Master thesis

Preparation of ultracold atomic sources
towards ground state polar molecules

Supervisor: Associate Prof. Shin Inouye

February 2008
Department of applied physics, School of engineering
The University of Tokyo

66509 Kiyotaka Aikawa

Acknowledgements
First and foremost, I would like to thank my advisor Dr. Shin Inouye for allowing me various chances tolearn what is important for an experimentalist and a physicist. His endless knowledge and clear way of thinking
always help me to understand underlying physics. I feel very fortunate to have had the opportunity to study
under such an excellent physicist.
I am also grateful to Dr. Tetsuo Kishimoto for his careful and unerring advises on experiments. Without his
advises, I would have spent much more time in doing anything. I learned from him the importance of
forecasting. I wish experiments at his new lab achieve significant results.
The post-docs in our lab have taught me a variety of new aspects of physics through repeated discussions. Dr.
Jun Kobayashi has taught me how atoms interact with light in various situations. His intuitive explanation
greatly helps my understanding. Dr. Daisuke Akamatsu has taught me the importance of interpreting every
phenomenon with fundamental principles. It has been my pleasure to discuss with him about any scientificproblem without any stereotype.
I also thank to the students in the lab, Kai Noda, Masahiro Hayshi, Yusuke Tanooka and Yousuke Fujikake
for supporting researchl activities.
Dr. Masahito Ueda, who is the project leader of ERATO, has given me many advices and encouragements on
research. I would like to be grateful for his insightful advices and also for financial supports. I also thank for Dr.
Mikio Kozuma and Dr. Takashi Mukaiyama who have taught me what I could not learn from papers, that is,details and essences on various experiments. It has been my good experience to discuss with them.
Discussions with the post-docs of other groups have given me various knowledge which I would not know
without them. I am grateful to Dr. Nobuyuki Takei, Dr. Makoto Takeuchi and Dr. Munekazu Horikoshi forteaching me details on their experiments. Thanks also go to the graduate students of our neighboring group,
Yasuhisa Inada, Taizo Miyato and Shuta Nakajima. Talking with them has also given me insights on physics.
Finally, I would like to thank Dr. Atsushi Yamaguchi and Dr. Youichi Enomoto for advising on research and
Mr. Toshihiko Sato, Ms. Minako Shioda and Ms. Ai Kouda for supporting office jobs.

Contents

Acknowledgment
Chapter 1

Introduction

1

1.1 Cold atoms………………………………………………………………………………………. 1
1.2 Cold molecules………………………………………………………………………………….. 1
1.3 This work………………………………………………………………………………………... 3
1.3.1 Our goal…………………………………………………………………………………….. 3
1.3.2 Overview of past work on laser cooling of potassium atoms………………………………. 4
1.3.3 Outline of this thesis………………………………………………………………………... 5

Chapter 2

Theoretical background6

2.1 Laser cooling ……………………………………………………………………………………. 6
2.1.1 Optical force on atoms……………………………………………………………………… 6
2.1.2 Doppler cooling…………………………………………………………………………….. 10
2.1.3 Magneto-optical trapping…………………………………………………………………… 11
2.2 Molecules………………………………………………………………………………………... 13
2.2.1 Internal structure……………………………………………………………………………. 13
2.2.2 Angular momenta in a molecule……………………………………………………………. 19
2.2.3 Interactions with light………………………………………………………………………. 22
2.2.4 Numerical investigations………………………………………………………………….... 24

Chapter 3

Experimental methods

29

3.1 Laser system……………………………………………………………………………………...29
3.1.1 ECDL……………………………………………………………………………………….. 29
3.1.2 Slave laser…………………………………………………………………………………... 32
3.1.3 Tapered amplifier…………………………………………………………………………… 34
3.1.4 Distributed Feed Back laser………………………………………………………………… 37
3.1.5 FM sideband locking……………………………………………………………………….. 39
3.1.6 Pulse laser…………………………………………………………………………………... 42
3.1.7 Optics……………………………………………………………………………………….. 43
3.2 Vacuum chamber………………………………………………………………………………… 45
3.3 Computer control………………………………………………………………………………... 46
3.4 Estimation of number of atoms with fluorescence……………………………………………… 47
3.5 Absorption imaging……………………………………………………………………………… 48
3.6 Ion counting with a microchannel plate (MCP)…………………………………………………. 50

Chapter 4

Results and analysis

51

39

4.1 Laser cooling of K at E1………………………………………………………………………. 51
4.1.1 Number of atoms measured with fluorescence……………………………………………... 51
4.1.2 Properties of the 2nd MOT…………………………………………………………………. 54
4.1.3 Efficiency of transferring atoms……………………………………………………………. 56

4.2 Laser cooling of 41K at E1………………………………………………………………………. 65
4.2.1 Number of atoms measured with fluorescence……………………………………………... 65
4.2.2 Characterization with absorption imaging………………………………………………….. 69
4.2.3 Time dependent MOT………………………………………………………………………. 74
4.3 Spectroscopy with REMPI at E2………………………………………………………………... 77
4.3.1 Dual species MOT of 41K and 87Rb………………………………………………………… 77
4.3.2 Ionization and detection of atoms and molecules…………………………………………... 79
4.3.3 Photoassociation spectroscopy of 87Rb2……………………………………………………. 83

Chapter 5Conclusion and outlook

85

Appendix A Circuits

87

Appendix B Box design

90

Appendix C

Design of ECDL

94

Appendix D

Design of slave laser

97

Appendix E Design of tapered amplifier

99

Appendix F Matlab code for the 6 level model

101

Appendix G Properties of Potassium

104

Bibliography

106

1

Chapter 1

Introduction

1.1 Cold atoms
Since the first realization of Bose-Einstein condensation (BEC) of dilute atomic gases [1, 2, 3], atomic
physics has been revealed variety of intriguing quantum behaviors of atomic gases. One of milestones among
them is observation of a superfluid to a Mott insulator transition in a BEC trapped in optical lattices [4]. In theirexperiment, it is clearly shown that a superfluid phase appears at a small repulsive interaction regime while a
Mott insulator phase appears at a large interaction regime. The observed critical point well coincides with thatof theoretical prediction. The importance of their work lies in that they showed a general framework thattrapped atomic gases is an ideal quantum matter to simulate a problem in condensed matter physics. In thispoint of view, observation of condensation of fermionic atomic pairs [5] is also remarkable. They developed amethod to project fermionic atomic pairs to molecules taking advantage of Feshbach resonance. This novel
technique enabled them to measure the momentum distribution of fermionic atom pairs and show condensationof fermionic atoms. Their work associated research on atomic gases with superconductivity. Now researches on
fermionic atoms form a new field called BCS-BEC-crossover and are rapidly growing both in theoretical andexperimental aspects.
To obtain cold atoms in a quantum degenerate regime, both laser cooling and evaporative cooling are
required. Laser cooling slows atoms by using the light pressure on atoms to around 1mK while evaporative
cooling eliminates only hot atoms to cool down atomic clouds below 1μK. Both techniques have been
established in the 90s and now they are routinely used in experiments.
What make experiments on atomic gases fascinating are techniques of optical lattices and Feshbachresonance: we can trap atoms in an ideal periodical potential made of light just as electrons are in condensed
matters, while we can tune the strength of interaction between atoms by utilizing Feshbach resonance.
The technique of optical lattices has its origin in an old proposal [6]. The idea is simple: atoms could be
confined in a standing wave by means of the dipole force arising from the light shift. It was first demonstrated
in 1987 with Cs atoms [7]. However, experiments with optical lattices are not much investigated until therealization of BEC, since loading atoms to optical lattices with a high filling fraction requires a high atomic
density. With quantum degenerate atomic gases, nearly unit filling fraction is easily attained. Now experimentswith one dimensional and two dimensional as well as three dimensional lattices are used to study many body
physics in periodic potentials.
Feshbach resonance is intuitively understood as follows: when colliding atom pairs is coupled to a molecularvibrational level by some interaction (spin-exchange interaction or magnetic dipole-dipole interaction in thecase of alkali atoms), they behave as if they are virtually bounded and their scattering property varies. As iseasily known from the second order perturbation theory, the change in the scattering property is most strongly
affected when the atomic energy level is degenerate with the molecular level. Though Feshbach resonance wasoriginally found in nuclear physics [8], it was theoretically predicted for atomic gases [9, 10, 11, 12, 13] andexperimentally observed [14, 15, 16].
Current interest of this field mainly lies in fermionic superfluidity and ordered phase in optical lattices. Since
they are too widespread to overview, here good reviews on these topics are quoted [17, 18].

1.2 Cold molecules
Cold molecules have been of great interest since they may reveal many novel physics not observable inatomic systems. For example, supersolid and checkerboard phases are predicted for the system of heteronuclear
molecules trapped in optical lattices [19]. A crystal phase is predicted for cold heteronuclear molecules in a twodimensional trap [20]. Here tuning anisotropic dipole-dipole interaction between molecules with microwave

2will give a novel tool to engineer strongly correlated quantum phases. An exotic chemical reaction based onbosonic stimulation is expected for the system of Bose-condensed atoms coupled to a diatomic molecular Bosegas [21]. Another possible application of cold molecules is to use their strong internal electric field to measureelectron electric dipole moment (eEDM) [22]. The result is expected to help to select or justify fundamental
theory of elementary particles. The application of cold molecules to quantum computation [23] is alsofascinating since they are expected to enable us to perform 105 C-NOT gate operations simultaneously.
However, obtaining cold molecules is not straightforward. Laser cooling which is successful for atoms is notapplicable to molecules due to their complicated rovibrational levels. Laser cooling requires a cycling transition
which repeats absorption of cooling lasers and spontaneous emission. If you want to cool molecules with lasers,
a few tens of extra repumping lasers would be necessary to avoid optical pumping to the rovibrational levels
irrelevant to the cooling transition. Preparing such a large number of lasers is impractical.
Early works on cold molecules utilizes supersonic expansion to cool molecules vibrationally and rotationally[24]. In this method, a high pressures gas such as Ar with molecules undergoes a rapid adiabatic expansion
from a small nozzle. During the expansion, rotational and vibrational motions of molecules are converted totranslational motion of molecules. The resulting gas stream contains molecules with their rotational andvibrational temperature of around 10K [25]. Though this method is quite powerful to make absorption lines
narrow in molecular spectroscopy, it is not suitable as a source for trapping molecules since it inevitably gives
molecules a large translational velocity. To circumvent this problem and trap cold molecules in supersonic
expansion, the Stark deceleration method is developed [26]. This method takes advantage of the fact that polar
molecules strongly interact with an electric field and converts their translational velocity to the potential energy.
With this method, 104 ND3 molecules are successfully loaded in a electrostatic trap with a density of 107cm-3 at
a temperature of 1mK [27].
Another promising approach for cooling molecules is to use cryogenic technology. A helium gas cooled toaround 300mK with a dilution refrigerator works as a coolant in this method. The kinetic energy of molecules isremoved through elastic collisions to the coolant helium gas and their vibrational and rotational motions arequenched through inelastic collisions. So far trapping 108 CaH molecules with a density of 8×107cm-3 at atemperature of 400±50mK had been demonstrated with this method [28].
An alternative approach to obtain cold molecules is to associate cold atoms. In this approach, laser cooled
and even evaporatively cooled atoms are used as an initial source. They are associated by photoassociation or
Feshbach resonance without heating. Thus it is easy to achieve a translational temperature well below 1mK
with this method. Since collisions between cold atoms occur for low partial waves, the rotational temperatureof formed molecules is already low. However, they are in their high vibrational levels. Though this approach islimited to molecules made of atomic species to which laser cooling is applicable, now it is regarded as the mostpromising way to achieve quantum degenerate molecular gases. Table 1.1 lists cold molecules obtained byassociation of cold atoms where PA means photoassociation and FR means Feshbach resonance. Here numbers
and lifetimes of molecules are given only for trapped molecules. Note that lifetimes largely depend on density
and trap depth and thus reference values are listed below.
Currently this approach has a problem that formed molecules have a short lifetime (1~100ms) and a small
dipole moment (10-7Debye [29]) due to their large molecular size. Efforts to transfer them to the lowest
vibrational level which is expected to have a long lifetime and a large dipole moment are now under way. So far
a small number of RbCs molecules are formed in their lowest vibrational level [30].

3Table 1.1 Cold molecules obtained by association of cold atoms
Molecule

MethodNumberTemperature [μK]

Lifetime [ms]Reference

Cs2

PA

NA

~300

NA

[31]

133

PA

NA

260±130NA[32, 33]

K2

PA

NA

~300

NA

[34]

87

PA

NA

90±50

NA

[35]

5

133
23

Na

Cs

39
85

Rb2, Rb2
87

PA

~10

~0.10

0.10~0.45

[36]

85

Rb2

FR

3

~10

NA

0.1

[37]

23

PA

NA

Rb2
Na2

6

Li2

40

K2

133

Cs2

~500

NA

[38]

FR

5

1.5×10

~0.14

3

10

[39]

FR

5

2.5×10

~0.15

1

[40]

FR

3

NA

[41]

3×10

-3

(40±3±2)×10 (v)-3(2±2±3)×10 (h)
23

5

Na2

FR

10

~0.03

5

[42]

39

85

K Rb

PA

NA

~150

NA

[43]

85

133

PA

NA

~100

NA

[44]

4

Rb Cs
133

Cs4

FR

10

0.25

NA

[45]

7

Li

133

PA

NA

~100

NA

[46]

40

87

FR

4

NA

20

[47]

Cs

K Rb

10

1.3 This work
1.3.1 Our goal
We are going to produce a BEC of heteronuclear molecules in their lowest vibrational level. As explained
above, association of cold atoms is a promising way for our purpose. A possible pathway to realize it is asfollows: First two fermionic or bosonic atoms are cooled to quantum degeneracy with laser cooling andevaporative cooling. Then they are associated by Feshbach resonance to form loosely bound heteronuclear
molecules. Finally they are transferred to the lowest vibrational level by means of stimulated Raman transition
without heating.
Formation of heteronuclear molecules with Feshbach resonance has demonstrated before [47, 48]. The mostuncertain process in the above procedure is what wavelength is the most suitable to transfer loosely boundmolecules to the lowest vibrational level. Though Sage et al. [30] had performed the transfer process
successfully, their operation is a pulsed one and therefore number of obtained molecules is limited. Thus thenext important task is to demonstrate a transfer from loosely bound molecules to the lowest vibrational levelwith a high efficiency. Search for the optimum excited state and demonstration of the transfer process are going
to be performed with molecules produced by photoassociation since formation of loosely bound molecules withphotoassociation is simpler than that with Feshbach resonance. The resultant small number of molecules is
detected by resonance enhanced multiphoton ionization (REMPI) which has been used for photoassociative
spectroscopy as a sensitive detection tool.
Our target molecule is 41K87Rb. 41K87Rb has several advantages over other alkali dimers:
1. There are many spectroscopic data on excited states. Scattering properties between 41K and 87Rb are also
known precisely through experiments on a 40K-87Rb system [49, 50].
2. Preparation of ultracold atomic sources is relatively easy because the resonant laser frequencies are close toeach other and both species can work with dispensers. It is also important that simultaneous condensation of4
41

K and 87Rb has already been demonstrated [51].

3. Photoassociation rate is high due to their large masses and a large van der Waals interaction between them[52].
4. Broad interspecies Feshbach resonances are predicted for the 41K-87Rb system [53].

1.3.2 Overview of past works on laser cooling of potassium atoms
Potassium is the last stable alkali atom to be successfully cooled and trapped in a MOT. The peculiar
hyperfine structure in the excited states of bosonic potassium isotope 39K and 41K prevents a conventional
approach for laser cooling, while extremely small natural abundance of fermionic isotope 40K(~0.012%) makes
it difficult to trap it. It was the first time that Williamson and Walker succeeded in a MOT of 39K and 41K in
1995 [54]. Cooling and trapping of 40K was successfully done even later in 1998 [55].
The peculiarity of bosonic potassium isotopes lies in the relatively narrow hyperfine splitting of excitedstates due to their small nuclear magnetic moments. In the case of 39K, the splitting between the F’=3 and F’=2
of P3/2 manifold is 21 MHz, which is comparable to the natural linewidth of 6.2MHz, while in the case of 41K
the corresponding splitting is even narrower and is only 13MHz. If one employ a conventional approach forother alkali atoms which detune MOT lasers by about 20MHz to the red of F’=3, ground state F=2 atoms will
experience a strong heating and optical pumping into F=1 state via excited F’=2 state of P3/2. To avoid theseeffects, Williamson and Walker took a large detuning for both trapping and repumping beam. This led to a first
MOT of potassium, though it required a relatively high intensity of around 500mW/cm2. Here both trapping
and repumping beam provided cooling and trapping forces and a large capture velocity of 30 m/s was obtained.
There are a few reports on further investigations on laser cooling of bosonic potassium. Wang et al. reported
a dark SPOT of 39K to increase the density of MOT by one order, which enabled them to observe
photoassociation spectra of 39K2 in a vapor cell [56]. Dark SPOT is originally developed for sodium [57] andthen extended for 87Rb [58], 133Cs [59] and 39K [56], but not for other species. It suppresses repulsion forces
between atoms due to repeated absorption and emission of resonant photons by reducing the intensity ofrepumping beam at the center of the MOT. Atoms in the center of the MOT fall into F=1 dark state and do not
feel any optical force. This results in an increase in density by two orders. Another approach for a high density
MOT is a compressed MOT (C-MOT) which changes the condition of MOT temporally and was first reported
for 87Rb [60]. This method utilizes the fact that a condition for a large number of atoms and that of a highdensity is different: first atoms are collected in a large number condition and then compressed and cooled inanother condition. A C-MOT of potassium was successfully performed by Prevedelli et al. for both 39K and 41K[61]. They observed significant decrease in temperature from a few mK to around 150μK when the detuningof both trapping and repumping light is decreased to hit around F’=0. However, the density after C-MOT
remained low compared with that of other alkali species by nearly two orders. A detailed experimental andtheoretical investigation on the cooling mechanism in a 39K MOT is given by [62]. They performed numericalsimulations of the optical Bloch equation and showed that a large detuning required to collect atoms
significantly decreases a friction force at the MOT, which leads to a high temperature and a low density. They
also showed that a sub-Doppler feature appears at small detuning and low intensity regime, where an attainable
minimum temperature was estimated to be 20μK.

51.3.3 Outline of this thesis
First basics on laser cooling are presented in chapter 2.1. Interaction between a two level atom and light isdiscussed by means of the optical Bloch equations. Two kind of forces exerted on atoms by light are derived.
Mechanism of the Doppler cooling is briefly explained by using derived formulae. An explanation onmagneto-optical trapping, which is an actual implementation of the Doppler cooling, is also given here. In
chapter 2.2, important topics on diatomic molecules are explained. The notion of potential energy between two
atoms is given. Coupling between various angular momenta is explained. Interaction between molecules andlight is reviewed. Then numerical examples are presented to clarify what theoretical calculations can predict.
Here calculated potential energy curves of KLi are shown since we had been planning to produce 40K6Li
molecules.
Next our experimental apparatuses are overviewed in chapter 3. Details on our laser system are given.
Structure and operation of external cavity diode laser, slave laser, DFB laser and tapered amplifier are
explained. Our vacuum chambers are briefly explained. Methods to estimate number, temperature and densityof atoms are shown. Then principle and actual operation to detect atoms and molecules with multiphoton
ionization is reviewed.
In chapter 4, our main results are presented. First, laser cooling of 39K atoms are studied. More than 109
atoms are trapped in a MOT. We found that number of atoms is enhanced when trapping and repumping lasers
satisfy a Raman condition. Then properties of a double MOT system are investigated. Longitudinal andtransverse velocities of pushed atoms are measured. We show that almost all atoms could be trapped in the 2nd
MOT. Second, laser cooling of 41K is studied in detail. With absorption imaging, we found that temperature of
atomic cloud is quite high (~10mK) and density is low (~109/cm3) at the Raman condition where the largest
number is obtained. Even if this condition is avoided, still temperature is high and density is low. We found thattime dependent MOT could overcome this difficulty. Finally phase space density of 10-6 is obtained after thewhole laser cooling processes. Lastly, we give our latest results on detection of atoms and molecules withmultiphoton ionization. A time-of-flight mass spectroscopy shows that we have certainly detected K+, Rb+ and
Rb2+. We give qualitative arguments on how atoms and molecules are ionized based on the observed signals.

6

Chapter 2

Theoretical background2.1. Laser CoolingIn this section, theory of laser cooling is briefly reviewed. Only the most important topics are covered here
since our purpose is not to present a thorough theory, but to provide some basic concepts which is necessary tounderstand the experimental results. First, basic quantum mechanical description of a two level systeminteracting with light is presented in 2.1.1 based on [63]. The subsequent section 2.1.2 explains the basic
principle of Doppler cooling with which a laser light cools atoms down to a sub-mK temperature based on asimple model.

2.1.1 Optical force on atoms
We consider the force F exerted on atoms by light in a semiclassical approach. We assume that force on
atoms is expressed as a time-derivative of the expectation value of momentum operator based on the analogy toclassical mechanics:

)
d )
F =
p
dt

(2.1)

According to the Ehrenfest theorem, we can rewrite this expression as

)
)
i ) )
∂p
F =
H, p +
∂t
h

[

]

(2.2)

where H is written as a sum of kinetic energy and interaction with light:)) p) 2H=+ H light
2m

(2.3)

Here the second term in (2.2) vanishes since p does not depend on time. We use the usual expression on the)
momentum operator p = −ih∇ to modify (2.2):

[

]

)
)
)
i )
)
F =
H light , p = H light ∇ − ∇H light
h

(2.4)

Noting that the second term in the bracket in (2.4) means that

(

)

)
)
)
∇H light = ∇H light + H light ∇(2.5)(2.4) becomes))
F = − ∇H light(2.6)

which says the force exerted on atoms by light is the negative gradient of the potential made by light. If weassume that the spatial variation of electric field is sufficiently large compared to the atom, (2.6) is simplified:))
F = −∇ H light(2.7)

We write down the Hamiltonian of the electromagnetic field under the electric dipole approximation as)
H light = −er ⋅ E(2.8)

where r is the electron coordinate, e is charge of electron. E is electric field defined as

E = E0 e
where E0 is the amplitude of light field and e is a unit vector containing spatial and time variation of electricfield.

7
Here we introduce the assumption that the atom is described as a two level system. Then we can define theRabi frequency as

Ω=

− eE0
Ψe r Ψgh(2.9)

where Ψg ,e denotes the electronic ground and excited wavefunction. In this notation, the integral on theelectron coordinate in (2.7) is expressed in terms of Ω and (2.7) becomes)
F = −h∇ Ωe(r, t ) ≡ −h∇ Ω′(2.10)

where r is the atomic coordinate and e = e . To calculate this integral, we introduce the density matrix)

ρ= Ψ Ψ

(2.11)

where the atomic wavefunction Ψ satisfies the Schrodinger equationih)∂Ψ =H Ψ
∂t(2.12)

Since ρ is a 2*2 matrix, we define the four components as⎛ ρ gg

ρ eg ⎞⎟
ρ ee ⎟⎠

ρ = ⎜⎜⎝ ρ ge(2.13)

Then (2.10) is written with the density matrix as)
F = h ρ eg* ∇Ω′ + ρ eg ∇Ω′*()

(2.14)

since the expectation value of an operator A is obtained with

A = Tr (ρA)

(2.15)and there are no diagonal terms in (2.10). Here we used rotating wave approximation (RWA) to neglect the highfrequency terms.
Next we split the derivative of Ω ′ into its real and imaginary part as∇Ω′ = (q r + iq i )Ω′(2.16)

Then (2.14) is expressed as a sum of two terms:

(

)

(

)
*
F = hq r Ω′ρ eg+ Ω′* ρ eg + ihq i Ω′ρ eg* − Ω′* ρ eg)

(2.17)To express this force with frequency and intensity of light, we have to obtain the explicit form of the density
matrix. The time dependence of the density matrix is derived from (2.1) as

ih

[

) )
⎡⎛ ∂
∂ )
⎞
⎛∂
⎞ ⎤ )) ) )ρ = ih ⎢⎜ Ψ ⎟ Ψ + ⎜ Ψ ⎟ Ψ ⎥ = Hρ − ρH = H , ρ
∂t
⎠
⎝ ∂t
⎠ ⎦
⎣⎝ ∂t

]

(2.18)

where H = H† is used. Taking account into the fact that the atom is a two level system and there isspontaneous emission with a constant rate Γ 2 , we obtain the optical Bloch equations (OBE) which describes
the time dependence of a two level atom interacting with light:

8

dρ gg

(

)

i *
Ω ρ eg − Ωρ ge e −i (ωl −ωa )t
dt
2
dρ ee
i
= −Γρ ee + Ωρ ge − Ω* ρ eg e −i (ωl −ωa )t
dt
2
dρ ge
Γ
i
= − ρ ge + Ω* (ρ ee − ρ gg )e i (ωl −ωa )t
dt
2
2
dρ ge
Γ
i
= − ρ eg + Ω(ρ gg − ρ ee )e i (ωl −ωa )tdt
22= Γρ ee +()(2.19)

where ω l and ω a denotes the frequency of light and atomic resonance, respectively. If we define thepopulation difference between the ground and excited states as

w = ρ gg − ρ ee
and use the optical coherenceρ ge = ρ eg*
the OBE (2.19) is simplified to a set of equations

dρ eg

iwΩ
⎛Γ
⎞
= −⎜ − iΔ ⎟ ρ eg +
dt
2
⎝2
⎠
dw
*
= −Γw − i Ωρ eg− Ω* ρ eg + Γ
dt

(

)

(2.20)

where Δ ≡ ω l − ω a denotes the detuning from the resonance. The steady state solutions of the OBE areobtained by setting the time derivatives of w and ρ eg to zero in (2.20):

w=

1
1+ s

iΩ
ρ eg =2(Γ 2 − iΔ )(1 + s )(2.21)

where
2

Ω 2
s0
s= 2
≡
2
Δ + Γ 4 1 + (2Δ Γ )2

(2.22)

is the saturation parameter. Here s0 is the on-resonance saturation parameter:

s0 ≡

2Ω 2
I
=
2
Γ
Is

(2.23)

where Is is the saturation intensity defined as

Is ≡

πhcΓ
3λ3

and depends on atomic species.
The population of the excited state is easily obtained as(2.24)

9ρ ee =I Is
1− w 1=2
2 1 + I I s + (2Δ Γ )2

(2.25)

An important expression for the scattering rate of light by atoms is obtained here. The population in the excitedstate decays with the rateΓρ ee =I Is
Γ
2 1 + I I s + (2Δ Γ )2

(2.26)

which is equal to the excitation rate in the steady state. Thus we can see that an atom scatters photons with therate (2.26).
Substituting (2.21) to (2.17), we obtain an explicit expression for the force exerted on atoms by light as asum of two kinds of forces [64]:

)
F =

I Is
hΓ ⎞
⎛
− q r hΔ + q i
⎟
2 ⎜
2 ⎠
1 + I I s + (2Δ Γ ) ⎝(2.27)

The first term is called the dipole force and the second term is called the scattering force or the dissipative force.
The dipole force arises when there is a gradient in the amplitude of the electric field while the scattering force
arises when there is a gradient in the phase of the electric field. The specific cases where these two forces exist
are a standing wave and a traveling wave. For a traveling wave with the electric fieldE(r , t ) = eE 0 cos(ωt − kr )(2.28)

qr,i of (2.16) is written as
q r = 0, q i = k(2.29)

Here we used RWA to neglect a high frequency term. Substituting (2.29) to (2.27), only the scattering forceisremaining:
)
I Is
Γ
hk
F =
2 1 + I I s + (2Δ Γ )2

(2.30)

This expression is interpreted intuitively as momentum of an absorbed photon hk exerts a force on atoms
with the scattering rate (2.26). Since this force is dissipative, it is possible to cool atoms with it. In contrast, fora standing wave made by two counterpropagating lasers with the electric fieldE(r , t ) = eE 0 cos kr cos ωt(2.31)

qr,i of (2.16) is written as
q r = −k tan kr, q i = 0(2.32)

which means that only the dipole force arises in this case. Substituting (2.32) to (2.27) and taking into accountthe spatial modulation of I/Is, the only remaining term is the dipole force written as)
2 I I s sin 2kr
F = hkΔ2
1 + I I s cos 2 kr + (2Δ Γ )(2.33)

This force is also exerted by momentum of a photon but here absorption from one laser and stimulated
emission to the other laser occur at the same time. Thus the dipole force is conservative and can be used to trap
atoms but cannot be used to cool atoms.

2.1.2 Doppler cooling
When an atom is moving with velocity v in the two counterpropagating lasers with the same intensity an the
opposite polarization, they feel the scattering force of the form (2.30) from both beams independently. The
forces exerted by both laser depend on the velocity of atoms due to the Doppler shift and written as

10
F± (v ) = ± hk

I Is
Γ
2 1 + I I s + [2(Δ m kv ) Γ ]2

(2.34)

where the upper/lower sign indicate the left/right laser. The total force felt by the atom is expressed as
F (v ) = hk⎤
Γ I Is ⎡
1
1
−
⎢
2
2 ⎥
2 ⎢⎣1 + I I s + [2(Δ − kv ) Γ ] 1 + I I s + [2(Δ + kv ) Γ ] ⎥⎦(2.35)

If velocity is small compared to Δ, Γ this force is approximated to be

F (v ) ≅[8hk 2 Δ I I s

Γ 1 + I I s + (2Δ Γ )

]

2 2

v

(2.36)

This dependence of the force on velocity means that it works as a damping force for a negative detuning. Thisis also confirmed from figure 2.1 which plot (2.35) against velocity. Atoms in this configuration are cooled in
the lasers as if they are moving in molasses. Thus this technique is called optical molasses. However, opticalmolasses is effective only for atoms with small velocity. We can see from figure 2.1 that the damping force
exists only for a limited velocity range:v<Γ≡ vc for I I s ≤ 1, Δ = −Γ 2k(2.37)

Here we introduced a new parameter, capture velocity vc. Only atoms with velocity smaller than vc are cooled in
the optical molasses. In most cases, vc is ranged in 10~30m/s. This is much smaller than the velocity of room
temperature atoms. Thus collecting atoms with optical molasses from room temperature gases is not efficient.
Note that the capture velocity depends on the intensity and detuning: higher intensity and larger detuning leadto a larger capture velocity, though both of them makes cooling of atoms less effective due to a high photon
scattering rate as explained below.
0.5
I/ Is=1, Δ=- 4Γ

Damping force [h̄kΓ/2]

0.4

I/ Is=1, Δ=- 0.5Γ

0.3

I/ Is=10, Δ=- 0.5Γ

0.2
0.1
0
- 0.1
- 0.2
- 0.3
- 0.4
- 0.5
- 30

- 20

- 10

0

10

20

30

Velocity [m/s]Fig.2.1 Velocity dependence of the damping force for one dimensional optical molasses.
Here Γ/2π=6.09MHz and k/2π=767nm are assumed. Theses values correspond to the case ofpotassium atom.
The temperature obtained in the optical molasses is, of course, not zero. There are two limitations. First,
atoms cannot have a smaller energy than the energy of a photon since cooling are performed by absorption andemission of photons. Second, spontaneous emissions in random directions cause heating of atoms. The firstlimitation is often represented as the recoil temperature

11
h2k 2
kB M

Tr =

(2.38)

where k is wavenumber of a photon, M is mass of atom and kB is the Boltzmann constant. The recoil
Btemperature is the lowest temperature obtained with optical transition except for a few cases such as Raman
cooling. The expression for the second limitation is obtained by setting the sum of cooling and heating rate iszero at the steady state. Cooling rate is obviously

Fv =

[

8hk 2 Δ I I s

Γ 1 + I I s + (2Δ Γ )

]

2 2

v2

(2.39)

while heating rate is4

I Is
h2k 2 Γ2m 2 1 + I I s + (2Δ Γ )2

(2.40)

Here the first numerical factor 4 means that there are two beams and an atom is kicked twice at each scatteringby absorption and emission of photons. Thus temperature at the steady state is written as
h 1 + I I s + (2Δ Γ )
4
2Δ Γ

2

T=

(2.41)

We can see that high intensity and large detuning cause heating. The minimum temperature at sufficiently low
intensity is obtained by setting Δ = −Γ 2 in (2.41):
TD =

hΓ
2

(2.42)

which is called the Doppler temperature and the lower limit obtained with the Doppler cooling.

2.1.3 Magneto optical trapping
Though optical molasses can slow atoms, it cannot confine atoms since there is no restoring force in opticalmolasses. It is possible to yield a restoring force by adding magnetic field gradient to the optical molasses if
there are magnetic sublevels in the excited states. This configuration is called magneto optical trap (MOT) andnowadays widely used to cool and trap atoms.
The mechanism of magneto optical trap is easily understood by adding a position dependent term due to the
Zeeman shift of magnetic sublevels to (2.35):
F (v, z ) = hk⎤
Γ I Is ⎡
1
1
−
⎢
2
2 ⎥
2 ⎢⎣1 + I I s + [2(Δ − kv + μ ′B ′z h ) Γ] 1 + I I s + [2(Δ + kv − μ ′B ′z h ) Γ] ⎥⎦(2.43)

where B’ is a magnetic field gradient at the center of magnetic field, z is distance from the center of magnetic()

field and μ ′ = g e M e − g g M g μ B is the difference of magnetic moments between ground and excitedmagnetic sublevels. When the Doppler and Zeeman shifts are small compared to the detuning, (2.43) isapproximated to be
F = −αz − βv

(2.44)The second term contributes to cool atoms as in optical molasses, whereas the first position dependent termworks as a restoring force which confines atoms to the center of the magnetic field.
Note that dependence of the restoring force on position has a similar shape to figure 2.1: there is a finite
capture range in which atoms feel confining forces. This capture range is easily estimated from the capture

12velocity as
zc ≈

hkvc
μ ′B ′(2.45)

which is usually ranged in 2~20mm. The radius of the MOT never exceeds the capture range. The capture range
is important since it determines the trap depth of MOT through1
W = αz c22(2.46)

which is typically around a few K. Heating processes such as fine structure changing collision easily exceed thetrap depth and lead to loss of atoms. The actual size of the MOT is approximately related to the typical velocity
of atoms v based on the virial theorem:
1 2 1 2αz = mv
2
2

where z is the typical radius of the atomic cloud. Since density and temperature of a MOT is limited by thenature of atoms and light, these relations indicate that maximum number captured in a MOT is also limited toaround the usual value 109~1010.
In reality, MOT is performed with a three dimensional configuration shown in figure 2.2. The anti-Helmholtz
coils produce a magnetic field gradient in all directions. Six laser beams are irradiated to produce three
dimensional confining forces. Here all beams are circularly polarized to manipulate magnetic sublevels of
excited states separately.

σ+
I

σ-

σ-

I

σ+

σ+

σFig.2.2 Configuration of MOT
Six circularly polarized lights are incident on the vacuum chamber. Anti-Helmholtz
coils make a magnetic field gradient at the middle of two coils.

2.2 Molecules
When atoms are bonded to form a molecule, rich new natures appear which is not originally contained in theconstituent atoms. Since our research goals is to form ultracold polar molecules, it is necessary to take into
account these new features in our discussions. Molecular physics is also important to treat interactions between
atoms in the physics of collisions between cold atoms. In this chapter, basics of molecular physics are presented

13
briefly. Though there are huge background works on the molecular physics, most of them are not necessarilyclosely related to our case. Therefore here we restrict our focus on the selected topics which are quite importantto understand the behavior of ultracold alkali diatomic molecules.

2.2.1 Internal structure
The main difference of a molecule from an atom lies in its internal structure: vibration and rotation. Thesemotions are based on the fact that interaction between its constituent atoms are not strong enough to constraint
them completely; if it is, there will be no novel behaviors compared to an atom. A moderate interaction between
atoms enables molecules to have a wide variety of observable behaviors. For example, molecules can transit
from one rovibrational state to another rovibrational state by collisions or emission of microwave without any
change in the electronic configuration. There are of course many excited states as in the case of atoms, which
are accessible with optical transitions.
To dealing with these behaviors simply, we adopt Born-Oppenheimer approximation: the motions ofelectrons in a molecule is fast enough to neglect the motion of the molecule itself. This is usually a good
approximation for cold atoms and molecules. However, Born-Oppenheimer breakdown (BOB) effects become
the problem in the case of a light atom. This consists of the next two factors: the isotopic difference in theelectronic potential energy and non-adiabatic coupling of electronic motion to nuclear motion [65, andreferences therein]. Though the latter could be neglected for cold atoms and molecules since it is proportional
to a rotational energy, the former might become a problem for light atoms even in the case of cold atoms andmolecules: for example, the isotopic difference in the potential depth of the order of 0.1cm-1 is reported for Li2[66]. For heavy atoms with mass number of A, the isotopic difference is expected to scale as A-2 and thus
negligible.
Most of the molecular properties are explained with the potential energy between its constituent atoms whichis expressed as a function of the interatomic distance. An example of the calculated potential energy curve(PEC) for the alkali dimer KLi is shown in figure 2.3. Labeling to these states are performed as follows. Firstthe ground state is labeled X. The states with the same spin multiplicity as X is labeled in alphabetical orderwith capital letters from the energetically lowest state except X to higher states. The states with the different
spin multiplicity from X are labeled in alphabetical order with lower case letters from the energetically lowest
state to higher states. These alphabetical labeling is followed by a Greek alphabet indicating the projection of
the orbital angular momentum to the interatomic axis. Σ, Π and Δ corresponds to L=0, 1 and 2,
respectively. A superior figure between the two letters denotes the spin multiplicity: 1 means singlet, while 3means triplet. It is noteworthy that another letter g or u is added after the Greek letters to indicate the symmetryof the state in the case of a homonuclear molecule. Though this type of labeling is easily comprehensible, there
is another type of labeling where the first alphabet is replaced by the number in parentheses which indicate theorder of the state in the same symmetry and spin multiplicity.

14

4

2

x 10

e3 Σ+
d3 Π

1.5

D1 Π
1 +
B1 Π C Σ

Energy [cm−1 ]

1

c3 Σ+

A1 Σ+

0.5

b3 Σ+
a3 Σ+

0

X1 Σ+

- 0.5

-1

2

3

4

5

6

7

8

9

10

11

12

Interatomic distance [Å]Fig.2.3 Potential energy curve of KLi calculated with MOLPROAs is easily known from the analogy to the harmonic oscillator, there are many bound states in this potential.
Classically we can interpret these states as follows. When two atoms in the particular state of the molecule
become close to each other, they feel a large repulsive force due to the overlap of electrons in the atoms andtheir motion is slowed. Finally they are reflected at the distance where their energy is equal to the potential
energy. The same motion occurs when their distance becomes large, because the van der Waals force between
two atoms becomes small at a large interatomic distance. Thus two atoms in a particular bound state are
oscillating in the region determined by their energy and the PEC. This is a vibrational motion of molecule.
Actually we can consider it quantum mechanically. The motion is described by a corresponding wavefunction
as the probability distribution over the internuclear distance. We cannot observe two atoms located in aparticular distance without using a very fast optical process which can resolve the motion of electrons.
There is another motion in a molecule: rotation. It is quite easy to understand classically since it is onlyrotating against an axis vertical to the interatomic axis. However, the rotational motion is also interpreted15
quantum mechanically as a bound state of the combined potential of the centrifugal term and the electronic
potential energy term. In practice we can consider the rotational levels as attached to each vibrational level,
since the level spacing between rotational levels is much smaller than that of vibrational levels. The vibrational
spacing is smaller for heavier molecules, which is easily understood as follows: If we assume that the potential
is approximated to be a harmonic oscillator near the bottom and written as
V (x ) =1 2
kx
2

Then the vibrational level spacing is described ashkμwhere μ is the reduced mass of constituent atoms.
In the case of room temperature molecule, there is a thermal distribution over rovibrational levels. Most of
the molecules are in the lowest vibrational state of the potential well except for a heavy molecule which has asmall vibrational level spacing, whereas most of the molecules are in the high rotational level since a rotational
level spacing is much smaller than the thermal energy of the room temperature. It is noteworthy that a
particular rotational state of rotational angular momentum J has multiplicity 2J+1. This leads to a thermal
distribution⎛⎞(2 J + 1) exp⎜⎜ − BJ ( J + 1) ⎟⎟
⎝

k BT

⎠

which has a peak aroundJ=

k BT 1−2B 2

where B is a rotational constant and defined asB=h2
1Ψ (R ) 2 Ψ (R )2μR

A thermal distribution over vibrational levels obeys the usual Maxwell-Boltzmann distribution and thus has apeak at the lowest energy level.
You can see several examples on how the PEC is useful. The depth of the ground state potential represents
the stability of chemical bonding between two atoms. If it is much higher than the thermal energy of aparticular temperature, then the molecule is stable under the temperature. The size of the room temperature
molecule is easily estimated by reading the equilibrium distance where the potential is minimum. Informationon the excited states enables us to predict which states are accessible from the ground state molecule and whatis the wavelength of that transition.
Here the analytic form of the potential energy is reviewed. The origin of the potential energy between two
atoms is usually explained as follows. For a small interatomic distance, overlapped electrons of atoms yield alarge repulsive force, while there is an attractive force for a long interatomic distance due to dipole-dipole
interaction caused by an instantaneous polarization in the electrons of atoms. However, there is a large force ina real molecule: spin exchange interaction caused by the electronic spins of atoms. The necessity that theelectronic wavefunction must be antisymmetric under exchange of electrons lower the energy of the singlet
states and higher that of the triplet states. This interaction exists whenever both two constituent atoms haveelectronic spins. Thus the analytical representation of the PEC is generally written as

16
V (R ) = −

Cn

Cn

Cn

± AR a exp(− bR )R
R
Rn
where R denotes the interatomic distance. The upper/lower sign refers to triplet/singlet potentials.
1

n1

−

2

n2

−

3

3

The first three terms account for the van der Waals interaction at long range and the repulsion at short range,
while the fourth term represents the exchange interaction of the outermost electrons. It is easily known that thesum of singlet and triplet potentials represents the van der Waals interaction, while their difference indicates theexchange interaction. The integer values ni (i=1,2,3) depend on the state of two atoms. In the case of groundstate alkali dimers, n1=6, n2=8 and n3=10. The r-dependency of the van der Waals terms is readily
comprehensive: the terms with C6, C8 and C10 represents dipole-dipole, dipole-quadrupole andquadrupole-quadrupole interaction between two atoms, respectively. The accurate values of the dispersion
coefficients have been obtained by ab initio calculations [67, 68, 69, 70, 71, 72, 73, 74], which are usually usedwithout any modification. The derivation of the above mentioned analytical form of the exchange interaction is
given in [75]. They showed that the values of a and b are written using the ionization energy of constituent
atoms, I1 and I2, as
a=

2
2I1

+

2
2I 2

−

1
−1
2(I 1 + I 2 )b = 2I1 + 2I 2

The value of A in the above expression is usually obtained by fitting the above function to an experimental
results to assure accuracy, though an analytical expression and calculated values are reported for several alkali
dimers [75, 76]. The above expression of the exchange interaction well coincides with experimentally attained
potentials for various alkali dimers [77].
Note that the above expression is accurate near the dissociation limit of the potential. Thus it gives accurate
results for the calculations of a long rage molecule, but not for a deeply bound molecule. If you have to deal
with a deeply bound state theoretically, it is necessary to fit the experimentally obtained potential to acomplicated function. Several accurate analytic functions have been developed, part of which is shown here[78]:
1. Generalized Morse Oscillator (GMO)⎛⎜ ⎞⎟ ⎛⎜⎡
R R − Re ⎞⎟⎠ ⎤−β
VGMO (R ) = De ⎢1 − e GMO ⎝ ⎠ ⎝
⎥
⎦
⎣

2

β GMO (R ) = β 0GMO + β 1GMO (R − Re ) + β 2GMO (R − Re )2 + L

2. Extended Morse Oscillator (EMO)⎛⎜ ⎞⎟ ⎛⎜⎡
−βz R − Re ⎞⎟⎠ ⎤EMO ⎝ ⎠ ⎝VEMO (R ) = De ⎢1 − e
⎥
⎣⎢
⎦⎥

2

β EMO (z ) = β 0EMO + β 1EMO z + β 2EMO z 2 + L

3. Modified Lennard-Jones Oscillator (MLJ)2n⎡⎤⎛ R ⎞ − β ⎛⎜ z ⎞⎟ z
VMLJ (R ) = De ⎢1 − ⎜ e ⎟ e MLJ ⎝ ⎠ ⎥⎢ ⎝ R⎠
⎥
⎣
⎦
MLJ
MLJ
MLJ 2
β MLJ (R ) = β 0 + β 1 z + β 2 z + L

where z ≡ (R − Re ) (R + Re ) .

It is important to recognize that all ‘analytic’ forms given here are not obtained by theoretical calculation but

17solely by fitting to the experimental results for the calculations of molecular properties such as transition
strength. In most cases, fitting procedure should be carried out with great care to reproduce experimental resultscorrectly changing the parameters one by one.
Next we would like to see details on experimentally obtainable potentials. As you can see from the previous
discussion, we cannot directly measure the potential energy itself since the wavefunction of a molecule in aparticular state has a form of the probability distribution over the interatomic distance. What we can observe are
only the energy levels of bound states in the potential. Thus a conversion process is required to acquire the PEC
from experimental results.
Now the most widely used procedure for this purpose is Rydberg-Klein-Rees (RKR) method [79, 80, 81, 82].
Based on a first order semiclassical approximation, it derives two key equations from the vibrational energy
levels G(v) and rotational constants B(v)2R2 (v ) − R1 (v ) =
1

1

β∫

dv ′

v

G (v ) − G (v′)

vmin

1

R (v ) R (v ) 2πβ ∫
−

1

=

2

B(v′)dv′

v

vmin

≡2f

G (v ) − G (v′)≡ 2gHere R1 and R2 denote the inner and outer classical turning points and v represents vibrational quantum numberin the semiclassical approach. β is a numerical constant. These two equations give the expression for R1 andR2 as

R1 (v ) =f2+ f g−f

R2 (v ) =f2+ f g+ f

Repeating this procedure for observed vibrational levels yields a potential energy function against theinteratomic distance. Now a reliable program package RKR1 is available for the whole calculations presented
above [83].
Tthere are two different methods to express experimentally obtained energy levels: Dunham Expansions andNear Dissociation Expansions (NDE’s). Based on the behavior of harmonic oscillator and rigid rotor, Dunham
expansions expresses the vibrational energy levels G(v) and the rotational constants B(v) as∑ Y (v + 1 2) = ω (v + 1 2) − ω x (v + 1 2) + ω y (v + 1 2) + L
B(v ) = ∑ Y (v + 1 2) = B − α (v + 1 2) + γ (v + 1 2) + L

G (v ) =

2

l

l ,0

e

e

3

e

e

e

l =1

2

l

l ,1

e

e

e

l =1

The coefficients Yl,m are used for the above calculations in RKR1.
NDE’s are based on the theoretically known behavior of simple inverse-power potential
V (R ) = De −Cn
Rnwhere the integer value n depends on the problem and Cn is the dispersion coefficient which is alreadymentioned above. It was proved [84, 85] that the vibrational levels and rotational constants in this form ofpotential are expressed as
G (v ) = De − K 0 (v )B(v ) = K 1 (v )[K m (v ) = X m (n ) μ n C n2

] (v − v )1 2− n2 n n−2−2 mD18
where vD denotes the non-integer effective vibrational number at dissociation and X m (n ) is a known
numerical constant depending on the physical constants and n. This type of representation of the energy levels
near the dissociation is particularly useful for dealing with the problem of cold atoms and molecules since allparameters except effective vibrational number vD is known from theoretical calculations and thus analysis ofspectra in this region is easily carried out with (2.2).
The use of Dunham expansion is appropriate near the potential minimum, while NDE’s should be used near
the dissociation limit. In the code RKR1, both types of calculation are available.
Though the RKR method gives an accurate potential enough to estimate most of the molecular properties
correctly, the vibrational energy levels calculated from the RKR potential typically differs from the original
experimental values by around 0.1cm-1 due to the semiclassical approximation mentioned above. To obtain amore accurate potential, an Inverse Perturbation Analysis (IPA) approach was developed [86]. Its basic idea is
as follows.
Consider the Schrodinger equation⎡ h2 2⎤∇ + V (R )⎥ψ n = E nψ n
⎢−
2
μ
⎣
⎦
where μ is the reduced mass, V is the unknown real potential and En is an experimentally obtained
eigenenergy. On the other hand, the inaccurate approximate potential V0 gives a different eigenenergy En0
according to the corresponding Schrodinger equation⎡ h2 2⎤∇ + V 0 (R )⎥ψ n0 = En0ψ n0
⎢−
⎣ 2μ
⎦
The difference in eigenenergy

ΔEn = En − En0

is due to the difference in the potential

ΔV (R ) = V (R ) − V 0 (R ) . A first order perturbation theory enables us to equate
ΔEn = ψ n0 ΔV (R )ψ n0
If you expand the difference potential ΔV (R ) in a set of appropriate basis functions g i (R ) , the equation
becomesΔEn =∑ c ψ g (R ) ψ
i

0
n

i

0
n

i

This set of linear equations for the unknown expansion coefficients ci is solved to construct a new potential
and the whole procedure is repeated until the convergence criteria is satisfied. In this approach the RKR
potential is usually used as a first approximate potential and gives an improvement of the accuracy by one
order.

The RKR and IPA potential are sufficiently accurate for many purposes, but they are usually available only
for limited states and only in the energy range near the potential minimum. The reason is obvious: they are
obtained with spectroscopy on the ground state molecules, which means that we can know the potential withthe levels accessible from the ground state molecules with optical transitions.
For the first estimate of the shape and depth of the potential for every state and in every energy range, it is19
useful to utilize the results of ab initio calculations. It gives purely numerical solutions of the Schrodinger (orDirac) equation of many electron systems. Since ab initio calculation of molecular potentials which is often
called computer chemistry itself is a tremendous research field, here we do not give a detailed description of it.
A numerical example for KLi molecule calculated with the program package MOLPRO [87] is presented laterin the section 2.2.4 with experimental and past works. In most cases, the potential obtained with ab initio
calculation differs from experimental results by 20 to 200 cm-1 depending on the calculation method and theatomic basis functions.

2.2.2 Angular momenta in a molecule
In a real molecule, many kinds of angular momenta are coupled with each other in various ways dependingon the energy scale of each angular momentum. There are four important angular momenta in a molecule:electronic orbital angular momentum, rotational angular momentum, electronic spin and nuclear spin. Coupling
between them makes the observed spectra complicated. For correct understanding of the experimental results, itis necessary to recognize which angular momenta are strongly coupled and which angular momenta are nearly
uncoupled in the current problem. Here the coupling schemes based on the Hund’s classification are given.
There are five coupling cases, (a) to (e), but the important ones are (a) and (c) in the case of cold atoms andmolecules. in most cases.
Hund’s case (a) is the most frequently encountered case in molecular physics. Here the energy of electronicinteraction mentioned in the section 2.2.1 is the largest and then spin-orbit interaction and rotational energy
follows. Hyperfine interaction due to nuclear spins is omitted here since it is much smaller than other angularmomenta. If you deal with a deeply bound molecule, then it corresponds to the case (a). The energy scale ofeach term is of the order of 1000cm-1, 100cm-1 and 0.1cm-1, respectively. In this case, states are labeled withorbital angular momentum and spin multiplicity as mentioned in the section 2.2.1.
Hund’s case (c) is a relatively rare case compared to case (a), but important in the treatment of cold atoms
and molecules. Here the energy of spin-orbit interaction is the largest and that of electronic interaction andhyperfine interaction follows. Rotational angular momentum is neglected here. The molecule in an excited state
formed by photoassociation corresponds to the case (c). In this case, the energy scale of each term is of theorder of 100cm-1, 10cm-1 and 0.1cm-1, respectively. Rotational energy in case (c) is much smaller than in case(a) because the size of the molecule near dissociation limit is large. The labeling of states is performed with Ωwhich denotes the projection of the sum of orbital angular momentum and electronic spin to the molecular axis.
Both of the two coupling schemes are illustrated in figure 2.4. Transition from case (a) molecule to case (c)molecule occurs around the internuclear distance of 20Å as shown in figure 2.5. More detailed discussions on
the various coupling schemes between angular momenta are found in [88].

20

J
J

N

N

Ω

Ω
Σ

Λ

L

S

L

Ja
S
case (c)case (a)Fig.2.4 Coupling of angular momenta in case (a) and case (c)L is orbital angular momentum, S is spin, J is total angular momentum and N is rotational angularmomentum. Λ and Σ are projection of L and S to molecular axis, respectively. Ω is a sum of Λ andΣ.

(3)3Σ+
(2)3Π

(2)1Π
(1)1Π
(3)1Σ+(2)3Σ+(2)1Σ+(1)3Π

(1)1Σ+(1)3Σ+

1Σ+

3Σ+

1Π

3Π

1Π

3Π

1Σ+

3Σ+

1Σ+

3Σ+

0+
01 2P3/2+2S1/2
2
1
1
0+ 2P1/2+2S1/2
02
1 2
2
0+ P3/2+ S1/2
100+
1 2P1/2+2S1/2
00+ 0- 1

2S +2S
1/2
1/2

Fig.2.5 Transition between case (a) and case (c) for alkali dimmers
Interatomic distance increases from left to right. As internuclear distance increases, the electronic binding
energy becomes smaller and spin-orbit interaction dominates.

Here we would like to see a particularly useful example of coupling between orbital, spin and rotational
angular momenta: loosely bound alkali diatomic molecules of excited states dissociating to S+P limit. Thissituation frequently becomes problem in the experiments on cold atoms and molecules since atoms are excited
to these states by photoassociation. The molecular Hamiltonian for this problem is described here based on thearguments in [89]. We do not argue detailed derivations on it and omit hyperfine interaction for simplicity.

21
Molecular Hamiltonian in the excited states without hyperfine interaction is written as
H = H K + H V (R ) + H rot (R )where each term represents kinetic energy, potential energy including spin-orbit interaction and rotationalenergy, respectively. We have to obtain the specific form of HV and Hrot to solve the problem. We take the basis
in case (a) for writing these Hamiltonians for convenience. Then HV is written in a block diagonal form where
each block corresponds toΩ=0,1,2 as follows:

H

Ω=0+
V

− 2Δ 2 (R ) ⎞Σ 0+ ⎛⎜ V1 Σ (R )⎟(R ) = 3 ⎜Π 0 ⎝ − 2Δ 2 (R ) V3 Π (R ) − Δ1 (R )⎟⎠

H

Ω=0−
V

2 Δ 3 (R ) ⎞Σ 0+ ⎛⎜ V3 Σ (R )⎟(R ) = 3 ⎜Π 0 ⎝ 2Δ 3 (R ) V3 Π (R ) − Δ1 (R )⎟⎠1

3

Σ 0+ ⎛V 3 Σ (R ) Δ 3 (R )Δ 3 (R ) ⎞
⎜
⎟
Ω =1
1
H V (R ) = Π ⎜ Δ 3 (R ) V1 Π (R ) − Δ 3 (R )⎟3Π 1 ⎜⎝ Δ 3 (R ) − Δ 3 (R ) V 3 Π (R ) ⎟⎠
3

Ω =2

HV

(R ) = V Π (R ) + Δ1 (R )3

where V denotes case (a) potential energy and Δi denotes a spin-orbit interaction function.
The rotational part is written as
⎛ x+2
⎜
⎜ 0
⎜
0
H rot (R ) ⎜= ⎜− 2 x
B (R )
⎜
⎜ 0
⎜ 0
⎜
⎜ 0
⎝

0

0

x+4

2 2

2 2

x+2

0

0

−2 x
−2 x
0

± 2x
− 2x
0

−2 x
0

0

0

−2 x

0
x

± 2x
0

−2 x
− 2x

0

x+2

0
0

0
2

x+2
2
− 2(x − 2 ) − 2(x − 2)

⎞
⎟
⎟
0
⎟
0
⎟
⎟
0
⎟
− 2(x − 2 ) ⎟− 2(x − 2 ) ⎟⎟x − 2 ⎟⎠
0where x = J(J+1) and J denotes the total angular momentum. The upper/lower sign indicates e/f parity levels (eand f parity means that total parity is equal to (-1)J and (-1)J+1, respectively). Here 1Σ+ appears only for e parity
while 3Σ0+ appears only for f parity.
There are three notable points in this Hamiltonian:
1. The problem is a coupled channel Schrodinger equation. Though originally it became problem in the theoryof atomic structure [90], now it appears in various fields of atomic, molecular and nuclear physics. In ourcase, it is well-known that the theory of cold collisions includes this type of Hamiltonian in general. To
solve it, a special care should be taken as presented later in the section 2.2.4.
2. Spin-orbit interaction depends on the interatomic distance. For a large interatomic distance, it should be
asymptotically equal to the atomic fine structure constant. However, recent works have been elucidated thatit has a significant dip at a small interatomic distance due to the reduced fraction of the excited P state in themolecular wavefunction [91, 92, 93, 89, 94, 95, 96, 97, 98]. This behavior is also confirmed for KLi in the
ab initio calculation with MOLPRO which is shown later in the section 2.2.4.
3. Rotational angular momentum mixes blocks with different Ω. The Hamiltonian associates two channels
for J=0 while five channels for J=1 and six channels for J>1.
More detailed discussions on the coupling between angular momenta are found in [99].

2.2.3 Interactions with light22
Manipulation of molecules is in most cases performed with light. In our case, molecules are formed by aphotoassociation laser from atoms and detected by photoionization with an intense pulse laser. Analysis ofexperimentally obtained spectra needs the knowledge on which transition is involved in the specific case. Thus
it is important to know how molecules interact with light. In this section, the most important topics related tothe optical transitions in a molecule are given.
First we see the selection rules for various cases. An optical transition occurs in the limited cases due torestrictions on the angular momentum: it can be interpreted as a composition of the angular momenta of amolecule and light. Here we focus only on electric dipole transitions since other transitions such as electric
quadrupole or magnetic dipole transitions are relatively small. There is a general rule for the quantum number Jof the total angular momentum

ΔJ = 0,±1 with the restriction that J = 0 ↔ J = 0 is forbidden
In addition, there are restrictions on the symmetry of molecule:+ ↔ − is allowed but + ↔ +,− ↔ − are forbidden
where +/- means that the sign of the total molecular wavefunction is changed/unchanged against spatial
inversion. For homonuclear molecules, an additional rule holds:g ↔ u is allowed but g ↔ g , u ↔ u are forbidden
Apart from these rules there are several rules which depend on how angular momenta are coupled in amolecule, that is, case (a) or (c). In case (a), the quantum number Λ should satisfyΔΛ = 0,±1
which means that Σ-Σ, Σ-Π transitions are allowed but transitions such as Σ-Δ are forbidden. Anotherrestriction comes from the electronic spin component:ΔΣ = 0which says singlet-singlet and triple-triplet transitions are allowed but singlet-triplet transitions are forbidden.
In case (c), there is a restriction for Ω asΔΩ = 0,±1
and also a rule for Ω=0 states

0 + ↔ 0 + ,0 − ↔ 0 − are allowed but 0 + ↔ 0 − is forbidden
holds. More detailed arguments on these topics are found in [88].
Note that the above rules for each case do not hold exactly in a real molecule since there are usually small
perturbation effects such as spin-orbit interaction in case (a) or rotation in case (c). Thus some of transitions
forbidden in case (a) or (c) are allowed in a real molecule. This fact enables us to convert a singlet molecule toa triplet molecule or vice versa with optical transitions via mixed excited states.
Though the above arguments on selection rules result from the angular parts of molecular wavefunction,
there is a kind of selection rule for the radial parts of the wavefunction. The rule is based on the Franck-Condon
principle, which says the electronic transition caused by light is so fast that the distance between two atoms
does not change during the transition. Then we can describe a selection rule for the radial parts of wavefunction
as follows: a transition occurs only when ground and excited state molecule has nearly the same size. In thequantum mechanical language, this rule is interpreted as follows: there should be a significant overlap between
ground and excited wavefunctions for the transition to occur. The rule is readily confirmed by considering thetransition dipole moment of the corresponding transition with quantum mechanics. Here the transition dipole
moments involve all electrons in the molecule. We can write the total electric dipole in the molecule as a sum of
dipole induced by electrons and nuclei

23

r

μ=

∑e r

(2.47)s sswhere es and rs indicates the charge and position of electrons and nuclei, respectively. The total wavefunctions
for both ground and excited states are written as products of electronic wavefunction and rovibrational
molecular wavefunction under the Born-Oppenheimer approximation
g ,e
(R )
Ψtotg ,e (r ) = ψ elg ,e (r; R )ψ molwhere R = R

and r and R denotes electronic and molecular coordinate, respectively. Here theBorn-Oppenheimer approximation corresponds to the approximation introduced above which says electronictransition is fast enough to neglect atomic motions. Note that the electronic wavefunction depends only on theinteratomic separation in the case of diatomic molecule. Taking account the expression of the dipole moment(2.47), the transition dipole moment between ground and excited molecular states is expressed asreg(R ) ψ ele (r; R ) es rs ψ elg (r; R ) Ψmol(R )d = Ψtote μ Ψtotg = Ψmol∑

(2.48)

s

Here we define the transition dipole moment at a particular molecular coordinate R as

M(R ) = ψ ele (r; R )∑ e r ψ (r; R)
s s

g
el

s

Then we can rewrite (2.48) in a simple form:eg(R ) M(R ) Ψmol(R )d = Ψmol

It is noteworthy that M(R) can be written as a product of radial and angular part
M (R ) = M (R )e(θ , ϕ )where e is a unit vector and determined by the relation between orientation of molecule and polarization of light.
M(R) is not a measurable value but obtained from theoretical calculations as shown later in the section 2.2.4.
Thus we can see that a transition rate between the ground and excited molecular states is proportional to the
product of radial and angular parteg(R ) M (R ) Ψmol(R )d = Ψmol22eg(θ ,ϕ ) e(θ ,ϕ ) Ψmol
(θ ,ϕ )
Ψmol

2(2.49)

If we assume that M(R) does not depend on R significantly and its typical value is M0, we can write the integralin (2.49) in a simpler form:eg(R ) Ψmol(R )d = M 02 Ψmol
2

2

e
g
(θ ,ϕ ) e(θ ,ϕ ) Ψmol
(θ ,ϕ )
Ψmol

2

The factoreg(R ) Ψmol
(R )
Ψmol

2

is called the Franck-Condon factor and represents an overlap between ground and excited molecular states. As
is clear from the above expression, the Franck-Condon factor does not contain any information on theelectronic structure of atoms and reflects only the radial size of ground and excited molecule. Note that
selection rules on the angular parts mentioned above are included in the transition dipole moment: M0 has afinite value only for an allowed transition.
In the end we can formulate the selection rule for the radial parts as follows: a transition occurs when theFranck-Condon factor is close to unity. However, any transition with small but non-zero Franck-Condon factor
is possible in reality just as in the case of the selection rules for the angular part. In most cases, there are manytransitions between ground and excited rovibrational levels.

24
Though knowledge on the Franck-Condon factor is quite useful to estimate transition strength, calculation of
the Franck-Condon factor is not so straightforward. Now an established program package LEVEL to calculate
the Franck-Condon factors as well as vibrational levels is available [100]. An example of calculated
Franck-Condon factors for various transitions is presented later in the section 2.2.4.

2.2.4 Numerical investigations
In this section, molecular properties mentioned above are numerically calculated for a specific case. First we
see an example of ab initio calculation of the PEC of KLi molecule. Calculated potentials are compared withexperimental results to show accuracy of the calculations. Next variation of transition dipole moments againstinteratomic distance is investigated for KLi. In the end, we studied an optimum transition to convert molecules
in a high vibrational level of the ground X state into the lowest vibrational level of the X state for 41K87Rb.
In figure 2.6 calculated PECs of KLi already given in figure 2.3 are shown with experimentally obtained
potentials. We can see that calculations are in good agreement with experiments. However, there is always afinite difference between them due to approximations made in the calculations to reduce computational tasks.
The deviations of the present calculations from experiments are explicitly plotted against energy levels in figure
2.7. Here bound states are calculated with LEVEL [101]. The deviations of the present calculations are ranged
in ±20cm-1. This value is reasonable compared to other published works and regarded as the limit of thecurrent calculation method.
To see the accuracy of the present work from another point of view, the Franck-Condon factors between thecalculated excited state energy levels and the experimental ground state energy level are compared in figure 2.8.
Though there are deviations, they are in good agreement with each other. Since the Franck-Condon factors
reflect the transition strength, this kind of calculations is very useful to interpret experimental results. Our
conclusion is that experiments are necessary to know exact energy levels but calculations are quite useful for
analysis of the observed spectra.

25

4

2

x 10

e3Σ+
1.5

d3 Π
D1Π
B1Π C1Σ+

Energy [cm−1]

1

c3Σ+

A1Σ+

0.5

b3 Π
a3Σ+

0

X1Σ+
-0.5

-1

2

3

4

5

6

7

8

9

10

11

12

Interatomic distance [Å]Fig.2.6 PEC of KLi
Blue, green, purple and red lines indicate 1Σ+, 3Σ+, 1Π and 3Π, respectively. Solid circles are RKR data,
while open circles are IPA data. The references of the presented experimental data are following: X [102],B IPA [103], B RKR [104], C IPA [103], C RKR [104], D IPA [105].

26

Deviation (Calc.-Exp.) [cm−1]

20
B
C
D

15
10
5
0
-5
- 10
- 15
- 20
1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5
4

Energy [cm−1]

x 10

Fig.2.7 Deviations of calculated levels from experimentally obtained levels
Each point indicates a bound state. Here atomic masses are assumed to be 40u for K and 6u for
Li where u is the atomic mass unit.

Franck-Condon factors

0.25
X(v=0)- B exp.
X(v=0)- B calc.
X(v=0)- C exp.
X(v=0)- C calc.
X(v=0)- D exp.
X(v=0)- D calc.

0.2

0.15

0.1

0.05

0
1.1

1.15

1.2

1.25

1.3

1.35
−1

Energy [cm

1.4

1.45

]

1.5
4

x 10

Fig.2.8 Franck-Condon factors between ground state (experiments) and excited state (calculations)
Here atomic masses are again assumed to be 40u for K and 6u for Li.
Next we investigated variations of transition dipole moments against interatomic distance for KLi molecule.
The expectation values of the corresponding operator are calculated with the molecular wavefunctions obtainedin the previous calculations on the potential energy. Figure 2.9 shows dependence of transition dipole moments.
For sufficiently large interatomic distance, the transition dipole moments well coincides with thecorresponding atomic transition dipole moments. This means that transitions between loosely bound molecules
can be treated with atomic transition dipole moments. However, transition between molecules with aninteratomic distance smaller than 10Å is no more than approximated to be an atomic dipole moment.

27
Note that spin-orbit interactions are not taken into account in this calculation. In actual cases, spin-orbit
interactions mix case (a) sates and thus transition dipole moments show a more complicated variation againstinteratomic distance [29].

Transition dipole moments [ea0]

5
4
3
2
X- A
X- B
X- C
X- D
a- b
a- c
a- d
a- e

1
0
-1
-2
2

4

6

8

10

12

14

16

18

20

Interatomic distance[Å]Fig.2.9 Calculated transition dipole moments of KLi
A, B, b and c states are dissociating to K(4P)+Li(2S) limit while C, D, d and e states are dissociating toK(4s)+Li(2P) limit. Here atomic transition dipole moments are 2.92ea0 for K(4P) and 2.35ea0 for Li(2P).
In the end, we give a consideration on the optimum transition to carry out a stimulated Raman transition for
41

K87Rb. Our goal is to form molecules in the lowest vibrational states (v=0) from loosely bound singlet

molecules.
The most important point in a transfer between two molecular states is that the pump and dump should havenearly the same Rabi frequency. If the transition dipole moments for pump and dump transitions are nearly thesame, the intensity needed for an efficient transfer between two ground states are also expected to be the same.
However, if one of the two transitions has quite a small transition dipole moment, the intensity for that weaktransition should be fairly large. This leads to unwanted excitations to closely lying excited sates and reduces
the overall transfer efficiency. Note that the effective Rabi frequency for the overall transition is a product of
the two Rabi frequencies but increasing the intensity of the strong transition also leads to unwanted excitations.
Thus the transition dipole moments between the excited levels and the two vibrational levels in the groundstates should be large.
As a first approximation, we can consider that the transition dipole moments are proportional to theFranck-Condon factors as mentioned above. Thus we can estimate the transition strength with the Franck
Condon factors. For precise estimations, we have to take into account variation of the transition dipole
moments against interatomic distance. Based on these arguments, we can find out the optimum transition where
both pump and dump transitions have large Franck-Condon factors. Figure 2.10 shows calculated
Franck-Condon factors between the vibrational levels of X state and four singlet excited states A, B, C and D.
It is clear that the C state is the most suitable for our purpose since crossing between blue and other dots
occurs at comparatively large Franck-Condon factors. If we select v=90 molecules (green dots) as a starting
point, the optimum transition is found to be 868nm for pump and 637nm for dump. With this transition, the28Franck-Condo factor of 0.01 for pump and that of 0.015 for dump is expected.
We note that the following two points:
1. In this argument, singlet-triplet mixing due to spin-orbit interaction is neglected. This effect is expected tobe large for KRb since Rb has a large spin-orbit interaction and vibrational level spacing is small. If themixing effect between excited singlet and triplet states is taken into account, it is possible to convert triplet
molecules to singlet molecules. There is a possibility that triplet molecule is more suitable for a starting
point as proposed in [106] and demonstrated for RbCs [30].
2. The loosely bound molecule with a binding energy of 0.01cm-1 (v=100) has fairly small Franck-Condon
factors. Even when the X-C transition is used, the Franck-Condon factor at the optimum transition is around10-5 which is much smaller than that of v=90 molecules. This case is particularly important since molecules
formed by Feshbach resonance have typically a similar binding energy. Though we do not know whether
the Franck-Condon factor of 10-5 is sufficient to achieve a high transfer efficiency, transfer of Feshbach
molecules to the lowest vibrational level might require two or three steps.
(a) X- A transitions

010(c) X- C transitions

0

10

-5

10

-5

10

FCF

FCF

- 10

- 10

10

10

- 15

10

- 15

10

- 20

10

- 20

10

6000

- 25

7000

8000

9000
−1

Energy [cm

10000

10

11000

]

10

- 10

10

]

1.3
4

x 10

-5

- 10

10

FCF

FCF

1.2

(d) X- D transitions

0

10

-5

- 15

10

- 20

- 15

10

- 20

10

10

- 25

1.05

1.1

Energy [cm

10

10

1

−1

(b) X- B transitions

0

10

0.9

- 25

1.1

1.15

1.2

1.25
−1

Energy [cm

]

1.3
4

x 10

10

1.15

1.2

1.25

1.3
−1

Energy [cm

]

1.35
4

x 10

Fig.2.10 Calculated Franck-Condon factors (FCF) of KRb singlet transitions
The Franck Condon factors between the vibrational levels in the ground X states and various excited states
are plotted against energy levels of the excited states. Here ab initio Potentials from [107] are used. Blue,green, orange, red and purple indicate v=0, 90, 92, 95 and 100, respectively. Formation of the v=0 molecule
is our goal while the loosely bound v=90~100 molecules are expected to be formed by photoassociation.
Here v=90, 92, 95 and 100 molecules are bounded by 17.12, 9.10, 2.54 and 0.01cm-1, respectively.

29

Chapter 3

Experimental methods

Experiments on cold atoms and molecules need a large amount of apparatus. Most of the time is spent tomake them and set them up. In this chapter, details on apparatuses used in our experiments are described. First,
we see our laser system to manipulate potassium and rubidium atoms. Then the design and operation of vacuum
chamber system is overviewed. In 3.3, a computer control system realized with a LabView program developedby Dr. T.Mukaiyama is shown.

3.1 Laser system
Here a whole laser system used in our experiments is presented. A home made external cavity diode laser(ECDL) locked to an atomic D2 line is used as a frequency stabilized light source. A portion of this master laser(~0.5mW) injects slave lasers and is amplified by around 20 times. Then a tapered amplifier (TA) amplifies theslave laser to obtain a high power around 0.6W.

3.1.1 ECDL
ECDL is a low cost tunable narrow band laser source and is widely used for experiments on atomic physics.
It utilizes the fact that a semiconductor laser has a wide gain spectrum and therefore its oscillation wavelengthis determined only by the length of cavity. Usually the front facet of a semiconductor laser has a reflectivity of
about 3% while the back facet has nearly unit reflectivity, and thus these two facets are making up a cavity tooscillate in a single frequency. A part of photons in the cavity is emitted from the front facet. However, if you
put another mirror in front of the front facet, the oscillation frequency is determined by the length of cavity
made of a new mirror and the back facet when the reflectivity of the mirror is higher than that of the original
facet. Thus it becomes possible to tune the oscillation frequency to the atomic absorption line by adjusting theposition of the external mirror with a piezo electric actuator.
This is the basic principle of ECDL, but it does not suffice for the use of atomic experiments as the linewidthof a usual semiconductor laser (a few GHz) is much broader than the natural linewidth of atoms (typically afew MHz). It is necessary to select the oscillation wavelength. For that purpose, usually a grating is used as anexternal mirror: a diffracted light is feed backed to the laser diode, whereas a reflected light is used forexperiments. Selection of wavelength is realized with the fact that the diffraction angle depends on wavelength.
By adjusting the angle of the grating, the wavelength of feed backed light can be tuned, which means that
oscillation wavelength can also be tuned. Tuning of wavelength is also possible by adjusting the position of thegrating since it means that the cavity length is changed. For more detailed information on these topics, there aregood reviews [108, 109].
In general, there are two types of ECDL: Littrow configuration and Littman configuration. Schematics ofthese two configurations are shown in Fig.3.1. The main difference lies in the output power and the linewidth.
In Littrow configuration, the feed backed light is diffracted one time. This leads to a high output power andbroad linewidth. In contrast, the feed backed light is diffracted two times and thus the output power is low andthe linewidth tends to be narrow in Littman configuration. However, a linewidth of 1MHz is easily obtained inboth configurations. In our case, a high power is required to trap many atoms. Thus we have decided to make
ECDL in a Littrow configuration.

30
MirrorOutput

Lens

Grating
Laser diode(a) Littrow configuration
Loss
MirrorOutputLens

Laser diode

Grating(b) Littman configuration
Fig3.1 Littrow and Littman configuration
In the Littrow configuration, the diffracted light is directly feed backed to the laser diode. In
contrast, in the Littman configuration the diffracted light is reflected at a mirror and then again
diffracted at the grating and return to the laser diode. Since a part of power is lost in the Littman
configuration, the output power of the Littman configuration is lower than that of the Littrow
configuration.

Schematics of our ECDL are shown in figure 3.2, which is a slightly modified version of [110]. A mirror isadded after the grating to compensate the change in the grating angle. With this modification, an optical path
after ECDL remains unchanged if the grating angle is changed. In this type of ECDL, the diffraction angle isused as a rough tuning knob, while the position of the grating works as a fine tuning knob.
Our standard grating for near-infrared light is a gold coated 1800 lines/mm holographic grating for UV range.
Here we employ a grating for UV light to suppress the diffraction efficiency and obtain high output power. The
diffraction efficiency at the grating is 3 or 5% for a S polarized light which depends on the direction of thegrating, though it is much higher (more than 20%) for a P polarized light.
We use an antireflection coated laser diode to suppress reflection at the front facet of the laser diode andensures stable operation, though it seems to have a difficulty that its lifetime is limited (1~1.5 year) comparedwith an ordinary laser diode. The temperature control of laser diode is inevitable to obtain stable operation as
the length of laser diode and the gain characteristics depends on the temperature. In our design, the temperatureof the whole mirror mount including a laser diode is stabilized with a peltier module under the mirror mount.
We use a commercial thermo electric controller (TEC) and laser diode current controller.

31
Mirror mount(Newport U100-P)

Grating mount

Piezo module(Single layer)

Collimation tube
(Thorlabs LT230P-B)

Holographic grating(1800 lines/mm for UV)
Laser diode
Adapter(Newport UPA-PA1)

1cm
Right angle mirror(Gold coated)

Fig.3.2 (a) Design of our ECDL

Fig.3.2 (b) Image of our ECDLThe black package is a commercial laser diode socket which contains a protection circuit (Thorlabs SR9B).

The procedure to tune this ECDL is as follows.
First the temperature should be set to a correct value: if the center wavelength of the laser diode at room
temperature is different from the wavelength you want by a few nm or more, the temperature of the laser diodeshould be changed such that the center wavelength lies within 1 nm from the value you want. Usually thecenter wavelength shifts about +0.2nm per Kelvin.

32
Then the current to the laser diode is set to a value at which you want to use, which is recommended to besmaller than an operation current (not max current) stated on the data sheet of the laser diode to have a longer
lifetime.
After these preparations the mirror mount angle is adjusted so that diffracted light goes back to the laser
diode. If you put a white paper with a pin hole just in front of the laser diode, the diffracted light which passed
through the pin hole will be seen on the paper near the pin hole. You have only to adjust the mirror mount sothat this diffracted light is incident on the pin hole. The pin hole should be placed at the center of the lightemitted from laser diode. Putting away the paper, the power after the grating will be much higher than theoriginal spontaneous emission if the diffracted light is correctly feed backed to the laser diode.
Next the wavelength is tuned within a few GHz from what you want by adjusting the horizontal angle ofmirror mount monitoring it with a wavemeter. It is usually impossible to match the wavelength to exact value
you want in this stage because the mirror mount is very sensitive.
At this stage the vertical angle of the mirror mount is adjusted so that operation should be stable. This iscarried out by lowering the threshold current as much as you can. When the power is monitored on an
oscilloscope and the current is modulated near the threshold current, you can see the threshold current where
the power begins to rise. By adjusting the vertical angle, you can obtain the lowest threshold current. This
ensures that the laser oscillation is the most stable at this wavelength.
The last stage is to set the wavelength to an exact value you want with adjusting the voltage applied to thepiezo electric module behind the grating.
The mode hop free range of this ECDL is around 5GHz. Its linewidth averaged for about 1s is 1MHz, whichis estimated with the width of the beat note of two independent ECDL. The linewidth is probably limited bymechanical vibration of mirror mount and fluctuation in the refractive index of air in the cavity due toconvection.

3.1.2 Slave laser
Since the role of a slave laser is only to amplify the master ECDL, it has a very simple structure as shown infigure 3.3. The collimation tube containing a laser diode and a collimation lens is mounted on a aluminum jig
whose temperature is stabilized with a inexpensive home made TEC. Usually laser diode used in a slave laser isthe same as that of maser ECDL.

Fig.3.3 Image of our slave laser (top view)

33
Injection locking to a slave laser is carried out in the configuration shown in figure 3.4. When a part ofmaster ECDL light is incident on the slave laser diode, it induces stimulated emission and the original seed lightis amplified. The gain depends on the current. A faraday rotator is necessary to separate the seed light and theamplified light; otherwise the amplified light will only go back to the maser laser.

Faraday rotator

PBS

PBS

Slave laser

OutputIsolator

Seed light(Master laser)

Fig.3.4 Injection locking
The seed light which originates from a master laser is amplified at the slave laser.
Although operation of a slave laser is very simple, there is a problem always you have to note. The output
power is expected to increase as you raise the current, but actually there is a bad point where the output power
abruptly decrease as shown in figure 3.5. At around this current, the oscillation of the slave laser is unstable andamplified spontaneous emission (ASE) increases. Thus you have to pay attention to the relation between
current and power whenever you start up the laser: sometimes the power may increase as you decrease thecurrent, which means that the output light contains undesirable ASE.

4035

Output power [mW]

30
25
20
15
10

Bad points

5
0

0

20

40

60
Current [mA]80

Fig.3.5 Output power of slave laser against current

100

34

3.1.3 Tapered amplifier
Tapered amplifier is another kind of semiconductor laser. The main difference from an ordinary laser diodelies in its tapered structure and both facets are antireflection coated. This enables us to obtain high poweraround 1W from the front facet when a seed light is incident on the back facet. In the case of an ordinary single
mode laser diode, maximum attainable power is limited by the fact that high optical power more than 300mW
breaks down the front facet.
The role of a tapered amplifier is also to amplify the seed light as a slave laser does, but a situation is quite
different: the heat released from a slave laser diode is less than 0.2W, whereas that from a tapered amplifier
usually exceeds 3W. Thus you have to take care of the released heat to be transferred effectively to some heatsink. We usually buy only a tapered amplifier chip and mount it on a home made jig made of copper. Figure 3.6
shows our design of a mount for a tapered amplifier devised by Dr. T.Kishimoto. In this design, collimationlenses are fixed to the copper heat sink which is cooled by a peltier module under the heat sink. The whole ofthese parts are placed in a box to avoid dusts attaching to the tapered amplifier chip.
Collimation lens holder
Output collimation lens(Thorlabs C390TM-B)

OutputScrew
Heatsink(Copper)

Top view
Tapered amplifier chip
Input

Input collimation lens(Thorlabs C230TM-B)

Peltier module

Side view

Pedestal
Heat

Optical table(Breadboard)

Fig.3.6 Tapered amplifier design and assembled image

35
A graphite sheet and aluminum foils are used to compensate roughness of the surface of the heat sink andobtain high thermal conductivity. We did not use thermal grease for this purpose as it might have a chemical
damage to antireflection coating of tapered amplifier.
Temperature of the heat sink is stabilized with a home made TEC which is a slightly modified version of the
one used for a slave laser. We usually set the temperature nearby the tapered amplifier around 20 degrees
Celsius. Longer lifetime is expected for lower temperature, though too low temperature leads to condensation
which may destroy the chip.
Alignment of collimation lenses are carried out with a mirror mount as illustrated in figure 3.7. Here a mirror
mount acts as a stage. It is possible to move the collimation lens in any direction by adjusting the three knob of
the mirror mount. Another cylindrical lens is required for the output beam as the beam divergence is different
for horizontal and vertical direction.

Fig.3.7 Alignment of collimation lenses
Three knobs of a mirror mount enable us to tune a collimation lens holder in any direction precisely. The
input lens is adjusted to obtain a maximum output power. The output lens is adjusted such that the output
beam is colinear in the vertical direction. The horizontal beam divergence is compensated at the cylindricallens just after the output collimation lens (see figure 3.9).

Typical characteristics on the power of tapered amplifier are shown in figure 3.8. You can see from thesefigures that higher power is obtained both by raising the current and with higher power of seed light. If it ispossible, the latter way is recommended because the heat generated at the chip is lower for the latter than theformer. We usually use about 10mW of seed light and 2.0A of current for 1W tapered amplifier, thus obtaining
600mW after an isolator.

36

600

500

Output power [mW]

400

300

200
2.3A
2.0A
1.8A

100

0

0

5

10

15

Injection power [mW]

Fig.3.8 (a) Output power of tapered amplifier against injection power
500
450
400

Output power [mW]

350
300
250
200
150
100
50
0

0.8

1

1.2

1.4

1.6
1.8
Current [A]2

2.2

Fig.3.8 (b) Output power of tapered amplifier against current

2.4

2.6

37
We construct an amplifying system on a small bread board whose input seed light is supplied with apolarization maintaining single mode (PMSM) fiber and output is coupled to a PMSM fiber, as shown in figure
3.9. Connecting a fiber from a slave laser to this system, you can obtain a mode shaped amplified light from theoutput fiber. Here acoustic optic module (AOM) serves as a fast switch of light. The diffraction efficiency at
AOM is typically 80%. Efficiency of coupling to the output PMSM fiber is limited to around 65% due to a badspatial mode of the output beam. Thus we can usually use about 200~300mW for experiments.
Cylindrical (vertical) Cylindrical (horizontal) Cylindrical (vertical)
f=50mm
f=30mmf=100mm

Input fiber
λ/2λ/2Taperedamplifier

f=500mm

Isolator

f=200mm

Photodiode(Power stabilization)
Output fiber

AOM

Breadboard(300× 450 mm)

Fig.3.9 Schematics of 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 before the tapered amplifier is adjusted to obtain a maximum output power.
The cylindrical lens after the tapered amplifier compensates the horizontal beam divergence. The
waveplate after the isolator is used to compensate the polarization rotated at the isolator.

3.1.4 Distributed Feed Back laser
Distributed Feed Back (DFB) laser is a kind of semiconductor laser. Though it appears the same as an
ordinary laser diode, it has subtle difference in its internal structure: the width of light emitting semiconductor
layer is periodically modulated. This structure works as a grating and select the oscillation wavelength just as
done in ECDL. Thus DFB laser itself has a very narrow linewidth (typically around a few Hz) compared to an
ordinary laser diode (a few GHz). It means that DFB laser alone can be used for atomic experiments as ECDL;
however, the largest advantage of DFB laser lies in a large mode hop free range of 500~1000GHz, which islarger than ECDL by two orders. We use this DFB laser for photoassociation spectroscopy which typicallyneeds tuning wavelength over a few hundred GHz.
Wavelength can be tuned both with current modulation and with temperature modulation. Typical change inwavelength is 25GHz/K and 1GHz/mA. As easily know from these values, stability of temperature and current
is necessary to utilize it for spectroscopy. We used commercial TEC unit and current source to ensure these
conditions.

38
Figure 3.10 shows wavelength characteristics of our DFB laser against temperature and current. A large
mode hop free range is clearly seen. Figure 3.11 shows saturated absorption spectroscopy of Rb D1 lines with
our DFB laser. It shows that the linewidth is narrow enough to carry out spectroscopy.

(a) Temperature dependence

795.8

Wavelength [nm]

795.6

795.4

795.2

795

794.8

794.6

20

25

30

35

40

45

50

◦

Temperature [ C](b) Current dependence

795.95795.9

Wavelength [nm]795.85795.8
795.75795.7
795.65795.6
795.55
20

22

24

26

28

30

32

34

36

38

Current [mA]
Fig.3.10 Wavelength of DFB laser against (a) temperature and (b) current

40

39

1.2

Photodiode voltage [V]

1.15
1.1
1.05

85Rb F=3

87Rb F=1

87Rb F=2

1
0.95
0.9

85Rb F=2

0.85
0.8
0.75
- 0.035- 0.03- 0.025- 0.02- 0.015- 0.01- 0.00500.005Time [s]Fig3.11 Saturated absorption spectrum of Rb D1 line observed with the DFB laserThis spectrum is taken by modulating the current. Here 10ms corresponds to around 2.7GHz.

3.1.5 FM sideband locking
To manipulate atoms with light, the light should have long term stability in frequency as well as a narrow
linewidth. An ECDL alone does not meet this demand. Thus it is necessary to lock the wavelength of ECDL toan atomic absorption line actively. There are several methods to realize it. Here our standard scheme based on[111] is described.
Figure 3.12 shows our scheme to lock ECDL to an atomic absorption line. The original saturated absorption
signal is differentiated with frequency modulation and lock-in and then sent to a servo loop which feed back the
difference between current point and set point to piezo voltage and laser current. Fluctuation in the laser
frequency locked in this way is typically less than 2MHz. This is sufficiently lower than the natural linewidth ofalkali atoms.
For both K and Rb, we heated gas cells to around 70 degrees Celsius to obtain large saturated absorption
signals.

40Function
generator

15MHz
4.5V

Power
splitterPhaseshifter
Gas cellPhotodiodeLockcircuit

Mixer
EOM

Current
Current source
feedback

ECDL

Optical table

Piezo
feedback

Fig.3.12 Schematics of FM sideband locking
0.25

39
0.2

K crossover

0.15

Voltage [V]

0.1

39

0.05

K F=1→F00- 0.05- 0.1

39

K F=2→F041- 0.15

K F=2→F0- 0.2- 0.25- 0.015- 0.01- 0.005

0

0.005

0.01

0.015

Time [s]Fig.3.13 Differentiated saturated absorption signal of potassium
The central sharp edge is the differentiated signal of 39K crossover used for our locking. Note thatthe hyperfine structure in the excited states is quenched in the above peaks.

41(a) 87 Rb Cooling (F=2→F0)
0.5

Crossover(F0 =1+F0 =3)

0.4

F0 =3

0

F =1 F0 =2

0.3

Voltage [V]

0.2
0.1
0
- 0.1
- 0.2
- 0.3

Crossover(F0 =1+F0 =2)- 0.4

Crossover(F0 =2+F0 =3)- 0.5
0.03640.03650.0366

0.0367

0.03680.03690.037

Time [s](b) 87 Rb Repumping (F=1→F0 )0.4

Crossover(F0 =0+F0 =2)
F0 =0
F0 =2

0.3
0.2

Voltage [V]

0.1
0
- 0.1
- 0.2

F0 =1
Crossover(F0 =0+F0 =1)

- 0.3
- 0.4

Crossover(F0 =1+F0 =2)

- 0.5
0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time [s]Fig.3.14 Differentiated saturated absorption signal of rubidium of (a) cooling and (b) repumping
Cooling is locked to the crossover of F=2→F’=2 and F=2→F’=3 transitions while repumping islocked to the crossover of F=1→F’=2 and F=1→F’=1 transitions.

42

3.1.6 Pulse laser
A pulse laser is not used for trapping and cooling atoms, but for ionization: its high intensity enables direct
multiphoton ionization of atoms and molecules. Resulting ions are detected with microchannel plates, whose
detailed description will be given in 3.6.
A dye laser pumped by a 532nm doubled Nd:YAG laser is directly shone to atomic clouds with a diameter ofaround 1mm. The intensity of the dye laser is easily controlled by a Q-Switch delay of the Nd:YAG laser asshown in figure 3.15. Though the maximum output energy of our dye laser is 30~100mJ depending on the
wavelength, we usually set the pulse energy to around 1mJ to avoid an undesirable ionization of atoms attachedto the surface of the chamber.
25Power [mJ]

20

15

10

5

0
150

200

250

300

350

400

450

Q switch delay [μs]Fig.3.15 Power of doubled Nd:YAG laser against Q-switch delay

500

43

3.1.7 Optics
Experiments on cold atoms need many kinds of laser. Here our setup on optics is briefly explained.
In general, we have two optical tables for one experiment: one prepares lasers and the other has a vacuum
chamber on it. All lasers are carried with polarization maintaining single mode fibers between the two tables.
Figure 3.16 and 3.17 shows optical setup of E1 and E2 experiments, respectively. All MOT beams are first
expanded to the diameter of around 25mm and then divided to provide 6 beams which are incident on thechamber. Cooling and repumping beams are overlapped after they are expanded.
1st MOT
Repumping

1st MOT
Cooling

Repumpingfor probe

E2

Optical pumping

Probe

Push

AOM
AOM

AOM
AOM

Slave
Wavemeter

2nd MOT
Repumping

PhotodiodeL/4 waveplate
L/2 waveplate

Master
ECDL

Prism pair

AOM

Fiber
Cylindrical lensSpherical lensMirrorSlave
2nd MOT
Cooling

EOM

Gas cell

PBS
Isolator

Figure 3.16 E1 optical layout of the potassium lock table
All lights are coupled to fibers and carried to the chamber table. Four MOT lights are amplified bytapered amplifiers at the chamber table. A part of master laser is amplified by a slave laser andcarried to the E2 optical table (see figure 3.15). A set of cylindrical lenses are used for compensating
an aspect ratio and the difference in the beam divergence.

44
To wavemeter

Rb lock table

Rb MOT
Repumping

MasterECDL for Rb(Repumping)

EOM

PhotodiodeL/4 waveplate
L/2 waveplate
Prism pair
Fiber
Cylindrical lensSpherical lensMirrorMasterECDL for Rb(Cooling)

EOM

PBS
Isolator

To E2 optical table
K MOT
Repumping

To TA for K
Repumping

Taperedamplifier(K repumping)

Taperedamplifier(K cooling)

K Probe
To Rb TADFB

To wavemeter

Rb
DepumpFrom
K lock table

To TA for K
CoolingFrom
Rb lock table
Rb Probe

K MOTCooling

To TA
for PA

E2 optical table

Figure 3.17 E2 optical layouts
The cooling light for rubidium is carried from the rubidium lock table.

E1 experiment adopts a double MOT system. Here the first MOT at the low vacuum side serves as an atomic
source while the second MOT at the high vacuum side collects atoms from the first MOT. A push beam which
is a slightly blue detuned light is focused to the center of the first MOT, which pushes atoms towards thesecond MOT.
E2 experiment needs lasers for two atomic species, K and Rb. These two lasers are overlapped with dichroic
mirrors. As the resonant frequencies of these two atoms are close to each other, waveplates for one frequency
are shared.

45

3.2 Vacuum chamber
Laser cooling and trapping of atoms are carried out under ultrahigh vacuum environment to avoid collisions
between cooled atoms and room temperature hot atoms. Thus construction of a vacuum chamber which canreach a vacuum of 10-9 Torr or higher is an important task.
The vacuum chamber of E1 experiment is an assembly of commercial vacuum components. It has two cells
where MOT is carried out and they are connected by metal components. Two ion pumps and a titanium
sublimation pump is utilized to obtain ultrahigh vacuum of the order of 10-11 Torr. The E2 chamber is based ona custom-made metal chamber. Only one ion pump is sufficient to obtain ultrahigh vacuum of the order of 10-9
Torr. Schematics of these two chambers are shown in figure 3.18.

(a) E1 chamber

Cloverleaf coils
1st cellPush

2nd cell(b) E2 chamber
CCD
MOT

Pulse laser
MOT

Channeltron
ICF70
Ion pump(75l/s)Dispenser

MCP
MOT
MOT

Probe

Fg.3.18 Schematics of the vacuum chambers of (a) E1 and (b) E2

117mm

46
These figures omit a pumping station which is a combination of a turbo pump and a scroll pump. A pumping
station is connected to the angle valve shown in the above figure and evacuate the chamber to 10-7 Torr before
an ion pump is turned on. If an ion pump begins to operate, the angle valve is closed and the pumping station isseparated as the vibration of scroll pump and turbo pump affects the experiments.
Usually we have to bake the chambers to reach an ultrahigh vacuum regime. The basic mechanism of thisoperation is simple: if you heat the chamber and keep on drawing a vacuum with a pumping station,
background atoms attached to the surface of the chamber fall off the chamber by the heat and eventually drawn
into a pumping station. The temperature of the chamber is monitored and kept constant around 200 degrees
Celsius. This operation removes background gases such as hydrogen, oxygen and nitrogen from the chamber.
An ion pump is turned on after baking. A few days of baking is sufficient for 10-10 Torr with an ion pump,though a titanium sublimation pump is necessary to reach 10-11 Torr.
Both E1 and E2 chambers have dispensers as an atomic source in them. A dispenser emits atoms through achemical reaction when it is heated via a current flowed to it. It is easy to control the amount of yield with thecurrent. As it is based on a chemical reaction, there is a threshold current at which atoms begin to yield. We use
a commercial dispenser for Rb and 39K atoms, while home made ones based on [112] are used for 41K atoms.

3.3 Computer control
Manipulation of atoms is carried out with a digital and analog control of many apparatuses within a shorttime. It is impossible to do this with a manual operation, which means that a computer control system isrequired for all experiments. We use a LabView code developed by Dr.T.Mukaiyama. An example of timing
control on a computer is shown in figure 3.19.

Fig.3.19 Image of LabView program47

3.4 Estimation of number of atoms with fluorescence
It is important to know number of atoms in a MOT as it reflects conditions on MOT light and the abundanceof trapped atoms and background junk atoms in the chamber. In the case of condensate experiments, estimationof number of atoms is also important because it is closely related to whether you can obtain a condensate fromthe laser cooled atoms or not.
The most simple and reliable way to estimate number of atoms in a MOT is to count number of photons
scattered by the atomic cloud with a photodiode. An atom in a MOT always scatters MOT light to any direction
with a rateI IsΓ2 1 + I I s + (2Δ Γ )2

(3.1)

where Γ denotes the natural linewidth of atoms, I denotes the intensity of MOT light at the atomic cloud, Is
denotes a saturation intensity of the atom and Δ denotes the detuning of MOT light from resonance. Thus a
MOT containing N atoms emits photos with a rate

I IsNΓ
2 1 + I I s + (2Δ Γ )2

(3.2)

Now consider a configuration to count number of photons emitted from a MOT as shown in figure 3.20. A partof photons are collected by a lens and focused to a photodiode. Multiplying the solid angle Ω to theexpression (3.2), number of photons incident on the photodiode within a unit time is expressed as

I Is
NΓΩ
2 1 + I I s + (2Δ Γ )2

(3.3)

On the other hand, you can know number of photons incident on the photodiode from the current flowed to
the resistance illustrated in figure 3.18. If the sensitivity of the photodiode at the wavelength of photons is
written as S in units of A/W, number of photons incident on the photodiode per unit time is described as
V
RShν

(3.4)

where hν is energy of a photon, V is the measured voltage and R is the output load. The value of S ismeasured at the corresponding wavelength as it largely depends on the wavelength. Equating (3.3) and (3.4)
yields the expression for number of atoms in the MOTN=

1 + I I s + (2Δ Γ )2VI IsΩRShνΓ

2

+Vcc

Lens
MOT

PhotodiodeOscilloscope
R

I

Fig.3.20 Counting number of atoms in a MOT with the fluorescence
The photodiode should have a larger area than the focused image of MOT.

48
Though number of atoms can also be estimated from an absorption imaging as explained in 3.5, estimation
with fluorescence from a MOT is more reliable: absorption imaging suffers from the maximum amount ofabsorption that can be detected with CCD as well as depends on the relation between the polarization of probe
light and the bias magnetic field. On the contrary, there is no unclear factor in the fluorescence method, andthus it gives a reliable estimate.
The above model is correct only for a two level system. It is a good approximation to deal with the opticaltransition between particular hyperfine states for most of alkali atoms, but not for 39K and 41K. We used a 6
level model which takes into account the effect of other hyperfine states for correct estimation of number ofatoms [113]. Compared to the 6 level model, we did not observe significant difference from the model that
F=2-F’=3 transition for the trapping light and F=1-F’=2 transition for the repumping light are separately closed
transitions.

3.5 Absorption imaging
Experimental results are obtained by taking images of atoms with CCD camera. The amount of absorption at
an atomic cloud is measured by irradiating a resonant probe beam to the atoms. Three images, one with atoms
and a probe light, one without atoms and with a probe light and one without atoms and a probe light, are taken
in each run and yield an image of optical density (OD) which shows the amount of absorption. Fitting to thisimage gives the size and density of atomic cloud, which are used to deduce total number of atoms. Temperatureof atoms is known by taking a several images changing the time of flight: the speed of spread of an atomic
cloud reflects its temperature. Here we briefly see how a temperature of atoms is derived from the measured
size of an atomic cloud. First we assume that the spatial distribution of trapped atoms has a Gaussian shape:f 0 (r ) = (2π )−3 2(σ σ σ ) exp(− x 2σ − y 2σ − z 2σ )
−1

x

y

2

z

2
x

2

2
y

2

2
z

(3.5)

We also assume that atoms obey the Maxwell-Boltzmann distribution⎛ 2πk B ⎞f MB (v ) = ⎜
⎟
⎝ m ⎠

−3 2

(T T T )

−1 2

x

y

z

⎡ m ⎛ v 2 v y2 v 2 ⎞⎤⎜ x + + z ⎟⎥
exp⎢−
⎜
2
k
Ty Tz ⎟⎠⎦⎥B ⎝ Tx⎣⎢This distribution means that the probability that atoms located at the origin at t = 0 are found at r after a time offlight of t is
⎛ 2πk B t 2 ⎞⎟⎟f MB (r, t ) = ⎜⎜
⎝ m ⎠

−3 2

(T T T )

−1 2

x

y

z

⎡
m ⎛⎜ x 2 y 2 z 2 ⎞⎟⎤
exp⎢−
+
+
⎥
2
⎢⎣ 2k B t ⎜⎝ Tx Ty Tz ⎟⎠⎥⎦

(3.6)

Taking a convolution of (3.5) and (3.6), we obtain the spatial distribution of atoms after time of flight of t:f (r, t ) = dx′dy ′dz ′f 0 (r ′) f MB (r − r ′, t )

∫

(3.7)

As is easily known from the form of (3.5) and (3.6), (3.7) is written as a product of x, y and z components. Here
we only deal with calculations for the x component in detail. The y and z components are also calculated in asimilar manner. Carrying out an integral on x’ of (3.7), we obtain an expression for the distribution in the x
direction:⎡ ⎛k T t 2 ⎞⎤f x (x, t ) = ⎢2π ⎜⎜ σ x2 + B x ⎟⎟⎥
m ⎠⎦⎥
⎣⎢ ⎝

−1 2

⎞
⎛
x2
⎟⎟
exp⎜⎜ −
22
⎝ 2σ x + 2k BTx t m ⎠(3.8)

If we assume that the measured atomic distribution after a time of flight of t has a similar form to (3.5), themeasured width of atomic cloud σis related to the temperature T as

49

σ i2 (t ) = σ i2 +k BTi 2tm

(3.9)

where i = x, y and z. The expression (3.9) is used to derive a temperature of atoms in the whole experiments.
Typical arrangement of the optical layout for an absorption imaging is shown in figure 3.21. The image ofatomic clouds is focused onto the CCD. It is noteworthy that the fluorescence scattered by atoms is also
focused to the CCD when the imaging is successful. It reduces OD of the atomic clouds when the MOT light is
not completely turned off at the time of the imaging. Thus it is important to reduce the fluorescence as far as
you can. We pay attention to the next two points. First, the first lens should be placed far from a MOT. This
contributes to reduce the amount of the fluorescence on the CCD quadratically as the distance between the
MOT and the lens increases. Second, an iris should be inserted at the focus of the probe beam. The probe beam
which has information on the OD of atomic clouds can pass through this iris, whereas most of the fluorescenceis cut off at this iris.
Iris
Probe light

CCD

Fig.3.21 Absorption imaging with an iris
The shadow of the atomic cloud is cast on the CCD. The probe light passes through the iris, whilemost of the MOT fluorescence indicated in the red line is blocked by the iris. Note that the shadow
is also focused at the CCD and not at the iris.
In actual experiments, we always have to note the value of OD: if it exceeds 3, then it might be not correct. It
is easily understood by considering the actual value recorded on the CCD. If you use 16 bit CCD, the recorded
value of the probe beam without atoms is kept around 10000. In the case of imaging of OD 4, the recorded
value of the transmitted light is 10000/e4 which is around 180. This value is close to the value of backgroundnoise. Thus too high OD causes a saturation of the absorption image. If OD seems to be too high, it is necessary
to detune the probe beam from the resonance. In that case, the actual OD is easily obtained by multiplying thefollowing factor:⎛ 2Δ ⎞
1+ ⎜
⎟
⎝ Γ ⎠

2

which is derived from (3.1) for sufficiently low intensity light.

50

3.6 Ion counting with a microchannel plate (MCP)
Ion counting offers a sensitive tool for detecting atomic and molecular ions. Even one atom or molecule canbe detected with an electron multiplier tube just as in the case of photon counting. Here its principle and basic
operation are briefly described.
An MCP has a lot of channels which work as electron multipliers independently. Figure 3.22 shows themechanism of detecting ions with an MCP. First ions are accelerated by an electric field applied between MCP
and the ground. They collide to MCP to generate secondary electrons with a quantum efficiency of around 50%[114]. These electrons are further multiplied by repeating collision to the surface of MCP and generation of
secondary electrons. Finally a single ion generates 105~106 secondary electrons in each tube which arecollected to an anode. The resulting signal is sent through a low pass filter to suppress fast noises and thenamplified with a home made preamplifier [115] and acquired by a computer.
Single channel(Diameter 12μm)

Chamber surface

Anode
Positive ion

Preamplifier

e-

-2kV

-0.1kV

GND

Fig.3.22 Detection of positive ions with an MCP
Here only one channel is shown. Actually there are many channels in an MCP.
Note that the output voltage of MCP is determined by a capacitance of MCP in a pulsed operation. When acharge Q is generated at a single shot, the output voltage V from MCP which has a capacitance of C is written
as
V =

Q
C

and does not depend on an output load. This is not the case for a continuous operation: MCP works as aconstant current source in that case and the output voltage of MCP is determined by the resistance connected tothe anode.

51

Chapter 4

Results and analysis

Since I was engaged in the two experiments, E1 and E2, here the results obtained at each experiment aredescribed separately. In 5.1, the result of the first experiment on potassium at our lab is presented. We tried
magneto-optical trapping of 39K as a first step and evaluated its properties. Especially we studied the efficiency
of a double MOT system of 39K, where atoms are first collected from dispensers at 1st MOT and thentransferred to the second MOT by a resonant laser to be recaptured and used for subsequent experiments,
because there are a few reports on the double MOT system for potassium. We showed that a continuous double
MOT system is also efficient for 39K as rubidium atoms.
Section 4.2 presents our main results on the laser cooling of 41K atoms. More detailed investigations on theproperties of a 41K MOT is carried out with absorption imaging than that of 39K. We confirmed that laser
cooling of 41K is quite less efficient than other alkali atoms, as is already reported. At a particular condition thattrapping and repumping light induces a Raman transition between two ground state hyperfine levels, we sawthat number of atoms in a MOT is quite large while density is low and temperature is high, where the phasespace density is quite low compared with other alkali atoms. However, it is found out that a time dependent
MOT can improve the phase space density of the original continuous MOT by three orders.
Section 5.3 describes the first result at E2 experiments. We have characterized a dual species MOT of 41K
and 87Rb. Atoms and molecules are ionized by a doubled pulsed Nd:YAG laser and then detected with a MCP.
Utilizing this method, we performed photoassociation spectroscopy of 41K87Rb molecules.

4.1 Laser cooling of 39K at E1
4.1.1 Number of atoms measured with fluorescence
As there is little information on laser cooling of 39K, first we measured number of atoms with fluorescencefrom the MOT as a function of intensity and detuning of the MOT light. Figure 4.1 and 4.2 shows therespective results.

52
9

2

x 10

1.8

Number of atoms

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

20

40

60

80

100

120

140

160

Total intensity of trap beam [mW/cm2]
9

2.2

x 10

2

Number of atoms

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

5

10

15

20

25

30

35

40
2

Total intensity of repump beam [mW/cm ]Figure 4.1 Number of atoms against intensity of trapping and repumping beam

53(a) Trapping dependency

9

2.5

x 10

half intensity
max intensity

Number of atoms

2

1.5

1

0.5

0
- 80

- 70

- 60

- 50

- 40

- 30

- 20

- 10

0

-10

0

Detuning [MHz](b) Repumping dependency

9

3

x 10

half intensity
max intensity

Number of atoms

2.5

2

1.5

1

0.5

0
-70

-60

-50

-40

-30

-20

Detuning [MHz]Figure 4.2 Number of atoms plotted against detuning of (a) trapping (b) repumping beam
As is clearly seen from figure 4.1, higher intensity of the MOT beam leads to larger number of atoms forboth trapping and repumping laser. Though trapping laser often shows this tendency for other alkali atoms, it isunusual for repumping laser among other alkali atoms. We consider it to be caused by an unusual high54
hyperfine pumping rate between ground state F=2 and F=1 levels as already suggested by [54]. It is noteworthy
that number of atoms actually trapped in the MOT is comparable to that of other alkali atoms, which isprobably due to our large beam diameter and large capture velocity of 39K which can compensate the low
friction force in our 39K MOT.
A clear dependency on the detuning of the MOT light in figure 4.2 shows there is a good condition in
number of atoms both for trapping and repumping light, though there is no real state in this region of the atomic
structure. Though we have not investigated this behavior in detail here, we can see that the width of the peak in
number of atoms decreases when the intensity of the MOT light is reduced. Later a similar behavior wasobserved also for 41K, and we specified that this phenomenon occurs when trapping and repumping laser
satisfies a Raman condition that connects ground state F=2 and F=1 levels. Actually the width of the peaks innumber of atoms in figure 4.2 well coincides with the linewidth of the corresponding Raman transition.
Although this kind of behavior has not yet reported for other atomic species, previous experimental works forpotassium atoms have probably used this resonance to enhance number of atoms and the theoretical calculation[62] seems to have suggested this phenomenon.

4.1.2 Properties of the 2nd MOT
Before detailed characterization and optimization of the push beam, we briefly studied the properties of the
2nd MOT. The situation is quite different from the 1st MOT: room temperature hot atoms are captured in the
1st MOT, while the 2nd MOT captures laser cooled atoms accelerated by a resonant laser. Here we changed
only the intensity of the MOT laser and did not change the detuning since the laser source is the same as the 1st
MOT. Figure 4.3 shows number of atoms in the 2nd MOT measured against the intensity of the 2nd MOT laser.
It is clear that high intensity is not necessary for the 2nd MOT to trap atoms from the 1st MOT. This means thateven accelerated atoms have the velocity within the capture range of moderate intensity. We can see that a large
capture range of 39K which is already reported in [54] is advantageous to a double MOT configuration.
8

18

x 10

16

Number of atoms

14
12
10
8
6
4
2
0

0

50

100

150

200

250

300

Power [mW]
Fig.4.3 Number of atoms in the 2nd MOT against MOT beam power

For estimation of the minimum diameter required to trap atoms at the 2nd MOT, we measured number ofatoms against the beam diameter of MOT light. The beam diameter is changed by inserting an iris into the55
MOT beam. Knowledge on a minimum diameter for the MOT was necessary to design and make a magnetic
trap which was used to cool atoms evaporatively to make a condensate. Figure 4.4 shows the change in numberof atoms when the diameter of all 6 MOT beams is changed equally. On the other hand, figure 4.5 shows thechange in number of atoms when the diameter of one of 6 MOT beams is changed. The experimental
configuration of figure 4.5 is explained in figure 4.6.
250

Number of atoms [a.u.]

200

150

100

50

0
15

20

25

30

35

40

45

50

MOT beam diameter [mm]Fig.4.4 Number of atoms against the beam diameter of the MOT light

Number of atoms [a.u.]

250

200

150

100

50

0

5

10

15

20

25

30

35

40

45

50

MOT beam diameter [mm]Fig.4.5 Number of atoms against the diameter of the MOT beam
Green circle points indicate the decelerating beam is modified and blue triangle
points indicate the beam not relevant to deceleration is modified.

56
Beam irrelevant to deceleration
Decelerating beamgravity

Pushed atoms

Beam irrelevant to deceleration

2nd cell

Decelerating beam

Fig.4.6 Experimental configuration of figure 4.5
The two left beams are not relevant to deceleration of pushed atoms, while two right beams
contribute to deceleration. Figure 4.5 shows the beam diameter of the right beams is important toefficiently catch atoms.
Figure 4.4 shows that the beam diameter of around 30mm is sufficient to capture atoms at the 2nd MOT. We
can confirm from Figure 4.5 that this minimum diameter is mainly determined by the decelerating beam, andthus other 4 beams could have a smaller diameter around 15mm.

4.1.3 Efficiency of transferring atoms
We have performed detailed investigations on a double MOT configuration of potassium atoms since there
are so far limited reports on it. Here the problem is that atoms confined in a MOT have comparatively hightemperature around mille Kelvin as reported in [61]. This reduces the efficiency of transferring atoms from the
1st MOT to the 2nd MOT since the atomic beam spread out at the 2nd MOT if the transverse temperature of theatomic beam is high. Actually they reported that a temporal cooling process at the 1st MOT before transfer
improves the transfer efficiency by one order. It is noteworthy that we first tried continuous pushing scheme
which gave a reasonable efficiency, while Prevedelli et al. mentioned that pulsed pushing is more efficient thancontinuous one. We have also tried pulsed pushing but it did not give a good result. Our difference from themprobably lies in the focus point of the push beam: we usually focus the push beam to the center of the 1st MOT,
while they focus it to the center of the 2nd MOT.
We first define a loading rate L of a MOT as follows:

L=

N
T

where N denotes a final number of atoms in the MOT and T denotes the exponential time constant of the MOT
loading. This value is later used as a measure of the efficiency of capturing atoms both in the 1st MOT and 2nd
MOT. In the case of the 2nd MOT, it also represents the flux of atomic beam from the 1st MOT if we assume
all the pushed atoms are captured at the 2nd MOT.
We have evaluated the velocity of atoms pushed by a resonant laser under various conditions. This is57
accomplished by measuring a time lag between the time at which the push beam is turned on and the time atwhich number of atoms in the 2nd MOT begins to increase. Dividing the length between the 1st MOT and the
2nd MOT by the time lag, we can know the velocity of the fastest atoms directly. Figure 4.7 shows a typicalresult obtained in this way.

0.035

6

0.03

4
Push TTL [V]

Number of atoms in the 2nd MOT [arb.u.]

Push TTL

Number of atoms in the 2nd MOT
0.025

2

0.02
-0.1

-0.05

0

0.05

0.1

0.15
Time [s]

0.2

0.25

0.3

0.35

0
0.4

Fig.4.7 Typical result of the velocity measurement
There is a time lag between the time the push beam is turned on and the timenumber of atoms begins to increase.

Using the above method, we measured the velocity of atoms and the loading rate of the 2nd MOT against thepower and the detuning of the push beam both for a colinear push beam and a focused push beam at the 1st
MOT. Here we have employed repumping light as a push beam and not trapping light, since trapping light did
not give an efficient transfer. Figure 4.8 and 4.9 show the detuning dependency and power dependency of thevelocity and the loading rate for a colinear beam whose diameter is 2mm, respectively.

58
7

x 10

30

8

24

6

18

4

12

2

6

0
-40

-30

-20

-10

0

10

20

Velocity [m/s]Loading rate [atoms/s]

10

0
30

Detuning [MHz]Fig.4.8 Velocity of pushed atoms (green triangles) and loading rate at the 2nd MOT against
detuning of push beam (blue circles)

7

30

8

24

6

18

4

12

2

6

0

0

0.5

1

1.5

2

Velocity [m/s]Loading rate [atoms/s]

10

x 10

0
2.5

Power [mW]
Fig.4.9 Velocity of pushed atoms (green triangles) and loading rate at the 2nd MOT against power ofpush beam (blue circles)

59
We can see following three things from these figures:
1. There is a general tendency that a nearly resonant push beam leads to a low loading rate, while there is apeak at a few tens of MHz detuning both to the red and to the blue. It is important that the peaks movetoward a large detuning as the power increases. These facts can be interpreted as follows. A resonant light
breaks the balance of the MOT light and thus reduces number of atoms and density in the 1st MOT, whichleads to a low flux. If you detune the push beam from the resonance, number of atoms and density recovers.
In this case, power broadening due to the high intensity of the push beam allows atoms in the MOT to beaccelerated by absorbing the push beam. This is easily confirmed by considering the actual value of thepower broadening. We can see that the peak shifts by about 10MHz when the power is increased from
1.28mW and to 2.75mW. These two values of the beam power correspond to around 24Is and 52Is,
respectively. These values give the difference of about 7MHz, which is consistent with the observed shift.
2. A loading rate increases as the velocity of the atoms increases except the push power is too low. This isbecause the transverse spread of the atoms at the 2nd MOT is small when they get there. Too low power ofthe push beam leads to a high loading rate due to its small effect on the beam balance of the MOT light.
3. A blue detuned light gives higher velocity and loading rate than a red detuned light. This is easily
understood as follows. If a red detuned light is shone to the atoms, they are accelerated to become off
resonance due to their Doppler shift. However, accelerated atoms become close to the resonance in the caseof a blue detuned light. Thus a blue detuned light can push atoms more than a red detuned light. Here theacceleration stops when atoms come out of the MOT light and undergo hyperfine pumping to the F=2 stateby the push beam which did not affect F=2 atoms.
The conclusion from the above arguments is that a light detuned to the blue by about 40MHz with the powerof around 3mW is the best for a colinear push beam.
Next the results of the similar experiment for a focused beam whose diameter is 0.15mm at the 1st MOT ispresented. Figure 4.10 and 4.11 show the detuning dependency and power dependency of the velocity and theloading rate, respectively, while figure 4.12 shows the detuning dependency of the loading rate for two different
values of the beam power.

60
8

x 10

20

3

15

2

10

1

5

0
-40

-20

0

20

40

Velocity [m/s]Loading rate [atoms/s]

4

0
80

60

Detuning [MHz]Fig.4.10 Velocity of pushed atoms and loading rate at the 2nd MOT against detuning of push beam
Here the push beam is focused to the 1st MOT. The power of push beam is kept to be 0.5mW.

8

25

3.2

20

2.4

15

1.6

10

0.8

5

0

0

0.5

1

Velocity [m/s]Loading rate [atoms/s]

4

x 10

0
1.5

Power [mW]
Fig.4.11 Velocity of pushed atoms and loading rate at the 2nd MOT against power of push beam
Here the push beam is focused to the 1st MOT. The detuning of the push beam is

61

8

4

x 10

Loading rate [atoms/s]

3.5
3
2.5
2
1.5
1
0.5
0
- 80

- 60

- 40

- 20

0

20

40

60

80

Detuning [MHz]Fig.4.12 Dependence of loading rates on detuning for two different powers
Green triangles and blue circles indicate the beam power of 0.5mW and 2.0mW, respectively.
Though a tendency similar to the previous result is observed, the observed values are different from theprevious results: the loading rate is higher and the velocity is lower. The low velocity seems to suggest that
atoms are moving in the transverse direction when they are accelerated, and thus higher velocity is obtainedwith the larger diameter of the push beam. The high loading rate is due to its small effect on the balance of the
MOT light.
Based on the above results, we have decided to adopt a focused beam for pushing atoms. We use a blue
detuned light by 20MHz with a moderate power around 2mW for later experiments. The polarization of thepush beam did not cause a significant change in the above results.
Here we define another nondimentional index which represents the efficiency of pushing atoms:
P = L2 L1
where L1 and L2 denotes the loading rate of the 1st MOT and the 2nd MOT, respectively. If this value reaches
unit, all the atoms collected at the 1st MOT are transferred and captured at the 2nd MOT. Of course, thissituation is unreal and the P value is usually ranged in 0.1~0.4. Since it also reflects the alignment of the pushbeam, hereafter we have used this factor as an index of the efficiency of the push beam. Figure 4.13 shows thelinearity of L1 and L2 against the loading rate of the 1st MOT. Here the alignment of the push beam was kept
unchanged and only the amount of the background potassium atoms was changed by increasing the dispenser
current.

62
8

14

x 10

Loading rate at the 2nd MOT [atoms/s]

12

10

8

6

4

2

0
0

0.5

1

1.5
2
2.5
3
3.5
Loading rate at the 1st MOT [atoms/s]

4

4.5

5
9

x 10

Fig.4.13 Comparison of the loading rates at the 1st and 2nd MOT
The slope gives the efficiency of pushing atoms. The efficiency depends on alignment, powerand detuning of the push beam. Here the slope is kept constant around 0.3 over wide range.
We checked the dependence of P on the push beam diameter as shown in the figure 4.14. We can see that
smaller diameter is advantageous. We employed the diameter of 0.35mm for later experiments.
0.4
0.35

Push efficiency

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.5

1

1.5
2
Diameter of push beam [mm]

2.5

3

3.5

Fig.4.14 Push efficiency against the diameter of push beam at the 1st MOT

63
Next we investigated the transverse distribution of the pushed atoms, which is important for designing thedifferential pumping tube inserted between the 1st MOT and the 2nd MOT. The differential tube should have asmaller diameter for obtaining a higher vacuum at the 2nd MOT, though it might limit the number of atoms
captured at the 2nd MOT if the spread of the atomic beam is comparable to the inner diameter of the tube. Thus
it was necessary to measure the spatial profile of pushed atoms, which is performed with a configuration
illustrated in figure 4.15.

1st cell (top view)

Pushed atoms

gravity

MOT
Push beam

Beam cutter(The height is changed)
Resonant blowing light

1st cell (side view)
Resonant blowing light(Partially blocked)
MOT

gravity

Push beam

Beam cutter(The height is changed)

Fig.4.15 Configuration to measure the spatial distribution of pushed atoms
The beam cutter blocks a part of the blowing light. Only atoms which pass through the region
where blowing light does not exist can reach the 2nd cell. Thus number of atoms captured at the
2nd MOT reflects the atomic flux in the blocked region. Changing the height of the beam cutter,
the spatial distribution is obtained.
First pushed atoms are completely blown out by a resonant beam which has a large diameter of around
15mm. At this time, there are no atoms in the 2nd MOT. Then a part of the blowing beam is blocked by a black
paper. This enables atoms which pass through the region where the blowing beam is absent to reach the 2nd
MOT. Measuring the number of atoms in the 2nd MOT yields the integrated distribution of the pushed atoms.
Actually we block the atomic beam by blocking the lower side of the blowing beam. Figure 4.16 shows themeasured results of the number of atoms collected in the 2nd MOT when the height of the blocking paper isgradually increased. We performed experiments at two different distances near the 1st MOT, as there is no
window to introduce the blowing beam near the tube. From the two observed distribution, we can estimate thetransverse spread of the atoms.

Number of atoms in the 2nd MOT [arb.u.]

Number of atoms in the 2nd MOT [arb.u.]

64

1000
800

5cm from MOT (Exp)5cm from MOT (Fit)2

600

Atomic beam (1/e =2.18mm)

400
200
0
95

96

97

98
99
Beam height [mm]

100

101

102

100

101

102

800
700
600

4cm from MOT (Exp)4cm from MOT (Fit)

500

Atomic beam (1/e =1.94mm)

2

400
300
200
100
0
95

96

97

98
99
Beam height [mm]Fig.4.16 Measured and fitted spatial distribution of pushed atoms
If we assume a Gaussian distribution for pushed atoms, the resulting curve is an error function. We fitted theexperimental results with an error function and obtained the width of 1/e2 radius of atomic beam. Since theexperiments are performed for the two different distances from the 1st MOT, the transverse temperature of
atoms could be estimated from the difference in the width. We deduced it as around 230 μK, which isconsistent with the temperature of an ordinary MOT. However, it is not consistent with [61]. They reported thatthe temperature of the 1st MOT of 39K exceeds 1mK, and thus a temporal cooling is effective to improve theflux of atoms. We have not yet ascertain this unexpected low transverse temperature, but there is a possibility
that atoms are cooled at the low intensity region of the MOT light when they are accelerated.
The above results enable us to estimate the width of the atomic beam at the 2nd MOT to be around 15mm.
Based on these values, we determine the diameter of the differential pumping tube and the size of the 2nd cell.

65

4.2 Laser cooling of 41K at E1
4.2.1 Number of atoms measured with fluorescence
We have measured number of atoms in the MOT of 41K against frequency of MOT light, intensity of MOT
light, magnetic field gradient and frequency of the push beam as done for 39K. The results are shown in figure
4.17 to 4.20. We measured only frequency dependence for the 1st MOT. All parameters related to the 1st MOT
are optimized to have a large number of atoms in the 2nd MOT. Though there are no detailed measurements,
the maximum push efficiency obtained is around 0.1, which is much lower than the case for 39K. This seems toindicate that the transverse temperature of pushed atoms of 41K is higher than that of 39K.
In the frequency measurements, we can see a similar tendency as previously observed for 39K. There is apeak where number of atoms is large. This frequency again corresponds to a Raman condition where thetrapping and repumping light hit the same energy level.
The main difference from the previous 39K experiments lies in the intensity dependence of number of atoms.
Figure 4.19 shows higher intensity leads to larger number of atoms even at the highest intensity. This behavior
is quite different from that of 39K, where maximum number of atoms is obtained at low intensity. This suggeststhat the velocity capture range of 41K is small compared to that of 39K. We consider that it is a result of a higher
hyperfine pumping rate between the ground state F=1 and F=2 due to even smaller hyperfine splitting of theexcited states of 41K compared to 39K.
The subsequent figures are used to optimize number of atoms in the 2nd MOT. These results for 41K shownearly the same tendency to 39K. The optimum magnetic field gradient which gives maximum number of atoms
largely depends on the optical alignment. A better alignment leads to shift the peak of number of atoms to lower
magnetic field gradient. Typical our value of magnetic field gradient is 10G/cm for the 1st MOT, whereas as
low as 3G/cm for the 2nd MOT.

66(a) Trapping dependency

8

14

x 10

Number of atoms

12

10

8

6

4

2

0
- 60

- 50

- 40

- 30

- 20

- 10

0

Detuning [MHz](b) Repumping dependency

8

16

x 10

14

Number of atoms

12
10
8
6
4
2
0
- 70

- 60

- 50

- 40

- 30

- 20

- 10

0

10

Detuning [MHz]Fig.4.17 Number of atoms in the 1st MOT against detuning of (a) trapping (b) repumping
The repumping frequency is fixed at a detuning of -22MHz in (a) while the trapping
frequency is fixed at a detuning of -32MHz in (b).

67(a) Trapping dependency

8

Number of atoms

15

x 10

10

5

0
- 60

- 50

- 40

- 30

- 20

- 10

0

Detuning [MHz](b) Repumping dependency

8

14

x 10

Number of atoms

12
10
8
6
4
2
0
- 40

- 35

- 30

- 25

- 20

- 15

- 10

-5

0

Detuning [MHz]Fig.4.18 Number of atoms in the 2nd MOT against detuning of (a) trapping (b) repumping
The repumping frequency is fixed at a detuning of -20MHz in (a) while the trapping
frequency is fixed at a detuning of -16MHz in (b).

689(a)Trap intensity dependency

x 10

2.5

Number of Atoms

2
1.5
1
Repump = 18mW//cm

0.5
0Repump = 10mW/cm
0

5

10
15
Trap intensity [mW/cm2]

9

2

20

2

2

25

(b) Repump intensity dependency

x 10

Number of Atomd

1.5

1

0.5

0
2

4

6

8
10
12
14
Repump intensity [mW/cm2]

Trap = 26mW/cm

2

Trap = 15mW/cm

2

16

18

20

Fig.4.19 Number of atoms in the 2nd MOT against intensity of (a) trapping (b) repumping
8

Number of atoms in the 2nd MOT

10

x 10

9
8
7
6
5
4
3
2
1
- 50

- 40

- 30

- 20

- 10

0

10

20

30

40

Detuning from F=1→F’=2

Fig.4.20 Number of atoms in the 2nd MOT plotted against detuning of push beam
Here number of atoms in the 2nd MOT reflects the atomic flux from the 1st MOT.

69

4.2.2 Characterization with absorption imaging
For subsequent experiments, we required data on the temperature and density of atomic clouds as well as
number. We utilized an absorption imaging to measure these properties. It is noteworthy that the measurementsof temperature and density with the fluorescence image of a MOT focused to the CCD did not give a goodresult, since it required shining MOT lasers to the atomic clouds intense enough to record an image on the CCD.
In our setup, this process has an effect on the atomic cloud: the laser necessary to obtain a fluorescence image
inevitably cools and confines atoms. Thus we did not use the fluorescence imaging in the later experiments.
As a preparation, we have checked the resonance of the cycling transition F=2→F’=3 for a correct imaging.
OD is measured against the frequency of the probe beam and shown in figure 4.21. The width of the peak wellcoincides with the atomic natural width, which means that our imaging is correct. The fitted center of the peakis defined to be the resonance and used in the later experiments.

4
Measured peak OD of 41K atomic cloud
Lorentzina fitting (center 89.44MHz, FWHM 4.2MHz)

OD

3

2

1

0
80

82

84

86
88
90
92
AOM frequency [MHz]

94

96

98

Fig.4.21 Optical depth against probe frequency
The resonant frequency is obtained by fitting the absorption spectrum with Lorenzian shape. The small
peak near 83MHz corresponds to F=2→F’=1,2 transition.
We investigated the properties of the 41K MOT in the steady state since temperature seemed to be too highand the density seemed to be too low. The results are shown in figure 4.22 to 4.25. In any condition, thetemperature is very high compared to other atomic species, while the density is very low. The resulting phasespace density of the order of 10-10 is much lower than other species by more than three orders. This means that
41

K is not suitable for a condensate experiment, as it needs a high phase space density at the stage of laser

cooling to reach quantum degeneracy.

70

2
1.5
1
0.5

0
- 50
5

- 30

- 20

- 10

0

- 40

- 30

- 20

- 10

0

0
- 50
10

- 40

- 30

- 20

- 10

0

8

x temperaturey temperature

- 20

- 10

0

−3
Temperature [mK] Density [cm ]Radius [mm]Number of atoms

9

x 10

4

- 40
x radius
y radius

3
2
1- 50 10
x 10
2

1.5
1
0.5

6
4
2
- 50

- 40

- 30

Detuning [MHz]Fig.4.22 Number, radius, density and temperature of atoms against trapping detuning

71

Radius [mm]Number of atoms

9

2
1.5
1
0.5

0
1
2

2

3

4

5

6

7

1.5
1

x radius
y radius

0.5
0Temperature [mK] Density [cm−3]

x 10

2

1
10
x 10

2

3

4

5

6

7

1

2

3

4

5

6

7

1.5
1
0.5
0
15
x temperaturey temperature

10
5
0

1

2

3

4

5

6

7
2

Intensity of trapping beam [mW/cm ]Fig.4.23 Number, radius, density and temperature of atoms against trapping intensity

72

Number of atoms

9

4

x 10

3
2
1

Temperature [mK] Density [cm−3]

Radius [mm]

0
- 30
5

- 25

- 20

- 15

- 10

-5
x radius
y radius

4
3
2

1- 30 9
x 10
10

- 25

- 20

- 15

- 10

-5

- 25

- 20

- 15

- 10

-5

8
6
4
2
- 30
40

x temperaturey temperature

30
20
10
0
- 30

- 25

- 20

- 15

- 10

-5

Detuning [MHz]Fig.4.24 Number, radius, density and temperature of atoms against repumping detuning

73

Radius [mm]Number of atoms

9

2

x 10

1.5
1
0.5
0
0
2.5

0.5

1

1.5

2

2.5

3

3.5

x radius
y radius

2
1.5
1

−3
Temperature [mK] Density [cm ]

0.5
2

0
10
x 10

0.5

1

1.5

2

2.5

3

3.5

0

0.5

1

1.5

2

2.5

3

3.5

1.5

2

2.5

3

1.5
1

0.5
8

x temperaturey temperature

6
4
2

0

0.5

1

3.5
2

Intensity of repumping beam [mw/cm ]Fig.4.25 Number, radius, density and temperature of atoms against repumping intensity
The notable feature observed in the frequency experiments is the peak ranged in the repumping detuning ofaround 20MHz and trapping detuning of around 20MHz. Here the temperature exceeds 10mK and the density
reaches down to around 109/cm3, while nearly tripled number of atoms is obtained compared to a normal
condition. There are no atomic energy levels in this frequency range. As already mentioned above, this resulted
from a Raman condition of the trapping and repumping laser. In practice, the peak in the experiment on the

74
detuning of trapping laser shifts when the detuning of repumping laser is shifted, and vice versa. An example ofthe image of atomic clouds at the Raman condition and at the normal condition is shown in figure 4.26. We cansee that density at the center of the MOT is low at the Raman condition, which is quite a peculiar behavior
among other atomic species.

Fig.4.26 Images of atoms at Raman condition (left) and non-Raman condition (right)
These images are taken at the same condition except for the frequency of repumping light. It is clearlyseen that atoms tend to escape from the center of the MOT at Raman condition.
We have tried to use this resonance to enhance number of atoms, but it leads to too high temperature and too
low density which cannot be compensated in the subsequent experimental stage. Thus we decided to employ a
non-Raman condition to obtain a higher phase space density. With this selection of the frequency, we can
improve the phase space density by one order at the cost of number of atoms. The intensity of each beam is
kept to be the maximum value as it is not so sensitive to the various properties of the MOT.

4.2.3 Time dependent MOT
The above results show that 41K cannot be used for the subsequent condensate experiments directly since it is
impossible to catch hot and low density atoms efficiently by another trap such as a magnetic trap or an optical
trap. Thus we tried a time dependent MOT which is originally developed for Rb atoms and investigated for 39K
and 41K atoms [61].
Our scheme consists of two stages: first atoms are compressed by raising the magnetic field gradient andthen cooled by changing the detuning of both trapping and repumping laser close to the resonance and lowering
the intensity of both trapping and repumping laser.
Optimization of each stage is carried out by changing one of the many parameters (intensity, frequency andmagnetic field gradient) to obtain a higher phase space density one by one. Thus we achieved an improvementin the phase space density by two orders. Figure 4.27 shows experimental time sequence for the achieved best
condition. Figure 4.28 shows comparison of images between the steady state MOT and time dependent MOT
for different time of flight. We can clearly see the effect of compression and cooling: images with 0.1ms TOF
show original density distributions while images with 3ms include thermal expansion. The condition of atoms
at each step including the steady state and the Raman condition is shown in the table 4.1.

75

Fig.4.27 Experimental time sequences of TDMOT

Fig.4.28 Images of atoms (a) after the steady state MOT and (b) after the time dependent MOT
The image of the steady state MOT after TOF of 10ms is not shown here since it is widespread. The
images with TOF 0.1ms reflect the original density distribution while those with a long TOF are mainly
determined by the temperature of the atomic cloud.

76
Table 4.1 Measured properties at each step
Number [109]Density [1010cm-3]

Temperature[mK]Phase space density [10-7]

Raman

3

0.3

30

0.012

Steady state

1.6

1.5

3

0.1

Compression

1.5

12

0.5

1.8

Cooling

1.1

5.1

0.1

11

Achieved phase space density after the cooling stage is comparable to that of other alkali species such as
sodium or rubidium. We conclude that 41K could be used for a condensate experiment if the conditions on the
MOT light and magnetic field are changed from the steady state condition which collects a large number ofatoms to the condition which compress and cool atoms within a sufficiently short time. Here the achieved
temperature is lower than the Doppler limit of 41K, 149μK. This suggests that the sub-Doppler coolingmechanism can work even when hyperfine pumping rate is high due to the small hyperfine splitting of excitedstates, which have already been suggested for the case of 39K [62].
There are three important points to achieve the above condition.
1. Cooling is the most efficient at a small detuning. This is easily understood as a result of the Doppler coolingwhich is most effective at the detuning of Γ/2 (see expression 2.41). However, the sub-Doppler coolingseems to work for a sufficiently low intensity. This behavior is contrary to the case of Na, Rb and Cs forwhich the sub-Doppler cooling works well and the lowest temperature is obtained at a large detuning ofaround 60MHz.
2. The attainable minimum temperature seems to be limited by large number of atoms. In the last coolingstage, both trapping and repumping lasers are close to the resonance, which means that most of the light are
scattered by the atoms located near the outer surface of the MOT. Thus the atoms located in the center ofthe MOT do not feel any cooling force, which leads to high temperature. It is important to fall atoms to thedark state (F=1) by lowering the intensity of the repumping laser at the last cooling stage. We confirmedthat more than 90% of atoms are in the dark F=1 state after the cooling stage, but even such a low
population of F=2 can cause decay of cooling force. It is possible to achieve even lower temperature forsmaller number of atoms.
3. Atoms are very sensitive to the optical force in the last cooling stage because they feel the large force ofnearly resonant light. If there is an imbalance between two counterpropagating MOT beams or the center ofthe magnetic field is not matched to the center of the MOT beams, atoms are easily blown up and disappear.
The compensation coils for three axes are necessary to compensate the earth magnetic field and precisely
tune the center of the magnetic filed. Even the best alignment could lose about 30% of atoms in our case.

77

4.3 Spectroscopy with REMPI at E2
4.3.1 Dual species MOT of 41K and 87Rb
Since number of molecules formed by photoassociation is proportional to the product of densities of 41K and87
Rb, it is crucial to achieve high density for both species to perform molecular spectroscopy with a highsignal-to-noise ratio. Since we have already known the properties of the MOT of 41K, we have measured how
number and density of atoms in the MOT depend on the frequency and intensity of MOT light for Rb as donefor 41K. Figure 4.29 shows frequency dependence, while Figure 4.30 shows intensity dependence. Dependence
on the frequency of the repumping light was not shown here since it is small. We fixed the repumping
frequency to F=1→F’=2 transition.
(a) Number of atoms

9

2

x 10

(b) Density

10

9

1.8

x 10

8

Density [atoms/cm3 ]

1.4
1.2
1
0.8
0.6

7
6
5
4

0.4
3

0.2
0
- 35

- 30

- 25

- 20

- 15

- 10

-5

2
- 35

- 30

- 25

Detuning [MHz]

- 20

- 15

- 10

Detuning [MHz]Fig.4.29 Frequency dependence of (a) number and (b) density for 87Rb cooling light

9

x 10

(a) Cooling (number)

14

Density [atoms/cm3 ]Number of atoms

2
1.5
1
0.5
0

0

10

20

30

40

50

60

9

(b) Repumping (number)

6
4
0

1.4
1.2
1
0.8
0

1

2

3

Total intensity [mW/cm2 ]

30

40

50

x 10

(b) Repumping (density)

0

1

10

9

1.6

0.6

20

8

60

70

Total intensity [mW/cm2 ]Density [atoms/cm3 ]

x 10

10

10

Total intensity [mW/cm ]1.8(a) Cooling (density)

12

2

70

10

x 10

2

Number of atoms

Number of atoms

1.6

4

8.5
8
7.5
7
6.5

2

3

4

Total intensity [mW-cm3 ]Fig.4.30 Intensity dependence of the MOT of 87Rb for (a) cooling and (b) repumping-5

78
We can see clear differences between 41K and 87Rb: density of 87Rb atoms is higher and much lower intensity
suffices for repumping compared with 41K. These differences result from the fact that 87Rb can be regarded as a
two level system while 41K behaves as a three level system. Figure 4.31 shows level diagrams of 41K and 87Rb.
In the case of 87Rb, the large splitting between F’=3 and F’=2 relative to the atomic natural width (~6MHz)
allows us to treat F=2→F’=3 transition to be a two level system.
F’=3

F’=3
13MHz

266MHz

F’=2

F’=2

F’=1
F’=0

F’=1
F’=0

Cooling

Repumping

F=2

Cooling

Repumping

F=2
6.83GHz

254MHz
F=1

F=1

41K

(Γ=6.1MHz)87Rb(Γ=6.1MHz)

Fig.4.31 Level diagram of 41K and 87Rb
For 41K we avoided Raman condition mentioned above to achieve high density. Intensity is kept to the
maximum value for both trapping (50mW/cm2) and repumping (33mW/cm2) to obtain a large number of atoms.
For 87Rb we had a large detuning of 30MHz for trapping laser to achieve high density while repumping light isresonant. Intensity per each beam is around 8mW/cm2 for trapping and 0.5mW/cm2 for repumping.
Although we did not observe a loss of 87Rb atoms due to coexistence with 41K atoms, we observed a small
loss of 41K atoms due to coexistence with 87Rb atoms as observed for 39K and 85Rb [116]. According to theirwork, the main loss mechanism of K-Rb system at low intensity regime is heteronuclear hyperfine changing
collision and is not radiative escape or fine structure changing collision.

79

4.3.2 Ionization and detection of atoms and molecules
We observed ionized atoms and molecules with time-of-flight (TOF) mass spectroscopy. A typical TOF
spectrum is shown in figure 4.32. This data was taken without shining the photoassociation laser and turning off
the MOT laser and magnetic field. The first large signal is due to the pulse laser scattered at the glass window
and the internal surface of the chamber. Though each peak should have more complicated structure just after
the MCP due to variation in the arrival time, we obtained an integrated single peak for each species because all
signals passed through a low pass filter. It is clearly seen that K+, Rb+ and Rb2+ are separately detected at thedelay time proportional to square root of the mass. We confirmed that each peak appears only when the MOT
light for the corresponding atomic species is irradiated. There are no background atoms or molecules. We also
confirmed that Rb2+ peak exists even when the MOT light is turned off 300μs before a pulse laser is irradiated.
This ensures that we have detected ground state molecules. These molecules are supposedly formed by thetrapping laser of 87Rb.

Pulse laser- 0.12

MCP signal [V]- 0.14- 0.16- 0.18- 0.2

41- 0.22

87Rb+
2

K+
87Rb+

- 0.24
0

1

2

3

4

Tims [s]

5

6
-5

x 10

Fig.4.32 Time-of-flight spectrum of laser-cooled K, Rb and Rb2.
The first signal is caused by the scattered pulse laser. The subsequent three peaks
correspond to K+, Rb+ and Rb2+, respectively. This signal is obtained without
photoassociation laser.

80
Observed relative ratios between three peaks aret Rb t K = 1.44
t Rb t Rb = 1.42
2

These values well coincides with square root of actual mass ratios [117]m Rb mK = 1.457
m Rb m Rb = 1.414
2Though the TOF ratios are in good agreement with the mass ratios, the observed TOF’s are much longer thanwe expected. If we assume that the elected field gradient between MCP and an atomic cloud is constant, TOF is
estimated to bet=

2m(a + b )beV

where a is distance between an atomic cloud and the surface of the chamber opposite to the MCP, b is distance
between atomic cloud and the MCP, e is charge of electron and V is the voltage applied between the MCP and
the chamber. This expression gives TOF time of 3.4μs for 41K, 4.9μs for 87Rb and 6.9μs for 87Rb2. These
values are much smaller than actually observed. This unexpected behavior seems to indicate that the actual
electric field gradient is rather small near the atomic cloud due to the proximity to the chamber surface.
Here we give considerations on how atoms and molecules are ionized. In the above experiment, the pulse
laser was operated at around 602.6 nm which is used in previous works on Rb2 [35, 118, 119, 120, 121, 122]
and KRb [43, 123, 124, 125]. The wavelength 602.6 nm is near resonant with two-photon transitions
connecting the ground state atoms to the Rydberg state Rb(16D) and K(8D) and thus they are finally ionizedwith absorbing an additional photon. This means that atoms are ionized with three photon transitions. Theintermediate state at the first two-photon process is far away from the atomic transition for both K and Rb andthus the whole ionization efficiency is expected to be small for both species. The situation is quite different forRb2 molecule. The wavelength 602.6nm lies in the Rb2 diffuse band [126, 127] which is connecting the groundstate a3Σu+ to the excited (2)3Πg and the ground state X1Σg+ to the excited (2)1Σu+ as shown in figure 4.3.6.
Using (2)3Πg and (2)1Σu+ as an intermediate state, Rb2 are ionized to (1)2Σg+ of Rb2+ potential. It is clear thatRb2 molecules are ionized with two photon transitions in contrast to atoms. Here the intermediate state is nearlyresonant to a vibrational level and has a significant overlap between both the ground state and the excitedionized state as seen from figure 4.33. Thus the whole ionization efficiency is expected to be large. These
differences in the ionization process give a qualitative explanation to the counterintuitive results that observed
numbers of atoms and molecules are of the same order: molecules formed by photoassociation are much lessthan atoms in a MOT but ionization efficiency is much higher for molecules than for atoms.

81

4

4

x 10

3.5

Rb+
2

3

(1)2Σ+g

Energy [cm−1]

2.5

602.6nm2(2)1Σ+u1.5(2)3Πg1602.6nm
0.5

a3Σ+
u

0

- 0.5

X1Σ+
g
4

6

8

10

12

14

Interatomic distance [Å]Fig.4.33 Potential energy curves of Rb2 and Rb2+ molecule. Rb2 potentials are calculated with MOPLRO
and only 18 states dissociating to 5S+5S, 5S+5P and 5S+4d are shown. Here Δ states are omitted. Rb2+
potentials are taken from [128].
Although we have not yet succeeded in observation of KRb molecules, we give a consideration on how KRb
molecules are ionized. Figure 4.34 shows the interatomic potentials of KRb [107] and KRb+ molecule [129].
When the pulse laser of the wavelength 602.6 nm is irradiated, the ground state singlet molecules are ionized toform loosely bound Rb2+ through the intermediate (4 ) Σ
1+state. This process is expected to work well since

there is a significant overlap between the ground and the intermediate states and also between the intermediate
and the excited ionized states as in the case of Rb2. In contrast, triplet molecules are considered to be ionized82
with lower efficiency since both intermediate states (4 ) Σ
3+and (3) Π have a relatively small overlap
3

between the ground states and the ionized states. In practice, it is reported that detection of triplet KRb
molecules with the wavelength of 602.6 nm requires higher laser intensity [125].

4

4

x 10

(2)2Σ+

K+

3.5

Rb+
3

(1)2Σ+

602.6nm

Energy [cm−1]

2.52(4)1Σ+
(3)3Π

1.5

1

602.6nm(4)3Σ+

0.5

a3Σ+

0

X1Σ+
- 0.5

2

4

6

8

10

12

14

Interatomic distance [Å]Fig.4.34 Potential energy curves of KRb and KRb+. KRb potentials are taken from [107] while KRb+
potentials are taken from [129]. Blue, green, red and purple lines indicate 1Σ+, 3Σ+, 1Π and 3Π,respectively. Here Δ states are omitted.
We have also tried ionization of atoms and molecule at the pulse laser wavelength of 532nm which isseparated from the doubled Nd:YAG laser but we did not detect any molecule and observed only K and Rb
atoms. This fact is attributed to the low ionization efficiency due to the poor overlap of wavefunction between
the ground and intermediate and also the intermediate and excited ionized state.

83

4.3.3 Photoassociation spectroscopy of Rb2
We performed a photoassociation spectroscopy of Rb2 molecules to confirm that our system has a capability
to produce molecules with an additional photoassociation laser. Since we know that laser-cooled 41K atoms
have a high temperature and a low density at a steady state, we first tried photoassociation of 87Rb which has
been well studied so far. Our photoassociation laser is derived from a DFB laser which enables us to scan awide frequency range without a mod hop. 2mW of DFB laser is amplified to 200mW through a slave laser anda tapered amplifier. The photoassociation laser is focused at the center of MOT to a diameter of around 450μm.
This corresponds to an intensity of 320W/cm2.
We had not been able to detect any molecule at a steady state in the presence of a photoassociation laser. Thismeans that molecules which had been observed previously is formed by the MOT laser and not by the pulse
laser since the pulse laser is irradiated in both cases. Supposedly the non-resonant photoassociation laser pumps
molecules formed by the MOT laser back to atoms.
To enhance the molecule formation rate, we performed a compressed MOT (C-MOT) for Rb continuously.
Figure 4.35 shows our experimental time sequences of C-MOT. The whole compression procedures shown
below are synchronized to the pulse laser and repeated at 10Hz. This procedure increases the density of MOT
for around 10ms by one order (2×1010/cm3→2×1011/cm3) and is expected to enhance the molecules formation
rate by two orders.
Pulse laser
10ms

10ms

10ms

3.6mW/cm2
Repumpingtotal intensity

26μW/cm2-18MHzCooling
Frequency
47mW/cm2
Cooling
total intensity-28MHz29mW/cm2

Fig.4.35 Experimental time sequences of Rb C-MOT
With a C-MOT, photoassociation spectra are easily observed. Figure 4.36 shows a typical spectrum.
Observed linewidth of each peak is around 10MHz which probably reflects the linewidth of the DFB laser. The
spectrum in figure 4.36 shows a clear rotational structure and does not show any complicated hyperfine
structure. This means that this state has a symmetry of 0g－or 0u+. From this figure, we can see that the molecule
formation rate is greatly enhanced only when the photoassociation laser is resonant to a molecular excited state.

84

Fig.4.36 Photoassociation spectrum of Rb2

85

Chapter 5Conclusion and outlook

We have studied a double MOT system with 39K atoms. Velocity of pushed atoms is measured directly withthe time difference between the time when push beams are irradiated and the time when number of atoms in the
2nd MOT begins to increase. Measured velocity is ranged in 10~20m/s which is smaller than the capture
velocity of MOT of 39K. The transverse distribution of pushed atoms is measured by blowing out a part ofatoms with a resonant light. The atomic distribution measured at two different points revealed that thetransverse temperature is 230μK. This transverse temperature enables us to estimate the spread of atomic
beam at the 2nd MOT is 15mm. These data on longitudinal and transverse velocity of atoms indicate that mostof pushed atoms are captured at the 2nd MOT in our double MOT system. A focused beam with a diameter of
0.3mm is the most suitable for pushing atoms from the 1st MOT since it does not affect the 1st MOT seriously.
Though the ratio between the loading rate of the 2nd MOT and that of the 1st MOT is ranged in 0.3~0.4, thelifetime of the 2nd MOT is longer than that of the 1st MOT by one order. Thus resulting number of atoms in the
2nd MOT is larger than that of the 1st MOT by a factor of 2~3.
We have investigated propertied of MOT of 39K and 41K atoms. For 39K, we measured dependence of numberof atoms on the detuning and intensity of MOT light with a fluorescence method. We showed that number ofatoms increases as intensity increases. A large detuning is required to suppress hyperfine pumping to F=1 state.
We found that number increases by a factor of three at a detuning of Raman condition for cooling andrepumping light compared to an ordinary non-Raman condition. With a high intensity and a large detuning, wewere able to trap more than 5×109 atoms.
For 41K, we studied dependence of temperature and density on the detuning and intensity of MOT light withabsorption imaging. As in the case of 39K, number of atoms is enhanced at the Raman condition also for 41K. 3
×109 atoms are captured in a MOT at this condition. It is found that temperature of atomic cloud is quite high(~10mK) and density is quite low (~109 atoms/cm3) at the Raman condition. We also found that temperature islower (~3mK) and density is higher (~1010 atoms/cm3) at a non-Raman condition. In this stage, phase spacedensity of atomic cloud is relatively low (~10-10) compared to other alkali atomic species. To achieve a higher
phase space density, we have performed a time-dependent MOT. By increasing the magnetic field gradient, wewere able to increase the density by a factor of 5. By lowering the detuning and intensity of MOT light, thetemperature is reduced by a factor of 30. With this time-dependent MOT, the phase space density is improvedby 3 orders. The resulting phase space density of 10-6 is comparable to other alkali species such as Na and Rb.
We conclude that we have established a method of laser cooling of 41K atoms. The next goal of this experimentis to make a condensate of 41K and 87Rb and to form loosely bound Feshbach molecules of 41K87Rb.
We built up another machine to perform spectroscopy on loosely bound heteronuclear molecules formed byphotoassociation. A dual species MOT of 41K and 87Rb is constructed. Density and number of atoms aremeasured with absorption imaging for both 41K and 87Rb to maximize number of molecules formed byphotoassociation. At a steady state condition, we have trapped 2×109 87Rb atoms with a density of 5×
10atoms/cm3 and 5×108 41K atoms with a density of 2×1010atoms/cm3. Shining an intense pulse laser to thedual species MOT, we observed K+, Rb+ and Rb2+ separately with a microchannel plate (MCP). Relative ratios
between time-of-flights of K+, Rb+ and Rb2+ are in good agreement with the square root of their mass ratios.
Observed number of each species is estimated to be around 100 for 41K, 20 for 87Rb and 2 for 87Rb2. Thesefigures mean that our apparatus is sensitive enough to detect even a few molecules. With this apparatus, we areable to perform a precise spectroscopy on loosely bound 41K87Rb molecules formed by photoassociation. We
are going to search an optimum transition to transfer the loosely bound molecules to the lowest vibrational levelof the ground state singlet potential with a two photon transition. Detecting a resultant few molecules is also
possible with our sensitive machine.

86
Our final goal is formation of quantum degenerate heteronuclear molecules in their lowest vibrational level.
Applying the optimum transition found in the spectroscopy experiment to Feshbach molecules of 41K87Rb, weare going to achieve a stable Bose-Einstein condensate of heteronuclear molecules. Properties of dipolarcondensates will be investigated in the presence of optical lattices. Tunability of intermolecular potential withmicrowave will be demonstrated. We are going to study an ordered phase of dipolar condensates in opticallattices by tuning interaction between molecules.

87

Appendix A Circuits(1) Thermoelectric controller for a slave laser(2) Thermoelectric controller for a tapered amplifier(3) AOM driver88(4) High speed photo detector for locking(5) Protection circuit for a laser diode89(6) Lock circuit

90

Appendix B

Box design(1) Thermoelectric controller for a slave laser

91(2) Thermoelectric controller for a tapered amplifier

92(3) VCO and RF amplifier

93(4) AOM driver

94

Appendix C

Design of ECDL

95

96

97

Appendix D

Design of slave laser

98

99

Appendix E

Design of tapered amplifier

100

101

Appendix F Matlab code for the 6 level model
Each code should be saved as an M-file. The values of R, CE, f_lens, d_lens and distance depend on an
experimental condition. The code for 41K differs from that for 39K only in the detuning of each transition.
(a) 39K
function y=slevel39(It,Ir,d1,d2,pdvoltage)% about potassium39
hv=1.617% energy of one photon in eV

Is=1.77% saturation intensity in mW/cm2

gamma=6.09% natural linewidth in MHz/2pi% about photodetector
R=200000% resistor of the photodetector in ohm

CE=0.42% conversion efficiency in A/W

Current=pdvoltage/(1000*R)% current flowed from PD in A

Nphoton=Current*10^19/(CE*hv*1.6)% observed number of photons in /s% experimental configuration
i1=6*It/0.70882% intensity of a trap beam measured by Thorlabs Power Meter in mW/cm2

i2=6*Ir/0.70882% intensity of a trap beam measured by Thorlabs Power Meter in mW/cm2

det1=(d1-103.9)*2
det2=(116.1-d2)*2
f_lens=40% focal length of the lensd_lens=25.4% diameter of the lensdistance=100% distance between the lens and the MOT

separation=(distance*f_lens)/(distance-f_lens)% distance between the lens and the PD

tanc=(d_lens/2)/separation% tangent of the angle

omega=(1-1/sqrt(1+tanc^2))/2% solid angle

cell_loss=0.04% loss on the surface of the cell (1 path)% atom number in 6 level model
b=[1 5/6 1/2 0;0 1/6 1/2 1]% branching ratio matrix

c=[1/3 1 5/3 7/3;1/5 3/5 1 7/5]/3% oscillator strength of the S1/2->P3/2 transition

Int=0
Int=[i2 i2 i2 0;0 i1 i1 i1]/Is% intensity matrix

Det=0
Det=2*[det2-12.5 det2-9.3 det2 0;0 det1-30.3 det1-21.0 det1]/gamma % detuning matrix
R=0
R=(pi*gamma*10^6)*(b.*c).*(Int./(1+(Det.^2))) % exitation-rate matrix
s1=0
s2=0
for F=1:4% sum over upper hyperfine statess1=s1+(R(2,F)+2*pi*gamma*10^6*b(2,F))*R(1,F)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)s2=s2+(R(1,F)+2*pi*gamma*10^6*b(1,F))*R(2,F)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)end
p2ovp1=s1/s2
spFovp1=0% population ratio of the ground states

102
for F=1:4% sum over upper hyperfine states

spFovp1=spFovp1+(R(1,F)+R(2,F)*p2ovp1)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)end
Pe=spFovp1/(1+p2ovp1+spFovp1)% excited-state fraction

y=Nphoton/(2*pi*gamma*10^6*Pe*omega*(1-cell_loss)^2) % number of atoms(b) 41K
function y=slevel41(It,Ir,d1,d2,pdvoltage)% about potassium41
hv=1.617% energy of one photon in eV

Is=1.77% saturation intensity in mW/cm2

gamma=6.09% natural linewidth in MHz/2pi% about photodetector
R=200000% resistor of the photodetector in ohm

CE=0.42% conversion efficiency in A/W

Current=pdvoltage/(1000*R)% current flowed from PD in A

Nphoton=Current*10^19/(CE*hv*1.6)% observed number of photons in /s% experimental configuration
i1=6*It/0.70882% intensity of a trap beam measured by Thorlabs Power Meter in mW/cm2

i2=6*Ir/0.70882% intensity of a trap beam measured by Thorlabs Power Meter in mW/cm2

det1=(89.8-d1)*2
det2=(210.1-d2)*2
f_lens=50% focal length of the lensd_lens=25.4% diameter of the lensdistance=75% distance between the lens and the PD

separation=(distance*f_lens)/(distance-f_lens)% distance between the lens and the MOT

tanc=tan((d_lens/2)/separation)% tangent of the angle

omega=(1-1/sqrt(1+tanc^2))/2% solid angle

cell_loss=0.04% loss on the surface of the cell (1 path)% atom number in 6 level model
b=[1 5/6 1/2 0;0 1/6 1/2 1]% branching ratio matrix

c=[1/3 1 5/3 7/3;1/5 3/5 1 7/5]/3% oscillator strength of the S1/2->P3/2 transition

Int=0
Int=[i2 i2 i2 0;0 i1 i1 i1]/Is% intensity matrix

Det=0
Det=2*[det2-3.5 det2-3.4 det2 0;0 det1-17 det1-13.6 det1]/gamma % detuning matrix
R=0
R=(pi*gamma*10^6)*(b.*c).*(Int./(1+(Det.^2))) % exitation-rate matrix
s1=0
s2=0
for F=1:4% sum over upper hyperfine statess1=s1+(R(2,F)+2*pi*gamma*10^6*b(2,F))*R(1,F)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)s2=s2+(R(1,F)+2*pi*gamma*10^6*b(1,F))*R(2,F)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)end
p2ovp1=s1/s2% population ratio of the ground states

103
spFovp1=0
for F=1:4% sum over upper hyperfine states

spFovp1=spFovp1+(R(1,F)+R(2,F)*p2ovp1)/(R(1,F)+R(2,F)+2*pi*gamma*10^6)end
Pe=spFovp1/(1+p2ovp1+spFovp1)% excited-state fraction

y=Nphoton/(2*pi*gamma*10^6*Pe*omega*(1-cell_loss)^2) % number of atoms

104

Appendix G Properties of Potassium(a) Vapor pressure
The vapor pressure Pvapof potassium at a temperature T is expressed as
Pvap = 760 × 10 A+ B T(G.1)

where A = 4.961 and B = -4646 for potassium [130]. Figure G.1 shows the formula (G.1).
2

10

0

10

-2

Vapor pressure [Torr]

10

-4

10

-6

10

-8

10

-10

10

-12

10

0

50

100

150
200
250
Temperature [℃]

300

350

400

Fig G.1 Vapor pressure of potassium(b) Fundamental physical properties
Table G.1 Fundamental Physical Properties of Potassium
Atomic numberZ

19

Total Nucleons

Z+N

39, 40, 41

Relative Natural Abundanceη

93.2581(44)% (39),[130]

0.0117(1)% (40),6.7302(44)% (41)
Nuclear Lifetime

τnstable (39, 41),

[130]

9

1.26×10 yr (40)
Atomic massm

38.9637069(3) u (39)[130]

39.96399867(29) u (40)40.96182597(28) u (41)Density at 20℃

ρm

0.862 g/cm3[130]

Melting pointTM

63.38 ℃

[130]

Boiling point

TB

759 ℃

[130]

Specific Heat Capacity

cp

0.757 J/g K[130]

Molar Heat CapacityCp

29.600 J/mol K[130]

Vapor pressure at 25℃

Pv-8

2×10 Torr[130]

Nuclear SpinI

3/2 (39, 41), 4(40)Ionization limit

EI

35009.8140cm-1

B

[131]

105
(c) Optical properties
Table G.2 Potassium D2 (42S1/2→42P3/2) Transition Optical Properties
Frequency

ω02π・391.01617003(12) THz (39)[132]

2π・391.016296050(88) THz (40)2π・391.01640621(12) THz (41)
Wavelength (Vacuum)

766.70092180(24) nm (39)λ766.70067470(17) nm (40)766.70045870(24) nm (41)Lifetimeτ

26.34(5) ns

Natural Line WidthΓ

2π・6.042(11) MHz

Recoil temperatureTr

836.117 nK (39)[133]

815.190 nK (40)795.332 nK (41)
Doppler shift (v = 1m/s)

Δωd2π・1.30429 MHz

Doppler Temperature

TD

145μK

Table G.2 Potassium D1 (42S1/2→42P1/2) Transition Optical Properties
Frequency

ω02π・389.286058716(62) THz (39)[132]

2π・389.286184353(73) THz (40)2π・389.286294205(62) THz (41)
Wavelength (Vacuum)λ770.108385049(123) nm (39)770.108136363(144) nm (40)770.107919192(123) nm (41)Lifetimeτ

26.69(5) ns

Natural Line WidthΓ

2π・5.963(11) MHz

Recoil temperatureTr

828.734 nK (39)[133]

807.992 nK (40)788.310 nK (41)
Doppler shift (v = 1m/s)

2π・1.29852 MHzΔωd(d) Hyperfine constants
Table G.3 Potassium D Transition Hyperfine Structure Constants
Magnetic Dipole Constant 42S1/2

A 2S1/2

461.720 MHz (39)[134]

-1285.79 MHz (40)254.014 MHz (41)2Magnetic Dipole Constant 4 P1/2

2

A P1/227.775(42) MHz (39)[132]

-34.523(25) MHz (40)
15.245(42) MHz (41)2Magnetic Dipole Constant 4 P3/2

2

A P3/2

6.093(25) MHz (39)[132]

-7.585(10) MHz (40)3.363(25) MHz (41)2Magnetic Dipole Constant 4 P3/2

2

B P3/22.786(71) MHz (39)-3.445(90) MHz (40)3.351(71) MHz (41)

[132]

106

Bibliography[1] M.H.Anderson et al., Science 269, 198(1995)[2] C.C.Bradley et al., Phys.Rev.Lett. 75, 1687(1995)[3] K.B.Davis et al., Phys.Rev.Lett. 75, 3969(1995)[4] M.Greiner et al., Nature 415, 39(2002)[5] C.A.Regal et al., Phys.Rev.Lett. 92, 040403(2004)[6] V.S.Letokhov et al., JETP Lett. 7, 272(1968)[7] C.Salomon et al., Phys.Rev.Lett. 59, 1659(1987)[8] H.Feshbach et al., Ann.Phys. 19, 287(1962)[9] E.Tiesinga et al., Phys.Rev.A 46, R1167(1992)[10] E.Tiesinga et al., Phys.Rev.A 47, 4114(1993)[11] A.J.Moerdijk et al., Phys.Rev.A 51, 4852(1995)[12] J.M.Vogels et al., Phys.Rev.A 56, R1067(1997)[13] H.M.J.M.Boesten et al., Phys.Rev.A 54, R3726(1996)[14] S.Inouye et al., Nature 392, 151(1998)[15] P.Courteille et al., Phys.Rev.Lett. 81, 69(1998)[16] J.L.Roberts et al., Phys.Rev.Lett. 81, 5109(1998)[17] S.Giorgini et al., cond-mat/0706.3360 (2007)[18] I.Bloch et al., cond-mat/0704.3011(2007)[19] K.Goral et al., Phys.Rev.Lett. 88, 170406(2002)[20] H.P.Buchler et al., Phys.Rev.Lett. 98, 060404(2007)[21] D.J.Heinzen et al., Phys.Rev.Lett. 84, 5029(2000)[22] M.G.Kozlov et al., J.Phys.B 28, 1933(1995)[23] D.DeMille, Phys.Rev.Lett. 88, 067901(2002)[24] A.Kantrowitz et al., Rev.Sci.Instr. 22, 328(1951)[25] R.E.Smalley et al., J.Chem.Phys. 61, 4363(1974)[26] H.LBethlem et al., Nature 406, 491(2000)[27] J.van Veldhoven et al., Phys.Rev.Lett. 94, 083001(2005)[28] J.D.Weinstein et al., Nature 395, 148(1998)[29] S.Kotochigova et al., Phys.Rev.A 68, 022501(2003)[30] J.M.Sage et al., Phys.Rev.Lett. 94, 203001(2005)[31] A.Fioretti et al., Phys.Rev.Lett. 80, 4402(1998)[32] J.P.Shaffer et al., Phys.Rev.Lett. 82, 1124(1999)[33] C.Haimberger et al., Phys.Rev.A 70, 021402(R) (2004)[34] A.N.Nikolov et al., Phys.Rev.Lett. 82, 703(1999)[35] Gabbanini et al., Phys.Rev.Lett. 84, 2814(2000)[36] R.Wynar et al., Science 287, 1016(2000)[37] E.Donley et al., Nature 417, 529(2002)[38] F.K.Fatemi et al., Phys.Rev.A 66, 053401(2002)[39] K.Strecker et al., Phys.Rev.Lett. 91, 080406(2003)[40] C.A.Regal et al., Nature 424, 47(2003)[41] J.Herbig et al., Science 301, 1510(2003)[42] K.Xu et al., Phys.Rev.Lett. 91, 210402(2003)[43] M.W.Mancini et al., Phys.Rev.Lett. 92, 133203(2004)

107[44] A.J.Kerman et al., Phys.Rev.Lett. 92, 153001(2004)[45] C.Chin et al., Phys.Rev.Lett. 94, 123201(2004)[46] S.D.Kraft et al., J.Phys.B. 39, S993(2006)[47] C.Ospelkaus et al., Phys.Rev.Lett. 97, 120402(2006)[48] J.J.Zirbel et al., cond-mat/0712.3889(2007)[49] F.Ferlaino et al., Phys.Rev.A 73, 040702(R)(2006)[50] A.Pashov et al., Phys.Rev.A 76, 022511(2007)[51] G.Modugno et al., Phys.Rev.Lett. 89, 190404(2002)[52] W.C.Stwalley et al., J.Chem.Phys. 108, 5767(1998)[53] C.Klempt et al., Phys.Rev.A 76, 020701(2007)[54] R.S.Williamson et al., J.Opt.Soc.Am.B 12, 1393(1995)[55] F.S.Cataliotti et al., Phys.Rev.A 57, 1136(1998)[56] D.Wang et al., Phys.Rev.A 53, R1216(1996)[57] W.Ketterle et al., Phys.Rev.Lett. 70, 2253(1993)[58] M.H.Anderson et al., Phys.Rev.A 50, R3597(1994)[59] C.Townsend et al., Phys.Rev.A 53, 1702(1996)[60] W.Petrich et al., J.Opt.Soc.Am.B, 11, 1332(1994)[61] M.Prevedelli et al., Phys.Rev.A 59, 886(1999)[62] C.Fort et al., Eur.Phys.J D 3, 113(1998)[63] H.J.Metcalf et al., Laser cooling and trapping, Springer(1999)[64] Gordon et al., Phys.Rev.A 21, 1606(1980)[65] R.J.Le Roy et al., J.Mol.Struc.(Theochem), 591, 175(2002)[66] J.A.Coxon et al., J.Mol.Spectr., 235, 235(2005)[67] M.Marinescu et al., Phys.Rev.A 49, 982(1994)[68] M.Marinescu et al., Phys.Rev.A 52, 311(1995)[69] M.Marinescu et al., Zeitschrift fur Physik D 36, 239(1996)[70] S.H.Patil et al., J.Chem.Phys. 106, 2301(1997)[71] A.Derevianko et al., Phys.Rev.Lett. 82, 3589(1999)[72] M.Marinescu et al., Phys.Rev.A 59, 390(1999)[73] A.Derevianko et al., Phys.Rev.A 63, 052704(2001)[74] S.G.Porsev et al., J.Chem.Phys. 119, 844(2003)[75] B.M.Smirnov et al., JETP 21, 624(1965)[76] G.Hadinger et al., J.Mol.Spectr. 175, 441(1996)[77] D.Zemke et al., J.Chem.Phys. 111, 4962(1999)[78] J.Y.Seto, University of Waterloo, Master thesis(2000), available at http://etd.uwaterloo.ca/jyseto2000.pdf[79] R.Rydberg, Z.Physik 73, 376(1931)[80] O.Klein, Z.Physik 76, 226(1932)[81] R.Rydberg, Z.Physik 80, 514(1933)[82] A.L.G.Rees, Proc.Phys.Soc.(London) 59, 998(1947)[83] R.J.Le Roy, A Computer Program Implementing the First-Order RKR Method for Determining Diatom
Potential Energy Curves from Spectroscopic Constants, University of Waterloo Chemical Physics Research
Report CP-425 (1992); available at http://leroy.uwaterloo.ca/programs/.
[84] R.J.Le Roy et al., Chem.Phys.Lett., 5,42(1970)[85] R.J.Le Roy et al., J.Chem.Phys., 52, 3869(1970)[86] C.R. Vidal et al., J.Mol.Spectr. 65, 46(1977)

108[87] MOLPRO, a package of ab initio programs designed by H.-J.Werner and P.J.Knowles, version 2002.1,
R.D.Amos, A.Bernhardsson, A.Berning, P.Celani, D.L.Cooper, M.J.O.Deegan, A.J.Dobbyn, F.Eckert,
C.Hampel, G.Hetzer, P.J.Knowles, T.Korona, R.Lindh, A.W.Lloyd, S.J.McNicholas, F.R.Manby, W.Meyer,
M.E.Mura, A.Nicklass, P.Palmieri, R.Pitzer, G.Rauhut, M.Schutz, U.Schumann, H.Stoll, A.J.Stone, R.Tarroni,
T.Thorsteinsson and H.-J.Werner.
[88] G.Herzberg, Molecular spectra and molecular structure 1.spectra of diatomic molecules, Krieger
Publishing Company, Malabar, Florida(1950)[89] T.Bergeman et al., J.Chem.Phys. 117, 7491(2002)[90] D.R.Hartree, Rep.Prog.Phys. 11, 113(1946)[91] M.R.Manaa, Int.J.Quant.Chem. 75, 693(1999)[92] V.Kokoouline et al., Phys.Rev.A 62, 032716(2000)[93] R.Ferber et al., J.Chem.Phys. 112, 5740(2000)[94] M.Tamanis et al.,J.Chem.Phys. 117, 7980(2002)[95] M.R.Manaa et al., J.Chem.Phys. 117, 11208(2002)[96] T.Bergeman et al., Phys.Rev.A 67, R050501(2003)[97] T.Bergeman et al., Eur.Phys.J. D 31, 179(2004)[98] P.Qui et al., J.Chem.Phys. 127, 044301(2007)[99] H.Lefebvre-Brion, Perturbations in the Spectra of Diatomic Molecules (Academic, New York, 1986)[100] R.J.Le Roy, LEVEL 8.0: A Computer Program for Solving the Radial Schrodinger Equation for Boundand Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663 (2007); available at
http://leroy.uwaterloo.ca/programs/.
[101] R.J.Le Roy, LEVEL 7.7: A Computer Program for Solving the Radial Schrodinger Equation for Boundand Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663 (2001);[102] F.Martin et al., J.Chem.Phys. 115, 4118(2001)[103] W.Jastrzebski et al., J.Mol.Spectr. 209, 50(2001)[104] A.Pashov et al., Chem.Phys.Lett. 292, 615(1998)[105] A.Grochola et al., Chem.Phys.Lett. 372, 173(2003)[106] W.C.Stwalley, Eur.Phys.J. D 31, 221(2004)[107] S.Rousseau et al., J.Mol.Spec. 203, 235(2000)[108] C.E.Wiemann et al., Rev.Sci.Instr., 62, 1(1991)[109] K.B.MacAdam et al., Am.J.Phys., 60, 1098(1992)[110] A.S.Arnold et al., Rev.Sci.Instr., 69, 1236(1998)[111] R.Drever et al., Appl.Phys.B 31, 97 (1983)[112] B.DeMarco et al., Rev.Sci.Instr., 70, 1967(1999)[113] R.S.Williamson, Ph.D. thesis (1997)[114] MCP Assembly Technical Notes (Japanese), Hamamatsu Photonics K.K., Shizuoka, 2006[115] E.Flaxer, Meas.Sci.Technol. 17, N37(2006)[116] L.G.Marcassa et al., Phys.Rev.A 63, 013413(2000)[117] J.Emsler, The Elements, Oxford Chemistry Guides (Oxford Univ. Press, New York, NY, 1995)[118] A.Fioretti et al., Eur.Phys.J. D 15, 189(2001)[119] R.S.Freeland, Ph.D. thesis, The University of Texas at Austin (2001),
available at http://www.ph.utexas.edu/~coldatom/pubs.html[120] A.R.L.Caires et al., Phys.Rev.A 71, 043403(2005)[121] J.Lozeille et al., Eur.Phys.J. D 39, 261(2006)[122] A.Fioretti et al., J.Phys.B. 40, 3283(2007)

109[123] D.Wang et al., a Eur.Phys.J. D 31, 165(2004)[124] D.Wang et al., Phys.Rev.Lett. 93, 243005(2004)[125] D.Wang et al., Phys.Rev.A 72, 032502(2005)[126] G.Pichler et al., J.Phys.B 16, 4619(1983)[127] R.Beuc et al., J.Phys.B 17, 739(1984)[128] A.Jraij et al., Chem.Phys. 290, 129(2003)[129] M.Korek et al., Int.J.Quantum Chemistry 92, 376(2003)[130] David R.Lide (Ed.), CRC Handbook of Chemistry and Physics, 83rd ed. (CRC Press, Boca Raton, 2002)[131] J.Sugar and C.Corliss, J.Phys.Chem.Ref.Data 14, Suppl. 2(1985)[132] S.Falke et al., Phys.Rev.A 74, 032503(2006)[133] H.Wang et al., Phys.Rev.A 55, R1569(1997)[134] E.Arimondo, M.Inguscio and P.Violino, Rev.Mod.Phys. 49, 31(1977)