the accurate potential of the (3)1 Σ+ state obtained in thisstudy and an experimentally determined potential of X1 Σ+ [108] are used. The Fracnk-Condon
factors are plotted with respect to the vibrational number of the ground state. This plot gives
a rough estimation on which vibrational levels in the ground state are populated as a result ofspontaneous decays from the v ′ = 41 level of (3)1 Σ+ .
of v ′′ = 70–85 should be much larger. These estimations imply that the scheme of detecting an
excitation with molecules formed by spontaneous decays should work well for the v ′ = 41 levelof (3)1 Σ+ .
In this scheme, it is crucial to ﬁnd wavenumbers which can eﬃciently ionize the newly formed
molecules. The ionization eﬃciency directly inﬂuences the signal-to-noise ratio of the spectrum.
Based on the knowledge that photoassociated molecules are eﬃciently ionized in the red regionof 16400–16600 cm−1 , we searched the ionization wavenumbers for the deeply bound molecules
at slightly higher wavenumbers. We found several good wavenumbers by scanning the pulsed
laser while the excitation laser at 875 nm was irradiated (Fig.6.7). The excitation laser isresonant to the transition v ′ = 41, J ′ = 1, (3)1 Σ+ ← v ′′ = 91, J ′′ = 2, X1 Σ+ . Therefore, thediﬀerences between the two spectra indicate deeply bound molecules formed by the excitationof photoassociated molecules in the v ′′ = 91, J ′′ = 2 level of X1 Σ+ . In terms of achieving
a high signal-to-noise ratio, of particular importance is that the pulsed laser does not ionize
photoassociated molecules. We selected the wavenumber of 16578 cm−1 which gives a high ion
signal for a deeply bound level and a low ion signal for photoassociated levels.
When the pulsed laser was ﬁxed to 16578 cm−1 and the excitation laser was scanned, we
obtained a spectrum indicating the increase of deeply bound molecules due to an excitationfrom photoassociated molecules. We named this new spectroscopic method as Spontaneousdecay Induced Double Resonance (SpIDR) spectroscopy. SpIDR spectroscopy enabled us toobserve a one-photon transition with a much lower intensity and a much shorter interaction timethan depletion spectroscopy. In depletion spectroscopy, a power of 100 µW and an interaction
time of 10 ms were required. In contrast, a power of 5 µW and an interaction time of 400 µs
were suﬃcient in SpIDR spectroscopy. As a result, the excited states could be observed withan unprecedented narrow spectral width, revealing previously unobserved structures (Fig.6.8).
Although the typical ion counts of a SpIDR spectrum (∼2) was much lower than that of adepletion spectrum (∼20), a high signal-to-noise ratio was achieved due to the low oﬀset ion
counts (∼0.6). In the present study, SpIDR spectra are averaged over 20 points for suppressing
the inﬂuence of the shot-to-shot ﬂuctuation of the signal due to the ﬂuctuation of the gain of
the MCP and the intensity of the pulsed laser.
When we increased the intensity, the spectral width broadened and ﬁnally the ﬁne structure
157

35

With 875nm
Without 875nm

Ion counts (arb.units)

30
25
20
15
10
5
0
16500

16520

16540
16560
-1
Ionization frequency (cm )

16580

16600

Figure 6.7: Ionization spectra obtained with and without the excitation laser at 875 nm. The
spectrum without irradiating the excitation laser arises solely from photoassociated molecules,
whereas that with the excitation laser includes the signals of deeply bound molecules formed byspontaneous decays from the excited state.
was smoothed out (Fig.6.9). The optimum intensity and interaction time were determined to be
4.5×102 µW/cm2 and 400 µs, respectively, such that saturation broadening was not signiﬁcant.
The minimum observed width was ∼400 kHz. If we consider the Doppler broadening of ∼100
kHz and the natural linewidth of ∼300 kHz, we obtain a crucial implication: SpIDR spectroscopy
enables us to observe a transition with negligible saturation broadening.
The signal-to-noise ratio of a SpIDR spectrum is dependent on the beam diameter of thepulsed laser (Fig.6.10). The highest signal-to-noise ratio is obtained when the diameter of thepulsed laser is close to that of the excitation laser. A too large or a too small diameter of thepulsed laser results in a relatively large oﬀset ion counts from photoassociated molecules.
The structures revealed with SpIDR spectroscopy are the hyperﬁne structures in both theground and the excited states. The hyperﬁne structure in the ground state arises from theinteraction between closely-lying singlet and triplet molecular states and is of the order of 10
MHz. The magnitude of splitting is determined by the nuclear dipole moments. The structureis labeled by the total angular momentum F ′′ = IRb + IK as F ′′ = 1, 2, 3, 4. By contrast, thehyperﬁne structure in the excited state arises from the interaction between rotational angularmomentum and nuclear spins and is of the order of 1 MHz. The magnitude of splitting isdetermined by the nuclear electric quadrupole moments. Although we can not neglect thenuclear spin of 41 K, the total angular momentum T ′ = J ′ + IRb is a good quantum number fordescribing the structure. The observed structure can be roughly labeled as T ′ = 5/2, 3/2, 1/2,
although the T ′ = 1/2 level appears to be further split by the nuclear spin of 41 K. The observed
hyperﬁne structure can be quantitatively understood with a formulation for a deeply boundsinglet molecule in the ground state [137] because the (3)1 Σ+ state is a purely singlet state.
Here the interaction between the rotational angular momentum and the electric quadrupole
moment is the dominant mechanism for the hyperﬁne structure. The matrix elements of the
Hamiltonian are given as follows:

158

2.0

12
10

1.5

8
6

1.0

4
2

0.5

0
-1280

SpIDR counts (arb.units)

Depletion spectrum
SpIDR spectrum
SpIDR ion counts (arb.units)

Ion counts in the v''=91 level (arb.units)(a) Wide range
14

-1260
-1240
-1220
Offset frequency of L1 (MHz)-1200(b) Central structure
2.0
1.5
1.0
0.5
-1240

-1238
-1236
-1234
Offset frequency of L1 (MHz)-1232

Figure 6.8: SpIDR spectrum of the transition v ′ = 41, J ′ = 1 of (3)1 Σ+ ← v ′′ = 91, J ′′ = 0
of X1 Σ+ . (a) Depletion spectrum and SpIDR spectrum in a wide range. (b) SpIDR spectrum
for the central structure. In depletion spectroscopy, a much larger intensity (4 mW) and amuch longer interaction time (10 ms) is required for observing the transition, resulting in amuch broader spectrum. Due to the low power (5 µW) and the short interaction time (400µs) required for SpIDR spectroscopy, the SpIDR spectrum has a much narrower width. SpIDR
spectroscopy reveals previously unobserved structures in both ground and excited states.

159

7

2

1.8 mW/cm , 10 ms
6
3.6 mW/cm
2.4 mW/cm

Ion counts (arb.units)5

1.8 mW/cm
4
1.6 mW/cm
1.3 mW/cm

2

2

2

2

2

3
2

1.0 mW/cm

20.78 mW/cm
220.48 mW/cm

20.19 mW/cm
120.10 mW/cm

-1240

-1238

-1236
-1234
-1232
Offset frequency of L1 (MHz)-1230

Figure 6.9: Power broadening of SpIDR spectrum. Except for the uppermost line, the interaction
time is taken as 400 µs. When we increase the intensity, the spectral width broadens. When
we reduce the intensity, the spectrum disappears at a width of ∼400 kHz. This is consistent
with the Doppler broadening of ∼100 kHz and the natural width of ∼300 kHz. The optimum
intensity is determined such that saturation broadening is not signiﬁcant.

160

Ion counts (arb.units)

1.8

BD 1mm
BD 1.5mm
BD 0.5mm

1.6
1.4
1.2
1.0
0.8
0.6
0.4

-1237.0 -1236.5 -1236.0 -1235.5 -1235.0 -1234.5 -1234.0
Offset frequency (MHz)Figure 6.10: SpIDR spectrum with a diﬀerent beam diameter for the pulsed laser. The signalto-noise ratio of a SpIDR spectrum depends on the beam diameter of the pulsed laser. The
highest signal-to-noise ratio is obtained at a diameter of ∼1 mm.

< I1 , I2 , J, F1 , F |HeQq |I1 , I2 , J ′ , F1′ , F ′ >√(eQq)1′(−1)J +J+F1 +I1 (2J + 1)(2J ′ + 1)
= δF F ′ δF1 F1′
4
{
}(
)(
)−1
′
I1 J F1J 2 J′
I1 2 I1
×J I1 20 0 0−I1 0 I1√
(eQq)2(−1)F +I1 +I2 +1 (2J + 1)(2J ′ + 1)(2F1 + 1)(2F1′ + 1)
+ δF F ′
4
{
}{
}(
)(
)
I2 F1′ F
F1 J I1J 2 J′
I2 2 I2
×F1 I2 2J ′ F1′ 2
0 0 0−I2 0 I2

(6.2)

where I1 and I2 denote the nuclear spins, (eQq)1 and (eQq)2 are the electric quadrupole constants for each nuclei, F1 = J + I1 , and F = F1 + I2 . Here we neglected the nuclear spin-spin
interaction terms with c1 , c2 , c3 , c4 because they are of the order of 1 kHz [137, 138]. The comparison between the observed spectra and the calculations is shown in Fig.6.11. We found agood agreement between the observed spectra and the calculated spectrum when the electric
quadrupole constants are set as (eQq)Rb = −6.8 MHz and (eQq)K = −2.8 MHz. These values
are approximately 4 times larger than those in the ground state X 1 Σ+ . The selection rule on
F , ∆F = 0, ±1, was also conﬁrmed from this comparison.
SpIDR spectroscopy enables us to systematically compare the structure and the population in
the initial levels. Photoassociation distributes molecules over multiple rotational and hyperﬁnelevels. Before we developed SpIDR spectroscopy, we could not measure the diﬀerence in thehyperﬁne structure and in the population between two rotational levels of J ′′ = 0 and J ′′ = 2.
Here a particularly interesting question is whether the J ′′ = 0 level and the J ′′ = 2 level havethe identical hyperﬁne structures. As explained in the section 4.3.5, we have assumed thatrotational angular momentum is decoupled from other angular momentum in a weakly boundmolecule. However, this assumption is not trivial. In fact, SpIDR spectroscopy shows thatrotational angular momentum is strongly coupled to the nuclear spin and forms complicatedhyperﬁne structures in the excited state (Fig.6.8). In order to conﬁrm that the assumption
is appropriate, we compare the SpIDR spectrum from the J ′′ = 0 level and that from the
J ′′ = 2 level (Fig.6.12). The observed structures are almost identical, indicating that rotational
angular momentum can be treated as independent from other angular momentum. However, the161Ion counts (arb.units)

2
0

2.5
F''=3
2.0

Total angular momentum

4

1.5
1.0

F''=2

0.5

F''=1
F''=0

0.0
-1268
-1264
-1260
Offset frequency of L1 (MHz)-1256

Figure 6.11: Assignment of the hyperﬁne structure in the excited state. SpIDR spectra for the
F ′′ = 0, 1, 2, 3 levels of the v ′′ = 91, J ′′ = 0 level are compared with a calculated spectrum (blacklines). The spectra for the F ′′ = 1, 2, 3 levels are shifted by -8.852, -27.089, and -55.960 MHz,
respectively. These values correspond to the separation between these levels and the F ′′ = 0 level
and are measured by STIRAP spectroscopy (section 6.6.3). The calculated spectrum is obtainedby setting the electric quadrupole constants as (eQq)Rb = −6.8 MHz and (eQq)K = −2.8 MHz.

162

Ion counts (arb.units)

Spectrum from the J''=2 level
Spectrum from the J''=0 level

2.5
2.0
1.5
1.0
0.5
-1280

-1260

-1240
-1220
-1200
Offset frequency of L1 (MHz)-1180

Figure 6.12: Comparison between the SpIDR spectrum for the J ′′ = 0 level and that for the
J ′′ = 2 level. The spectrum for the J ′′ = 2 level is shifted by -119.08 MHz such that thelargest peak in the F ′′ = 2 level coincides with the F ′′ = 2 level in the spectrum for the J ′′ = 0
level. The hyperﬁne structures for the J ′′ = 0 level and those for the J ′′ = 2 level are identical,
suggesting that coupling of the rotational angular momentum to the nuclear spins is negligible
for the weakly bound level.
diﬀerence in the population among the hyperﬁne levels implies the possibility of the coupling
between rotational angular momentum and the nuclear spins.
Thus far, we have focused on the v ′′ = 41, J ′ = 1 level of (3)1 Σ+ . The successful results on
this level raise a new question: is SpIDR spectroscopy available for other states? This is a nontrivial question because SpIDR spectroscopy relies on the existence of deeply bound molecularstates to which most of the excited molecules decay. Our answer to the question is yes for somestates, but no for the other states. When we applied SpIDR spectroscopy to the (2)3 Σ+ statefor investigating the hyperﬁne structures of this state, we found that SpIDR spectra could beobtained for the v ′′ = 44 level, but could not be obtained for the v ′′ = 55 level. We have notchecked the availability for the other states.
The SpIDR spectroscopy for the (2)3 Σ+ state was realized with a completely diﬀerent ionization wavenumber from that for the (3)1 Σ+ state. Fig.6.13 shows the ionization spectra ataround 15250 cm−1 with and without the excitation laser resonant to the v ′′ = 44 level of(2)3 Σ+ . This spectra were taken with molecules in the v ′′ = 91, J ′′ = 2 level of X1 Σ+ . Due tothe simple vibrational structure of the observed spectra, we could deﬁnitely assign the spectrum.
The spontaneous decays from the v ′ = 44 level of (2)3 Σ+ formed molecules in the v ′′ = 7–11
levels of a3 Σ+ . These molecules were ﬁnally ionized through a resonant excitation to the v ′ = 15level of (3)3 Σ+ .
As in the case of the study for singlet states, the SpIDR spectroscopy for the triplet states
revealed previously unobserved hyperﬁne structures. The details on the results have alreadygiven in the section 4.3.5. In summary, we observed the hyperﬁne structure of the order of 10
MHz induced by the nuclear spin of 41 K for both the (2)3 Σ+ state and the a3 Σ+ state. The
minimum observed width was ∼7 MHz, which is in good agreement with the expected naturallinewidth of the (2)3 Σ+ state.
Another interesting application of SpIDR spectroscopy is to directly show the existence of
closely-lying singlet and triplet states in the ground state. The closely-lying two states with
diﬀerent symmetries are of great interest in terms of the precision measurement of the electronto-proton mass ratio [28]. Although KRb molecule is not necessarily suited for this purpose, it isimportant to show the existence of such levels. The SpIDR spectroscopy clearly shows that thev ′′ = 91 level of X1 Σ+ and the v ′′ = 22 level of a3 Σ+ are closely-lying with each other (Fig.6.14

163

Ion counts (arb.units)v''=8Without 901nm
With 901nm

4

v''=7

3
v''=11
2v''=10

v''=9

1
0
15220

15230

15240
15250
15260
15270
-1
Ionization wavenumber (cm )15280

Figure 6.13: Ionization spectrum at around 15250 cm−1 with and without the excitation laser.
Molecules in the v ′′ = 91, J ′′ = 2 level of X1 Σ+ are excited to the v ′ = 44, J ′ = 1, T ′ = 5/2
level of (2)3 Σ+ . The excited molecules decay mainly to the v ′′ = 7–11 level of a3 Σ+ , which are
ﬁnally ionized through a resonant excitation to the v ′ = 15 level of (3)3 Σ+ .
in the section 4.3.6). The triplet component included in the v ′′ = 91 level of X1 Σ+ allows us
to observe the transitions to the (2)3 Σ+ state. The measured separation between the v ′′ = 91,
J ′′ = 2 level of X1 Σ+ and the v ′′ = 22, N ′′ = 2 level is 2.21 GHz. These structures can befurther investigated via microwave spectroscopy.

6.3

Two-photon dark resonance

SpIDR spectroscopy enabled us to observe a molecular transition with negligible saturation
broadening. This fact implies the possibility of observing a two-photon dark resonance bydetecting excitations with the SpIDR scheme. We directly tried the experiments of a two-photon
dark resonance to the rovibrational ground state with a setup shown in Fig.6.15. Fig.6.16
shows the schematic diagram of the relevant molecular levels. Actually this method worked
well; we could observe a dip indicating the two-photon resonance when we added the laser 2
with an intensity of ∼1 W/cm2 during the interaction time of 400 µs for the SpIDR detection(Fig.6.17(a)). The measured biding energy of the v ′′ = 0, J ′′ = 0 level with respect to thev ′′ = 91, J ′′ = 0 level is -124955.92(4) GHz, where the accuracy was limited by the uncertaintyof our wavemeter. Taking into account the binding energy of the v ′′ = 91, J ′′ = 0 level withrespect to the atomic threshold FK = 1+FRb = 1, -374.75(3) GHz, the binding energy ofthe v ′′ = 0, J ′′ = 0 level with respect to the atomic level without the hyperﬁne structure
was determined to be -125335.11(5) GHz. In order to ensure that the observed signal deﬁnitely
indicated the rovibrational ground, we also observed the two-photon resonance for the rotational
excited v ′′ = 0, J ′′ = 2 level. From the separation between the J ′′ = 0 level and the J ′′ = 2
level, we obtained a rotational constant of 1095.4(1) MHz. The uncertainty arose from theunknown hyperﬁne structure of the J ′′ = 2 level. We can compare the measured constant withthe previous experiments on 40 K87 Rb [139], which gave a rotational constant of 1113.950(5)MHz. Considering the dependence of rotational constant on the reduced mass B ∝ µ−1 , weobtain a calculated value for 41 K87 Rb as 1095.362(5) MHz. Thus, our measurement agrees withthe previous experiments to ﬁfth digits, giving the direct evidence of observing the rovibrational
ground state of 41 K87 Rb.
The two-photon dark resonance allowed us to determine the precise frequency of the twophoton resonance with an accuracy of 10 kHz. Another important aspect of the two-photon darkresonance is the spectral width, which gives the transition strength between the rovibrational
164

SpIDR counts (arb.units)

20
10

2

0
1

0
(a)

1

2

3
4
Offset frequency of L1 (GHz)5SpIDR counts (arb.units)

20
10

(b)

0
1

0

6

7
8
9
Offset frequency of L1 (GHz)

Depleton counts in v''=91 (arb.units)

3

2

Depleton counts in v''=91 (arb.units)

3

10

Figure 6.14: SpIDR spectrum near the v ′′ = 91 level of X1 Σ+ and the v ′′ = 22 level of a3 Σ+ (redline). The excited state is the Ω = 1, v ′′ = 44 level of (2)3 Σ+ . For comparison. the spectrum
obtained with the v ′′ = 21 level and the depletion spectrum for the v ′′ = 91 level of X1 Σ+ are
shown in blue and black lines, respectively. The initial states are the J ′′ = 0, 2, v ′′ = 91 level of
X1 Σ+ and the T ′′ = 5/2, N = 2, v ′′ = 22 level of a3 Σ+ .

165

RF switch(2 stages)

DBM

RF amplifier

80MHz

Signal Generator

Laser1(IR)

AOM

Laser2(Red)

AOM

to chamber

Splitter

RF amplifier
RF switch(2 stages)

DBM

Figure 6.15: Optical setup for the Raman lasers. The Raman lasers pass through the AOM forcontrolling the intensity. After passing through optical ﬁbers independently, the two lights areoverlapped with a dichroic mirror. The light is incident on the chamber without being focusedsuch that the Raman lasers cover the wide-spread molecular clouds. The 1/e2 diameter of theRaman lasers is 1.5 mm. The RF switches before the AOM are crucial in a STIRAP experimentin order to avoid the leakage in the AOM from killing the photoassociated molecules.

Excited state(3)1Σ+,

875 nmLaser 1 (L1)
Probe light

641 nmLaser 2 (L2)Control light

X1Σ+,Initial state=91, =0,=2

Final stateX1Σ+, =0, =0
Figure 6.16: Level diagram of relevant molecular states for two-photon dark resonance spectroscopy. The v ′′ = 91, J ′′ = 0, F ′′ = 2 level of X1 Σ+ is coupled to the rovibrational groundstate of X1 Σ+ via a two-photon transition mediated by the v ′ = 41, J ′ = 1, T ′ = 5/2 level of(3)1 Σ+ . The laser 1 (875 nm) couples the upward transition and is called a probe light, whereas
the laser 2 (641 nm) couples the downward transition and is called a control light. The intensityof the laser 1 is 4.5×102 µW/cm2 and that of the laser 2 is 1 W/cm2 .

166

SpIDR counts (arb.units)

2.0
1.5
1.0
0.5
0.0

(a) J = 0

SpIDR counts (arb.units)

1735

1736

1737
1738
1739
Offset frequency of L2 (MHz)

1740

1.5
1.0
0.5
0.0

(b) J = 2

-4837

-4836
-4835
-4834
Offset frequency of L2 (MHz)-4833

Figure 6.17: Two-photon dark resonances observed with the SpIDR scheme. (a) Signal for thev ′′ = 0, J ′′ = 0 level. (b) Signal for the v ′′ = 0, J ′′ = 2 level. The laser 1 is ﬁxed to theresonance and the laser 2 is scanned. In both measurements, the master laser is ﬁxed and theoﬀset frequency between the master laser and the slave laser is scanned. The constant oﬀset ion
counts indicate a weak excitation of molecules in the initial state with the laser 1. However, theexcitation is suppressed with the formation of a dark state when the laser 2 is on two-photon
resonance.

167ground state and the intermediate excited state. The spectral shape of a dark resonance f (∆)
is written as a function of the detuning ∆ by1(

f (∆) ∝
1+

Ω2c
4Γ∆

)

(6.3)

where Ωc is the Rabi frequency of the control light and Γ is the natural linewidth of the excitedstate. The full-width-at-the-half-maximum (FWHM) of the spectrum is thus given as Ω2c /2Γ.
The observed spectrum was in good agreement with the theoretical function (6.3) (Fig.6.18).
The derived width showed a linear dependence on the intensity. From the slope of this plot at
low intensities, we determined the transition dipole moment from the slope to be 0.0098(3) ea0 .
In this derivation, we required an assumption that the natural linewidth is 300 kHz because thetransition dipole moment included in Ωc does not coincide with that included in Γ due to its
R-dependence. We have to deal with Γ as an independent variable.
At high intensities, the width became broader than expected from the linear dependence at
low intensities. This behavior should be interpreted as a result of a low transparency arisingfrom the scattering of undesirable frequency components in the laser 2 (the dye laser). In otherwords, the decoherence process in the two-photon transition becomes signiﬁcant at intensities
of higher than 3 W/cm2 . The coherence properties of the two-photon transition will be later
discussed in detail in the section 6.5.
At low intensities, we observed a small oﬀset width of ∼40 kHz. This should be interpreted asa two-photon Doppler eﬀect arising from the diﬀerence in the two-photon resonance for moving
molecules. If we assume a temperature of molecules as 100µK, the diﬀerence in the Doppler
shift for the two wavelengths (641 nm and 875 nm) is 2π×34 kHz and is in good agreement withthe observed oﬀset width.
The obtained transition dipole moment of 0.0098 ea0 is much smaller than that of atomic
transitions, but is comparable to that of the (2)3 Σ+ state, which was used for the study on
40 K87 Rb. Therefore, we reach an important conclusion that the (3)1 Σ+ state can be a goodintermediate state for a STIRAP transfer. If we take into account the fact that the (3)1 Σ+ state
has a much narrower linewidth (∼300 kHz) than the (2)3 Σ+ state (∼7 MHz), the (3)1 Σ+ stateshould give a higher transfer eﬃciency than the (2)3 Σ+ state under the same laser intensities
and laser linewidth.
The observation of a two-photon dark resonance enabled us to verify an important hypothesisin this study. Up to now, we have been working with a hypothesis that saturation broadening
in the one-photon spectroscopy makes a two-photon dark resonance unobserved. We could
readily conﬁrm this hypothesis just by observing the dependence of transparency on intensity(Fig.6.19). In this measurement, the intensity ratio of the control laser to the probe laser was
kept constant (103 ) such that the dark state has the same form. As expected, the depth of thetwo-photon resonance decreased with increasing the intensity. A clear dip was obtained at an
intensity of lower than 1 mW/cm2 . In contrast, the typical intensity required for observing aclear depletion spectrum was 10 mW/cm2 . Thus, we can see that it is almost impossible toobserve a two-photon dark resonance by detecting the excitation with the population of theinitial photoassociated molecules.
Fig.6.19 includes the important information on saturation intensity. A clear two-photon
resonance can be observed when the intensity of the probe laser is suﬃciently lower than saturation intensity. If we assume the line shape of the one-photon transition without saturation broadening as f (∆) ∝ 1/(1 + (2∆/Γ)2 ) and that with saturation broadening as fs (∆) =I/Is /(1 + I/Is + (2∆/Γ)2 ), we obtain an approximate representation of the relation between
transparency T and intensity I:
T ≈1
1 + I/2IS
168

(6.4)

1.8
1.6
1.4
1.2
1.0
0.8
0.6

(a) Fit to the spectrum

0.8

1737.01737.51738.0
Offset frequency of L2 (MHz)1738.5(b) Intensity dependence

0.6

2

FWHM (Ω /2Γ) (MHz)1736.5

0.4
0.2
0.0
0

1
2
3
4
2
Peak intensity of the control light (W/cm )5

Figure 6.18: Measurement of the transition strength. (a) Fit to the observed spectrum. (b)Dependence on intensity. The width of a two-photon dark resonance is related to the Rabi frequency of the downward transition. By ﬁtting the observed spectrum with a theoretical function(6.3), we can derive the quantity Ω2c /2Γ as a full width at half maximum (FWHM), where Ωc is
the Rabi frequency of the control light and Γ is the natural linewidth of the excited state. If weassume the natural linewidth as 300 kHz, the slope of this plot, 98.3(6.9) Hz/(mW/cm2 ), gives
a transition dipole moment of 0.0098(3) ea0 .

169

0.6

Transparency

0.5

f(I) = 1+(1+I/2Is)2
Is = 0.30(1) mW/cm

0.4
0.3
0.2
0.1
0.0
0

1
2
3
4
2
Intensity of the probe laser (mW/cm )5

Figure 6.19: Disappearance of a two-photon dark resonance with saturation broadening. When
we increase the intensity of the probe laser, the two-photon dark resonance disappears. For
observing a clear peak, the intensity should be lower than 1 mW/cm2 .
where IS denotes saturation intensity. Here we assume that the broadening of the spectral width
due to the long interaction time of 400 µs in the present experiment is included in the above
expression for fs . By ﬁtting the observed dependence with the relation (6.4), we determined thesaturation intensity of the transition v ′ = 41, J ′ = 1, (3)1 Σ+ ←v ′′ = 91, J ′′ = 0, X1 Σ+ to be
0.30(1) mW/cm2 . In this ﬁt, the rightmost point was neglected because the measurement of thewidth indicated that transparency can be reduced by laser-induced decoherence at an intensityof higher than 3 W/cm2 for the control light. The obtained value for Is is in good agreementwith the observation in Fig.6.9, in which we observed that the spectrum was broadened at an
intensity of higher than 1 mW/cm2 .
In the previous studies on Feshbach molecules, the spectrum of a two-photon dark resonancehas often been given by scanning the probe laser. This is because the one-photon spectrum of
the excited state gives the information on the natural linewidth and saturation broadening. In
contrast, we have been obtained the spectrum of a two-photon dark resonance by scanning thecontrol laser for a reason speciﬁc to our study. We deal with the purely singlet (3)1 Σ+ state,
which has a hyperﬁne structure of the order of 1 MHz. If we take into account the nuclear spinof 41 K, there should also be splitting of the order of 10 kHz. We cannot resolve these structures
because there is no magnetic ﬁeld. Therefore, ﬁtting to the spectrum obtained by scanning theprobe laser requires a careful treatment of the hyperﬁne structures in addition to the two-photon
and one-photon characteristics. If we scan the control light, we can simplify the problem anddeal with only the two-photon resonance.
However, we can take a spectrum by scanning the probe laser just for showing the well-known
dark resonance (or electromagnetically induced transparency) shape (Fig,6.20). This spectrum
was taken solely for showing the two-photon resonance appearing in the one-photon spectrum
and was not useful for some other purposes in the present study.

6.4

STIRAP transfer

With the observation of a two-photon dark resonance, most of the uncertain quantities wereprecisely determined: the frequency of the one-photon transition and the two-photon resonance,
and the transition dipole moment of the downward transition. The remaining uncertainty was
the wavenumber of the pulsed dye laser for ionizing the rovibrational ground-state molecules.
The wavenumber should be included in the range 15100–16800 cm−1 covered by the combination of the DCM dye and the Kiton Red 620 dye. These dyes are required for obtaining the170

Ion counts (arb.units)

SpIDR (L1 only)
Dark resonance (L1 & L2)

3.0
2.5
2.0
1.5
1.0
0.5
-1238

-1237

-1236 -1235 -1234 -1233
Offset frequency of L2 (MHz)-1232

Figure 6.20: Two-photon dark resonance in a one-photon spectrum. The control light is ﬁxed
to the transition v ′ = 41, J ′ = 1, T ′ = 5/2 of (3)1 Σ+ → v ′′ = 0, J ′′ = 0 of X1 Σ+ . The probe
laser is scanned over the hyperﬁne structure of the v ′ = 41, J ′ = 1 level of (3)1 Σ+ .
wavenumbers for locking the photoassociation laser (depletion spectroscopy at 16543 cm−1 ) andfor locking the Raman lasers (SpIDR spectroscopy at 16578 cm−1 ). Thus, we can use only
red lights for detecting the rovibrational ground state. Here a diﬃculty arises from the large
required energy for ionization. In contrast to the ionization of photoassociated molecules, theionization of the rovibrational ground-state molecules requires at least three photons in the redregion. Another possible path is a two-photon ionization with a combination of the red lightand the green light for pumping the dye laser (532 nm).
In any scheme, it is crucial to avoid the contamination of ion counts from photoassociatedmolecules. We focused on the range 15200–15600 cm−1 from the atomic level in which there are
no bound states (Fig.6.21). Due to the absence of bound states in this range, photoassociatedmolecules cannot be resonantly ionized (Fig.6.22). By contrast, the rovibrational ground-state
molecules can be resonantly ionized because there are various bound states from the vibrational
ground state in this range.
Among the ﬁve symmetries in this range from the vibrational ground state, the (1)1 Π state isparticularly promising as an intermediate state of REMPI ionization due to its large transition
strengths with the X1 Σ+ state. The RKR potential curve of the (1)1 Π state was previously
reported [140, 141]. Hence, we can predict possible ionization wavenumbers with an accuracy ofseveral cm−1 by calculating the Franck-Condon factors, which are also shown in Fig.6.22. The
uncertainty in the calculation arises from the spin-orbit coupling with other states.
On the basis of these considerations, we took an ionization spectrum after a STIRAP transfer,in which we could ﬁnd several signals which were absent for photoassociated molecules (Fig.6.23).
The assignment on theses signals will be discussed in the section 6.6.2. A naive interpretationof these signals was the ionization of the rovibrational ground-state molecules produced by a
STIRAP transfer. However, we should not neglect the possibilities that they were not from therovibrational ground state and they were not with a STIRAP process. In order to ensure thatthe signal indicates the rovibrational ground-state molecules obtained by a STIRAP transferof photoassociated molecules, we carried out two measurements at the wavenumber giving thelargest signal (15407.5 cm−1 ). The ﬁrst one was the spectrum obtained by scanning the Raman
laser (Fig.6.24) and the second one was the time variation of the ion counts during the multiple
transfers (Fig.6.25). The Raman spectrum was taken for both the J ′′ = 0 level and the J ′′ = 2
level with the same ionization wavenumber. These spectra were in excellent agreement with thetwo-photon dark resonance spectra, indicating that the observed increase in ion counts was due tothe ionization of the rovibrational ground-state molecules. Further, the time variation in the ion
171

3 +

(4) Σ

16000

-1

Energy (cm )

16200

15800

1 +

(4) Σ

15600
15400
3 +

(3) Σ

15200
15000
3

4

5
6
7
Internuclear separation (Å)

89Figure 6.21: Window region of 15200–15600 cm1 where there are no bound states. Photoassociated molecules cannot be ionized in this range, whereas the rovibrational ground-state molecules
can have numerous transitions in this wavenumber range.

60

0.30

v'=1

0.25

Rb 10D

Rb 8D

40

v'=0

v'=3
v'=4

0.20

v'=5

30

0.15

Rb 9D
3 +

20

(3) Σ

0.10
1 +

(4) Σ

10

0.05

0
15000

0.00
15200

15400
15600
15800
16000
-1
Ionization wavenumber (cm )16200

Figure 6.22: Ionization spectrum indicating the window region. The REMPI ionization of photoassociated molecules to the (4)1 Σ+ state, the (4)3 Σ+ state, and the (3)3 Σ+ state is clearlyobserved, thereby indicating the window region where photoassociated molecules cannot be ionized. In this window region, the vibrational ground state v ′′ = 0 of X1 Σ+ has strong transitions
to the (1)1 Π state. The Franck-Condon factors for these transitions are shown in the right axis.

172

Franck-Condon factors

50
Ion counts (arb.units)

3 +

(4) Σ

Ionization spectra for PA molecule
1 +Franck-Condon factors from v=0,J=0 of X ΣIon counts (arb.units)

10

Only PA molecules
After STIRAP

8
6
4
2
0
15200

15300
15400
15500
-1
Ionization wavenumber (cm )15600

Figure 6.23: Ionization spectrum after a STIRAP transfer. This spectrum is taken only withthe red light. New ion signals appear after a STIRAP transfer. A naive interpretation ofthese signals is the ionization of the rovibrational ground-state molecules formed by a STIRAP
transfer, but we have to conﬁrm it with other measurements.
counts followed the variation in the intensity of the Raman laser, indicating that the rovibrational
ground-state molecules were produced via a STIRAP transfer. This measurement enabled us todetermined the one-way transfer eﬃciency to be 73 % because 53 % of the population recovered
with round-trip transfers.
The measured width of the two-photon line shape was between 100 kHz and 500 kHz depending on the intensity of the Raman lasers and duration. A simple theoretical model (seesection 6.7) gives an approximate representation of the width asΩ(6.5)
δSTIRAP ≈ 2.4 √Γτ
Substituting the typical Rabi frequency of 2π×600 kHz, the duration of 10µs, and the natural linewidth of 2π× 300 kHz to the expression (6.5), we obtain the typical width of 2π× 300
kHz. This is in good agreement with the observed widths. In the present study, the rovibrational ground-state molecules are ionized by three-photon transitions with only the red light.
This should be much more ineﬃcient than two-photon ionization since three-photon ionization
includes an oﬀ-resonant transition. In order to improve the signal-to-noise ratio we tried toobserve two-photon ionization with the red light and the green light. However, we did not observe any enhancement of the signal by adding the green light. Instead, the green light induced
oﬀ-resonant multi-photon ionization, thereby making oﬀset ion counts and degrading the signalto-noise ratio. Thus, we decided not to use the green light and to use only the red light for thepresent study.
The TOF spectrum for the rovibrational ground-state molecules was quite diﬀerent fromthat for photoassociated molecules (Fig.6.26). We observed a large amount of atomic K+ ionsas well as KRb+ ions. This behavior is understood as a result of the dissociative ionizationof the rovibrational ground-state molecules to the (2)2 Σ+ state. The potential energy curvesof the relevant molecular states are shown in Fig.6.27. We can observe KRb+ signals whenmolecules are excited to one of the bound states in the (1)2 Σ+ state. However, three red
photons have a much higher energy than required for a transition to the (1)2 Σ+ state, resulting
in a low ionization eﬃciency. Three red photons can readily excite the rovibrational groundstate molecules to the (2)2 Σ+ state, which immediately dissociate to K+ ions and Rb atoms.
173

1.5
1.0
SpIDR
STIRAP

0.5
0.0

3
2

(a) J = 0

1
1735

1736
1737
1738
1739
Offset frequency of L2 (MHz)

1740

1.5
1.0
SpIDR
STIRAP

0.5

4
3
2

0.0

(b) J = 2

-4837

1
-4836
-4835
-4834
Offset frequency of L2 (MHz)STIRAP counts (arb.units)

SpIDR counts (arb.units)

4

STIRAP counts (arb.units)

SpIDR counts (arb.units)

2.0

-4833

Figure 6.24: Two-photon line shape of the STIRAP ion counts. (a) J”=0. (b) J”=2. Ion counts
are plotted with respect to the oﬀset frequency of the laser 2 (641 nm) between the master andthe slave lasers. The ionization wavenumber is ﬁxed to 15407.5 cm−1 . An increase in the ion
counts is observed exactly at the same frequencies as two-photon dark resonances. This is the
direct evidence of observing the rovibrational ground state.

174

0.5
4
0.0
Ion counts (arb.units)

3

Ion counts
L1 intensity
L2 intensity2

Laser intensity (arb.units)

1.0

1

0
0

10

20

30
Time (µs)

40

50

60

Figure 6.25: Time variation in the STIRAP ion counts during multiple transfers. The variationin the intensity of Raman lasers is shown in the upper lines. The ion counts follow the variationin the intensity of the Raman laser. This is the direct evidence of producing molecules in therovibrational ground state with a STIRAP process. The one-way transfer eﬃciency is determined
to be 73 % from this plot.
This is why we observe a much larger number of K+ ions than KRb+ ions. In the present study,
only ion counts in the KRb+ peak were used for monitoring the population in the rovibrational
ground state.
In the above experiments, the initial state for a STIRAP transfer was chosen as the J ′′ = 0
level. We also observed a STIRAP transfer of photoassociated molecules in the J ′′ = 2 level
which are more abundant than those in the J ′′ = 0 level. However, the number of molecules
produced in the rovibrational ground state was slightly smaller than that from the J ′′ = 0 level.
This is presumably because molecules in the J ′′ = 2 level are distributed over multiple sublevels
and are not eﬃciently transferred to the J ′′ = 0 level.
The number of produced molecules is estimated to be ∼103 if we assume the typical ionizationeﬃciency as 1 %, which is an upper limit for three-photon ionization. The temperature andthe density are ∼100 µK and ∼106 cm−3 , respectively. The most signiﬁcant diﬀerence fromthe previous methods based on magnetoassociation is the time required for the formation: werequired only several tens of ms in total for compressing atomic samples (∼40 ms), applying aphotoassociation laser (∼10 ms), and applying Raman lasers (∼10 µs) (Fig.6.28).
Finally we note a peculiar line shape of the J ′′ = 2 spectrum in Fig.6.24. The clear diﬀerencein the line shapes between J ′′ = 0 and J ′′ = 2 originates from the coupling between rotational
angular momentum and the nuclear spins [138]. For the rovibrational ground state, the hyperﬁnesplitting arises only from spin-spin interaction, which is of the order of 1 kHz and cannot
be resolved. By contrast, the rotational excited J ′′ = 2 level has a strong coupling between
rotational angular momentum and the nuclear spins, resulting in the hyperﬁne structure of
the order of 100 kHz. However, we could not completely resolve the complex structure in thismeasurement even if we narrowed the spectral width by decreasing the intensity and increasing
the transfer duration.

175

Ion signal (arb.units)

0.20
0.18
0.16
0.14
0.12
0

10

20
30
Time (µs)

40Figure 6.26: TOF spectrum for the rovibrational ground-state molecules. Due to the dissociative
ionization of the rovibrational ground-state molecules, a large amount of atomic K+ ions areobserved.

5

4

2 +

(2) Σ

+

K +Rb

2 +

(1) Σ

4

-1

Energy (10 cm )

+

K+Rb

3

2
1

(1) Π

4S+5P

1 +

4S+5S

1
649 nm

XΣ

0

2

4

6
8
10
Internuclear separation (Å)

12Figure 6.27: Potential energy curves of the molecular states relevant to the ionization of therovibrational ground state molecules. The potential curves of KRb are taken from Ref. [105]and those of KRb+ are taken from Ref. [106]. The rovibrational ground-state molecules are ﬁrst
excited to the (1)1 Π state and then oﬀ-resonantly ionized to the (1)2 Σ+ state. The molecules
also undergo a three-photon transition to the (2)2 Σ+ state, which immediately dissociates toK+ ions and Rb atoms.
176

40 ms
C-MOT

MOT

MOT

Photoassociation laser
STIRAP lasers
Pulsed laser
Figure 6.28: Time sequence of the STIRAP experiments. After compressed and cooled in the
C-MOT process, 41 K and 87 Rb atoms are photoassociated and form weakly bound molecules.
They are transferred into the rovibrational ground state via STIRAP. Finally molecules aredetected via REMPI ionization with the pulsed laser.

6.5

Coherence properties

Now that we found a good ionization wavenumber for the rovibrational ground state, we canprecisely investigate the optimum values for the parameters of a STIRAP process: there arethree parameters, intensities of both Raman lasers and the duration. A particularly useful
method for optimizing the STIRAP process is to study eﬃciency with respect to duration. Hereduration indicates the time diﬀerence between the two pulses. A simple theoretical model on athree-level system gives an analytical expression for a transfer eﬃciency as
)
(
π2Γ
(6.6)
P = exp − 2 − DτΩ τ
where Ω is the Rabi frequency averaged over two lights, Γ is the natural linewidth of theintermediate state, D is decoherence rate, and τ is the duration, respectively. Fig.6.29 shows themeasured eﬃciency with respect to duration. By ﬁtting this plot with the analytical expression(6.6), we obtain both the Rabi frequency and decoherence. The inﬂuence of the two-photond
Doppler shift due to the ﬁnite temperature of the molecules T can be taken into account by
multiplying a factor f (T ) to P , where f (T ) is given as
1

f (T ) = √(6.7)

kB T Γτ1+(k1 − k2 )2mΩ2
by using the wavenumbers of the two lights k1 and k2 , and the mass of molecules m.
The derived Rabi frequency of 2π×600–800 kHz agreed with the measurement of transition
strengths with a two-photon dark resonance. However, the measured decoherence rate of 2π×2
kHz was unexpectedly high. The high decoherence rate is the main limitation on the transfer
eﬃciency, although the two-photon Doppler shift of ∼40 kHz can also limit the eﬃciency byseveral %. Before this study, we had the following estimations on the decoherence rate. Our
Raman lasers have a linewidth of below 10 Hz. Therefore, decoherence due to the laser linewidthcannot be observed with our typical STIRAP transfer duration of tens of µs. Instead, thedecoherence rate should be limited by spin precession arising from the Zeeman splitting ofmagnetic sublevels of the v ′′ = 91 level of X1 Σ+ , which is of the order of kHz at a residual
magnetic ﬁeld of 0.1 G. These estimations suggested the possibility that the observed highdecoherence rate was due to residual magnetic ﬁelds.
177

3.5
Γ = 300 kHz (hold)D = 1.6(1) kHz
A = 3.9(1)Rabi = 602(34) kHz

Ion counts (arb.units)

3.0
2.5
2.0
1.5
1.0
0.5
0.0
20

40

60

80
Time (µs)

100

120

140

Figure 6.29: Eﬃciency vs duration. The ion counts in the v ′′ = 0 level are plotted with respect
to the duration of a STIRAP transfer. The oﬀset ion counts arising from oﬀ-resonant multiphoton ionization of photoassociated molecules are subtracted. Roughly speaking, the increaseof the eﬃciency at short duration depends on the Rabi frequency, whereas the decrease of theeﬃciency at long duration depends on decoherence such as laser linewidth. By ﬁtting this plot
with an analytical expression, we can derive both the Rabi frequency and the decoherence ratefrom a single plot. In this ﬁtting, the inﬂuence of the two-photon Doppler shift is taken into
account by assuming the temperature of the molecules to be 100 µK.
A good method for checking if the residual magnetic ﬁeld has an inﬂuence on the decoherence
rate is to begin with the more deeply bound v ′′ = 90 level of X1 Σ+ . The v ′′ = 90 level has amuch smaller triplet component of 2 % than the v ′′ = 91 level because the v ′′ = 90 level isseparated from the nearby triplet v ′′ = 21 level by more than 0.4 cm−1 . Although the initial
population of the v ′′ = 90 level was approximately half of that of the v ′′ = 91 level, we could
produce the rovibrational ground-state molecules with a STIRAP transfer from the v ′′ = 90
level.
However, the study with the v ′′ = 90 level gave a result similar to the v ′′ = 91 level (Fig.6.30).
We observed similar decoherence rates for both the v ′′ = 91 level and the v ′′ = 90 level, indicatingthat the observed decoherence rates were not limited by the residual magnetic ﬁelds. We also
suspected the inﬂuences of the leakage of the MOT light and the photoassociation light, butneither had a signiﬁcant inﬂuence on the coherence properties.
At this stage, the observations indicated that the high decoherence rate should be related
to the Raman laser itself. By observing the decoherence rate at various conditions, we foundthree interesting behaviors. First, the decoherence rate strongly depended on the intensity ratio
between the infrared light (875 nm) and the red light (641 nm). Fig.6.31 shows a comparison
between two measurements with diﬀerent Rabi frequency ratios. These two measurements gave
approximately the same Rabi frequencies, but the decoherence rate was much lower when theinfrared laser had a higher Rabi frequency than the red laser.
The second interesting fact is recognized by lowering the intensity with keeping the Rabi
frequency ratio as 1 (Fig.6.32). The observed decoherence rate decreased as the intensities of
the Raman lasers were decreased. The lowest observed value was ∼ 100 Hz, implying that there
are no other decoherence mechanisms down to ∼100 Hz in this system. This fact suggests thatmolecules are at the center of magnetic ﬁeld gradients and the residual DC magnetic ﬁeld at theposition of molecules is smaller than 10 mG. In addition, we can conﬁrm that our laser system
has indeed the linewidth of below 100 Hz.
The third interesting fact is related to the polarization of the Raman lasers. With circularly

178

3.5

v''=91
Γ = 300 kHz (hold)D = 1.6(1) kHz
A = 3.9(1)Rabi = 602(34) kHz

Ion counts [arb.units]

3.0
2.5
2.0
1.5

v = 91v = 90v''=90Γ = 300 kHz (hold)D = 1.8(7) kHz
A = 1.7(5)Rabi = 690(300) kHz

1.0
0.5
0.0
20

40

60

80
Time [s]

100

120

-6

140 160x10

Figure 6.30: Comparison of the coherence properties during the STIRAP transfer from thev ′′ = 91 level and from the v ′′ = 90 level. The ion counts in the vibrational ground state
are plotted with respect to the duration of a STIRAP transfer. Nearly identical behaviors areobserved for the v ′′ = 91 level and the v ′′ = 90 level.

1.6ΩDye/ΩIR = 1.1ΩDye/ΩIR = 0.53

Ion counts (arb.units)

1.4
1.2
1.0

Ω = 2π × 740 kHz
D = 2π × 2.1 kHz

0.8
0.6
0.4
Ω = 2π × 650 kHz
D = 2π × 4.1 kHz

0.2
0.0
0

20

40

60
80
Time (µs)

100

120

140

Figure 6.31: Coherence properties of the STIRAP transfer measured with a diﬀerent intensity
ratio between the infrared light and the red light. The decoherence rate increases with increasing
the intensity of the red laser. This fact indicates that the red light itself has a strong inﬂuenceon the decoherence rate.

179Decoherence rate/2π (kHz)6
Circularly polarizedLinearly polarized (P)

5
4
3
2
1
0
0

100

200
300
400
500
Rabi frequency/2π (kHz)

600

700

Figure 6.32: Dependence of the observed decoherence rate on the Rabi frequency. Both the Rabi
frequency and a decoherence rate are obtained by ﬁtting a plot on the eﬃciency with respect
to duration. A circularly polarized light gives a much higher decoherence rate than a linearlypolarized light.
polarized lights, we obtained much higher decoherence rates as also shown in Fig.6.32. We
recognized this fact before this measurement through an experiment on a two-photon darkresonance: we could not observe any resonance with circularly polarized lights.
We have not yet completely understood these behaviors. Here we present a possible interpretation of these behaviors. The dependence on the intensity ratio between the red laser andthe infrared laser can be explained in terms of the diﬀerence in the regulation of the laser: theinfrared light (a diode laser) is locked with a current feedback, whereas the red light (a dyelaser) is locked with an AOM. Due to the limited bandwidth of the AOM, the red light has amuch higher pedestal than the infrared light by more than 10 dB within the frequency range of 2
MHz (Fig.5.41). The pedestal in this range can be the origin of a high decoherence rate throughscattering because of the natural linewidth of ∼300 kHz and the small hyperﬁne structure of 1
MHz in the excited state. The decrease in a decoherence rate with a decrease in intensity canalso be understood in terms of the scattering of the pedestal in the red light within a few MHz.
Similar behaviors are also observed by other groups [142, 143].
However, the signiﬁcant increase in the decoherence rate with circularly polarized lights cannot be understood in terms of scattering. The dependence on polarization should be attributed
to a spin-dependent phenomenon. The AC stark shifts arising from the Raman lasers can be
the origin of decoherence: due to a diﬀerence in the Clebsch-Gordan coeﬃeicnts, each spin state
undergoes a diﬀerent AC stark shift from the Raman lasers, causing an inﬂuence similar tomagnetic ﬁelds. Circularly polarized lights are expected to have a larger inﬂuence than linearlypolarized lights because of a larger diﬀerence in the Clebsch-Gordan coeﬃcients. There are various candidates for the relevant transitions. One possibility is the inﬂuences of the carrier lighton far-oﬀ-resonance transitions, such as transitions at ∼641 nm from weakly bound levels andthose at ∼875 nm from the rovibrational ground state. The other possibility is the inﬂuences ofthe pedestal lights on near-resonance transitions.

6.6

Studies with the rovibrational ground-state molecules

The production of the rovibrational ground-stat molecules opens up a wide variety of applications. In this section, we present three interesting applications. The ﬁrst application is the
spectroscopy of the various excited state which is accessible from the rovibrational ground state.
180

We observed a number of expected transitions to the (3)1 Σ+ state and the (1)1 Π state (section 6.6.1). The second application is the measurement of the transition strength from theweakly bound level to the excited state and the natural linewidth of the excited state. The
dark resonance spectroscopy using the molecules in the rovibrational ground state enables us
to determine the transition dipole moment of the upward transition (section 6.6.2). The thirdinteresting application is the precision spectroscopy of the hyperﬁne structure in the weakly
bound molecules. Our previous studies on weakly bound level suggested the existence of thehyperﬁne structure, but we could not precisely determine the structure. The STIRAP transfer
into the rovibrational ground state enables us to map the initial population onto the groundstate, revealing the hyperﬁne structure in an unexplored region (section 6.6.3).

6.6.1

Molecular transitions from the rovibrational ground state

In this study, we made great eﬀorts for ﬁnding the intermediate states for STIRAP, identifying
the symmetry, and assigning the observed structures (chapter 4). These eﬀorts were necessary
because we did not know whether the observed state has a strong transition to the rovibrational
ground state.
With the successful creation of the rovibrational ground-state molecules, we obtained a goodalternative method for ﬁnding the excited states which can be coupled to the rovibrational
ground state. Just by scanning the pulsed laser after a STIRAP transfer, we can readily obtaina spectrum indicating possible transitions from the rovibrational ground state over tens of nm.
Fig.6.33 shows such a spectrum. Although this spectrum is essentially identical to the previously
given spectrum (Fig.6.23), the optimization of the STIRAP process as well as averaging gives
much more rich spectrum.
A part of the observed lines can be understood from calculations. In Fig.6.33, the calculated
Franck-Condon factors for both the (1)1 Π state and the (3)1 Σ+ state are also shown. Further,
we plot the observed energy levels of the Ω = 1, (2)3 Σ+ state in the same ﬁgure assuming theFranck-Condon factors as 0.01.
The previously reported RKR potential curve of the (1)1 Π state [140, 141] gives a good
estimation on the Franck-Condon factors. Although the strong spin-orbit mixing is expected toshift the calculated energy levels by a few cm−1 , the observed v ′ = 5–11 levels shows fairly good
agreements with calculations. The (1)1 Π state shows a broad structure of several cm−1 for each
vibrational level. One of the reasons for this feature is saturation broadening of ∼1 cm−1 due tolarge transition dipole moments. However, we do not have a good explanation on the observed
splitting of each vibrational level.
For the (3)1 Σ+ state, most of the levels in this range (v ′ = 31–41) are observed, whichshow quite good agreements with the calculations based on the accurate potential energy curve
obtained with our study (section 4.2.6). The v ′′ = 41 level used for a STIRAP transfer is also
clearly observed. Note that the v ′′ = 31–40 levels are not explicitly observed with depletion
spectroscopy and thus are not included in the direct potential ﬁt analysis. Nevertheless, ourpotential gives a good prediction for these levels, showing the validity of our method for obtaining
an accurate potential curve.
For the (2)3 Σ+ state, we do not have a precise value for the Franck-Condon factors. They can
only be obtained through experiments. However, we can conﬁrm that the observed ionization
spectrum is in good agreement with our previous spectroscopic studies particularly in the range
15400–15600 cm−1 .
There are still numerous unassigned lines. These lines are expected to be mainly the Ω = 1
component of the (1)3 Π state, which also has a large transition strength with the rovibrational
ground state.

181

5

1 +

FCF to (3) Σ-1(a) 15200-15400 cm

1

FCF to (1) Π
3 +FCF to (2) Σ

3

-1

10

-2

10

2
-3

10

Franck-Condon factors

Ion counts (arb.units)

4

1
-4

10
0
15200

15250
15300
15350
-1
Ionization wavenumber (cm )

15400
1 +

5

FCF to (3) Σ-1

1

FCF to (1) Π(b) 15400-15650 cm

3 +FCF to (2) Σ

3

-1

10

-2

10

2
-3

10

Franck-Condon factors

Ion counts (arb.units)

4

1
-4

10
0
15400

15450

15500
15550
-1
Ionization wavenumber (cm )15600

Figure 6.33: Ionization spectrum for the rovibrational ground state. All the lines in this range
are the transitions from the rovibrational ground state since there are no transitions from photoassociated molecules in this range. We can assign a part of the observed lines with calculations.
For the (1)1 Π state and the (3)1 Σ+ state, calculated Franck-Condon factors are given in the ﬁgure since we have reliable estimations on these states. For the (2)3 Σ+ state, the Franck-Condon
factors are assumed as 0.01 for all vibrational levels.

182

6.6.2

Transition strength from a weakly bound level

In the previous sections, we have determined most of uncertain quantities related to the Raman
transition via SpIDR spectroscopy and two-photon dark resonance spectroscopy. However, twoimportant quantities are still not measured: the transition dipole moment of the upward transition and the natural linewidth of the excited state. Thus far, the transition dipole moment of
the upward transition is roughly known from the width of depletion spectra, whereas the naturallinewidth has been assumed as 300 kHz throughout this study. In this section, we show thatthese two quantities can be experimentally estimated by using molecules in the rovibrational
ground state. In addition, the knowledge on the transition strengths for both the upward andthe downward transitions enables us to predict the absolute values of the transition strengths
for each vibrational level.
The transition dipole moment of the upward transition can be accurately measured by meansof a two-photon dark resonance spectroscopy for the rovibrational ground state. Here the situation is opposite to the previous measurements on the downward transition. First we prepare
molecules in the rovibrational ground state via a STIRAP transfer. Then we apply SpIDR
spectroscopy to detect weak excitations from the rovibrational ground state. When an intense
infrared laser is resonant to the upward transition, the excitations from the rovibrational groundstate are suppressed. For this experiment, the experimental setup is slightly modiﬁed (Fig.6.34).
In this experiment, the oﬀset frequencies between the master laser and the slave laser are kept
constant and the frequency of the AOM is scanned for a few MHz.
Fig.6.35 shows a typical dark resonance spectrum for the rovibrational ground state and thedependence of the width on the intensity of the infrared laser. The widths are derived by ﬁtting
the spectrum with an analytical expression (6.3) as implemented previously in the section 6.3.
The slope of this plot gives a transition dipole moment of 0.035(2) ea0 , which is approximately
3 times larger than that for the downward transition. At low intensities, we again observed an
oﬀset width of ∼40 kHz, indicating the broadening due to the two-photon Doppler shift.
Previously we have assumed that the natural linewidth of the (3)1 Σ+ state is Γ ∼ 2π × 300
kHz, which was derived from the ab initio calculation of transition dipole moments [109]. If
the actual natural linewidth is signiﬁcantly diﬀerent from the assumed value, the measurements
presented in this study are not consistent with each other. In such a case, the estimationon the Rabi frequencies from two-photon dark resonance spectroscopy is inconsistent with theestimation based on depletion spectroscopy in the section 4.2.6. In addition, the two-photon
linewidths observed as a two-photon dark resonance (section 6.3) and as a STIRAP transfer
eﬃciency (section 6.4) does not coincide with theories (section 6.7).
The minimum width observed in SpIDR spectroscopy (∼ 400 kHz) indicates that the naturallinewidth is below 2π × 400 kHz. However, we should not neglect the possibility that the naturallinewidth is much smaller than this value because the hyperﬁne interaction in the excited state
can broaden the spectral width.
A STIRAP transfer into the rovibrational ground state enables us to directly measure thenatural linewidth by monitoring the time decay of the population in the rovibrational groundstate with an intense resonant light. The time sequence of the experiment is given in Fig.6.36.
Here the important point is that the dye laser at 641 nm gives the Rabi frequency of 2π×700 kHz,
which is suﬃciently larger than the expected natural linewidth. In such a situation, molecules
decay with approximately the half of the natural linewidth. Therefore, we can estimate thenatural linewidth just by monitoring the time constant of the decay. The width of the pulsed laser
for ionization is below 10 ns, which is much smaller than the expected time decay of molecules. In
principle, the same measurement should be possible with photoassociated molecules. However,the diﬃculty in observing a single hyperﬁne state with a high signal-to-noise ratio makes it
diﬃcult to perform the same measurement for photoassociated molecules. Although the SpIDR
detection of the excitation of photoassocaited molecules should also enable this measurement,a high signal-to-noise ratio for detecting the rovibrational ground state makes it ideal for this183

Signal Generator

STIRAP Dark resonance(10µs)(400µs)80MHz(sweep)DBM

RFswitchRF switch(2 stages)
RF amplifier

80MHz(constant)Signal Generator

Laser1(IR)

AOM

Laser2
(Red)

AOM

Red
IR
Time variation in the intensities
of Raman lasers

to chamber

Splitter

RF amplifier

DBMRF
switchRF switch(2 stages)

80MHz(constant, -32dBm)

Signal Generator
Figure 6.34: Optical and electrical setup for measuring the transition strength of the upward
transition from the rovibrational ground-state molecules. The frequencies of both Raman lasers
are kept constant during a STIRAP transfer. After the transfer, a signal generator is switched tothe other signal generators. By sweeping the RF frequency for the AOM controlling the infraredlaser, we obtain a two-photon dark resonance spectrum. The signal generator for the red laser
is required to obtain a low intensity during the two-photon dark resonance.

184

Ion counts (arb.units)

2.6
2.4
2.2
2.0
1.8
(a) Typical spectrum
1.6

0.4

2.0

(b) Dependence of the width on intensity.

0.3

2

FWHM(Ω /2Γ)(MHz)

0.5
1.0
1.5
AOM Frequency -79 (MHz)0.2

fit(x) = a + bx
a = 0.0337(23)b = 0.00123(18)

0.1
0.0
0

50

100
150
200
250
2
Peak Intensity (mW/cm )

300

350

Figure 6.35: Two-photon dark resonance observed with the rovibrational ground-state molecules.
(a) Typical spectrum. (b) Dependence of the width on intensity. In (a), the observed spectrum
is given in red line, whereas a ﬁt to the observed spectrum is given in blue line. From the slope of
the plot in (b), 1.23(18) kHz/(mW/cm2 ), the transition dipole moment of the transition v ′ = 41,
J ′ = 1, (3)1 Σ+ ← v ′′ = 91, J ′′ = 0, F ′′ = 2, X1 Σ+ is determined to be 0.035(2) ea0 .

185

(3)1Σ+Ionization ( =0)
Excitation time
L1
875 nmL2
641 nm

X1Σ+,

=91,

=0

STIRAP
X1Σ+,=0,

Excitation with L2=0

Figure 6.36: Time sequence of the measurement of the natural linewidth of the excited state.
After a STIRAP transfer into the rovibrational ground state, the produced molecules are destroyed by the resonant light at 641 nm for a STIRAP transfer. The time constant of the decay
is determined by the natural linewidth of the excited state because the Rabi frequency for the
641 nm light is higher than the expected natural linewidth.
measurement.
Fig.6.37 shows the measured time decay of the population in the rovibrational ground state.
From the observed time constant of the decay of 1 µs, we determined the natural linewidth ofthe v ′ = 41 level of (3)1 Σ+ state to be 2π×3.2(3)×102 kHz, which is in good agreement withthe previously assumed value. Although the obtained value gives a lower limit because the Rabi
frequency is not inﬁnitely larger than the natural linewidth, the expected deviation due to theﬁnite Rabi frequency is ∼10 % because the saturation parameter 2Ω2 /Γ2 is approximately 10.
Thus, we reach a conclusion that all the measurements given in this study are consistent witheach other.
Finally we add a supplementary result on the transition strengths for each vibrational levelof the (3)1 Σ+ state. In the section 4.2.6, we presented a prediction on the transition strengths
based on the Franck-Condon factor (Fig.4.19). However, this is not an accurate prediction on thetransition strengths. Now we have the measured values of the transition dipole moment between
the v ′ = 41 level of (3)1 Σ+ and the v ′′ = 91 level of X1 Σ+ and that between the v ′ = 41 level of(3)1 Σ+ and the v ′′ = 0 level of X1 Σ+ . If we combine these results with our potential curve of the(3)1 Σ+ state, we can systematically compare the transition dipole moments for each vibrational
level. Fig. 6.38 shows our prediction for the transition dipole moments of the (3)1 Σ+ state withthe least bound state and for those with the lowest rovibrational level (v ′′ = 0). The v ′ = 41
level, which was used for the STIRAP transfer of weakly bound singlet molecules (v ′′ = 91), hasfavorable wavefunction overlaps with both the weakly bound and the lowest rovibrational levels.
In addition, other levels such as v ′ = 38 and v ′ = 39 can potentially serve as an intermediate
state for the STIRAP transfer of weakly bound molecules to the rovibrational ground state.

6.6.3

Precision spectroscopy of weakly bound molecules

As a ﬁnal result of this study, we present a precision spectrum of the hyperﬁne structure of
weakly bound molecules. Before this study, depletion spectroscopy on photoassociated moleculessuggested the existence of the hyperﬁne structure of the order of 10 MHz, although we could not
resolve the structure. With the development of SpIDR spectroscopy, we could clearly resolve
the hyperﬁne structures in both the ground and the excited state, but the hyperﬁne structure in
the excited state made it diﬃcult to measure the structure of weakly bound molecules precisely.

186

Ion counts (arb.units)

1.0
0.8

2

Γ = 2π·3.2(3)×10 kHz

0.6
0.4

0

2

4

6
8
10
Excitation time (µs)

1214Figure 6.37: Time decay of the population in the rovibrational ground state with the resonant
light at 641 nm. The population in the v ′′ = 0 level is plotted with respect to excitation time.
The light used for a STIRAP transfer destroys the produced molecules. The observed time
constant of the decay of 1.0(1) µs corresponds to the twice of the lifetime in the excited state.

Transition dipole moment (ea0)0.1

Least bound level
Lowest vibrational level

7
6
5
4
3
2

0.01
7
6
5
4

36

38

40
42
44
Vibrational number

46

48

50

Figure 6.38: Transition dipole moments of the (3)1 Σ+ state with the X 1 Σ+ state calculated fromthe experimentally obtained accurate potential. The absolute values for weakly bound levels arecalibrated by dark resonance spectroscopy for the transition v ′ = 41, (3)1 Σ+ ← v ′′ = 91, X 1 Σ+
whereas those for the rovibrational ground state are calibrated by dark resonance spectroscopy
for the transition v ′ = 41, (3)1 Σ+ ← v ′′ = 0, X 1 Σ+ .

187

A STIRAP transfer into the rovibrational ground state opens up a new route for a precision
molecular spectroscopy. By mapping the initial population onto the rovibrational ground state,
we can precisely measure the hyperﬁne structure of photoassociated molecules. Hereafter we callthe new spectroscopic method as STIRAP spectroscopy. Fig.6.39 shows a schematic representation of the three spectroscopic methods, depletion, SpIDR, and STIRAP. The small hyperﬁne
structure in the rovibrational ground state (∼ a few kHz) enables us to precisely determine thepeaks (Fig.6.40). The observed energy levels of the F ′′ = 3, 2, 1 levels with respect to the F ′′ = 0
level are 55.960(9), 27.089(7), and 8.852(7) MHz, respectively. The values of these energy levels as obtained by a coupled channel calculation are 56.180, 27.176, 8.870 MHz, which are in
reasonable agreement with our observations.
The present method, based on mapping the population into a single quantum state with aRaman transfer, can be extended to other molecular species as well as other molecular states
for precision spectroscopy. Combined with a optical frequency comb, a Raman spectroscopy ofthe energy diﬀerence between a weakly bound level and a deeply bound level is expected to givean accurate measurement of the time variation of the electron-to-mass ratio [27].

6.7

Theoretical description of a two-photon dark resonance anda STIRAP transfer

In this section, we introduce a simple three-level model for describing coherent two-photon
phenomena such as a two-photon dark resonance (or electromagnetically induced transparency,
EIT) and STIRAP. We ﬁrst derive the time-dependent Schrodinger equation which can begenerally used for the later discussions (section 6.7.1). Then we consider the two importantphenomena relevant to the present study separately. In the section 6.7.2, we derive the width ofa two-photon dark resonance. In the section 6.7.3, we discuss the two-photon width as well as
the transfer eﬃciency in a STIRAP experiment. The obtained expressions are used for analyzing
the experimentally observed behaviors.

6.7.1

General discussions for a three-level system

We consider a three-level system shown in Fig.6.41 and begin with the dressed state Hamiltonian
for a two-level system in the rotating wave approximation:
(
)
~ 0 Ω
(6.8)
H=
2 Ω 2∆
where ~ is the Planck constant, Ω is the Rabi frequency, and ∆ is the laser detuning from theatomic resonance frequency. The Hamiltonian for the three-level system is obtained by using
Eq.(6.8) twice:


0
Ωp
0
~

Ωs
(6.9)
H = Ωp 2∆p20Ωs 2(∆p − ∆s )Assuming that the system is in a steady state and the two-photon detuning δ = ∆p − ∆s iszero, we obtain the three eigenvalues of the time-independent Schrodinger equation.

)
√
1(

2 + Ω2 + Ω2

ω
=
∆
+
∆
p

p
p
s
 + 2
ω0 = 0()√
ω− = 1 ∆p − ∆2 + Ω2 + Ω2pps2Here we deﬁne three variables θ, ϕ, and Ω0 as
188

Excited state(3)1Σ+,

Excited state(3)1Σ+,

KRb+

Monitor

KRb+
Spectroscopy

Spectroscopy
MonitorInitial state=91, =0

Initial state=91, =0

X1Σ+,Initial state=91, =0

X1Σ+,

X1Σ+,
(a) Depletion spectroscopy(b) SpIDR spectroscopy

Excited state(3)1Σ+,

KRb+
Spectroscopy

MonitorInitial state=91, =0

X1Σ+,Final state=0, =0

X1Σ+,(c) STIRAP spectroscopy
Figure 6.39: Schematic diagram of three spectroscopic methods. (a) Depletion. (b) SpIDR. (c)STIRAP. In depletion spectroscopy, the initial population is directly monitored with REMPI
ionization. In SpIDR spectroscopy, the initial population is mapped onto the excited state and ismonitored by the REMPI ionization of the deeply bound molecules. In STIRAP spectroscopy,
the initial population is mapped onto the rovibrational ground state and is monitored with
REMPI ionization.

189

10
6

5
4

F''=2
STIRAP (v''=0, J''=0)
SpIDR (v'=41, J'=1)
Depletion (v''=91)

0

2
F''=1

Depletion ion counts (arb.units)

STIRAP and SpIDR Ion counts (arb.units)

15

F''=3

F''=0

-1260

-1240
-1220
Offset frequency of L1 (MHz)-1200

Figure 6.40: Spectra indicating the hyperﬁne structure of photoassociated molecules. (a) Depletion. (b) SpIDR. (c) STIRAP. In depletion spectroscopy, saturation broadening makes it
diﬃcult to resolve the structure. In SpIDR spectroscopy, the hyperﬁne structures in both theground and the excited states are resolved, but the hyperﬁne structure in the excited state makes
it diﬃcult to clearly determine the peak position. In STIRAP spectroscopy, the population ismapped to the rovibrational ground state, which has a small hyperﬁne structure of the order of
1 kHz

|2>
p
s

¶ p

¶ s

|1>
|3>
Figure 6.41: Level diagram of a three-level system. The probe light couples the initial state |1⟩
and the excited state |2⟩, whereas the control light couples the excited state |2⟩ and the ﬁnal
state |3⟩.
190


Ωp

tan θ =



√ Ωs

Ω0 = Ω2p + Ω2s



Ω

tan 2ϕ = 0∆p(6.10)

With these variables, the eigenstates of the eigenvalues in Eq.(6.10) are written as a linear
combination of the original bases without light |1⟩, |2⟩, and |3⟩:
|a+ ⟩ = sin θ sin ϕ|1⟩ + cos ϕ|2⟩ + cos θ sin ϕ|3⟩|a0 ⟩ = cos θ|1⟩ − sin θ|3⟩
|a− ⟩ = sin θ cos ϕ|1⟩ − sin ϕ|2⟩ + cos θ cos θ|3⟩

(6.11)The eigenstate |a0 ⟩ does not include the excited state |2⟩ and thus this state is called asa dark state. This dark state plays a crucial role in the coherent phenomena in the following
manners. We observe a two-photon dark resonance in the one-photon spectrum of the transition|1⟩–|2⟩ when the initial state is close to the dark state and the probe light is not scattered.
When we adiabatically change the intensity ratio, we can eﬃciently transfer the population in
the initial state |1⟩ to the ﬁnal state |3⟩ by keeping the population in the dark state.
In the real experiments of these phenomena, we always have to take care how the two-photon
detuning δ and the time variation in θ have inﬂuences on the coherence of the system. Both δand θ can mix the excited state |2⟩, thereby inducing the absorption of lights. The absorptionof lights reduces the transparency of a two-photon dark resonance and the transfer eﬃciency
for STIRAP. In the following discussions, we will separately argue the two cases, with a ﬁnite
two-photon detuning, and with a time variation in θ.
For better understanding, we ﬁrst derive the eigenstates with δ = 0 and then consider theinﬂuence of a ﬁnite two-photon detuning. The time-dependent Schrodinger equation for thethree-level system for δ = 0 is given asi~∂
C = HC∂t(6.12)




0
Ωp
0
0
sin θ
0
~
~Ω
0 
Ωp 2∆p Ωs  =sin θ 2/ tan 2ϕ cos θ
H=
2
2
0
Ωs
0
0
cos θ
0
 
C1
C = C2 
C3where C1 , C2 , and C3 denote the coeﬃcients for the three eigenstates without light. We rewrite
the expression (6.13) with the bases given in Eq.(6.11):
i~U −1∂
U C̃ = U −1 HU C̃
∂t
 
C+

C̃ = C0 
C−(6.13)(6.14)

where the C+ , C0 , and C− denote the coeﬃcients for the eigenstates with light. U is the
diagonalization matrix of H and is given by
191




sin ϕ sin θ cos θ cos ϕ sin θ0− sin ϕ 
U =  cos ϕ
sin ϕ cos θ − sin θ cos ϕ cos θ(6.15)

With the eigenvalues in Eq.(6.10), the right-hand side of Eq.(6.14) can be written as


ω+ 0
0
U −1 HU = ~  0 ω0 0  ≡ ~ω
(6.16)
00 ω−The diﬀerentiation with respect to t in the left-hand side of Eq.(6.14) results in the following
general equation.
)(∂C̃ = ~ω C̃
i~ B +
∂t


0
−θ̇
−ϕ̇
∂U
= θ̇ sin ϕ
B ≡ U −10θ̇ cos ϕ∂tϕ̇−θ̇ cos ϕ0(6.17)

where θ̇ and ϕ̇ denote the diﬀerentiation of θ and ϕ with respect to time t.
Thus, we ﬁnd that a ﬁnite two-photon detuning introduces the following term for both sidesof Eq.(6.7.1).





0 0 0
U −1 0 0 0  U = ~δD
0 0 ~δ

(6.18)

sin2 ϕ cos2 θ− sin ϕ sin θ cos θ sin ϕ cos ϕ cos2 θsin2 θ
− cos ϕ sin θ cos θD ≡ − sin ϕ sin θ cos θ2
sin ϕ cos ϕ cos θ − cos ϕ sin θ cos θ
cos2 ϕ cos2 θ
By using ω, D, and B introduced in Eqs.(6.16, 6.7.1, 6.7.1), we can write the time-dependent
Schrodinger equation of a three-level system in the following simple form.
∂C̃ = − (iω + iδD + B) C̃
(6.19)∂tNote that the inﬂuence of the spontaneous decays from the excited state |2⟩ can be phenomenologically taken into account by the following permutation in Eq.(6.9). In this case, ϕ inEq.(6.10) is a complex number.
∆→∆−i

6.7.2

Γ
2

(6.20)

Two-photon dark resonance

The most important quantity characterizing a two-photon dark resonance is the width of thetransparency peak. Here we derive an analytical expression for the width on the basis ofEq.(6.19) assuming that there is no decoherence mechanism other than a ﬁnite two-photon
detuning δ. In the real experiments, magnetic ﬁelds, the polarization of lights, and the relative
linewidth between two lights can induce decoherence, thereby limiting the minimum observablewidth. In the absence of these decoherence mechanisms, the spectral width of a two-photon
dark resonance is determined by the Rabi frequency and the natural linewidth of the excitedstate.

192

The basic idea for deriving the spectral width of the transparency peak is as follows. The
transparency indicates the extent to which the probe light is not scattered. Therefore, we have
only to obtain the fraction of the excited state included in the dark state under a ﬁnite twophoton detuning. The deviation in the eigenvalue ω0 is a good measure of the excited state
fraction because the population is originally in the dark state |a0 ⟩.
The eigenvalue equation for deriving the steady state with a ﬁnite two-photon detuning andwithout a time variation in θ is given by
λI + iω + iδD = 0(6.21)

which can be explicitly written as()()
λ λ + iω+ + iδ sin2 ϕ cos2 θ λ + iω− + iδ cos2 ϕ cos2 θ+ iδ sin2 θ (λ + iω+ ) (λ + iω− ) + λδ 2 sin2 ϕ cos2 ϕcos4 θ = 0 (6.22)
Here we note that the deviation from Eq.(6.10) due to the ﬁnite δ is quite small because of
the condition θ ≪ 1 kept in a two-photon dark resonance. The eigenvalue ω0 is expected to be
suﬃciently smaller than 1 even under a ﬁnite δ because ω0 is originally zero at δ = 0. Thus, we
can neglect higher order terms than O(λ) in Eq.(6.22) and obtain the following expression for
the eigenvalue.
λ=−iδω+ ω− sin2 θ)ω+ ω− + ω− δ sin2 ϕ + ω+ δ cos2 ϕ cos2 θ + δ sin2 θ (ω+ + ω− )((6.23)The validity of the assumption made for deriving this expression is conﬁrmed by considering
the magnitude of θ. Experimentally sin θ < 10−2 is ensured, indicating that λ is suﬃciently
smaller than 1.
If we take into account the inﬂuence of spontaneous emissions by the permutation (6.20),
the eigenvalues ω± can be written as√−iΓ/2 ± i (Γ/2)2 − Ω20(6.24)
ω± =2Thus, we obtain a simple expression for the eigenvalue from Eq.(6.23:
λ=−iδΩ20 sin2 θΩ20 + 2iδΓ

(6.25)

Here we assumed Γ > Ω0 because we consider a situation where the light intensity is weak.
The fraction of the excited state in the dark state is approximately given by the absolute valueof Eq.(6.25).
|δ|Ω2 sin2 θ|λ| = √ 4 0Ω0 + 4δ 2 Γ2

(6.26)

Eq.(6.26) indicates that the excited state is not mixed at δ = 0 and increases with δ. From
this expression, we obtain a simple expression for the full-width-at-half-maximum (FWHM) of
the transparency peak:Ω2Ω2
δdark = √ 0 ≈ 0.58 0(6.27)
Γ3Γ
The condition θ ≪ 1 indicates Ω0 ≈ Ωs . Thus, the Rabi frequency of the control light canbe obtained by measuring the width of the transparency peak of a two-photon dark resonance.
A more rigorous treatment [144] gives the line shape of a two-photon dark resonance in thefollowing form:
193

1
(

f (∆) ∝
1+

Ω2c
4Γ∆

)

(6.28)

Here the probe laser is ﬁxed on resonance and the control laser is scanned. This expression
gives the representation for the FWHM width of Ω2c /2Γ, which is in good agreement with theabove simple consideration. In this study, Eq.(6.28) was used for ﬁtting the observed spectra.
In the above discussions, we neglected the inﬂuence of decoherence such as magnetic ﬁelds
and laser linewidths. If the width of the transparency becomes smaller than such decoherence,
the transparency peak itself disappears because the dark state is destroyed.

6.7.3

STIRAP transfer

In this section, we derive an analytical expression for the STIRAP transfer eﬃciency and see
how the transfer duration τ , the natural width Γ, and the two-photon detuning δ can inﬂuenceon the eﬃciency. We ﬁrst deal with the case with δ = 0 for intuitive understanding and thenderive the expression for a ﬁnite δ. Further discussions with a ﬁnite one-photon detuning ∆ andwith higher order terms can be found in Ref. [145].
The transfer eﬃciency is expected to be higher when the excited state fraction in the darkstate is smaller. Focusing on Eq.(6.19) describing the time evolution of the system, we ﬁndthat B appearing with the time variation in θ and ϕ can mix the eigenstates in Eq.(6.11).
The population cannot be kept in the dark state via the two-photon transition with a too rapid
variation in intensity, resulting in the loss of a part of the population through spontaneous decays
from the excited state. From Eq.(6.19), the adiabaticity condition imposed on this rapidity isgiven by the following forms:|ω+ | ≫ θ̇ sin ϕ(6.29)|ω− | ≫ θ̇ cos ϕ(6.30)

Here we neglected ϕ̇ because ϕ does not change signiﬁcantly in an ordinary STIRAP experiment. From Eq.(6.30), we obtain an explicit expression for the adiabaticity condition.
Ω0 τ ≫√

1 + 2∆τ(6.31)

Under the condition ∆ = 0 and Γ = 0, Eq.(6.31) gives a simple form
Ω0 τ ≫ 1(6.32)The expression (6.32) is not valid under real experiments because of the ﬁnite natural widthof the excited state. Later we will discuss this point in detail.
By solving Eq.(6.19), we ﬁnd that the transfer eﬃciency is close to unity if the adiabaticity
condition is fulﬁlled. In order to simplify the following discussions, we assume the time variationof light intensity as{Ωp = Ω0 sin (πt/2τ )Ωs = Ω0 cos (πt/2τ )(6.33)

which indicates three variables are constant:
√


Ω
=
Ω2p + Ω2s = const
0


π
θ̇ =
= const2τϕ̇ = 0194(6.34)

Under this assumption, Eq.(6.19) can be written in a simple form−iω+θ̇ sin ϕ0∂ C̃= −θ̇ sin ϕ0−θ̇ cos ϕ C̃∂t0θ̇ cos ϕ −iω−

(6.35)

Substituting C0 (t = 0) = 1 and C± (t = 0) = 0 and assuming that the time variation in C0
is negligible, we obtain the solution for C± under the adiabaticity condition:θ̇ sin ϕ[1 − exp(−iω+ t)] C0
C+ (t) =iω+θ̇ cos ϕC− (t) =[1 − exp(−iω− t)] C0iω−(6.36)

By substituting Eq.(6.36) to the equation for C0 and neglecting the high frequency oscillation
term exp(−iω± t) due to the adiabaticity condition, we obtain a simple expression for C0 at t = τ
as
(
)
2 ∆
C0 (τ ) = exp −iπ(6.37)
τ Ω20This expression implies that the transfer eﬃciency is given as|C0 (τ )|2 = 1(6.38)and does not depend on ∆, τ , and Ω0 as long as the adiabaticity condition is satisﬁed. However,
the eﬃciency decreases with the ﬁnite natural linewidth of the excited state:
(
)
π2Γ
2
P = |C0 (τ )| = exp − 2(6.39)τ Ω0Thus, we require a condition on τ for achieving a high transfer eﬃciency as
Ω20 τ
≫ π2
(6.40)
Γ
when the natural linewidth of the excited state is ﬁnite. Although this expression corresponds
to an adiabaticity condition based on Eq.(6.30), we have to take care of the following two points.
First, a factor of π 2 is added to Eq.(6.32). Second, Eq.(6.40) gives a more strict condition when
the Rabi frequency is smaller than the natural linewidth.
Next we consider a situation with a ﬁnite δ. From the viewpoint of implementing STIRAP
experiments, this consideration is important because it gives an expression for the two-photon
linewidth as well as estimation on the required linewidth for Raman lasers.
We solve Eq.(6.19) by assuming the adiabaticity condition and substituting C0 (t = 0) = 1
and C± (t = 0) = 0 as implemented for the case of δ = 0. If we consider a situation with a smallδ and assume δ ≪ ω± , we can neglect terms including δC± and obtain the following equations.

∂C+





∂t



∂C

 −
∂t

∂C0




∂t






(
)
∼
= −iω+ C+ + θ̇ sin ϕ + iδ sin ϕ sin θ cos θ C0()∼
= −iω− C− + θ̇ cos ϕ + iδ cos ϕ sin θ cos θ C0()
= iδ sin ϕ sin θ cos θ − θ̇ sin ϕ C+()
+ iδ cos ϕ sin θ cos θ − θ̇ cos ϕ C− − iδ sin2 θC0

195(6.41)

Assuming that the time variation in C0 is suﬃciently slow, we can solve the ﬁrst and thesecond equations in Eq.(6.41).
iδ sin ϕ sin θ cos θ + θ̇ sin ϕ[1 − exp(−iω+ t)] C0
C+ (t) =iω+iδ cos ϕ sin θ cos θ + θ̇ cos ϕC− (t) =[1 − exp(−iω− t)] C0iω−(6.42)

Substituting Eq.(6.42) to the third equation of Eqs.(6.41) and neglecting the rapidly oscillating term exp(−iω± t), we obtain the equation describing the time evolution of C0 .
][() ( sin2 ϕ cos2 ϕ )
∂C0
2
2
2
2
2
+
− δ sin θ̇t C0(6.43)= i δ sin θ̇t cos θ̇t + θ̇∂tω+ω−Thus, we obtain the coeﬃcient for the dark state at t = τ C0 (τ ) by integrating Eq.(6.43)
from t = 0 to t = τ :
[ ( 2
)
]
)( 2
δ
sin ϕ cos2 ϕδτ2
C0 (τ ) = exp i
+
+ θ̇
−i
(6.44)8ω+ω−
2Here we use a general relation
sin2 ϕ cos2 ϕ
4∆
+
=− 2
ω+
ω−
Ω0
for deriving a simple and general expression for the coeﬃcient of the dark state:[]()τ ∆ δ2 π2δτ
C0 (τ ) = exp −i 2
+ 2 −iτ2Ω0 2(6.45)(6.46)

If we assume that the one-photon detuning ∆ is zero but the natural linewidth is ﬁnite,which we encounter the most frequently, we obtain a simple and useful expression for the transfer
eﬃciency:
(
)
(
)
π2Γ
Γτ 2
P = |C0 (τ )| = exp − 2
exp − 2 δ
(6.47)Ω0 τ
2Ω0
The ﬁrst term indicates the adiabaticity condition and appears also in Eq.(6.39), whereas
the second term appears with a ﬁnite detuning and gives the two-photon width as√Ω0Ω0δSTIRAP = 2 2 ln 2 √≈ 2.4 √
(6.48)ΓτΓτOn the basis of the discussions thus far, we can estimate a requirement for Raman lasers witha realistic experimental condition. Typically we have the natural width and the Rabi frequencyof{Γ ≈ 2π × 300kHzΩ0 ≈ 2π × 600kHz(6.49)

for a molecular transition in our experiment. The adiabaticity condition restricts the transfer
duration asτ ≫ 1.3µs
which indicate that we can set τ as

196

(6.50)τ ≈ 10µs(6.51)

With these values, the two-photon width of a STIRAP transfer is given as
δSTIRAP ≈ 2π × 300kHz(6.52)

Thus, Raman laser should be stable within 100 kHz for a day-to-day reproducibility. The
requirement for the laser linewidth D is obtained by assuming the typical transfer eﬃciency as
90 % in the expression for a decrease in eﬃciency P = exp(−Dτ ):exp(−Dτ ) > 0.9(6.53)

D < 2π × 2kHz

(6.54)

which imposes the condition on D as

Thus, we need Raman lasers with a linewidth narrower than 1 kHz for our experiment.
We note that the required Rabi frequency depends on the natural width of the excited state.
For clarifying this point, we rewrite the duration τ satisfying the adiabaticity condition asτ = 50ΓΩ20(6.55)

which gives the two-photon width asΩ2
δSTIRAP = √ 050Γ

(6.56)The natural linewidth is proportional to the square of the transition dipole moment, whereas
the Rabi frequency is proportional to the transition dipole moment. Therefore, the required
Rabi frequency for ensuring an identical two-photon width with an identical transfer duration
becomes smaller with a smaller natural linewidth. Another important implication for a moleculartransition in Eq.(6.55) and Eq.(6.56) is that the Franck-Condon factor, rather than transition
dipole moments, should be large. With the small Franck-Condon factor, the transfer duration
should be long whereas the width decreases, giving more stringent conditions for both thelinewidth and the long-term stability of Raman lasers.

197

Chapter 7

Conclusion and outlook
In this thesis, I described a new method for producing ultracold 41 K87 Rb molecules in therovibrational ground state. I showed that a rotationally and vibrationally pure sample of therovibrational ground-state molecules can be achieved in a MOT via photoassociation followed
by a STIRAP transfer. Although photoassociated molecules are much more hot and dilute than
Feshbach molecules used in the previous studies, they can be transferred to the rovibrational
ground state with a high STIRAP transfer eﬃciency of more than 70 %.
The present all-optical approach has two major signiﬁcances. First, our results open up
the new possibility of populating a wide variety of molecular species, which cannot be formed
via magnetoassociation, in an arbitrary rovibrational level. Magnetoassociation is available
only for paramagnetic atoms such as alkali atoms or a highly magnetic Cr atom. In contrast,
photoassociation has been observed for various atomic species, including alkaline-earth and raregas atoms. The all-optical approach is expected to enable producing the molecular species in
the rovibrational ground state from these atomic species. Second, the high repetition rate of ourapproach makes it ideal for precision measurements. Magnetoassociation has been applied toquantum degenerate atoms obtained by evaporative cooling, which typically takes more than 10
s. In contrast, photoassociation can be directly applied to laser-cooled atoms without trappingand evaporative cooling. Due to such simplicity, we can produce the rovibrational ground-state
molecules with an unprecedented rapidity.
What I want to emphasize here is the usefulness of photoassociation. In the long historyof photoassociation, coherent phenomena such as a two-photon dark resonance and a STIRAP
transfer have not been observed for photoassociated molecules. Further, the hyperﬁne structures have been neglected in most of the previous studies of photoassociated molecules. Now
most of the cold molecules experiments appear to focus on Feshbach molecules. However, ourstudy showed that photoassociated molecules are still an intriguing subject to be investigatedin detail. Fig.7.1 summarizes the physical quantities obtained for the ﬁrst place in this study.
In addition, photoassociation gives a good starting point for various applications which do not
require quantum degeneracy. For example, quantum computation [25], precision spectroscopyof physical constants [27, 28], and ultracold chemistry [29, 54] are within reach.
Despite the intensive eﬀorts in this study, there are still several unknown quantities. One of
the important studies is to reveal the hyperﬁne structure in the vibrational ground state. There
are numerous previous studies on the microwave spectroscopy of room-temperature molecules (athorough review is found in Ref. [137]). These studies revealed the hyperﬁne structures of deeply
bound levels with an accuracy of 1 kHz. The hyperﬁne structure of KRb in the rovibrational
ground state should also be elucidated with a similar method.
The improvement of the STIRAP transfer eﬃciency is another important task. In terms ofmaking a quantum degenerate molecular gas, it is crucial not to loose the phase-space density
during the transfer. Although I gave possible interpretations of the present high decoherence
rate limiting the eﬃciency, we have not yet clearly conﬁrm the observed behaviors. I expectthat replacing the dye laser with a diode laser would improve the eﬃciency, but I do not know198

Rotational constant
B=618.2(1) MHz
Natural linewidth
Γ/2π=3.2(3)x102 kHz

Hyperfine structure
1.58(1) MHz

Excited state(3)1Σ+,

1.17(1) MHz
0.47(1) MHz

Transition strength
d=0.035(2) ea0
Transition strength
d=0.0098(3) ea0

Hyperfine strucutre: 8.852(8) MHz
Rotational constant
B=229.78(5) MHz
Initial state1X Σ+, =91, =0: 18.237(4) MHz: 28.871(5) MHz

Rotational constant
B=1095.4(1) MHz
Final stateX1Σ+, =0, =0

Figure 7.1: Physical quantities measured in this study. SpIDR spectroscopy, two-photon darkresonance spectroscopy, and STIRAP spectroscopy reveal most of the important molecular properties relevant to the present study. The rotational constant of the excited state is measured more
precisely with SpIDR spectroscopy for the v ′′ = 91,J ′′ = 2 level of X1 Σ+ than with depletion
spectroscopy.

199

what will be observed then. In such a situation, the high frequency noises in diode lasers may
also cause considerable decoherence and limit the eﬃciency.
As a possible future work with the present setup, I would like to point out the importance ofultracold chemistry. Ultracold chemistry for KRb molecules, 2KRb→K2 +Rb2 , has been explored
for the fermionic 40 K87 Rb molecules [54,139]. These studies exhibit an excellent agreement witha theory, implying that the underlying physics is well understood. However, in these studies,
an important aspect remains unobserved: the reaction products. In terms of investigatingchemistry, it is imperative to observe the reaction products, because one of the most important
aspects of chemistry is to produce new chemical substances. The present study based on the
REMPI detection of molecules is expected to provide a unique opportunity to detect the reaction
products. For this purpose, we will require optical trapping of molecules such that a high reaction
rate is obtained.

200

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Appendix A

Publication: Toward the productionof quantum degenerate bosonic polar
molecules, 41K87Rb
K. Aikawa, D.Akamatsu, J. Kobayashi, M. Ueda, T. Kishimoto,and S. Inouye

New Journal of Physics 11, 055035 (2009).

A1

New Journal of Physics
The open–access journal for physics

Toward the production of quantum degeneratebosonic polar molecules, 41K87Rb
K Aikawa1 , D Akamatsu2 , J Kobayashi2 , M Ueda3,4 , T Kishimoto5and S Inouye2,4,6
1
Department of Applied Physics, University of Tokyo, Bunkyo-ku,
Tokyo 113-8656, Japan
2
Institute of Engineering Innovation, University of Tokyo, Bunkyo-ku,
Tokyo 113-8656, Japan3
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan
4
ERATO Macroscopic Quantum Control Project, JST, Bunkyo-ku,
Tokyo 113-8656, Japan
5
University of Electro-Communications, 1-5-1 Chofugaoka, Chofu 182-8585,
JapanE-mail: inouye@sogo.t.u-tokyo.ac.jp
New Journal of Physics 11 (2009) 055035 (12pp)

Received 26 December 2008
Published 14 May 2009
Online at http://www.njp.org/
doi:10.1088/1367-2630/11/5/055035

We review our experimental progress toward achieving the
production of quantum degenerate bosonic polar molecules, 41 K87 Rb. Our
primary goal is to transfer quantum degenerate Feshbach molecules into theabsolute ground state via the stimulated Raman adiabatic passage (STIRAP).
We have achieved a dual-species Bose–Einstein condensate of 41 K and 87 Rb
in a magnetic trap, which consists of NK = 1.4 × 104 potassium atoms and
NRb = 3.5 × 104 rubidium atoms. An optimum optical transition for STIRAP isbeing identified by a separate experiment based on photoassociation in a dualspecies magneto-optical trap of 41 K and 87 Rb.
Abstract.

6

Author to whom any correspondence should be addressed.

New Journal of Physics 11 (2009) 055035
1367-2630/09/055035+12$30.00© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2Contents

1. Introduction
2. Key issues in the experimental design
3. Double-species Bose–Einstein condensate of 41 K and 87 Rb
3.1. Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Production of a 41 K Bose–Einstein condensate . . . . . . . . . . . . . . . . . .
3.3. Quantum degenerate mixture of 41 K and 87 Rb . . . . . . . . . . . . . . . . . .
4. Spectroscopy toward STIRAP
4.1. Selecting the optimum path for STIRAP . . . . . . . . . . . . . . . . . . . . .
4.2. Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Spectroscopy of the (3)1 6 + state . . . . . . . . . . . . . . . . . . . . . . . . .
5. Conclusion
Acknowledgments
References

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1. Introduction

Quantum degenerate gases of polar molecules are regarded as one of the most exotic quantum
many-body systems for the anisotropic and long-range nature of dipole–dipole interaction.
Novel quantum phases, such as crystal [1] and supersolid [2], are predicted for a Bose–Einsteincondensate of polar molecules. The dipole–dipole interaction between molecules in an opticallattice is expected to pave the way for simulating a spin system [3], while the controllability of
the anisotropic interaction via a microwave [4] will offer a new tool to explore the physics of aquantum many-body system.
To achieve quantum degenerate polar molecules, many approaches have been devised andimplemented [5]. Recently, significant advance has been made in the ‘indirect approach’, which
is based on associating ultracold atoms to form ultracold diatomic polar molecules. Here theprocess involves two steps: the first is to form loosely bound molecules by photoassociation ormagneto-association at a Feshbach resonance of ultracold atoms and the second is to transfer
them to their lowest rovibrational level via optical transitions such as stimulated emission
pumping (SEP) or stimulated Raman adiabatic passage (STIRAP) [6]. The second process isrequired to cope with the following two unfavorable characteristics of loosely bound molecules:
firstly, loosely bound molecules are unstable against collisions (vibrational quenching), andsecondly, they have a negligible dipole moment since atoms in the molecule are well separated.
Thus, transferring loosely bound molecules to the absolute ground state is very important.
This two-step scheme was originally demonstrated at Yale for laser-cooled 85 Rb and 133 Cs
atoms to form ultracold 85 Rb133 Cs molecules in their absolute ground state [7]. In that pioneering
work, a combination of photoassociation and SEP was employed. However, a further quest
for atomic cooling and improvement in the transfer efficiency is needed for realizing quantum
degeneracy with polar molecules.
Here we employ an alternative strategy that combines magneto-association at a Feshbachresonance and STIRAP. Magneto-association can transfer atoms to a single molecular levelat nearly 100% efficiency. STIRAP realizes a conversion efficiency of 100% since it uses a
two-photon dark state where population between two molecular states can be controlled by the
New Journal of Physics 11 (2009) 055035 (http://www.njp.org/)3
relative intensity of two lasers. Moreover, there are no heating processes in these steps. Thus,
this combination should enable us to convert quantum degenerate atoms to quantum degeneratepolar molecules. Intensive efforts to obtain quantum degenerate molecules in this scheme havebeen made for bosonic Rb2 [8] and Cs2 [9] molecules at Innsbruck and for fermionic 40 K87 Rb
molecules at JILA [10].
We are aiming at achieving a Bose–Einstein condensate of polar molecules. To produce
bosonic polar molecules from atoms, there are two possible choices for constituent atoms,
i.e. two fermionic species or two bosonic species. Choosing two fermionic species forconstituent atoms has an advantage in that Feshbach molecules are expected to have a long
lifetime due to the Pauli exclusion principle. There are not many combinations of atomic species
left for this approach because (a) it is natural to choose an alkali–alkali combination to make useof a magnetically tuned Feshbach resonance and (b) known stable fermionic alkali species are
only 6 Li and 40 K. The possibility of performing efficient STIRAP for LiK is yet to be explored,since knowledge about the molecular potentials of LiK is currently limited. An experiment inthis approach is ongoing in Munich [11].
In contrast, there are several possible combinations for choosing two bosonic species.
Molecular potentials of both ground states and excited states are already well known for someof the combinations. Among others, 41 K87 Rb has the advantage that Feshbach resonances havebeen reported at low magnetic fields [12] and all the associated scattering lengths (aRb , aK
and aKRb ) are positive [13]. In addition, a high photoassociation rate of KRb [14] enables us
to explore molecular potentials using photoassociated molecules. A drawback of using two
bosonic species is that Feshbach molecules tend to have a short lifetime because there is nosuppression of collisions [15]. This problem can be circumvented by putting atoms in an opticallattice before applying magneto-association [16]. We note that a combination of 87 Rb and 133 Cs
is yet another attractive possibility since molecular potentials in the excited states have been
investigated in detail [17] and many Feshbach resonances have been reported [18].
Here we report our progress toward achieving a quantum degenerate gas of 41 K87 Rb
molecules based on magneto-association and STIRAP. Our efforts involve the production of a
mixture of 41 K and 87 Rb Bose–Einstein condensates and the search for an optimum intermediate
state for STIRAP via spectroscopy of 41 K87 Rb molecules.
In the following, section 2 discusses important parameters that set the ultimate efficiency
for magneto-association and STIRAP. Section 3 describes our experimental set-up to obtaina double-species Bose–Einstein condensate of 41 K and 87 Rb. Section 4 presents our spectroscopic studies on photoassociated KRb molecules. Section 5 summarizes the obtained results
and discusses future prospects.
2. Key issues in the experimental design

STIRAP is expected to acheive 100% transfer efficiency when the whole population is held in adark state. However, the efficiency is decreased when the dark state is perturbed. The efficiency
of a STIRAP transfer at a two-photon resonance is given by [19]
π 2 0 DτP = exp − 2 −
,
(1)
2
0 τ
where 0 is the natural linewidth of the excited state, τ is the pulse duration, D is the
linewidth associated with the frequency difference between the two lasers and 0 is the reduced
New Journal of Physics 11 (2009) 055035 (http://www.njp.org/)

4qRabi frequency, which is defined as 0 = 2P + 2S , where P (S ) is the peak Rabi frequencyof the pump (Stokes) field7 . Here the first term describes the reduction of efficiency due todiabatic processes, while the second term describes the limitation due to decoherence. Fromequation (1), we can derive the necessary condition for an efficient transfer:201  D.
(2)2
π 0τ
Thus, reducing the linewidth D helps reduce the intensity required for a successful transfer.
The Rabi frequency in equation (2) depends on the Franck–Condon factor, which is ameasure of the wavefunction overlap between the ground and excited states:2|he|gP(S) i|2 .
2P(S) = d 2 E P(S)(3)

Here d is a transition dipole moment, E P(S) is an electric field of the pump (Stokes) laser and|he|gP(S) i|2 is a Franck–Condon factor between an excited state and a loosely bound level (thelowest rovibrational level). We assume that the transition dipole moment does not depend on theinternuclear distance. To achieve a high transfer efficiency, one must identify a transition with alarge Franck–Condon factor. Thus, it becomes imperative to perform a molecular spectroscopy
to find the details of interatomic potential.
Now let us examine if one can satisfy condition (2) in the actual experiment using theparameters for 41 K87 Rb. The differential frequency of two independent lasers can be stabilized towithin D ∼ 2π × 10 kHz by locking them to a stable high-finesse cavity. To satisfy the conditionof D and τ in (2), the pulse duration should be τ ∼ 1 µs. The time variation of laser intensityof the order of 1 µs is easily achieved by an acousto-optical modulator. The condition of 0 , 0and τ in (2) then gives a limitation on the Rabi frequency:
20
 107 s−1 .
0(4)

For the KRb molecule, the (3)1 6 + is a good candidate for the intermediate state as explained
in section 4. This state has a transition dipole moment of ∼1ea 0 [20], which corresponds toa natural linewidth of ∼2π × 500 kHz. The Franck–Condon factors between this state and theground state are of the order of 10−2 [21]. Substituting these values into the inequality (4), we
obtain a required Rabi frequency 0 ∼ 2π × 3 MHz and a required intensity I ∼ 600 mW cm−2 .
Since the typical size of a Bose–Einstein condensate is less than 100 µm, the power necessary
for an efficient transfer is less than 100 µW, which is easily obtained by using a semiconductor
laser.
Another key issue that affects the efficiency of STIRAP is the closed-channel fraction of
Feshbach molecules, which contributes dominantly to the Franck–Condon factor. In general,the closed-channel fraction of Feshbach molecules away from the resonance is larger than thatclose to the resonance. However, the absolute amount of the closed-channel fraction depends
strongly on the width (or the coupling strength) of the resonance [22]: molecules formed at a
broad resonance are strongly perturbed by an atomic channel even away from the resonance,
while those formed at a narrow resonance are dominated by the closed channel except very
near the resonance. Thus, it is beneficial for us to use a narrow Feshbach resonance for the
preparation of molecules with a large Franck–Condon factor. However, there is a drawback in
7

In this expression, P ' S is assumed.

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5
using closed-channel-dominated molecules: they can easily decay via vibrational quenching due
to a large wavefunction overlap with low-lying molecular levels. To avoid this problem, atoms
should be loaded into an optical lattice before they are converted to molecules.
To date, two Feshbach resonances have been reported for 41 K87 Rb at 35 G with a width of
37 G and at 79 G with a width of 1.2 G [12, 13]. Apparently, the former is a broad resonance,
while the latter is a narrow resonance. In fact, coupled channel calculations indicate that aclosed-channel fraction of Feshbach molecules formed at the 79 G resonance exceeds 90%
below 70 G [21]. Based on the above discussions, we conclude that the resonance at 79 G isappropriate for the preparation of molecules with a large closed-channel fraction.
3. Double-species Bose–Einstein condensate of 41 K and 87 Rb

3.1. Experimental set-up
Our experimental apparatus for the production of a Bose–Einstein condensate of 41 K and 87 Rb
is based on a double MOT system. Two-stage differential pumping was employed to realize ahigh vapor pressure (∼10−8 Torr) for both potassium and rubidium and an ultra-high vacuum of
3 × 10−11 Torr. The first cell contained home-made enriched 41 K dispensers and commercial Rb
dispensers. The cell was kept at 50 ◦ C in order to increase the loading efficiency of potassium.
For evaporative cooling of both potassium and rubidium atoms, the cloverleaf trap with amagnetic field gradient of 292 G cm−1 and a curvature of 67 G cm−2 was constructed aroundthe second cell. Typically, 109 41 K atoms and 109 87 Rb atoms were simultaneously collected ina MOT in the first cell and continuously pushed to the second cell by resonant laser beams. The
pushing laser for potassium was resonant with the F = 1 to F 0 = 2 transition, while that for
rubidium was resonant with the F = 2 to F 0 = 3 transition. The pushed atoms had a velocity ofaround 20 m s−1 and were captured again by the MOT in the second cell. The loading efficiencyof the second MOT depends on the diameter and intensity of both cooling and repumping beamsof the second MOT. For potassium, a loading rate of 3 × 108 atoms s−1 was obtained at an e−2
beam diameter of about 20 mm and a total intensity of about 50 mW cm−2 .
3.2. Production of a 41 K Bose–Einstein condensate
Bose–Einstein condensation of 41 K atoms was first realized by sympathetic cooling withrubidium atoms [23]. After that, we reported the successful production of a Bose–Einsteincondensate of 41 K with a large number of atoms by means of direct evaporative cooling [24].
Here, we briefly describe two key issues for achieving a Bose–Einstein condensate of 41 K
without sympathetic cooling.
The first issue is the complexity in laser cooling of 41 K. It originates from the fact that thehyperfine splitting between F 0 = 3 and F 0 = 2 in the 43 P3/2 state is as small as 13.6 MHz. This isof the same order as the natural linewidth of this excited state (6.2 MHz). Thus, it is practically
impossible to form a closed transition for laser cooling since off-resonance excitations to nearby
levels such as F 0 = 2 and F 0 = 1 are not negligible. To cope with this problem, we have preparedalmost the same amount of laser power for both cooling and repumping beams and detuned bothlights below the whole hyperfine manifold by about 20 MHz [25]. Although this configuration
enabled us to trap a large number of atoms, it led to a high temperature of about 5 mK and a low
density of about 1 × 1010 cm−3 . The obtained phase-space density at this stage was about 10−9 .
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6
(a)

(b)

(c)

Integrated OD(arb. units)

2.0
1.0
0.0

Figure 1. Realization of Bose–Einstein condensation of 41 K atoms by direct

evaporation in a magnetic trap [24]. Absorption images of 41 K atoms after 40 ms
expansion show (a) a thermal cloud at T = 750 nK, (b) a bimodal distribution at
T = 580 nK and (c) an almost pure condensate of 3 × 105 atoms. The evaporative
cooling was performed by driving rf-transition between hyperfine levels of 41 K
atoms. No coolant was involved in the experiment.
By introducing additional stages for spatial compression and Doppler cooling [26], we wereable to reach a phase-space density of 10−6 , which was a good starting point for evaporative
cooling.
The second key issue is the necessity of using the hyperfine transition for evaporative
cooling. We could not obtain a Bose–Einstein condensate of 41 K atoms by driving transitions
between Zeeman sublevels. This was presumably because the rate constants of inelastic
processes involving atoms in the F = 2, m F = 1 state [27, 28] were higher for 41 K than for 87 Rb.
We confined atoms in the F = 2, m F = 2 state in a magnetic trap and performed evaporative
cooling by driving the hyperfine transition to the F = 1, m F = 1 state. Typical frequencies
needed for evaporation were less than 400 MHz, since the hyperfine splitting of 41 K atoms
in the ground state is only 254 MHz. By evaporating atoms in the F = 2, m F = 2 state whileconstantly removing atoms in the F = 2, m F = 1 state, it was possible to achieve Bose–Einsteincondensation as shown in figure 1 [24]. The typical number of atoms in a pure condensate
was 3 × 105 .
3.3. Quantum degenerate mixture of 41 K and 87 Rb
We have also succeeded in producing a quantum degenerate mixture of 41 K and 87 Rb in amagnetic trap. We first loaded almost the same number of 41 K and 87 Rb atoms into a magnetic
trap and performed evaporative cooling of both 41 K and 87 Rb atoms by driving the F = 2,m F = 2 to F = 1, m F = 1 transition simultaneously. However, we observed significant losses ofatoms for both species. Thus, we switched to sympathetic cooling of 41 K by 87 Rb [23]. So far,
we have succeeded in achieving a dual-species Bose–Einstein condensate with up to 1.4 × 104
atoms for 41 K and 3.5 × 104 atoms for 87 Rb (figure 2).

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7

Figure 2. Images of dual Bose–Einstein condensates of 41 K and 87 Rb atoms

after 20 ms of expansion. The evaporative cooling was performed by driving theF = 2, m F = 2 to F = 1, m F = 1 transition for the 87 Rb atom. 41 K atoms
were sympathetically cooled as in [23]. The numbers of atoms in these images
were 3.5 × 104 and 1.4 × 104 for Rb and K atoms, respectively.
4. Spectroscopy toward STIRAP

4.1. Selecting the optimum path for STIRAPThe optimum state for STIRAP should have large Franck–Condon factors with both weakly
bound levels and the lowest rovibrational level. To understand which state can be used as
an intermediate state, knowledge of the excited states of molecules is required. Here webriefly explain the excited states of alkali diatomic molecules with electronic orbital angularmomentum L of 1. There are extensive spectroscopic data for these states that facilitate ourspectroscopy. Since an alkali dimer has a total spin S = 0 (singlet) or S = 1 (triplet), there are
four excited states labeled as 1 6 + , 1 5, 3 6 + and 3 5 in Hund’s case (a). Figure 3 shows thecalculated potential energy curves of the ground and excited states of KRb molecule in Hund’scase (a) [29]. In Hund’s case (c), these states are coupled by the spin–orbit interaction to yieldfour components:  = 0+ (1 6 + and 3 5), 0− (3 6 + and 3 5), 1 (1 5, 3 6 + and 3 5) and 2 (3 5), whereE is a projection of total angular momentum JE = LE + S.
The Franck–Condon factor depends not only on the spatial overlap but also on the spincomponent of the wavefunction. Since a Feshbach molecule has both spin triplet and singlet
components, there are two possible paths to transfer it to the absolute ground state: (a) Feshbach
molecule (triplet component) → intermediate state (spin mixed) → the absolute ground state(singlet) or (b) Feshbach molecule (singlet component) → intermediate state (singlet) → theabsolute ground state (singlet).
The first scheme uses a spin mixing due to the spin–orbit interaction in the intermediate
state. This scheme was first pointed out for the  = 1 manifold consisting of (1)1 5, (2)3 6 + and(1)3 5 [17] and demonstrated for RbCs [7] and for 40 K87 Rb [10]. In this scheme, the (2)3 6 +state couples to the loosely bound molecules, whereas the (1)1 5 state couples to the lowest
rovibrational level.
In contrast, in the second scheme we can focus on a spin singlet state with large spatial
overlaps of the wavefunctions. In fact, the experiment for Cs2 at Innsbruck has been performed
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8

Σ

Σ

Π

Σ

+

+

+

Σ

+

Π

Σ

+

Σ

+

Figure 3. Ab initio potential energy curves of KRb [29]. Here only potentials
related to our experiments are shown. Singlet and triplet potentials are indicatedby blue and red curves, respectively.

based on this scheme [30]. For the case of KRb it was pointed out that the (3)1 6 + state hasfavorable wavefunction overlaps both with loosely bound levels and the lowest level [31].
We are currently pursuing the second scheme using the (3)1 6 + state. The (3)1 6 + state has apurely singlet character; it can potentially couple with states dissociating to the K(4P) + Rb(5S)
limit, but they are so far away from the (3)1 6 + state that the spin–orbit coupling between
them is negligible. The accurate potential energy curve of the (3)1 6 + state has been found in afrequency range from 9778 cm−1 (potential minimum) to 10 412 cm−1 by means of conventional
spectroscopy of 39 K85 Rb molecules [32, 33]. Since the best candidate for the intermediate statefor STIRAP lies between 11 400 and 11 800 cm−1 , the potential in this region has to be more
precisely determined.
4.2. Experimental set-up
For spectroscopic studies, we prepared another experimental apparatus that can produce looselybound molecules via photoassociation of laser-cooled 41 K and 87 Rb atoms following a similar
experiment by the University of Connecticut group [31]. This set-up enables us to take aspectrum over a wide range of frequencies at a rate of 10 Hz, which is much faster than taking aspectrum with Feshbach molecules. Figure 4 shows our experimental set-up for spectroscopy ofloosely bound 41 K87 Rb molecules. First, 1 × 108 K atoms and 2 × 108 Rb atoms were cooled
and trapped in a double-species MOT. Compressed MOT (C-MOT) was applied for both
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9

Figure 4. Schematic diagram of the experimental set-up used for the depletion

spectroscopy of ultracold 41 K87 Rb molecules. 41 K and 87 Rb atoms were cooled
and trapped simultaneously in a MOT. The photoassociation laser was applied for
10 ms to form loosely bound X1 6 + and a3 6 + molecules, whereas the depletion
laser was always turned on. Molecules in a specific vibrational level of the X1 6 +state were ionized by an intense pulse laser and detected with a Channeltron
ion detector.
clouds to cool and compress further to obtain number densities of 2 × 1011 cm−3 for K and4 × 1011 cm−3 for Rb and temperatures of 400 µK for K and 100 µK for Rb. Both K and
Rb atoms were prepared in their lower hyperfine states F = 1 during the C-MOT process.
Then, a photoassociation laser with an intensity of 8.5 W cm−2 irradiated K and Rb atoms
for 10 ms to form ground state molecules8 . The frequency of the photoassociation laser wasusually fixed at about 12 570.134 cm−1 , exciting the J = 1 level of the  = 0+ state. A tunable
CW Ti:S laser with an intensity of 13 W cm−2 was simultaneously applied to molecules todeplete the population in a specific rovibrational level. Resultant molecules were detectedwith a Channeltron (Burle 4869) after they were ionized via resonance-enhanced multi-photon
ionization induced by a pulsed dye laser with an intensity of 3 × 106 W cm−2 . Signals fromthe Channeltron were amplified with a homemade pre-amplifier [34] and sent to a high-speed
digitizer (National Instruments PCI-5112) to count the number of molecules. Scanning thefrequency of the Ti:S laser yielded a depletion spectrum that can resolve rotational structures ofboth ground and excited states.
The levels used for the two-photon ionization are indicated in figure 3. Molecules werefirst excited to the v 00 = 43 level of the (4)1 6 + state9 and were then immediately excited to theionization level. By selecting an appropriate wavelength for the ionizing pulse laser, we wereable to monitor molecules in a specific vibrational level though the rotational structure could not
be resolved due to a broad linewidth of the pulse laser (∼0.1 cm−1 ). We often used molecules in
8

Typical 1/e2 beam diameter was 0.7 mm, which was nearly the same as the size of the MOT.
The numbering given here has an uncertainty of a few quanta since the potential minimum of the (4)1 6 + state
has not yet been identified. This assignment of the vibrational number is based on the work on 39 K85 Rb by D Wang.
9

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10

Figure 5. A depletion spectrum showing rotational structures of cold molecules.

The spectrum was obtained by scanning the depletion laser while monitoring
molecular ion counts. Each dip corresponds to a resonant excitation ofmolecules in the monitored level. Since we used the J = 1 rotational levelfor photoassociation, subsequent decays yielded ground state molecules in the
J 00 = 0 and 2 levels based on a selection rule 1J = ±1. The same selection rulealso holds for an excitation from X1 6 + to (3)1 6 + states. Thus, three transitions
were observed for a single upper vibrational level, namely, the J 0 = 3 ← J 00 = 2,
J 0 = 1 ← J 00 = 2 and J 0 = 1 ← J 00 = 0.
the v 00 = 91, J = 2 level of the X 1 6 + state for spectroscopy of the excited state as they were themost abundant. The binding energy of this level was determined to be −12.454(1) cm−1 relative
to the FK = 1 + FRb = 1 threshold by comparing a depletion spectrum and a photoassociation
spectrum for the same state.
4.3. Spectroscopy of the (3)1 6 + state
Before taking a depletion spectrum, we performed a spectroscopy of the (3)1 6 + states witha pulse laser as a rough guide in the frequency range from 11 340 to 11 740 cm−1 , whereconventional spectroscopic studies have not been reported. Since the infrared photon does nothave enough energy to ionize molecules via a two-photon transition, we simultaneously applied
an additional 532 nm pulse laser derived from a pump laser for the pulsed dye laser. In thisscheme, molecules were first excited by the infrared laser and immediately ionized by the green
laser. To suppress a three-photon transition with the infrared laser and a two-photon transition
with the green laser, we had to keep the intensity of the infrared and green lasers below 5 × 105
and 1 × 105 W cm−2 , respectively. Based on the obtained spectrum, we performed depletion
spectroscopy for the (3)1 6 + state. A typical spectrum for the v 00 = 48 level of the (3)1 6 + state isshown in figure 5. The narrow width of the spectrum (∼100 MHz) enabled us to locate absoluteenergy levels of the excited state with much higher accuracy than using an ab initio potential,
which typically has an error of the order of 10 cm−1 (∼300 GHz).
To understand the observed spectrum, we need to consider the selection rule for rotational
levels, i.e. 1J = ±1 [36]. Spontaneous decays from the photoassociation level of J = 1 yield
the ground state molecules of J = 0 and 2. Molecules in J = 0 can only be excited to J = 1 of
New Journal of Physics 11 (2009) 055035 (http://www.njp.org/)

11
the (3)1 6 + state, whereas those in J = 2 can be excited to J = 1 and 3 of the (3)1 6 + state. Thus,
we obtain three transitions as shown in figure 5. From this spectrum, rotational constants of thev 00 = 91 level of the X1 6 + state and the v 00 = 48 level of the (3)1 6 + states were determined tobe 0.00763(8) and 0.02017(1) cm−1 , respectively. The measured constant of the X1 6 + state wasin good agreement with that calculated from the experimentally determined accurate potential
for the ground state [37] within the experimental error. We compared the measured constantof the (3)1 6 + state with that obtained from the ab initio potential [29] and found that the
ab initio potential gave a larger value by an amount of about 1%. This deviation has alreadybeen reported in [31] and should be attributed to the inaccuracy of the ab initio potential. We
observed similar spectra for ten vibrational levels (v 00 = 41–50) that were used to construct
an accurate potential that can explain the measured energy levels and rotational constants [38].
Based on this potential, we conclude that the optimum intermediate state lies around the v 00 = 42level, which has Franck–Condon factors of about 10−2 with both the loosely bound level andthe lowest rovibrational level [21].
5. Conclusion

In this paper, first we discussed key issues in the experimental design of quantum degeneratepolar molecules. Then, first experimental results toward the realization of a Bose–Einsteincondensate of 41 K87 Rb molecules were reported. A dual-species Bose–Einstein condensate of41
K and 87 Rb was realized by evaporative cooling of rubidium atoms and sympathetic cooling ofpotassium atoms. Excited states of KRb molecules were studied using depletion spectroscopy
and photoassociation. We plan to load a dual-species Bose–Einstein condensate of 41 K and87
Rb into an optical lattice to form Feshbach molecules. A STIRAP transfer of loosely boundmolecules will be studied using photoassociated 41 K87 Rb molecules. Finally, a STIRAP transfer
will be applied to Feshbach molecules to produce a Bose–Einstein condensate of 41 K87 Rb
molecules in the absolute ground state. The list of physics we would like to pursue with quantum
degenerate polar molecules includes resonant features in collisions [39], the stability of themolecule [40] and the supersolid phase [2].
Acknowledgments

We acknowledge fruitful discussions with P Naidon. KA and DA acknowledge support from theJapan Society for the Promotion of Science.
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New Journal of Physics 11 (2009) 055035 (http://www.njp.org/)

Appendix B

Coherent transfer of photoassociatedmolecules into the rovibrational
ground state
K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Kobayashi,
P. Naidon, T. Kishimoto, M. Ueda, and S. Inouye.

Physical Review Letters 105, 203001 (2010).

B1

PRL 105, 203001 (2010)

PHYSICAL REVIEW LETTERS

week ending
12 NOVEMBER 2010

Coherent Transfer of Photoassociated Molecules into the Rovibrational Ground State
K. Aikawa,1,* D. Akamatsu,2,† M. Hayashi,1,‡ K. Oasa,1 J. Kobayashi,2 P. Naidon,3 T. Kishimoto,4M. Ueda,3,5 and S. Inouye2,31
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Institute of Engineering Innovation, The University of Tokyo, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
3
JST, ERATO, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
4
Center for Frontier Science and Engineering, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
5
Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received 16 July 2010; published 10 November 2010)2

We report on the direct conversion of laser-cooled 41 K and 87 Rb atoms into ultracold 41 K87 Rb
molecules in the rovibrational ground state via photoassociation followed by stimulated Raman adiabatic
passage. High-resolution spectroscopy based on the coherent transfer revealed the hyperfine structure of
weakly bound molecules in an unexplored region. Our results show that a rovibrationally pure sample ofultracold ground-state molecules is achieved via the all-optical association of laser-cooled atoms, opening
possibilities to coherently manipulate a wide variety of molecules.
DOI: 10.1103/PhysRevLett.105.203001

PACS numbers: 37.10.Mn, 33.15.Pw, 42.50.Ct, 82.80.Ms

Progress in cooling and manipulating atoms has led todiverse applications. Even more possibilities will open up
if molecules are cooled and prepared in the rovibrational
ground state because then the translational, vibrational,
and rotational degrees of freedom of the molecules can be
manipulated with greater accuracy. The potential applications range from novel quantum phases to ultracold chemistry, quantum computation, and precision measurements [1].
Various schemes to achieve cold molecules have been demonstrated thus far. Direct cooling of molecules has been
limited in terms of temperature [2]. Recently, quantum
gases of molecules in the rovibrational ground state
have been achieved [3–5] with a coherent optical transfer
of weakly bound molecules produced via the magnetoassociation [6] of quantum degenerate atoms. However, thisscheme is only applicable to atomic species for which magnetoassociation is available. An alternative general method
for producing weakly bound molecules from ultracold atoms
is photoassociation (PA) [7]. Up to now, only the incoherent
formation of molecules in the vibrational ground state via
photoassociation has been reported [8–10]. Although thedirect photoassociation yielded molecules in the v00 ¼ 0,
J 00 ¼ 2 level predominantly [10], the state-selective photoassociative formation of cold molecules in the rovibrational
ground state is yet to be achieved.
Here, we report that a rotationally and vibrationally puresample of ultracold 41 K87 Rb molecules in the rovibrational
ground state can be directly produced in a magneto-opticaltrap of 41 K and 87 Rb via photoassociation followed bystimulated Raman adiabatic passage (STIRAP) [11]. In
spite of the low laser intensity due to the large volume ofour molecular gas, we have reached a STIRAP transfer
efficiency of more than 70% by employing an intermediate
state with a narrow natural linewidth. In order to identify
the two-photon resonance precisely, we developed a new
0031-9007=10=105(20)=203001(4)detection scheme, spontaneous-decay-induced double
resonance (SpIDR), which provided the high-resolution
spectra of one-photon transitions from PA molecules andallowed us to observe a two-photon dark resonance [12].
Our results show that a rovibrationally pure sample
of ultracold molecules in the ground state is achieved
via the all-optical association of laser-cooled atoms, opening novel possibilities to coherently manipulate a wide
variety of cold molecules that can be produced via
photoassociation.
Figure 1 shows the schematic representation of the coherent transfer of ultracold 41 K87 Rb molecules. The potential
curves are taken from Ref. [13]. Details of the experimental
setup for the production and detection of weakly boundmolecules are found elsewhere [14]. For the present study,
we used 41 K87 Rb molecules in the v00 ¼ 91, J 00 ¼ 0 level of
X1 þ produced via photoassociation of 41 K and 87 Rb atoms
in a magneto-optical trap. The typical numbers, densities,
and temperatures of atoms were 1  108 , 2  1011 cm3 ,and 400 K for 41 K and 2  108 , 4  1011 cm3 , and100 K for 87 Rb, respectively. The PA laser resonant to
the J 0 ¼ 1 level of  ¼ 0þ (wave number 12570 cm1 ,intensity 1  103 W cm2 ) was applied for 10 ms.
Molecules were ionized through resonance enhanced multiphoton ionization by a pulsed dye laser and detected by
microchannel plates. The entire experimental procedure
was operated at 9 Hz.
Despite the successful demonstrations on quantum
gases, the STIRAP transfer into the rovibrational groundstate was not straightforward for PA molecules. Any residual magnetic field could cause decoherence for Raman
coupling between two molecular states since our molecularsample was not spin polarized. In addition, untrapped,
high-temperature PA molecules occupied a volume larger( 1 mm3 ) than that occupied by the trapped quantum

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Ó 2010 The American Physical Society

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PRL 105, 203001 (2010)

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FIG. 1 (color online). Relevant molecular potential energy
curves of KRb. Weakly bound molecules were produced by
the photoassociation of laser-cooled 41 K and 87 Rb atoms.
Molecules in the v00 ¼ 91, J00 ¼ 0, F00 ¼ 2 level of X 1 þ
were coupled to the v00 ¼ 0, J00 ¼ 0 level of X 1 þ by a twophoton transition with wavelengths near 875ðL1Þ and641ðL2Þ nm via the v0 ¼ 41, J 0 ¼ 1 level of ð3Þ1 þ . For depletion spectroscopy and SpIDR spectroscopy, two-photon ionization through the ð4Þ1 þ state was used. For the detection of therovibrational ground-state molecules, a three-photon transition
mediated by the v0 ¼ 6 level of ð1Þ1  was used.

gases ( 104 mm3 ). This limited the available laser
intensity for driving the transition.
These difficulties were circumvented by starting fromsinglet molecules which were insensitive to magnetic
fields. Utilizing singlet excited states with a narrow naturallinewidth as an intermediate state, we realized an efficient
Raman transfer of PA molecules into a singlet ground state.
This was in contrast with previous works on the twophoton transfer of heteronuclear molecules [3,8], whichstarted from triplet molecules and transferred the PA molecules into the singlet ground state via the  ¼ 1 state (thespin-mixed state of ð2Þ3 þ , ð1Þ3  and ð1Þ1 ). A transition
strength comparable to the  ¼ 1 state was obtained withthe v0 ¼ 41 level of ð3Þ1 þ [14,15], whereas the narrow
natural linewidth of this state (  2  300 kHz, calculated from Ref. [16]) allowed us to obtain an efficienttransfer with a relatively low laser intensity.
Our optical system for the Raman transition consistedof four lasers. Two diode lasers (875 nm and 641 nm)
were locked to an ultralow expansion cavity with a dualwavelength coating. These lasers worked as master lasers.
Two slave lasers, a ring dye laser (641 nm) and a diodelaser (875 nm), were then locked to master lasers throughan optical phase-locked loop (OPLL) [17] and used for the

10

20

30
40
Time (µs)

5060FIG. 2. STIRAP transfer from the v00 ¼ 91, J 00 ¼ 0, F00 ¼ 2
level to the v00 ¼ 0 level of X1 þ . (a) Transfer to the J00 ¼ 0
level. The population in the v00 ¼ 0 level of X 1 þ is plottedagainst the frequency of the down transition laser (L2).
(b) Transfer to the J 00 ¼ 2 level. The asymmetry observed in
the J 00 ¼ 2 spectrum is expected to be a result of the hyperfine
structure induced by the nuclear electric quadrupole moments of
87 Rb. (c) Time evolution of the population in the v00 ¼ 0, J 00 ¼ 0
level of X1 þ during the multiple STIRAP transfer process. The
error bars indicate the standard deviation. (d) Time variation of
the intensities of the Raman lasers during the multiple-transfer
process. Both intensities are normalized to unity. During thewaiting time of 20 s set between the transfers, the remaining
population that is not transferred by STIRAP is removed by theresonant Raman lasers.

experiments. The scanning of the Raman lasers was realized by sweeping the microwave source for OPLL. In thisreport, the microwave frequency for OPLL was noted as anoffset frequency. The typical short-term linewidth of theRaman lasers was less than 10 Hz, whereas the long-term
stability of the ultralow expansion cavity was estimated tobe less than 100 kHz by monitoring the Cs D2 transition.
The 1=e2 diameter of the Raman lasers was set to 1.5 mm
which was larger than the size of atomic clouds. For the
STIRAP transfer, the power of the laser L1 and the laser L2
was set to 9 mW and 25 mW, respectively.
Figure 2(a) shows the population in the v00 ¼ 0 level of1 þX  after the STIRAP transfer into the J 00 ¼ 0 level as afunction of the frequency of the down transition laser (L2).
The observed two-photon linewidth of 200 kHz was consistent with a simple three-level theoretical model [11]when we took into account the Rabi frequency (2 
400 kHz) and the transfer duration (10 s). In order toconfirm that Fig. 2(a) indicated the rovibrational groundstate, we also performed a STIRAP transfer into the v00 ¼ 0,
J 00 ¼ 2 level [Fig. 2(b)] and determined the rotational
constant to be 1095:4ð1Þ MHz  h (h is the Planck’s

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PHYSICAL REVIEW LETTERS

PRL 105, 203001 (2010)

constant). The uncertainty was due to the unresolved hyperfine structure of the J 00 ¼ 2 level. The obtained value
was in good agreement with the calculated value of1095:362ð5Þ MHz  h obtained from the mass scalingof the experimental value for 40 K87 Rb [18]. The typicalnumber, density, and temperature of the produced groundstate molecules were 103 , 106 cm3 , and 130 K, respectively. The production rate in the rovibrational ground state
was as high as 104 s1 .
The direct evidence of the STIRAP transfer was obtained by studying the time evolution of the population in
the ground state [Fig. 2(c)] during the multiple-transfer
process [Fig. 2(d)]. The ground-state molecules after asingle transfer were used for measuring the efficiency oftransfer. We typically retrieved 53% of the ground-state
molecules after the double-transfer process, which implieda single-step efficiency of 73%. Currently, the efficiency islimited by the difference in the Doppler shift for the twolasers.
The STIRAP transfer into the rovibrational ground state
can be used as new precision spectroscopy for weakly
bound molecules in an unexplored region (Fig. 3). By
mapping the population in the initial hyperfine levels onto
the ground state, we could measure the hyperfine splitting of
PA molecules with an accuracy of 10 kHz  h. A higher
resolution could be achieved by using a longer transfer
duration and smaller intensities for the Raman lasers. The
observed energy levels of the F00 ¼ 3, 2, 1 levels withrespect to the F00 ¼ 0 level were 55.960(9), 27.089(7), and8:852ð7Þ MHz  h, respectively. The values of these energy levels as obtained by a coupled channel calculation
were 56.180, 27.176, and 8:870 MHz  h, which were in
reasonable agreement with our observations.
The rest of this report is devoted to SpIDR spectroscopy,
which is an essential improvement of the one-photon
spectroscopy of PA molecules. This improvement wasnecessary for identifying the structure of the excited statesand observing a two-photon dark resonance. To date,

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Offset frequency of L1 (MHz)FIG. 3. STIRAP spectroscopy of weakly bound molecules (thev00 ¼ 91, J00 ¼ 0 level of X 1 þ ). An efficient transfer into therovibrational ground state enables the precision spectroscopy ofthe hyperfine structure of photoassociated molecules in an unexplored region with a typical accuracy of 10 kHz. The labeling isbased on the total angular momentum including the nuclear spin.

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depletion spectroscopy has been widely used for analyzing
the transition from PA molecules [15]. A continuous-wave(cw) laser depletes the number of PA molecules, while theloss is monitored with resonance enhanced multiphoton
ionization. Although this spectroscopy was straightforward, we encountered a serious problem related to thesignal-to-noise ratio when we attempted to record data at
a low intensity in order to avoid saturation broadening. At arelatively high intensity, where we could observe a large
signal, the spectral width broadened to tens of megahertz,
washing out all the structure from the hyperfine levels. In
addition, saturation broadening smoothed out the transparency peak of the two-photon dark resonance and made itdifficult to identify the two-photon resonance.
In order to improve the signal-to-noise ratio of depletion
spectroscopy, we developed a method to monitor not the
depletion but the creation of molecules in the excited state.
In practice, we monitored the population in one of thevibrational levels in the ground state to which a significant
fraction of excited molecules decays. Thus, we could
obtain a background-free spectrum, thereby increasing
the signal-to-noise ratio to observe the narrow linewidthof the ð3Þ1 þ state.
Figure 4(a) shows the SpIDR spectrum of the v0 ¼ 41,
0
J ¼ 1 level of ð3Þ1 þ . A considerably smaller power(5 W) and shorter time (400 s) than those requiredfor depletion spectroscopy (> 100 W, 10 ms) were sufficient to detect the excitation. This led to a significantly
narrow linewidth close to the natural linewidth. The coarse
splitting resulted from the hyperfine structure in the groundstate, whereas the fine splitting implied the existence of thehyperfine structure in the excited state. For the sake ofcomparison, the depletion spectrum in the same frequency
range is shown in Fig. 4(b).
The SpIDR spectroscopy enabled us to observe a twophoton dark resonance [Fig. 4(c)]. The dip indicatedthe emergence of the two-photon dark state with the laser
L2 [12]. By comparing the two frequencies at the dip, we
determined the binding energy of the v00 ¼ 0, J 00 ¼ 0 levelof X1 þ with respect to the v00 ¼ 91, J 00 ¼ 0 level of
X1 þ to be 124 955:92ð4Þ GHz  h, where the accuracy
was limited by the uncertainty of our wave meter. Taking
into account the binding energy of the v00 ¼ 91, J 00 ¼ 0
level with respect to the threshold FK ¼ 1 þ FRb ¼ 1,
374:75ð3Þ GHz  h, the binding energy of the v00 ¼ 0,
J 00 ¼ 0 level with respect to the atomic level without the hyperfine structure was determined to be125 335:11ð5Þ GHz  h. From the width of the transparency peak, we determined the transition dipole moment
for the down transition to be 0.0098(7) ea0 , assuming thenatural linewidth of the excited state to be 2  300 kHz.
When we locked the laser L2 to the dip frequency andscanned the laser L1, we obtained a well-known darkresonance spectrum [Fig. 4(d)].
In conclusion, we realized an efficient STIRAP transfer
of PA molecules into the rovibrational ground state. Highresolution spectroscopy of the hyperfine structure of PA

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repetition rate makes our method a promising candidate
for high-precision spectroscopy. The precision measurement of the electron-to-proton mass ratio is within reachas microwave spectroscopy [20] or optical spectroscopy
combined with a frequency comb [21].
We thank M. Kozuma for helpful discussions, andM. Falkenau, Y. Tanooka, and K. Mori for experimental
assistance in the early stages of the experiment. K. A. and
D. A. acknowledge support from the Japan Society for the
Promotion of Science.

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Offset frequency of L1 (MHz)FIG. 4 (color). SpIDR spectroscopy of photoassociated molecules. (a) SpIDR spectrum of the v0 ¼ 41, J 0 ¼ 1 level ofð3Þ1 þ taken by exciting molecules in the v00 ¼ 91, J 00 ¼ 0
level of X 1 þ . Molecules formed by spontaneous decays fromthe excited state are counted against the offset frequency of thelaser L1. The coarse splitting results from the hyperfine structureof the v00 ¼ 91, J00 ¼ 0 level of X1 þ . The fine splitting in eachline implies the existence of hyperfine structure in the excitedstate. For the two-photon transition, the largest peak (F00 ¼ 2) is
employed as an initial state. (b) Depletion spectrum. In orderto observe any signal, we had to increase the laser intensity by
3 orders of magnitude as compared to (a). Transitions from thev00 ¼ 91, J 00 ¼ 0 level are barely observable since the ion counts
are dominated by molecules in the v00 ¼ 91, J 00 ¼ 2 level.
(c) Two-photon dark resonance between the v00 ¼ 91, J00 ¼ 0,
F00 ¼ 2 level and the v00 ¼ 0, J 00 ¼ 0 level of X1 þ . The downtransition laser (L2) is scanned while the up transition laser (L1)is held on resonance. The dots indicate experimentally obtained
data and the solid curve is a fit to the data points. (d) Two-photon
dark resonance obtained by scanning the laser L1. The laser L2
is held on resonance. The blue dashed curve indicates a SpIDR
spectrum for the F00 ¼ 2 level, while the red solid one indicatesa spectrum obtained with the laser L2.

molecules in an unexplored region was demonstrated using
coherent transfer. We developed a highly sensitive detection scheme which enabled us to take a high-resolution
spectrum of one-photon transitions and observe a twophoton dark resonance for PA molecules. Our method isreadily extended to other atomic species for which laser
cooling and photoassociation are available. Trapping
molecules into an optical dipole trap [19] is the next key
issue. The trapped sample of molecules will constitutea test tube for ultracold chemistry. Moreover, the high*ka_cypridina@atomtrap.t.u-tokyo.ac.jp†
Present address: National Metrology Institute of Japan,
Tsukuba 305-8563, Japan.
‡
Present address: Nikon Corporation, Kanagawa 252-0328,
Japan.
[1] R. V. Krems, W. C. Stwalley, and B. Friedrich, Cold
Molecules: Theory, Experiment, Applications (CRC,
Boca Raton, 2009); L. D. Carr, D. DeMille, R. V. Krems,
and J. Ye, New J. Phys. 11, 055049 (2009).
[2] J. D. Weinstein et al., Nature (London) 395, 148 (1998);
H. L. Bethlem, G. Berden, and G. Meijer, Phys. Rev.
Lett. 83, 1558 (1999); M. S. Elioff, J. J. Valentini, andD. W. Chandler, Science 302, 1940 (2003); M. Gupta and
D. Herschbach, J. Phys. Chem. A 103, 10 670 (1999);
S. A. Rangwala et al., Phys. Rev. A 67, 043406 (2003).
[3] K.-K. Ni et al., Science 322, 231 (2008).
[4] F. Lang et al., Phys. Rev. Lett. 101, 133005 (2008).
[5] J. G. Danzl et al., Nature Phys. 6, 265 (2010).
[6] T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys.
78, 1311 (2006).
[7] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne,
Rev. Mod. Phys. 78, 483 (2006).
[8] J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Phys.
Rev. Lett. 94, 203001 (2005).
[9] M. Viteau et al., Science 321, 232 (2008).
[10] J. Deiglmayr et al., Phys. Rev. Lett. 101, 133004 (2008).
[11] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod.
Phys. 70, 1003 (1998); N. V. Vitanov, M. Fleischhauer,
B. W. Shore, and K. Bergmann, Adv. At. Mol. Opt. Phys.
46, 55 (2001); P. Kral, I. Thanopulos, and M. Shapiro, Rev.
Mod. Phys. 79, 53 (2007).
[12] S. E. Harris, Phys. Today 50, No. 7, 36 (1997); M. D.
Lukin, Rev. Mod. Phys. 75, 457 (2003); M.
Fleischhauer, A. Imamoglu, and J. P. Marangos, ibid. 77,
633 (2005).
[13] S. Rousseau, A. R. Allouche, and M. Aubert-Frecon, J.
Mol. Spectrosc. 203, 235 (2000).
[14] K. Aikawa et al., New J. Phys. 11, 055035 (2009).
[15] D. Wang et al., Phys. Rev. A 75, 032511 (2007).
[16] R. Beuc et al., J. Phys. B 39, S1191 (2006).
[17] L. Ricci et al., Opt. Commun. 117, 541 (1995).
[18] S. Ospelkaus et al., Phys. Rev. Lett. 104, 030402 (2010).
[19] E. R. Hudson et al., Phys. Rev. Lett. 100, 203201 (2008).
[20] D. DeMille et al., Phys. Rev. Lett. 100, 043202 (2008).
[21] T. Zelevinsky, S. Kotochigova, and J. Ye, Phys. Rev. Lett.
100, 043201 (2008).

203001-4

Appendix C

Predicting and verifying transition
strengths from weakly boundmolecules
K. Aikawa, D. Akamatsu, M. Hayashi, J. Kobayashi,
M. Ueda, S. Inouye.

Physical Review A (in press); arXiv:1011.1319 (2010).

C1

Predicting and verifying transition strengths from weakly bound molecules
K. Aikawa,1, ∗ D. Akamatsu,2, † M. Hayashi,1, ‡ J. Kobayashi,2 M. Ueda,3, 4 and S. Inouye2, 3
1
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Institute of Engineering Innovation, The University of Tokyo, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
3
JST, ERATO, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
4
Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Dated: February 18, 2011)

We investigated transition strengths from ultracold weakly bound 41 K87 Rb molecules produced via
the photoassociation of laser-cooled atoms. An accurate potential energy curve of the excited state(3)1 Σ+ was constructed by carrying out direct potential fit analysis of rotational spectra obtained via
depletion spectroscopy. Vibrational energies and rotational constants extracted from the depletion
spectra of v ′ = 41–50 levels were combined with the results of the previous spectroscopic study,
and they were used for modifying an ab initio potential. An accuracy of 0.14% in vibrational levelspacing and 0.3% in rotational constants was sufficient to predict the large observed variation intransition strengths among the vibrational levels. Our results show that transition strengths fromweakly bound molecules are a good measure of the accuracy of an excited state potential.
PACS numbers: 34.20.Cf,33.20.Sn,37.10.Mn,82.80.Ms∗ ka cypridina@atomtrap.t.u-tokyo.ac.jp

address:
National Metrology Institute of Japan,
Tsukuba 305-8563, Japan‡ Present address: NIKON CORPORATION, Kanagawa 2520328, Japan(4)1Σ+

1.5
4P+5S
(1)1Π

(3)1Σ+-1

Ultracold molecular gases are prominent candidates
for realizing novel distinctive applications in physics andchemistry, including precision measurements, quantum
computation, ultracold chemistry, and novel quantum
phases [1, 2]. Thus far, the production of ultracold
molecules in the vibrational ground state has been dependent on the optical transfer of weakly bound molecules
formed via either photoassociation [3] or magnetoassociation [4]. Among the previously proposed methods [5–10],
stimulated Raman adiabatic passage (STIRAP) [11–13]
of weakly bound molecules is the most efficient method
for preparing a molecular sample in a single quantum
state.
In the STIRAP transfer, an excited state is used as
an intermediate state. It is important to select an excited state having large transition strengths from both
weakly bound and deeply bound levels. In general, thestrengths of molecular transitions from low vibrational
levels are readily predicted on the basis of the FranckCondon factors (FCFs) calculated from a potential energy curve or molecular constants [14]. The small numberof nodes in the radial wavefunction for a low vibrational
level indicates a low sensitivity to deviations in the radial direction, thereby enabling us to explain the intensity distribution over vibrational levels from molecularconstants. However, it is difficult to predict transition
strengths from weakly bound molecular levels because
weakly bound levels have a large number of nodes in theirwavefunctions; hence, the FCFs are quite sensitive to the† Present

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INTRODUCTION

4S+5P

1.0
(2)3Σ+

4

I.

Energy (10 cm )arXiv:submit/0199405 [physics.atom-ph] 18 Feb 2011

2

PA

Depletion

(1)3Π

0.5

Decay

REMPI
(Depletion)a 3Σ +

0.0

4S+5S
X1Σ+

-0.5
2

4

6

8

10

FIG. 1. (Color) Relevant potential energy curves of KRb.
Weakly bound molecules were produced by the photoassociation of laser-cooled 41 K and 87 Rb atoms. The (3)1 Σ+
state was investigated via the spectroscopy of molecules in thev ′′ = 91 level of X 1 Σ+ . The intermediate state for REMPI
ionization was the (4)1 Σ+ state.

wavefunctions of excited states. In this letter, we showthat transition strengths from weakly bound levels canbe predicted on the basis of an accurate potential energy
curve of the excited state, constructed via direct potential fit (DPF) analysis [15] of both vibrational energies

2
and rotational constants.
We focus on the (3)1 Σ+ state of KRb in the range
11400–11800 cm−1 with respect to the ground atomic
threshold 4S + 5S, which was proposed as a potential candidate for the STIRAP transfer of weakly boundmolecules to the rovibrational ground state [16]. An RKR
potential curve of the (3)1 Σ+ state was reported from thepotential minimum up to 10400 cm−1 [17]. An ionization
spectrum for 39 K87 Rb obtained by using a pulsed laser
and a depletion spectrum near 11700 cm−1 were provided in Ref. 16. Recently, we have realized the STIRAP
transfer of weakly bound molecules (v ′′ = 91, J ′′ = 0 of
X 1 Σ+ ) to the rovibrational ground state (v ′′ = 0, J ′′ = 0
of X 1 Σ+ ), mediated by the v ′ = 41 level of the (3)1 Σ+
state [18].
In this article, we report on the spectroscopic studies
on the (3)1 Σ+ state carried out in the range 11400–11800
cm−1 , which reveals 10 vibrational levels (v ′ = 41–50).
We find that the width of the observed spectra is highly
dependent on the vibrational levels of the (3)1 Σ+ state.
By introducing an analytical representation for power
broadening, we extract the transition strengths for each
vibrational level. The other electronic states that correlate with the (3)1 Σ+ state via spin-orbit interaction are
far away; hence we assume that most of the perturbations from these electronic states are negligible. Thus,
we can analyze the experimentally obtained spectra on
the basis of a single potential curve. In addition to theprevious spectroscopic work near the bottom of the potential [17], vibrational energies and rotational constants
extracted from the spectra are used to construct an accurate potential via DPF analysis. Using the modified
potential, it is possible to explain the variation in transition strengths among the vibrational levels of the (3)1 Σ+
state in terms of the FCFs.

II.

DEPLETION SPECTROSCOPY

Previously, our experimental setup for the spectroscopy of ultracold 41 K87 Rb molecules was describedin detail [19]. We provide a brief summary herein. We
start with a dual-species magneto-optical trap (MOT)of 1 × 108 41 K atoms and 2 × 108 87 Rb atoms. A compressed MOT (C-MOT) procedure is applied for 40 ms tocompress and cool the 41 K and 87 Rb atoms. The typicaldensities and temperatures of 41 K and 87 Rb at the endof C-MOT are 2 × 1011 cm−3 and 400 µK and 4 × 1011
cm−3 and 100 µK for 87 Rb, respectively. A photoassociation (PA) laser (wavenumber, 12570.13 cm−1 ; intensity, 1 × 103 W/cm2 ) is applied for 10 ms at the endof the C-MOT process. The produced molecules are detected using micro-channel plates (MCP) after they are
ionized via resonance enhanced multi-photon ionization(REMPI) using a pulsed dye laser (wavenumber, 16543
cm−1 ; intensity, 3×106 W/cm2 ).
Depletion spectra are obtained by monitoring ion
counts in the v ′′ = 91 level of X 1 Σ+ , while a

CW Ti:Sapphire laser (Sirah Matisse TX; intensity, 50
W/cm2 ; beam waist, 350 µm) is continuously applied andscanned. In the present study, we analyze spectra for thev ′′ = 91, J ′′ = 2 level, whose binding energy with respect
to the atomic threshold FK =1+FRb =1 was measured as−12.454(1) cm−1 . The frequency of the Ti:Sapphire laser
is monitored using a Fabry-Perot cavity which is locked
to 87 Rb D2 line. The cavity transmission signal is used tocalibrate the variation in the scanning speed and to measure the relative frequency of the laser with respect tothe cavity transmission peak with a precision of 3 MHz.
The absolute frequency is measured using a commercial
wavemeter (Wavelength WS-7; accuracy, 60 MHz). The
accuracy of measurements for rotational constants is limited by the spectral width, which is of the order of 100
MHz, whereas that for vibrational energies is limited byboth the spectral width and the wavemeter.
Fig. 2 shows a depletion spectrum for the v ′ = 41 levelof the (3)1 Σ+ state, which was used as an intermediate
state for the STIRAP transfer from the v ′′ = 91, J ′′ = 0
level of X 1 Σ+ to the v ′′ = 0, J ′′ = 0 level [18]. We can extract transition strengths as well as rotational constantsfor both ground and excited states from the spectra. In
the appendix, we show that an approximate representation of the full-width-at-half-maximum (FWHM) of adepletion spectrum is given by√2∆ ∼ 0.79Ω Γτ .
(1)where Ω is the Rabi frequency; Γ, the natural width ofthe excited state; and τ , the interaction time. Roughly
speaking, the width increases not only with the light intensity but also with the interaction time. In our case,τ was estimated as a few milliseconds on the basis of
the temperature of the molecules and the beam diameter of the depletion laser. The natural width of the(3)1 Σ+ state has not been precisely known, but it wasestimated as 2π × 300 kHz from an ab initio calculation[20]. In the following discussion, we derive the Rabi frequency from the observed spectrum
by assuming τ = 2√
ms, Γ = 2π × 300 kHz, and Γτ = 60.
III.

ANALYSIS

Although the ab initio potential of Ref. 21 enabled us
to identify the symmetry of the observed spectra without
any ambiguity, it gave rotational constants that exceeded
the experimental ones by approximately 1% (Fig. 3).
There are two reasons for this deviation. First, the potential minimum exceeds the minimum of the RKR curve by
97 cm−1 . Second, the outer turning point is at a shorter
internuclear separation than the RKR potential. We findthat the variation in the observed transition strengths
can not be understood with the ab initio potential. The
observations can not be attributed to the small change
in the vibrational quantum number, which corresponds
to the energy difference between the minima of the RKR
and the ab initio potentials. These facts indicate that the1-1

Rotational constants (10 cm )

3.0

-2

30

1 +

Ion counts in v"=91, X Σ (arb.units)

3J' = 1←J" = 0
20

J' = 3←J" = 2
J' = 1←J" = 2

10011428.911429.011429.1
11429.2-1
Depletion wavenumber (cm )11429.3

ab initio potential is not sufficiently accurate. In orderto obtain an accurate potential, we carried out a DPF
analysis, whereby a potential is iteratively modified until
its eigenvalues coincide with those determined from theexperimental spectra [15]. With the aid of the phiFIT
program code [22], we first construct an analytical Extended Morse Oscillator (EMO) potential of the (3)1 Σ+
state on the basis of (1) the RKR curve, (2) a few points
from the inner curve of the ab initio potential, and (3)a few points around 12500 cm−1 from the outer curveof the ab initio potential. The potential curve of theground state X 1 Σ+ is also required for the calculations.
We use an EMO potential fitted to an accurate, experimentally determined potential [23]. Then, the analyticalpotential of the (3)1 Σ+ state is modified to reproduce
our data by using the DPotFit program code [24]. A
good convergence is achieved when we modify the potential significantly by manually moving the points from theab initio potential. This procedure is repeated until the
eigenvalues of the potential are within 0.05 cm−1 of theobserved levels, i.e., only 0.14% of the vibrational levelspacing. The remaining deviations are presumably due
to the incomplete analytical function used to represent
the potential, although perturbations from other states
can also be the origin of the deviations for some levels.
Table I lists the final potential parameters. These parameters are used to represent the potential energy curve inthe following form:
V (R) =Vmin + De (1 − e−φ(R)(R−Re ) )2
φ(R) =12X

φi y(R, Re )ii=03y(R, Re ) =R − Re3.
R3 + Re3(2)

Fig. 4 shows the RKR potential, ab initio potential andmodified potential. With the new potential, rotational

1 +

3 +

(2) Σ

(2) Σ

Observed

2.6
2.4
2.2
2.0
1.8
1.6
11400
11500
11600
11700
11800
1 +
-1
Transition wavenumber from v"=91, X Σ (cm )′FIG. 2. (Color online) Depletion spectrum of v = 41 levelof (3)1 Σ+ taken with molecules in v ′′ = 91 level of X 1 Σ+ .
The spectrum gives rotational constants of both ground andexcited states. In addition, the width of the spectrum enables
us to estimate the transition strength.

(1) Π

1 +

(3) Σ

2.8

3

(1) Π

FIG. 3. (Color online) Comparison between calculated andobserved rotational constants. Calculated rotational constants of five molecular states in this range are plotted againstenergy levels from the ground atomic threshold. The observed
rotational constants are in good agreement with those obtained from the ab initio potential of the (3)1 Σ+ state. Thereis a systematic deviation of ∼1% between the observed values
and ab initio values, which indicates the inaccuracy of the abinitio potential.

constants are reproduced within 0.3% of the observed
values (Table II). We excluded the v ′ = 44 level fromthe analysis because the characteristics of this level areanomalous; a much larger transition strength, a muchsmaller rotational constant, and a much larger deviation
in a vibrational energy were observed at this level. Thesefeatures indicate that this level is coupled to the (2)1 Σ+
state which was observed at ∼1 cm−1 above the v ′ = 44level.
The most important implication of this analysis is thatthe variation in the transition strengths with respect tothe vibrational levels of the excited state can be accurately explained on the basis of the new potential. Fig. 5
shows a plot of the Rabi frequencies derived from theobserved spectra against those calculated from the corrected potential. For comparison, the same plots basedon the ab initio potential are also shown. The calcu-

TABLE I. Parameters for EMO potential obtained via DPF
analysis of the observed spectra. The units are cm−1 for Vmin
and De , Å for Re , and Å−1 for φi .
Parameter
Vmin
De
Re
φ0
φ1
φ2
φ3
φ4

Value
9777.6963
3246.0363
5.25904119
0.449794066
0.200265883
0.406840126
0.207884795-0.476349301

Parameter
φ5
φ6
φ7
φ8
φ9
φ10
φ11
φ12

Value
-0.804673587
3.00474008
11.3877324-2.140717462-39.73110586-17.01400311
50.93279647
44.83137419

4
TABLE II. Comparison between observed, ab initio, and modified values for vibrational energies (E) and rotational constants(B). Vibrational energies are energy levels of the J ′ = 1 level. Error bars in vibrational energies are 2 × 10−3 cm−1 , whereas
those in rotational constants are 5 × 10−5 cm−1 . Observed vibrational energies are reproduced within 5 × 10−2 cm−1 , whereas
observed rotational constants are reproduced within 7 × 10−5 cm−1 . The vibrational numbering for ab initio values is shiftedby 2 because the potential minimum lies below that of the correct potential.
v′
41
42
43
44
45
46
47
48
49
50

Obs.
11428.965
11466.644
11504.193
11541.484
11578.919
11616.036
11652.982
11689.751
11726.308
11762.685

E (cm−1 )ab initio - Obs.
Mod. - Obs.(×10−3 )
-1.5
2.5
-1.6
29.9
-1.8
42.6
-1.9
164.2
-2.2
-11.4
-2.4
-28.0
-2.6
-37.3
-2.7
-37.5
-2.7
1.2
-2.7
43.0

1.20

4

-1

Energy (10 cm )

1.15

1.10

1.05
modified
ab initio
RKR

1.00

0.95
4

5
6
7
Internuclear separation (Å)

8FIG. 4. (Color) Potential energy curves of (3)1 Σ+ state. The
modified potential obtained in the present study is comparedwith the ab initio potential and the experimental RKR potential. The shaded area denotes the region where depletion
spectra were obtained.

lated values are calibrated on the basis of our recent
measurement of the transition dipole moment between(3)1 Σ+ , v ′ = 41 and X 1 Σ+ , v ′′ = 91, 0.035(2)ea0, determined via dark resonance spectroscopy of the rovibrational ground-state molecules. Weakly bound molecules
have more than 90 nodes in their wavefunction; hence,
the FCFs are highly dependent on the wavefunction of
the excited state. In other words, the FCFs can serve
as a sensitive measure of the accuracy of wavefunctions.
Our results show that accuracies of 0.14% in vibrational
level spacings and 0.3% in rotational constants are sufficient to predict the FCFs from weakly bound levels;
these values are justified by considering the typical sizeof nodes in the radial wavefunction. On one hand, the

Obs.
2.067
2.056
2.05
1.99
2.038
2.037
2.027
2.017
2.008
2.005

B (10−2 cm−1 )ab initio - Obs.(×10−2 )Mod. - Obs. (×10−3 )
1.9
-7.0
2.3
-1.7
2.4
-1.5
7.8
52.6
2.4
-1.5
1.8
-6.8
2.2
-3.2
2.5
0.3
2.9
2.5
2.5
-1.4

weakly bound level v ′′ = 91 in the ground state X 1 Σ+
has an outer turning point of ∼10Å and an inner turning point of ∼2Å in the internuclear distance. Withinthese two points, there are 91 nodes; hence each node
has a typical size of ∼0.1Å. Therefore, the required accuracy for representing the wavefunction is ∼10−2 Å. On
the other hand, in the present analysis, an accuracy of∼0.3% in rotational constants or ∼0.15% in internuclear
distance is obtained for the (3)1 Σ+ state because therelation between the rotational constant B and the internuclear distance R is given by B ∝ R−2 . Assuming
the typical size of molecules in the v ′ = 41–50 levels of(3)1 Σ+ as 6Å, we can derive the accuracy of the modified
potential in the radial direction as ∼10−2 Å; this value isin good agreement with the required accuracy for representing the weakly bound level.

Now that we obtained an accurate potential curve as
well as the absolute values of the transition dipole moment of the v ′ = 41, (3)1 Σ+ level with the v ′′ = 0 andv ′′ = 91 levels of X 1 Σ+ , we can predict the transition
dipole moment for each transition. Fig. 6 shows our prediction for the transition dipole moments of the (3)1 Σ+
state with the least bound state and for those with thelowest rovibrational level (v ′′ = 0). The v ′ = 41 level
used in Ref. 18 has favorable wavefunction overlaps withboth the weakly bound and the lowest rovibrational levels; however other levels such as v ′ = 38 and v ′ = 39 canpotentially serve as an intermediate state for the STIRAP
transfer of weakly bound molecules to the rovibrational
ground state. The potential presented herein can enable
an accurate prediction for other isotopic combinationsof KRb. Further, the present method for achieving an
accurate potential and verifying its accuracy can be extended to other molecular states that exhibit significant
spin-orbit mixing by evaluating eigenvalues via coupledchannel calculations including spin-orbit interaction.

615
500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)

10

5

0
42

Observed Rabi frequency (2π MHz)

(a)

15

44
46
Vibrational number

15

48

50

(b)

10

Transition dipole moment (units of ea0)

Observed Rabi frequency (2π MHz)

5

5
4
3

2

0.01

9
8
7
6

Least bound level
Lowest vibrational level

5
4

36

38

40
42
44
46
Vibrational number

48

50

5

10

0
0

5

5

10

15

500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)

FIG. 6. (Color online) Transition dipole moments of the(3)1 Σ+ state with the X 1 Σ+ state calculated from the experimentally obtained accurate potential. The absolute values forweakly bound levels are calibrated by dark resonance spectroscopy for the transition v ′ = 41, (3)1 Σ+ ← v ′′ = 91, X 1 Σ+
whereas those for the rovibrational ground state are calibrated by dark resonance spectroscopy for the transitionv ′ = 41, (3)1 Σ+ ← v ′′ = 0, X 1 Σ+ .

0
0

5
10
15
Calculated Rabi frequency (2π MHz)

FIG. 5. (Color online) Observed transition strengths. (a)Variation among the vibrational levels. (b) Comparison between the observed values and the calculated values. Observed values are determined from the spectra obtained via
a depletion laser at 104 mW (red circle) and 500 mW (bluerectangle). For the v ′′ = 44 and the v ′′ = 47 levels, the spectra are taken with low powers of 11 mW (v ′ = 44) and 26 mW(v ′ = 47) in order to extract the peaks correctly. Calculations
based on the modified potential can explain the variation intransition strengths with respect to the vibrational levels interms of the Franck-Condon factors. The v ′ = 44 level is expected to be mixed with other states, and thereby shows asignificantly larger transition strength than other levels. The
inset in (b) shows the same plot based on the ab initio potential.

IV.

threshold. We developed a theoretical model that related the spectral width with the Rabi frequency, which
was used to compare the transition strengths for each vibrational level. Rotational constants extracted from theobserved spectra showed a systematic 1% deviation fromthose calculated using an ab initio potential. By carryingout a DPF analysis, we constructed an accurate potential that reproduced energy levels with an accuracy of
0.14% in vibrational level spacing and 0.3% in rotational
constants. The variation in transition strengths among
vibrational levels can be understood in terms of FCFs
calculated with the modified potential. Our results indicate that the transition strengths from weakly bound levels serve as a sensitive measure of wavefunctions, which
can be used to test the accuracy of the potential curve.
In general, the proposed procedure can be adopted forconstructing an accurate potential and verifying its accuracy on the basis of rotational spectra for weakly boundmolecules.

CONCLUSION

The (3)1 Σ+ state of KRb was investigated via the depletion spectroscopy of ultracold molecules formed by
the photoassociation of laser-cooled 41 K and 87 Rb atoms.
The spin-orbit mixing of other electronic states with the(3)1 Σ+ state was negligible; hence, we could assume thisstate is well-described by a single potential. The simplicity of the (3)1 Σ+ state enabled us to modify the potential to reproduce our observations as well as to assign the
spectra. We observed 10 vibrational levels in the range
11400–11800 cm−1 with respect to the 4S + 5S atomic

Appendix: Analytical expression for the line shapeof a depletion spectrum

We consider a general situation, as shown in Fig. 7, andwe assume that radiative transitions occur much fasterthan the time variation of the population in each molecular level because of the photoassociative creation ofmolecules. We first consider the evolution of the population in the ground and excited states of a single molecule.
The optical Bloch equation for this system can be written

6

Excited state

Here, we assume that the time taken by the mean value ofthe population ratio w/s to attain a constant value z afteris greater than the typical time for radiative transitions.
By using the relation w = zs, Eqs.(A.3) give the following
equations:

PA level

∆

Ω
Γ2-Γ1

Γ1

ds
1−z
=−(Γ2 − Γ1 )
s
dt
2
1−z
2Ω2 Γ2ds
zs=(Γ2 + Γ1 )
s− 2
z
dt
2
Γ2 + 4∆2

APA

(A.4)

Atoms
Weakly boundlevel

Substituting the first equation in the second equation, we
obtain a time-independent equation for z:

Other levels
FIG. 7. Energy levels relevant to depletion spectroscopyof photoassociated molecules. Weakly bound molecules areformed at a rate of APA . A laser having Rabi frequencyof Ω detuned from the resonance by ∆ excites ground state
molecules. Excited molecules spontaneously decay to the initial level and other levels at rates of Γ1 and Γ2 − Γ1 , respectively. We mainly consider a situation where Γ2 > Γ1 issatisfied.

as
s−w
ds
=−(Γ2 − Γ1 )
dt
2
s−w
dw
=(Γ2 + Γ1 )− 2ΩIm(ρf
eg )
dt
2
Γ2
i
ρf
eg
=−(
− i∆)ρfeg + wΩ
dt
2
2
−i∆t
ρf
eg ≡ρeg eiΩw
Γ2 − 2i∆4Ω2 Γ2zΓ22 + 4∆2

(A.5)

This equation is readily solved, and it gives the following
expression for z.
p(1 + k 2 )Γ22 + 2Γ1 Γ2 k − (kΓ2 + Γ1 )z=
Γ2 − Γ1
2Ω2
k= 2
(A.6)
Γ2 + 4∆2This expression is used in the following discussion. Next,
we derive rate equations for the population in the groundand excited molecular levels for the number of molecules:(A.1)

where s = ρgg + ρee , w = ρgg − ρee , and |e > and |g >denote an excited state and a weakly bound level, respectively. ∆ and Ω denote the detuning frequency andthe Rabi frequency, respectively. The decay rates fromthe excited state to the initial weakly bound level andto other levels are given by Γ1 and Γ2 , respectively. The
photoassociation rate is denoted by AP A . The width ismuch larger than the Rabi frequency in the experiment;
hence, we can assume that the time evolution of s andw is much slower than that of ρfeg . Thus, we can setdρfeg /dt = 0 and obtain the following expression for ρf
eg :
ρf
eg =

(1 − z) [Γ2 (1 + z) + Γ1 (1 − z)] =(A.2)

Substituting Eq.(A.2) in Eqs.(A.1), we obtain alternativeequations for s and w ass−w
ds=−(Γ2 − Γ1 )dt
2s−w
2Ω2 Γ2 w
dw=(Γ2 + Γ1 )− 2
dt2
Γ2 + 4∆2(A.3)

dN=APA − (Γ2 − Γ1 )Ne
dt
1−z
N
Ne =
2
1+z
Ng =
N
2

(A.7)

where Ng and Ne denote the number of molecules in the
ground and excited states, respectively, and N = Ng +Ne
is the total number of molecules. In these rate equations, a typical timescale is of the order of 1 ms, and it isgoverned by APA . This is much longer than the typicaltimescale for radiative transitions in most cases (≤1 µs).
The time evolution of Ng is given by
1+z
1−z
dNg
=
APA −
(Γ2 − Γ1 )Ng
dt
2
2

(A.8)

Thus, the solution for Ng is given by



1−z
1 − exp −(Γ2 − Γ1 )τ
.
2
(A.9)
This expression gives the line shape of a depletion spectrum for an interaction time τ . Assuming (Γ2 −Γ1 )τ ≫ 1,
Eq.(A.9) gives
APA 1 + z
Ng (τ ) =
Γ2 − Γ1 1 − z

Ng (τ ) →APAΓ2 − Γ1(A.10)

7
where W is the Lambert W function. Thus, the parameter k in Eqs.(A.6) is given by

500

Width (MHz)

400
300

k=

200
100
0
0

10

20
30
2
Peak intensity (W/cm )40

50

FIG. 8. (Color online) Width of the depletion spectrum vsintensity. The widths of the depletion spectra for the v ′ = 42level are plotted with respect to the intensity of the depletion
laser. The errors indicate standard deviations in the fits tothe spectra.

on resonance (∆ = 0 and w → 0), whereas Ng at an
infinite detuning (∆ → ∞ and w → 1) is given as
Ng (τ ) → APA τ(A.11)

Ng at an infinite detuning is much larger than Ng on resonance. Thus, the width of the spectrum is determinedby finding w such that it satisfies
Ng (τ ) =1
APA τ.
2

3.18724(1 − 1.59362/Γ2τ )(Γ2 − Γ1 )τ − 3.18724(A.15)

Assuming (Γ2 − Γ1 )τ ≫ 1, we obtain the following expression for the FWHM:p2∆ ≈ 0.79Ω (Γ2 − Γ1 )τ

(A.16)

When decays from the excited state to the initial state
are negligible (Γ2 ≫ Γ1 ), we obtain a simple relation (1).
The expression (1) was experimentally confirmed by observing the relation between the width and the intensity(Fig.8).
It is difficult to evaluate the numerical factor√Γτ precisely; therefore, the width of a depletion spectrum cannot serve as an accurate measure of the transition strength. However, the expression (1) enables us tosystematically compare transition strengths for different
vibrational levels.

(A.12)

Substituting Eq.(A.9) in Eq.(A.12) and rewriting theequation with a new variable x ≡ 1 − z(≪ 1), we obtain the following equation:h xi Γ − Γ22
1
1 − exp − (Γ2 − Γ1 )τ =
τx
2
2

(A.13)An analytical solution of this equation is given by

ACKNOWLEDGMENTS(A.14)We thank P. Naidon and T. Kishimoto for insightful
discussions, and K. Oasa, Y. Tanooka, and K. Mori for
their assistance with the experiment. K. A. and D. A.
acknowledge the support of the Japan Society for the
Promotion of Science.

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