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Department of Physics and Astronomy
University of Heidelberg
Diploma thesis
in Physics
submitted by
Bastian Höltkemeier
born in Bielefeld
2011
2D MOT as a source of a
cold atom target
This diploma thesis has been carried out by
Bastian Höltkemeier
at the
Physikalisches Institut
in Heidelberg
under the supervision of
Prof. Matthias Weidemüller
2D MOT as a source of a cold atom target
This thesis reports on the development, assembly and characterization of a two-dimensional magneto-optical trap (2D MOT). The 2D MOT is a particularly wellsuited source for cold atom targets due to a high brilliance with low ion background.We present the creation of a two-dimensional quadrupole field using permanentmagnets which, in combination with the use of opical fiber couplers leads to acompact and modular design. Our 2D MOT provides a well collimated beam of 4×109 atoms/s with a divergence of only 26mrad (FWHM), resulting in a brilliance of8×1012 atoms/(s · rad). The atoms have a mean velocity of 14m/s. The dependenceof the resulting velocity distribution and flux on the laser settings, magnetic fieldconfiguration and the use of a pushing laser beam is investigated. Furthermore, wepresent numerical simulations which reproduce the measured dependencies and thetotal atom flux.
2D MOT als Atomquelle für ein kaltes Target
In dieser Arbeit wird der Aufbau und die Realisierung einer zweidimensionalenmagneto-optischen Falle (2D MOT) als Quelle ultrakalter Atome beschrieben. Auf-grund hoher Brillanz und geringem Hintergrund eignet sich diese Quelle insbeson-dere zum Laden einer dreidimensionalen magneto-optischen Falle (3D MOT), welcheals Target für Ion-Atom Stöße dient. Gezeigt wird unter Anderem die Realisierungeines magnetischen Quadrupolfelds durch Permanentmagneten. In Kombinationmit optischen Faserkopplern führt dies zu einem kompakten und modularen Auf-bau. Die 2D MOT erzeugt einen kollimierten Atomstrahl mit 4 × 109 Atomen/sund einer Divergenz von 26mrad (FWHM). Dies entspricht einer Brillanz von8× 1012 Atomen/(s · rad). Die mittlere Geschwindigkeit der Atome beträgt 14m/s.Die Abhängigkeit der Geschwindigkeitsverteilung und des Atomflusses von denLaserparametern, dem Magnetfeld und einem Pusher-Laserstrahl wird untersucht.Es wird ein theoretisches Modell vorgestellt, welches die gemessenen Abhängigkeitengut beschreibt.
Contents
1 Introduction 1
2 2D MOT as a source of cold atoms 5
2.1 Magneto optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Interaction of light and atoms . . . . . . . . . . . . . . . . . . . 52.1.2 Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Atom beam sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Double MOT systems and LVIS . . . . . . . . . . . . . . . . . . 142.2.4 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Model of a 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Model for the light force . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Atom trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Realization of the 2D MOT 23
3.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Cooling laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Repumping laser . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.3 Laser cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 2D MOT setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Bellow with differential pumping tube . . . . . . . . . . . . . . . 293.2.2 Glascell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Magnetic field design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.3 Setup of the magnets . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Characterization of the atom beam 41
4.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.1 Measurement of the velocity distribution . . . . . . . . . . . . . 434.1.2 Measurement of the flux . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Beam divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Dependence on laser settings . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Detuning cooling laser . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 Power cooling laser . . . . . . . . . . . . . . . . . . . . . . . . . 49
I
Contents
4.3.3 Repumping laser . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.4 Pushing laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.5 Counter-propagating pushing beam . . . . . . . . . . . . . . . . 59
4.4 Influence of partial pressure and length of cooling volume . . . . . . . . 604.5 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.1 Magnetic field gradient . . . . . . . . . . . . . . . . . . . . . . . 634.5.2 Compensation coils . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Beam size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Conclusion and Outlook 67
List of Figures 70
Bibliography 74
II
1 Introduction
State of the art experiments on ultracold quantum gases involve a three-dimensional
magneto-optical trap (3D MOT) as a first cooling procedure which is a typical starting
point for further experiments such as evaporative cooling or trapping in optical lattices.
These schemes require favorable initial conditions such as low starting temperatures,
high atom numbers and long lifetimes of the atoms in the trap. A typical limitation
is the collisions with background gas atoms, making it necessary to perform these
experiments under ultra-high vacuum (UHV) conditions. However, low vapor pressure
results in a very inefficient loading from background vapor. Instead, cold atom beams
can be used as a source to achieve short loading times and high atom numbers in the
MOT without compromising UHV conditions.
Since the first demonstration of a laser cooled atom beam by Philipps and Metcalf
[1], new methods for producing high brilliance cold atom beam sources have been
under investigation. One of the most common techniques is to decelerate a thermal
atom beam with the radiation pressure from a resonant laser beam. The atoms are
kept on resonance either by chirping the laser frequency (chirped slower) [2] or by
spatially varying magnetic fields (Zeeman slower) [3]. These kind of setups produce
bright atom beams at low temperatures, but require a great engineering effort. The
main drawback is that the atom beams have a large transverse velocity spread since
the atoms are only cooled in longitudinal direction. In order to avoid the increasing
beam divergence at low longitudinal velocities, more sophisticated schemes for these
slowers have been developed [4].
Beams with even lower mean velocities that are both longitudinally and transversely
cooled can be provided by MOT sources. For most alkali metals they are commonly
used alternatives providing advantages such as a compact setup, well collimated beams
and low laser power. In a seperate chamber atoms are trapped in a vapor cell MOT
from where they can be extracted. The first breakthrough of this method was realized
by Lu et al. [5] who extracted the atoms by a power imbalance in two of the MOT’s
cooling lasers. This low velocity intense source (LVIS) provides an bright atom beam
at very low temperatures, with only a small thermal background.
Another approach is the two-dimensional magneto-optical trap (2D MOT). Atoms
are cooled and trapped in two dimensions, leading to a well collimated cold atom beam
in the third dimension. The atoms are transferred to the science chamber through a
1
CHAPTER 1. INTRODUCTION
small aperture which at the same time is used for differential pumping making this
system perfectly suited for loading a 3D MOT under UHV conditions. During the last
15 years, 2D MOTs for rubidium [6, 7], potassium [8], cesium [9, 10, 11], sodium [12]
and lithium [13] have been developed and the characteristics of the produced beams
have been investigated in experiment and theory. Most of the 2D MOTs’ features can
be qualitatively explained on the basis of simple models, yet some details are still not
understood to a satisfying degree.
In this thesis we report on the setup of a 2D MOT which will be used to load a cold
target for the investigation of atoms-ion collisions. The experiment will focus on the
dynamics in multiple electron transfer between neutral atoms and highly charged ions.
For this purpose the setup will be implemented in the HITRAP beamline at GSI where
ions up to Ur92+ will be available. The momentum transferred during these collisions
is investigated using “Recoil Ion Momentum Spectroscopy” (RIMS) [14]. Depending
on the projectile energy, the collision of a rubidium atom and a highly charged ion
can either lead to ionization of the target atom or to charge transfer. Using RIMS,
the measured three-dimensional momentum distribution reveals the dynamics during
the collision. The ion’s longitudinal momentum is proportional to the total change in
binding energies of the transfered electrons (Q-value) while the transverse momentum
contains information about the scattering angle of the collisions. The resolution of this
technique depends strongly on the initial velocity of the target atoms as the momentum
transfer between the ion and the atom is very small. Therefore, the target has to be
cold. In the case of light noble gases (He, Ne, Ar), the targets are cooled by supersonic
expansion. Cold targets with other species can be provided by 3D MOTs [15]. Using
the atomic beam from a 2D MOT might also present an alternative approach.
In our setup, rubidium atoms are trapped in a MOT which is positioned at the
center of a RIMS detector. The cross sections of the investigated collisions are in the
order of 10−15 cm2, depending on the number of electrons transfered [16]. Hence, a
large number of target atoms is needed to achieve adequate statistics in reasonable
run-times. At the same time, the background should be small during the measure-
ments. Therefore, the implementation of a 2D MOT as a source for the 3D MOT
serves multiple purposes. It allows to keep a very low pressure in the experimental
chamber, provides a large number of cold atoms leading to short loading times and no
background ions.
We present in this work the development, assembly and characterization of a com-
pact and modular 2D MOT setup. The efficiency of different loading schemes for 3D
MOTs will be discussed in the first part of chapter 2. Section 2.1 will give a short
introduction on the working principle of a MOT. Based on this section 2.2 will present
2
different atom beam sources and discuss their advantages and disadvantages for load-
ing a MOT. In section 2.3 a model that can describe some of the key features of a 2D
MOT will be presented.
The third chapter will give a detailed description of the construction and assembly
of the 2D MOT including a new laser system (section 3.1) and a compact and modular
design of the 2D MOT itself (section 3.2).
After assembly, the 2D MOT was characterized on a test chamber (section 4.1)
where the dependency on the atom flux and the velocity distribution on various pa-
rameters was investigated. These dependencies are best understood when comparing
the measurements to a numerical simulation. All the measurements and results from
the model are presented and discussed in sections 4.2 to 4.6. In particular, we inves-
tigate in detail the influence of an additional laser beam in the direction of the atom
beam (section 4.3.4 and 4.3.5).
3
2 2D MOT as a source of cold
atoms
In the first part of this chapter the principle of a magneto optical trap (MOT) will be
explained, followed by an overview of some of the most common atom beam sources
which are used to load such a trap. A 2D MOT is one of these beam sources and
in the last part of this chapter a theoretical model to describe a 2D MOT will be
introduced. It can be used to simulate the dependence of the atom flux on various
important parameters of the 2D MOT.
2.1 Magneto optical trap
In a MOT atoms are cooled to temperatures of a few hundred µK by using a force of
near resonant laser light which arises form spontaneous scattering of photons by the
atoms. In the first section an overview over the interaction of light and atoms will
be given. Based on this, section 2.1.2 will show how this interaction can lead to a
cooling of the atoms. This can be used to trap the atoms at the zero point a magnetic
quadrupole field (section 2.1.3).
2.1.1 Interaction of light and atoms
The interaction of light and atoms can be described as a periodically perturbed two-
level system [17]. This seems to be a strong simplification at first glance since all atoms
have a very complicated inner structure far from being a two-level system. But if the
light only couples to two of the atom’s energy levels as in the case of monochromatic
laser light, the two-level approximation becomes far more realistic.
Let us consider the simple case where a two level atom is in the ground state and at
rest. It is located in the light field of a monochromatic laser in resonance with the
atomic transition and is propagating in ~ez direction. Three processes can occur when
the atom interacts with the light field of a laser: spontaneous emission, stimulated
emission and absorption. Absorbing a photon, the atom receives the photon’s initial
momentum of ~~k giving it a kick in the direction ~ez of the laser. The atom is now in
5
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
the excited state and there are two different channels to decay back into the ground
state. It can either interact with the light field of the laser again and emit a photon
back in ~ez direction receiving a kick in the opposite direction (stimulated emission) or
it can spontaneously decay back to the ground state where the direction of the emitted
photon is random (spontaneous emission).
In the first case, the atom has the same momentum before and after the absorption
cycle. The momentum only differs while it is in the excited state. Nevertheless, it
is possible to create a setup where this process creates a net force on the atom. If
the light field has an intensity gradient and is far detuned from the atomic transition,
there is a force pointing towards the intensity minimum (in case of blue detuning)
or the intensity maximum (in the case of red detuning) of the light field [18]. This
phenomenon is used in dipole traps where atoms are directly trapped in the light field
of a laser.
If the atom spontaneously decays back into the ground state instead, the light is
scattered isotropically. Averaging over many cycles where the atom absorbs photons
in ~ez direction and emits spontaneously in all directions, there is a net force ~Fspon
acting on the atom in the direction of the laser ~ez. This is the so called spontaneous
or scattering force which can be used to cool and in combination with a magnetic field
also trap atoms.
2.1.2 Laser cooling
The spontaneous force acts on the atoms in the propagation direction of the laser
beam. If the laser is on resonance with the atomic transition, this affects only atoms
at rest or very slow atoms. Due to the Doppler shift atoms at higher velocities are
no longer in resonance with the laser light. They see the laser at a frequency of
ω = ωres − ~k~v. Here ωres is the resonance frequency of the atomic transition, ~k is
the wave vector of the laser and ~v is the atoms velocity. Analogously, if the laser is
detuned from the transition by δ, atoms with a velocity of ~v = δ/~k are on resonance.
This means that the total detuning is given by
δtot = δ + ~k~v . (2.1)
But also atoms that are not precisely on resonance can interact with the light field
because of the finite natural linewidth of the transition. The velocity classes which
are still affected by the light field of the laser depend on the lasers intensity and its
detuning [19]:
~Fspon(~v) = ~~kΓ
2
I/I0
1 + I/I0 + (2(δ − ~k~v/Γ))2(2.2)
6
2.1. MAGNETO OPTICAL TRAP
00.10.20.30.40.50.60.70.80.9
1
-60 -40 -20 0 20 40 60scatt
eri
ng
rate
γ(Γ/2)
velocity v (m/s)
I = 200 I0I = 20 I0I = 5 I0I = I0
Figure 2.1 Scattering rate as a function of the atom velocity for different laser inten-sities. The scattering rate is shown as a fraction of the maximum scattering rateΓ/2. At high intensities the scattering rate saturates at γ = Γ/2.
where Γ is the natural linewidth of the transition, I is the laser intensity and I0 is
the saturation intensity. The saturation intensity is based on the fact that the rate
at which the atoms scatter photons (scattering rate) only increases linearly at low
intensities. At higher intensities, stimulated emission is the dominant process and
the spontaneous emission levels out at a maximum scattering rate of γ = Γ/2. The
saturation intensity of an atomic resonance is defined as the laser intensity where the
scattering rate is half the maximum scattering rate γ(I0) = Γ/4. The scattering rate as
a function of the atoms velocity is shown in figure 2.1 where the laser is on resonance.
The saturation effect can be seen as the scattering rate for atoms at rest has almost
reached the maximum of Γ/2 at I = 20× I0. The only difference at higher intensities
is the widening of the peak which means that also atoms at higher velocities will be
affected by the light field. In order to understand the concept of laser cooling we now
want to consider an one dimensional system with counter propagating laser beams
along the x-axis. Both beams are shifted by the same detuning δ. The force of each
laser beams is given by:
~F±(~v) = ±~~kΓ
2
I/I0
1 + I/I0 + (2(δ ∓ ~k~v/Γ))2(2.3)
where ~F+/~F− are the forces resulting from the beam traveling in positive/negative
x-direction. The resulting total force ~Ftot = ~F+ + ~F− can be expanded in powers of~k~v/Γ:
~Ftot(~v) =8~k2δ
Γ
I/I01 + I/I0 + (2δ/Γ)2
~v + O((~k~v/Γ)4) . (2.4)
7
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
-0.1-0.08-0.06-0.04-0.02
00.020.040.060.080.1
-40 -20 0 20 40
Forc
eF
(N)
velocity v (m/s)
F+/F−
Ftot
Figure 2.2 Spontaneous force on atoms in the light field. Shown are the forces actingon the atoms by the single laser beams F+/F− and the resulting total force Ftot.The detuning is δ = 2Γ and I/I0 = 20.
For δ < 0, this describes a damping force with a positive damping coefficient thus
slowing down the atoms that are close to resonance. Figure 2.2 shows the complete
behavior of Ftot without neglecting any higher order terms. To slow down the atoms
they have to scatter multiple photons. An atom with an initial velocity of about 20
m/s has to scatter about 10000 photons to loose its momentum.
It is not possible to stop the atoms completely. So far, we assumed that the mo-
mentum gain due to spontaneous emission averages to zero because of its isotropic
nature. However, considering only the single momentum transfers due to spontaneous
emissions of single photons the atom will undergo a random walk in momentum space
with step size ~k. At low temperatures, this heating process competes against the
cooling mechanism due to absorption. At the so called Doppler temperature both
processes are balanced out. This temperature is given by [19]:
TDoppler =~Γ
2kB. (2.5)
The first temperature measurements on optically cooled and stored atoms already
contradicted this result by showing temperatures below the Doppler limit [20]. It was
shown that this discrepancy of theory and experiment is caused by the too rigorous
approximation of a two-level system. The physical limit for the temperature achievable
with laser cooling techniques is the recoil temperature of a single photon which is well
below the Doppler temperature.
8
2.1. MAGNETO OPTICAL TRAP
2.1.3 Magneto-optical trap
Figure 2.3 Setup of a 3D MOT. The atoms (red cloud in the center), are trappedin the light field of six orthogonal laser beams and the magnetic quadrupole fieldfrom a pair of magnetic coils. (Picture adapted from [21])
In order to trap atoms one has to find a way to make the velocity dependent spon-
taneous force spatially dependent as well. In a MOT this is achieved by adding a
magnetic field to the system. To understand how this can be used to trap atoms, a
two level system is considered once more. A linear magnetic field B(x) = B0 x with
a constant gradient B0 is applied lifting the degeneracy of the atom’s magnetic sub-
levels. In the simplest case, the ground state has an angular momentum of J = 0 and
the excited state has J = 1. This means that the excited state will split up into three
Zeemann levels (mJ = −1, 0, 1) with an energy of:
Ezeeman(mJ) = E0 + gJ µB mJ B0x (2.6)
with E0 being the energy of the inperturbed excited state, gJ the Lande factor, µB the
Bohr magneton and mJ the magnetic quantum number. This splitting of the excited
state’s energy levels also affects the spontaneous force. The overall detuning of the
laser from the atomic resonance is now given by
δtot = δ + ~k~v +gJµBmJB0x
~, (2.7)
where the last term takes the position dependent energy splitting of the excited state
into account. In this setup, the spontaneous force depends on the atomic sublevel the
9
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
Figure 2.4 Phasespace of neutral Rb atoms in the light field of a 3D MOT. Onlyatoms that are moving slowly enough can be captured in the MOT. The parame-ters for this simulation are δ = 2Γ and I/I=20. All heating effects are neglected.
atom is excited to:
~F±(~v, x) = ±~~kΓ
2
I/I0
1 + I/I0 + (2(δ ∓ ~k~v ± gJµBmJB0x/~)2. (2.8)
For the excitation to mJ = 0 this force is the same as in equation 2.3, but for the
excitation to mJ = 1 and mJ = −1 the force depends on the atoms’ position. This
can be used to trap the atoms in a similar way to the red detuning that was used to
cool the atoms. The red detuning causes the atoms to always interact more strongly
with the beam that is traveling against their own direction of propagation.
A similar effect occurs if a σ+/σ− polarization is applied to the beam coming from
the left/right so that the light selectively pumps the atoms to the mJ = −1/mJ = +1
state. This leads to a configuration where atoms at x < 0 interact more strongly
with the beam coming from the left and atoms at x > 0 with the beam from the
right. This effectively pushes the atoms to x = 0 where the different Zeeman levels are
degenerated again. Thus the atoms interact equally with both beams and the total
force vanishes.
Figure 2.4 shows the atoms trajectories in phase space. Both, the confinement in
10
2.2. ATOM BEAM SOURCES
velocity and in position space are clearly visible. The simulation also shows, that there
are certain limits to the capturing of the atoms. If they travel faster than a certain
velocity (capture velocity), they cannot be trapped. There is also a limitation on the
volume from which atoms can still be pulled into the trap but this cannot be seen in
the one dimensional phasespace since it is hard to miss something in a one dimensional
world.
In higher dimensions, the situation is far more complicated but the general principle
still holds (In section 2.3 an heuristic equation which can be used to model the spon-
taneous force in three dimensions as well, will be presented). In order to trap atoms in
our three dimensional world two more pairs of counter propagating laser beams along
with a magnetic field gradient have to be applied in the other two dimensions. Nowa-
days this technique is one of the standard tools in AMO physics and is successfully
used in labs all around the world.
The limited capture velocity makes the loading from background gas quiet inefficient.
At room temperature, the mean velocity of rubidium gas is about 280m/s. Typical
capture velocities are well below that [22] which means that most atoms in the gas are
too fast to be captured by the trap. For a capture velocity of 40m/s, only 0.33% of
the rubidium atoms are actually slow enough to be trapped. Another limitation when
loading from background vapor are losses due to collisions with background gas. Even
a single collision of a trapped atom with an atom at room temperature is sufficient
to remove it from the trap. For that reason, experiments with ultra-cold atoms have
to be carried out under ultra high vacuum (UHV) conditions. On one hand, it is
favorable to have a high partial pressure of the atoms one wants to trap leading to a
higher loading rate. On the other hand a higher partial pressure leads to an increased
loss rate due to collisions which outweighs the increase in the loading rate at high
pressures. This is why there has been a lot of research on creating slow atom beams
to overcome this problem.
2.2 Atom beam sources
To load a 3D MOT efficiently a high flux of slow atoms is needed. A good benchmark
for an atom beam is the total flux of atoms below the capture velocity of a 3D MOT.
It also has to be taken into account that the cooling volume of a 3D MOT is limited.
Therefore, the divergence of the atom beam is of great importance as well. In the
following some common cold atom sources are presented and their advantages and
disadvantages are briefly discussed.
11
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
(a) (b)
Figure 2.5 Atom beam from an oven. (a) Flux through area dA. All the atoms inthe gray cylinder will reach the area dA during the time dt. (b) Beam collimationwith an aperture.
2.2.1 Oven
One way to produce an atomic beam is to use an oven. This is basically just a reservoir
filled with rubidium gas with a hole through which atoms can leave the oven.
This process is called effusion and the flux of such an oven can easily be calculated
if the diameter of the exit-hole is much smaller than the mean free path length of
the atoms in the reservoir [23, 24]. In this case, the Maxwell-Boltzmann velocity
distribution can be used to describe the movement of the atoms because the hole is
too small to perturb the equilibrium in the reservoir. First, the number of atoms that
passes an infinitely small area dA which is part of the hole will be calculated. Let the
z-axis be the normal to this surface, then within the time interval [t, t + dt] from all
atoms with a velocity between [v, v+dv] and an azimuth angle between [θ, θ+dθ] and
[φ, φ + dφ] only those which are closer to the surface than vdt will reach dA. These
are all the atoms in a cylinder of volume dAvdt cos θ (see figure 2.5a). The number
of atoms within the considered velocity interval is given by fMB(v)d3v, where fMB(v)
is the Maxwell-Boltzmann distribution. Dividing by dA and dt gives the flux per unit
area:
Φ(v)d3v = cos θ v fMB(v) d3v . (2.9)
Integrating over all velocities with vz > 0 and over the area of the hole gives us the
total flux of atoms:
Φ = A
∫
vz>0
cos θ v fMB(v) d3v . (2.10)
This means that the atoms which leave the oven through the exit hole are not emitted
isotropically, but with p(θ) ∼ cos θ. Moreover, the mean velocity of the atoms in the
beam is not the same as the mean velocity of the atoms in the reservoir. The Maxwell-
Boltzmann distribution has a velocity dependence of fMB(v) ∼ v2 expmv2/(2kBT ) with
kB being the Boltzmann-constant, while in the beam fbeam(v) ∼ v3 expmv2/(2kBT ).
At a temperature of T = 300K a gas of Rubidium atoms has a mean velocity of
12
2.2. ATOM BEAM SOURCES
Figure 2.6 Zeeman Slower. A thermal atomic beam is decelerated by a counterpropagating resonant laser beam. In order to keep the laser on resonance a nonlinear magnetic field has to be applied.
vmean ≈ 273m/s whereas Rubidium atoms that leave the oven have a mean velocity
of vmean ≈ 321m/s.
The oven emits atoms into a solid angle of 2π. In order to get an atom beam with
a smaller divergence, one can place an aperture with radius rap at a distance l from
the exit hole (figure 2.5b). This way, only atoms that are emitted in a solid angle
of θ = arctan(rap/l) participate in the atom beam. As mentioned before, only atoms
that are emitted in a solid angle of [0, θmax] which covers the cooling volume of the 3D
MOT, can be trapped. The total flux of atoms emitted in this angle is given by
Φ = A
∫ θmax
θ=0
∫ 2π
φ=0
∫
∞
v=0
cos(θ) sin(θ) v fMB(v) dvdφdθ . (2.11)
For typical MOT sizes, this leads to a flux of a few 109 atoms/s, at room temperature.
The number of atoms that can be trapped is a lot smaller, since the mean velocity of
the oven beam is about 321m/s, whereas typical capture velocities are in the range
of 25-40m/s. This means that only about 0.05% of the atoms are slow enough to be
trapped, leading to a loading flux of a few 106 atoms/s into the MOT. The following
section describes more efficient techniques for providing high fluxes at low velocities
2.2.2 Zeeman slower
A Zeeman slower uses a beam of thermal atoms as a starting point and decelerates
them. The thermal beam can be provided by an oven like the one just discussed. In
order to decelerate the atoms a counter propagating laser beam which is on resonance
with the atomic transition is used. When the atoms slow down due to the spontaneous
force from the light field, the resonance frequency changes because of the changing
Doppler shift. This is compensated by a magnetic field that keeps the atoms on
resonance due to the Zeeman shift of the atoms magnetic sublevels. The challenge is
to find a magnetic field configuration where the Zeeman splitting exactly compensates
13
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
the Doppler shift at any position on the beam path.
Typical fluxes achieved with this method are some 1010 atoms/s at a mean velocity
of 60m/s [3, 25]. One problem of the Zeeman slower is that there is no cooling in
transverse direction. The divergence of the beam is given by
α = arctan(vtrans/vlong) = vtrans/vlong − O((vtrans/vlong)3) , (2.12)
which means that it will increase at slower longitudinal velocities. This effect becomes
critical for the very low longitudinal velocities towards the end of the Zeeman slower.
Another difficulty is that either the laser is on resonance or the magnetic field has to
be non zero at the end of the zeeman slower. This means that the MOT that normally
sits at this position is either disturbed by the resonant light or the magnetic field of the
slower. To overcome this problem, more sophisticated schemes have been developed
[4].
2.2.3 Double MOT systems and LVIS
Another source of cold atom beams are double MOT systems. The first MOT is loaded
from background gas at a high partial pressure. The atoms are then transfered to the
second trap that can be placed in a chamber with better vacuum conditions.
There are different ways to transfer the atoms between the two MOTs. The atoms
can be dropped in the earth’s gravitational field [26] or pushed out of the MOT with
an additional laser beam [27, 28]. The first continuous beam source of this kind was
developed by Lu et al. [5] called “Low Velocity Intense Source” (LVIS) where one of the
mirrors for the counter propagating beams of the low vacuum MOT has a little hole.
The hole creates a shadow in the backreflected beam which results in an imbalance of
the light forces at this position. This imbalance pushes the atoms through the hole
into the UHV chamber with the second 3D MOT.
Typical fluxes obtained with such a setup are a few 109 atoms/s with a very low
mean velocity of about 15m/s or less. This means that all the atoms in the beam can
be trapped in a 3D MOT.
2.2.4 2D MOT
A 2D MOT can provide the same flux of cold atoms as an LVIS by using less laser
power [6]. The working principle is the same as in a three-dimensional MOT. The only
difference being that there are neither trapping beams nor a magnetic field gradient
in the third dimension (from now on denoted as the z′-direction). The atoms leak out
of the trap producing an atom beam that is cooled in transverse direction. Actually,
14
2.2. ATOM BEAM SOURCES
Figure 2.7 Setup of a 2D MOT. The atoms are cooled in two dimensions and leavethe 2D MOT through a small aperture in the third dimension. (Picture adaptedfrom [21])
two atom beams are produced one in +z′ and one in −z′ direction. To ensure that
only trapped atoms leave the 2D MOT the beam is passing a small aperture at the
end of the trap, minimizing the backgound from thermal atoms. One of the nice
features of this setup is the fact that even though there is no direct cooling in the
longitudinal direction the mean velocity of the produced atom beam is well below
room temperature. To understand this, a simple model that was developed in [7, 29]
can be used.
First the concept of a capture velocity, already discussed for the one-dimensional
trap, will be applied on the 2D MOT. Here the situation is a bit more complicated.
The capture velocity strongly depends on where the atom enters the 2D MOT and
on both its velocity components (transversal and longitudinal). The atoms have to
stay in the cooling volume of the 2D MOT for a sufficient amount of time in order to
be trapped by the cooling beams. Therefore the size of the effective cooling volume
has to be known. Assuming a cylindrical shape of the cooling volume, let ltrap be its
length (in z′ direction) and rtrap the radius in transverse direction. The radius of the
trap is the volume where the detuning of the laser is greater than the Zeemann shift
of the magnetic sublevels:
|δ| > |gjµBmJB0r
~| =⇒ rtrap <
~δ
gjµBmJB0≈ 0.6mm . (2.13)
The value of rtrap < 0.6mm is calculated for the dimensions of our trap. For larger r,
the atoms see the laser to be blue detuned thus no longer being decelerated. In the
following, it will be assumed that only atoms that can be stopped in this volume will
contribute to the atom beam.
15
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
In order for an atom to stop it has to scatter
Nstop =mvr~k
(2.14)
photons, where m is the atomic mass and vr its transversal velocity. The number of
photons an atom actually scatters depends on the time spent in the cooling volume
and the scattering rate. The time the atoms spend in the cooling volume can be
limited by either their longitudinal or by their transversal velocity:
Nscattered =Γ
2
2rtrapvr
or Nscattered =Γ
2
ltrapvz
(2.15)
with vz being the atom’s longitudinal velocity. It was assumed that the atoms scatter
photons with the maximum scattering rate of Γ/2 all over the trapping volume. This
is a quite severe approximation since the actual scattering rate is a function of the
atoms velocity and position as well as the laser intensity (see equation (2.8)). So it
cannot be expected to get any quantitative results from this model but it should still
describe the main principles of the 2D MOT.
Now a prediction about the velocity distribution of the atom beam can be made.
As mentioned before, the longitudinal velocity of the atoms in the beam is well below
room temperature even though there is no direct cooling in this dimension. The reason
is that the trapping of an atom is limited by its transversal as well as the longitudinal
velocity. In this model the atoms are trapped if
Nstop < Nscattered (2.16)
where Nscattered is given by one of the two expressions above depending on which one is
smaller. From equations (2.14), (2.15) and (2.16) the capture velocities in transversal
and in longitudinal direction can be derived:
vr,max =
Γ2
~km
ltrapvz
for ltrapvz
< 2rtrapvr
√
Γ2
~k 2rtrapm
≈ 36m/s for ltrapvz
> 2rtrapvr
(2.17)
At low longitudinal velocities vz, the transverse cooling time limits the capture velocity.
Only atoms that are slower than vr = 36m/s (for our trap) can be trapped. In the
case of larger longitudinal velocities, the capture velocity falls off as 1/vz. How fast the
capture velocity drops with vz also depends in the length of the MOT. For bigger ltrapthe longitudinal velocity should be less critical. These predictions could be verified
in the experiment. For the results see chapter 4.1. In the next section an advanced
16
2.3. MODEL OF A 2D MOT
model of the 2D MOT will be presented.
2.3 Model of a 2D MOT
There are different approaches to model the characteristics of a 2D MOT. In the pre-
vious section, the dependence of the transverse capture velocity vtrans,max on the MOT
length ltrap and the longitudinal velocity vz was already discussed. The cooling time
for a specific atom depends on the z-position where the atom enters the cooling vol-
ume. Dieckmann et al. [6] used this to calculate the velocity distribution of a 2D MOT
by using a rate equation model that was first developed for a vapor cell MOT in [30].
The rate of trappable atoms entering the cooling volume at a position z, is calculated
from a Maxwell-Boltzmann distribution which is truncated at the capture velocity
vtrans,max(z, vz). By integrating over a surface surrounding the trapping volume, the
total flux and the velocity distribution of the beam was obtained.
They also investigated the dependence of the total flux on the vapor pressure. On
one hand, the trapping rate increases linearly with the number of atoms available. On
the other hand a higher pressure leads to a higher loss rate from the beam due to
collisions with the background gas. This loss rate is described by an exponential loss
term which competes with the capture rate R of the trap:
Φ =R
1 + Γbeam/Γoutexp (−Γbeamtout) . (2.18)
Here Γout is the rate of atoms exiting through the hole, tout is the average time it
takes the atoms to leave the vapor cell and Γbeam is the collision rate with atoms
from background gas. The collision rate is given by Γbeam = σeff nRb v with σeff being
the effective collision cross section, nRb the density of background rubidium atoms
and v being their mean velocity. In the presence of near resonant light, as in the
case of the 2D MOT, very large cross sections σeff can occur. These are caused by a
strong dipole-dipole interaction which follows a C3/R3 potential [31, 32]. By fitting
their experimental data with equation (2.18) Dieckmann et al. obtained a value of
σeff = 2.3× 10−12 cm2.
A similar result of σeff = 1.8× 10−12 cm2 was obtained by Schoser et al. [7] where the
same model has been used to investigate the dependence of the atom flux on the total
MOT length.
In order to investigate the atom flux as a function of the magnetic field gradient,
laser power and laser detuning, a different model was developed by Wohlleben et al.
[28]. It is used to describe the light forces in a 3D MOT from which a continuous jet
17
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
of rubidium atoms is extracted by a thin laser beam. The model was extended by
Catani et al. [8] to a 2D MOT where a heuristic expression for the light forces in the
trap is used. Trajectories of atoms starting at random initial positions are calculated
by integrating their equation of motion. These trajectories are used to determine the
fraction of atoms that participate in the atom beam with respect to the total number
of atoms. In the following an advanced model which is a combination of the models
by Wohlleben et al. and Dieckmann et al., will be presented.
2.3.1 Model for the light force
For the light force a heuristic equation from [28] is going to be used. The equation
can be motivated as following:
In a two level system with a Ji = 0 → Jf = 1 transition, the light of a single laser
beam with polarization σ± along a magnetic field B only couples to one of the magnetic
sublevels (mJ = −1, 0, 1). The spontaneous force is given by equation (2.8) which can
be written as:~F = ~~k
Γ
2
s(~r, ~v)
1 + s(~r, ~v)(2.19)
with the saturation parameter
s(~r, ~v) =I
I0
Γ2
Γ2 + (4(δ − ~k~v ± µB/~))2. (2.20)
In general, the light couples to all three Zeeman sublevels (mj = −1, 0, 1). This is the
case if either ~B is not parallel to ~k or the light is elliptically polarized. In this case,
a coupling between the different transitions occurs which is described by the optical
Bloch equations. In the low intensity limit (I ≪ I0), the coupling can be neglected
which leads to the following approximation:
~F = ~~kΓ
2
∑
m=−1,0,1
sm(~r, ~v) (2.21)
with
sm(~r, ~v) =ImI0
Γ2
Γ2 + (4(δ − ~k~v + mµB/~))2. (2.22)
Here the laser beam with mixed polarization is represented by three beams with inten-
sities Im that drive the mi = 0 → mf = −1, 0, 1 transitions. This can be interpreted
as a separation into the beam’s σ+, π and σ− components. For N laser beams, we get
18
2.3. MODEL OF A 2D MOT
a total force of
~F =N∑
n=1
Fn =N∑
n=1
~~knΓ
2
∑
m=−1,0,1
sn,m(~r, ~v) (2.23)
with
sn,m(~r, ~v) =In,mI0
Γ2
Γ2 + (4(δj − ~kj~v + mµB/~))2. (2.24)
The light field in a MOT is far beyond the low intensity regime. In order to avoid
solving the coupled Bloch equations Wohlleben et al. use a heuristic equation. The
equation should fulfill the following conditions:
• In case of a plain wave with circular polarization along the magnetic field, equa-
tion (2.19) should be recovered.
• In the low intensity limit equation (2.21) should be recovered.
• The force should never exceed ~~k Γ/2.
The simplest equation of this kind is:
~F =
N∑
i=1
~~kiΓ
2
∑
m si,m(~r, ~v)
1 +∑
n,m sn,m(~r, ~v). (2.25)
A special case of this equation was also used by Phillips et al. [33] to account for
saturation effects. In the following, this equation is going to be used to calculate the
trajectories of atoms in the glass cell.
2.3.2 Atom trajectories
A 2D MOT is normally connected to the main chamber via a differential pumping
tube which makes it possible to keep a high partial pressure in the 2D MOT without
affecting the UHV in the 3D MOT chamber. An atom can only leave the 2D MOT if
the following two conditions are fulfilled:
• The atom has to hit the exit hole at z = 0.
• Behind the exit hole, the angle between the atoms trajectory and the z-axis has
to be smaller than αmax, the maximum divergence of the beam which is set by
the geometry of the differential pumping tube (see section 4.2).
In order to calculate the total flux of atoms that leave the 2D MOT, the trajectories of a
large number of atoms is calculated and then sorted into “good” and “bad” trajectories.
A trajectory is considered “good” if the atom leaves the 2D MOT. In order to calculate
19
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
Figure 2.8 Trajectories in the 2D MOT. The shown trajectories are calculated forthe dimensions of our 2D MOT setup.
the trajectories of an atom, its equation of motion which arises from equation (2.25), is
integrated numerically. In case of a 2D MOT, the four cooling beams are propagating
along the direction of the quadrupole magnetic field and have circular polarization.
This means that the sum in equation (2.25) only consists of the four cooling beams
with m = ±1 respectively, the pushing beam is neglected for now. The beams have a
Gaussian profile truncated to the diameter of the wave plate holders which are placed
in every beam path to give the beam its circular polarization.
For the simulation, a set of initial positions ~r0 and velocities ~v0 is needed. The
surface of the 2D MOT chamber is going to be used as starting points ~r0 for the
trajectories assuming that the atoms last scatter on one of the walls before being
trapped. This assumption is valid as long as the mean free path of the atoms is
larger than the dimensions of the glass cell. At higher pressures the trajectories can
also start from within the glass cell due to collisions. This effect will be neglected in
the simulation. Furthermore, it will also be assumed that the atoms are reflected at
the walls. The number of atoms which are emitted from the walls can be calculated
analogously to equation (2.10). The flux from an area of size A is given by
Φ = A
∫
vz>0
cos θ v fMB(v) d3v (2.26)
with the Maxwell-Boltzmann distribution
fMB(v) = 4π
(
m
2πkBT
)3/2
v2 exp
(
− mv2
2πkBT
)
. (2.27)
The factor cos(θ) v takes into account that the scattering rate of an atom with one of
the walls depends on the atom’s velocity v and its angle of incidence θ. Thus faster
atoms and atoms that fly directly towards the wall scatter more often.
Equation (2.26) is used to calculate a set of N trajectories which start on a surface
area A. This is done the following way: First a random vector ~rrandom with an isotropic
distribution is computed. To pick a random vector, it would be incorrect to compute
two random numbers φ ∈ [0, 2π] and θ ∈ [0, π] and use them to calculate the vector in
20
2.3. MODEL OF A 2D MOT
spherical coordinates. This would lead to an accumulation of vectors close to the poles
and less vectors on the equator since the area element dΩ = cos(θ)dθdφ is a function
of θ. A fast way to compute ~rrandom = x, y, z is to choose two random numbers
r1 ∈ [0, 2π] and r2 ∈ [−1, 1]. The random vector is directly given by:
x =√
1− r22 sin(r1)
y =√
1− r22 cos(r1)
z = r2 . (2.28)
This method is based on a quite unintuitive fact: The projection of randomly dis-
tributed points on a unit sphere onto the z-axis leads to a uniform distribution between
−1 and 1.
In order to get vectors with the distribution function p(θ) = cos(θ), a rejection
method [34] can be applied to the isotropically distributed vectors. For every ~rrandom
another random number rtest ∈ [0, 1] is computed and compared to p(θ) of ~rrandom. If
rtest < p(θ) the vector is accepted, if rtest > p(θ) the vector is rejected. This leads to
a set of vectors with a distribution of p(θ).
The rejection method can also be used to calculate random velocities with p(v) =
v fMB(v). Multiplying every ~rrandom with a random velocity, one gets a set of vectors
that fulfills the emission characteristics given by equation (2.26). In the simulation
a set of N of these vectors is used as initial directions for N atoms which start from
every unit area dA on the surface of the glass cell. The total flux Φ of atoms that
leave the glass cell is proportional the the number of “good” trajectories Ngood divided
by the total number of trajectories Ntotal = N ·A/dA:
Φ = ηNgood
Ntotal. (2.29)
Here η is the total number of emitted atoms form the surface of the glass cell which
can be obtained by integrating equation (2.26) over all velocities [23]:
η = Ap√
2πmkBT. (2.30)
Besides the total atom flux, also the atoms’ velocity distribution can be derived from
the simulation. It is simply the distribution of the final velocities of all “good” tra-
jectories. So far, collisions have been neglected from the model. This results in an
overestimation of atoms with very low longitudinal velocities (v < 5m/s). In the
experiment these atoms are not observed. Due to their long dwell time in the MOT
21
CHAPTER 2. 2D MOT AS A SOURCE OF COLD ATOMS
they are very likely to collide with atoms from background gas before they can exit
through the hole. To account for these collisions, the exponential loss term from equa-
tion (2.18) can be used. In the simulation the total time an atom stays in the glass
cell is used to calculate its collision probability:
pcol(tout) = exp(−σeffnRbv tout) . (2.31)
In chapter 4 the results from this simulation will be compared to the results of the
measurements.
22
3 Realization of the 2D MOT
Our setup consists of an optical table with the laser setup and the 2D MOT itself
which can be connected to the main chamber on a CF16 flange. The 2D MOT has a
very modular and compact design which consists of a glass cell which is connected to
an optics module via a midpiece. The optics module can easily be seperated from the
rest of the setup.
In the first subsection the laser system with the beam paths for the 2D MOT will
be covered and in the second subsection a detailed description of the 2D MOT setup
will be given.
(a) (b)
Figure 3.1 Picture of the setup. (a) All the optics is placed on a solid cage thatsurrounds a glass cell with rubidium dispensers. (b) Picture of the optical table.
3.1 Laser system
To operate the 2D MOT, two lasers are necessary which are stabilized at different
hyperfine transitions of the trapped atoms. In the experiment 85Rb is used which
has a nuclear spin of I = 5. Due to hyperfine coupling, the ground state 52S1/2
splits up into two sub-states F = 2, 3, the excited state 52P3/2 splits up into four
23
CHAPTER 3. REALIZATION OF THE 2D MOT
Figure 3.2 Hyperfine level scheme of Rb with cooling scheme for 2D MOT. Thecooling laser is red detuned from the F = 3 → F ′ = 4 transition. The repumperpumps the atoms that fall into the dark state back into the cooling cycle. Theprobe beam is resonant with the F = 3 → F ′ = 4 transition.
substates F = 1, 2, 3, 4. The energy splittings of the hyperfine substates of the D2
line 52S1/2 → 52P3/2 used for cooling of the rubidium atoms are shown in figure 3.2.
In order to cool the atoms a cooling and a repumping laser are needed. The cooling
laser drives the F = 3 → F ′ = 4 closed cooling transition, but since the energy splitting
of the F ′ = 3 and F ′ = 4 sub-states is only 121MHz some of the atoms will unavoidably
be pumped into the F ′ = 3 state. From there they can decay spontaneously into the
F = 2 ground state which cannot be addressed by the cooling laser and is therefore a
dark state. This happens approximately once every thousand cooling cycles. Therefore
we need a repumping laser that pumps the atoms from the F = 2 dark state back into
the cooling cycle. To operate the 2D MOT and to characterize the atom beam, we
need three different laser beams:
• For the cooling beams we need light that is red detuned from the F = 3 → F ′ = 4
cooling transition and resonant light on the F = 2 → F ′ = 3 transition to pump
back the atoms that fall into the dark state.
• An enhancement of the normal 2D MOT is the so called 2D+ MOT [6] where
an additional pair of laser beam is put in the direction of the atom beam. These
beams are also red detuned from the F = 3 → F ′ = 4 transition and can be
used to modify the velocity distribution of the atoms.
• Finally, to characterize the atom beam, we need a probe beam with resonant
light on the cooling as well as the repumping transition (see section 4.1).
24
3.1. LASER SYSTEM
Figure 3.3 Locking scheme of the cooling laser. The TA is locked on the F’=3/4crossover, but it runs on the F’=2 transition due to the AOM before of the spec-troscopy. The cooler, pusher and probe beam are then shifted seperately by doublepass AOM’s to their desired frequencies.
3.1.1 Cooling laser
The cooling laser is a Toptica TA Pro. This laser is based on a diode laser (DL Pro)
that is coupled into a tapered amplifier chip. The beam is amplified in a single pass
through the chip preserving the spectral properties of the beam and then coupled into
a single mode fiber. The tapered amplifier chip makes it possible to reach output
powers as high as 1.3W (730mW after the fiber) which would destroy the facet of a
normal laser diode. In order to protect the diode laser from retro reflected light a 60dB
optical isolator is placed between the DL Pro and the TA chip. Between the isolator
and the tapered amplifier, a test beam is split off that is used for the spectroscopy to
stabilize the laser frequency.
The laser is mounted on an optical table that also contains the beam paths for the 2D
MOT. Only the spectroscopy is placed on a separate breadboard that is placed under
the optical table and set on a Sorbothane sheet to damp vibrations. The test beam
is transfered to the spectroscopy in a single mode polarization maintaining fiber. On
the spectroscopy board (figure 3.4), the beam first passes an acousto-optical modular
(Crystal Technology 3000 Series) that is set up in a double pass configuration and
shifts the laser frequency by 2 × 62MHz. Afterwards the beam is split up, one part
passes an 1 : 3 telescope and is send through a Rubidium vapor cell to be used for
Doppler free saturation spectroscopy. The other part is coupled into a Fabry Perot
interferometer to exhibit the mode profile of the laser. Using FM spectroscopy, the
laser is locked on the F ′ = 3/4 crossover which provides the largest signal. This means
that the laser is running on the F = 3 → F ′ = 2 transition (figure 3.3).
On the optical table (figure 3.5), the beam is split up into cooling, pushing and probe
25
CHAPTER 3. REALIZATION OF THE 2D MOT
Figure 3.4 Doppler free spectroscopy for laser lock. The Doppler free spectroscopyboard includes two spectroscopy branches to lock both lasers. Both lasers are alsocoupled into a Fabry Perot interferometer and can be shifted in frequency by anAOM in double pass configuration.
Figure 3.5 Beam Paths for the 2D MOT. All beams are shifted in frequency usingAOMs. Afterwards the beams are coupled into fibers to transfer them to the 2DMOT setup.
26
3.1. LASER SYSTEM
Figure 3.6 Locking scheme of the repumper. The diode laser is locked on the F’=2/3crossover, but it runs 142MHz higher due to the AOM before of the spectroscopy.The repumper beam is then shifted by a single pass AOM to the F’=3 transition.
beam by polarizing beam splitters (PBS) in combination with half-wave plates. All
beams are shifted to their desired frequencies with acousto-optic modulators (AOM).
Thereby the detuning of the cooler and pusher beams can easily be optimized by
changing the AOM frequencies. In a single pass configuration of the AOM, this would
cause the beam position to shift when changing the AOM’s frequency. Since all beams
are coupled into fibers to transfer them to the 2D MOT, this would cause intensity
fluctuations as already a small shift of the beam position significantly decreases the
coupling efficiency. Therefore, all the beams apart from the repumper are set up in
a double pass configuration [35] where the beam passes the AOM twice. Thus the
frequency can be adjusted without any significant changes in the beam position. A
disadvantage of passing the AOM twice is that we loose about 15−20% of laser power
every time the beam passes the AOM. After passing the AOMs, all the beams are
coupled into optical fibers.
3.1.2 Repumping laser
The repumper is a home build diode laser in Littrow configuration [36]. For a detailed
description of the diode laser setup see [37]. The laser beam passes an optical isolator
to protect the diode laser from back reflections. Then a small fraction of the light is
split off to be used for the FM spectroscopy which is identical to the one of the cooling
laser. The beam is shifted by 2 × −71MHz before being send to the spectroscopy
where it is locked on the F ′ = 2/3 crossover.
On the optical table, the beam is split up into two beams both of which are shifted
by 110MHz to be on resonance to the F = 2 → F ′ = 3 transition. One beam is then
superimposed on the cooling beam using a PBS which leads to a crossed polarization
of the two beams. The other beam is superimposed with the probe beam.
27
CHAPTER 3. REALIZATION OF THE 2D MOT
3.1.3 Laser cubes
Before the light is transfered to the 2D MOT setup, the trapping beam, as well as
the pusher beam are divided into two beams. This is done using Schäfter+Kirchhoff
laser cubes. In these cubes, the trapping beam is coupled out of the fiber and equally
split up into two beams. Since the two beams are coupled into the fiber with crossed
polarizations this can be done with a polarizing beam splitter in combination with
an adjustable half-wave plate. Also a small fraction (about 1%) is split off the beam
using a glass plate and then detected on a photodiode. This can be used to monitor
the laser power in the beam which is particularly useful when coupling the beam into
the fiber on the optical table. At the end all four beams are coupled back into four 15
meter long polarization maintaining fibers, to transport the light to the 2D MOT.
3.2 2D MOT setup
Figure 3.7 Design of the 2D MOT. A cage with all the optics is placed around theglass cell which is filled with rubidium gas.
Our 2D MOT consists of a glass cell that is surrounded by a metal cage holding
all the optics for three cooling regions and the pusher beams (figure 3.7). The glass
28
3.2. 2D MOT SETUP
Figure 3.8 Center piece of the 2D MOT setup. The center piece connects all thedifferent components of the 2D MOT.
cell provides perfect optical access from all directions and is filled with rubidium from
dispensers.
The 2D MOT was designed to be as modular and compact as possible and ensure a
high flux of cold atoms in order to load a 3D MOT as fast as possible. The complete
2D MOT setup has a size of 20 × 20 × 40 cm and can be mounted onto the main
chamber in one piece. The only connections needed are four optical fibers for the
different laser beams and electrical connections for the compensation coils and the
dispensers. The center piece of our design is a 82 × 74 × 24mm stainless steel part
(figure 3.8), that connects all the different components of the setup. On one side, a
bellow is connected on a CF40 flange to mount the setup on the UHV chamber. On
the other side, another CF40 flange connects the mid piece with the glass cell. To the
sides of the mid piece, there are two CF16 flanges, that are equipped with electrical
feedthroughs to connect the dispensers. Also on the sides, one finds several threaded
holes where the cage with the optics can be mounted. In the following, the different
components will be discussed in more detail.
3.2.1 Bellow with differential pumping tube
The bellow mounts the 2D MOT on the main vacuum chamber and is closed on its
CF40 side. The only connection between the two chambers is a 13 cm long differential
pumping tube with an inner diameter of 8mm which has an 800µm hole on the 2D
MOT side. It enables us to keep the rubidium pressure in the glass cell at around
10−7mbar, whereas the pressure in the main chamber should be 10−10mbar or less.
The end of the tube is polished and tilted by 45°, serving as a mirror for a counter-
propagating pusher beam.
29
CHAPTER 3. REALIZATION OF THE 2D MOT
Figure 3.9 Alignment of the differential pumping tube. Once the horizontal laserbeam is reflected horizontally again, the adjustment is complete.
The pumping tube is directly screwed into the bellow and fixed with a counter-
screw. Since the position cannot be changed after assembly, it is important that the
polished surface is oriented towards the outcoupler of the pusher beam. To align the
tube, a horizontal laser beam and two irises are used (see figure 3.9).
Since the bellow is the only fixed connection between the 2D MOT and the rest of
the setup, it can be used to change the relative orientation of the 2D MOT to the main
chamber. This is done with three threaded rods that are connected to the bellow on
one side. On the other side, three threaded tubes are placed in recesses so they can
be rotated and screwed onto the rods which then works like three telescope bars. This
allows us to fine adjust the atom beam for optimal loading of the three dimensional
trap.
3.2.2 Glascell
Figure 3.10 Technical drawing of the glass cell (Japan Cells).
The glass cell (figure 3.10) is made form Schott Tempax Borofloat glass with an AR
coating for 780nm on the outside of the cell. We decided to leave the inside of the cell
uncoated because the coating can act as a getter material for rubidium, which could
create a metal mirror on the surface of the cell. Some groups successfully use coatings
30
3.2. 2D MOT SETUP
Figure 3.11 Connection of the dispensers. The dispensers are connected to thefeedthrough with barrel connectors and sit to the side of the differential pumpingtube.
on the inside of the cell, but we did not want to take any chances. Our cell provides
us with 140mm of coated glass (figure 3.10), which is enough for three cooling regions,
each of which taking up about 35mm. Inside the glass cell up to four dispensers
contain the necessary rubidium. Two electrical feedthroughs that are placed on CF16
flanges on the sides of the mid piece provide the electrical connection to the outside
of the vacuum. The wires of the feedthroughs are bend at 90° so that the ends point
towards the rear side of the glass cell. The dispensers are directly connected to the
feedthrough using barrel-connectors (figure 3.11). The dispensers we used (Alvatec
AS-3-Rb-20-F), contain 20mg of rubidium in a cylinder shaped canister that is sealed
with indium. To open the dispensers, there is a special procedure [38] which includes
several steps at different currents, in order to break the seal and start the emission
of rubidium vapor. The activation is preferably done during a bake out, but can also
be done under normal vacuum conditions. After activation, the dispensers produce a
directed jet of rubidium vapor, in our case pointing to the far side of the glass cell.
The rubidium pressure can be controlled by changing the current of the dispensers.
3.2.3 Optics
All the laser light is transfered from the optical table to the 2D MOT via polarization
maintaining single mode fibers. At the 2D MOT, the fiber outcouplers are placed
in matched fittings, that form a bow over the outcoupler and are open on one side
(figure 3.12). At this opening a M4 screw is used to slightly compress the holder, just
enough to lock the outcoupler into position. The holder is connected to the mid-piece
by two screws and two positioning pins. Those pins allow for removal of the entire
31
CHAPTER 3. REALIZATION OF THE 2D MOT
Figure 3.12 Holder for fiber outcouplers. The holder is positioned using two pins andthen fixed by two screws. This way the fiber coupler can be removed withoutlosing the beam alignment.
holder without risking a misalignment of the beam. This was tested using a position
sensitive detector (Thorlabs PDQ 90S1) to measure the position of the outcoupled
beam after removing the entire holder numerous times from the mid piece. During
this measurement, the position of the beam varied by less then 0.1%. The same result
was obtained when only removing the outcoupler from the fitting. The outcouplers for
the two cooling/repumping fibers produce elliptical beams with a long axis of 22mm
and a short axis of 11mm (1/e2 values). Both beams are then split up into three
beams by a sequence of two polarizing beam splitters. All optics is placed as compact
as possible, in order to move the three cooling regions as close together as possible.
This is done using cube shaped modules (figure 3.13) that have a PBS in the middle
and are mounted onto mirror holders. The wave plates are glued into copper rings
that are placed into special fittings on the faces of the cube which are open at the top.
This way, the copper rings containing the quarter- and half-wave plates can still be
rotated after the faces of the cube are enclosed with two aluminium plates, to prevent
the rings from falling out of the fitting. The rings containing the wave plates can be
fixed in their final positions with M2 set screws. The power distribution among these
three beams can be controlled with half-wave plates, that are placed in front of every
PBS. Every beam also passes through a quarter-wave plate to achieve the circular
polarization needed for the trapping of the rubidium atoms. Then the beams pass the
glass cell and are retro-reflected by a mirror placed behind another quarter-wave plate
(figure 3.14). All three cooling regions add up to a total MOT length of up to 66mm.
The elliptical profile of the beams was chosen to ensure saturation of the cooling
transition, all over the cooling region. The two pushing beams have a diameter of
32
3.3. MAGNETIC FIELD DESIGN
8mm and are aligned to the center line of the glass cell. As mentioned above, the front
face of the differential pumping tube serves as a mirror for the counter-propagating
pusher beam. The next section deals with the magnetic field setup that is also part
of the cage with all the optics.
Figure 3.13 Splitting of the trapping beam. These compact optics modules are usedto split up the beam for the three cooling regions and to polarize all beamscircularly.
Figure 3.14 Optics module with glass cell in the center (top view)
3.3 Magnetic field design
For cooling of the rubidium atoms in two dimensions, a magnetic field gradient of
about 15G/cm is needed along these two directions and no field component in z-
33
CHAPTER 3. REALIZATION OF THE 2D MOT
direction. Most commonly, this field is produced by quadrupole coils. We decided to
use permanent magnets instead (figure 3.14).
Figure 3.15 Optics module with glass cell in the middle (side view). In a plane witha 45° angle to the laser beams sit the magnets.
3.3.1 Permanent magnets
Permanent magnets are made from materials with a magnetic dipole moment on mi-
croscopic scale. Usually these moments are orientated randomly averaging out any
magnetic moment on macroscopic scale. In a permanent magnet all the magnetic
dipole moments are aligned which can create quite strong fields on a macroscopic
scale as well. The permanent magnets we used are 25 × 3 × 10mm bars made from
neodymium (RS Components No.434-6877). The axis connecting the magnetic north
and south pole coincides with the short axis (3mm long) of the magnet. This axis
will be denoted as the y-axis in the following. The magnetic field of a cubical shaped
magnet like ours is given by [39]:
Bx(x, y, z) = −µ0M
4π
1∑
i,j,k=0
(−1)i+j+k arcsinh
(
z − zk√
(x− xi)2 + (y − yj)2
)
By(x, y, z) = +µ0M
4π
1∑
i,j,k=0
(−1)i+j+k arctan
(
(x− xi)2(z − zk)
2/(y − yj)2
√
(x− xi)2 + (y − yj)2 + (z − zk)2
)
Bz(x, y, z) = −µ0M
4π
1∑
i,j,k=0
(−1)i+j+k arcsinh
(
x− xk√
(z − zi)2 + (y − yj)2
)
(3.1)
Here Bx, By and Bz are the three Cartesian components of the magnetic field, µ0 =
4π×10−7Vs/Am is the magnetic constant, M is the magnetization which is a material
specific parameter with unit A/m. Finally (x0, y0, z0) = ~rmin and (x1, y1, z1) = ~rmax
34
3.3. MAGNETIC FIELD DESIGN
are two opposite corners of the magnet, that define its spatial position.
3.3.2 Magnetization
Figure 3.16 Measurement of the magnetization of a permanent magnet. Measure-ment a is taken at the center of the magnet, measurement b along the side ofthe magnet.
To determine the magnetization of the permanent magnets, the magnetic field along
the axis at the center of the magnet was measured using a hall probe (SMT Teslamter
907) mounted on a three dimensional translation stage (figure 3.16). This measurement
would have been enough to determine the magnetization. In this case the data could
have been fitted by taking the magnetization and a relative z offset between translation
stage and magnet as fitting parameters. But it turned out that the accuracy could be
increased by taking a second measurement with an offset in x-direction (in our case
x = 1.26 cm) to determine the relative z offset separately. This result is then used
to fit the data of the first measurement with the magnetization being the only fitting
parameter (figure 3.17). The increase in accuracy when taking two measurements is
due to the fact, that the second measurement is more sensitive for the the relative
z position towards the magnet, since the field over a much longer distance could be
measured on both sides of the magnet. The Hall probe over-saturated when getting
too close to the magnet on the center line. The final result was a magnetization of
M = (8.7± 0.2)× 109A/m .
The limiting factor for the accuracy of the measurement was the error of the Hall
probe signal and the relative positioning of magnet and translation-stage. Also the
deviation of the magnetization for different magnets was measured. The measurement
was carried out by putting the magnets in a special fitting, the size of a magnet
and placing the hall probe above the center of the magnet. Now we could place
different magnets in the holder, ensuring that the distance between the magnets and
35
CHAPTER 3. REALIZATION OF THE 2D MOT
(a) (b)
Figure 3.17 Magnetic field of the permanent magnet. (a) Fit of the magnetic fieldwith the hall probe moving along the center line of the magnet and (b) with adisplacement of x = 1.26 cm to the side
the hall probe was always the same. The measurement shows (table 3.1), that the
magnetization of the magnets varies by approximately 3%. Now we have completely
characterized the magnets and can combine the fields of several magnets to create a
two dimensional quadrupole field.
Table 3.1 Magnetization of different magnets. The magnetization of the permanentmagnets varies by approximately 3%.
Number Magnetic field (G) Magnetic field (G)magnet front side rear side
1 40 -39.42 39.3 -38.73 38.5 -39.74 39 -39.75 39.3 -39.26 40.2 -39.5
3.3.3 Setup of the magnets
In the following there are two different coordinate systems we are going to use, one
being the system of the magnets and the other one being the system of the glass cell
and the lasers. The system of the lasers is going to be denoted ~r′ = (x′, y′, z′), the
system of the magnets will be denoted ~r = (x, y, z). The z axes of both systems are
36
3.3. MAGNETIC FIELD DESIGN
Figure 3.18 Design of the magnet holder. On both sides of the glass cell, an arrayof six permanent magnets (blue) is mounted onto a holder. The position of themagnets was optimized to get a smooth magnetic field gradient. Therefore, themagnets in the middle are moved further to the back. On both sides the fifthmagnet is placed on a translation stage. This way the zero line of the magneticfield can be shifted at the position of the exit hole.
the same, being the symmetry axes of the two rows of magnets, as well as the center
line of the glass cell. For our 2D MOT we use two rows of six magnets each. The
magnets are glued onto a holder that is placed under a 45° angle with respect to the
faces of the glass cell (figure 3.18). To calculate the magnetic field of this setup, one
has to add the magnetic field (equation (3.1)) of all twelve magnets
~Btotal =
12∑
n=1
~Bn , (3.2)
Bn being the field of the nth magnet. The magnetic field of all twelve magnets is
shown in figure 3.19 and figure 3.20. Note that the dipole moments of the magnets in
the upper row are oriented in the opposite direction to the magnets in the lower row.
This means that the magnetic field of both cancel each other at the center. One gets
an almost perfect two-dimensional quadrupole field with the zero line of the magnetic
field along the center of the glass cell. On a length of about 8 cm on the two axes
coinciding with the cooling beams, the gradient is almost constant. In order to get the
same gradient at all positions of the z axis too, the inner magnets have to be moved
further away from the glass cell. The reason is, that in the middle, the field of the
neighboring magnets adds up, which leads to a steeper gradient in the center then at
the sides. The position of the magnets was optimized for a field gradient of 15G/cm.
37
CHAPTER 3. REALIZATION OF THE 2D MOT
(a) (b)
Figure 3.19 (a) Vector field of the permanent magnets in the z=0 plane. (b) Zoominto the center of the field where the atoms are trapped.
Other gradients can also be obtained by moving the magnet holders to a different
position in the x′-direction. The holder can be moved to five different positions where
it is fixed with special positioning screws. The five accessible gradients are 10.6,
12.6, 15.2, 18.5 and 23.0G/cm. To position the magnets on the holder a template
with a pit at the position of each magnet is used. The magnets are glued into these
pits with two-component adhesive (Uhu endfest 300). After the glue hardens within
a day, the template is left in position because it adds some extra stability to the
magnets. To see how good the magnetic field matched the calculations, the field of
one configuration was measured with a three dimensional hall probe. The result is
shown in figure 3.16. There is a slight deviation of the magnetic field gradient from
the calculations on the right side. But this error is within the range of the fluctuations
of the magnets’ magnetization mentioned earlier. These small fluctuations are not
very important concerning the slight change of the gradient, since this is not a very
sensitive parameter for capturing the atoms, at least on this scale. But it is critical
when it comes to the zero line of the magnetic field. The trapped atoms are going
to follow that zero line and already a small offset can cause the atom beam to miss
the 800µm exit hole in the differential pumping tube. Already the earths magnetic
field (about 400mG in Heidelberg) causes a shift of about 270µm at the 15G/cm
configuration. Another error source is the position of the magnets. Therefore the last
but one magnet was placed on a small translation stage. Moving this magnet changes
the magnetic field at the end of the last cooling region before the atoms head towards
the exit hole. The idea is that with this degree of freedom the atom beam could be
38
3.3. MAGNETIC FIELD DESIGN
Figure 3.20 Magnetic field gradient in the plane of the laser beams. On the left acontour plot of the magnetic field in the plane of the laser beams is shown. Onthe right side B(z′) along the x′ = 0 axis is shown. The obtained gradient isclose to constant for −5 < z′ < 5 and −8 < z′ < 8.
4
6
8
10
12
14
-10 -5 0 5 10
field
gra
die
nt∇B
(G/cm)
z position (cm)
calculationmeasured points
Figure 3.21 Magnetic field gradient. Every point shown in the graph consists of fivedata points that were taken at different x-positions and then used to fit thegradient at that point. The solid line shows the calculated curve.
39
CHAPTER 3. REALIZATION OF THE 2D MOT
coupled into the pumping tube.
The experiment showed that only one degree of freedom is not enough to accomplish
that. In order to adjust the beam through the hole it is necessary to have full control
over the magnetic field in both x- and y-direction. It would be beneficial to move
the whole holder with all magnets rather than only bending the field at the last
cooling region, because the straighter the zero line of the magnetic field, the easier
the atoms can follow the field. To get both degrees of freedom needed two pairs of
compensation coils in x′- and y′-direction were installed which add a constant field
along their symmetry axis thus shifting the zero line in that direction. Now the atom
beam can be moved in two dimensions in order to pass through the differential pumping
tube. To see how critical the alignment of the magnetic field really is the atom flux
behind the pumping tube as a function of the compensation field was measured (see
section 4.5.2).
40
4 Characterization of the atom
beam
Before connecting the 2D MOT to the main vacuum chamber it is optimized on a test
chamber. Good optical access enables us to take time of flight measurements with a
sensitive photodiode. This way the atom beam could fully be characterized and the
flux could be optimized for our later requirements. The velocity distribution of the
beam is particularly interesting in order to load the 3D MOT because only atoms
under a certain capture velocity can be captured.
In the first section details of the test setup are shown and the way different pa-
rameters were measured is explained. Afterwards the results of the measurements are
presented. This includes the measurement of the velocity distribution and the total
flux as a function of the various parameters that are accessible in our setup. For all
measurements only one parameter was changed and all the other settings were put on
their optimum values. The results will also be compared to results from our simula-
tion and other experiments. Table 4.1 gives on overview of the beam characteristics
of different rubidium atom sources.
experiment mean beam total flux luminosityvelocity (m/s) divergence (mrad) (s−1) (atoms s−1 rad−1)
Dieckmann [6] 15 43 5× 109 3.4× 1012
Dieckmann [6] 8 43 9× 109 6.2× 1012
Schoser [7] 60 32 5× 1010 6.2× 1013
Wohlleben [28] 14 10 1× 108 1.3× 1012
Utfeld [40] - 30 3× 108 4.2× 1011
our result 14 26 4× 109 7.5× 1012
Table 4.1 Beam charcteristics of other Rb 2D MOTs.
4.1 Test setup
In order to characterize the atom beam, the 2D MOT is flanged onto a test chamber
(figure 4.1) that consists of two CF40 six-way crosses and one four-way cross. The
41
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
two detection regions
Figure 4.1 Test setup. The beam passes through two six way crosses where resonantlaser light is shone onto the atoms. The fluorescence can be measured at twodetection regions.
beam passes through both six-way crosses that are equipped with viewports to all
sides providing good optical access. The only flanges with no viewports are the one
the 2D MOT is mounted on and the one next to it where the setup is connected to a
four-way cross with an ion pump and a Bayard Alpert gauge. Also connected to the
four-way cross is a turbo molecular pump and a roughing pump that can be separated
from the chamber with a gate valve once the pressure is low enough to operate the ion
pump. After evacuating the chamber, it was baked for five days at 180 C resulting
in a final pressure of 5 × 10−8mbar. For a chamber of this size, one would expect
a much lower end pressure, but we were limited by the fact that our ion pump was
contaminated with lithium oxide.
To analyze the atom beam, resonant light is shone onto the atoms from the top.
Then the fluorescence is either detected with a sensitive photodiode (Thorlabs PDA36)
or a CCD camera (Allied Visions, Guppy Cam F038b) that is placed in front of a
viewport at a 90° angle with respect to the direction of the probe beam. The signal
can be increased by adding some repumper light to the probe beam and by retro-
reflecting the beam. Without the retro-reflected beam, the atoms are pushed out of
the detection region of the photodiode/CCD-camera.
42
4.1. TEST SETUP
-0.3
-0.28
-0.26
-0.24
-0.22
-0.2
-0.18
0 0.01 0.02 0.03 0.04
PD
sig
nal(V
)
time v (s)
(a)
10 20 30 40
0
0.1
0.2
0.3
ato
mflux
velocity v (m/s)
(b)
Figure 4.2 Measurement of the velocity distribution. (a) Fluorescence signal on thephotodiode. At t = 0 the cooling laser was switched to a blue detuning (−8MHz)in order to stop the atom flow. The velocity distribution (b) is obtained bydifferentiating the fluorescence signal.
4.1.1 Measurement of the velocity distribution
The velocity of the atoms is especially important for the loading efficiency of the 3D
MOT. Typical capture velocities vcap are in the range of 25-40m/s. This is why we
are not only interested in the total flux of atoms, but also in the velocity distribution
of the beam. In particular the flux of atoms below the capture velocity Φ(v < vcap).
For this measurement, the sensitive photodiode was used at the first detection region
in order to get a time resolved fluorescence signal. To collect light from a bigger solid
angle, a combination of three plano-cenvex lenses (two inch) was placed in front of the
photodiode. First a f = 100mm lens with the atom beam sitting in the focus of the
lense collects the light, followed by two f = 60mm lenses to focus the light onto the
photodiode. Two f = 60mm lenses are used because lenses with a smaller focal length
are not available with a two inch diameter. The lenses were placed in a lens-tube with
an iris and the photodiode placed at the end of the tube to minimize the background
signal from stray light.
To take time-of-flight measurements, the beam has to be switched on and off. In
order to prevent systematic errors this has to be done at an exactly known location.
Simply blocking the cooling beam for instance does not produce a good signal as all
atoms that are already trapped could still exit the hole. Switching the cooling beams
to a blue detuning blows away already trapped atoms, thus effectively cutting of the
beam at the end of the last cooling region in front of the exit hole. This is done by
43
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
switching the AOM for the cooling beams between the normal detuning of δ = 12MHz
to a blue detuning of δ = −8MHz.
The atom beam is switched off at t = 0. It takes an atom ∆t = d/v to reach
the detection region, d being the distance to the probe beam and v being the atoms
longitudinal velocity. For t < d/v atoms with a velocity of v still fly through the
detection region. At time t = d/v the last atoms with velocity v enter the detection
region and the flux stops. For all times t > d/v this velocity class is missing from the
photodiode signal. This means that the signal U(t) is proportional to the fluorescence
of all atoms with 0 < v < d/t:
U(t) ∼∫ v=d/t
v=0
N(v′)dv′ .
N(v) is the atom-distribution and d is the distance from the 2D MOT to the detection
region. Substitution of t′ = d/v′ leads to
U(t) ∼∫ t=d/v
t=∞
N(d/t′)d
t′2dt′
⇒ ddt
U(t) ∼ ddt
∫ t=d/v
t=∞
N(d/t′)d
t′2dt′ = N(v)
v2
d
⇒ N(v) ∼ d
v2dUdt
.
The atom-flux-distribution is Φ(v) = N(v)× v. This gives us
Φ(v) = ξd
v
dUdt
(4.1)
The proportionality factor ξ converts the signal of the photodiode U(t) into the
number of atoms in the detection region. Note that there is a difference between
the number of atoms in the detection region and the atom distribution N(v). The
atom-distribution already accounts for the fact that slow atoms scatter more photons
because they need longer times to pass the detection region.
The atoms scatter light with Γ2= 1
τwhere Γ = 6× 2 πMHz is the natural linewidth
of the F = 3 → F ′ = 4 transition with a lifetime of τ = 27 ns. But we only detect
the light in the solid angle of the lenses placed in front of the photodiode. With a two
inch lens at a distance of 10 cm we get a solid angle of Ωpd = 1.6%. We also have to
take into account that the viewports are not coated and we loose approximately 10%
of power on both surfaces of the glass. The quantum efficiency of the photodiode was
44
4.1. TEST SETUP
measured with a laser by measuring the voltage drop as a function of the laser power.
This gave us a ratio of κ = 1.0 × 106V/W. Taking everything into account we get a
proportionality factor of
ξ = 0.81 κΓ
2Ωpd hν dprobe
Here h is Planck’s constant, ν is the frequency of the atomic transition and dprobe is the
width of the probe beam which is the length of the detection region. Now we can use
equation 4.1.1 to calculate the velocity distribution from the signal of the photodiode
(figure 4.2b). In order to minimize noise the photodiode signal was averaged over 16
measurements.
4.1.2 Measurement of the flux
To calculate the total flux of atoms, the atom-flux-distribution can be used. The
height of the fluorescence signal is not a good indicator for the total flux. As mentioned
before, the number of photons a single atom scatters is proportional to its longitudinal
velocity. This means a slow atom beam would result in a much higher fluorescence
signal than a fast atom beam. This is why we need the velocity distribution of the
beam to put a number on the total flux.
The total atom flux simply is the integral over the atom-flux-distribution:
Φtot =
∫
∞
0
Φ(v) dv
For our experiment an even better benchmark is the flux of atoms with velocities below
the capture velocity of the 3D MOT:
Φcap =
∫ vcap
0
Φ(v) dv
The area under the curve was calculated by using a box integral.
Measuring the flux by taking the velocity distribution some systematic errors occur:
• In the calculation we assumed that all atoms scattered into the solid angle of the
lense in front of the photodiode are detected. Due to misalignment of the imaging
lenses it is likely that we loose a good portion of those photons because they miss
the detecting area of the photodiode. This means that we underestimate the flux.
• The opposite effect occurs as photons are also scattered into the detection angle
from the walls of the chamber. We only want to detect atoms that are comming
directly from the atom beam but some of the photons that are scattered in
another directions might be reflected back onto the photodiode.
45
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
• We also assumed that we detect all the atoms that are within the detection
region of length dprobe. But since the diameter of the probe beam is smaller than
the width of the atom beam at the first detection region, we only illuminate the
center of the beam.
With some effort, it would be possible to further diminish the systematical errors just
mentioned, but the most reliable result would still be obtained by using the beam to
load a three dimensional MOT and to measure the loading curve. For the results in
the following section this means that the total atom flux shown in the graphs has a
quite large systematical error. This error should be the same for all measurements
taken and the error bars shown in the graphs only relate to the statistical error due
to fluctuations during the measurements.
4.2 Beam divergence
(a) (b)
Figure 4.3 (a) Atom beam on the CCD camera. The atom beam was made visiblewith a 6mm probe beam and then detected with a CCD camera at a 90 angle. (b)CCD image of a template with 0.5mm lines that we used to calibrate the camera.
The divergence of the atom beam is a very important parameter. It tells us how big
the beam is going be at the position of our 3D MOT which can be compared to the
size of the capture region of the trap. If the area, where we can effectively trap the
atoms is smaller than the atom beam, atoms are lost.
There is an upper limit for the divergence of the atom beam that is set by the
geometry of the differential pumping tube (figure 4.4). For the dimensions of our
tube, this limit is αmax = 64mrad. To measure the divergence of the atom beam an
image of the beam is taken with the CCD-camera. The camera’s exposure time is
40ms and an Lab-View interface is used to read out the charge state of every pixel
on the camera. To measure the divergence, the exact distance to the exit hole of the
46
4.2. BEAM DIVERGENCE
Figure 4.4 Geometry of the differential pumping tube. The geometry of the pumpingtube sets an upper limit for the divergence of the atom beam
2D MOT has to be known thus a small beam is favorable. On the other hand, the
fluorescence decreases when using a smaller beam. A beam with a width of 2mm at a
distance of 30 cm from the 2D MOT is used. By adding up all the pixels of every row
along the propagation axis of the beam, the profile of the atom beam is obtained (figure
4.5). The camera is calibrated using a template with 0.5mm thick lines which can be
used to calculate the conversion factor from pixels to meters for our images. The data
is fitted with a Gaussian which has a beam diameter of FWHM = (7.8 ± 0.5)mm.
Assuming a beam diameter of zero at the exit hole, the divergence of the beam is:
α = (26± 4)mrad ,
α being the full opening angle. This means that the atom beam is not limited by
the geometry of the pumping tube, since the divergence is well below the upper limit
αmax.
The profile of the beam seems to be slightly asymmetric, the slope on the upper
side of the profile is steeper than at the bottom. This can be explained by the grav-
itational force on the atoms that fly horizontally through the test chamber. It takes
the atoms about 20ms to reach the first detection region in which time they gather
a vertical velocity of 0.2m/s due to gravity. This is on the same scale as the atoms
initial vertical velocity. To understand how this leads to an asymmetric beam profile,
one has to understand the vertical velocity distribution within the beam. When the
beam leaves the 2D MOT the beam size is less than 800µm. Without external forces,
the atoms just fly on straight trajectories depending on their initial direction when
leaving the MOT. Only looking at the vertical velocity component, this means that
the fast atoms can be found on the outside of the beam and all the atoms with small
vertical velocities stay close to the center of the beam. This way the projected CCD
image can also be interpreted as the vertical velocity distribution of the beam.
The beam is assymmetric because all the atoms on the upper side of the beam move
against the gravitational field and are slowed down. This effectively decreases the
divergence on the upper side of the beam. On the lower side, the opposite happens re-
sulting in the asymmetric beam profile or actually an asymmetric velocity distribution.
47
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18
norm
alize
dsi
gnalhig
ht
x (mm)
projection of the CCD imagegauss fit
Figure 4.5 Beam profile of the atom beam at the first view port. The profile wasobtained by projecting the CCD image. To make the atom beam visible we useda retro reflected probe beam with a diameter of 2mm.
4.3 Dependence on laser settings
In order to change the detuning and the power of the different beams the AOMs placed
in every beam path are used. Changing these two parameters is not fully decoupled.
When the frequency of the AOM is changed the diffraction efficiency changes as well
which results in a shift in beam power. Also the position of the beam changes leading to
a change in the coupling efficiency into the optical fiber. This effect was minimized by
setting up the AOMs in a double pass configuration but it can not fully be suppressed.
In order to change the detuning of the beams without changing its power the AOM
power was adjusted for every setting of the frequency as well.
4.3.1 Detuning cooling laser
The highest flux of atoms was measured for a detuning of δ = 12MHz = 2Γ (figure
4.6). In figure 4.6 also the results of our simulations (green triangles) is shown which
reproduce the measured data quiet nicely. To match the data, the fluxes from the
simulations are scaled down by a factor of two. The detuning of the cooler is an
important parameter for every magneto-optical trap:
• Together with the magnetic field gradient it defines the size of the trapping
48
4.3. DEPENDENCE ON LASER SETTINGS
volume. The smaller the detuning of the cooling laser, the smaller the volume
where the atoms are cooled.
• The detuning influences the shape of the trapping potential. For large detunings
we only get a weak confinement for the already trapped atoms.
Because of the smaller radius of the trapping volume at small detunings, the cooling
time of the atoms is strongly limited by their transverse velocity whereas their longi-
tudinal velocity is less critical. At higher detunings the longitudinal velocity becomes
the limiting factor of the cooling time leading to a reduction of fast atoms in the beam.
This can be seen in figure 4.7 where the number of fast atoms is larger for smaller
detunings.
4.3.2 Power cooling laser
For different powers of the cooling beams we see an almost linear increase of the flux
at low powers and a plateau at high beam powers (figure 4.8). At low powers the
scattering rate increases linearly with the laser power which leads to a higher capture
velocity and an increasing flux. As soon as the transition is saturated all over the
cooling volume and the scattering rate is close to its maximum of γ = Γ/2, the flux
levels out. For our setup with three cooling regions the plateau is reached at a power
of P ≈ 55mW in each of the two beams (x and y beam). This corresponds to a
saturation of I/I0 = 12 at the center of the cooling beams.
The simulation can reproduces the data quiet good again. The flux was scaled by
the same factor of two as in figure 4.6 to be comparable.
The peak velocity of the beam is slightly shifted to lower velocities with increasing
cooler power (figure 4.9). This is in contrast to the results in [7] where an increase
of the peak velocity at higher powers was observed. It was argued that since the
transverse cooling works more efficiently at high intensities, also atoms with higher
longitudinal velocity can participate in the beam. In [7] a conventional 2D MOT
without a pushing beam was used. In our case the peak in the velocity distribution is
mainly caused by atoms that first travel away from the exit hole and are then turned
around by the pushing beam (see section 4.3.4). This is a totally different situation
where the argument in [7] no longer holds.
Table 4.2 shows the cooler powers and detunings of other rubidium 2D MOTs. The
beam sources are divided into 2D MOTs (no pusher), 2D+ MOTs (with pusher) and
a MOT + pusher. The last one is a regular 3D MOT where the atoms are pushed
out by an additional laser beam. For a better comparability also the I/I0 value in the
49
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
0.5
1
1.5
2
2.5
3
3.5
2 4 6 8 10 12 14 16 18
ato
m-fl
uxΦ
(109
s−1)
detuning cooler δ (MHz)
Figure 4.6 The total atom flux as a function of the cooler’s detuning. The highestflux was observed for a detuning of δ = 12MHz. The green triangles show theresult of the simulation.
0
0.05
0.1
0.15
0.2
0.25
0.3
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
δ = 16MHzδ = 14MHzδ = 10MHzδ = 8MHz
Figure 4.7 Velocity distribution for different detunings of the cooler. The number offast atoms is increased at small detunings.
50
4.3. DEPENDENCE ON LASER SETTINGS
0.5
1
1.5
2
2.5
3
3.5
4
10 20 30 40 50 60 70 80 90
ato
m-fl
uxΦ
(109
s−1)
power cooler P (mW)
Figure 4.8 The total atom flux as a function of the cooler power. The atom fluxsaturates at a power of P ≈ 55mW. The green triangles show the result of thesimulation.
0
0.05
0.1
0.15
0.2
0.25
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
P = 23.4mWP = 29.0mWP = 42.6mWP = 62.0mW
Figure 4.9 Velocity distribution for different cooler powers. For higher cooling pow-ers, the peak in the velocity distribution is shifted to higher velocities.
51
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
center of the cooling beams is listed.
experiment beam total cooler detuning I/I0source power (mW) cooler (MHz)
Dieckmann [6] 2D MOT 16.4 10.2 7.8Dieckmann [6] 2D+ MOT 15 18 7.1
Schoser [7] 2D MOT 320 11.4 12.7Wohlleben [28] MOT + pusher 60 18 5.8
Fuchs [41] 2D+ MOT 35 10 5.6Utfeld [40] 2D+ MOT 150 7 10our result 2D+ MOT 110 12 12
Table 4.2 Laser settings of other Rb 2D MOTs. The total power is either the max-imum power available or the the power where the flux saturated. The saturationparameter I/I0 refers to the center of the cooling beam.
4.3.3 Repumping laser
The total atom flux also depends on the power of the repumper. Normally a few mW
of repumping power are sufficient since the atoms only fall into the F = 2 dark state
once every thousand cooling cycles. The repumping power was varied over a range of
1.5-3.5mW where the flux increased linearly with the repumping power (figure 4.10).
With the powers that were available during the time of the measurement no saturation
of the flux could be observed, so a further increase of the power should lead to a larger
flux. In [40] they had a similar setup and saw a saturation of the flux at a power of
P = 6mW.
The mean velocity of the beam is shifted to higher velocities with increasing re-
pumping power. The peak velocity is reduced by about 2m/s for a doubling of the
repumping power from 1.5mW to 3.1mW. This is quiet interesting since an increase
in the cooling power leads to lower velocities. To understand these dependences one
has to take a closer look at the pushing beam.
4.3.4 Pushing laser
In the 2D MOT two atom beams are produced, one in z and one in −z-direction. Due
to the symmetry of the setup there should be the same number of atoms traveling in
both directions. Without the pushing beam all the atoms traveling in −z direction
are lost. With a red detuned pushing beam on the axis of the atom beam it should be
possible to turn around some of the atoms traveling the “wrong” way. In this simple
52
4.3. DEPENDENCE ON LASER SETTINGS
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
1 1.5 2 2.5 3 3.5
ato
m-fl
uxΦ
(109
s−1)
power repumper P (mW)
Figure 4.10 The total atom flux as a function of the repumper power. The fluxincreases linearly with the power and there is no saturation of the flux withinthe measured range.
0
0.05
0.1
0.15
0.2
0.25
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
P = 1.5mWP = 2mW
P = 2.5mWP = 3.1mW
Figure 4.11 Velocity distribution for different repumper powers. For higher repump-ing powers the velocity distribution is shifted to lower velocities.
53
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
picture the maximum gain expected from the pushing beam would be a doubling of
the flux if all the atoms initially traveling in −z-direction can be turned around. This
is exactly what is observed in the experiment. Without the pushing beam we have
a total flux of Φ ≈ 1.9 atoms/s and with the pushing beam we can reach a flux of
Φ ≈ 3.7 atoms/s at a detuning of 40MHz and a power of 0.8mW (figure 4.12 and
4.13).
To get a qualitative understanding of the influence of the pushing beam, a simple
model will be used. The pushing beam can be described by the force in equation
(2.25) with B = 0 (there is no magnetic field component in z-direction). It has to be
considered that the pushing beam does not contain any repumping light. Therefore it
only acts on the atoms as long as they stay within the cooling volume of the 2D MOT
where they are pumped back from the dark state.
Figure 4.14 shows a phase space diagram of the atoms which are initially traveling
in −z-direction. Outside the cooling region of the 2D MOT the force is zero, therefore
the atoms travel with a constant velocityA. After the atoms enter the cooling region
they are slowed down by the pusher. The red trajectories represent atoms which pass
the cooling region before the pusher can stop them. The green trajectories represent
atoms which can be turned around by the pusher and are then accelerated back in z
direction. While the atoms get faster the force on them decreases because they are
further detuned from the pusher. This is why all green trajectories accumulate around
the same velocity where the atoms are out of resonance with the pushing beam.
To calculate the velocity distribution of the atoms after they are turned around,
their initial velocity distribution has to be known. The distribution should be the same
(with opposite signs) as the velocity distribution of the atoms traveling in z direction
(green curve in figure 4.15). The red curve shows a simulation of the atoms’ velocity
distribution after the pushing beam (with δ = 40MHz and P = 1mW) interacted
with the atoms. The simulation shows that most of the atoms change their direction
and pushed out of the cooling volume in z-direction. Only atoms which travel faster
than 50m/s (red trajectories in 4.14) leave the cooling region in −z-direction.
The blue line shows the measured velocity distribution with the pusher present.
The long tail of fast atoms can be interpreted as the atoms initially traveling in z
direction (green curve). The peak of the velocity distribution is roughly at 14m/s
which is reproduced quiet good in the simulation, whereas the width and hight are
different in the simulation. In the simulation all the atoms enter the cooling volume
at the same z value whereas they should start from points distributed all over the
cooling region. This could be accomplished by implementing the pushing beam into
the numerical simulation of the 2D MOT and should lead to a widening of the peak as
54
4.3. DEPENDENCE ON LASER SETTINGS
1.5
2
2.5
3
3.5
4
-20 -10 0 10 20 30 40 50 60
ato
m-fl
uxΦ
(109
s−1)
detuning pusher δ (MHz)
Figure 4.12 The total atom flux as a function of the pusher’s detuning. The highestflux is obtained at a detuning of δ = 40MHz.
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
0 0.5 1 1.5 2 2.5
ato
m-fl
uxΦ
(109
s−1)
power pusher P (mW)
Figure 4.13 The total atom flux as a function of the pusher power. Increasing thepusher power from 0mW (pusher off), to 1mW leads to a doubling of the totalflux.
55
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
Figure 4.14 Phasespace of atom beam with pusher. The phasespace of the atomsinitally travelling in the −z direction is shown. Red trajectories represent atomsthat are too fast to be turned around, green trajectories represent atoms thatare reflected by the pusher. Most of these trajectories accumulate around thesame final velocity. On the right the force F (v) on the atoms is shown.
in the measured curve. For an improved simulation the distribution of points where
the atoms enter the cooling volume has to be known. Nevertheless the simulation can
explain most of the measured dependences.
The velocity at which the atoms get out of resonance with the beam depends on
the pusher’s detuning and power:
• For higher powers the peak is shifted to higher velocities due to the power broad-
ening of the pushing beam.
• For larger detunings the peak is shifted to lower velocities.
This was also observed experimentally (see figures 4.17 and 4.16). For a detuning of
δ = 14MHz the peak is at a velocity of 11m/s. When increasing the detuning to
δ = 54MHz the peak’s velocity is shifted by about 50% to 11m/s. A change from
P = 0.5mW to 2.3mW in pushing power results in a shift of about 30% from 11m/s
to 15.5m/s.
As mentioned earlier, there is also a dependence of the peak position on the cooling
56
4.3. DEPENDENCE ON LASER SETTINGS
0
0.1
0.2
0.3
0.4
0.5
0.6
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
pusher onpusher offsimulation
Figure 4.15 Simulation of the pushing beam. The solid lines show the data takenwith (blue) and without (green) the pushing beam. The dashed line shows asimulation of the velocity distribution of the −z atom beam which is turnedaround by the pusher. The parameters used for the simulation are δpusher =40MHz and Ppusher = 1mW.
and repumping powers:
• The peak is shifted to lower velocities for larger cooling powers. The atoms
scatter more likely with the cooling laser the higher its power. Thus they scatter
less likely with the pushing beam which diminishes the force of the pusher.
• The peak is shifted to higher velocities for larger repumping powers because the
atoms are pumped back into the cooling cycle more efficiently. This increases
the force of the pusher.
The total flux of atoms that can be reached with the pushing beam depends on how
many atoms from the −z beam can be addressed by the pusher. In order to address a
lot of atoms it is favorable to have an atom beam with a narrow velocity distribution
to begin with. This can be illustrated by comparing the results by Dieckmann et al.
[6] and Schoser et al. [7].
The experiment performed by Dieckmann et al. produces an atom beam with a
width of the velocity distribution (FWHM) of about 25m/s at a mean velocity of
15m/s. They report an increase of the flux from 5 × 109 atoms/s without a pushing
beam, to 9× 109 atoms/s with a pushing beam. This is again close to a factor of two
as in our case.
57
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
10 15 20 25 30
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
δ = 54Hzδ = 40Hzδ = 32Hzδ = 14Hzδ = 0Hz
Figure 4.16 Velocity distribution for different detunings of the pusher. For biggerdetunings the peak in the velocity distribution is shifted to higher velocities.
0
0.05
0.1
0.15
0.2
0.25
0.3
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
P = 0.5mWP = 0.8mWP = 1.1mWP = 2.3mW
pusher off
Figure 4.17 Velocity distribution for different pusher powers. For higher pusher pow-ers the peak in the velocity distribution is shifted to higher velocities.
58
4.3. DEPENDENCE ON LASER SETTINGS
In the second experiment by Schoser et al. they have an atom beam with a width of
the velocity distribution (FWHM) of about 75m/s at a peak velocity of 50m/s. They
can only achieve a small increase of the flux by applying a pushing beam. Only the
slowest atoms of the atom beam can be adressed by the pusher since the width of the
velocity distribution is too large. It should also be mentioned that they used resonant
light for the pushing beam whereas our results suggest that it is beneficial to use far
red detuned light.
The last setting of the laser system that was investigated is the influence of a counter-
propagating pushing beam.
4.3.5 Counter-propagating pushing beam
A similar beam was used by Dieckmann et al. to shape the velocity distribution of the
atom beam. Without a significant change in the total flux they could lower the mean
velocity from 15m/s (without any pushing beams) to 8m/s, with the two counter-
propagating beams. The width of the velocity distribution dropped from 25m/s to
only 3.3m/s. The two beams had to be unbalanced to get the best result. More power
was used in the beam towards the exit hole.
Our measurement was taken for various power balancings at a number of different
powers and detunings. In all cases the flux was always better without the counter-
propagating pushing beam. Also in most cases the mean velocity of the beam was in-
creased rather than decreased. Figure 4.18 shows the velocity distribution for three set-
tings with the counter-propagating beam turned on and off. The counter-propagating
beam decreases the number of slow atoms. This might be interpreted as a reduction
of the others pushing beam’s efficiency.
The same effect was observed in [29] where they also saw a decrease in the number
of slow atoms due to the second pushing beam. Catani et al. [8], who investigated
a potassium 2D MOT, report a small decrease of about 20% in the mean velocity of
the beam.
In the experiment the detunings of the two pushing beams could not be adjusted
separately. It might be beneficial to detune the second pushing beam even further to
only address the fast atoms.
In the next two subsection the influence of the total MOT length, the pressure and
the magnetic field gradient will be discussed.
59
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
0
0.05
0.1
0.15
0.2
0.25
0.3
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
retro on δ = 44MHzretro on δ = 34MHzretro on δ = 24MHzretro off δ = 48MHzretro off δ = 34MHzretro off δ = 24MHz
Figure 4.18 Velocity distribution with/without a counter-propagating pushing beam.The solid lines show a setting where 820µW are in the pushing beam and 400µWin the counter-propagating beam. A second setting with no counter propagatingbeam is represented by dashed lines.
4.4 Influence of partial pressure and length of
cooling volume
So far all the measurements shown were taken with three cooling regions. To investi-
gate the influence of the MOT length the last one or two cooling regions were blocked,
giving all power to the active cooling regions. The simple model in section 2.2.4 pre-
dicted that an increase of the MOT length should lead to the capture of faster atoms.
This could be verified in the experiment where the mean velocity increases from 13m/s
with two cooling regions to 16m/s with all three cooling regions (figure 4.19). The
dependence of the total flux on the number of cooling regions is shown in figure 4.20.
With only one cooling region the flux drops by more than one order of magnitude.
With two cooling regions the flux is about 60% smaller as the flux with all three
cooling regions. Our simulation predicts that adding a fourth region would lead to
another 60% increase in the total flux.
The “coupling efficiency“ between the different cooling regions has been investigated
by Ramirez-Serrano et al. [9]. They measured the dependence of the flux on the
spacing between the cooling regions. They found that even at large spacings the
atoms in one trapping region can efficiently be transfered into the neighboring ones.
60
4.4. INFLUENCE OF PARTIAL PRESSURE AND LENGTH OF COOLING
VOLUME
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
8 10 12 14 16 18 20 22 24
ato
m-fl
ux
dis
trib
uti
on
(r.u
.)
velocity v (m/s)
2 cooling regions3 cooling regions
Figure 4.19 Velocity distribution with 2 and 3 cooling regions. The velocity distri-bution is shifted to higher velocities for more cooling regions.
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3
ato
m-fl
uxΦ
(109
s−1)
number of cooling regions
Figure 4.20 The total atom flux with a different number of cooling regions. Withone cooling region the flux is more than one order of magnitude smaller thanwith all three cooling regions.
61
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
Figure 4.21 Origin of the atoms on the glass-cell. This is the result of our numer-ical simulation. It shows the origin of all atoms which passed through the exithole. Clearly visible are three areas in the distribution where most of the atomsoriginate from. These are the areas close to the three cooling regions..
Faster atoms that cannot be trapped in a single region can be trapped by passing
through multiple regions. In our simulation we obtain a similar picture. Figure 4.21
shows the origin of all the “good” trajectories on the side-face of the glass-cell. The
influence of the three cooling regions is clearly visible. Most of the atoms are trapped
in the last cooling region and are then further cooled in the following ones.
The total MOT length also influences the optimum rubidium pressure. Collisions
with background gas atoms limit the flux when the length of the MOT is increased.
The longer the way of the atoms through the MOT the more likely they get lost from
the beam due to collisions. Therefore the optimum pressure decreases with increasing
MOT length [7].
In general the flux first raises linearly with the pressure until a critical point. At
this critical point the free mean path length of the atoms becomes comparable to the
cooling time. The critical pressure seems to differ depending on the usage of a pushing
beam. For the operation without a pushing beam Dieckmann et al. [6] measured an
increase of the flux all the way up to 5 × 10−7 mbar, the vapor pressure of rubidium
at room temperature. Schoser et al. [7] went to even higher pressures by heating the
rubidium cell and could measure the critical point to be at 1.8× 10−6 mbar.
In the presence of a pushing beam the critical point is already reached at much
lower pressures. The reason seems to be the progressive absorption of the relatively
weak pushing beam with increasing pressure. The critical pressure in this case was
measured by Dieckmann et al. to be around 1.5 × 10−7. This is in good agreement
with the value of 2.1× 10−7 measured by Catani et al. for potassium.
In our experiment we could not measure the pressure in the glass-cell but the flux was
very sensitive to the dispenser current. In Table 4.3 the pressure and the magnetic field
gradient of several 2D MOTs is listed. The pressure corresponds to either the optimum
pressure or if marked (*) to the maximum pressure accessible in the experiment. The
influence of the magnetic field is going to be addressed in the following.
62
4.5. MAGNETIC FIELDS
experiment beam pressure magnetic fieldsource (mBar) gradient (G/cm)
Dieckmann [6] 2D MOT 5× 10−7 * 17.7Dieckmann [6] 2D+ MOT 1.5× 10−7 12.6
Schoser [7] 2D MOT 1.8× 10−6 17Wohlleben [28] MOT + pusher 7.6× 10−9 15
Catani [8] 2D+ MOT (K) 2.1× 10−7 17Fuchs [41] 2D+ MOT 3.8× 10−9 * 21.6Utfeld [40] 2D+ MOT 1× 10−7 19our result 2D+ MOT - 18
Table 4.3 Pressure and magnetic field gradient of other 2D MOTs.
4.5 Magnetic fields
As described in section 3.3 a two-dimensional quadrupole field is used to trap the
atoms. The atoms travel along the zero line of this field which is in the center of
the glass-cell. In order to get the beam through the exit hole the zero line can be
fine adjusted with two sets of Helmholtz coils. With these coils the atom beam can
continuously be moved in two dimensions.
4.5.1 Magnetic field gradient
In the experiment five different settings of the magnetic field gradient were accessible.
The atom flux at these five settings is shown in figure 4.22. Up to about 15G/cm
the flux raises linearly with the gradient. The optimum field gradient is somewhere
between 15G/cm and 18G/cm which is in good agreement with the results obtained
in other experiments. For higher gradients the flux starts to decrease again. The
same trend can be observed in the simulation where the highest flux is obtained at a
gradient of 18G/cm.
4.5.2 Compensation coils
By changing the current of the compensation coils it can be determined how critical
the alignment of the atom beam is with respect to the exit hole. Figure 4.23 shows
the flux as a function of the displacement of the atom beam. A displacement of zero
corresponds to the optimum setting of the compensation coils. The displacement can
be deduced from the compensation coils’ current Icomp by d = Icomp/(∇B) where ∇B
is the gradient of the quadrupole field. The measured flux drops by approximately
50% when the beam is displaced by 0.4mm. This matches the size of the exit hole of
0.8mm because at a displacement of 0.4mm exactly half of the beam should be cut.
63
CHAPTER 4. CHARACTERIZATION OF THE ATOM BEAM
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
8 10 12 14 16 18 20 22 24
ato
m-fl
uxΦ
(109
s−1)
magnetic field gradient ∂B (G/cm)
Figure 4.22 Dependence of the total atom flux on the magnetic field gradient. Thehighest flux is obtained with a magnetic field gradient between 15G/cm and18G/cm. The green triangles show the result of the simulation.
0
0.5
1
1.5
2
2.5
3
3.5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
ato
m-fl
uxΦ
(109
s−1)
displacement of the MOT I (mm)
x-axisy-axis
Figure 4.23 The total atom flux as a function of compensation coils current. Withthe compensation coils the atom beam can be misaligned with respect to theexit hole. This way the beam profile at the exit hole is obtained.
64
4.6. BEAM SIZE
It can also be deduced that almost no atoms are lost due to the size of the exit hole
since at a displacement of 0.8mm the flux almost drops to zero.
4.6 Beam size
The elliptical beams used in the experiment have a long axis of 22mm and a short
axis of 11mm (1/e2 values). The size of the cooling beams can not be changed as
easily as the laser settings or the magnetic fields. For every beam size a different
set of lenses has to be used and normally all the beams in the 2D MOT have to be
realigned as well. Instead of optimizing the ellipticity of the beams in the experiment,
the numerical model can be used as well. So far all the measured dependences could be
reproduced by the model adequately. Therefore one can expect to get reliable results
for the dependence on the beam diameters as well. Figure 4.24 shows the results of the
numerical simulations which show an increase of the flux up to a diameter of 20mm.
Even larger beams do not lead to higher fluxes because the beam is limited by the
diameter of the wave plate. This means that the diameters of our cooling beams are
too small. In the simulation an increase from 11mm to 20mm leads to a gain in the
atom flux of roughly 80%.
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
4 6 8 10 12 14 16 18 20 22 24
ato
m-fl
uxΦ
(109
s−1)
diameter of the short axis (mm)
Figure 4.24 The total atom flux as a function of the beam diameter. The diameter ofthe short axis was changed in the simulation. The flux increases up to a diameterof 20mm.
65
5 Conclusion and Outlook
In this thesis we reported on the development, assembly and characterization of a
2D MOT as a source of a cold atom beam which will be used to load a cold atom
target. With a mean velocity of 14m/s and a divergence of 26mrad, the produced
beam is well suited to load our 3D MOT. The achieved flux of 4×109 atoms/s is in the
range of similar setups and is sufficient for later experiments where we expect a similar
loading flux. The setup is modular and compact, including permanent magnets for
the creation of the 2D magnetic quadrupole field. To compensate for stray magnetic
fields, two pairs of coils still had to be implemented. In a 2nd generation setup these
coils can be avoided by making the permanent magnets adjustable in all dimensions.
After assembly, the 2D MOT was characterized on a test setup and most of the
measured dependences could be reproduced by our numerical simulations. We found
that it is favorable to detune the cooling and the pushing beam separately since for
the latter a rather high detuning of 40MHz results in the best flux. The best im-
provement which can be accomplished by using a pushing beam was a doubling of the
flux. This can be interpreted as a reversal of the trapped atoms initially moving away
from the exit hole. In our setup we could not see an improvement of the flux or the
mean velocity by using a counterpropagating pushing beam as reported in [7]. The
situation might be different if the two pusher beams can be detuned separately. We
also used our theoretical model to test the influence of parameters which are not easily
accessible in the experiment such as the diameter of the laser beams or the number
of cooling regions. Our model suggests that a fourth cooling region leads to a 60%
increase of the atom flux. Furthermore, circular cooling beams should lead to another
increase of 80% . All of these improvement could be implemented in a future setup.
Alongside the development and realization of the 2D MOT the MOTRIMS experi-
ment has been redesigned. During the last year a new laser system was set up and
tested by creating a dispenser loaded MOT in the old setup. Afterwards the main
vacuum chamber has undergone a complete reconstruction, starting with the imple-
mentation of a newly designed high resolution spectrometer in combination with a
new ion detector. Also new water cooled quadrupole coils were added as the old coils
did not leave enough room for the new spectrometer. As a last step the 2D MOT was
connected to the chamber before it was finally closed and is now under UHV again.
67
CHAPTER 5. CONCLUSION AND OUTLOOK
Figure 5.1 New design of the MOTRIMS experiment. The atom beam from the 2DMOT is used to load a 3D MOT at the center of the new spectrometer. In collisionexperiments atoms from the MOT will be ionized and accelerated onto a positionsensitive ion detector.
Currently final adjustments are made to load the 3D MOT from the atom beam
provided by the 2D MOT. Also, the possibility of using the atom beam generated
from the 2D MOT as a new target will be explored. After successful commissioning,
the setup will be transported to GSI. In first collision experiments, charge transfer
between Rb and Ar16+ generated by an EBIT, will be investigated As a next step, the
setup will be implemented in the HITRP beamline, where slow highly charged ions up
to bare uranium will be accessible.
68
List of Figures
2.1 Scattering rate for different laser intensities . . . . . . . . . . . . . . . . 7
2.2 Spontaneous force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Setup of a 3D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Phasespace of atoms in a three dimensional light field . . . . . . . . . . 10
2.5 Atom beam from an oven . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Setup of a 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Trajectories in the 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Magnetic field of a single permanent magnet . . . . . . . . . . . . . . . 23
3.2 Hyperfine level scheme of Rb with cooling scheme for 2D MOT . . . . . 24
3.3 Locking scheme of the cooling laser . . . . . . . . . . . . . . . . . . . . 25
3.4 Doppler free spectroscopy for laser lock . . . . . . . . . . . . . . . . . . 26
3.5 Beam Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Locking scheme of the repumper . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Design of the 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Center piece of the 2d MOT setup . . . . . . . . . . . . . . . . . . . . . 29
3.9 Alignment of the differential pumping tube . . . . . . . . . . . . . . . . 30
3.10 Drwing of the glascell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Connection of the dispensers . . . . . . . . . . . . . . . . . . . . . . . . 31
3.12 Holder for fiber outcouplers . . . . . . . . . . . . . . . . . . . . . . . . 32
3.13 Splitting of the trapping beam . . . . . . . . . . . . . . . . . . . . . . . 33
3.14 Optics Module Top View . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.15 Optics Module Side View . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.16 Measurement of the magnetic field of a permanent magnet . . . . . . . 35
3.17 Magnetic field of a single permanent magnet . . . . . . . . . . . . . . . 36
3.18 Design of the magnet holder . . . . . . . . . . . . . . . . . . . . . . . . 37
3.19 Magnetic field of a single permanent magnet . . . . . . . . . . . . . . . 38
3.20 Magnetic field gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.21 Measured field gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
69
List of Figures
4.1 Test setup for beam characterisation . . . . . . . . . . . . . . . . . . . 42
4.2 Measurement of the velocity distribution . . . . . . . . . . . . . . . . . 43
4.3 Magnetic field of a single permanent magnet . . . . . . . . . . . . . . . 46
4.4 Geometry of the differential pumping tube . . . . . . . . . . . . . . . . 47
4.5 Profile of the atom beam . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 The total atom flux as a function of the cooler’s detuning . . . . . . . . 50
4.7 Velocity distribution for different detunings of the cooler . . . . . . . . 50
4.8 The total atom flux as a function of the cooler power . . . . . . . . . . 51
4.9 Velocity distribution for different cooler powers . . . . . . . . . . . . . 51
4.10 The total atom flux as a function of the repumper power . . . . . . . . 53
4.11 Velocity distribution for different repumper powers . . . . . . . . . . . 53
4.12 The total atom flux as a function of the pusher’s detuning . . . . . . . 55
4.13 The total atom flux as a function of the pusher power . . . . . . . . . . 55
4.14 Phasespace of atom beam with pusher . . . . . . . . . . . . . . . . . . 56
4.15 Simulation of the pushing beam . . . . . . . . . . . . . . . . . . . . . . 57
4.16 Velocity distribution for different detunings of the pusher . . . . . . . . 58
4.17 Velocity distribution for different pusher powers . . . . . . . . . . . . . 58
4.18 Velocity distribution with counter propagating pushing beam . . . . . . 60
4.19 Velocity distribution with 2 and 3 cooling regions . . . . . . . . . . . . 61
4.20 The total atom flux for a different number of cooling regions . . . . . . 61
4.21 Origin of the atoms on the glass-cell . . . . . . . . . . . . . . . . . . . . 62
4.22 Dependence of the total atom flux on the magnetic field gradient . . . . 64
4.23 The total atom flux as a function of compensation coils current . . . . 64
4.24 The total atom flux as a function of the beam diameter . . . . . . . . . 65
5.1 New design of the MOTRIMS experiment . . . . . . . . . . . . . . . . 68
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Erklärung:
Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als
die angegebenen Quellen und Hilfsmittel benutzt habe.
Heidelberg, den 15.11.2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Danksagung
Als erstes möchte ich mich bei Matthias Weidemüller für die nette Aufnahme in seiner
Forschungsguppe bedanken. Vielen Dank das Du es mir ermöglicht hast meine Diplo-
marbeit in diesem spannenden Bereich der Physik zu schreiben. Ich habe im letzten
Jahr viel gelernt, sowohl bei zahlreichen Diskussionen bei Kaffee und Pizza oder auch
bei zahlreichen Vorträgen und Konferenzen.
Vielen Dank auch an Herr Prof. Quint, sowohl für die Übernahme der Zweitkorrektur
dieser Arbeit, als auch die gute Zusammenarbeit im Bezug auf das MOTRIMS Ex-
periment.
Mein ganz besonderer Dank gilt Simone, die mich im letzten Jahr durch meine Diplo-
marbeit begleitet hat. Ich habe im letzten Jahr viel von dir lernen können und die
gemeinsame Arbeit, sowohl im Labor, als auch im Büro hat mir sehr viel Spaß gemacht.
Vielen Dank auch an Thomas, der uns leider in Richtung Hamburg verlassen hat, aber
mir immer eine große Hilfe war, als wir noch in einem Büro gesessen haben.
Als nächstes möchte ich mich ganz herzlich bei Dominic bedanken: Die Zeit die wir
zusammen im und außerhalb des Labors verbracht haben habe ich sehr genossen.
Ein großer Dank geht auch an Christoph, der mich immer unterstützt hat und mit dem
ich viel zusammenarbeiten konnte, insbesondere bei der Planung und dem Aufbau der
2D MOT.
Ganz herzlichen Dank auch an Hannes, der mich in den letzten Tagen meiner Diplo-
marbeit sehr unterstützt hat.
Allen anderen Mitgliedern der QD Gruppe möchte ich natürlich auch ganz herzlich
für die schöne Zeit in und ausserhalb der Uni danken.
Als letztes geht ein ganz großes Dankeschön an die Werkstätten und Konstruktion
des Physikalischen Instituts, ohne die diese Arbeit nicht möglich gewesen wäre. Mein
besonderer Dank gilt Kevin, der uns die Planung unseres Aufbaus von Anfang an
begleitet und die Realisierung erst möglich gemacht hat.