Trapping metastable neon atoms
Tempelaars, J.G.C.
DOI:
10.6100/IR543676
Published: 01/01/2001
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Tempelaars, J. G. C. (2001). Trapping metastable neon atoms Eindhoven: Technische Universiteit Eindhoven
DOI: 10.6100/IR543676
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Trapping Metastable Neon Atoms
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN
T ECHNISCHE U NIVERSITEIT E INDHOVEN , OP GEZAG
R ECTOR M AGNIFICUS , PROF. DR . M. R EM , VOOR
EEN COMMISSIE AANGEWEZEN DOOR HET C OLLEGE
VOOR P ROMOTIES IN HET OPENBAAR TE VERDEDIGEN
OP DONDERDAG 26 APRIL 2001 OM 16.00 UUR
DE
VAN DE
DOOR
J EFFREY G ODEFRIDUS C ORNELIS T EMPELAARS
GEBOREN TE
B ERGEN OP Z OOM
D IT PROEFSCHRIFT IS GOEDGEKEURD
DOOR DE PROMOTOREN :
PROF. DR . H. C. W. B EIJERINCK
EN
PROF. DR .
B. J. V ERHAAR
COPROMOTOR :
DR . IR .
E. J. D. V REDENBREGT
Druk: Universiteitsdrukkerij Technische Universiteit Eindhoven
Ontwerp omslag: Astrid van den Hoek
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN
Tempelaars, Jeffrey Godefridus Cornelis
Trapping Metastable Neon Atoms / by
Jeffrey Godefridus Cornelis Tempelaars. Eindhoven: Technische Universiteit Eindhoven, 2001. - Proefschrift. ISBN 90-386-1759-3
NUGI 812
Trefw.: atomaire bundels / laserkoeling / atoombotsingen / atomen; wisselwerkingen /
laserspectroscopie / neon
Subject headings: atomic beams / laser cooling / optical cooling of atoms; trapping /
interatomic potentials and forces / neon
aan mijn ouders
The work described in this thesis was carried out at the Physics Department of the
Eindhoven University of Technology and was part of the research program of the
‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’ (NWO).
Contents
1 Introduction
1
Cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Bose-Einstein condensation of metastable neon atoms . . . . . . . . . . .
3
Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Laser cooling and trapping
1
Introduction . . . . . . . . . . .
2
Light forces . . . . . . . . . . .
3
Applications of laser cooling
4
Cold atomic collisions . . . .
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3 Intense beam of cold metastable Ne(3s) 3 P2 atoms
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
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Experimental setup . . . . . . . . . . . . . . . . . . . . .
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Source and collimator . . . . . . . . . . . . . . . . . . . .
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Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . .
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Magneto-optical compressor and sub-Doppler cooler
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Beam characteristics . . . . . . . . . . . . . . . . . . . .
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Concluding remarks . . . . . . . . . . . . . . . . . . . . .
4 Metastable neon atoms in
1
Introduction . . . . . .
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Trap dynamics . . . .
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Experimental setup .
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Trap volume . . . . .
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Loading rate . . . . . .
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Trap population . . .
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Temperature . . . . .
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Trap decay . . . . . . .
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Concluding remarks .
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5 Photoassociation spectroscopy of 85 Rb2 0+
u states
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2
Bound state calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1
2
Contents
3
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Application to the rubidium dimer
Photoassociation experiment . . .
Model calculations . . . . . . . . . .
Concluding remarks . . . . . . . . .
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Summary
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Samenvatting
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Dankwoord
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Curriculum Vitae
120
Chapter 1
Introduction
1 Cold atoms
Since the demonstration of the use of laser light to cool and confine neutral atoms
in the late 1980s [1, 2], the cooling and trapping field has made an enormous development. Several technological applications of cold atoms have been realized. Laser
cooled atoms in atomic beams can be focused to a very small scale: because of the
small de Broglie wavelength of the cold atoms, very high resolution can be obtained
in the fabrication of nanostructures [3]. The use of laser cooled atoms in atomic
clocks gave a breakthrough in time-metrology: the application of fountains of cold
atoms [4] increased the accuracy in the atomic frequency standard by two orders of
magnitude [5]. Aside from these practical applications, cold and ultracold collisions
are of interest in themselves as fundamental quantum mechanical phenomena. The
increasing use of atom traps as an experimental and technological tool increased the
interest in collisions at temperatures below 1 mK [6]. The questions how “bad” collisions lead to trap loss, and how “good” collisions could be used in cooling schemes,
fascinated atom trappers constantly [7].
The insights obtained led to the experimental realization in 1995 of Bose-Einstein
condensation (BEC) in cold, dilute samples of alkali-metal atoms [8–10]. Bose-Einstein
condensation in atomic samples occurs when the mean inter-particle distance is of
the same order as the thermal wavelength. In formula:
nΛ3dB = n(2π 2 /mkB T )3/2 ≥ 2.61,
(1.1)
where n is the number density, ΛdB the thermally averaged de Broglie wavelength
at temperature T , m the atom’s mass, and kB Boltzmann’s constant. This condition
requires densities of the order of 1014 atoms/cm3 and ultra low temperatures of the
order of 1 µK. It was a different cooling technique, called evaporative cooling [11,
12], which made the final step to the quantum degenerate regime possible. Since
1995, BEC’s of alkali-metal atoms were produced all over the world, and in 1998
also atomic hydrogen was Bose-condensed [13, 14]. Other atomic species which are
candidates for BEC are metastable rare gas atoms, with helium and neon the prime
possibilities [15–18], as well as alkali-earth atoms [19].
3
4
Chapter 1
The attainment of BEC in weakly interacting gases opened a whole new area
in physics. Quantum-statistical effects have been observed in BEC’s which are related to superconductivity phenomena and superfluidity of liquid helium [20]. It has
been shown that the speed of light can be reduced to several meters per second
in BEC’s [21], and consequently, experiments have been suggested to create atomic
samples behaving like optical black holes [22]. Furthermore, experiments showed
that the coherence of the atoms can be conserved when atoms are coupled out of a
BEC with optical or microwave pulses. Because of the analogy to the coherence of
laser light, this may be considered the birth of the atom laser [23].
2 Bose-Einstein condensation of metastable neon atoms
Experimentally, a Bose-Einstein condensate is created in roughly three steps. First,
atoms are loaded into a magneto-optical trap (MOT) in which the combination of
a laser field with a magnetic quadrupole field confines the atoms in the trapping
center. Second, the atoms are transferred from the MOT to a magneto-static trap.
Finally, the atoms are cooled down in the magneto-static trap by a forced evaporative
cooling process until BEC is reached. Until now, BEC has only been observed in dilute
samples of alkali-metal atoms and atomic hydrogen. In our group we are trying to
produce a BEC with metastable neon atoms [18]. A BEC of metastable rare gas atoms
is attractive for a number of reasons. First of all, real time diagnostics can be used
such as Auger deexcitation of escaping atoms at a surface and UV photon detection
following optical pumping to a non-metastable state. Second, tight magnetic traps
are easy to achieve for metastable atoms, due to their large magnetic moment. Third,
different isotopes with integer and half-integer angular momentum are available,
which potentially allows experiments with stable and unstable condensates as well
as Fermi gases. Furthermore, the high internal energy, e.g. 17 eV for Ne(3s)3 P2 ,
makes single atom detection possible, which opens new ways to look at the statistics
and formation of BEC’s. Finally, future experiments with atom lasers of electronically
excited atoms would be possible.
The experimental conditions are less favorable for metastable rare gas atoms.
First of all, efficient beam-brightening techniques are necessary to overcome the low
efficiency of ∼ 10−4 with which rare gas atoms in the metastable state are produced
in commonly used sources. Second, trap lifetimes and densities are limited by the
process of Penning ionization. And finally, the finite lifetime of the metastable state,
e.g. 24 s for the Ne(3s)3 P2 state, limits the time scale of the experiments. Despite all
those experimental difficulties, the prospects for creating a BEC of metastable neon
atoms are rather good [18]. Calculations done by Doery et al. [24] showed that, by
spin-polarizing the atoms, which happens spontaneously in a magneto-static trap,
the ionization of metastable neon atoms may be suppressed by four orders of magnitude. Consequently the trap lifetime due to ionizing losses can be increased to
values much larger than the natural lifetime of the metastable state [18]. However,
those calculations were based on modified sodium potentials, since accurate interaction potentials of metastable neon atoms are not available at all. A full theoretical
Introduction
5
calculation based on new, ab initio short-range [25] and semiempirical long-range
potentials [26], could therefore still alter these expectations.
3 Scope of this thesis
The road we are following to reach Bose-Einstein condensation with metastable neon
atoms, is analogous to that of sodium [27], with the exception that we have to load
our magneto-optical trap with a brightened atomic beam, instead of from a vapor
cell or slowed atomic beam. In this thesis the first successful steps towards BEC of
metastable neon atoms are described.
Chapter 2 gives a general introduction into the physics of cooling and trapping
neutral atoms. In chapter 3 we describe how we applied several laser cooling techniques to create an intense beam of cold metastable Ne(3s)3 P2 atoms. The beam, with
a flux of 5 × 1010 atoms/s, can be used for a variety of cold collision experiments.
In chapter 4 we describe how we used the bright beam setup to create a magnetooptical trap containing almost 1010 metastable neon atoms. A large initial number
of atoms trapped in the MOT facilitates the evaporative cooling phase in a magnetostatic trap, necessary to reach the quantum degenerate regime. Recently 109 atoms
were transferred from the magneto-optical trap to a magneto-static trap [28], developed in our lab [29, 30].
Knowledge of the interaction parameters for cold colliding atoms was crucial
for understanding the properties of a BEC in the case of alkali-metal atoms. The
technique of photoassociative spectroscopy (PAS) has been used, with considerable
success, to determine collisional properties of the alkali atoms [31], and is equally
appropriate for studies of the rare gases [32, 33]. This technique, explained in more
detail in chapter 2, is based on the formation of bound excited-state molecules by
exciting two slowly colliding ground state atoms with a probe laser. In chapter 5
of this thesis it is shown by an experiment with rubidium that PAS can be used to
investigate the coupling between two molecular states.
Recently, photoassociation experiments on metastable helium atoms were reported by Herschbach et al. [33]. From PAS spectra they estimated long range excited
state interaction parameters for metastable helium. In the future we plan to apply
the PAS technique to metastable neon atoms to obtain experimental data on longrange ground-state potentials [32] as well as ionization properties [24]. The latter
will give information about the suppression of ionization of spin polarized samples.
References
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48 (1985).
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Lett. 59, 2631 (1987).
6
Chapter 1
[3] J.J. Mcclelland, R.E. Scholten, E.C. Palm, and R.J. Celotta, Science 262, 877 (1993).
[4] M.A. Kasevich, E. Riis, S. Chu, and R.G. DeVoe, Phys. Rev. Lett. 63, 612 (1989).
[5] P. Lemonde, Phys. World, January, 39 (2001).
[6] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag
New York, 1999.
[7] J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71(1), 1, (1999).
[8] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995).
[9] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn,
and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[10] C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75, 1687
(1995).
[11] H. F. Hess, Phys. Rev. B. 34, 3476 (1986).
[12] N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytak, H.F. Hess, and
G.P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[13] T.C. Killian, D.G. Fried, L. Willmann, D. Landhuis, S.C. Moss, T.J. Greytak, and D.
Kleppner, Phys. Rev. Lett. 81, 3807 (1998).
[14] D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and
T.J. Greytak, Phys. Rev. Lett. 81, 3811 (1998).
[15] S. Nowak, A. Browaeys, J. Poupard, A. Robert, D. Boiron, C. Westbrook, and A.
Aspect, Appl. Phys. B 70, 455 (2000).
[16] N. Herschbach, P.J.J. Tol, W. Hogervorst, and W. Vassen, Phys. Rev. A 61,
050702(R) (2000).
[17] M. Zinner, C. Jentsch, G. Birkl, and W. Ertmer, private communication.
[18] H.C.W. Beijerinck, E.J.D. Vredenbregt, R.J.W. Stas, M.R. Doery, and J.G.C. Tempelaars, Phys. Rev. A 61, 023607 (2000).
[19] T. Ido, Y. Isoya, and H. Katori, Phys. Rev. A 61, 061403(R) (2000).
[20] F. Chevy, K.W. Madison, and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000).
[21] L. Vestergaard Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, Nature 397(6720),
594 (1999).
[22] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000).
Introduction
7
[23] M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W.
Ketterle, Phys. Rev. Lett. 78, 582 (1997).
[24] M.R. Doery, E.J.D. Vredenbregt, S.S. Op de Beek, H.C.W. Beijerinck, and B.J. Verhaar, Phys. Rev. A 58, 3673 (1998).
[25] S. Kotochigova, E. Tiesinga and I. Tupitsyn, Phys. Rev. A 61, 042712 (2000).
[26] A. Derevianko and A. Dalgarno, Phys. Rev. A 62, 062501 (2000).
[27] M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W.
Ketterle, Phys. Rev. Lett. 77, 416 (1996).
[28] S.J.M. Kuppens, V.P. Mogendorff, J.G.C. Tempelaars, E.J.D. Vredenbregt, and
H.C.W. Beijerinck, to be published.
[29] R. Stas, Laser cooling and trapping of metastable neon atoms, internal report,
Eindhoven University of Technology (1999).
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University of Technology (2000).
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Verhaar, Phys. Rev. A 57, 3603 (1998).
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8
Chapter 1
Chapter 2
Laser cooling and trapping
1 Introduction
The momentum carried by a beam of light can be used to manipulate atomic trajectories. This was already demonstrated in the early thirties by a small deflection of
a beam of sodium atoms illuminated by a resonant lamp [1]. Since the development
of the laser, intense and highly directional light sources became available which enhanced the efficiency of the momentum transfer from the light field to the atoms
drastically. In 1985, laser light was used to cool and trap neutral atoms for the first
time [2]. In section 2 of this chapter we discuss the origin of optical forces and how
they are used in the laser cooling and trapping field. In section 3 we describe the
principles of the neutral atom traps used in the experiments described in chapters 4
and 5 of this thesis. Since in this thesis experiments with both metastable neon
atoms and rubidium atoms are described, in section 4 we give a short description
of their atomic structure and discuss the physics of low temperature collisions for
both atoms.
2 Light forces
2.1 Principles
Let us consider a two-level atom at rest in a classical electro-magnetic field with elec R,
t) = E
0 (R)[cos(ω
tric field component E(
The optical force exerted on
L t − kL · R)].
the atom can be of two types: a dissipative, spontaneous force and a conservative,
dipole force [3]. The spontaneous force arises from the absorption and spontaneous
emission of a photon from the light field. Absorbing a photon from the light field
L to the atom in the direction of the light
results in a transfer of a momentum k
propagation. If the decay of the atom, after absorbing a photon, is spontaneous, the
emission of the photon is in a random direction, so that over many events the corresponding recoil averages to zero effect on the momentum of the atom. As a result
the net force is in the direction of the light propagation, as shown schematically in
Fig. 2.1. The dipole force arises from the interaction of the dipole moment, induced
9
10
Chapter 2
Scattered light
Incoming light
Net force
Atom
Figure 2.1: Principle of the spontaneous radiation force. Photons are absorbed from
the direction of the laser beam and reemitted spontaneously in random direction.
The net force is in the direction of the light propagation.
by the oscillating electric field, with the gradient of the electric field amplitude. This
becomes clear when we look more quantitatively at the character of the forces, as
done in detail by Cohen-Tannoudji et al. [4].
The expression for the interaction Hamiltonian of an atom at rest in the electric
R,
t) is given by
field E(
R,
t),
H = −
µeg · E(
(2.1)
eg the transition dipole moment. The light force on the atom is then given
with µ
by [5]
R,
t)
,
R H
= ∇
R F = −∇
µeg · E(
(2.2)
where we neglected the spatial variation of the electric field over the size of the atom.
The two types of forces arise from the gradient operator working on the electric field:
the spontaneous force Fsp is related to the spatial dependence of the phase of the
electric field while the dipole force Fdip is related to the spatial dependence of the
amplitude of the electric field. The expression for the two types of forces arise from
the steady-state solutions of the optical Bloch equations [5]:
F = Fsp + Fdip
=
Ω2
Γ /2
2
(∆L )2 + (Γ /2)2 + Ω2 /2
Ω2
∆L
−∇R |E0 |
,
2
(∆L )2 + (Γ /2)2 + Ω2 /2
L · R)
R (k
∇
(2.3)
with ∆L = ωL − ω0 the detuning of the optical field from the atomic transition
frequency ω0 , Γ the natural width of the atomic transition; Ω reflects the strength of
the coupling between field and atom and is called the Rabi frequency,
Ω=−
eg · E0
µ
.
(2.4)
The spontaneous force is a dissipative force and can be used to cool atoms. This
is why it is sometimes called the cooling force. We can express this force as
L Γ s k
Fsp =
,
(2.5)
2
1+s
Laser cooling and trapping
11
with the saturation parameter s defined as
s=
Ω2 /2
.
(∆L )2 + (Γ /2)2
(2.6)
L times the scattering
The cooling force is simply the momentum of a photon k
L Γ /2 and
rate, as mentioned earlier. The maximum cooling force equals Fmax = k
is reached for high saturation parameters. By defining the on-resonance saturation
parameter s0 as
I
2Ω2
= 2 ,
(2.7)
s0 =
I0
Γ
with I the light intensity and I0 the saturation intensity depending on the atomic
transition involved, we write for the cooling force
Fsp =
L Γ
k
s0
.
2 1 + s0 + (2∆L /Γ )2
(2.8)
The dipole force can be written in terms of the gradient of the light intensity ∇I:
s ∆L ∇I
∆L ∇I
s0
Fdip = −
.
(2.9)
=−
2I
1+s
2I 1 + s0 + (2∆L /Γ )2
A red-detuned light beam, i.e. ∆L < 0, produces a force that attracts the atoms
to the intensity maximum, while a blue-detuned light beam repels the atoms from
the intensity maximum. The dipole force is sometimes called the trapping force,
because, for example, a focused, red-detuned laser beam can be used for trapping
neutral atoms (as described in section 3.2 of this chapter).
2.2 Velocity dependence
Cooling atoms with laser light requires a velocity dependent force. The dissipative,
spontaneous force described above can be used for this purpose. Considering an
atom moving in the direction of a slightly red-detuned laser beam, the effective detuning ∆eff from resonance contains not only the laser detuning ∆L , but also contains
L · v
due to the Doppler effect. Therefore the force given
a contribution ∆D = −k
by Eq. (2.8) becomes velocity dependent by replacing ∆L by the effective detuning
∆eff = ∆L + ∆D . This is the basis for the standard laser cooling mechanism known as
“Doppler cooling” or “optical molasses” [6].
Let us now consider a standing light wave produced by two counterpropagating,
slightly red-detuned laser beams with the same frequency. An atom moving in the
direction of one of the laser beams experiences, due to the Doppler effect, a slightly
blue-shifted laser beam. As a consequence it will absorb more photons per unit of
time from the counterpropagating laser beam than from the copropagating beam.
Hence the atom experiences a net force opposite to its own motion. Around v = 0
the force is proportional to the velocity and can be seen as a damping force. For
s0 1 the force can be approximated by [7]
8k2L s0 (∆L /Γ )v
≡ −βv,
F=
[1 + (2∆L /Γ )2 ]2
(2.10)
12
Chapter 2
0.4
0.2
0.2
F/Fmax
F/Fmax
a
0.4
0.0
-0.2
-0.4
-2
b
0.0
-0.2
-vc
-1
vc
0
kLv/Γ
1
2
-0.4
-2
-xc
-1
xc
0
1
µ' Gx/h Γ
2
Figure 2.2: a: Velocity dependence of the cooling force for optimal conditions ∆L =
−Γ /2, s0 = 1. The dashed lines indicate the capture velocity vc = −∆L /kL . b: Position
dependence of the cooling force on an atom at rest (v = 0) for optimal conditions
∆L = −Γ /2, s0 = 1. The dashed lines indicate the capture range xc = −∆L /(µ G).
with β the damping coefficient. Figure 2.2a shows the damping force for optimal
conditions (∆L = −Γ /2 and s0 = 1) as a function of the atomic velocity. Atoms with
a velocity below the capture velocity vc = −∆L /kL feel the damping force given by
Eq. (2.10). In chapter 3, we describe how we use standing light waves to capture
metastable neon atoms and collimate them into an parallel atomic beam.
Although the velocity-dependent force can be used to cool atoms, it can not be
used to (spatially) trap atoms, since there is no position-dependent force to drive
the atoms to an equilibrium point in space. However, the spontaneous force can be
made position dependent by applying a spatially varying magnetic field.
2.3 Position dependence
the internal energy of an atom in a magnetic sublevel mi with
In a magnetic field B
changes by an amount ∆E = −
= µB Bgmi , with µ
the magnetic
respect to B
µ·B
moment of the atom, µB the Bohr magneton, and g the Landé factor. The transition
frequency for an atom with ground state g and excited state e is shifted by ∆B =
µ B/, with
(2.11)
µ ≡ (ge me − gg mg )µB ,
the effective magnetic moment of the transition [5]. Figure 2.3 shows the onedimensional situation of a hypothetical atom with atomic transition Jg = 0 → Je = 1,
in an inhomogeneous magnetic field B = B(x) ≡ Gx. The atom is illuminated by two
counterpropagating laser beams, each detuned slightly to red of the zero magnetic
field atomic transition. The laser beams are opposite circularly polarized, the beam
Laser cooling and trapping
13
B
σ+
σ−
0
me= -1
Je=1
σ−
Jg=0
σ+
B< 0
Force
0
σ−
x
+1
σ+
B=0
Force=0
σ−
σ+
B> 0
Force
Figure 2.3: Principle of the position dependent radiation force for a Jg = 0 → Je = 1
atomic transition. The inhomogeneous magnetic field splits the Zeeman levels of
the exited state. As a consequence, an atom positioned at the left of the origin is
more resonant with the σ + -polarized beam and feels a resulting force directed to
the origin. An atom positioned at the right of the origin, is also pressed to the origin
because it is more resonant with the σ − -polarized beam.
running in the +x direction has σ + polarization and the beam running in the −x
direction has σ − polarization. If the atom is located at a position left of the origin,
the atom is, because of the Zeeman splitting of the Je = 1 state, more resonant with
the σ + polarized beam and feels a force directed to the origin. An atom positioned
on the right of the origin will be pushed back to the origin by the σ − polarized beam.
The position-dependent force on the atom is analogous to the damping force in
velocity space, described in the previous section. Figure 2.2b shows the force on an
atom in an inhomogeneous magnetic field B = B(x) ≡ Gx. Analogous to the velocitydependent force, a capture radius can be defined by xc = −∆L /(µ G). Positiondependent forces are used to trap atoms in magneto-optical traps, like described in
section 3.1. In chapter 3 we describe two applications of position-dependent forces
in an atomic beam: Zeeman-compensated slowing to decelerate atoms in an atomic
beam and a two-dimensional version of a magneto-optical trap to funnel atoms into
a narrow beam.
2.4 Cooling limits
Cooling atoms with laser light also introduces heating effects caused by the randomness of the momentum steps undergone by the atom with each emission or absorption. The motion of the atom can be compared to a random walk in momentum
space caused by the randomness in direction of the spontaneous emitted photons
and the uncertainty in the number of absorbed photons from the light field. The
heating corresponding to this random walk process can be expressed in terms of a
14
Chapter 2
momentum diffusion coefficient D [5].
In the laser cooling and trapping field it is convenient to define the temperature
of an atomic sample for each degree of freedom separately by kB Ti /2 = mvi2 /2,
with kB Boltzmann’s constant and m the atomic mass [5]. The equilibrium temperature which can be reached by laser cooling can be found by comparing the heating
rate Ėheat = D/m, introduced by the random walk process, with the cooling rate
Ėcool = Fsp vi = βvi2 of the damping force:
D
2∆L
Γ
Γ
+
kB Ti =
=−
.
(2.12)
β
4 2∆L
Γ
The minimum temperature is reached for ∆L = −Γ /2: kB TD ≡ Γ /2, and is called
the Doppler limit. For neon the Doppler
temperature and corresponding Doppler
velocity equal TD = 196 µK, and vD = v 2 = 0.29 m/s, respectively.
Most applications of laser cooling described in this thesis consider only Doppler
cooling with corresponding temperatures given by Eq. (2.12). However, experiments
have shown that, by using polarization gradients or a homogeneous magnetic field,
laser cooling below the Doppler limit is possible [7–11]. A new limit may be considered caused by the exchange of the recoil energy ER = 2 k2L /(2m) with the absorption or emission of a single photon. The recoil velocity equals vR = 0.031 m/s for
metastable neon, which corresponds to a recoil temperature defined by TR = 2ER /kB
of TR = 2.3 µK. By eliminating the spontaneous emission process, cooling below the
recoil limit is possible. This can be achieved in two ways. One uses optical pumping
into a velocity-selective dark state (VSCPT) [12], while another scheme uses velocity
selective Raman transitions within a Zeeman multiplet [13].
Even much lower temperatures can be reached with a different cooling technique,
called evaporative cooling. This technique, first described by Hess [14], is based on
the removal of the hottest particles from an atomic sample and was first used on
atomic hydrogen [15]. With this technique temperatures below 1 µK can be reached.
At such low temperatures, and high enough atomic densities, Bose-Einstein condensation (BEC) can occur [16].
3 Applications of laser cooling
3.1 Magneto-optical trap (MOT)
The most common neutral atom trap makes use of both optical and magnetic fields
to form a magneto-optical trap (MOT), which was first demonstrated by Raab et al.
[17]. The principle of the MOT is shown schematically in Fig. 2.4. Three orthogonal
pairs of counter propagating, circularly polarized laser beams intersect at the center
of a magnetic quadrupole field, generated by two anti-Helmholz coils. The force on
the atoms is velocity and position dependent and is for each direction ξ = x, y or
z given by F (ξ, vξ ) = F+ (ξ, vξ ) + F− (ξ, vξ ), with F± (ξ, vξ ), analogous to Eq. (2.8)
written as
L Γ
k
s0
F± (ξ, vξ ) = ±
,
(2.13)
2 1 + s0 + (2∆± (ξ, vξ )/Γ )2
Laser cooling and trapping
15
σ−
σ−
σ−
σ+
B
σ+
z
σ+
x
y
Figure 2.4: Schematic view of a magneto-optical trap. Two anti-Helmholz coils generate a spherical quadrupole magnetic field, indicated with the small arrows. Three
pairs of counter propagating, opposite polarized, laser beams intersect at the origin
of the quadrupole field.
with the effective detuning for each beam given by ∆± (ξ, vξ ) = ∆L ∓kL vξ ±µ B/. For
small displacements and velocities the force can be expanded analogous to Eq. (2.10),
resulting in
F = −βvξ − κξ,
(2.14)
with β the damping coefficient defined in Eq. (2.10), and κ the spring constant given
by
µG
κ=
β,
(2.15)
kL
with µ the effective magnetic moment given by Eq. (2.11) and G the magnetic field
gradient. Note that the magnetic field gradient, produced at the center of the trap,
has in the direction along the coil axis (z direction) twice the strength of that in the
radial direction (x or y direction). The same holds for the spring constant for which
we write κ ≡ κz = 2κx = 2κy .
The motion of the atoms in the MOT is characterized by that of aharmonic oscillator with damping rate γ = β/m and oscillation frequency ω = κ/m [5]. For
typical values of the magnetic field gradient G ≈ 0.1 T/m, the oscillation frequency
is a few kHz, much smaller than the damping rate which is a few hundred kHz.
This leads to an overdamped atomic motion with a damping time 2γ/ω2 less than a
millisecond.
The temperature of the atoms trapped in a MOT is the same as the temperature
of optical molasses [5]. When the atomic density is low enough, the spatial and
momentum distribution of the atoms is Gaussian and the cloud can be characterized
by three rms radii σx , σy , σz , and one temperature T [18]. The equipartition theorem
gives then for each degree of freedom a relation between the temperature and the
16
Chapter 2
radius:
1
1
κi σi2 = kB T .
2
2
(2.16)
Townsend et al. [18] call this regime the temperature limited regime, since the radius
is limited by the temperature. At higher atomic densities the interaction between the
atoms has to be taken into account. Then the density is limited by repulsive forces
between the atoms caused by reabsorption of emitted photons. Densities which
can be reached in MOT’s are limited to ∼ 1011 atoms/cm3 [18]. Ketterle et al. [19]
proposed a scheme in which the photon-induced repulsion is reduced by pumping
the atoms in the center of the trap into a dark ground state that is not sensitive to
the trapping light. Using this scheme, known as the dark spontaneous-force optical
trap (dark SPOT), much higher densities could be reached. Because of the lack of a
dark state, this scheme could not be applied on metastable atom MOT’s. The atomic
density in the metastable neon MOT described in chapter 4 of this thesis is low
enough that it operates in the temperature limited regime.
3.2 Far-off resonance trap (FORT)
Magneto-optical traps have the disadvantage that the frequency of the trapping
beams must be near resonance. This causes density limitations by trap loss due
to excited state collisions and the reabsorption of scattered photons. Dipole traps,
such as a far-off resonance trap (FORT), make use of the dipole force instead of the
spontaneous force. This has the advantage that it can be operated far from resonance, resulting in negligible excited state population [3]. The simplest form of a
FORT consists of a single, linearly polarized, strongly focused Gaussian laser beam.
Since the dipole force given by Eq. (2.9) is conservative, the trapping potential Udip
can be found by integrating the force:
Udip = −
∆L
s0
Fdip dR =
,
ln 1 +
2
1 + (2∆L /Γ )2
(2.17)
which for large laser detunings and high intensities becomes
Udip s0 Γ
.
8∆L /Γ
(2.18)
Because the trap depth is proportional to the light intensity, and inversely proportional to the laser detuning, the trap can be very deep even at large detunings when
high intensities are used. Consequently the fraction of atoms in the excited state
is very low since this fraction is proportional to s0 Γ /(2∆L /Γ )2 . The FORT used for
the photoassociation experiments described in chapter 5 of this thesis consists of a
focused laser beam with a waist w 10 µm and a total power of P = 1 W, resulting
in a trap depth Udip /kB 6 mK. Typical densities in a FORT are ∼ 1012 atoms/cm at
a temperature of a few hundred µK [20]. Because FORT’s are rather difficult to load,
the amount of trapped atoms is normally very low ∼ 104 . However, experiments
have shown that it is even possible to load up to 107 atoms into a FORT [21].
Laser cooling and trapping
17
4 Cold atomic collisions
4.1 Principles
In the experiments described in this thesis, collisions between laser cooled atoms
play an important role. In this section we give a short overview of the physics
involved in atomic collisions at low temperatures. Collisions between two structureless particles are generally described by the technique of potential scattering, in
which the particles interact through a potential V (R) depending only on their relative coordinate R [22]. The effect of the collision is expressed by the total scattering
cross section, which is found by solving the problem quantum mechanically. In the
so-called partial-wave analysis the incoming wavefunction, describing the relative
motion of the particles, is expanded in spherical waves, each with an angular momentum 1. The scattering of each partial wave is found by solving the radial part
of the time-independent Schrödinger equation. For large internuclear distances the
solutions evolve to simple oscillatory functions, and the total elastic scattering cross
section can be written as [22]
σ (k) =
∞
4π (21 + 1) sin2 δ1 (k),
k2
(2.19)
1=0
with k the wave number and δ1 (k) the phase shift of the scattered wave relative to
the incident wave.
The number of partial waves that contribute to the collision depends on the collision energy E = 2 k2 /2µ, with µ the reduced mass. At low energy, the number
is reduced by the rotational barrier created by the rotational energy part Vrot of the
interaction potential:
U (R) = V (R) + Vrot = V (R) +
2 1(1 + 1)
.
2µR 2
(2.20)
If the collision energy is lower than the 1 = 1 barrier, only central collisions (1 = 0)
contribute to scattering. In this so-called s-wave scattering regime the phase shift
varies as δ0 (k) = −ka, with a the s-wave scattering length defined by
a = − lim
k→0
tan δ0 (k)
.
k
(2.21)
For such ultracold collisions the total elastic scattering cross section for two identical particles approaches
k→0
σ (k) = 8π a2 ,
(2.22)
where the factor of 8 instead of factor 4 occurs due to identical particle symmetry [3].
The elastic scattering cross section determines the thermalization time of a dense
sample of atoms. This means that evaporative cooling of a cold sample of atoms with
a large scattering length, e.g. sodium, is easier than cooling of a sample of atoms
with a small scattering length, e.g. hydrogen [5]. The sign of the scattering length is
18
Chapter 2
important for the stability of a Bose-Einstein condensate. A positive scattering length
provides a stable condensate while for a negative scattering length a condensate is
only stable when it contains a small number of atoms. The scattering length can be
estimated in several ways: one can measure the thermalization time of a cold sample
of atoms or precisely determine the ground state interaction potential. The latter can
be done by photoassociation spectroscopy, described later in this chapter. To find
an expression for the potentials describing the interaction between two colliding
particles, i.e. Eq. (2.20), first the atomic structure must be analyzed. Since in this
thesis we describe experiments with rubidium atoms as well as metastable neon
atoms, we describe briefly the atomic structure of alkali-metal atoms and rare-gas
atoms below.
4.2 Atom-atom interactions
Alkali-metal atoms
The electron configuration of a ground state alkali-metal atom, consists of a number of closed shells, called the core, with one valence electron. Since the orbital
momentum of the core is zero, the orbital angular momentum of the atom is fully
determined by the angular momentum of the valence electron. The latter consists of
an angular part l and a spin part s which are coupled to j = l + s. The electron angu
lar momentum is coupled to the nuclear spin i by the hyperfine interaction to form
As an example of the fine and hyperfine structure
total angular momentum f = j+ i.
of alkali-metal atoms we consider the isotope 85 Rb with nuclear spin i = 5/2. The
85
Rb ground state, 2 S1/2 in Russel-Saunders notation, is split by the hyperfine interaction into f = 2 and f = 3 states. Exciting the valence electron to the 5p orbital,
the 2 P1/2 and 2 P3/2 states are formed with f = 2, 3 and f = 1, 2, 3, 4 hyperfine states,
respectively. Figure 2.5 shows a schematic diagram of the 85 Rb energy levels. The
S1/2 (f = 3) → P3/2 (f = 4) transition, which is used for laser cooling, can be pumped
with ordinary diode lasers.
The interaction between two ground state alkali-metal atoms is given by the central, electronic and hyperfine interaction. At long range the central electronic interaction is written as V (R) = −Cn /R n . The dispersion coefficient Cn is for ground
state collisions (S − S) given by the van der Waals interaction with n = 6. At very
long internuclear separation, the hyperfine-interaction dominates and three different atomic hyperfine ground state potentials can be distinguished. If one atom is in
the excited state, the central interaction is dominated by the ±C3 /R 3 dipole-dipole
interaction, which for large internuclear separation is comparable to the spin-orbit
coupling. Neglecting the hyper-fine interaction sixteen S − P -potentials are distinguished. In chapter 5 of this thesis we investigate the spin-orbit coupling between
two of those potentials for the 85 Rb2 -dimer. The spin-orbit coupling can cause finestructure changing collisions.
Inelastic collisions such as fine-structure-changing and hyperfine-changing collisions are important contributions to heating and loss in alkali-metal traps. Trap loss
measurements were done intensively for alkali-metal traps [3]. Typical loss rates,
Laser cooling and trapping
19
f
4
3
2
1
3
5 2P3/2
5 2P1/2
D2
D1
5 2S1/2
2
λ0= 780 nm
3
2
Figure 2.5: Schematic energy diagram of 85 Rb with on the left the fine structure
states and on the right the hyperfine splitting. The D1 (795 nm, 5 S1/2 →5 P1/2 ), D2
line (780 nm, 5 S1/2 →5 P3/2 ), and the transition used for laser cooling are indicated.
usually expressed in cm3 /s, are for hyperfine-changing collisions of the order of
10−11 cm3 /s, and for fine-structure-changing collisions of the order of 10−15 cm3 /s.
This means that a trap with an atomic density of 1011 atoms/cm3 will self-destruct
within one second when hyperfine-changing collisions are not suppressed.
Rare-gas atoms
The rare gases differ from the alkalis in that, as a general rule, the most abundant isotopes have total nuclear spin equal zero, so that hyperfine structure is not
present [23]. On the other hand, singly excited rare-gas atoms are complicated by
extensive fine structure, which we illustrate by considering neon. Figure 2.6 shows
a part of the energy level scheme of neon. A ground state neon atom, 1 S0 in RusselSaunders notation, has an electronic configuration {(1s)2 (2s)2 (2p)6 }. Excitation of
the rare-gas atoms is not easily possible via optical transitions from the ground state
since this requires extreme-ultraviolet lasers. Fortunately, two of the first-excited
states are metastable and can be used as effective ground states in laser cooling
processes. When exciting one electron to a 3s orbital, four fine-structure states in
the {(1s)2 (2s)2 (2p)5 (3s)} configuration are distinguished; two of them, the 3 P0 and
3
P2 states, are metastable and have a calculated natural lifetime τ = 430 s and
τ = 24.4 s, respectively [24]. The second set of ten excited states, numbered by {αi }
in order of decreasing energy, has an electronic configuration {(1s)2 (2s)2 (2p)5 (3p)}
and are all short-lived (lifetime ≈ 20 ns). The lifetime of the {α9 } (3 D3 ) state is
τ = 19.4 ns [25] and the corresponding linewidth Γ = 8.2 (2π )MHz. For laser cooling
and trapping of metastable neon the closed Ne(3s) 3 P2 ↔(3p) 3 D3 optical transition at
a wavelength λ0 = 640.2 nm is used. This wavelength regime can be reached with
20
Chapter 2
αi
19.0
1
2
5
7
3
E (eV)
18.6
4
6
8
9 3D3
10
18.2
λ0= 640.2 nm
1
16.9
3
P0
P1
3
P1
3
P2
16.5
1
S0
0
J=0
J=1
J=2
J=3
Figure 2.6: Partial energy level scheme of neon. For the 3s fine-structure multiplet
we use the Russel-Saunders notation; the 3p multiplet is numbered by αi in order
of decreasing energy. The (3s) 3 P2 ↔(3p) 3 D3 closed level transition, used for laser
cooling, is indicated.
dye-lasers.
The long-range interaction between two metastable rare-gas atoms is given by a
quadrupole-quadrupole term and the Van der Waals interaction: V (R) = −C5 /R 5 −
C6 /R 6 . The quadrupole-quadrupole term is caused by the quadrupole moment of the
unfilled core, and is rather small compared to the Van der Waals interaction because
the core is tightly bound [26]. The S − P interaction is again dominated by the C3 /R 3
dipole-dipole interaction; however, the quadrupole-quadrupole term for the heavier
rare-gas atoms cannot be fully neglected [26].
The internal energy of the rare-gas metastable states is always large enough to
allow for ionization during a collision. The internal energy of the neon 3 P2 state
equals 16.6 eV which has to be compared to the neon ionization energy of 21.6 eV.
The process of ionization for the neon 3 P2 state is given by
Ne(3 P2 ) + Ne(3 P2 ) → Ne(1 S0 ) + Ne+ + e− (PI)
−
→ Ne+
2 (v) + e
(AI).
(2.23)
The first reaction is called Penning ionization (PI) and results in an ion and a ground
state atom, both carrying an energy of 100 to 500 K. In the second reaction, called
associative ionization (AI), a molecular ion is formed which contains the internal
energy in the form of vibrational energy (vibrational level labeled with v in Eq. (2.23)).
Both reactions produce an electron with an energy of the order of 12 eV.
Laser cooling and trapping
21
The process of ionization is a major loss process in metastable atom traps.
This loss process can be introduced in the scattering potential V (R) by adding
a complex term: W (R) = V (R) − iΓ (R)/2, with Γ (R) the so-called autoionization
width. Doery et al. [26] used such potentials to calculate the ionization rate for
metastable neon in an energy range of ≤ 1 mK, and found for an unpolarized sample
Ki = 8×10−11 cm3 /s. This means that a cold sample of metastable neon atoms, with
a typical density of 1010 atoms/cm3 , has an ionization lifetime τi of approximately
one second. Fortunately, the ionization rate can be suppressed by spin-polarizing
the atoms. By spin-polarizing the atoms, the initial state in Eq. (2.23) has total electron spin S = 2, while the final state can only have electron spin S = 0, 1. Since
the ionization process conserves electron spin, ionization is prohibited [26]. Calculations show that in the case of metastable neon the ionization rate may be suppressed by approximately four orders of magnitude [26]. In a magneto-static trap
the ionization lifetime is then approximately 200 s, much higher then the natural
lifetime of the neon 3 P2 state. The calculations done by Doery et al. [26] were based
on modified Na2 potentials, since no experimental data is available for metastable
neon. An experimental technique to derive potential parameters is photoassociation
spectroscopy, which is applied frequently on alkali-metal atoms [27–29] but can also
be done in metastable atom traps [30, 31].
4.3 Photoassociation spectroscopy
A powerful experimental technique to study cold atomic collisions is photoassociation spectroscopy (PAS). The principle of PAS is shown schematically in Fig. 2.7.
Atoms trapped in a MOT or FORT are illuminated by a probe-laser beam with a certain frequency ν. During a collision of two ground-state atoms, one of them can be
excited at a certain internuclear distance and an excited molecular state is formed.
The molecule dissociates again while the molecule decays back to the ground state.
Normally the kinetic energy which is gained during this process is higher than the
trap depth which results in trap loss. This trap loss is measured as a function of the
frequency ν of the probe laser beam.
An excited diatomic molecule can only be formed when the frequency of the
probe laser is resonant with the transition from the ground state to a bound vibrational state v. Scanning the probe laser frequency results in a series of vibrational
energies, which is indicated on the left in Fig. 2.7. Since the collision energy E is of
the order of a few mK, the resolution of the PAS spectrum is of the order of 10 MHz.
From such spectra a lot of information is gained of the collision dynamics.
The spacings between the vibrational levels is determined by the long-range part
of the S − P -potential, i.e., the C3 /R 3 part of the central interaction. The relative intensity of the peaks in the spectrum gives information about the long range part of
the S − S potential, i.e., the C6 /R 6 part of the central interaction, since the transition
rate is determined by the overlap between the ground and excited state wave functions. Knowing the long-range ground state potential the s-wave scattering length
can be determined with only limited knowledge of the short-range potentials [32].
When the collision energy E is higher than the rotational energy barrier for 1 = 1,
22
Chapter 2
v
v-1
v-2
S+P
Signal
ν2
ν1
E
V(R)
S+S
R
R2
R1
Figure 2.7: Principle of photoassociation spectroscopy. During a binary S+S collision
one of the atoms is excited to the P -state and a diatomic molecule is formed. When
the excited molecule decays, the atoms gain kinetic energy and can leave the trap.
This trap loss is measured as a function of the frequency ν of the probe laser beam.
also the rotational energy spacing can be measured. This spacing gives information
about the shape of the S − P potential, as well as possible couplings to other excited state potentials. In Chapter 5 we describe a photoassociation experiment in
which we measured the rovibrational spacings of two excited state potentials of the
85
Rb2 dimer. From the photoassociation data, a qualitative picture of the spin-orbit
coupling between the two states was developed.
References
[1] R. Frisch, Z. Phys. 86, 42 (1933).
[2] S. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,
48 (1985).
[3] J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71(1), 1, (1999).
[4] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-photon interactions:
basic processes and applications, Wiley New York, 1992.
[5] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag
New York, 1999.
Laser cooling and trapping
23
[6] See the special issue on laser cooling and trapping of atoms in J. Opt. Soc. Am.
B 6, 2084 (1989).
[7] P.D. Lett, W.D. Phillips, S.L. Rolston, C.E. Tanner, R.N. Watts, and C.I. Westbrook,
J. Opt. Soc. Am. B 6, 2084 (1989).
[8] P. Lett, R. Watts, C. Westbrook, W.D. Phillips, P. Gould, and H. Metcalf, Phys. Rev.
Lett. 61, 169 (1988).
[9] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[10] M.D. Hoogerland, H.C.W. Beijerinck, K.A.H. van Leeuwen, P. van der Straten, and
H.J. Metcalf, Europhys. Lett. 19, 669 (1992).
[11] B. Sheehy, S-Q. Shang, P. van der Straten, and H.J. Metcalf, Phys. Rev. Lett. 64,
858 (1990).
[12] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji,
Phys. Rev. Lett. 61, 826 (1988).
[13] M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
[14] H. F. Hess, Phys. Rev. B. 34, 3476 (1986).
[15] N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytak, H.F. Hess, and
G.P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[16] M.H. Anderson et al., Science 269, 198 (1995); K.B. Davis et al., Phys. Rev. Lett.
75, 3969 (1995); C.C. Bradley et al., Phys. Rev. Lett. 75, 1687 (1995); D.G. Fried
et al., Phys. Rev. Lett. 81, 3811 (1998).
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2631 (1987).
[18] C.G. Townsend, N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P.
Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).
[19] W. Ketterle, K.B. Davis, M.A. Joffe, A. Martin, and D.E. Pritchard, Phys. Rev. Lett.
70, 2253 (1993).
[20] R.A. Cline, J.D. Miller and D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994).
[21] K.L. Corwin, S.J.M. Kuppens, D. Cho, and C.E. Wieman Phys. Rev. Lett. 83, 1311
(1999).
[22] C.J. Joachain, Quantum Collision Theory, North-Holland, Amsterdam, 1975.
[23] Argon, Helium, and the Rare Gases, edited by G.A. Cook (Interscience Publishers, New York, 1961).
[24] N. Small-Warren and L. Chin, Phys. Rev. A 11, 1777 (1975).
24
Chapter 2
[25] S.A. Kandela and H. Schmoranzer, Phys. Lett. 86a, 101 (1981).
[26] M.R. Doery, E.J.D. Vredenbregt, S.S. Op de Beek, H.C.W. Beijerinck, and B.J. Verhaar, Phys. Rev. A 58, 3673 (1998).
[27] L.P. Ratliff, M.E. Wagshul, P.D. Lett, S.L. Rolston, and W.D. Phillips, J. Chem. Phys.
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Chapter 3
Intense beam of cold metastable
Ne(3s) 3P2 atoms
1 Introduction
The proposal of using resonant laser light to cool and trap neutral atoms, stems
already from 1975 [1, 2]. Since then the cooling and trapping field has made a huge
development starting with the first experimental data by Chu et al. [3] in 1985. Cooling and trapping of atoms made it possible to study collisions between cold atoms.
As a consequence, a lot more is known about the collision dynamics of cold atoms. A
recent review of experiments and theory in cold and ultracold collisions is given by
Weiner et al. [4]. This review includes advances in experiments with atomic beams,
light traps, and purely magnetic traps. Most of the experiments are dealing with
alkali-metal atoms, while collision experiments with metastable rare-gas atoms have
been somewhat neglected [5–9]. This is due to the relative ease of building and operating optical traps for alkali-metal atoms. In the case of rare-gas atoms it is not possible to excite atoms from the ground state, since this requires extreme-ultraviolet
lasers. So no traps can be produced within a vapor cell, like in the alkali-metal case.
Fortunately, two of the first excited states of the rare-gas atoms, with the exception of helium, are metastable and one of them can be used for laser cooling (helium
has only one metastable first excited state). The production of these metastable
states, however, is very inefficient: at best, only about 1 in 103 atoms, excited in a
plasma discharge source, are in the metastable state [10]. Furthermore, high-energy,
background atoms, produced by such a plasma discharge, limit the trap density if
the source is situated near the trapping region. That is why rather complicated
beam-brightening techniques must be applied to be able to reach high trap densities.
In this chapter we describe the experimental techniques to produce an intense
beam of cold metastable Ne(3s) 3 P2 atoms. The beam, with a flux of 5×1010 atoms
per second, has a transverse and longitudinal beam temperature of 285 µK and
28 mK, respectively. This beam can be used for a variety of cold collision experiments, e.g., to study collision rate constants as a function of velocity, alignment and
25
26
Chapter 3
Transverse
Doppler
Cooler
Transverse
Doppler
Cooler
Transverse
Sub-Doppler
Cooler
y
z
Source
Collimator
Bright Beam
Zeeman
Slower
Magneto-Optical
Compressor
Figure 3.1: Schematic view of the neon beam line. Six laser cooling sections are indicated starting from the source with the collimator, the first transverse Doppler
cooler, the Zeeman slower, the second transverse Doppler cooler, the magnetooptical compressor and the sub-Doppler cooler.
orientation of the colliding partners, as was done by Wang and Weiner [11] for thermal alkali-metal collisions and by Thorsheim et al. [12] for subthermal alkali-metal
collisions. In the last decade, in several other places metastable atom beams were
developed [13–17] and were used for various experiments [18–23]. We show in chapter 4 that our intense neon beam is ideal for loading a magneto-optical trap (MOT),
as has been done by Woestenenk et al. [21] and Herschbach et al. [23] for metastable
helium.
Figure 3.1 shows a schematic overview of the neon beam line, with the six laser
cooling sections necessary to create an intense beam of cold atoms. In this chapter
we describe the operation of the different laser cooling techniques which are applied
in the beam line, starting with an overview of the complete setup in section 2. In section 3 a description of the laser collimator and the first transverse Doppler cooling
stage is given.
The collimator captures metastable Ne(3s) 3 P2 atoms from a discharge-excited
supersonic expansion operated with LN2 cooling and collimates them into a parallel
beam. An extra transverse Doppler-cooling section reduces the divergence of the
beam to a few times the Doppler limit. In the Zeeman slower, described in section 4, the atoms are axially slowed from 500 to 100 m/s. Again, an extra transverse
Doppler-cooling section, positioned in between the two Zeeman solenoids, reduces
the divergence of the slowed atomic beam. In section 5, we describe the magnetooptical compressor (MOC), which captures the slowed atoms and funnels them into
a narrow beam. In addition sub-Doppler cooling is achieved in the last portion of
the beam-line. The effects of the sub-Doppler cooler are also shown in section 5.
Finally, in section 6 we give an overview of the beam characteristics and show that
the intense beam of cold neon atoms is very useful for various experiments, e.g., for
loading a magneto-optical trap (MOT).
Intense beam of cold metastable Ne(3s) 3 P2 atoms
27
Table I: Characteristic quantities of the neon atom and the laser cooling transition
Ne(3s) 3 P2 ↔Ne(3p) 3 D3 .
Quantity
Symbol
Value
mass
internal energy of 3 P2 state
wavelength
wavevector
spontaneous decay rate
natural lifetime of 3 D3 state
saturation intensity σ light
saturation intensity π light
recoil velocity
Doppler limit, velocity
Doppler limit, temperature
m
Ei
λ0
k = 2π /λ0
Γ
τ = 1/Γ
I0,σ
I0,π
vr ec = k/m
vD = (Γ /2m)1/2
TD = Γ /2kB
33.2×10−27 kg
16.6 eV
640.225 nm
9.81 × 106 m−1
8.20 (2π ) MHz
19.42 ns
4.08 mW/cm2
7.24 mW/cm2
0.031 m/s
0.29 m/s
196 µK
2 Experimental setup
2.1 Laser setup
All laser-cooling sections use the Ne(3s) 3 P2 ↔(3p) 3 D3 optical transition, which was
already indicated in Fig. 2.6 of section 4.2 in chapter 2. The most important parameters for this transition are given in Table I. An overview of all the optical
components, necessary to prepare the laser light for the different laser cooling sections, is given in Fig. 3.2. All laser cooling sections are operated by using a single
continuous-wave single-frequency ring dye laser, (Coherent, type 899-21). The dye
laser typically has an output power of 700 mW while pumped with 7 W of light from
a argon ion laser, type Coherent Innova 300. The dye laser is locked to a frequency
that is Zeeman shifted almost two line widths (∆L = ωL − ω0 = −1.8Γ ) to the red
of the Ne (3s) 3 P2 ↔(3p) 3 D3 transition by using saturated absorption spectroscopy.
The linewidth of the laser is about 1 MHz. Acousto optic modulators (AOMs) are
used to shift the laser frequency for the collimator and the Zeeman slower. Optical
telescopes are used to expand the laser beams to the required sizes. The laser beam
characteristics of each laser cooling section are given in Table II. The average laser
intensity I
, given in the table, is found by dividing the total laser power P by the
area 2wx,y × 2wz , with the waist radius taken to be the 1/e2 intensity drop of a
Gaussian laser beam.
2.2 Beam diagnostics
Figure 3.3 shows an artist’s impression of the beam line. The total length of the
beam line, taken from the nozzle of the discharge, is approximately 3 m. Besides
28
Chapter 3
700 mW
λ/2
PCB
λ/2
PCB
λ/2
PCB
CT
CT
250 mW
T
400 MHz
AOM
130 mW
90 mW
T
40 mW
λ/2
PCB
λ/2
PCB
Argon Ion Laser
Dye-Laser
T
10 mW
80 MHz
AOM
T
CT
λ/4
T
10 mW
T
80 mW
50 mW
CT
λ/4 λ/2
PD
PD
Ne* Cell
λ/4
CB
Compressor Sub-Doppler
Cooler
Doppler
Cooler
Collimator
Doppler
Cooler
Zeeman
Slower
Figure 3.2: Schematic view of the optical components necessary for the different
laser cooling sections of the neon beam line. All laser cooling sections are operated
with a single dye-laser which is locked 1.8Γ to the red of the Ne (3s) 3 P2 ↔(3p) 3 D3
transition by using saturated absorption spectroscopy. T: optical telescope , CT:
cylindrical optical telescope, CB: cubic beam splitter, PCB: polarizing cubic beam
splitter, PD: photo diode, AOM: acousto optic modulator, λ/2, λ/4: half and quarter
waveplates.
Table II: Laser beam characteristics for the different laser cooling sections.
Section
P
2wx,y × 2wz or 2wr
(mW)
(mm×mm or mm)
s
= I
/I0 1
∆L /Γ
collimator2
4 × 20
51 × 17
0.3
+8
1st Doppler cooler
50
51
0.6
-1.8
Zeeman slower
90
51
1.5
-50
2nd Doppler Cooler
40
85 × 34
0.6
-1.8
MOC3
250
30 × 90
1
-1.8
sub-Doppler cooler
10
5 × 18
1.6
-1.8
1
For the Zeeman slower and MOC circularly polarized light is used, I0,σ =
4.08 mW/cm2 , for the other sections linear polarized light is used, I0,π =
7.24 mW/cm2 (Table I).
2
The collimator laser beam is split in two pairs of laser beams, one pair
for cooling in the x direction and one pair for cooling in the y direction,
respectively.
3
For the MOC not the waist of the laser beam but the dimensions of the
interaction area are given.
Intense beam of cold metastable Ne(3s) 3 P2 atoms
29
Wire Scanners
Source
Collimator
Zeeman Slower
0
1
Compressor
3 z (m)
2
Figure 3.3: Artist’s impression of the neon beam line. The main laser cooling sections and the wire scanners for beam diagnostics are indicated. Table III gives a
detailed overview of the position and the length of the different parts of the setup.
the four main laser cooling sections, also four wire scanners, positioned along the
beam line for beam diagnostics, are indicated. Table III gives detailed information
about the position and length of the different laser cooling sections and the position
of the wire scanners. We take the z axis along the beam axis; relative positions, e.g.,
the length of the laser cooling sections, are indicated with z .
A wire scanner consists of a stainless steel wire that can be moved transversely
through the atomic beam by a stepping motor. When a metastable atom hits the
wire it transfers its internal energy, 16.6 eV for the 3 P2 state, to the metal, thereby
emitting an electron from the wire. This Auger process has a quantum efficiency of
almost unity [24] since the work function of an electron in the metal is about 5 eV. By
scanning the wire in transverse direction through the atomic beam and measuring
the number of emitted electrons from the wire, a one-dimensional beam profile is
generated. The beam profile results from integrating the two-dimensional density
distribution of the atomic beam along the length of the wire:
d/2 l/2
I(x) = ηA e
−d/2
−l/2
l/2
Φm (x, y)dydx ≈ ed
−l/2
Φm (x, y)dy,
(3.1)
with ηA ≈ 1 the quantum efficiency of the Auger process, e the elementary charge, d
and l the diameter and length of the wire, respectively, and Φm (x, y) the transverse
spatial distribution function of the beam of metastable atoms. Note that Eq. (3.1)
is a lower limit for the beam signal since the exact value of the quantum efficiency
of the Auger process for Ne(3 P2 ) atoms on stainless steel is not known; the review
article of Hotop [24] reports values between ηA = 0.3 − 0.91.
We use two types of wire scanners, one type consisting of a single wire with a
diameter of 1 mm, and a second type, a so-called crossed-wire scanner, consisting
of two perpendicular wires with a diameter of 0.1 mm. Behind the collimator we
use a set of two wire scanners of the first type, scanning in the x and y direction,
respectively. A second set of such wire scanners is used 580 mm downstream from
30
Chapter 3
y-Scan
y
y'
y'-Scan
x'
φ=π/4
x
y'-Wire
y-Wire
x'-Wire
x-Wire
Scanner 3
Detection
plate
Scanner 4
Figure 3.4: Crossed-wire scanners downstream of the sub-Doppler cooler used for
beam diagnostics and beam alignment. The x (x ) and y (y ) wires measure independently the beam profile in two perpendicular directions while a scan in the y (y) direction is made. Below the wires of scanner 4, a circularly shaped, optically
transparent detection plate is situated.
Table III: Axial position and length of different components of the beam line.
Device
axial position1 z
length z
(mm)
(mm)
nozzle
0
collimator
43
150
st
1 wire scanner (x,y)
210
st
1 Doppler cooler
230
50
beam stop
300
2nd wire scanner (x,y)
790
1st Zeeman solenoid
1230
850
nd
2 Doppler cooler
2260
40
nd
2 Zeeman solenoid
2370
150
magneto-optical compressor
2670
90
sub-Dopper cooler
2775
18
3rd wire scanner (x,y)
2800
4th wire scanner (x ,y )
2950
Zeeman mirror
3000
1
The axial position indicates the beginning of the different parts of the setup, measured from the nozzle position
(z = 0).
Intense beam of cold metastable Ne(3s) 3 P2 atoms
31
the first set (see Fig. 3.3 and Table III). Just behind the sub-Doppler cooler, and
150 mm downstream of the sub-Doppler cooler, we use a crossed-wire scanner. The
advantage of the crossed-wire scanners, shown schematically in Fig. 3.4, is that by
making a scan in the y (or y) direction, information in the x and y (or x and
y ) direction is obtained. The last crossed-wire scanner also contains an optically
transparent detection area, consisting of a plate of glass covered with a conducting
Indium-Tin-Oxide layer, with which the total atom flux can be measured.
Typical values for the measured normalized wire current I/d are in the range 1
to 100 nA/mm, and can be translated into a 1D-intensity I by using the relation
1 nAmm−1 = 6.25 × 109 atoms s−1 mm−1 .
(3.2)
The total amount Ṅ of atoms in the beam can be found by integrating the beam
profile given by Eq. (3.1) over the scan direction x, or can be approximated by multiplying the height of the beam profile I(0) by the beam diameter dbeam , for which we
take the Full Width Half Maximum (FWHM) of the beam profile:
(3.3)
Ṅ = I(x)dx ≈ I(0)dbeam .
By comparing beam profiles taken with two successive wire scanners, information
about the divergence of the atomic beam can be found. Experimentally the atomic
beam divergence is found by dividing the difference in FWHM beam diameters by
the distance between the two wire scanners, in formula Θ = (d2 − d1 )/(z2 − z1 ). The
beam divergence can also be expressed in terms of the transverse and longitudinal
velocity distribution of the atoms moving in the atomic beam. In these terms we
define the divergence Θ of an atomic beam by the ratio between the FWHM of the
transverse velocity
distribution and the mean longitudinal velocity distribution, in
formula Θ = 2 2 ln(2)σv⊥ /v̄|| , with σv⊥ the rms value of the transverse velocity
distribution and v̄|| the mean of the longitudinal velocity distribution of the atoms.
In section 6 we give an overview of the beam characteristics obtained from the beam
scans made with the wire scanners. In the following sections each of the laser cooling
stages will be discussed in detail, as well as the diagnostics and beam properties after
each stage.
3 Source and collimator
3.1 Metastable atom source
The beam line starts with a discharge-excited supersonic expansion. The source,
schematically shown in Fig. 3.5, produces a beam of metastable Ne(3s) 3 P2 atoms in
a DC discharge that runs through the nozzle of a supersonic expansion [25]. The
nozzle has a diameter of 150 µm, the source pressure is typically 5 mbar, and the
discharge current 7 mA at a voltage of ≈ 400 V. The axial velocity of the atoms is
reduced from approximately 1000 m/s to 500 m/s by cooling the source with liquid
nitrogen. The resulting center-line intensity of metastable 20 Ne atoms is Im (0) =
32
Chapter 3
400 V
LN2
Ne
LN2
Nozzle Skimmer
Figure 3.5: The discharge excited metastable neon source. The boron-nitride nozzle
plate is pressed to a reservoir filled with liquid nitrogen. The discharge is drawn
from the cathode through the nozzle to the skimmer.
Table IV: Characteristics of the metastable neon source.
nozzle diameter
source pressure
discharge current
discharge voltage
center-line metastable intensity Im (0)
mean velocity
vFWHM
excitation fraction
20
Ne isotope fraction
3
P2 fraction
150 µm
5 mbar
7 mA
400 V
2 × 1014 s−1 sr−1
480 m/s
100 m/s
10−4 -10−5
0.91
0.83
2 × 1014 s−1 sr−1 . Other high-energy products emerging from the source are atoms in
the metastable 3 P0 state, metastable atoms of the isotope 22 Ne, and UV photons. In
Table IV characteristics of the source are given.
The atomic beam, produced by the source, has a divergence of Θsource = 350 mrad,
meaning that, at one meter distance from the source the beam has a diameter
dbeam = 350 mm! This results in a very low beam intensity, which is proportional
to (dbeam )−2 . This is the reason why immediately behind the source a collimation
section is placed, which cools the atoms in transverse direction into a nearly parallel
beam.
3.2 Collimating section
After leaving the source, the atoms enter the collimator in which they are captured
and collimated by a two-dimensional optical molasses. For the molasses beams we
use, analogous to Hoogerland et al. [26], curved wave fronts to achieve a large capture angle while keeping the cooling time as short as possible. The curved wave
Intense beam of cold metastable Ne(3s) 3 P2 atoms
33
βn
60 mm
Θc/2
v||
v⊥
α /2
β0
150 mm
Figure 3.6: Schematic view of the collimation section. One pair of mirrors and the
path of one laser beam is shown. The mirrors with a length of 150 mm, are separated
60 mm from each other and make an angle of α = 1.7 mrad. The laser beams
are injected at β0 = 130 mrad, corresponding to a capture angle Θc = 88 mrad.
After each reflection the angle between the laser beam and the atomic beam axis is
reduced with amount α, to keep the atoms in resonance with the laser light.
fronts are produced by using the zig-zag method, which is shown schematically in
Fig. 3.6. The laser light, with detuning ∆L = +65 (2π )MHz, is injected at an angle
β0 = 130 mrad with respect to the plane perpendicular to the atomic beam axis.
With each reflection, the angle β is reduced by an amount α = 1.7 mrad, the angle
between the mirrors. The Doppler shift ∆D seen by the atoms in the collimator is
given by
∆D = −k · v = −kv|| sin β + kv⊥ cos β ≈ −kv|| β + kv⊥ ,
(3.4)
with v|| and v⊥ the longitudinal and transverse velocity of the atoms respectively,
and β the angle with respect to the plane perpendicular to the atomic beam axis.
With the condition ∆D + ∆L = 0 we find for the resonant transverse velocity of the
atoms,
v⊥ (z) = −∆L /k + v|| β(z).
(3.5)
Because β(z) changes while the atoms move downstream through the collimator,
also the transverse velocity v⊥ (z) changes through the collimator. At the entrance
of the collimator this velocity is, for β0 = 130 mrad and ∆L = +65 (2π )MHz, v⊥ (z =
0) = 21 m/s, resulting in a capture angle Θc = 2v⊥ /v̄|| = 88 mrad. At the end of
the collimator at z = 150 mm, i.e., after n = 12 reflections of the laser beam, the
resonance velocity will be v⊥ (z = 150 mm) = 11 m/s, corresponding to a beam
divergence Θcol = 46 mrad.
The advantage of the multiple laser-beam reflections, used in the zig-zag method,
is the reduction in required laser power. The total laser power used for the collimator
is 80 mW, i.e., 20 mW per laser beam (Table II). Other methods, e.g., by using a broad
34
Chapter 3
Current (nA/mm)
60
50
40
30
77 mrad
20
-20
-10
0
10
20
Detector Position (mm)
Figure 3.7: Line-integrated profile of the atomic beam just behind the collimator,
showing Θc = 77 mrad capture range. Full line: beam profile with collimation laser
on; broken line: beam profile with collimation laser off.
converging laser beam [16], would require more than 300 mW of laser power for
achieving the same capture angle.
Figure 3.7 shows experimental beam profiles taken in the x direction with the
first wire scanner which is placed just behind the collimator. The effect of the
collimator is clearly visible. The atoms are captured and cooled in transverse direction, resulting in an atomic beam with a diameter of dcol = 9 mm containing
Ṅcol = 1.1 × 1012 atoms/s in the 3 P2 state. Measured beam characteristics are listed
in Table VI. From the beam profiles in Fig. 3.7 the capture angle Θc of the collimator
can be estimated at Θc = 77 mrad. This is approximately the same as found with
Eq. (3.5).
The large effective detuning kv⊥ at the end of the collimator results in a transverse velocity of almost 11 m/s, as mentioned above. Measurements show a beam
divergence Θcol = 23 mrad, i.e., a transverse velocity of 5.5 m/s, which is still much
more than the Doppler velocity (Table I). This is why an extra transverse Doppler
cooling stage is positioned immediately behind the collimator. Extra transverse
Doppler cooling reduces the divergence of the atomic beam. This can be seen in
Fig. 3.8, which shows beam profiles taken in the x direction with the second wire
scanner, 580 mm downstream of the collimator. Measurements have shown that
with extra transverse Doppler cooling the divergence of the beam reduces to approximately 10 mrad, corresponding to seven times the Doppler limit.
Switching the collimation and Doppler cooling laser off, the beam flux decreases
drastically, as shown by the dashed line in Fig. 3.8. This beam profile contains,
besides atoms in the 3 P2 state, also parasitic products of the source, i.e., atoms in the
metastable 3 P0 state, metastable atoms of the isotope 22 Ne, and UV photons. Those
products could disturb the collision experiments that the atomic beam is used for.
Current (nA/mm)
Intense beam of cold metastable Ne(3s) 3 P2 atoms
35
10
5
0
-20
-10
0
10
20
Detector Position (mm)
Figure 3.8: Influence of extra transverse Doppler cooling behind the collimator. The
beam profiles are taken 580 mm downstream of the collimator; full line: collimation
and Doppler cooling laser on; dash-dotted line: collimation laser only; dashed line:
source only.
Therefore a beam stop, consisting of a disc with a diameter of 3 mm, is positioned
in the center of the beam just behind the transverse Doppler cooler. The beam stop
keeps the parasitic products from reaching the end of the setup, while most of the
atoms in the 3 P2 state can pass due to parabolic character of the beam leaving the
collimator.
4 Zeeman slower
The collimated and transversely cooled atoms enter a midfield-zero Zeeman slower
in which they are decelerated by a counterpropagating laser beam. The changing
Doppler shift ∆D is compensated by making use of the spatial varying Zeeman-effect
in the slower [27]. For this slowing process circularly polarized σ + light is used,
which pumps the atoms to the 3 P2 |mg = +2
↔3 D3 |me = +3
magnetic sublevel
system. The laser beam is coupled into the Zeeman slower through a mirror with
a 3 mm orifice in the center. The mirror is positioned behind the magneto-optical
compressor (see Fig. 3.1) and transmits the atoms which are captured and molded
into a narrow beam by the magneto-optical compressor (section 5).
Using σ + polarized light, the resonance condition for an atom in the |mg = +2
ground state, moving with longitudinal velocity v(z ), can be written as
kv(z ) = −∆L + ∆B = −∆L +
µB
B(z ),
(3.6)
where kv(z ) is the Doppler shift, ∆L the laser detuning, ∆B = (µB /)B(z ) the
Zeeman shift due to the magnetic induction B(z ), and µB the Bohr magneton.
36
Chapter 3
Table V: Characteristics of the Zeeman slower.
length
field strength/current
current
∆L
mean capture velocity
mean final velocity
Solenoid 1
Solenoid 2
850 mm
7.39 × 10−3 T/A
3.7 A
−400 (2π ) MHz
480 m/s
150 mm
−2.56 × 10−3 T/A
6.8 A
−400 (2π ) MHz
98 m/s
Slowing atoms with initial longitudinal velocity vi to final velocity vf , with uniform deceleration a = ηamax = ηkΓ /(2m), η ≤ 1, requires a distance
∆z =
zf
−
zi
=
vi2 − vf2
2a
,
which gives for the longitudinal velocity v(z )


2
− z 
v
z
f
i
1− 2 .
v(z ) = vi 1 − zf − zi 
vi 
(3.7)
(3.8)
Combining this expression with the resonance condition, Eq. (3.6), the magnetic field
B(z ) must have the following shape to get uniform deceleration


2
− z 
v
z
k
f
i
1− 2 .
(3.9)
B(z ) =
∆L +
vi 1 − µB
µB
zf − zi 
vi 
The required field is produced by two solenoids producing fields in opposite
direction. In between the solenoids the magnetic field is zero. In the first solenoid
the atoms are slowed down from v(z = zi1
) = vi1 to v(z = zf 1 ) = vf 1 , over
a distance ∆z1 = zf 1 − zi1
. In the second solenoid they are slowed further from
v(z = zi2 ) = vi2 = vf 1 to v(z = zf 2 ) = vf 2 over a distance ∆z2 = zf 2 −zi2
. Knowing
from the resonance conditions in between the two solenoids that ∆L = −kvf 1 we can
write the magnetic field of the two coils as



2 

v
z − zi1
vi1 f1 

1 − zi1
B(z ) = B0 1 −
≤ z ≤ zf 1 ,
(3.10)
1 −
2 
vf 1
zf 1 − zi1
vi1



2 

vf 2 
z − zi2

B(z ) = B0 1 − 1 − ≤ z ≤ zf 2 ,
(3.11)
zi2

1 −
z − zi2
v2 
f2
with B0 = ∆L /µB .
f1
1.0
0.6
0.8
0.4
0.6
0.2
∆B
0.4
0.2
0.0
0.0
kvi
|∆L|
kv(z)
-0.2
kvf
0.0
37
∆B/2π (GHz)
kv/2π (GHz)
Intense beam of cold metastable Ne(3s) 3 P2 atoms
0.5
1.0
z (m)
-0.4
1.5
Figure 3.9: Schematic representation of the Zeeman slowing process. Full line: Zeeman shift caused by the magnetic field of the Zeeman slower as a function of the
distance along the beam line (z = 0 indicates the beginning of first magnet of the
Zeeman slower); dashed line: resonance velocity kv(z) of an atom on its trajectory
through the Zeeman slower, entering the Zeeman slower with velocity vi = 490 m/s
and leaving it with velocity vf = 100 m/s. Here ∆L = −400(2π ) MHz indicates the
detuning of the slowing laser.
To keep the atoms in resonance during the slowing process, the change in magnetic field must not exceed the changing Doppler shift [28]
dB
k a
k ηkΓ
≤
=
.
dz
µB v(z )
µB 2mv(z )
(3.12)
To fulfill to this expression we choose the total length of the solenoids such that the
atoms can follow the magnetic field with a fraction of one tenth of the maximum
radiation force, i.e., η = 0.1. In table V the characteristics of the Zeeman slower are
given. The advantage of a midfield-zero Zeeman slower is that the longitudinal velocity spread of the atoms leaving the slower is minimized because the high magnetic
field at the end of the slower forces the atoms to get very abruptly out of resonance.
A second advantage of a midfield-zero Zeeman slower is the possibility of putting a
vacuum pump in between the solenoids.
The Zeeman slowing process is schematically shown in Fig. 3.9, which depicts
the trajectory of an atom at resonance condition as given by Eq. (3.6). The longitudinal velocity of the atoms leaving the Zeeman slower is measured by a standard
Time-of-Flight (TOF) technique. Figure 3.10 gives the velocity distribution of unslowed, partially slowed (second Zeeman solenoid operated with reduced current),
and fully slowed atoms. The fully slowed atoms have a final velocity of approximately 100 m/s. Measurement show that approximately 50% of the atoms, entering
the Zeeman slower are slowed to 100 m/s.
38
Chapter 3
Signal (arb. units)
20
15
8 m/s
10
80
100
120
5
0
0
200
400 600 800
Velocity (m/s)
1000
Figure 3.10: Measured longitudinal velocity distributions. Dashed line: initial velocity distribution, average velocity 500 m/s; open circles: only slowing in the first
magnet of the Zeeman slower, average velocity 200 m/s; closed circles: slowing in
the full Zeeman slower, average velocity 100 m/s. Inset: Close-up of 100 m/s velocity
distribution showing 3.4 m/s rms width, i.e., 8 m/s FWHM.
Current (nA/mm)
10
8
6
4
2
0
-2 -1
0
1
2
Detector Position (mm)
Figure 3.11: Line-integrated beam profile just behind the magneto-optical compressor showing the influence of extra transverse Doppler cooling between the two
solenoids of the Zeeman slower. Full line: beam profile with extra Doppler cooling; dashed line: beam profile without extra Doppler cooling.
Intense beam of cold metastable Ne(3s) 3 P2 atoms
39
The atoms are not only slowed in the Zeeman slower but also cooled in the longitudinal direction. The temperature of the atoms can be derived from the width of the
velocity distributions, using the relation kB T|| = mσv2|| , with σv|| the rms width of the
longitudinal velocity distribution. The 100 m/s velocity distribution has a 3.4 m/s
rms width, corresponding to a longitudinal beam temperature of T|| = 28 mK; this
has to be compared with the temperature 4.4 K of the 500 m/s atoms.
During the slowing process the transverse velocity of the atoms increases due to
the randomness in the direction of the spontaneously emitted photons. To counteract this effect, an extra transverse Doppler-cooling stage is put in between the two
solenoids of the Zeeman slower. Transverse Doppler-cooling is possible because the
magnetic field is almost zero between the two solenoids, as can be seen in Fig. 3.9.
Reducing the divergence of the atomic beam increases the flux of atoms within the
capture range of the magneto-optical compressor. Extra transverse cooling increases
the beam flux with approximately 50%, this can be seen in Fig. 3.11 which shows the
beam profile behind the magneto-optical compressor.
5 Magneto-optical compressor and sub-Doppler cooler
Due to the slowing process in the Zeeman slower, the diameter and divergence of the
atomic beam has increased significantly. This results in a small phase space density
(see section 6). A common way to compress an atomic beam in phase space, is to
use a two-dimensional version of a magneto-optical trap [29], a so-called magnetooptical compressor (MOC). Below we give a description of the MOC, analogous to the
description of the magneto-optical trap in section 3.1 of chapter 2.
5.1 Compression of atomic beam
The principle of the MOC is given in Fig. 3.12, showing two orthogonal pairs of
circularly polarized laser beams, intersecting at the center of a magnetic quadrupole
field produced by four permanent magnets. To explain the operation of the MOC we
consider a hypothetical atom with a J = 0 → J = 1 transition. This simplification is
chosen because for the case of the J = 2 → J = 3 transition for neon, the ongoing
redistribution over the five magnetic sublevels of the lower state complicates the
process [30].
The force on the atom is given by the sum of the spontaneous forces F± exerted
by the σ + and σ − beams [29]
F (x, vx ) = F+ (x, vx ) + F− (x, vx ),
kΓ
s0
with F± (x, vx ) = ±
,
2 1 + s0 + (2∆± (x, vx )/Γ )2
µB ∂B
∆± (x, vx ) = ∆L ∓ kvx ∓
x,
∂x
(3.13)
(3.14)
(3.15)
where s0 is the saturation parameter for zero detuning, ∆± (x, vx ) the effective detuning with ∆L the laser detuning, kvx the Doppler shift due to transverse velocity
40
Chapter 3
σ+
σ+
B
σ-
σFigure 3.12: Principle of a two dimensional magneto-optical trap. The solid arrows
indicate the magnetic quadrupole field resulting from the four permanent magnets.
The open arrows indicate the light field, consisting of two pair of counterpropagating
σ + − σ − polarized laser beams. The atoms move perpendicular to the plane of the
laser beams, at the center of the quadrupole field.
15
x (mm)
10
5
0
-5
-10
-15
-100 -75 -50 -25
z' (mm)
0
Figure 3.13: Monte-Carlo simulation of some atomic trajectories in the MOC, showing
compression of the atomic beam.
Intense beam of cold metastable Ne(3s) 3 P2 atoms
41
y
B
B
σx
σ+
B
B
Figure 3.14: Schematic view of the magneto-optical compressor. The mirror section
provides counterpropagating σ + -σ − laser beams in the x and y direction. The
shaded section indicates the area where the atoms, moving perpendicular to the
plane of the laser beams, interact with the laser light.
vx , and (µB /)(∂B/∂x)x the Zeeman shift due to the local transverse magnetic field.
The sign difference in the Zeeman term is caused by the excitation of the different
magnetic subtransitions, mg = 0 → me = 1 for the σ + -beam and mg = 0 → me = −1
for the σ − -beam.
For each position x there exists an equilibrium velocity vx,eq = −(µB /k)(∂B/∂x)x
towards the beam axis, for which the forces F± are balanced. As a consequence,
the atoms are cooled to this equilibrium velocity and will approach the axis like an
overdamped harmonic oscillator; the cooling force can, for small displacements and
velocities, be written as [31]
F (x, vx ) = −β(vx − vx,eq ) = −βvx − κx,
(3.16)
which is similar to Eq. (2.14) of chapter 2. In Fig. 3.13 some simulated atomic trajectories through the MOC are depicted. Atoms captured within the spatial capture
radius xc ≈ 10 mm are dragged inwards to the atomic beam axis.
Figure 3.14 shows a schematic view of the MOC implemented in the beam-line.
The two pairs of σ + − σ − laser beams are produced by recycling a single σ + laser
beam by a mirror section. The transverse size of the MOC equals an area of 2wx,y ×
2wz = 30 × 90 mm2 , the laser beam has an average saturation parameter of s
=
I
/I0,σ = 1 and has a detuning of ∆L = −1.8Γ (see Table II).
The magnetic quadrupole field of the MOC is generated by four permanent magnets mounted outside of the vacuum system. The magnets, which are made of a
Nd-Fe-B alloy, have dimensions 60×60×30 mm3 and a magnetization of 1.15 T. At
the position of the Sub-Doppler cooler four much smaller magnets are positioned
(10×5×5 mm3 , magnetization 1.15 T). These magnets, which can individually be
42
Chapter 3
∂B/∂x (T/m)
0.6
SG-effect
0.4
SD
0.2
MOC
0.0
-100
0
100
z' (mm)
200
Figure 3.15: The gradient of the radial quadrupole magnetic field due to the MOCmagnets and the compensating sub-Doppler magnets. The arrows indicate the interaction regions of the MOC and the sub-Doppler cooler, and the region from where
the Stern-Gerlach force plays a role (Section 5.2). The position z = 0 corresponds to
the end of the interaction area, i.e., the center of the magnets of the magneto-optical
compressor.
dB/dx=0.6 T/m
0.6
dB/dx=0.3 T/m
F/Fmax
dB/dx=0.1 T/m
0.3
0.0
-0.3
-0.6
-15 -10 -5
0
5
10
15
x (mm)
Figure 3.16: Position dependency of the transverse force on the atoms in the MOC
for vx = 0 m/s, ∆L = −1.8Γ , and s0 = 1. Full line: force at the end of the MOC
(z = 0 mm), field gradient ∂B/∂x = 0.6 T/m; dashed line: force half way in the
MOC (z = −50 mm), field gradient ∂B/∂x = 0.3 T/m; dash-dotted line: force at the
entrance of the MOC (z = −90 mm), field gradient ∂B/∂x = 0.1 T/m.
Intense beam of cold metastable Ne(3s) 3 P2 atoms
43
Current (nA/mm)
7
6
5
4
1 mm
3
2
1
0
-20
-10
0
10
20
Detector Position (mm)
Figure 3.17: Line-integrated profile of atomic beam, taken with the vertical wire
of the 3rd wire scanner, positioned 40 mm behind the MOC. Full line: data with
compression laser on; dashed line: compression laser off. The dip, left of the peak
in the beam profile with compression laser on, is caused by electrons which are
emitted from the second wire of the crossed-wire scanner.
moved in radial direction, are situated much closer to the beam axis and compensate locally the magnetic field generated by the MOC magnets. Figure 3.15 shows
the magnetic field gradient of the MOC and sub-Doppler cooler. At the entrance
of the MOC the quadrupole field gradient has a strength of 0.1 T/m and increases
linearly to almost 0.6 T/m at the end of the MOC. The field gradient drops to approximately zero at the position of the sub-Doppler cooler, due to the influence of
the compensation magnets.
The transverse force on the atoms, for vx = 0 ms−1 and ∆L = −1.8Γ at three
different longitudinal positions in the MOC is given in Fig. 3.16. The curve depicting
the force at the entrance of the MOC shows a spatial capture radius of approximately
xc = 10 mm, which was also shown in Fig. 3.13.
Figure 3.17 shows a beam profile measured with the vertical wire of the third wire
scanner. The effect of the MOC is visible: the atoms are molded into a narrow beam
with a diameter of dMOC = 1 mm, containing ṄMOC = 5 × 1010 atoms/s. This results
2
in a particle density of n = Ṅ/(π x⊥
v̄|| ) = 6.4 × 108 atoms/cm.
Measurements show that a 0.25 fraction of the slowed atoms are captured by
the MOC. A beam profile measured with one of the wires of the fourth wire scanner
is given in Fig. 3.18, showing a beam profile with a diameter of 2 mm. From the
beam profiles given in Fig. 3.17 and Fig. 3.18, the divergence of the atomic beam was
measured to be equal to 8 mrad which approximately corresponds to the transverse
Doppler velocity: vD ≈ v⊥ = 0.35 m/s. This results in a transverse beam temperature
T⊥ = 285 µK. In Table VI the characteristics of the atomic beam are given.
44
Chapter 3
Current (nA/mm)
5
4
3
2
2 mm
1
0
-5
0
5
Detector Position (mm)
Figure 3.18: Line-integrated profile of atomic beam 190 mm behind the MOC. Full
line: data with compression laser on; dashed line: compression laser off.
5.2 Sub-Doppler cooling of the atomic beam
To decrease the divergence of the atomic beam to a value less than the Doppler limit,
a transverse sub-Doppler cooling stage is situated 15 mm downstream of the MOC.
Two orthogonal pairs of linearly polarized, counterpropagating laser beams give rise
to a two-dimensional molasses with polarization gradients. This cooling technique,
usually referred to as Sisyphus cooling [32], is expected to give a transverse beam
velocity of 0.11 m/s, i.e., three times less than the Doppler limit.
Steering
We systematically investigated the effects of sub-Doppler cooling on the atomic
beam [33]. Figure 3.19 shows beam profiles measured with the fourth wire scanner. Measuring the beam profile with this wire scanner gives a distribution in the x and y direction. Each profile is measured with sub-Doppler cooling laser on and off.
We found that the sub-Doppler cooling stage has only steering effects on the atomic
beam: no decrease of the atomic beam divergence was observed at all. The steering
effects are clearly visible in Fig. 3.19; here, the sub-Doppler forces push the atomic
beam upwards by giving the atoms a transverse velocity vy = 3 m/s. We found that
those steering effects can be explained very well in terms of velocity-selective resonances (VSR) [33–35]. The VSR-model says that a resonance velocity of vy = 3 m/s,
for example, corresponds to a local magnetic field in the sub-Doppler cooler of 6 G,
which is possible when the atomic beam enters the sub-Doppler cooler not exactly
in the middle, where the magnetic field is compensated.
We applied the steering effects for aligning the atomic beam through the orifice
Intense beam of cold metastable Ne(3s) 3 P2 atoms
45
Current (nA/mm)
4
3
x'
y'
2
1
0
0 5 10 15 20 25
Detector Position (mm)
Figure 3.19: Influence of the sub-Doppler cooler on the atomic beam. Beam profiles
in the x and y -direction are obtained by scanning crossed-wire scanner number 4
in the y-direction. Full lines: sub-Doppler laser on; dashed line: sub-Doppler laser
off.
in the Zeeman mirror. This can be done by changing the radial positions of the
compensation magnets of the sub-Doppler, thereby changing the magnetic field at
the position of the sub-Doppler light field and, selecting different resonance velocity
conditions.
Stern-Gerlach effect
The fact that no sub-Doppler velocities where observed can be explained in terms
of Stern-Gerlach forces which the remaining magnetic field downstream of the subDoppler cooling stage exert on the atoms [33], as first reported by Koolen [36]. With
inhomogeneous magnetic fields it is possible to deflect atomic beams [37]. By creating quadrupole or hexapole magnetic fields it is even possible to focus atomic beams,
as described by Kaenders et al. [38]. The internal energy of an atom with magnetic
changes by
in an external magnetic field B
dipole moment µ
Vdip = −
µ · B.
(3.17)
Considering a moving atom that experiences only small variations in the external
field, i.e., so-called adiabatic motion, the projection, mg , of the atom’s magnetic
dipole moment onto the external field remains constant. The corresponding SternGerlach force is then given by
= −µB gg mg ∇|B|,
B
FSG = (
µ · ∇)
with µB the Bohr magneton and gg (=1.5 in our case) the Landé factor.
(3.18)
46
Chapter 3
The Stern-Gerlach force can be ignored in the MOC and sub-Doppler cooler, because there the optical forces dominate. In the region downstream of the subDoppler cooler, however, there is no light field but there is still a strong radial
magnetic field gradient from the MOC-magnets (Fig. 3.15).
Let us consider the influence of the Stern-Gerlach force on the atoms moving
in the region indicated in Fig. 3.15, which shows the radial magnetic field gradient.
Solving Newton’s second law, we find that an atom in the |mg magnetic substate is
accelerated to
µB gg mg
|∂B/∂r | dz,
(3.19)
∆v⊥,SG = −
mv̄||
with m the atomic mass, v̄|| the longitudinal velocity of the atoms, and ∂B/∂r the
remaining radial magnetic field gradient as indicated in Fig. 3.15. Due to the dependence of the Stern-Gerlach force on the magnetic substate, the atoms are deflected
in 2J + 1 different directions. From the measured magnetic field gradient given in
Fig. 3.15 we estimate that ∆v⊥,SG = 0.2 m/s for the |mg = 2
magnetic substate.
This leads to an additional beam divergence of 4 mrad. Presumably this contributes
to the measured transverse velocity spread v⊥ = 0.35 m/s, which equals a beam
divergence of 8 mrad.
6 Beam characteristics
In the previous sections we showed that, by using several sets of wire scanners, it
is possible to estimate the characteristics of the atomic beam such as the flux Ṅ
of atoms, the beam diameter and divergence. In atomic beam physics often the
brightness and brilliance are given to characterize a beam of atoms. Below we follow
Lison et al. [39] to give explicit definitions for the brightness and brilliance of an
atomic beam.
6.1 Brightness and brilliance
2
The flux density or intensity of a circularly shaped atomic beam is defined as Ṅ/π x⊥
,
where x⊥ is the beam radius. The geometrical solid angle occupied by the atoms in
a beam is Ω = π (v⊥ /v̄|| )2 , with v⊥ and v̄|| the transverse and average longitudinal
velocity of the atoms, respectively. Then the brightness is defined as the flux density
per solid angle, and is given by
Ṅ
R=
.
(3.20)
2
π x⊥
Ω
In analogy with the frequency spread of optical beams, i.e., coherence length,
atomic beams can be characterized by their longitudinal velocity spread σv|| . The
spectral brightness or brilliance of an atomic beam, is then given by
B=R
v̄||
.
σv||
(3.21)
Intense beam of cold metastable Ne(3s) 3 P2 atoms
47
Note that both R and B have the same dimensions as flux density, usually given in
terms of s−1 mm−2 mrad−2 , and that for thermal atomic beams R and B have similar
values because v̄|| /σv|| 1.
Furthermore, the brilliance is related to the Liouville phase-space density Λ,
which is the number of particles N per unit of phase-space volume. The relation
between B and Λ can easily be found by splitting the spatial and momentum coordinates in transverse and longitudinal components, which results for a circularly
shaped atomic beam in
1
N
Λ=
.
(3.22)
2
π (x⊥ p⊥ ) x|| p||
Writing, Ṅ = N v̄|| /x|| , this can be can be expressed as
Λ=B
π
,
m3 v̄||4
(3.23)
with m the atomic mass. The quantum limit occurs when the quantity defined by
Λ̃ = Λh3 , with h Planck’s constant, is unity.
6.2 Overall characteristics
From the beam characteristics, measured with the different sets of wire scanners,
we calculated the brightness, brilliance and phase space density of our atomic beam,
by using Eqs. (3.20), (3.21), and (3.23), respectively. In Table VI the measured beam
characteristics, such as the beam flux Ṅ, the beam diameter d = 2x⊥ , the longitudinal velocity and the rms value of the transverse velocity are given. With these values
the local values of the brightness, brilliance and phase space density were calculated,
i.e., the values at the axial position given in the second column of the Table VI (the
Table VI: Characteristics of the atomic beam. All quantities are measured locally at
the positions z (second column) of the different sections mentioned in the first column. S: source, C: collimator, D: Doppler cooler, Z: Zeeman slower, MOC: magnetooptical compressor, and SD: sub-Doppler cooler.
Section
S
C
D
Z
MOC
SD
1
R and
z
Ṅ
(mm)
×1010
(s−1 )
0
210
790
2800
2800
2950
B given
d
(mm)
v̄||
(m/s)
σv ⊥
R1
B1
Λh3
×1020
×1020
×10−13
288
0.55
1.27
1.02×10−3
15.0
3.5
2733
5.3
12.0
2.5×10−2
370
85.0
1285
2.5
5.6
6.7
1×105
2.3×104
(m/s)
3500
0.15
480
71
110
9
480
4.8
110
15
480
1.9
19.2
45
98
1.9
4.8
1.0
98
0.35
4.8
2.1
98
0.35
−1 −2 −1
in terms of s m sr .
48
Chapter 3
positions of the different wire scanners). Those calculated values are given in the
last three columns of Table VI.
Looking at the measured beam flux, Ṅ, it is clear that, moving from the collimator
to the end of the setup, more than a factor of 20 in beam flux is lost. This is partially
caused by the slowing process in the Zeeman slower. The Zeeman slower slows only
50% of the atoms to 100 m/s, and a significant fraction of the slowed atoms do not
reach the entrance of the MOC, because the divergence of the atomic beam increases
due to the slowing process. Furthermore, only 25% of the atoms which reach the
entrance of the MOC are captured by the MOC.
As mentioned in section 2.2, the beam divergence can be found by comparing
beam profiles taken with two successive wire scanners. From this it is found that
the divergence of the atomic beam decreases from 350 mrad at the source to 8 mrad
downstream of the magneto-optical compressor. This increases the brightness and
brilliance of the atomic beam drastically, which can be seen more clearly in Table VII,
which gives the values of the brightness, brilliance and phase space density at the
end of the setup for different operation modes of the atomic beam machine, going
from limited operation (only source on) to full operation (all laser cooling sections
on). All values are given relative to the full operation values, which are given in the
last row of Table VI. A graphic representation of this is given in Fig. 3.20 which shows
a plot of the brightness and brilliance as a function of the phase-space density, for
different operation modes of the beam machine.
From Table VII and Fig. 3.20 it is clear that the brightness, brilliance and phasespace density increase a lot when going from less to full operation. Switching on the
Zeeman slower, decreases the brightness and brilliance, due to the increasing beam
diameter and divergence and the loss-processes mentioned earlier. The magnetooptical compressor compensates this loss in brightness and brilliance. Comparing
the operation mode with only the source on with the full operation mode, we see that
the brightness and brilliance increase six orders of magnitude and the phase-space
Table VII: Brightness, brilliance and phase-space density at the end of the setup
(z = 2950 mm) relative to the full operation values. RAll = 3.5 × 1020 s−1 m−2 sr−1 ,
BAll = 8.5 × 1021 s−1 m−2 sr−1 , and Λ̃All = 2.3 × 10−9 (Table VI). S: only the source
on, S+C: source and collimator on, S+C+D: source, collimator and Doppler cooler
on, S+C+D+Z: source, collimator, Doppler cooler and Zeeman slower on, All: full
operation: all laser cooling sections on.
Section
R/RAll
B/BAll
Λ̃/Λ̃All
S
S+C
S+C+D
S+C+D+Z
All
1×10−6
3×10−3
8×10−2
6×10−5
1
7×10−7
1×10−3
3×10−2
6×10−5
1
1×10−9
2×10−6
5×10−5
6×10−5
1
Intense beam of cold metastable Ne(3s) 3 P2 atoms
R ,B (s -1m-2sr-1)
1022
Brightness
Brilliance
1020
49
All
S+C+D
S+C
1018
S+C+D+Z
1016
S
1014
10-18 10-16 10-14 10-12 10-10 10-8
Λ h3
Figure 3.20: Plot of the brightness R (open circles) and the brilliance B (solid circles) vs phase-space density at the end of the setup (z = 2950 mm). S: source only;
S+C: source and collimator on; S+C+D: source, collimator and Doppler cooler on;
S+C+D+Z: source, collimator, Doppler cooler and Zeeman slower on; All: full operation: all laser cooling sections on.
0.5
Current (nA/mm)
20
Ne
2
1
0
-2 -1
0
1
2
Detector Position (mm)
Current (nA/mm)
3
0.4
22 Ne
0.3
0.2
0.1
0.0
-2 -1
0
1
2
Detector Position (mm)
Figure 3.21: Beam profiles of a bright beam of the two bosonic neon isotopes
and 22 Ne.
20
Ne
50
Chapter 3
density nine orders of magnitude. The latter is still nine orders of magnitude from
the quantum limit (BEC).
By locking the dye-laser on the other bosonic isotope of neon, 22 Ne with natural
abundance 9%, we can also produce a cold and intense beam of this isotope. This
can be seen in Fig. 3.21, which shows atomic beam profiles of both isotopes. The
22
Ne atomic beam has the same width and divergence as the 20 Ne atomic beam, and
contains, seven times less atoms than the 20 Ne beam. The 22 Ne beam contains more
atoms than would be expected from the natural abundance of 9%, which is probably
caused by more efficient cooling of the laser cooling sections.
7
Concluding remarks
We produced a cold and intense beam of metastable Ne(3s) 3 P2 atoms. The cold
beam, with a diameter of 1 mm, has a flux of 5 × 1010 atoms per second. The beam
divergence of 8 mrad corresponds to a transverse beam temperature of T⊥ = 285 µK,
which is equal to the Doppler temperature. The longitudinal temperature of the
beam is T|| = 28 mK. The particle density immediately behind the magneto-optical
compressor is n = 6 × 108 atoms/cm3 , and the phase-space density Λh3 = 1 × 10−8 .
While the bright neon beam of Schiffer et al. [18], has a much larger phase-space
density Λh3 = 2 × 10−6 (due to the fact that they cool their beam in transverse
direction to sub-Doppler temperatures), their beam contains 103 fewer atoms.
Our cold intense atomic beam can be used for a whole range of cold collision
experiments, e.g., Penning electron energy spectroscopy [40] and photo-association
spectroscopy [41]. The beam can also be used for loading atom traps. In chapter 4 of
this thesis we show that the bright neon beam is an excellent candidate for loading
a magneto-optical trap (MOT). Thanks to the high loading rate, created by the beam,
we can easily make a MOT containing more than 109 metastable neon atoms.
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Niehaus, Eur. J. Phys. D 4, 131 (1998).
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Verhaar, Phys. Rev. A 57, 3603 (1998).
Chapter 4
Metastable neon atoms in a
magneto-optical trap
1 Introduction
The successful Bose-Einstein condensation (BEC) of alkali atoms has stimulated several groups [1–3] to extend the range to other species such as metastable rare gas
atoms in particular, with helium and neon being the prime candidates [4]. These
metastable atoms distinguish themselves from the alkalis by their immense internal energy of 4 to 20 eV, for Xe to He, respectively. This makes them particularly
interesting new candidates for BEC. The internal energy can be used for efficient
detection schemes leading to real time diagnostics [4] of the phase transition and
collective deexcitation phenomena in the UV range. The internal energy also makes
it a challenge because Penning ionization leads to high inelastic collisional loss rates
already at relatively low densities as compared to alkali atoms.
In our group we aim at reaching BEC with metastable neon atoms. The first
step is to trap large samples of metastable neon atoms in a magneto-optical trap
(MOT), which can then be transferred to a magnetic trap and evaporatively cooled.
Evaporative cooling is greatly facilitated when the initial number of atoms is high,
i.e., 109 −1010 atoms. In this chapter we show how such large amounts of metastable
neon atoms can be trapped in a MOT.
In contrast to alkali atoms a vapor cell MOT is not possible with metastable
atoms. Our MOT is loaded with the bright and slowed atomic beam described in
chapter 3. This allows for MOT loading rates of over 1010 atoms/s. The axial velocity
of the atoms exiting from the beam machine is still 100 m/s, so that for efficient
trapping an additional Zeeman slower is necessary. This introduces an additional
laser beam to the MOT which cannot be neglected in the description of the MOT’s
dynamics, i.e., a seven beam MOT is created. This seems awkward at first but nevertheless allows for a MOT containing 9 × 109 atoms, to our knowledge the largest
MOT of metastable atoms reported. In this chapter we describe the characteristics
of such a MOT.
This chapter is organized as follows. We start in section 2 with a description of
the dynamics of a metastable atom MOT. In particular we develop a model for a MOT
53
54
Chapter 4
disturbed by a seventh laser beam. The experimental setup is described in section 3.
In the sections 4 to 8 we describe the experiments we did to investigate the behavior
of the MOT. We end in section 9 with concluding remarks.
2 Trap dynamics
In this section we describe the physics necessary to understand the behavior of the
metastable neon MOT that is described in this chapter. The MOT is loaded with
atoms from the bright atomic beam, which is described in chapter 3. The atoms
are decelerated just before the MOT. This has the consequence that the laser beam,
necessary for the slowing process, intersects with the trapping beams of the MOT.
This results in a seven beam MOT instead of an ordinary six beam MOT, during the
loading of the MOT. The influence of this seventh laser beam on the trap dynamics
is described in this section. We start with a description of the population of atoms
in a metastable atom MOT.
2.1 Trap population
The evolution of the number of atoms N(t) in the MOT fulfills the equation
N(t)
Ṅ(t) = R −
− β n2 (r, t)d3 r .
τ
(4.1)
Here, R is the loading rate, τ the time constant representing loss processes which
are independent of the number of trapped atoms (e.g., collisions with background
gas atoms), β the rate constant that describes the density dependent losses (which
arise from two-body collisions between trapped atoms) and n(r, t) is the density
distribution.
The optical pumping process in the trap has the effect that a small fraction of the
trap population is in the excited P -state. The collision between two particles can be
a collision of three types: a S − S, S − P or P − P collision. The two-body loss rate β
is thus written as a function of the populations ΠS(P ) in the S (P ) state:
β = 2(ΠS2 KSS + 2ΠS ΠP KSP + ΠP2 KP P ),
(4.2)
with KSS(SP ,P P ) the rate coefficients for S − S (S − P , P − P ) collisions, respectively.
Experiments with metastable helium atoms [5–8] show that KSP is two orders of
magnitude larger than KSS and that the KP P can be neglected [5]. Mastwijk et al. [9],
however, report a KSP which is only a factor of 20 larger then KSS , for metastable
helium. No experimental results are reported for metastable neon atoms but there
is no reason to expect a larger ratio between KSP and KSS for metastable neon.
Moreover, our trap is operated with relatively low laser beam intensities, where the
fraction of atoms in the excited state is less then one percent. In the detuning range
where we operate the trap, the excited state population can be neglected, so that
from now on we write β ≡ βSS = 2KSS .
Metastable neon atoms in a magneto-optical trap
55
Generally, the density in metastable atom MOT’s is low enough to assume that
the spatial distribution is independent of the number of trapped atoms, i.e., n(r, t) =
n(r)f (t) [8]. In this so-called temperature limited regime [10] it is convenient to
assume a Gaussian density distribution with central density n0 and rms radius σρ in
radial direction and σz in axial direction, the density distribution is given by
ρ2
z2
n(r) = n(ρ, z) = n0 exp −
−
.
(4.3)
2σρ2 2σz2
Here the central density is related to the volume V via N = n0 V , with the volume of
the atom cloud given by
V = (2π )3/2 σρ2 σz .
(4.4)
By defining an effective trap volume [11] given by
Veff = N2
= 23/2 V ,
2
3
n (r)d r
(4.5)
Eq. (4.1) can be written as
Ṅ(t) = R −
N(t) βN 2 (t)
.
−
τ
Veff
(4.6)
The evolution of the number of trapped atoms during the loading phase can be
found by solving Eq. (4.6) with N(0) = 0 [8],
N(t) = NS
1 − exp(−t/τ0 )
,
1 + (NS2 β/Veff R) exp(−t/τ0 )
with
τ0 =
τ
,
(1 + 4βRτ 2 /Veff )1/2
(4.7)
(4.8)
and NS the steady state number of trapped atoms. An expression for the steady
state number of atoms can easily be found from Eq. (4.6) by substituting Ṅ = 0,
resulting in


1/2
Veff 
4Rβτ 2
NS =
− 1 .
(4.9)
1+
2βτ
Veff
For losses dominated by two-body trapped atom collisions, i.e., 2βτ/Veff 1, Eq. (4.9)
simplifies to
NS (RVeff /β)1/2 .
(4.10)
In this case the steady state number of trapped atoms is proportional to the square
root of the loading rate.
The trap decay can be found from Eq. (4.6) by substituting R = 0, resulting in
N(t ) =
NS exp(−t /τ)
,
1 + (βNS τ/Veff )[1 − exp(−t /τ)]
(4.11)
where NS = N(0) the initial number of trapped atoms. The dimensionless term
βNS τ/Veff is a measure for the relative importance of the losses caused by two-body
56
Chapter 4
collisions [11]. Substituting β = 0 gives a purely exponential trap decay with lifetime
τ, while for βNS τ/Veff 1 Eq. (4.11) can be written as
N(t ) =
NS
.
1 + (βNS /Veff )t (4.12)
The initial rate of decay is then proportional to the square of the number of atoms:
Ṅ(0) = βNS2 /Veff . We define a decay time determined by two-body losses, i.e., ionization, by τi = Veff /(NS β).
Now the expressions for the trap population are known, we can calculate them for
different trapping conditions, to find the best operating conditions. To do that we
first need to know more about the distribution of the atoms in the MOT. Therefore
the trapping potential is needed. We express the trapping potential in a simple
Doppler model, described below.
2.2 Doppler model
The MOT is operated in the temperature limited regime, as mentioned earlier. The
thermal motion of the atoms in the trap causes them to spread out to an rms radius
σρ in radial direction and σz in axial direction as
1
1
1
kB T = κρ σρ2 = κz σz2 ,
2
2
2
(4.13)
with κρ and κz the spring constant of the trap in radial and axial direction, respectively (Eq. (2.15)) and T the temperature of the atoms determined by the balance
between cooling and diffusion as described in chapter 2 (Eq. (2.12)). The volume
of the atom cloud, given by Eq. (4.4), can then be expressed in terms of the spring
constants and the temperature:
V = (2π kB T )3/2 1
κρ2 κz
.
(4.14)
Knowing that for large detunings of the trapping beams ∆M , the spring constant
κi scales as κi ∼ (∆M )−3 , and that the temperature is proportional to the detuning,
T ∼ ∆M , the volume of the cloud should be proportional to the detuning to the power
of six, V ∼ (∆M )6 .
The steady state number of trapped atoms can be calculated using Eq. (4.9) or,
assuming that the internal trap loss process dominates over the external loss process, by using Eq. (4.10). Since the volume is proportional to the detuning to the
power six, the steady state number of atoms should be proportional to the detuning
cubed, assuming that R and β are independent of the detuning. There is no reason
to assume that the loading rate depends strongly on the detuning of the trapping
beams since the slowing process is fully controlled by the seventh laser beam. As
mentioned earlier the loss rate β is only determined by S − S collisions and as a
consequence independent of the laser field. This model states that the number of
atoms is fully determined by the volume of the atom cloud, and because the volume
Metastable neon atoms in a magneto-optical trap
57
Table I: Slower and MOT characteristics.
slower
MOT
field maximum Bmax
100 G
laser detuning ∆S
1/e2 beam diameter
intensity sS ≡ IS /I0,σ
−6.7 Γ
∼ 20 mm
0.25
field gradient G ≡ ∇z B = 2∇ρ B
0.33 G/Acm1
laser detuning ∆M
-1Γ to -8Γ 1
2
1/e beam diameter
35 mm
total intensity sM ≡ 6IM /I0,σ
0.25 to 81
1
In the experiments described in this chapter we varied
the detuning and intensity of the trapping beams and
varied the magnetic field gradient.
increases as a function of the laser detuning, the number of atoms increases as well.
However, the increase of the volume cannot continue forever: for a certain detuning
the diameter of the atom cloud will be as large as the diameter of the trapping laser
beams! The volume, and therefore the number of trapped atoms, will not increase
anymore. Before this happens another process will occur, described in the next section, which will cause atoms to leak from the trap. The parameters used in the model
and, later on, in the experiments are given in Table I.
2.3 Trapping potential: geometrical trap loss
In our setup, described in detail in section 3, the laser beam for the extra Zeeman
slower, crosses the trapping center along the x-axis, while the radial trapping beams
intersect with the x-axis under an angle of 45◦ . The slower laser beam influences the
dynamics of the MOT. This can be seen if we look at the potential of the MOT created
by combination of the MOT trapping beams and the slower beam. The potential UMOT
MOT on the atoms, with F the trapping
of the MOT is related to the force F = −∇U
force described in section 3.1 of chapter 2. Considering only the direction of the
slower laser beam, the force is given by the sum of the spontaneous forces caused
by the radial MOT beams and the force caused by the slower beam.
The resulting potential is given in Fig. 4.1 for different detunings of the MOT
beams. The detuning ∆M of the MOT laser beams is expressed in units of the natural
line width Γ = 8.2(2π ) MHz of the Ne 3 D3 state. The intensities of the MOT laser
beams we express in terms of the total saturation intensity sM , which is defined by
six times the saturation intensity of one trapping beam sM = 6(IM )/(I0,σ ), with I0,σ =
4.08 mW/cm2 the saturation intensity of the Ne(3s) 3 P2 (mJ = 2) ↔Ne(3p) 3 D3 (mJ =
3) transition. We define the gradient G of the MOT magnetic field as G ≡ ∂B/∂z =
2∂B/∂ρ, and we write G in units of G/cm (Table I). The intensity and detuning of
the slower laser we keep fixed at sS = IS /I0,σ = 0.25 and ∆S = −6.7Γ . The force of
58
Chapter 4
a
1.5
b
25
∆M=-Γ -2Γ -3Γ
UMOT (mK)
UMOT (K)
20
1.0
0.5
15
10
5
0.0
-50
-25
0
25
x (mm)
0
-10
50
-5
0
5
x (mm)
10
Figure 4.1: a: Spatial profile of the confining potential of the MOT for three different
detunings of the trapping beams. The minimum of the potentials is fixed to zero
Kelvin. b: Detail of the deepest point of the potentials. The slower beam shifts
the deepest point of the well in radial direction (−x direction). Trapping conditions:
G = 10 G/cm, sM = 0.5.
a
sM=0.1 0.2 0.5 1
10
2
b
4
10
8
x0 (mm)
x0 (mm)
8
6
4
2
0
x0+σx
6
4
x0
2
0
2
4
6
-∆M(Γ)
8
10
0
0
2
4
-∆M(Γ)
∆C
6
Figure 4.2: a: Shift of the trap minimum x0 as a function of the laser detuning
for different laser beam intensities, going from left to right: sM = 0.1 to sM = 4
(G = 10 G/cm). b: Shift of the trap minimum x0 and the shift of the edge of the atom
cloud x0 + σx as a function of the laser detuning (sM = 0.5 and G = 10 G/cm). For
a detuning of ∆M = ∆C = −4.3Γ the edge of the atom cloud reaches the edge of the
axial trapping beams dbeam = 9 mm.
Metastable neon atoms in a magneto-optical trap
59
the slower beam causes a tilt of the MOT potential, as can be seen in Fig. 4.1a, which
shows the MOT potential for three different laser detunings. This tilt has the effect
that the minimum of the potential shifts away from the center of the trap, which is
shown in Fig. 4.1b. The depth of the potential is of the order of 1 K, as can be seen
in Fig. 4.1a. This is much more than the expected cloud temperature of 1 mK. If we
assume that the atom cloud is centered at the position of the potential minimum, we
expect also a shift of the atom cloud, when the laser detuning of the trapping beams
is changed.
We can calculate the shift of the deepest point of the potential for different
trapping conditions. Figure 4.2a shows this shift for a MOT with field gradient
G = 10 G/cm and trapping beam intensities corresponding to sM = 0.1 to sM = 4.
For low trapping beam intensities, the trap minimum shifts very rapidly away from
the trapping center while for high intensities this happens more slowly. Figure 4.2b
shows the shift of the center of the atom cloud x0 together with the shift of the edge
of the atom cloud x0 + σx for a MOT with field gradient G = 10 G/cm and trapping
beam intensity sM = 0.5. The rms radius in radial direction is calculated using the
expression given in Eq. (4.13). At a certain critical detuning ∆M ≡ ∆C of the trapping
laser beams, the shift of the cloud center together with the increase in the volume
is so large that the edge of the atom cloud hits the boundary of the axial trapping
beams, which have a diameter of approximately 18 mm. When this happens the
volume cannot increase anymore while the shift of the cloud center continues, for
increasing laser detuning. This causes a cut-off of the atom cloud which results in
a decrease of the volume and therefore a decrease of the number of trapped atoms.
Furthermore the trap starts to leak atoms. We call this phenomenon geometric trap
loss.
Knowing the relation between the number of trapped atoms and the volume of
the cloud (Eq. (4.10)), we can calculate the number of trapped atoms as a function of
the detuning of the trapping beams. For the loading rate and two-body loss rate we
substitute values which are independent of the laser intensity and detuning, namely
R = 1 × 1010 s−1 and β = 5 × 10−10 cm3 /s. Figure 4.3 shows the calculated number
of atoms as a function of the detuning of the trapping beams for three different
trapping conditions: (sM , G) = (0.2, 10 G/cm), (0.5, 10 G/cm), and (0.5, 15 G/cm).
Indicated in the figure is the critical detuning ∆C determined for the different trapping conditions in the same way as defined in Fig. 4.2b. We define the maximum
number of trapped atoms as N(∆M ) = N(∆C ) ≡ Nmax . We see that Nmax is larger for
higher laser beam intensities and higher magnetic field gradients. From this calculation we see that it is possible to trap more than 109 atoms in a trap with field gradient
G 10 G/cm, laser detuning ∆M −5Γ and laser beam intensities corresponding to
sM 0.5.
60
Chapter 4
G=10 10 15 G/cm
2.0
sM=0.2
N (109)
1.5
0.5
Nmax
1.0
0.5
0.0
∆C
0
2
4
6
8
10
-∆M(Γ)
Figure 4.3: Calculated number of trapped atoms as a function of the detuning of
the trapping beams for three different trapping conditions. The critical detuning
∆C , and the corresponding maximum number of trapped atoms Nmax , is larger for
higher laser beam intensities and higher magnetic field gradients. The number of
trapped atoms is calculated by using Eq. (4.10). For the loading rate and two-body
loss rate we substitute R = 1 × 1010 atoms/s and β = 5 × 10−10 cm3 /s, respectively.
Both loading and loss rate are taken independent of the detuning and intensity of
the trapping beams.
3 Experimental setup
3.1 Trapping chamber
Figure 4.4 shows an artist’s impression of the trapping chamber, while Fig. 4.5 shows
a schematic side view of the chamber. On the left the last part of the beam-machine
described in chapter 3, is visible. The trapping chamber is connected to the vacuum
system of the beam machine via a bellows, which makes it possible to align the center
of the trapping chamber with the atomic beam axis. A detector, consisting of a piece
of glass covered with a conducting Indium-Tin-Oxide layer, can be moved onto the
atomic beam axis to measure the flux of atoms entering the trapping region. Behind
the detector a flow resistance is positioned to provide a pressure drop between the
vacuum system of the beam-machine and an extra differential pumping chamber.
The differential chamber is pumped with an ion-getter pump (type Varian StarCell
Vaclon, pumping speed 20 l/s), and a Ti-sublimation pump (pumps not visible in
Fig. 4.4). Again, a flow resistance is positioned between the differential pumping
chamber and the trapping chamber, and a valve separates the trapping chamber
from the rest of the setup. The stainless steel trapping chamber has an octagonal
shape and has seven anti-reflection coated windows. One port of the chamber is
Metastable neon atoms in a magneto-optical trap
61
Beam
machine
Detector
Ion-getter pump
Diff.
pumping
chamber
y
z
Valve
x
Compensation
coils
Ti-sub.-pump
Figure 4.4: Artist’s impression of the trapping chamber. On the left the last part
of the neon beam-machine is visible (chapter 3). The amount of atoms entering the
trapping chamber can be detected with a movable detection plate (detector). Not
shown in this figure is a valve between the chamber of the beam machine and the
differential chamber.
used for the connectors to a channeltron situated above the trapping center. The
trapping chamber is also pumped with an ion-getter pump (type Ultek 202-2000,
pumping speed 20 l/s), and a Ti-sublimation pump. Ideally the pressures drops
from 10−8 mbar in the vacuum system of the beam-machine via ∼ 10−9 mbar in the
differential pumping chamber to ∼ 10−10 mbar in the trapping chamber1 .
3.2 Magnetic field coils
The MOT quadrupole field is produced by anti-Helmholz coils made of 4 mm diameter hollow tubing. Each coil consist of 24 turns: three layers of eight turns
each. The magnetic field gradient of the coil configuration was measured to be
G ≡ ∇z B = 2∇ρ B = 0.33 G/Acm [12] (see Table I). The coils can carry a maximum
current of 200 A, and are cooled by flowing water through the copper tubing.
To compensate for disturbing magnetic fields (e.g., the Earth’s magnetic field)
three pairs of Helmholz coils are positioned in the x, y and z-plane around the
trapping chamber (see Fig. 4.4). At the center of the trap, the coils produce a magnetic field of B0 = 4.2 G/A in the x and z direction and B0 = 3.3 G/A in the y
direction, respectively [12]. The maximum current through the coils is 3 A, so magnetic fields up to 10 G can be compensated. The compensation coils can also be used
for positioning the atom cloud in the middle of the trapping region, i.e., at the center
of the MOT quadrupole field. This is important when the atoms are transferred from
the MOT to a magneto-static trap: then the centers of both traps must overlap to
1
We cannot measure the pressure in the differential pumping and trapping chamber, the iongetter pump currents of both chambers indicate a pressure below 10−8 mbar.
62
Chapter 4
Detector
Slower-coil
Slower-coil
MOT-coils
Valve
Bellows
Slower beam
Atoms
7
100
10
100
Valve
Flow
resistance Radial trapping
beams
Beam
machine
y
x
Diff.-pumping
chamber
0
200
Trapping
chamber
400
600
800
1000 mm
Figure 4.5: Schematic side view of the trapping chamber.
minimize the loss in phase-space density. The compensation coils can also be used
for reducing the influence of the slower laser beam on the trap dynamics.
3.3 Extra slower
The atoms entering the trapping chamber have an average velocity of 100 m/s (see
chapter 3, section IV). Typical capture velocities of MOT’s are 50 m/s, so the atoms
need to be slowed down further to be captured by the MOT. This is done in an extra
slower positioned close to the center of the trap. It uses two coils producing opposite magnetic fields. The first coil, positioned 120 mm downstream of the trapping
center, consists of 100 turns and has a diameter of 80 mm. A second coil with 50
turns and a diameter of 300 mm is positioned 190 mm upstream of the trapping
center, see Fig. 4.5. The second coil is necessary to null the magnetic field produced
by the first coil in the center of the MOT. The magnetic field produced by the first
coil is at its maximum Bmax = 100 G, at 6 A current.
3.4 Laser setup
Figure 4.6 gives an overview of all the optical components necessary for preparing
the MOT trapping beams and the laser beam for trap diagnostics (section 3.5). The
light for the trapping beams comes from a continuous-wave single-frequency ring
dye laser (Spectra Physics, type 380D). The dye laser typically has an output power
of 120 mW while pumped with 4 W of light from a argon ion laser, type Coherent
Innova 70. The dye laser is locked to a frequency that can be Zeeman shifted between
one and twelve line widths to the red of the Ne (3s) 3 P2 ↔(3p) 3 D3 transition by using
Metastable neon atoms in a magneto-optical trap
63
Ne* Cell CB
CCD Camera
40 MHz
AOM
PD
Slower-beam T
T
PD
λ/4
λ/4
30-50 MHz
AOMs
Fiber
Radial trapping
beams
λ/4
CB
λ/4 T
λ/2
PCB
Dye
Probe-beam
Axial trapping
beams
T
S
Ar+
λ/2
PCB
λ/4
L
λ/2
PCB
Figure 4.6: Top view of the trapping chamber with schematically a view of the optical
components of the magneto-optical trap. The light for the six trapping beams all
come from a single dye laser which is locked to the Zeeman-tuned Ne(3s)3 P2 ↔(3p)3 D3
transition by saturated absorbtion. The light for the extra Zeeman slower and the
probe-beam come from the dye laser used for preparing the atomic beam (chapter
3). L: single lens, T: optical telescope, CB: cubic beamsplitter, PCB: polarizing cubic
beam splitter, PD: photo diode, S: beam-shutter, AOM: acousto optical modulator,
λ/2, λ/4: half and quarter wave plates.
saturated absorption spectroscopy. This Zeeman shifting is done by varying the
current through a coil producing a DC magnetic field in the saturated absorption
cell. The linewidth of the laser is about 1 MHz.
The total laser power to the MOT can be regulated by a half-wave plate and a
polarizing beam splitter cube. The light can be switched on and off within 2 ms with
a mechanical beam shutter (type Uniblitz VMM-T1). By using optical telescopes the
laser beam is expanded to a waist diameter of ∼ 35 mm (1/e2 intensity drop). The
necessary three pairs of counterpropagating σ + −σ − polarized beams are created by
splitting the main trapping beam into three beams of equal intensity, and reflecting
each one after passing appropriate quarter wave plates and the trapping center, back
to itself (see Fig. 4.5 and Fig. 4.6). The maximum available laser power results in an
intensity of IM = 5.5 mW/cm2 for each trapping beam, corresponding to a total
saturation parameter (i.e., sum of the six trapping beams) sM = 8. The trapping laser
beams are cut-off by a diaphragm resulting in a diameter of approximately 18 mm.
The laser light for the extra slower and for a probe-beam used for trap diagnostics, are derived from the dye laser used for preparing the atomic beam (see
chapter 3, section 2a). This dye laser is locked almost two line widths to the red of
64
Chapter 4
the Ne (3s) 3 P2 ↔(3p) 3 D3 transition; acousto optic modulators (AOMs) are used to
shift the beams to the right frequencies.
For the slower beam, 10 mW of laser light is split off and is shifted by a 40 MHz
AOM to a laser detuning of ∆S = −6.7Γ . The beam is expanded by two optical telescopes to a 1/e2 waist diameter of ∼ 20 mm and a quarter wave plate is used to make
the light σ + -polarized. The laser intensity is IS = 1 mW/cm2 which corresponds to a
saturation intensity of sS = 0.25 (Table I).
For trap diagnostics using absorption imaging (see next section) a laser beam
with variable intensity and detuning is necessary. We prepare this laser beam by
using two AOMs to split off a variable amount of laser power from the dye laser
used for preparing the atomic beam. This can be done because the probe beam is
normally only used when the loading of the trap is stopped, so that in principle all
the laser power otherwise needed to prepare the atomic beam can be used. We use
two tunable 40 MHz AOMs, the first one shifting the beam to the blue and the second
one to the red. In this way a detuning of a few line widths to the red or blue can be
generated. The probe-beam is coupled into a single-mode polarization maintaining
fiber and expanded by an optical telescope. The fiber provides spatial filtering of
the wavefront. The beam then crosses the trapping center in axial direction and is
imaged on a CCD-camera (Fig. 4.6).
3.5 Diagnostics
To investigate the dynamics of the atoms in the MOT we use a few techniques to
diagnose the cloud of atoms. Figure 4.7 is an artist’s impression of the trap showing
the detectors to measure the cloud characteristics such as number of atoms, volume
and temperature. The number of trapped atoms is determined by measuring the
power of the fluorescence emitted by the atoms. The fluorescence, emitted within
a solid angle of Ωdet = (7.6 ± 0.3) × 10−4 , is focused on a photo-diode via a lenssystem, in which two apertures prevent background light of reaching the photodiode. The measured efficiency of the system of lenses and photo-diode equals
ηdet · Tlenses = 0.125 ± 0.005 A/W. The loading and decay rate of the MOT can also
be derived from the fluorescence power by studying time-dependent phenomena
resulting from switching the loading on and off.
The volume of the atom cloud can be measured by analyzing fluorescence images taken with a CCD-camera positioned below the trapping chamber, see Fig. 4.7.
By fitting the intensity profile of the CCD-frames with Gaussians, the volume can be
estimated as described in section 4. The measured relation between camera-image
pixels and real sizes is for the x and z direction identical and equals 13.1 ± 0.5 pixels/mm. Both the number of atoms and the volume of the atom cloud can be measured by using an absorption imaging technique, which is described in detail in
section 7.1. In this technique a probe laser beam passes the trapping area in axial direction and gives information about the position (x and y) and the number
of atoms. The relation between CCD-camera pixels and real sizes was for the x
direction: 16.1 ± 0.5 pixels/mm and for the y direction: 9.1 ± 0.5 pixels/mm. By
taking images of the cloud time-dependently also the temperature of the atoms can
Metastable neon atoms in a magneto-optical trap
MOT beams
65
Channeltron
Slower beam
Slower-coils
Atomic beam
CCD-camera
MOT-coils
Beamsplitter
y
z
Lenses
Apertures
x
Photo-diode
Figure 4.7: Artist’s impression of the detection setup. The probe laser beam used
for absorption imaging, moves along the z-axis and is not shown in this figure.
be measured with the absorption imaging technique as described in section 7.1. The
temperature of the cloud can also be measured by measuring a time-of-flight (TOF)
metastable atom signal with the channeltron positioned 4 cm above the trapping
center (see Fig. 4.7). In front of the entrance of the channeltron (type Galileo 4039)
a grid is placed which can be put on a positive voltage to measure only metastable
atoms, or on a negative voltage to measure ions and metastable atoms.
In the next sections we describe the measurements we did to systematically investigate the behavior of our MOT, starting with the cloud volume in the next section.
4 Trap volume
4.1 Fluorescence technique: spatial distribution
The volume of the cloud of atoms in the MOT is measured by fitting CCD picture
frames containing a fluorescence image of the cloud in the x − z-plane. Figure 4.8a
shows such an image with a plot of the horizontal and vertical intensity profile
through the center of the cloud. These intensity profiles are fitted with Gaussians
resulting in rms widths in the x- and z-directions, as shown in Fig. 4.8b.
Assuming that the spatial distribution in the x and y direction are identical, we
write σx = σy ≡ σρ , with ρ the radial coordinate. The volume V of the cloud is
then calculated using Eq. (4.4). The relative uncertainty in the measured volume is
66
Chapter 4
a
b
Pixel Contents
80
z
x
60
40
20
0
-8 -6 -4 -2 0
2
4
6
8
x (mm)
Figure 4.8: a: CCD image of the trapped cloud of atoms in the MOT. Horizontal and
vertical intensity profiles through the center of the image are shown. b: Gaussian fit
through the horizontal (x) intensity profile which is shown in a.
approximately 12%, and is determined by the uncertainty in the Gaussian fits and
the uncertainty in the conversion from pixels to real sizes.
4.2 Volume measurements
The volume of the atom cloud fully determines the maximum number of atoms that
can be trapped. As mentioned in section 2.2 the volume itself is expected to depend
very strongly on the detuning of the MOT laser light, i.e., (∆M )6 . Here we investigate
how V depends on the trap parameters sM , ∆M , and G.
The strong detuning dependence of V is illustrated by the data shown in Fig. 4.9a.
Different sets of data are shown taken for different values of sM at a fixed G =
23 G/cm. Varying ∆M from −2Γ to −5Γ the volume typically increases by a factor
of 20 to 50, depending on sM . At higher intensities the volume doesn’t increase as
much. Note that the volume can become as large as 0.5 cm3 . It is at these large
volumes that the largest number of atoms is trapped. At the critical detuning the
number of trapped atoms was Nmax = 6 × 109 , for the sM = 0.4 data set. The number
of atoms was measured with the fluorescence technique described in the next section. We checked that the volume is independent of the number of atoms in the trap.
This was found by decreasing the loading rate and measuring the volume again.
Knowing the volume of the cloud and the number of atoms in the cloud we can
estimate the central density n0 = N/V . With V increasing rapidly as a function of
∆M the density remains low. This is illustrated by the data in Fig. 4.9b, where the
central density n0 is plotted again as a function of detuning. At the detunings where
the largest samples are trapped, i.e., at the critical ∆C , the density is only about
Metastable neon atoms in a magneto-optical trap
a
b
∆C
60
0.4
1.0
2.0
0.2
0
2
4
-∆M(Γ)
sM=0.4
sM =0.4
n0 (109 cm-3)
V (cm3)
0.6
0.0
67
6
∆C
40
20
8
0
0
2
4
-∆M(Γ)
6
8
Figure 4.9: a: Volume of the cloud of atoms in the MOT as a function of the MOT
laser detuning for different intensities of the trapping beams sM , the magnetic field
gradient is fixed G = 23 G/cm. b: Central intensity n0 = N/V as a function of the
detuning of the trapping beam, for sM = 0.4 and G = 23 G/cm. In both graphs the
critical detuning is indicated for the sM = 0.4 trapping condition.
1 × 1010 atoms/cm3 .
In order to gain a full understanding of the behavior of the MOT volume we
develop the Doppler model mentioned in section 2.2 in detail here. The basic assumption of the model is that the volume is given by the equipartition of kinetic
and potential energy, resulting in Eq. (4.14). The Doppler temperature is given by
Eq. (2.11) in Chapter 2. The spring constant is taken to be the derivative of the
photon scattering force (Eq. (2.12) of chapter 2). Inserting these into Eq. (4.14) one
arrives at
3
1
(1 + s0 + 4(∆2M /Γ 2 ))2 2
V = V0 (2∆M /Γ +
,
(4.15)
)
2(∆M /Γ )
(∆M /Γ ) s0 (G/G0 )
with
3
π2
V0 =
32
Γ
kL µB G0
3
2
,
(4.16)
where G0 = 10 G/cm is a reference value, kL is the wavenumber of the laser, ∆M is
1
the MOT laser detuning in units of the linewidth, and s0 = 6 sM is the single beam
on-resonance saturation parameter. The value of V0 = 2.5 × 10−9 cm3 . Note that all
units are included in V0 by the introduction of G0 .
Comparing Eq. (4.15) to the data we find that the measured volumes are much
larger than the model predicts. However, by using V0 as a single fit (scaling) parameter we obtain the solid curves shown in Fig. 4.9a. Clearly, the model predicts the
correct dependence of the volume on detuning. Moreover, data sets taken for different intensities (sM = 0.4 to 2.0) and magnetic field gradients (10 to 23 G/cm) all
68
Chapter 4
10
x0,z0 (mm)
8
x0
z0
6
4
2
0
-2
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.10: Shift of the center of the atom cloud in radial (x0 ) and axial (z0 ) direction
as a function of the detuning of the trapping beams. The black dots indicate the
measured shift in radial direction and the open circles indicate the measured shift
in axial direction. The solid line shows the predicted shift by the Doppler model
(see section 2.3). The first data point is fixed to the theoretical curve. Trapping
conditions: sM = 0.4, G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .
result in the same value of V0fit = (1.1 ± 0.2) × 10−8 cm3 . Here the uncertainty refers
to the spread of the fits.
The discrepancy between V0 and V0fit can be explained partly from the fact that
the model is based on the Doppler force for a simple two level atom. In reality the
force becomes smaller because the Clebsch-Gordan coefficients of the different sublevels have to be included. If we take care of this and substitute 0.5s0 for s0 the
fit parameter V0fit agrees with V0 . This correction for the effective Clebsch-Gordan
coefficient is larger then we expect, as described in section 5. However, in the model
calculations we use s0 → 0.5s0 . Radiation trapping can be neglected because of the
rather low density. This is confirmed by the fact that the volume is independent of
the number of atoms in the trap.
From the CCD-camera images it is also possible to obtain the position of the atom
cloud. Figure 4.10 shows the coordinate of the cloud center (x0 , z0 ) as a function of
the detuning of the trapping beams for the sM = 0.4 data set of Fig. 4.9a. We see that,
going from ∆M = −2Γ to −5Γ , the cloud center moves 6 mm in radial (x) direction,
while in axial direction (z) the cloud stays more or less fixed. The solid line in the
figure indicates the radial shift calculated with the Doppler model of section 2.2.
We see that the Doppler model predicts a correct dependence of the radial shift on
the detuning however more rapidly than the measurements show. The shift, fully
determined by the balance between the radial trapping beams and the slower beam,
follows the measurements better when the intensity ratio between trapping beams
Metastable neon atoms in a magneto-optical trap
69
2.0
σx/σz
1.5
1.0
1
2
2
0.5
0.0
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.11: Ratio between the radial and axial rms width of the atom cloud σx /σz
as a function of the detuning of the trapping beams. Trapping conditions: sM = 0.4,
G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .
and slower beam is decreased.
This could also be the case in the experiments since the ratio
√ between the rms
width in radial and axial direction σx /σz was higher then 1/ 2, as predicted by
Eq. 4.13. This can be seen in Fig. 4.11 which shows the ratio σx /σz as a function of
the detuning of the trapping beams, corresponding again with
√ the sM = 0.4 data set
1
of Fig. 4.9a. As can be seen the ratio is always larger then 2 2, and increases as a
function of the detuning. This means that during the measurements the intensity of
the radial trapping beams was larger than the intensity of the axial trapping beams,
i.e., sM,x > sM,z . We cannot explain the increase of this ratio with the detuning.
Apparently the atom cloud moves through a region where the balance between the
intensity of the axial √
and radial trapping beams changes. The fact that the ratio is
always higher then 12 2 means that the intensity of the radial trapping beams was
higher then one sixth of the total intensity IM which we measured. This explains
why the calculated shift of the atom cloud increases more rapidly with ∆M than the
measured one.
In conclusion, we find that the volume of the cloud of atoms in the MOT can
indeed be described qualitatively very well by a simple Doppler model. However,
quantitative agreement is only obtained to within a factor of four. The shift of the
atom cloud with the detuning is described very well by the geometrical loss model.
Finally, Fig. 4.12 shows the CCD-camera images of the atom cloud for different laser
detunings. The radial shift and the increasing of the volume is clearly visible as well
as the changing aspect ratio of the cloud.
70
Chapter 4
∆M =-4.5Γ
16
x (mm)
12
-4.0Γ
-3.3Γ
8
4
-2.7Γ
-2.1Γ
0
-4
Figure 4.12: CCD-images of the trapped cloud of atoms corresponding to some of
the data points of Fig. 4.10.
5 Loading rate
5.1 Fluorescence technique: number of atoms
A standard way to determine the amount of atoms in a MOT is to measure the power
of the fluorescence emitted by the atoms in the trap [13]. The total power emitted
by one atom can be written as
Pscat = ωL Γ Π(e) ,
(4.17)
with ωL the laser frequency and Π(e) the population of the excited states and Γ
the decay rate. For a two-level atom illuminated by a single travelling wave of Rabi
frequency Ω and detuning ∆L , this can be written as
Pscat = ωL
Γ
Ω2 /2
.
2 ∆2L + Γ 2 /4 + Ω2 /2
(4.18)
In a MOT, however, one must calculate an average over all the transitions between
the various Zeeman sublevels between the ground and excited states. In this case
the total scattering rate per atom can be approximated [13]
Pscat ωL
2
CΩtot
/2
Γ
,
2
2
2 ∆L + Γ 2 /4 + CΩtot
/2
(4.19)
2
where Ωtot
is determined by six times the average light intensity of one trapping
beam, and C is a phenomenological parameter containing an average of ClebschGordan coefficients. Townsend et al. [10] found an experimental value of C = 0.7 ±
0.2 for a Cs MOT, much larger than the average squared Clebsch-Gordan coefficients
which yields C = 0.4. In the case of the Ne 3 P2 →3 D3 transition the average over the
Metastable neon atoms in a magneto-optical trap
71
Clebsch-Gordan coefficients equals 0.46; following Townsend et al. [10] we take in
our experiments C = 0.7 ± 0.2.
In our setup the atoms are slowed by an extra Zeeman slower in order to be
captured by the MOT (section 3.3). The extra slower laser beam influences the population of the excited states. To correct for this seventh laser beam in the MOT we
write for the population of the excited states
(e)
Πtot
(e)
(e)
= ΠM + ΠS
1
CsS
CsM
=
+
2
2
2 1 + CsM + 4∆M /Γ
1 + CsS + 4∆2S /Γ 2
,
(4.20)
with ∆M and ∆S the detuning of the MOT beams and slower beam, respectively, and
sM = IM /I0,σ with IM six times the intensity of one trapping beam, I0,σ = 4.1 mW/cm2
the saturation intensity for the used transition and sS = IS /I0,σ with IS the intensity
of the slower beam.
The fluorescence power emitted by the atoms is measured with a photo-diode (see
section 3.5). The current Idet from the photo-diode is proportional to the amount of
atoms N trapped in the MOT:
Idet = ηdet · Tlenses · Ωdet · Pscat · N.
(4.21)
Substituting Tdet = ηdet · Tlenses we write for the number of atoms
N=
Idet
.
Tdet · Ωdet · Pscat
(4.22)
When the uncertainty in the averaged phenomenological parameter C is neglected
the relative uncertainty in the number of trapped atoms is about 10%. The largest
contribution to this uncertainty is produced by the 2 MHz, i.e., 0.25Γ , uncertainty in
the frequency of both trapping and slower laser. The phenomenological parameter C
is not known for our trap, but since the averaged squared Clebsch-Gordan coefficient
for the 3 P2 →3 D3 transition is much smaller than C = 0.7, Eq. (4.20) gives a upper limit
to the population of the excited states.
5.2 Measurements
Figure 4.13 shows an example of a loading and decay curve for a trap with sM = 0.4
and G = 23 G/cm. At t = 0 the loading of the MOT is started by switching on the
atomic beam. This is done by unblocking the laser beams which go to the laser
cooling section of the beam-machine, as well as the extra Zeeman slower. As can
be seen in the figure, within one second the MOT reaches its steady-state number of
atoms NS . At t = 2.1 s the loading of the trap is stopped by blocking the laser beams
again. Again also the extra slower laser beam is switched off.
Figure 4.14a shows a detail of the loading process. To determine the loading
rate from this curve, we fit the curve with a straight line at t = 0, as indicated
in the figure. In this measurement, where ∆M = −4.4Γ , the loading rate is R =
72
Chapter 4
5
N (109)
4
3
2
1
0
0
1
2
3
t (s)
Figure 4.13: Example of a loading and decay curve. At t=0 the loading of the trap
is started, i.e., the atomic beam is switched on, at t = 2.1 s the loading is stopped.
Trapping conditions: ∆M = −4.4Γ , sM = 0.4 and G = 23 G/cm.
(1.2 ± 0.2) × 1010 atoms/s. We can measure the atom flux to the trapping chamber
with the detection plate at the entrance of of the trapping chamber, see Fig. 4.5.
The atom flux will be lower at the trapping center since the atomic beam has a
diameter of more than 6 mm at the entrance of the trapping chamber and has to
pass two flow resistance with a diameter of only 7 mm, see Fig. 4.5. The atomic
beam will be cut off by the flow resistances. This was also measured by taking a
CCD-camera image of the fluorescence of the atomic beam at the trapping center. In
this example the atom flux was measured to be 2.8 × 1010 atoms/s. Thus a loading
rate of R = (1.2 ± 0.2) × 1010 atoms/s means a loading efficiency of 43%.
Figure 4.14b shows the loading rate as a function of the detuning of the trapping
beams ∆M . As can be seen, the loading rate increases as a function of the laser detuning, until the critical detuning is reached, after which the loading rate decreases
again. The uncertainty in the loading rate is estimated by the uncertainty in the
number of atoms and the uncertainty in the fit. The maximum loading efficiency was
measured at the critical detuning and equals 54%. We cannot explain the increase of
the loading rate with the detuning; perhaps the shift in radial direction of the cloud
causes a change in the loading rate. The decrease of the loading rate at the critical
detuning can be explained by the shifting of the cloud center. Because the cloud is
shifted to the edge of the trapping beams probably less atoms can be captured. This
detuning dependence of the loading rate was not so clear at other trapping conditions, i.e., higher laser beam intensities and lower field gradients, then the loading
rate stays more or less constant and decreases at the critical detuning.
In conclusion, we obtain loading rates of up to 1.5 × 1010 atoms/s which is more
than 50% of the atomic flux of the bright atomic beam, described in chapter 3.
Metastable neon atoms in a magneto-optical trap
b
4
20
3
15
R (109 s-1)
N (109)
a
73
2
1
10
5
0
0.00
0.25
t (s)
0.50
0
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.14: a: Detail of the loading process shown in Fig. 4.13. The loading rate
R is estimated by fitting the loading curve at t = 0 with a straight line. b: Loading
rate of the MOT as a function of the laser detuning of the trapping beams. Trapping
conditions: sM = 0.4, G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .
6 Trap population
We measured the amount of atoms in the MOT with the fluorescence-detection
method described in section 5.1. Figure 4.15a shows the number of trapped atoms
as a function of the laser detuning for different laser beam intensities. In each graph
the number of trapped atoms increases until a certain critical detuning ∆C , beyond
which the number of atoms suddenly decreases. For the measured critical detuning
we take the detuning at which the measured amount of trapped atoms is maximum.
We see that this critical detuning shifts to larger values for higher laser beam intensities.
In Fig. 4.15b the measured critical detuning ∆C as a function of the intensity of
the trapping beams is shown together with the predicted critical detuning, calculated
with the geometric loss model of section 2.3. We see that the measured critical
detunings agree very well with the geometric loss model.
As shown in section 2.3 and Fig. 4.3, the model predicts an increase of Nmax with
the intensity of the trapping beams. This is not what we see in the measurements
shown in Fig. 4.15a. Here the maximum number of atoms Nmax , measured at the
critical detuning, decreases from Nmax = 6.0 × 109 to Nmax = 2.5 × 109 when going
from sM = 0.25 to sM = 4. We do not have an explanation for this decrease. The
steady state number of trapped atoms is determined by the loading rate, the volume
and the loss rate, as given by Eq. (4.10). Assuming a constant loading rate and
volume at ∆C , the decrease of Nmax with the intensity is determined by the two-body
loss rate β. For the highest laser beam intensity sM = 4 the fraction of atoms in
74
Chapter 4
a
8
b
8
sM= 0.25
0.20
4
6
0.5
-∆C (Γ)
N (109)
6
1
0.13
2
4
2
0
4
2
0
2
4
-∆M(Γ)
6
8
0
0
1
2
3
4
5
sM
Figure 4.15: a: Measured number of trapped atoms N as a function of the laser
detuning ∆M , for different intensities of the trapping beams sM . The magnetic field
gradient is fixed to G =10 G/cm. b: Critical detuning ∆C as a function of the intensity
of the trapping beams sM . Black dots: measurements, solid line: predictions of the
Doppler model.
the excited state is only 1.2% at the critical detuning. If we assume that KSP is not
more than two orders of magnitude larger than KSS , as mentioned in section 2.1, the
contribution of S − P -collisions to the two-body loss rate β is of the same order, or
less than the contribution of S − S-collisions to the two-body loss rate. Further on in
this section we show some experimental evidence for the assumption that β = 2KSS .
For higher laser beam intensities the drop in the amount of atoms beyond the
critical detuning is more gradual, as can be seen in Fig. 4.15a. This is also predicted
by the geometrical loss model as can be seen in Fig. 4.2a: the lines predicting the shift
of the center of the cloud are more gradual at the critical detuning for higher laser
intensities. This means that the leaking process for higher laser beam intensities
increases more slowly with ∆M , this is also what we see in the measurements.
We also changed the gradient of the magnetic field and measured again the number of atoms as a function of the detuning of the trapping beams, the intensity of the
trapping laser beams was fixed to sM = 0.4. The result of this measurement is shown
in Fig. 4.16a. Again the same behavior is visible. The critical detuning shifts to larger
values for higher field gradients. This is also what the geometrical loss model predicts as can be seen in Fig. 4.16b which shows the measured and predicted critical
detuning. Again the model predicts the values of the critical detuning very well.
From these measurements it is clear that there is an optimum magnetic field gradient for which the number of trapped atoms is maximum. This optimum gradient
is about G = 20 G/cm; the number of trapped atoms then equals Nmax = 9 × 109 ,
almost 1010 atoms! The atom cloud has a volume of more than 2 cm3 at this opti-
Metastable neon atoms in a magneto-optical trap
a
17
6
26 G/cm
G=10
6
-∆C (Γ)
N (109)
20
13
8
4
4
2
2
0
b
8
10
75
0
2
4
6
8
0
0
- ∆ Μ ( Γ)
10
20
G (G/cm)
30
Figure 4.16: a: Measured number of atoms N as a function of the laser detuning ∆M ,
for different magnetic field gradients G. The intensity of the trapping beams is fixed
sM = 0.4. b: Critical detuning ∆C as a function of the magnetic field gradient G. The
measured critical detuning corresponding to each trapping condition is indicated by
the black dots, solid line: predictions of the Doppler model.
mum trapping condition! The existence of an optimum magnetic field gradient is,
however, not predicted by the Doppler model.
We checked the fluorescence method by measuring the number of atoms with the
absorption imaging method described in section 7.1. Both methods gave the same
values within 20%.
We can calculate the two body-loss rate β from the measured steady state number
of atoms, the volume, and the loading rate by rewriting Eq. (4.9) to
β=
R − NS /τ
Veff .
NS2
(4.23)
The result of this is shown in Fig. 4.17, which shows the calculated two-body loss
rate. The trapping conditions are analogous to those corresponding to Fig. 4.14b, i.e.,
sM = 0.4 and G = 23 G/cm. For the lifetime corresponding to background collisions
we substitute τ = 3.75 s, which was found from a decay measurement discussed in
section 8. With the exclusion of the last two data points the two-body loss rate is
found to be constant as a function of the laser detuning. The average loss rate is
found to be β = (5.1 ± 1.4) × 10−10 cm3 /s. The much larger values found for the
last two data points in Fig. 4.14b are due to the uncertainty in the measured volume
and the number of trapped atoms caused by the leaking process of the trap. Going
from ∆M = −2Γ to the critical detuning ∆C , the fraction of atoms in the excited state
decreases from 1.4% to 0.3%. Since the two-body loss stays more or less constant as
a function of the laser detuning, we can conclude that the measured loss rate could
be ascribed purely to S − S collisions. Doery et al. [14] found a theoretical value
76
Chapter 4
2.0
β(10-9 cm3/s)
1.5
1.0
0.1
0.5
0.0
-0.5
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.17: Two-body loss rate calculated from the measured number of atoms, the
volume of the atom cloud and the loading rate. The solid line indicates the average
of the data points without the last two points and is found to be β = (5.1 ± 1.4) ×
10−10 cm3 /s.
of β = 2KSS = 1.6 × 10−10 cm3 /s, which is a factor three lower than the value we
measured. In section 8 we show direct measurements of β from decay curves.
By locking both the laser for the beam machine and the laser for the MOT beams
to the 22 Ne isotope, we trapped N = 1 × 109 atoms of this isotope. For the trapping
conditions used this was three time less atoms than for the 20 Ne isotope. So by
optimizing the trap it should also be possible to trap at least 3 × 109 atoms of the
22
Ne isotope.
In conclusion, we trapped almost 1010 20 Ne atoms. This was achieved under
rather unconventional low intensity of the trapping beams, i.e., sM 0.25. The
optimum trapping field is around G = 20 G/cm. At the optimum trapping conditions
the cloud has a extraordinarily large volume of more than 2 cm3 . The simple Doppler
model and the picture of the seven beam MOT discussed in section 2 explains the
trap behavior quite well.
7
Temperature
In the previous sections we described how we measured most of the trapping characteristics necessary to understand the working of the metastable neon MOT. The
remaining quantity is the temperature of the atom cloud. In this section we describe
two experimental methods to estimate the temperature of the atom cloud, namely
via absorption imaging and from metastable atom time of flight (TOF) signals.
Metastable neon atoms in a magneto-optical trap
77
OD
2.0
1.0
OD
1.0
y (mm)
0.0
15
2.0
10
5
5
10
15
x (mm)
20
Figure 4.18: CCD-camera image of the optical density distribution of the atom cloud.
In this example the central optical density equals 1.7.
7.1 Absorption imaging
Spatial distribution
With the absorption imaging technique described below it is not only possible to
measure the spatial distribution of the atom cloud but also the velocity distribution
of the atoms. For both we use a probe laser beam which is tunable in intensity and
detuning (see section 3.4). The probe beam, with a 1/e2 waist diameter of ∼4 cm,
passes the trapping chamber in axial direction as can be seen in Fig. 4.6. The polarization of the laser light is chosen linear. After passing through the trapping
chamber the probe beam is focused in such way on a CCD-camera, that an image of
the intensity distribution of the probe beam, at the position of the trapping center, is
made with a camera. The atoms in the MOT absorb laser light from the probe beam,
which gives information about the number of atoms and the position of the atoms.
While passing through the trapping chamber the probe beam is attenuated by the
atom cloud. Satisfying Beer’s law we can write for the intensity distribution of the
probe-beam
z
I(ρ, z) = I(ρ, −∞) exp −σa
n(ρ, z )dz ,
(4.24)
−∞
with σa the optical absorption cross section and n(ρ, z) the spatial density distribution of the atom cloud. The optical density OD is defined as
I(ρ, ∞)
OD(ρ) ≡ − ln
.
(4.25)
I(ρ, −∞)
Assuming a Gaussian density distribution given by Eq. (4.3), the optical density of
the atom cloud is given by
(4.26)
OD(ρ) = 2π σa σz n(ρ, 0).
78
Chapter 4
2.0
OD
1.5
1.0
0.5
0.0
-2
-1
0
1
2
∆probe(Γ)
Figure 4.19: The optical density of the atom cloud as a function of the detuning of
the probe laser beam.
Figure 4.18 shows an example of a CCD-image of the part of the intensity distribution of the probe beam which contains the atom cloud. By fitting the intensity
distribution with Gaussians, the radial density distribution of the atom cloud can be
obtained.
Number of atoms
The number of trapped atoms can be found by integrating the optical density given
by Eq. (4.26) over the radial coordinate ρ
√ σρ
N=4 π
σa
OD(ρ)dρ,
(4.27)
where the absorption cross section σa can be calculated knowing the detuning of the
probe beam [15]. Figure 4.19 shows the central optical density as a function of the
detuning of the probe laser beam. The MOT, containing 3 × 108 atoms, was operated
with a high intensity sM = 8 of the trapping beams, the detuning was ∆M = −4Γ
and the magnetic field gradient G = 10 G/cm. The intensity of the probe beam was
sprobe = 0.03 and the exposure time 30 µs, small enough not to influence the trap
population. The characteristic Lorentzian profile, with a FWHM-width of approximately one linewidth, is clearly visible.
Velocity distribution
Switching off the MOT beams, the atom cloud expands ballistically. The radial velocity distribution of the atom cloud can be measured by comparing absorption images
Metastable neon atoms in a magneto-optical trap
79
taken at different delay times after switching off the MOT beams. After a delay time
t the rms radius in radial direction is given by
σρ2 (t) = σρ2 (0) +
kB T 2
t ,
m
(4.28)
with σρ (0) the rms radius on t = 0, and T the temperature of the atoms.
7.2 Metastable atom TOF
With the channeltron in the trapping chamber (see subsection 3.5) we can measure
metastable atoms escaping from the cloud. Switching off the MOT beams and measuring the time dependence of the metastable atom signal, the velocity distribution
of the trapped cloud can also be obtained. Assuming that the velocity distribution of
the trapped atoms satisfies a Maxwell-Boltzman distribution we write for the velocity
distribution
vy2
vx2
vz2
P (vx , vy , vz ) = P0 exp(−
−
−
).
(4.29)
2σx2 2σy2
2σz2
After switching off the trapping beams at t = 0 the metastable atom signal on the
detector can be written as
2
vy,0
(vy,0 − gt)
Ṅ(t) ∝
exp(−
).
t4
2σy2
(4.30)
Here, vy,0 is the initial velocity of the atoms hitting the detector at time t
1
vy,0 =
yd + 2 gt 2
t
,
(4.31)
with yd = 4 cm the distance from the channeltron to the trapping center, and g the
acceleration caused by gravity.
7.3 Measurements
We measured the temperature of the atom cloud by using the absorption imaging
technique as described above. The trapping conditions were sM = 1.1, ∆M = −2.7Γ
and G = 7.6 G/cm at which the MOT contains N = 6.5 × 108 atoms. After the MOT
was loaded the MOT beams were switched off and the atom cloud was exposed by
the probe beam after a delay time t. A CCD-camera image of the absorption profile
was taken after a exposure time of 30 µs. The detuning of the probe beam was taken
∆probe = −Γ /2, and the intensity sprobe = 0.03. This measurement cycle was repeated
for increasing delay times from 0 to 7 ms in steps of 1 ms.
The result of the measurements are given in Fig. 4.20, which shows absorption
images of the atom cloud as a function of the delay time. The corresponding rms
widths in radial (x and y) direction are plotted as a function of the delay time. A
fit with Eq. (4.28) through the data points is shown, resulting in a horizontal and
80
Chapter 4
σx, σy (mm)
5
4
σx
σy
3
2
1
0
0
2
4
t (ms)
6
8
Figure 4.20: Absorption images of the atoms cloud for different delay times after
switching of the MOT beams. Black dots: horizontal distribution, open circles vertical
distributions. Corresponding temperatures Tx = 0.78 ± 0.15 mK and Ty = 0.85 ±
0.13 mK.
a sM= 0.8
b sM= 0.4
0.8
−2.7Γ
0.4
∆M =−4.6Γ
NTOF (arb. units)
NTOF (arb. units)
∆M =−3.9Γ
0.8
−3.3Γ
0.4
0.0
0.0
0
25
50 75 100 125
t (ms)
0
25
50 75 100 125
t (ms)
Figure 4.21: a: Metastable atom TOF-spectra for a MOT operated with laser beam
intensity sM = 0.8. b: TOF-spectra for a MOT operated with laser beam intensity
sM = 0.4. Both graphs show spectra for a small and large detuning of the trapping
beams. The temperature is larger for smaller detunings, as can be seen by the shift
of the curves to smaller flight times. The spectrum for large detuning showed in
Fig. b shows a fast peak around 20 ms.
Metastable neon atoms in a magneto-optical trap
81
2.0
T (mK)
1.5
1.0
0.5
0.0
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.22: Temperature as a function of the laser detuning for a MOT operated
with laser beam intensity sM = 0.8. The solid line shows the corresponding Doppler
temperature. The data point indicated with the open circle shows the result of the
temperature measurement done with the absorption imaging technique, showed in
Fig. 4.20. Trapping conditions for that measurement were different: sM = 1.1 and
G = 7.6 G/cm, indicated is the temperature T = Tx = Ty = 0.8 mK.
vertical temperature of Tx = 0.78 ± 0.15 mK and Ty = 0.85 ± 0.13 mK, respectively.
This is both slightly higher then the expected Doppler temperature at these trapping
condition which equals T = 0.55 mK.
A much easier way to determine the temperature of the atom cloud is the metastable
atom TOF technique, described above. Not a series of measurements need to be
done to determine the temperature, like in the case of the absorption imaging technique, but a single measurement directly gives the temperature. We systematically
measured the temperature of the atom cloud for different trapping condition, i.e.,
different detuning of the trapping beams.
Figure 4.21 shows, for two trapping conditions, the time-of flight signals, both for
a small laser detuning and for a large laser detuning (indicated in the figure). From
those TOF-signals it is clear that, for both trapping conditions, the temperature of
the cloud increases for larger laser detunings. For large laser detunings the TOFsignal of the trap operated with low laser beam intensities (sM = 0.4) shows an
additional fast peak (Fig. 4.21b). This makes it impossible to fit the TOF-signals with
the expression given by Eq. (4.30). This fast peak is probably caused by the influence
of the slower beam on the trap dynamics.
The TOF-signals of the trap using large laser beam intensities (sM = 0.8) were
fitted with Eq. (4.30). The fits are shown in Fig. 4.21a by the dashed lines. From those
fits the temperature of the atom cloud was estimated for different laser detunings.
The result of this is given in Fig. 4.22, which shows an increase of temperature while
82
Chapter 4
using larger laser detunings. The temperature of the cloud is close to the Doppler
temperature, which is indicated in Fig. 4.22 by the solid line. Around the critical
detuning the temperature starts to increase probably due to the geometrical loss
process or through a larger influence of the slower laser beam.
In conclusion, we found that the temperature of the atom cloud satisfies the
Doppler temperature.
8 Trap decay
In section 6 we showed how we estimated the two-body loss rate from the loading
rate and the steady state number of trapped atoms. In this section we show how we
estimated the two-body loss rate directly form trap decay curves.
An example of the load and decay process of the MOT was already shown in
Fig. 4.13. At t = 0 the MOT is loaded by switching on the atomic beam, i.e., switching
on the laser light to the cooling sections of the beam machine. After a time t = 2.1 s
the loading of the MOT is stopped by switching off again the laser light of the beam
machine. By switching off this laser light also the light to the extra slower is switched
off.
Figure 4.23a shows a detail of the decay process of the MOT. At t = 2.1 s the
number of atoms first seems to drop rapidly from 4 × 109 to 2.5 × 109 , after which
the number of atoms decay gradually, as can be seen in the figure. This sudden drop
is caused by the fact that also the slower beam is switched off at t = 2.1 s. This
has two effects; first, the fluorescense caused by absorption from the slower beam
immediately falls away, and this is not corrected for in this figure. Correcting for this
artifact results in 20% more atoms after t = 2.1 s. A second effect is that switching
off the slower beam results in a new situation: a trap with six trapping beams instead
of six trapping beams and a slower beam. Since the forces in a six beam trap have
a different equilibrium from the forces in a seven beam trap, the cloud in the trap
suddenly jumps to a different position while switching off the slower beam, which
can cause losses of trapped atoms.
We fitted the decay curve with Eq. (4.11), for τ we substitute τ = 3.75 s, which
was found from a measurement where the decay of the MOT was measured during
a longer time. At the steady state situation the volume of the atom cloud was measured by taking CCD-camera images, as described in section 4. Figure 4.23b shows
the two-body loss rate β as a function of the laser detuning. The trapping conditions
are the same as used at the measurement to estimate the loading rate, as described
in section 5, i.e., sM = 0.4 and G = 23 G/cm. From this it is clear that β increases
drastically near the critical laser detuning. This is not to be expected, as described in
section 2 the two-body loss rate should be independent of the laser detuning while
using low laser beam intensities. Moreover, the estimated values are much higher
than the theoretical value of 1.6 × 10−10 cm3 /s, which Doery et al. [14] calculated.
The extraordinarily high decay rate we measured is due to the kick to the atom
cloud caused by switching off the slower laser beam. This has two effects. First of
all, because of the kick atoms will be lost from the trap. And secondly, the volume
Metastable neon atoms in a magneto-optical trap
5
a
25
1
0
2.0 2.2 2.4 2.6 2.8 3.0
t (s)
β(10-9 cm3/s)
N (109)
2
b
30
4
3
83
20
15
10
5
0
∆C
0
2
4
-∆M(Γ)
6
8
Figure 4.23: a: Detail of the decay process shown in Fig. 4.13. The decay curve is
fitted with the expression given by Eq. (4.11). b: Decay rate of the MOT as a function
of the laser detuning. Again the critical detuning is indicated.
and temperature of the atom cloud will change because of the sudden changing of
the trapping parameters. The volume was measured during a steady state condition
of the trap, i.e., a seven beam MOT, while the decay of the MOT was measured for a
unbalanced six beam MOT. This will cause a misinterpretation of the volume of the
cloud, and consequently a discrepancy in the estimated decay rate.
We also measured a load and decay curve for trapping conditions at which the
influence of the slower beam is less important. This has the effect that the kick to
the atom cloud was negligible when the slower beam was switched off. We chose
higher laser beam intensities of the trapping beams and a slightly lower intensity
of the slower laser beam. Furthermore we set the detuning of the trapping laser
beams below the critical detuning. The trapping conditions we chosen were: sM =
0.9, sS = 0.16, ∆M = −4Γ and G = 13 G/cm. The number of trapped atoms was
N = 7.6 × 108 atoms and the volume of the cloud was V = 0.29 cm3 .
Under these conditions we measured the load and decay curve shown in Fig. 4.24.
The loading curve, showed in detail in Fig. 4.24a, was fitted with both the expression
of Eq. (4.7) and the straight line at t = 0 s. Both result in a loading rate of R =
(1.2 ± 0.1) × 109 atoms/s. The decay curve, showed in detail in Fig. 4.24b, was
fitted with the expression given in Eq. (4.11). From that a two-body loss rate of
β = (6.8 ± 1.0) × 10−10 cm3 /s and a background loss rate of τ = 3.75 ± 0.10 s was
found. This direct measurement of the two-body loss rate corresponds with the
indirect measurements discussed in section 6. Calculating again the two-body loss
rate from the loading rate and the steady state number of trapped atoms by using
the expression given by Eq. (4.23), a value of β = (6.1 ± 1.0) × 10−10 cm3 /s is found.
In conclusion, the trap decay measurements we did were spoiled by the influence
84
Chapter 4
a
8
10
b
N (108)
10
4
8
N (108)
N (108)
6
2
6
1
4
2
0
0
0.0
0.5
0
5
1.0
t (s)
10
15
1.5
20
2.0
0.1
5
7
9
11
t (s)
Figure 4.24: a: Loading and b: decay behavior of the MOT. On t = 0 s the loading
starts, i.e., the atomic beam is switched on, at t = 5 s the loading is stopped. The
inset in Fig. a shows the complete loading-decay curve. The loading process fitted
with both the expression given in Eq. (4.7) and the straight line indicated in the
graph. The decay of the MOT, shown on a log-scale, is fitted with Eq. (4.11).
of the slower beam on the trap dynamics. This effect was counteracted by choosing
trap conditions were the influence of the slower beam can be neglected. Again a
two-body loss rate of β = (6 ± 1) × 10−10 cm3 /s is found.
9 Concluding remarks
We trapped almost 1010 metastable neon atoms in a MOT. To our knowledge this is
the largest MOT of metastable atoms reported. So far this large number of atoms is
reached because of the high loading rates we can obtain with the bright atomic beam
described in chapter 3. Furthermore, this optimum number is obtained under rather
unconventional conditions, e.g., very low intensity of the trapping beams. Under
this conditions the volume becomes very large, thereby reducing the density and
consequently reducing the two-body loss rate.
Another remarkable phenomenon of our MOT is the influence of the slower beam
on the trap dynamics. As a consequence we do not have an ordinary six beam MOT
but a seven beam MOT. However, this seven beam MOT satisfies a simple Doppler
model, developed in this chapter, very well. The influence of this seventh MOT beam
can be regulated easily by choosing different trapping parameters, such as higher
intensities of the trapping beams or different compensating magnetic fields.
The temperature of the atom cloud was measured and found to be in the Doppler
limited regime, i.e., temperatures around 1 mK. For the highest measured density,
i.e., 4 × 1010 atoms/cm3 , this corresponds to a phase space density of nΛ3 10−7 .
Metastable neon atoms in a magneto-optical trap
85
Furthermore, the two-body loss rate was measured to be β = 2KSS = (5 ± 1) ×
10
cm3 /s, which is three times higher than calculated by Doery et al. [14].
A MOT containing 1010 atoms is a good starting point on the road to BEC, as
described by Beijerinck et al. [4]. Apart from a MOT of 20 Ne we can also make a MOT
containing more than 109 atoms of the 22 Ne isotope.
−10
References
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050702(R) (2000).
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[5] F. Bardou, O. Emile, J.-M. Courty, C.I. Westbrook, and A. Aspect, Europhys. Lett.
20, 681 (1992).
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A. 60, R761 (1999).
[8] A. Browaeys, J. Poupard, A. Robert, S. Nowak, W. Rooijakkers, E. Arimondo, L.
Marcassa, D. Boiron, C.I. Westbrook, and A. Aspect, Eur. Phys. J. D 8, 199 (2000).
[9] H.C. Mastwijk, J.W. Thomsen, P. van der Straten, and A. Niehaus, Phys. Rev. A
80, 5516 (1998).
[10] C.G. Townsend, N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P.
Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).
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Foot, J. Phys. B: At. Mol. Opt. Phys. 31, L321 (1998).
[12] V.P. Mogendorff, Towards BEC of metastable neon, internal report, Eindhoven
University of Technology (2000).
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J. Opt. Soc. Am. B 6, 2084 (1989).
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86
Chapter 4
Chapter 5
Photoassociation spectroscopy of 85Rb2
0+
u states
The work described in this chapter was a collaboration of the Ultracold Atomic Physics
Group of the University of Texas at Austin, the Atomic Physics Division of the National
Institute of Standards and Technology at Gaithersburg, and the Atomic Physics and
Quantum Electronics group at the Physics Department of the Eindhoven University of
Technology. The experimental part of the work has been done at the University of
Texas at Austin.
1 Introduction
Photoassociation spectroscopy has been shown to be a very powerful tool in cold
atom physics. Thanks to this technique, the long-range behavior of ground and
excited state cold collisions is now well-understood. From photoassociation data interaction parameters such as ground state scattering lengths could be derived [1, 2].
By using the photoassociation technique it was also possible to predict [3, 4] and
measure [5] Feshbach resonances in collisions of ultracold Rb atoms. Recently, by
combining photoassociation data with Feshbach resonance field data, the sodium
scattering length could be estimated with improved accuracy [6]. By calculating photoassociation spectra and comparing them with measured spectra, it is even possible
to obtain an estimate of the long range behavior of the hyperfine interaction [7, 8].
Photoassociation spectroscopy is also very useful for studying excited state potentials. Cline et al. [9] measured the rotationally resolved bound levels of the purely
long-range Rb2 0−
g state and estimated a number of excited state parameters such as
the C3 coefficient. From their spectra they also found an indication of predissociation of the Rb2 0+
u states, and estimated the probability of the Landau-Zener transition to the potential that connects asymptotically to the 5 2 S1/2 +5 2 P1/2 limit. Due
to this kind of measurement, much was learnt about the Rb excited state potentials.
Far less, however, is known about the couplings between them.
In this chapter we use photoassociation spectroscopy to study the coupling be3
tween the 85 Rb A 1 Σ+
u and b Πu states. We compare coupled channel bound state
87
88
Chapter 5
calculations with experimental photoassociation spectroscopy data, and develop a
3
qualitative picture of the spin-orbit coupling between the A 1 Σ+
u and b Πu states. In
section 2 we describe the Fourier Grid Hamiltonian method we used for our coupled
channel bound state calculations. Section 3 gives the results of those calculations
3
applied to the coupled 85 Rb A 1 Σ+
u and b Πu states. A description of the photoassociation experiment and the experimental results are given in section 4. In section
5 we compare the results of the coupled channel bound state calculations with the
experimental photoassociation data. As a reference we also describe the coupling
3
between the A 1 Σ+
u and b Πu states with a simple Landau-Zener formula.
2 Bound state calculations
With the photoassociation spectroscopy technique it is possible to measure rovibrational states over a wide range of energies. To compare experimental photoassociation spectra with theory, it is necessary to calculate bound state eigenvalues
over an equally wide energy range. Standard methods using iterative procedures,
e.g., the Numerov integration method, often involve time consuming calculations
and are therefore only useful for a relatively small energy range. Moreover, because of perturbations, e.g., spin-orbit coupling, bound states can be very close
together. This implies that small iterative steps are necessary as well as complex
basis-transformations to represent the coupling schemes over the whole internuclear range. Dulieu and Julienne [10] showed that a noniterative Fourier grid method
is a very powerful tool to calculate bound states of coupled systems. They applied a
Fourier Grid Hamiltonian (FGH) code to calculate the 150 lowest rovibrational states
3
of the coupled A 1 Σ+
u and b Πu potentials for two extreme cases: the light Na2 dimer
and the heavy Cs2 dimer. In this chapter we extend their approach to the case of Rb2 .
First we briefly discuss the main steps of the FGH method analogous to Monnerville
and Robbe [11].
2.1 Fourier grid method
The radial part of the eigenvalue problem of a diatomic molecule is written as
HΨ (R) = [T + V(R)]Ψ (R) = EΨ (R),
(5.1)
with T and V the kinetic energy and potential operator, respectively. To make a
discrete variable representation (DVR) approach we need to discretize the Hilbert
space. We do this, analogous to Colbert and Miller [12], by considering the coordinate
R restricted to the interval (R0 , RN ) and apply, in a first approach, a grid Ri with
uniform spacing ∆R = (RN − R0 )/N:
Ri = R0 + (RN − R0 )i/N,
i = 1 to N − 1.
(5.2)
The Hamiltonian, Eq. (5.1), is in the coordinate representation given by
Hii = Ri |H|Ri = Ri |T|Ri + Ri |V|Ri .
(5.3)
Photoassociation spectroscopy of
85
Rb2 0+
u states
89
Here the basis functions |Ri satisfy the orthonormal and completeness relationships,
(5.4)
Ri |Ri = δii ,
IR =
N−1
|Ri Ri |.
(5.5)
i=1
The discrete scalar product between two functions φn and φn in the discrete space
is defined by
N−1
φn (Rm )φn (Rm )∆R.
(5.6)
φn |φn =
m=1
Assuming orthonormal wave functions φn (R) satisfying the completeness relationship,
N−1
Iφ =
|φn φn |,
(5.7)
n=1
the Hamiltonian becomes [13],
Hii = Ri |H|Ri = Ri |T|Ri + Ri |V|Ri = Ri |T|Iφ |Ri + Ri |V|IR |Ri = Ri |T|
N−1
|φn φn |Ri + Ri |V|
n=1
N−1
|Rn Rn |Ri n=1
N−1
N−1
2 = −
Ri |φn φn |Ri +
V (Rn )Ri |Rn Rn |Ri 2µ n=1
n=1
= −
N−1
2
∗
φ
∆R
n (Ri )φn (Ri ) + V (Ri )δii
2µ
n=1
(5.8)
with, in the last step, the scalar products Ri |φ
n and φn |Ri like in Eq. (5.6),
written as,
Ri |φn =
N−1
Ri (Rm )φn (Rm )∆R =
m=1
δ
√ im φn (Rm )∆R = φn (Ri ) ∆R.
∆R
m=1
N−1
(5.9)
We take wave functions that vanish at the endpoints R0 and RN , i.e., particle-in-abox eigenfunctions,
φn (R) =
2
RN − R0
1/2
nπ (R − R0 )
sin
,
RN − R0
n = 1 to N − 1.
(5.10)
With these functions the Hamiltonian Eq. (5.8) becomes,
Hii
2
=−
2µ
π2
RN − R0
2
N−1
2 2
nπ i
nπ i
n sin
sin
+ V (Ri )δii ,
N n=1
N
N
(5.11)
90
Chapter 5
and working out the summation we obtain
!
"
1
π 2 2
1
i−i
−
i ≠ i : Hii =
(−1)
,
4µ(RN − R0 )2
sin2 [π (i − i )/2N] sin2 [π (i + i )/2N]
(5.12)
!
"
2 2
2
π 2N + 1
1
−
+ V (Ri ).
i = i : Hii =
2
2
4µ(RN − R0 )
3
sin (π i/N)
For an arbitrary number of spin channels j, the Hamiltonian for our eigenvalue problem is represented by
H{ij},{i j } = Tii δjj + Vjj (Ri )δii .
(5.13)
The advantage of the FGH approach follows from Eq. (5.12): No numerical matrix
element calculations have to be performed, since an analytical formulation of the
kinetic energy part of the Hamiltonian is available. Numerical accuracy demands that
the number of grid points be large enough to reproduce the maximum momentum
range. In other words, the stepsize S = (RN − R0 )/N, with R0 and RN the position of
the inner and outer turning points, must be small enough to represent correctly the
oscillations in the wavefunction at the internuclear separation where the potential
V is deepest (at that point the radial wave function oscillates most rapidly) [14]. In
formula:
π
S ≤ λmin /2 = ,
(5.14)
2µ
[E
−
V
]
min
2
with λmin the wavelength of the radial wave function at the position where the potential is deepest, E the binding energy (the energy of the bound state) and Vmin
the minimum of the potential. Rovibrational states extending to a large internuclear
separation involve a large gridlength RN − R0 , and therefore a large number N of grid
+
points. For example, calculating all vibrational states of the Rb2 A 1 Σ+
u (0u ) potential
2
2
4
up to the 5 S1/2 + 5 P1/2 dissociation limit requires N ≈ 10 grid points to fulfill the
criterion given by Eq. (5.14). This results in huge matrices with a huge demand on
computer memory, especially when several spin states are involved.
For large internuclear separations, however, the local wavelength of the radial
wave function is much larger than at the point where the potential is deepest. In this
region it makes sense to use a larger stepsize to reduce N [14, 15]. This gives a local
limitation of the allowed stepsize S(R) as given by:
S(R) ≤ λ(R; E)/2 = π
2µ
[E
2
− V (R)]
.
(5.15)
However, changing the gridsize also changes the expression for the kinetic energy,
as we discuss below.
2.2
Variable gridsize
To calculate the Rb2 bound states up to the 5 2 S1/2 + 5 2 P1/2 dissociation limit it
is necessary to make the grid spacing S(R) variable in such way that the grid size
Photoassociation spectroscopy of
85
Rb2 0+
u states
91
matches the local wavelength fairly closely. Tiesinga et al. [14] did this by applying
a non-linear coordinate transformation given by ρ = f (R), where f is a monotonic,
invertible function of R with a derivative that decreases with increasing R. Therefore, by defining a homogeneous grid in the transformed ρ coordinate, the distance
between two grid points in the original coordinate space grows with increasing R.
The relation between the variable grid spacing S(R) and the constant grid spacing
∆ρ is given by
∆ρ
S(R) = .
(5.16)
f (R)
Comparing this with Eq. (5.15), we adopt a transformation function given by [15]:
∞
12
dr
1 ∞ 2µ
[E − V (R)] dr
(5.17)
=
ρ = f (R) = 2
π R 2
R λ(r ; E)
with λ(r ; E) given by Eq. (5.15). An analytic expression for the transformation function f (R) is based on the long-range behavior V (R) = −Cn /R n of the potential.
Using this long-range behavior and the approximation E = 0, Eq. (5.17) leads to
n
1 ∞ 2µ Cn 1/2
1 2
2µCn 1/2 1 2 −1
f (R) =
dr =
.
(5.18)
π R 2 r n
π n−2
2
R
Deviations from a pure Cn /R n behavior at small internuclear distance and the position of the deepest point of the potential can make it necessary to add a term to the
transformation to provide a better mapping.
Transforming the coordinates also modifies the kinetic energy operator of the
Schrödinger equation according to [14]
!
"
2 d2
2
d
d2
T =−
=−
+ q(ρ)
p(ρ)
(5.19)
2µ dR 2
2µ
dρ 2
dρ
with
p(ρ) =
1
F (ρ)
and
q(ρ)
=
−
,
(F (ρ))2
(F (ρ))3
(5.20)
where F (ρ) is the inverse of f (R). We introduce a wavefunction φ(ρ) which is
normalized with respect to ρ if Ψ is normalized with respect to R
φ(ρ)
Ψ (R) = .
F (ρ)
(5.21)
With this substitution and Ṽ (ρ) = V (F (ρ)) the Schrödinger equation becomes
!
"
2
1
1
d2
− + V̄ (ρ) φ(ρ) = Eφ(ρ),
(5.22)
2µ
F (ρ) dρ 2 F (ρ)
where the potential term
2
V̄ (ρ) = Ṽ (ρ) +
2µ
!
3 (F (ρ))2
1 F (3) (ρ)
+
−
2 (F (ρ))3 4 (F (ρ))4
"
(5.23)
in which only the first part, given by Ṽ (ρ), is spin dependent. Equation (5.12) can be
used to discretize the second derivative term d2 /dρ 2 in Eq. (5.22).
92
Chapter 5
f
4
3
2
1
3
5 2P3/2780 nm
5 2P1/2795 nm
121 MHz
63.4 MHz
29.3 MHz
362 MHz
D2
2
D1
5 2S1/2
3
2
3.04 GHz
Figure 5.1: Schematic level diagram for 85 Rb with the first three fine structure states.
Indicated are, on the left, the D1 (795 nm), 5 S1/2 →5 P1/2 and D2 line (780 nm),
5 S1/2 →5 P3/2 and on the right the hyperfine intervals.
3 Application to the rubidium dimer
3.1
Coupled potentials
The formation of molecules by photoassociation takes place at rather large internuclear distances. The outer vibrational turning points of such molecules lie between
20 and 400 a0 for Rb. At these internuclear distances the overlap of the electron
clouds is negligible so that we can describe these molecules as two separate atoms
who interact via electrostatic interactions [14]. The atomic ground state Hamiltonian, which describes a single ground state atom, only involves a hyperfine interaction term which couples the nuclear spin i to the electron spin s to form total
The isotope 85 Rb has a nuclear spin i = 5/2, so the
angular momentum f = s + i.
2
ground state atom in a S1/2 fine structure state can have total angular momentum
f = 2 or f = 3. The atomic Hamiltonian for the excited 2 P states involves also the
strong spin-orbit interaction, which couples the electronic orbital angular momentum la to the electron spin s to form the electron angular momentum ja = la + s.
The hyperfine interaction then couples again the electron angular momentum to the
nuclear spin; for the 2 P1/2 state this gives total angular momentum f = 2, 3 and for
the 2 P3/2 state total angular momentum f = 1, 2, 3, 4. In Fig. 5.1 a schematic level
diagram for 85 Rb is shown.
For two atoms in the ground state the molecular Hamiltonian consists of the
atomic hyperfine interaction and the central, electrostatic interaction. The latter is,
for atoms at intermediate and large internuclear separation, given by the dispersion
and exchange interactions, forming together the singlet and triplet potentials. For
very large internuclear separation the central interaction becomes negligible compared to the atomic hyperfine interaction [16].
The molecular Hamiltonian for the interaction between a ground state atom and
an excited state atom consists, for large internuclear separation, of the atomic hy-
Photoassociation spectroscopy of
85
Rb2 0+
u states
93
3
85
Table I: Potential parameters for the 1 Σ+
Rb2 dimer, as used
u and Πu potentials for
for the calculations of the vibrational states.
1
C3 1 (a.u.)
C6 2 (103 a.u.)
C8 2 (105 a.u.)
Σ+
u
17.872
12.050
28.050
3
Πu
8.936
8.047
4.203
D1 3 (cm−1 )
12578.906
3
−1
D2 (cm )
12816.469
−1
∆F S (cm )
237.563
1
Value extracted from experimental long-range 0−
g analysis [23]. Values
1 +
given in Reference [22]: C3 ( Σu ) = 18.400 a.u. and C3 (3 Πu ) = 9.202 a.u.
2
Reference [22]
3
Reference [20]
perfine interaction, the spin-orbit interaction and the resonant electric dipole interaction. For the internuclear separations we are interested in, i.e., the internuclear
range between 20 a0 and 400 a0 , the hyperfine interaction is negligible for excited
states. The resonant electric dipole interaction has a 1/R 3 radial dependence, and
for large internuclear separations it is comparable to the spin-orbit interaction. Both
the resonant electric dipole interaction and the spin-orbit interaction conserve the
projection of the total electronic angular momentum j = ja + jb on the molecular
axis, therefore the so called Hund’s case (c) forms a convenient basis for this molecular Hamiltonian [17]. Following Movre and Pichler [18], the states are labeled by
Ωσ± , with Ω = |M| the absolute value of the projection of the total electron angular
momentum j on the molecular axis and σ = g or u the electronic parity. The superscript ±, which is only present for Ω = 0, indicates the symmetry with respect
to reflection in a plane containing the internuclear axis. Considering this, the total
Hamiltonian for all combinations of the 2 S ground state atom and the 2 P excited
state atom, is given by a 24×24 matrix which, because of the conservation of the
projection of total angular momentum, consists of three submatrices corresponding
to Ω =0, 1 and 2 [19]. Diagonalisation of those submatrices results in the 16 longrange adiabatic Movre-Pichler potential curves which for Rb2 are given in Fig. 5.2. For
infinite internuclear separation the curves go to the S1/2 +P1/2 and S1/2 +P3/2 limits,
from now on called the D1 and D2 dissociation limits. For 85 Rb2 these limits are at
D1 = 12578 cm−1 and D2 = 12816 cm−1 relative to the S1/2 +S1/2 limit [20] (Table I).
For intermediate internuclear separation, the resonant electric dipole interaction
becomes, because of its 1/R 3 radial dependence, much larger than the spin-orbit
interaction. In this region, called the uncoupled region by Movre and Pichler [18],
also the total spin S = sa + sb and Λ, the projection of total orbital angular mo = la + lb on the internuclear axis are conserved. Therefore the Hund’s
mentum L
case (a) labels 2S+1 |Λ|±
σ are used, resulting in four non-degenerate potential curves
94
Chapter 5
V (103 cm-1)
13.0
12.8
0u- 1u 0+g
2g
1g
1u
0g-
D2
+
0g
2u
1g
0+u
1u 0u-
12.6
12.4
D1
0g1g
20
0+u
40
60
R (units of a0)
80
Figure 5.2: The adiabatic long-range potential curves for the 2 S1/2 +2 P1/2,3/2 first excited states of 85 Rb2 labeled in Hund’s case (c). The two 0+
u curves, belonging to the
2
2
P1/2 dissociation limit D1 , and to the P3/2 dissociation limit D2 , respectively, are
indicated by arrows.
Σu
V (103 cm-1)
3
15
3
10
1
Πg
1
Πu
1
Σg
3
Σg
3
5
5
Πg
Πu
1
Σu
10
15
R (units of a0)
20
Figure 5.3: The eight adiabatic Born-Oppenheimer (ABO) potential curves for Rb2 . At
long, range all eight curves result in the same limit because spin-orbit interaction is
not included. The term adiabatic refers here to the electron motion.
Photoassociation spectroscopy of
85
Rb2 0+
u states
95
14
12
V (103 cm-1)
D2
b3Πu
A1Σ+u
D1
10
0+u(D2)
8
0+u(D1)
6
8
0
9
10
10
20
30
R (units of a 0)
11
40
3
Figure 5.4: Potential curves for the A 1 Σ+
u and b Πu states of Rb2 . The inset shows
the curves in their crossing region. Broken lines: diabatic potentials without spinorbit interaction; full lines: adiabatic 0+
u potentials with spin-orbit coupling included,
resulting for infinite internuclear separation in the two dissociation limits D1 and D2
(Table I).
corresponding to the Λ = 1 (Π) states with a ±C3 /R 3 radial dependence and the
Λ = 0 (Σ) states with ±2C3 /R 3 radial dependence. So for intermediate internuclear
distance the 16 adiabatic potential curves (Fig. 5.2) change over to four distinct electric dipole potentials [18]. At small internuclear separation, however, the exchange
interaction splits each of the four potentials into two separate branches, resulting in
the eight adiabatic Born-Oppenheimer (ABO) potentials shown in Fig. 5.3 [14]. The
Born-Oppenheimer approximation states that the electrons move adiabatically with
respect to the slow nucleus. The term adiabatic refers to the inclusion of Ψel |T |Ψel in the potential. For large internuclear separation the eight ABO potentials go to to
the same limit.
+
3
+
We are interested in the A 1 Σ+
u (0u ) and b Πu (0u ) excited states and the coupling
between them. We use the terms diabatic and adiabatic with respect to the spinorbit interaction. In this context diabatic means that the spin-orbit interaction is not
included, and adiabatic means that spin-orbit interaction is included. In Fig. 5.4 the
3
potential curves for the A 1 Σ+
u and b Πu states of Rb2 are shown over the complete
1 +
internuclear range. The adiabatic 0+
u potentials follow from the diabatic A Σu and
b 3 Πu potentials by including the spin-orbit interaction. More details of the potentials are given in the next section, in which we calculate the rovibrational states of
+
3
+
the coupled A 1 Σ+
u (0u ) − b Πu (0u ) system.
96
Chapter 5
25
Bv (10-3 cm-1)
20
15
b3Πu
0+u(D1)
0+u(D2)
10
A1Σ+u
5
D1 D2
0
6
8
10
12
Ev (103 cm-1)
14
Figure 5.5: Rotational constant Bv (J=0) in Rb2 . Broken lines: vibrational levels of
3
the A 1 Σ+
u and b Πu diabatic potentials, disregarding the spin-orbit interaction; full
lines: vibrational levels of the 0+
u adiabatic potentials. The data points (+) indicate
3
the results of the coupled bound state calculation, using the A 1 Σ+
u and b Πu diabatic
potentials as basis and the spin-orbit interaction as the coupling.
3.2
+
3
+
Coupled A 1 Σ+
u (0u ) − b Πu (0u ) system
We will use the FGH method described in section 2 to calculate the rovibrational
+
3
+
energies of the coupled 85 Rb A 1 Σ+
u (0u ) and b Πu (0u ) states. By implementing the
coordinate transformation described in section 2.2 in the FGH computer code used
by Dulieu and Julienne [10] it was possible to calculate the Rb2 rovibrational states
up to the D1 dissociation limit. In our case we are dealing with two spin channels,
3
the 1 Σ+
u and Πu states, so the total Hamiltonian given by Eq. (5.13) can be expressed
in terms of square matrices of order 2(N − 1),
HAA HAB
HBA HBB
=
T 0
0 T
+
VA VAB
VAB VB
,
(5.24)
where VA , VB , and VAB are diagonal (N − 1) × (N − 1) matrices and T is a nondiagonal
(N − 1) × (N − 1) matrix.
3
The A 1 Σ+
u and b Πu potentials we used are ab initio potentials calculated by
Foucrault et al. [21]. For the long-range part of the potentials we used the dispersion
coefficients given by Marinescu and Dalgarno [22], except for the C3 coefficient. For
that we used a value extracted from an experimental long-range 0−
g analysis [23].
1 +
Table I gives the most important potential parameters of the A Σu and b 3 Πu po3
tentials. We assume the spin-orbit coupling between the A 1 Σ+
u and b Πu potentials
to be R-independent. This means that we take a constant value for the coupling
term VAB in Eq. (5.24), namely the value of the spin-orbit interaction VSO at infinite
Photoassociation spectroscopy of
85
Rb2 0+
u states
97
Bv (10-3 cm-1)
6
0+u(D2)
4
v=437
v=443
2
D1
0+u(D1)
0
12.54
12.56
3
12.58
-1
Ev (10 cm )
Figure 5.6: Calculated rotational constants Bv for the vibrational states close to the
D1 dissociation limit. The data points are for the case of coupled potentials. The
arrows indicate a strongly perturbed vibrational state v = 437 and a vibrational
state, v = 443, whose Bv is close to the unperturbed adiabatic value. The solid lines
correspond to Bv for the case of the uncoupled adiabatic potentials.
internuclear separation:
VAB (R) = VSO (∞) = 2 · ∆F S /3,
(5.25)
in which ∆F S is the fine structure splitting (Table I).
By taking N = 1000 grid points and a physical grid extending from 3.5 a0 to
550 a0 , it is possible to calculate all rovibrational states of the coupled A 1 Σ+
u and
b 3 Πu potentials up to the D1 dissociation limit. Knowing that the coupled A 1 Σ+
u and
3
b Πu potentials contain 542 rovibrational states, labeled from v = 0 to v = 541,
it is remarkable that with only N = 1000 grid points, so less than 2 grid points per
node for the highest rovibrational states, we can reach an accuracy in the vibrational
energy better than 10−4 cm−1 [15]!
Analogous to Dulieu and Julienne [10], we calculate both the rovibrational ener2
gies and the rotational constant Bv = 2µ Ψv (R)|1/R 2 |Ψv (R)
for rotational quantum
number J = 0. Because of its 1/R 2 dependence, the rotational constant is very sensitive to the effective excursions of the vibrating molecule in the radial direction. Due
to the different radial range over which the two diabatic (or adiabatic) potentials
extend, the rotational constant is very sensitive to the coupling between the two potentials. The vibrational levels are sensitive to the shape of the potentials involved.
Moreover, the rotational constant is a parameter which can be readily measured by
high resolution photoassociation spectroscopy (see section 4). By comparing the
measured rotational constants to the calculated values, detailed insight is obtained
3
in the coupling of the A 1 Σ+
u and b Πu potentials.
In Fig. 5.5 we show the calculated rotational constant as a function of the binding
3
energy Ev , for three cases: (1) for the eigenstates of the individual A 1 Σ+
u and b Πu
98
Chapter 5
0+u(D2)
Ev (103 cm-1)
12.58
D1
0+u(D1)
12.56
12.54
Bv
12.52
12.50
20
40
60
80
R (units of a 0)
100
+
Figure 5.7: Vibrational levels of the adiabatic 0+
u (D1 ) and 0u (D2 ) potentials, together
with the corresponding rotational constants. The maxima in Bv (Ev ) coincide approximately with the vibrational levels of the 0+
u (D2 ) potential, with its wider vibrational
spacing and its smaller internuclear separation.
potentials, diabatic regarding to spin-orbit interaction; (2) for the eigenstates of the
1 +
3
adiabatic 0+
u potentials, and (3) for the coupled A Σu and b Πu channels. In this
figure we see that the results of the coupled channel calculations follow neither the
diabatic (broken) lines nor the adiabatic (solid) lines but oscillate. This oscillatory
behavior of the rotational constant is also found by Kokoouline et al. [15], who did
the same kind of calculations as described in this chapter. The oscillations in the
rotational constant can be seen more clearly when we look at the levels close to
the D1 dissociation limit, as shown in Fig. 5.6. The arrows indicate the vibrational
states v = 437 and v = 443 with a maximum and a minimum rotational constant,
respectively.
The oscillations in the rotational constant are due to perturbations caused by
the interleaving of the relatively sparse 3 Πu vibrational levels in the much denser
spectrum of the 1 Σ+
u vibrational levels dissociating to the D1 limit. In Fig. 5.7 we
see schematically that the position of the maxima of the peaks corresponds approximately to the values of the adiabatic 0+
u (D2 ) vibrational energies.
The perturbations of the 0+
(D
)
levels
in the spectrum of the 0+
2
u
u (D1 ) levels are
also visible in the vibrational wave functions. Figure 5.8 shows the vibrational wave
function of the v = 437 and v = 443 states. We see that the wave function corresponding to the vibrational state with the large rotational constant (v = 437) is
strongly perturbed around R = 20 a0 , the position of the outer turning point of the
0+
u (D2 ) potential.
Photoassociation spectroscopy of
|Ψv(R)|2
0.04
85
Rb2 0+
u states
99
v=437
0.02
0.00
0
0.08
20
40
R (units of a 0)
60
20
40
R (units of a 0)
60
|Ψv(R)|2
v=443
0.04
0.00
0
Figure 5.8: Vibrational wavefunctions |Ψv (R)|2 of the coupled potentials for the
strongly perturbed v = 437 vibrational state (top panel) and the only slightly perturbed v = 443 vibrational state (lower panel) (Fig. 5.6).
4 Photoassociation experiment
4.1
Experimental setup
We study photoassociation of 85 Rb atoms by measuring trap loss from a far-off resonance trap (FORT). The FORT we use consists of a single, linearly polarized, focused
Gaussian laser beam. Like described in chapter 2, the trapping force is the optical
dipole force, a conservative force based on the interaction between the gradient of
the electric field of the laser light and the induced atomic dipole moment. This interaction has been described by Dalibard and Cohen-Tannoudji [24] in a dressed state
picture, and in terms of optical Bloch equations by Gordon and Ashkin [25]. Assuming that the laser detuning ∆L = ωL − ω0 , with ωL the laser frequency and ω0 the
atomic transition, is much larger than the Rabi frequency Ω and the spontaneousemission rate of the atom Γ , the trap depth U0 due to the dipole force is, analog to
Eq. (2.18), given by [24, 25]
U0 = Ω02 /4∆L ,
(5.26)
with Ω0 the Rabi frequency in the waist of the FORT laser. The advantage of a FORT
over a magneto-optical trap (MOT) is the absence of near-resonance light, which
drastically reduces the spontaneous-emission rate γs = Γ Ω02 /4∆2L and thereby the
100
Chapter 5
Repumper
Molasses beams
Dipole
PMT
PA
Probe
Figure 5.9: Schematic view of the laser beams in the FORT setup.
heating by photon recoils [26]. The wavelength of the FORT laser beam, produced
by a Coherent Model 899-01 Ti:sapphire laser, can be tuned between 4 and 67 nm to
the red of the 85 Rb 5 2 S1/2 -5 2 P1/2 transition. Typical values for the trap depth are
1-10 mK.
In Fig. 5.9 a schematic view of the laser beams for the photoassociation spectroscopy (PAS) experiment are given. The FORT is loaded from a vapor cell MOT
consisting of three pairs of counterpropagating σ + − σ − beams, which intersect at
right angles inside a Rb vapor cell at the zero-field point of a magnetic spherical
quadrupole field [26]. When the FORT is loaded the photoassociation (PA) beam is
turned on and after a while the amount of atoms in the FORT is measured with laser
induced fluorescense.
Figure 5.10 gives a complete overview of the optical components necessary to
run the experiment. The laser beams for the MOT are derived from a high-power
diode laser producing 40 mW of laser light which is injection-locked to a gratingtuned DBR diode laser. The grating-tuned DBR diode laser is locked to the 85 Rb 5
2
S1/2 (f=3) ↔ 5 2 P3/2 (f =4) cross-over saturated absorption peak. One beam from
the laser (26 mW) is detuned with a 80 MHz acousto optic modulator (AOM) to 10
MHz (1.7 Γ ) to the red of the 85 Rb 52 S1/2 (f=3) ↔ 52 P3/2 (f =4) transition and is used
for all the MOT beams. A second beam from this laser is detuned with a 100 MHz
AOM and is used as a probe beam to measure the amount of atoms in the FORT with
laser-induced fluorescence. A second grating-tuned diode laser is locked with a 130
MHz offset to the 85 Rb 5 2 S1/2 (f=2) ↔ 5 2 P3/2 (f =3) cross-over saturated absorption
peak to be used as a repumper beam, preventing optical pumping into the 5 2 S1/2
(f=2) state. Figure 5.11 schematically shows the different laser transitions in the 85 Rb
level diagram.
For the PA laser we use a narrowband Ti:Sapphire laser (Coherent 899-21, 1 MHz).
The wavelength of this laser can be measured by a wavemeter with a resolution of
30 MHz. The stabilized HeNe laser of this wavemeter was calibrated by locking the
Ti:Sapphire laser to a Rb saturation absorption peak of known frequency [20]. During a PAS experiment, as described in the next section, the wavelength of the PA
laser is scanned and the amount of atoms in the FORT is measured. At the beginning and end of each scan, usually over 20 GHz, the wavelength is measured within
Photoassociation spectroscopy of
85
Rb2 0+
u states
101
Ti:Sapphire
Coherent 899-21
Argon Ion Laser
AOM
300 MHz
PA etalon
AOM
Ti:Sapphire
Coherent 899-01
AOM
AOM
FORT
AOM
Wavemeter
DL
Repumper
MOT
AOM
Rb Cell
Probe
Rb Cell
DL
DBR
Figure 5.10: Schematic view of the optical components used for the photoassociation
experiment.
f
4
52P3/2 780 nm
3
2
1
52P1/2 795 nm
c
d
e
a
b
52S1/2
3
2
Figure 5.11: Energy level diagram for 85 Rb showing the different laser frequencies
used in the photoassociation experiment: (a) FORT laser beam, (b) PA laser beam, (c)
MOT beams, (d) repumper, and (e) probe laser beam.
102
Chapter 5
PA rate (arb. units)
J=012 3 4 5
0+u
12.544
0+u 0-g
1g
12.545
12.546
Wavelength (10 3 cm-1)
Figure 5.12: Part of the experimental photoassociation spectrum, showing a rovibra−
tional transition of each of the 0+
u , 0g and 1g molecular states. The inset shows the
rotational structure of the vibrational level of the 0+
u state in more detail, including
the assignment of the rotational quantum number J.
0.001 cm−1 . During a scan, frequency markers are derived from the transmission
peaks of a stable 300 MHz (0.01 cm−1 ) etalon. By fitting this etalon signal, and using
the begin and end readings from the wavemeter, it is possible to assign the PA laser
frequency within 0.002 cm−1 .
4.2
Measuring routine
The experiments we did are similar to those described by Cline et al. [9]. Fully loaded,
the MOT contains 106 atoms at a density of 1010 cm−3 . From the MOT we loaded
approximately 104 85 Rb atoms with a temperature of several hundred µK into the
FORT. The density in the FORT was about 1012 cm−3 . The FORT laser, with a waist
of 11(1) µm and a power of 1.6 W, is tuned to 12288 cm−1 , which is between two
well-resolved photoassociation resonances. To avoid power broadening and a shift
of the photoassociation resonances by the FORT laser, the FORT and PA beams were
chopped out of phase with respect to each other at 200 kHz. By chopping the MOT
beams in phase with the FORT beam the atoms were kept in the f=2 ground state.
The time-averaged well depth of the FORT is U0 = 5(1) mK. After a photoassociation time of 500 ms, the remaining atoms were probed using laser-induced fluorescence. This measurement cycle was repeated for a succession of PA frequencies,
during which the intensity of the PA laser is changed in several discrete steps from
2 kW/cm2 for low PA frequencies to 1 W/cm2 for high frequencies close to the dissociation limit. We measured from the D1 dissociation limit (12579 cm−1 ) down to
12500 cm−1 , to obtain an overlap with earlier measured spectra by Miller et al. [26]
and Cline et al. [27].
Photoassociation spectroscopy of
85
Rb2 0+
u states
103
6
Bv (10-3 cm-1)
5
4
3
2
1
0
12.51 12.53 12.55 12.57 D1
Ev (103 cm-1)
Figure 5.13: Experimental values of the rotational constants Bv close to the D1 dissociation limit. The open circles on the horizontal axis indicate vibrational states for
which it was not possible to determine the rotational constant. Due to overlap of
lines some vibrational states are missing.
4.3 Experimental results
Figure 5.12 shows a small part of the spectra we obtained. In this figure we clearly
+
see the separate peaks from the 1g , 0−
g and 0u rovibrational series. We separated
−
the 0+
u rovibrational series from the 1g and 0g series by comparing the spectra with
−
1g and 0g spectra that were measured and assigned earlier [26, 27]. The rotational
structure of these 0+
u rovibrational states (see inset Fig. 5.12) has been analyzed in
great detail. We found more than a hundred 0+
u rovibrational states in the region
from 12579 cm−1 to 12500 cm−1 . From 63 of these states the rotational structure
could be resolved. From the rotationally resolved 0+
u spectra, we determined the
rotational constant Bv of the rovibrational states by analyzing the position of the
rotational lines with the relation Ev (J) = Bv J(J + 1).
We also looked for 0+
u states in spectra measured and assigned earlier by Miller
et al. [26] and Cline et al. [27]. Miller et al. obtained a well-resolved 85 Rb2 photoassociation spectrum from 11600 cm−1 to 12528 cm−1 , so their data overlap with ours.
Unfortunately, the resolution of their spectrum is not always high enough to observe
rotational structure. Moreover, they were more interested in the rotational structure
of the 0−
g and 1g rovibrational states so they only did high-resolution measurements
around areas where they expected those states. Nevertheless, it was still possible to
assign 16 0+
u vibrational states and determine their rotational constant in the region
from 11900 cm−1 to 12500 cm−1 .
In Fig. 5.13 we display the measured rotational constants for the highest rovibrational states as a function of the binding energy. The uncertainty in the measured
vibrational energy and estimated rotational constants is smaller than the size of the
104
Chapter 5
open circles, since the experimental resolution of the binding energy is better than
0.002 cm−1 . The uncertainty in the rotational constants follows from the accuracy
of the fits, and is less than 10%. The open circles on the energy axis in the figure indicate the vibrational states for which it was not possible to determine the rotational
constant. For some vibrational states it was not possible to determine the rotational
constant, either because the spectrum was not clear enough or because the 0+
u state
+
overlaped with a 0−
or
1
vibrational
state.
For
the
0
vibrational
states
close
to the
g
g
u
dissociation limit, the rotational constant could not be determined because it was
too small.
5 Model calculations
5.1 Coupled bound state calculations
Figure 5.14 shows the measured rotational constants for the highest rovibrational
states as a function of the binding energy, together with the calculated values. We see
that both the calculated and measured rotational constants of the 0+
u rovibrational
states oscillate as a function of their binding energy. We also see that the maxima of
the calculated oscillations are shifted in energy with respect to measurements. Also,
the period of the oscillations in the theoretical case is larger than in the experimental
case. From Fig. 5.7 and the description of section 3.2, it is clear that the period of
the oscillations is determined by the spacing of the vibrational levels of the 0+
u (D2 )
potential. Because of the shift in energy and the larger spacing in the theoretical
case, we can conclude that the 3 Πu potential used is not fully correct.
Because the spacing between the vibrational levels of the 0+
u (D2 ) potential in the
theoretical case is too large, we can conclude that the 0+
(D
)
2 potential is too steep
u
in the internuclear range corresponding to the energy range we are looking at. The
rovibrational levels of the 0+
u (D2 ) potential with a binding energy between 12500
cm−1 and the D1 dissociation limit have outer turning points around 20 a0 . This is
roughly the range where the ab initio 3 Πu potential is connected to the long-range
part by a spline-function. The vibrational level spacing close to dissociation is determined mostly by the long-range part of the potential, i.e., the dispersion coefficients
C3 , C6 and C8 . The C3 coefficient of the 0+
u (D2 ) potential can not be changed independently from the C3 coefficient of the 0+
u (D1 ) potential, since they both follow from
the dipole-dipole interaction theory and differ by a factor 2 (Table I). The higher
order dispersion coefficient, C6 and C8 , however, are different for both potentials,
but for large internuclear separation the influence of these higher order terms is
small. Therefore probably the semi-empirical connection between the ab initio 3 Πu
potential and the long-range part is not fully correct.
The width and the height of the peaks is determined by the strength of the spinorbit coupling at the position of the crossing of the two potentials. This can be
3
seen when we vary the coupling between the 1 Σ+
u and Πu potentials by changing
the spin-orbit interaction VSO at the position Rx of the crossing. We introduce a
modified coupling at the crossing Rx = 9.5 a0 of the potentials by subtracting a
Photoassociation spectroscopy of
6
5
5
4
3
2
Bv (10 -3 cm-1)
Bv (10 -3 cm-1)
a
6
1
0
85
Rb2 0+
u states
105
b
4
3
2
1
12.51 12.53 12.55 12.57 D1
Ev (103 cm-1)
0
12.51 12.53 12.55 12.57 D1
Ev (103 cm-1)
Figure 5.14: Rotational constant for the highest 0+
u vibrational states. Dots/full lines:
coupled channel bound state calculations; open circles/dashed lines: measurements.
a: Calculated with R-independent spin-orbit coupling (Eq. (5.25)); b: Calculated with
modified spin-orbit coupling (Eq. (5.27)).
Gaussian function with a width σ = 1 a0 and a height such that the coupling at the
crossing point Rx is 60% of its asymptotic value:
(R − Rx )2
VSO (R) = VSO (∞)[1 − 0.40 exp −
].
(5.27)
2σ 2
In this way we obtained a smaller value of the coupling at the crossing point of the
two potentials, in accordance with the calculations done by Dulieu and Julienne [10]
for the Cs2 dimer. Because the Gaussian function is broader than the width of the
avoided crossing, we are sure that the coupling is decreased by the same factor over
the full range of interest. Figure 5.14b shows the results of calculations done with
the smaller spin-orbit coupling defined by Eq. (5.27).
The influence of this smaller coupling is clear: the peaks in Fig. 5.14b are higher
and narrower than those in Fig. 5.14a, and resemble the experimental data more
closely. Due to the smaller spin-orbit coupling the modulation depth of the rotational constant as a function of the binding energy becomes larger. In other words,
the rotational constant of a vibrational state of one potential is less disturbed by the
other potential when the coupling strength is decreased. As expected, the position of
the peaks and the spacing between the maxima of the oscillations is not influenced
by the modified coupling, since they are determined solely by the spacing between
the vibrational levels of the 0+
u (D2 ) potential (Fig. 5.7).
As mentioned in section 4.3 we also looked for 0+
u vibrational states in spectra
measured and assigned earlier by Miller et al. [26] and Cline et al. [27]. We found
in the energy range from 11900 cm−1 to 12500 cm−1 16 0+
u vibrational states and
we determined their rotational constant. Figure 5.15 shows the measured and cal-
106
Chapter 5
14
b3Πu
Bv (10-3 cm-1)
12
10
8
6
0+u(D2)
A1Σ+u
0+u(D1)
4
2
0
11.9
12.1
12.3
12.5
Ev (10 cm )
3
-1
D1
Figure 5.15: Rotational constant for the vibrational 0+
u states with binding energy Ev
between 11900 cm−1 and the D1 dissociation limit. Dots/full lines: coupled channel
bound state calculations, done with the modified spin-orbit coupling. Broken lines:
3
vibrational levels of the A 1 Σ+
u and b Πu diabatic potentials, disregarding the spinorbit interaction; full lines: vibrational levels of the 0+
u adiabatic potentials. Open
circles: measurements.
culated rotational constant as a function of the binding energy in the region from
11900 cm−1 to the D1 dissociation limit. We see that, over the complete energy
range, the measured rotational constants fall in the region enclosed by the coupled
channel calculations and that they neither follow the diabatic values nor follow the
adiabatic values, but oscillate. The coupled channel calculations were done with the
modified spin-orbit coupling.
In Fig. 5.15 we also see that the values of the rotational constant coming from
the coupled bound state calculations for some vibrational levels with binding energy
below 12500 cm−1 fall below the values of the rotational constant for the diabatic
1 +
Σu potential and the adiabtic 0+
u (D1 ) potential. This means that the wave functions
of those vibrational states in the coupled case are disturbed in such a way that more
of probability lies at larger internuclear separation than in the purely diabatic or
adiabatic case.
Another thing which can be seen in Fig. 5.15 is that the lines indicating the rotational constants for the diabatic 3 Πu potential and the adiabtic 0+
u (D2 ) potential
shows a strange hump around 12300 cm−1 . This could also be an indication that the
semi-empirical connection between the ab initio 3 Πu potential and the long-range
part is not fully correct, as mentioned earlier.
We tried to assign the measured rovibrational levels by comparing the measured
binding energies with the calculated ones in two different ways, first we labeled the
measured vibrational levels such that the energy difference between the experimental and calculated binding energy is minimum, i.e., we labeled a measured vibrational
level with the vibrational wave number v of the calculated vibrational level which
Photoassociation spectroscopy of
85
Rb2 0+
u states
107
Evexp(103 cm-1)
Evexp-Evcalc (cm-1)
1
12.51
12.53
12.55
12.57 D1
0
-1
-2
400 420 440 460 480 500 520
v
exp
Figure 5.16: Energy difference between the experimental vibrational energy Ev
and the calculated vibrational energy Evcalc as a function of the vibrational number v
exp
(lower horizontal axis) and the experimental vibrational energy Ev (top horizontal
axis). The line with the dots indicates a vibrational number assignment where a
minimum energy difference is chosen, the line with the open circles indicates a more
physical vibrational number assignment (see text).
is closest to the measured one. The result of this labeling routine can be seen in
exp
Fig. 5.16, which shows the difference between the measured vibrational energy Ev
and the calculated vibrational energy Evcalc as a function of the assigned vibrational
number v and the experimental binding energy. From this we found that the highest
measured vibrational state has vibrational number v = 515. From the coupled channel calculations we found that the vibrational level closest to the D1 dissociation
limit has vibrational number v = 541. We see in the figure that from v = 515 down
to v = 470 the energy difference shows a smooth behavior, but starts to oscillate
from v = 470. This oscillation comes from the fact that for some vibrational levels
exp
the energy difference from an experimental level with energy Ev to a calculated
calc
level with energy Ev−1
is smaller than the energy difference to a calculated level
calc
with energy Ev . In this way the labeling routine skips a vibrational level. A more
physical way to assign the levels is also shown in Fig. 5.16: here we demand that the
energy difference increases with decreasing energy. This comes from the uncertainty
in the potentials used: from the dissociation limit to the bottom of the potential the
uncertainty accumulates. The jumps in the energy difference are probably an effect
of the coupling between the potentials or come from the fact that it is not always
clear which experimental level belongs to which calculated level. We conclude that
the coupling between the potentials, the uncertainties in the potentials, and the fact
that not all vibrational levels were measured, makes it impossible to assign all the
108
Chapter 5
vibrational levels.
5.2 Landau-Zener description
We can also describe the coupling between the potentials with the simple LandauZener formula [28]. In the adiabatic representation there is a finite chance to cross
from one adiabatic curve to the other, due to the finite value of the radial velocity of
the atoms. The probability of a single diabatic crossing is given by
Px = exp (−vref /vx ) ,
(5.28)
with vx the velocity of the atoms at the crossing radius Rx . The characteristic velocity vref is given by
vref =
V2AB
|
d∆V
|
dR
,
(5.29)
with d∆V /dR the derivative of the difference potential ∆V = VA − VB at Rx , with
VA and VB the diabatic potentials. From Eq. (5.28) we see that the probability to
jump from one adiabatic curve to the other is small for a large value of vref and
is large for a small value of vref . Decreasing the coupling potential VAB implies a
decrease of the crossing parameter vref , i.e., an increase of the probability to jump
from one adiabatic curve to the other. This is also what we see in Fig. 5.14: with
a large spin-orbit coupling the oscillations in the rotational constant are large, the
peaks are broad and the value of the rotational constant goes to the adiabatic case.
In conclusion, a large spin-orbit coupling, i.e., large coupling potential VAB , results
in a small probability to jump from one adiabatic potential curve to the other, the
so-called adiabatic case. Conversely, a small spin-orbit coupling results in a large
probability to jump from one adiabatic potential curve to the other, the so-called
diabatic case.
5.3 Alkali dimer systems
With the Landau-Zener transition picture we can also easily explain differences between Na2 , Rb2 and Cs2 . Dulieu and Julienne [10] found that, due to the small spin3
orbit coupling for the Na2 dimer, the A 1 Σ+
u and b Πu states act purely diabatical;
for the Cs2 dimer, due to a large spin-orbit coupling, the behavior is purely adia3
batic. We found that in the case of the Rb2 dimer the A 1 Σ+
u and b Πu act neither
diabatic nor adiabatic. We can determine the transition probability Px for the three
alkali dimers by calculating the velocity vx and the characteristic velocity vref at the
crossing Rx of the two potentials.
In Table II the potential parameters and the transition probabilties for the three
alkalis are given. We see that this estimate corresponds with the results of the
bound state calculations described in this chapter and those done by Dulieu and
Julienne [10]: a large transition probability for Na2 , a small one for Cs2 and a value
Photoassociation spectroscopy of
85
Rb2 0+
u states
109
Table II: Potential parameters and Landau-Zener probabilities for Na2 , Rb2 and Cs2
for an unmodified coupling VAB = VSO (∞), for Rb2 also for a modified coupling
VAB = 0.6 · VSO (∞).
Coupling
parameter
M
|V (Rx )| ( cm−1 )
∆F S (cm−1 )
d∆V
dR
VAB = VSO (∞)
VAB = 0.6 · VSO (∞)
(cm−1 a−1
0 )
vref (m/s)
vx (m/s)
Na2
23
8255
17
Rb2
85
5808
238
1
850(250)
$
#
0.6 +0.3
−0.2
4100
1
620(160)
# $
200 +70
−40
1800
$
#
Px
1
0.89
Px
-
0.96
+0.02
#−0.03 $
+0.01
−0.01
Cs2
133
5431
351
1
590(70)
# $
450 +60
−50
1400
$
#
0.72
+0.02
−0.03
-
1
The derivative of ∆V is determined by hand from a potential graph, this causes
d∆V
the given uncertainties in dR and therefore the uncertainties in vref and Px .
in between for Rb2 . In Table II also the transition probability for the modified spinorbit interaction in the Rb2 case is given: a 40% smaller spin-orbit coupling results
in an 8% higher probability to jump to the other adiabatic potential curve.
Cline et al. [9] estimated from broadened 0+
u vibrational levels, belonging to the
+
0u (D2 ) potential, the probability of the Landau-Zener transition, and found P =
2Px (1 − Px ) = 0.14 ± 0.02, with Px the probability of a single diabatic crossing given
by Eq. (5.28). Using this expression we find from Table II, P = 0.19+0.05
−0.03 using the
+0.02
unmodified coupling, and P = 0.08−0.02 using the modified coupling.
6 Concluding remarks
In summary, we have shown experimentally and theoretically that in the case of the
3
Rb2 dimer the A 1 Σ+
u and b Πu states neither act purely diabatically nor act purely
adiabatically. To obtain a better quantitative agreement, the shape of the A 1 Σ+
u and
b 3 Πu potentials has to be modified. From a qualitative comparison between the
experimental and calculated results, we conclude that the spin-orbit coupling at the
coupling radius is smaller than its asymptotic value, a trend that is in agreement
with the results for Cs2 found by Dulieu et al. [10].
Improved potentials are being developed by Stevens et al.: they are calculating
3
new Rb2 A 1 Σ+
u and b Πu with the same techniques they used to calculate the
3 +
Σu and 3 Σ+
g potentials for Cs2 and other alkali dimers [29]. In these calculations,
the spin-orbit interaction between the potentials is also rigorously taken into account [30]. In this way hopefully a more quantitative picture of the coupling between
the potentials can be formed.
A limitation of the model we used is that we only took the coupling between the
110
Chapter 5
3
A 1 Σ+
u and b Πu states into account and not the coupling to other states. Much
larger eigenvalue problems would have to be solved in that case [14].
References
[1] C.C. Tsai, R.S. Freeland, J.M. Vogels, H.M.J.M. Boesten, B.J. Verhaar, and D.J.
Heinzen, Phys. Rev. Lett. 79, 1245 (1997).
[2] J.R. Gardner, R.A. Cline, J.D. Miller, D.J. Heinzen, H.M.J.M. Boesten, and B.J. Verhaar, Phys. Rev. Lett. 74, 3764 (1995).
[3] J.M. Vogels, C.C. Tsai, S.J.J.M.F. Kokkelmans, B.J. Verhaar, and D.J. Heinzen,
Phys. Rev. A 56, R1067 (1997).
[4] F.A. Van Abeelen, D.J. Heinzen, and B.J. Verhaar, Phys. Rev. A 57, R4102 (1998).
[5] Ph. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar,
Phys. Rev. Lett. 81, 69 (1998).
[6] F.A. Van Abeelen and B.J. Verhaar, Phys. Rev. A. 59, 578 (1999).
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[8] X. Wang, H. Wang, P.L. Gould, W.C Stwalley, E. Tiesinga, and P.S. Julienne, Phys.
Rev. A. 57, 4600 (1998).
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[10] O. Dulieu and P.S. Julienne, J. Chem. Phys. 103, 60 (1995).
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[12] D.T. Colbert and W.H. Miller, J. Chem. Phys, 96, 1982 (1992).
[13] The author likes to thank F.A. Van Abeelen for his help with the derivation of
Eq. (5.8).
[14] E. Tiesinga, C.J. Williams, and P.S. Julienne, Phys. Rev. A 57, 4257 (1998).
[15] V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws, J. Chem. Phys.,
110(20), 9865 (1999).
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Molecules, 2nd ed. (Van Nostrand Reinhold Co., New York 1950).
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[19] M. Marinescu, A. Dalgarno, Z. Phys. D., 36, 239 (1996).
Photoassociation spectroscopy of
85
Rb2 0+
u states
111
[20] G.P. Barwood, P. Gill, and W.R.C. Rowley, Appl. Phys. B 53, 142 (1991).
[21] M. Foucrault, Ph. Millie and J.P. Daudey, J. Chem. Phys. 96 (1992).
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[24] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
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[28] E.E. Nikitin and S.Ya. Umanskii, Theory of Slow Atomic Collisions (SpringerVerlag, Berlin, 1984).
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[30] W.J. Stevens (private communication).
112
Chapter 5
Summary
113
Summary
Since the early eighties a variety of laser cooling techniques have been applied to
cool and trap neutral atoms. The increasing knowledge of trap loss mechanisms
has led to the confinement of larger and colder samples of atoms. The trapping of
atoms in purely magnetic traps and the technique of forced evaporative cooling were
responsible for the attainment of Bose-Einstein condensation (BEC) in cold, dilute
gases. In this quantum regime the atomic gases have typically a number density of
more than 1014 atoms/cm3 and a temperature of the order of 1 µK. Till now this
regime has only been reached with alkali-metal atoms and atomic hydrogen. In this
thesis we describe the first experimental steps towards a BEC of metastable neon
atoms. This research was carried out in the Atomic Physics and Quantum Electronics
group at the Physics Department of the Eindhoven University of Technology.
The usual scheme to create a BEC starts with the trapping of a large sample of
cold atoms in a magneto-optical trap (MOT). This first step is already complicated
for metastable rare gas atoms, since the efficiency of producing rare gas atoms in
the metastable state is very low, i.e., ∼ 10−4 . However, by applying several laser
cooling techniques we succeed in creating a cold and intense beam of metastable
Ne(3s) 3 P2 atoms. The atoms leaving the source are collimated, slowed, and finally
compressed to a beam with a diameter of 1 mm and a flux of 5 × 1010 atoms/s. The
transverse and longitudinal temperature of the bright beam are T⊥ = 285 µK and
T|| = 28 mK, respectively, while the maximum brilliance of the beam equals 4 × 1022
s−1 m−2 sr−1 . Apart from a bright beam of the 20 Ne isotope, it is easy to create a
beam of the 22 Ne isotope, with the only difference that the flux of the latter is seven
times smaller. Our cold intense atomic beam can be used for a whole range of cold
collision experiments, e.g., photoassociation spectroscopy. In this thesis it is shown
that the bright neon beam is also an excellent source for loading a magneto-optical
trap.
The beam flux of the atomic beam provides a loading rate of up to 1.5 × 1010
atoms/s of the MOT developed in our lab. Because of this high loading rate we were
able to trap almost 1010 metastable neon atoms in the MOT; to our knowledge this
is the largest number of trapped metastable atoms reported. This optimum number
is obtained under rather unconventional conditions. First of all, we do not have an
ordinary MOT consisting of six trapping beams but a seven-beam MOT. The seventh
laser beam is introduced by the extra slower necessary for efficient capturing the
atoms from the bright beam. Furthermore, the optimum number of trapped atoms
is obtained using a very low intensity of the trapping beams. Under these conditions
the trap volume becomes extraordinarily large, thereby reducing the number density
and consequently reducing the trap losses caused by ionization.
We developed a simple Doppler model which describes the trapping conditions of
our MOT very well. Even the influence of the seventh MOT beam satisfies the Doppler
model. Our MOT is operated in the Doppler limited regime, with corresponding temperatures around 1 mK. For the highest measured density, i.e., 4 × 1010 atoms/cm3 ,
114
Summary
this corresponds to a phase space density of 10−7 . The lifetime of our MOT (∼ 0.2 s)
is dominated by the two-body loss rate mostly caused by ionization, which we found
to be β = 2KSS = (5 ± 1) × 10−10 cm3 /s. In the regime we operate our MOT in, the
ionization losses are fully determined by ground state collisions.
A MOT containing 1010 atoms is a good starting point on the road to BEC. Apart
from a MOT of 20 Ne we can also make a MOT containing more than 109 atoms of
the 22 Ne isotope. This opens perspectives when the 20 Ne fails to meet the requirements for easily reaching BEC, e.g. a large and positive scattering length. Besides
the Penning ionization loss rate, almost no experimental collision data is available
for metastable neon. That is why in our group photoassociation spectroscopy experiments are planned to obtain those collision parameters.
In this thesis we describe how we used this technique to obtain the coupling between two excited states of the 85 Rb2 dimer. That work, described in chapter 5 of this
thesis, was a collaboration of the Ultracold Atomic Physics Group at the University
of Texas at Austin, the Atomic Physics Division of the National Institute of Standards
and Technology at Gaithersburg, and the Atomic Physics and Quantum Electronics
group at the Physics Department of the Eindhoven University of Technology. By using photoassociation spectroscopy of laser-cooled 85 Rb atoms, we studied the 85 Rb2
−1
0+
of the 52 S1/2 + 52 P1/2 dissociation limit, with a resu states that lie within 80 cm
3
olution better than 0.002 cm−1 . These levels arise from both the A 1 Σ+
u and b Πu
electronic states. We found that the rotational constants of the 0+
u levels oscillate as
a function of their vibrational number. We compared our measurements with closecoupled bound state calculations, from which we gained a qualitatively understanding of the mechanism causing the oscillations. We conclude that nuclear motion on
3
the A 1 Σ+
u and b Πu potentials is strongly coupled so that the Born-Oppenheimer
approximation does not provide a valid zero-order description of these states. The
3
coupling between the A 1 Σ+
u and b Πu potentials in the case of Rb2 is neither adiabatic, as in the case of Cs2 , nor diabatic, as in the case of Na2 . To obtain more
3
quantitative results, more has to be known about the 1 Σ+
u and Πu potentials and the
radial dependence of the spin-orbit coupling between them.
We showed in this thesis that in our group a setup was developed to study cold
collisions of metastable neon atoms. In the future, both the bright atomic beam and
the magneto-optical trap can be used for a variety of collision experiments. Already
some experience with photoassociation spectroscopy was obtained by studying the
85
Rb2 dimer. In the near future the road to Bose-Einstein condensation of metastable
neon will be continued by optimizing the loading of a magneto-static trap.
Samenvatting
115
Samenvatting
Sinds het begin van de jaren tachtig zijn tal van laserkoelingstechnieken toegepast
om neutrale atomen te koelen en op te sluiten. Een beter begrip van de verliesmechanismen die in atoomvallen optreden, heeft geleid tot het opsluiten van grotere en
koudere samples van atomen. Het opsluiten van atomen in puur magnetische vallen
en de techniek van gedwongen afdampkoelen waren verantwoordelijk voor het bereiken van Bose-Einstein condensatie (BEC) in koude, ijle gassen. In dit quantum regime
hebben de atomaire gassen typisch een dichtheid van meer dan 1014 atomen/cm3 en
een temperatuur in de orde van 1 µK. Tot nu toe is dit regime alleen bereikt met
alkali-metaal atomen en atomair waterstof. In dit proefschrift beschrijven we de eerste experimentele stappen in de richting van BEC van metastabiele neon atomen. Dit
onderzoek is uitgevoerd in de capaciteitsgroep Atoomfysica en Quantumelektronica
van de faculteit Technische Natuurkunde aan de Technische Universiteit Eindhoven.
Het gebruikelijke schema om een BEC te creëren, begint met het opsluiten van
een grote hoeveelheid koude atomen in een magneto-optische val (MOT). Deze eerste
stap is reeds gecompliceerd voor metastabiele edelgas atomen omdat de efficiëntie
waarmee edelgas atomen in de metastabiele toestand worden geproduceerd erg laag
is, ongeveer van de orde 10−4 . Door het toepassen van verschillende laserkoelingstechnieken zijn we er echter in geslaagd een koude, intense bundel van metastabiele
Ne(3s) 3 P2 atomen te creëren. De atomen die de bron verlaten, worden gecollimeerd,
afgeremd en uiteindelijk gecomprimeerd tot een bundel met een diameter van 1 mm
en een flux van 5 × 1010 atomen/s. De transversale en longitudinale temperatuur
van de bundel zijn respectievelijk T⊥ = 285 µK en T|| = 28 mK, terwijl de maximale
spectrale helderheid van de bundel 4 × 1022 s−1 m−2 sr−1 bedraagt. Naast een heldere
bundel van de 20 Ne isotoop is het eenvoudig een bundel van de 22 Ne isotoop te maken, met als enige verschil dat de flux van deze laatste zeven keer kleiner is. Onze
koude, intense atoombundel kan voor een skala van koude-botsings experimenten
gebruikt worden zoals bijvoorbeeld voor fotoassociatie spectroscopie. In dit proefschrift laten we zien dat de heldere neon bundel ook gebruikt kan worden als bron
voor het laden van een magneto-optische val.
De bundelflux van de atoombundel maakt het mogelijk laadsnelheden van de in
ons lab ontwikkelde MOT, van 1.5 × 1010 atomen/s te bereiken. Dankzij deze hoge
laadsnelheid waren we in staat bijna 1010 metastabiele neon atomen op te sluiten in
de MOT, wat naar ons inziens het grootste aantal opgesloten metastabiele atomen
is wat is gerapporteerd. Dit optimale aantal wordt bereikt onder enigszins onconventionele omstandigheden. Ten eerste beschikken wij niet over een gebruikelijke
MOT bestaande uit zes opsluitlaserbundels, maar over een zeven bundel MOT. De zevende bundel is afkomstig van de extra slower, nodig voor het efficient invangen van
atomen uit de heldere bundel. Verder wordt het optimum aantal gevangen atomen
bereikt door gebruik te maken van opsluitbundels met een erg lage intensiteit. Onder deze omstandigheden wordt het volume van de atoomwolk uitzonderlijk groot
116
Samenvatting
waardoor de atomaire dichtheid wordt gereduceerd en derhalve ook de verliezen
veroorzaakt door ionisatie.
We hebben een simpel Doppler model ontwikkeld dat de condities waaronder
de atomen opgesloten zitten in onze MOT goed beschrijft. Zelfs de invloed van
de zevende MOT bundel voldoet aan het Doppler model. Onze MOT is werkzaam
in het Doppler-gelimiteerde regime, welke correspondeert met temperaturen rond
1 mK. Voor de hoogst gemeten atomaire dichtheid, 4 × 1010 atomen/cm3 , komt
dit overeen met een faseruimte dichtheid van 10−7 . De levensduur van onze MOT
(∼ 0.2 s) wordt gedomineerd door twee-deeltjes verliezen welke voor het grootste deel bepaald worden door ionisatie; de gevonden verliezen waren gelijk aan
β = 2KSS = (5 ± 1) × 10−10 cm3 /s. In het regime waarin onze MOT werkt worden de
ionisatieverliezen volledig bepaald door grondtoestands botsingen.
Een MOT die 1010 atomen bevat is een goed startpunt op de weg naar BEC. Naast
een MOT van 20 Ne kunnen we ook een MOT maken die meer dan 109 atomen van het
22
Ne isotoop bevat. Dit biedt perspectieven wanneer 20 Ne niet voldoet aan de eisen
om gemakkelijk BEC te bereiken, zoals bijvoorbeeld het hebben van een grote en
positieve verstrooiingslengte. Naast de Penningionisatie verlies-snelheid is er bijna
geen experimentele botsingsdata beschikbaar over metastabiel neon. Dat is de reden waarom er in onze groep fotoassociatie experimenten zijn gepland om deze
botsingsparameters te bepalen.
In dit proefschrift beschrijven we hoe we deze techniek gebruikt hebben om koppelingsparameters tussen twee toestanden van de 85 Rb2 dimeer te bepalen. Dit werk,
beschreven in hoofdstuk 5 van dit proefschrift was een samenwerking tussen de Ultracold Atomic Physics Group van de University of Texas in Austin, de Atomic Physics Division van het National Institute of Standards and Technology in Gaithersburg
en de Atoomfysica en Quantumelektronica groep van de faculteit Technische Natuurkunde aan de Technische Universiteit Eindhoven. We hebben de 85 Rb2 0+
u toestanden
−1
2
2
bestudeerd die binnen 80 cm van de 5 S1/2 + 5 P1/2 dissociatie limiet liggen. Deze
toestanden werden bepaald met een resolutie beter dan 0.002 cm−1 door gebruik te
maken van fotoassociatie spectroscopie van laser-gekoelde 85 Rb atomen. Deze ener3
gieniveau’s behoren tot de elektronische A 1 Σ+
u en b Πu toestanden. We vonden dat
+
de rotatiekonstanten van de 0u toestanden oscilleren als een functie van het nummer
van hun vibratieniveau. We hebben de metingen vergeleken met een gekoppeldegebondentoestanden berekening, waaruit we een kwalitatief beeld hebben gevormd
van wat de oscillaties veroorzaakt. We concluderen dat de kernbeweging van de
3
A 1 Σ+
u en b Πu potentialen sterk gekoppeld is zodat de Born-Oppenheimer benadering geen goede nulde orde beschrijving oplevert. De koppeling tussen de A 1 Σ+
u
en b 3 Πu potentialen is voor Rb2 noch adiabatisch zoals bij Cs2 het geval is, noch
3
diabatisch zoals bij Na2 het geval is. Meer kennis van de 1 Σ+
u en Πu potentialen is
nodig om meer kwantitatieve resultaten te verkrijgen.
In dit proefschrift hebben we laten zien dat in onze groep een opstelling is ontwikkeld om koude botsingen tussen metastabiele neon atomen mee te bestuderen.
In de toekomst kan zowel de heldere atoombundel als de magneto-optische val gebruikt worden voor een verscheidenheid aan botsingsexperimenten. Met het bestuderen van de 85 Rb2 dimeer is al enige ervaring opgedaan met fotoassociatie spectro-
Samenvatting
117
scopie. In de nabije toekomst wordt de weg naar Bose-Einstein condensatie verder
vervolgd door het optimaliseren van het laden van een magneto-statische val.
118
Dankwoord
Dankwoord
Een promotieonderzoek doe je zeker niet in je eentje. Dit proefschrift was dan ook
niet tot stand gekomen zonder de hulp van vele anderen. Daarom wil ik hierbij
iedereen bedanken die bijgedragen heeft aan de totstandkoming van dit proefschrift
en voor de plezierige tijd die ik de afgelopen vier en een half jaar heb gehad.
De meeste dank ben ik verschuldigd aan mijn copromoter Edgar Vredenbregt. Edgar
heeft mij alle facetten van het laserkoelvak bijgebracht. Dat je drukregelaars subtiel
moet behandelen, werd me de eerste dag in het lab al door hem duidelijk gemaakt.
Geen enkel probleem kon ik tegen het lijf lopen of Edgar had er een oplossing voor.
Zonder zijn hulp was het GEMINI-project zeker niet zo ver gekomen. Edgar, bedankt
voor alles.
Mijn promotor Herman Beijerinck dank ik, naast alle hulp op het gebied van de fysica, ook voor zijn persoonlijke adviezen en begeleiding. Zijn onuitputtelijke stroom
nieuwe ideeën werkte zeer stimulerend.
Mijn tweede promotor Boudewijn Verhaar, die mij tijdens mijn afstuderen al enthousiast maakte voor de atoomfysica, dank ik voor alle heldere discussies op het gebied
van de botsingsfysica.
Veel dank ben ik ook verschuldigd aan de technische staf van de AQT-groep. Ik
ben de tel kwijt geraakt van het aantal modificaties aan de GEMINI-opstelling en
de vele malen dat we samen met Rien de Koning de verschillende vacuum kamers
opengehaald hebben (en dankzij Rien ook weer netjes dichtgekregen hebben). Naast
alle ontwerpen, tekeningen en constructies ben ik Rien ook dankbaar voor het steeds
weer oppeppen van de laser. Voor Louis van Moll kan een klusje (en bijbehorende
promovendus) niet gek genoeg zijn of hij bedenkt er de juiste oplossing voor. Altijd
stond hij voor het GEMINI-team klaar. Hiervoor ben ik hem zeer dankbaar. Jolanda
van de Ven heeft de taken van Rien vrij snel overgenomen. Haar dank ik o.a. voor de
mooie 3D-tekeningen.
Het enthousiasme en doorzettingsvermogen van Simon Kuppens hebben er voor gezorgd dat we in korte tijd een neon MOT tot stand gebracht hebben. De discussies
met Simon over de MOT en de rest van mijn proefschrift waren bijzonder nuttig.
Veel dank hiervoor.
In het lab hebben we ook veel kunnen lachen dankzij de afstudeerders John Oerlemans, Roland Stas, Veronique Mogendorff, Bert Claessens en Eric van Kempen. Naast
de leuke tijd dank ik hen ook voor het vele werk dat zij verricht hebben. Hetzelfde
geldt voor de stagiairs Wilbert Mestrom en wederom Bert Claessens aka d’n Bertrallius.
Ton van Leeuwen dank ik voor al zijn waardevolle opmerkingen tijdens werkbesprekingen en zijn hulp bij computerproblemen. Tevens dank ik Ton voor de leuke tijd
tijdens de colleges en instructies N3 voor T.
Onze secretaresses Rina Boom en Marianne van den Elshout dank ik voor de hulp bij
alle administratieve klusjes.
Dankwoord
119
De leden van het AQT “AIO-overleg” orgaan, te weten, De Dikke, De Kale, De Lullo
en Esther (Roel Knops, Armand Koolen, Roel Bosch en Patrick Sebel) bedank ik voor
alle (vaak iets te) gezellige bijeenkomsten en de leuke tijd binnen de groep.
Herman Batelaan dank ik voor de vele waardevolle discussies op de gang en in het
lab. De gesprekken met Jo Hermans leverden altijd weer een nieuwe, heldere kijk op
vele fysicaproblemen. Peter van der Straten bedank ik voor de gezellige donderdagen
en de discussies over de inhoud van dit proefschrift.
De theoreten Frank van Abeelen, Servaas Kokkelmans en Johnny Vogels ben ik dankbaar voor de dicussies en leuke tijd op conferenties.
I would like to thank Misha Kurzanov for all the work he did on the MOT and for
being such a nice colleague. I thank Marek Synowiec for all the laughing in the lab
during his visit at our group.
I am very thankful to Dan Heinzen, Riley Freeland, Philippe Courteille, Jean-Brice
Combebias and Kanny Ly for the great time during my visit at the University of
Texas. I also like to thank Paul Julienne, Carl Williams, and Eite Tiesinga for the
helpful discussions about the Rb2 problem. I like to thank Marya Doery for the nice
time during my visit at NIST Gaithersburg.
Naast bovengenoemde mensen ben ik ook veel dank verschuldigd aan vrienden, kenissen en familie.
Louis Selen dank ik voor de gezellige tijd in huis en voor de tien kilo die ik, mede
dankzij zijn bijzondere kookkwaliteiten, de afgelopen jaren ben aangekomen.
Quint Videler dank ik voor de avonturen tijdens het zeilen en de andijviestamp met
spekavonden.
Hugo de Jong, net zo verslaafd aan het goede leven als ik, dank ik voor alle ongein
tijdens het uitgaan en op party’s.
De bijzondere leden van Fysisch Genootschap Nwyvre dank ik voor alle, per definitie,
veel te gezellige tijden.
De Vastenavendvrienden JdeJ & LdeP, Bart & Miss Thole, Henk & Miranda, JJ en Stoffels dank ik voor alle leut tijdens het dweile.
DJ Stijn en DJ Robob ben ik erg dankbaar voor het aanleveren van al het moois op
vinyl.
Mijn ouders en de rest van de familie dank ik voor alle steun en liefde die ze mij al
heel mijn leven geven.
Tenslotte dank ik mijn lieve Aaf voor alle liefde en het zo gelukkig maken van me.
120
Curriculum Vitae
Curriculum Vitae
9 december 1971
Geboren te Bergen op Zoom
1984-1990
HAVO,
Roncalli Scholengemeenschap te Bergen op Zoom
1990-1991
Propadeuse Technische Natuurkunde,
Hoge School Eindhoven
1991-1996
Studie Technische Natuurkunde,
Technische Universiteit Eindhoven,
doctoraal examen augustus 1996
1996-2001
Onderzoeker-In-Opleiding bij Stichting FOM,
werkgroep AQ-E-a,
Experimentele Atoomfysica en Quantumelektronica,
Faculteit Technische Natuurkunde,
Technische Universiteit Eindhoven
april-augustus 1998
Werkbezoek Ultracold Atomic Physics Group,
University of Texas, Austin, TX, USA